Foundation Numeracy in Context

acr001_fn02 11/5/06 11:33 AM Page 1 MathsWorks for Teachers Series editor David Leigh-Lancaster Foundation Numeracy...

Author: David Tout | Gary Motteram

34 downloads 1564 Views 2MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

acr001_fn02

11/5/06

11:33 AM

Page 1

MathsWorks for Teachers Series editor David Leigh-Lancaster

Foundation Numeracy in Context David Tout and Gary Motteram Foundation Numeracy in Context describes an approach to teaching mathematics based on applied and contextual learning principles. This means that the teaching and learning of mathematics proceeds from a contextual, task-based and investigative point of view—where the mathematics involved is developed from a modelled situation or practical task. Practical investigations and projects are principle vehicles for student learning in such an approach. This text is written for teachers working with students who have become disengaged from learning mathematics during the middle to latter years of secondary schooling, and will likely have had limited success with mathematics. The approach used will be helpful for teachers of students who need a practical rather than formal mathematical background for their everyday life skills and further education, training or career aspirations. The text illustrates how this approach works through some sample contexts such as cars and driving, sport, cooking and catering, and draws together mathematics from the areas of number, measurement, space, data and statistics, and algebra.

Series overview

MathsWorks for Teachers has been developed to provide a coherent and contemporary framework for conceptualising and implementing aspects of middle and senior mathematics curricula. Titles in the series are: Functional Equations David Leigh-Lancaster Contemporary Calculus Michael Evans Matrices Pam Norton Foundation Numeracy in Context David Tout & Gary Motteram Data Analysis Applications Kay Lipson Complex Numbers and Vectors Les Evans

Foundation Numeracy in Context David Tout and Gary Motteram

MathsWorks for Teachers

120506•Foundation Nu#B2ACD.indd i

16/6/06 11:59:17 AM

First published 2006 by ACER Press Australian Council for Educational Research Ltd 19 Prospect Hill Road, Camberwell, Victoria, 3124 Copyright © 2006 Dave Tout, Gary Motteram, David Leigh-Lancaster All rights reserved. Except under the conditions described in the Copyright Act 1968 of Australia and subsequent amendments, no part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the written permission of the publishers. Edited by Ruth Siems Editorial Services Cover design by FOUNDRY Typography, Design & Visual Dialogue Text design by Robert Klinkhamer Typeset by Desktop Concepts P/L, Melbourne Printed by Shannon Books Pty Ltd National Library of Australia Cataloguing-in-Publication data: Tout, Dave, 1950– . Foundation numeracy in context. ISBN 0 86431 516 3. 1. Numeracy. I. Motteram, Gary. II. Title. (Series: MathsWorks for Teachers). 372.72044 Visit our website: www.acerpress.com.au

120506•Foundation Nu#B2ACD.indd ii

16/6/06 11:59:19 AM

CONTENTS

Introduction vii About the authors

1

viii

Teaching mathematics in context 1 Introduction 1 Why do we need a different approach? 2 Connections 4 Teaching in context and applied learning 10 So what does this mean for classroom practice? 12 Covering mathematics curriculum content and designing a program 16 Does it work? 17 Conclusion 19

2

Designing teaching programs 23 Introduction 23 How to plan 23 The teacher role 30 Information and communication technology (ICT)

3

Making meaning

33

35

Introduction 35 Mathematics areas covered 37 Some examples of using texts 42 Possible starting points 46 Resources 48

120506•Foundation Nu#B2ACD.indd iii

16/6/06 11:59:19 AM

Contents

4

Driving away 50 Introduction 50 Mathematics areas covered 50 Possible starting points 52 Example of mathematics skills covered 56 Resources 59

5

Making algebra work 61 Introduction 61 A common approach to teaching algebra at school Introducing and connecting with algebra 63 Mathematics areas covered 64 Extensions 66 Resources 69

6

62

Cooking mathematics up 72 Introduction 72 Mathematics areas covered 72 Possible starting points 74 Example of mathematics skills covered 78 Resources 82

7

A sporting opportunity 84 Introduction 84 Mathematics areas covered 85 Possible starting points 88 Examples of negotiated investigations Resources 97

120506•Foundation Nu#B2ACD.indd iv

92

16/6/06 11:59:20 AM

Contents

8

Curriculum connections 99

Appendices 105 A B C D E F G

Planning grid 105 Possible contexts 107 Sample brainstorm 110 Mathematics skills example 111 Resources 112 Contract for the outline of the investigation Sample worksheets 118

Notes

120506•Foundation Nu#B2ACD.indd v

117

144

16/6/06 11:59:20 AM

120506•Foundation Nu#B2ACD.indd vi

16/6/06 11:59:20 AM

INTRODUCTION MathsWorks is a series of teacher texts covering various areas of study and topics relevant to senior secondary mathematics courses. The series has been specifically developed for teachers to cover helpful mathematical background, and is written in an informal discussion style. The series consists of six titles: • An Introduction to Functional Equations • Contemporary Calculus • Matrices • Data Analysis Applications • Foundation Numeracy in Context • Complex Numbers and Vectors Each text includes historical and background material; discussion of key concepts, skills and processes; commentary on teaching and learning approaches; comprehensive illustrative examples with related tables, graphs and diagrams throughout; references for each chapter (text and web-based); student activities and sample solution notes; and a bibliography. The use of technology is incorporated as applicable in each text, and a general curriculum link between chapters of each text and Australian state and territory as well as and selected overseas courses is provided. A Notes section has been provided at the end of the text for teachers to include their own comments, annotations and observations. It could also be used to record additional resources, references and websites.

vii

120506•Foundation Nu#B2ACD.indd vii

16/6/06 11:59:20 AM

ABOUT THE AUTHORS David Tout is one of the most experienced adult numeracy educators in Australia. He is particularly interested in language and mathematics, and making mathematics relevant, interesting and fun for all students particularly those students who are disengaged from the subject. David has written a wide range of teaching, curriculum, assessment and professional development materials and resources, and has presented on a wide range of topics at numerous local, regional, national and international conferences and events, including in Sweden, Japan, Spain, the Netherlands, the USA, Canada and the UK. Gary Motteram has over 25 years of classroom teaching experience. He has a particular interest in developing programs to support numeracy development in vocational and real-world contexts. Gary has contributed to curriculum and policy development at local and state levels and has provided professional development activities for teachers on the incorporation of learning technologies in classrooms.

viii

120506•Foundation Nu#B2ACD.indd viii

16/6/06 11:59:20 AM

CHAPTER

1

T E AC H I N G M AT H E M AT I C S I N CONTEXT INTRODUCTION This text describes an approach to teaching mathematics based on applied or contextualised learning principles. This approach means that teachers teach mathematics from a contextual and task-based or investigative point of view, one where mathematics is developed from an actual or modelled situation or task in which the mathematics is embedded. Investigations and projects are used as vehicles for learning. Teachers are encouraged to take into account students’ interests and knowledge, and their own informal ways of doing mathematics, allowing the understandings and strategies learnt in and out of school to serve as valid resources. While primarily targeted at teachers of students who have become disengaged and/or disenfranchised from learning mathematics successfully during the middle to latter years of secondary schooling, this approach is also suitable for students who don’t need formal mathematics as part of their further education, training or career aspirations. Often these two groups overlap. Taking an applied learning approach with an emphasis on problem solving can enable most students to succeed and progress, with students able to work from the same starting point but then proceed in different directions, learning and applying different mathematical skills to varying levels of sophistication. It can be used to engage and motivate students in order for all of them to succeed in learning and applying mathematics to whatever level they are capable of—it does not have to be just for students who are considered to be unsuccessful at mathematics. It will support students to become numerate and engaged citizens who are capable of using, transferring and adapting their mathematical knowledge and skills in this era of life-long learning and change.

1

120506•Foundation Nu#B2ACD.indd 1

16/6/06 11:59:20 AM

MATHSWORKS FOR TEACHERS Foundation Numeracy in Context

This chapter provides a rationale for teaching mathematics in context, and explains what this means and what the potential benefits are for all students. Following chapters outline how teachers can develop, plan and teach in this way and provide models for this approach and ideas to use as a starting point.

WHY DO WE NEED A DIFFERENT APPROACH? For some time now there has been evidence of dissatisfaction with what children are learning—or not learning—in mathematics classes, especially in the middle to latter years of secondary schooling. It appears there are too many students in these years of schooling who feel that mathematics is not relevant to them, and have lost interest in studying or succeeding in mathematics. For example, in Australia the introduction of the ‘New Basics’ curriculum program in Queensland has typified some of the Australian-based issues and concerns about education and reform initiatives. As part of an initial longitudinal study (Lingard & Ladwig, 2001) into the quality of classroom learning (from Year 6 to Year 11) the study found that students learned only superficially and were mainly involved in trivial short-answer activities. This is supported by the results from the Trends in International Mathematics and Science Study (TIMSS) (Hollingsworth, Lokan & McCrae, 2003), an international study based on videos of mathematics classroom practice. As reported in this study, McIntosh describes a typical Australian Year 8 mathematics lesson as follows: The teacher talks a lot, the students mainly reply with very few words, most of the time the students work, using only paper and pencil, on a repetitive set of low level problems, most presented via the board or textbooks or worksheets; discussion of solutions is mainly limited to giving the right answer or going through the one procedure taught. There is little or no opportunity for students to explain their thinking, to have a choice of solution methods or to realise that alternative solution methods are possible, and very few connections are drawn out between mathematical ideas, facts and procedures. (McIntosh, 2003, 108)

This phenomenon is not confined to Australia. Similar TIMSS results are found in other western countries. In the US, Grouws and Cebulla (2000) reported that: data from the Third International Mathematics and Science Study (TIMSS) video study show that over 90% of mathematics class time in the United

2


16/6/06 11:59:21 AM

CHAPTER 1 Teaching mathematics in context

States eighth grade classrooms is spent practicing routine procedures, with the remainder of the time generally spent applying procedures in new situations … In contrast, students at the same grade level in typical Japanese classrooms spend approximately 40% of instructional time practicing routine procedures, 15% applying procedures in new situations, and 45% inventing new procedures and analyzing new situations. (Grouws & Cebulla, 2000, 17)

In 1990, as part of a national publication about US mathematics education, Davis, Maher and Noddings (1990) described the situation as follows: By now nearly everyone has probably read, or at least heard of, the recent spate of reports showing that students in the United States are not doing very well in mathematics … This leaves the United States with what might be called a war on two fronts. There is first, the fact of unsatisfactory results. But the second front is perhaps even more threatening: there is major disagreement on how to proceed in order to make things better. One school of thought would argue for ‘more’ and ‘more explicit.’ That is to say, they would argue that the United States needs more days of school per year, or more hours of mathematics instruction per week, or more homework, or all of the above, together with a highly explicit identification of the knowledge that we want students to acquire, and a sharply directed emphasis on precisely this knowledge. Prescriptions in this direction usually suggest more frequent testing, and making more—perhaps even teachers’ salaries—dependent upon the outcome of this testing. A different diagnosis and prescription might be said to tend in nearly the opposite direction … These recommendations argue for making mathematics more natural, fitting it better into the context of children’s lives, conceivably even moving toward less testing. (Davis, Maher & Noddings, 1990, 1)

Nearly ten years later, Forman and Steen (1999) expressed similar sentiments: Despite mathematics’ reputation as an ancient subject consisting of indisputable facts, mathematics education has recently become the source of passionate debate. At stake is nothing less than the fundamental nature of school mathematics: its content (what should be taught), pedagogy (how it should be taught), and assessment (what should be expected) … At the risk of oversimplifying, this debate can be characterized as a clash between ‘traditionalists’ who expect schools to provide the kind of well-focused mathematics curriculum that colleges have historically

3


16/6/06 11:59:21 AM


expected and ‘reformers’ who espouse a broader curriculum that incorporates uses of technology, data analysis, and modern applications of mathematics. (Forman & Steen, 1999, 2)

What seems to be agreed is that many current and past methods of teaching mathematics to children have been less than completely successful. In secondary school classrooms in Australia, and in many other Western countries, when teachers teach ‘maths’, it is often the case that there is a lot of ‘talk and chalk’, students use a textbook and do lots of repetitive practice, they prepare for tests and exams, and they learn formal rules, often by rote. There is little consideration of why and how the skills they are expected to learn can be put to use in the ‘real’ world. The limitations of this style of mathematics instruction have now been acknowledged for some time in mathematics education, and a range of alternative strategies have been proposed including those based on a constructivist view of mathematics. However, this traditional style of mathematics teaching does still seem to be the main approach used in secondary schools (see below for a brief description of constructivism, and for a summary about teaching strategies see Grouws & Cebulla, 2000). Given these factors, and the knowledge that many students are disengaged from mathematics, a different approach to this traditional style of teaching of mathematics is needed if more students are to succeed in mathematics in these crucial years of schooling and become numerate adults. The following section makes links and connections with other areas of research into mathematics learning and teaching to demonstrate how an approach to teaching mathematics based on applied or contextualised learning principles can help to achieve these aims.

CONNECTIONS Several areas of research into mathematics knowledge, acquisition, learning and teaching have challenged the traditional approach to the teaching of mathematics described above. These include constructivism, ethnomathematics, functional maths, numeracy education and gender issues. Constructivism As mentioned above, one of the major influences in mathematics education over the last few decades has been based around alternatives to the traditional perspectives on what it means to learn and know mathematics, centred largely

4


16/6/06 11:59:21 AM


on the philosophy of constructivism as opposed to positivism (and variations and interpretations of this philosophy such as critical and social constructivism). Constructivism has helped mathematics educators reflect and think about what mathematical knowledge is, how it is acquired, and what the implications are for teaching and learning (for example, see: Davis, Maher & Noddings, 1990; Ernest, 1998; Malone & Taylor, 1993). Under a positivist philosophy, teachers are seen to act as the experts, and they transmit their knowledge directly to their students (a transmission model of teaching). Knowledge is seen as objective, and learning is about receiving the information handed down, absorbing the facts, and reproducing them. On the other hand, a constructivist philosophy sees knowledge as internal and subjective, where learning and understanding need to be constructed by the learner, and where the individual learner makes sense of the mathematics. Teaching plays a facilitating and questioning role rather than a predominantly transmitting role. In social constructivism, learning is also seen as an activity where shared mathematical meanings are constructed with others and drawn from the learning environment. This also highlights the critical role and importance of language in the teaching and learning of mathematics—being able to talk about and share understandings of mathematical concepts and topics. Recent cognitive theories also hold that knowledge is constructed and restructured under a variety of constraints or conditions that either facilitate or limit what can be learned. Some interpretations of the key implications of constructivism for classroom practice, adapted from Hatano (1996) are: 1. Mathematical knowledge is acquired by construction; therefore, students should be given the opportunity to actively participate in the learning process rather than be forced to swallow large amounts of information. 2. Cognitive restructuring is necessary to advance mathematical knowledge; to that end, instruction should induce successive restructurings of mathematical knowledge. 3. Mathematical knowledge is constrained by internal factors (cognitive, such as innate and early understandings and previous knowledge) and external factors (sociocultural, situated in contexts, such as peers, teachers, tools, and artefacts); it follows that each collection of factors may either facilitate or limit mathematical learning. (Hatano, 1996, 211–13)

5


16/6/06 11:59:22 AM


Ethno-mathematics Often referred to as street maths, ethno-mathematics is about how mathematics is used in everyday community and work situations outside the formal mathematics classroom and across different cultures (for example, see Gerdes, 1994; Harris, 1991; Nelson, Joseph & William, 1993; Nunes, Schliemann & Carraher, 1993; Powell & Frankenstein, 1997). Bishop (1994) wrote that mathematics has generally been: assumed to be culture-free and value-free knowledge; explanations of ‘failure’ and ‘difficulty’ in relation to school mathematics were sought either in terms of the learner’s cognitive attributes or in terms of the quality of the teaching they received … ‘social’ and ‘cultural’ issues in mathematics education research were rarely considered. (Bishop, 1994, 15)

Ethno-mathematics has a number of lessons for mathematics education. For example, Zaslavsky (1994a) stated the following: Why is it important to introduce ethnomathematical perspectives into the mathematics curriculum? Students should recognize that mathematical practices and ideas arose out of the real needs and interests of human beings … Students should learn how mathematics impacts on other subject areas—social studies, language arts, fine arts, science. Most important, they should have the opportunity to see the relevance of mathematics to their own lives and to their community, to research their own ethnomathematics. (Zaslavsky, 1994a, 6)

Zaslavsky goes on to recommend how an ethno-mathematical perspective could be incorporated into school mathematics: Rather than a curriculum emphasizing hundreds of isolated skills, mathematics education will embody real-life applications in the form of projects based on themes and mathematical concepts. (Zaslavsky, 1994a, 7)

There are a number of messages to mathematics educators that emerge from ethno-mathematical research. One is the acknowledgment that formal, or school-based, mathematics is not the only mathematics—mathematical knowledge is acquired via both formal and informal learning—and that informal learning is as valuable as formal school-based learning. Students should be encouraged to build on their ‘real life’ mathematics experiences while also learning the conventions and practices of formal mathematics.

6


16/6/06 11:59:22 AM


Functional mathematics In the US, functional mathematics, where teaching and curriculum are connected to ‘real world’ applications, is part of a related argument of how to improve school mathematics, especially at the secondary school level. Forman and Steen (1999) describe the need for a functional mathematics curriculum: Any mathematics curriculum designed on functional grounds … will emphasize authentic applications from everyday life and work … By highlighting the rich mathematics embedded in everyday tasks, this approach … can dispel both minimalist views about the mathematics required for work and elitist views of academic mathematics as an area with little to learn from work-based problems. Neither traditional college-preparatory mathematics curricula nor the newer standards-inspired curricula were designed specifically to meet either the technical and problem solving needs of the contemporary workforce or the modern demands of active citizenship. (Forman & Steen, 1999, vi)

Forman and Steen then proceed to explain why and how such a functional mathematics curriculum could work to cater to both the traditional and reformist views of mathematics while at the same time making the learning of mathematics relevant and meaningful to all students. Numeracy The development and conceptualisation of numeracy has been another important influence on the teaching of mathematics. Numeracy is a relatively recent and new term, first attributed to the UK Crowther Report in 1959, where numeracy was described as the mirror image of literacy (Crowther, 1959). The concept of numeracy is quite closely related to that of functional mathematics, where numeracy is often described as applying mathematics in context. In Australia there has been ongoing discussion and debate in the adult education sector about defining the relationship between mathematics and numeracy and also to the concept of ‘critical’ numeracy. Johnston (1994) has argued that numeracy in fact incorporates, or should incorporate, a critical aspect of using mathematics. She argues: To be numerate is more than being able to manipulate numbers, or even being able to ‘succeed’ in school or university mathematics. Numeracy is a critical awareness which builds bridges between mathematics and the real world, with all its diversity. (Johnston, 1994, 34).

7


16/6/06 11:59:22 AM


She continues: In this sense ... there is no particular ‘level’ of Mathematics associated with it: it is as important for an engineer to be numerate as it is for a primary school child, a parent, a car driver or a gardener. The different contexts will require different Mathematics to be activated and engaged in. (Johnston, 1994, 34).

So numeracy here is described as making meaning of mathematics and sees mathematics as a tool to be used efficiently and critically for some social purpose. It is this broader view of numeracy that has driven much of the teaching, professional development and curriculum development in adult numeracy education in Australia since the 1990s. Examples of where it has been used in curriculum development are described below in the section ‘More to the point— does it work?’. An international view of the concept of numeracy is also behind the 2002 redesign of the 1992 International Adult Literacy Survey (IALS), an international large-scale comparative survey. The Adult Literacy and Lifeskills Survey (ALL) is intended to assess the distribution of basic skills in the adult populations of participating countries and is expanding the domains assessed. It includes for the first time an assessment of numeracy skills based on a similar broad view of numeracy (Gal et al., 2005). The view of numeracy as making meaning of mathematics has also being used as a way of pushing for change in the teaching of adult basic mathematics in the USA (Schmitt, 2000; Tout & Schmitt, 2002). As Schmitt writes: Adult basic education and GED [General Educational Development] mathematics instruction should be less concerned with school mathematics and more concerned with the mathematical demands of the lived-in world: the demands that adults meet in their roles as workers, family members, and community members. Therefore we need to view this new term numeracy not as a synonym for mathematics but as a new discipline defined as the bridge that links mathematics and the real world. (Schmitt, 2000, 4)

It is this broad view of numeracy as the bridge that links mathematics and the real world that lies behind the applied learning approach recommended in this text. Mathematics is an important tool in its own right, but there also needs to be a bridge that enables individuals to use mathematical skills and knowledge to solve problems embedded within social contexts and purposes.

8


16/6/06 11:59:22 AM


School systems, especially in the primary school years, and increasingly in the middle years of schooling, have also been introducing numeracy programs to support their mathematics curriculum and teaching. This interest has again been partly to address concerns with the success of traditional mathematics teaching: There is also an increasing interest in numeracy, reflecting both a concern that Mathematics teaching is not succeeding, and also a desire to have a more relevant and context-related mathematics curriculum in schools. (Bishop, 2000)

Gender Much has been written about gender and mathematics (for example, see Harris, 1997; Walkerdine, 1989; Willis, 1989), and much of the research in this area is linked with the ethno-mathematics movement. A quote from a UK report by Harris demonstrates how these fields overlap in their view that informal, or ‘real life’ mathematical knowledge is as valuable as that gained through formal instruction: Throughout the world it is women and girls who underachieve in mathematics. Mathematics is the study above all others that denotes the heights of intellect. Throughout the world, the activity that most clearly denotes the work of women, in both the unpaid, domestic sphere and in paid employment, is work with cloth. Work with cloth symbolizes women as empty-headed and trivial. Yet constructing cloth, decorating it during construction and converting it into garments, is work that cannot be done without involving spatial and numerical concepts that are the foundations of mathematics. (Harris, 1997, 191)

The United States did much in the 1970s and 80s to address the issue of the lack of success of girls in mathematics. Most notable was the work and publications of the Lawrence Hall of Science in Berkeley, California, through the EQUALS project, which from 1977 developed programs that promoted equity for under-represented groups in mathematics. Australia followed suit. These approaches to teaching challenged many traditions in mathematics education, and promoted alternative approaches that in many cases are attractive not only to girls but also to the many boys who struggle with learning mathematics in a traditional classroom. Apart from including both gender and culturally inclusive models and materials, teaching approaches promoted included working cooperatively, incorporating discussion within the

9


16/6/06 11:59:23 AM


mathematics classroom, sharing ideas, and using hands-on materials. These have much in common with the approaches to learning mathematics promoted by ethno-mathematicians. There is much in common between the approaches and beliefs about mathematics teaching and learning arising out of the work and research from ethno-mathematics, functional mathematics, numeracy education and gender, backed up by a constructivist philosophy of knowledge and learning. They all argue for an approach that moves away from aspects of the traditional transmission model of teaching and learning mathematics described earlier and that incorporates engaging in different ways with a wide range of mathematics content and skills that are embedded within social contexts and purposes.

TEACHING IN CONTEXT AND APPLIED LEARNING These arguments about teaching in context connect with many other ideas and approaches to teaching and learning that have developed over a long period of time. This approach has been variously called applied learning, place-based learning, contextual learning, experiential learning or problem solving, and also connects to discussions about learning to learn and learning styles (for a recent summary, see Long, 2004). This also overlaps and relates very closely to aspects of adult learning (for example, see Foley, 1995; Kidd, 1978; Mackeracher, 1996; Rogers, 1996). Whether labelled place-based, or contextualised teaching and learning, or something else, every learner needs the opportunity to see mathematics at work in his or her life. In many respects contextual teaching and learning supports what effective educators have always done. For example, Howey (1998) claims that contextual learning emphasizes higher-level thinking, knowledge transfer, collecting, analysing, and synthesizing information and data from multiple sources and viewpoints. Such opportunities are crucial to the success of students at risk because of cultural, ethnicity, geography or economics. (Long, 2004, 450)

Lambdin and Lester (2004) describe and discuss a new approach to problem solving that is more than the games and puzzles approach that seems to have often been labelled as problem solving. The main goal is for students to develop a deep understanding of mathematical concepts and methods. The key to understanding is the

10


16/6/06 11:59:23 AM


engagement of students in trying to make sense of problematic tasks in which the mathematics to be learned is embedded. In addition to the mathematics that is the residue of the work on the tasks, the kind of sense making and problem solving in which students engage involved doing mathematics. As students attempt to solve rich problem tasks, they come to understand the mathematical concepts and methods, become more adept at mathematical problem solving, and develop mathematical habits of mind that are useful ways to think about any mathematical situation. (Lambdin & Lester, 2004, 191)

Lambdin and Lester then go on to argue that there are at least six benefits of teaching mathematics through problem solving (2004, 192–4). These benefits were related to improved understanding of mathematics, and include the notion that: • understanding is motivating • understanding promotes more understanding • understanding helps memory • understanding enhances transfer • understanding influences attitudes and beliefs • understanding promotes the development of autonomous learners Within Australia there have also been moves to promote a more comprehensive and applied learning approach to teaching mathematics, as an alternative to both traditional classroom teaching and the puzzles and games style of problem solving. Lovitt (2003) discusses the failure of this problemsolving approach and argues for a rich, balanced and appropriate mathematics curriculum, through what he calls an open-ended investigative approach. It seems like forever we have been trying to say school mathematics has two equally important major dimensions — a thinking and reasoning side to go alongside the skill development. The rise of the term Investigative thinking or Open-ended Investigations is the latest in what is beginning to be a long line of terms which attempt to capture and describe this thinking, reasoning side of our subject. (Lovitt, 2003)

Lovitt continues to present a case for a structured investigative approach to teaching mathematics in the classroom. What are the characteristics of this approach? In summary, characteristics that are seen as fundamental to these applied learning or investigative approaches to teaching mathematics would include:

11


16/6/06 11:59:23 AM


• Make connections and apply knowledge—the contexts used in the teaching of the curriculum should be connected to, and start with, the interests and needs of the learner, and connect with real-life experiences and communities. • Allow learners to take control of their learning—use flexible and openended tasks and questions to promote understanding, thinking and reasoning skills. • Integrate learning—don’t separate learning into a separate set of unconnected skills. Consider the whole task and the whole person. • Use and build on the skills of the learners—all learners bring with them knowledge and experiences to the learning situation. • Accept and promote a diversity of learning styles and methods—learners learn in different ways.

S O W H AT D O ES T H I S M E A N FO R C L A S S RO O M PRACTICE? Therefore, from several different aspects of educational research and knowledge (including constructivism, ethno-mathematics, functional mathematics, numeracy education, gender, and applied or adult learning), there is a strong call for approaches that incorporate teaching a wide range of mathematics skills that are related to real-life experiences, contexts and purposes and that is based on an investigative, open-ended style of classroom practice. This involves an integrated, cohesive approach to teaching mathematics where knowledge and skills are developed through understanding, thinking and reasoning skills, and interest and application. In such an approach, the curriculum and associated teaching becomes task-oriented. Students become engaged in problem solving via investigations or projects involving open-ended and/or ‘real life’ mathematics. Teachers plan and develop tasks or investigations that are of interest to their students, and students then go about solving the problems posed. Mathematics skills are taught that are connected to, or arise out of, the tasks being investigated. Students are able to work from the same starting point but go off in different directions, learning and applying different mathematical skills to varying levels of sophistication. The traditional ‘chalk and talk’ and textbook approach described earlier does not suit this style as a predominant mode of discourse, as the learning involved requires students to work actively on projects or investigations, not to work their way through a sequence of sums or word

12


16/6/06 11:59:23 AM


problems. And the assessment that tends to follow from this approach is not typically test-based. But is it ‘vegie’ maths?—theory/skill versus application/context It is often argued that such a functional and applied learning approach, compared to a more skills-based approach, ‘dumbs down’ the mathematics skills and standards being taught, and that the development of more formal mathematics skills and knowledge is not an explicit or important aspect. It is assumed that the focus is solely on practical and hands-on applications and knowledge. However, in an applied learning or investigative approach there is equal importance put on both theory/skills and application. The difference with an applied learning approach will be that the starting point will often be the context and application (projects, investigations, and the like)—not the theory and abstract skills. The theory and skills behind what is being learnt are drawn out and developed from the practice and context, explained and taught, and then returned to an application or context for reinforcement and further practice or for transfer to another context. This approach helps the transferability of skills to new contexts, as shown in Figure 1.1. Context and application

Theory and skills Figure 1.1: The interrelationship between theory and context

It is the task of the teacher to pull together and teach the mathematics content and skills behind the contexts being investigated. This contrasts to most traditional classroom practice, where the mathematics skills are often taught first, and then may be applied via a classroom or textbook based word problem, if at all. Mathematical skills and different types of mathematical knowledge An important discussion from a mathematics education perspective involves the mathematical knowledge that students need to develop during their school years. What types of mathematical knowledge do students need to learn and

13


16/6/06 11:59:24 AM


what capabilities do they need, and how do these relate to how we teach mathematics? In particular, how do different forms of mathematical knowledge relate to how we teach mathematics in schools? In Table 1.1 Paul Ernest distinguishes six different types of mathematical knowledge and capabilities (Ernest, 2004, 317). These are not intended to be seen as mutually exclusive, but as a set of different foci for mathematics education. Table 1.1 Different types of mathematical knowledge and capabilities

Type of knowledge Associated capabilities or capability

1. Utilitarian knowledge

To be able to demonstrate useful mathematical and numeracy skills adequate for successful general employment and functioning in society

2. Practical, workrelated knowledge

To be able to solve practical problems with mathematics, especially industry and work centred problems

3. Advanced specialist knowledge

To have an understanding and capabilities in advanced mathematics, with specialist knowledge beyond standard school mathematics (including advanced high school specialist study of mathematics to knowledge of university and research mathematics)

4. Appreciation of mathematics

To have an appreciation of mathematics as a discipline including its structure, subspecialties, the history of mathematics and the role of mathematics in culture and society in general

5. Mathematical confidence

To be confident in one’s personal knowledge of mathematics, to be able to see mathematical connections and solve mathematical problems, and to be able to acquire new knowledge and skills when needed

6. Social empowerment through mathematics

To be empowered through knowledge of mathematics as a highly numerate critical citizen in society, able to use this knowledge in social and political realms of activity Source: Ernest, 2004, 317

14


16/6/06 11:59:24 AM


Of these six types of mathematical knowledge and capabilities, the main aim and focus of much of traditional senior school mathematics (and the years leading up to it) seems to be to enable some students to attain Category 3: Advanced specialist knowledge. However, an applied learning or investigative approach to teaching mathematics does not preclude this from happening—as Forman and Steen (1999) argue in relation to a functional mathematics curriculum, there is rich and complex mathematics embedded in everyday tasks and work-related tasks. In contrast, the other five categories are all compatible with and all fit in with the arguments and aims discussed above about an applied learning or investigative approach to teaching mathematics, and can often get ‘left aside’ in a traditional mathematics classroom. Utilitarian knowledge would seem to be a basic and minimal requirement for all school leavers, but many students don’t achieve this because of their disengagement from mathematics and their lack of ability to relate and transfer their classroom knowledge to everyday situations. Practical, workrelated knowledge is similar to Category 1, but more specific to training and work-related skills, and is often not a focus of school maths, except maybe as part of vocational programs. However, under an applied learning approach this could be incorporated if students were interested in particular workrelated skills. The final three categories are key aims of an applied or investigative approach: Appreciation of mathematics, Mathematical confidence and Social empowerment through mathematics. These should be aims for all students at all levels—but too many students are disengaged and uninterested as a consequence of traditional mathematics education practices for this to happen. When is the history and culture of mathematics taught in a traditional mathematics classroom? Mathematical confidence is core and crucial for anyone to experience success in the other five types of mathematical knowledge—but it is something that is often lacking in students when they leave school. Category 6, Social empowerment through mathematics, relates to the concept of critical numeracy described earlier, but again often the lack of success in the other categories means this category is not achieved for many students. Ernest’s six categories of mathematical knowledge and capabilities can be used as a critical tool to see how an applied learning approach could encompass a wider range of mathematical knowledge and skills than typically occur in a traditional approach.

15


16/6/06 11:59:24 AM


COV E R I N G M AT H E M AT I C S C U R R I C U L U M CONTENT AND DESIGNING A PROGRAM How can teaching mathematics using an applied learning and investigative approach enable a teacher to cover all the different content areas of the school mathematics curriculum? Won’t there be gaps, especially in areas such as algebra? And won’t more capable students be bored and not challenged? How can mathematics content be mapped against curriculum content? The next chapter addresses these issues in more detail, and gives descriptions and ideas for how this can be achieved. Suffice to say here that mathematics content and understanding is embedded within most everyday tasks and activities—but often people don’t realise what mathematics they are using, and they ignore or avoid addressing the mathematics they do. Often people cannot connect their informal uses of mathematics with the formal mathematical skills taught in the school classroom. It is often assumed that ‘mathematics’ is only or mainly a higher-level abstract discipline used by professional mathematicians in their university or research offices, or is the domain of professionals such as engineers and scientists. As such, this higher-level formal and abstract mathematical skill and ability is often treated as something special and only achievable by the talented few. School systems and their mathematics curriculum, especially in the middle and upper years of schooling, are often designed to educate all students for the possibility that they might need this level of formal mathematical knowledge, even if only a minority end up needing or using such mathematics. However, many students do not see this type of mathematics as relevant or important, and we also we teach this higher-level mathematics at the expense of teaching and learning about the wide range of practical mathematics skills that most people use—in their everyday lives, at home, in their community and at work. Many occupations and trades use a wide range of mathematics every day—shopkeepers, builders, electricians, plumbers, dressmakers, chefs, carpenters, mechanics, gardeners and landscapers, sportspeople, drivers, bookkeepers, asphalters and pavers, and so the list continues. Where in traditional secondary school mathematics classrooms do we learn about and connect with the mathematics people use every day and in these occupations? An applied learning and investigative approach provides a coherent and sound basis for addressing many of these issues.

16


16/6/06 11:59:24 AM


DOES IT WORK? Applied learning curriculum in practice Ongoing work with teaching mathematics to adults through, for example, the Certificates in General Education for Adults (CGEA) in Australia seems to indicate that an adult/applied learning approach works with students who have been (generally) unsuccessful at school mathematics. Yet it also includes some more formal school mathematics—called ‘Numeracy for knowledge’ (for example, see Tout, 2004). In the Netherlands there has been a longstanding tradition of using such an applied learning approach, called Realistic Mathematics Education (RME), which was initiated in the Netherlands during the 1970s. RME starts from the assumption that students should be given the opportunity to reinvent mathematics for themselves and that the subject matter should be ‘real’ for them. It focuses on functional contexts and applications, values problem solving and different mental models of thinking, and encourages interactions between students (Gravemeijer, 1994; Matthijsse, 2000). In Australia, the Victorian Certificate in Applied Learning (VCAL) certificate was introduced in 2003, and is explicitly based on an applied learning approach. It also seems to be having success with senior secondary school students who were not succeeding in mainstream mathematics education. Outcomes from Jo Boaler’s research into school practices Probably the most explicit research into the success or otherwise of an applied learning or investigative approach to teaching mathematics in secondary schools comes from a comprehensive longitudinal study of years 9 to 11 students done in the UK (Boaler, 1997). Boaler’s research indicates that such an applied learning/investigations/problem-solving approach to teaching mathematics at these crucial years can make a significant difference. Two schools in England were the focus for her research. In one of them, the teachers taught mathematics using whole-class teaching and textbooks and the students were tested frequently. Students were taught in tracked groups, there were high standards of discipline and the students worked hard. The second school was chosen because its approach to mathematics teaching was completely different. The students worked on open-ended projects in heterogeneous groups in every mathematics lesson, teachers used a variety of methods and a relaxed approach was taken in classroom management.

17


16/6/06 11:59:25 AM


Over a three-year period Boaler monitored a year cohort of about 300 students at both schools, from when they were 13 to when they were 16. She observed about 100 lessons at each school, interviewed the students, gave out questionnaires, conducted various different assessments of their mathematical knowledge and analysed the students’ responses to the British National school leaving examination in mathematics. At the beginning of the research period, the students at the two schools had experienced very similar mathematical approaches and, at that time, they demonstrated comparable levels of mathematical attainment on a range of tests; there were also no significant differences in terms of sex, ethnicity or social class between the two groups of students. At the end of the three-year research period the students had developed in very different ways. Main results/outcomes The main results of the research indicated that the applied learning or investigative approach gave the following benefits to students: • Positive attitudes to maths. There was a lack of mathematics anxiety, and they enjoyed their maths and had fun. These students appreciated and valued mathematics much more. • Transferability of mathematical skills. These students understood and could apply their mathematics inside and outside the classroom. In realworld situations the students from the traditional environment were unable to use the mathematics they had learned, because they could not adapt it to fit unfamiliar situations. They also could not see the relevance of the mathematics knowledge they had acquired in school for situations outside the classroom—they thought that success in mathematics involved learning, rehearsing and memorising standard rules and procedures. • Skills remained with them. Students at the two schools attained similar grades on short written tests immediately after finishing work, but although students at the textbook school soon forgot what they had learned, the project students did not. • Little or no underachievement or anxiety for girls. The project-based approach was also more equitable, with girls and boys attaining the different grades in equal proportions. • Still successful academically on tests. One of the more surprising results was that students at the project-based school attained significantly higher grades in the national mathematics examination. Boaler’s thesis is that differences in the performances of the students at the two schools were not to do with good teaching at one school and bad teaching

18


16/6/06 11:59:25 AM


at the other, they were to do with different mathematical approaches and the effectiveness of an open-ended, investigative and varied teaching for broad mathematical competence.

CONCLUSION The approach described in this text asks that teachers teach mathematics from a contextual and task-based or investigative point of view, where mathematics is developed from an actual or modelled situation or task in which the mathematics is embedded. This chapter has argued that teaching mathematics this way promotes not only an understanding of mathematical knowledge, but also the ability to transfer that knowledge and apply it and use it productively outside the classroom, empowering more students to succeed, progress and use mathematics constructively and critically. It can engage and motivate students in order for all students to succeed in learning and applying mathematics to whatever level they are capable of. It is a way to achieve the aim of empowering as many students as possible to become more numerate citizens and use their mathematics skills to contribute to their communities now and in the future. This approach supports the view about mathematics learning in schools that is now frequently espoused in the aims of mathematics education: Mathematics … has a fundamental role in enabling cultural, social and technological advances and empowering individuals as critical citizens in contemporary society and for the future. (VCAA, 2005, 53)

References Bishop, A 1994, ‘Cultural conflicts in mathematics education: Developing a research agenda’, For the learning of mathematics, vol. 14, no. 2. Bishop, A 2000, ‘Overcoming obstacles to the democratization of mathematics education’, in H Fujita (ed.), Abstracts of plenary lectures and regular lectures, ICME9, International Congress on Mathematical Education, Japan. Boaler, J 1997, Experiencing school mathematics, Open University Press, Buckingham, England. Bynner, J & Parsons, S 1997, Does numeracy matter? Evidence from the National Child Development Study on the Impact of Poor Numeracy on Adult Life, Basic Skills Agency, London. Crowther, G 1959, 15–18: Report of the Central Advisory Council of Education (England), vol. 1, HMSO, London.

19


16/6/06 11:59:25 AM


Davis, R, Maher, C & Noddings, N (eds) 1990, Constructivist views on the teaching and learning of mathematics, National Council of Teachers of Mathematics, Reston, VA. Ernest, P 1998, Social constructivism as a philosophy of mathematics, State University of New York Press, Albany, NY. Ernest, P 2004, ‘Relevance versus utility: Some ideas on what it means to know mathematics’, in B Clarke, DM Clarke, G Emanuelsson, B Johansson, DV Lambdin, FK Lester, A Wallby & K Wallby (eds), International perspectives on learning and teaching mathematics, National Centre for Mathematics Education, Göteborg University, Göteborg, Sweden. Foley, G (ed.) 1995, Understanding adult education and training, Allen & Unwin, Sydney. Forman, SL & Steen, LA 1999, Beyond eighth grade: Functional mathematics for life and work, National Center for Research in Vocational Education, University of California, Berkeley, CA. Gal, I, van Groenestijn, M, Manly, M, Schmitt, MJ & Tout, D 2005, ‘Adult numeracy and its assessment in the ALL survey: A conceptual framework and pilot results’, in TS Murray, Y Clermont & M Binkley (eds), Measuring adult literacy and life skills: New frameworks for assessment, Statistics Canada, Ontario, Canada, viewed 24 February 2006, (http://www.statcan.ca/cgi-bin/downpub/listpub. cgi?catno=89-552-MIE2005013) Gerdes, P 1994, ‘Reflections on ethnomathematics’, For the Learning of Mathematics, vol. 14, no. 2, 19–22. Gravemeijer, KPE 1994, Developing realistic mathematics instruction, Freudenthal Institute, Utrecht, Netherlands. Grouws, DA & Cebulla, KJ 2000, Improving student achievement in mathematics, International Academy of Education and International Bureau of Education, Brussels. Harris, M 1991, Schools, mathematics and work, Falmer Press, Basingstoke. Harris, M 1997, Common threads: Women, mathematics and work, Trentham Books, Stoke on Trent, Staffordshire. Hatano, G 1996, ‘A conception of knowledge acquisition and its implication for mathematics education’, in L Steffe et al (eds), Theories of mathematical learning, Lawrence Erlbaum Associates, Mahwah NJ. Hollingsworth, H, Lokan, J & McCrae, B 2003, Teaching mathematics in Australia: Results from the TIMSS 1999 video study, ACER, Melbourne. Johnston, B 1994, ‘Critical numeracy’, Fine print, vol. 16, no. 4, Summer, VALBEC, Melbourne. Kidd, JR 1978, How adults learn, Prentice-Hall, New Jersey. Lambdin, DV & Lester, FK 2004, ‘Teaching mathematics through problem solving’, in B Clarke, DM Clarke, G Emanuelsson, B Johansson, DV Lambdin, FK Lester, A Wallby & K Wallby, International perspectives on learning and teaching mathematics, National Centre for Mathematics Education, Göteborg University, Göteborg, Sweden. Lingard, B & Ladwig, J 2001, Queensland school reform longitudinal study: Final report, Education Queensland, Brisbane.

20


16/6/06 11:59:25 AM


Long, VM 2004, ‘Adding “place” value to your mathematics instruction’, in B Clarke, DM Clarke, G Emanuelsson, B Johansson, DV Lambdin, FK Lester, A Wallby & K Wallby, International perspectives on learning and teaching mathematics, National Centre for Mathematics Education, Göteborg University, Göteborg, Sweden. Lovitt, C 2003, Investigations as a central focus for a mathematics curriculum, AAMT Virtual Conference 2003. McIntosh, A 2003, ‘A typical Australian year 8 mathematics lesson?’, in H Hollingsworth, J Lokan & B McCrae, Teaching mathematics in Australia: Results from the TIMSS 1999 video study, ACER, Melbourne. Mackeracher, D 1996, Making sense of adult learning, Culture Concepts, Toronto. Malone, P & Taylor, P (eds) 1993, Constructivist interpretations of teaching and learning mathematics, Curtin University of Technology, Perth Matthijsse, W 2000, ‘Adult numeracy at the elementary level: Addition and subtraction up to 100’, in I Gal (ed.) Adult numeracy development: Theory, research, practice, Hampton Press, Cresskill, NJ. Nelson, D, Joseph, GG & William, J 1993, Multicultural mathematics: Teaching mathematics from a global perspective, Oxford University Press, Oxford. Nunes, T, Schliemann, A & Carraher, D 1993, Street mathematics and school mathematics, Cambridge University Press, Cambridge. Powell, AB & Frankenstein, M (eds) 1997, Ethnomathematics: Challenging Eurocentrism in mathematics education, State University of New York Press, Albany, NY. Rogers, A 1996, Teaching adults, Open University Press, Buckingham, UK. Schmitt, MJ 2000, ‘Developing adults’ numerate thinking: Getting out from under the workbooks’, Focus on BASICS, vol. 4, issue B, Sept 2000, National Center for the Study of Adult Learning and Literacy, Harvard University, Boston, MA. Tout, D 2004, ‘Curriculum frameworks and change’, in B Clarke, DM Clarke, G Emanuelsson, B Johansson, DV Lambdin, FK Lester, A Wallby & K Wallby (eds), International perspectives on learning and teaching mathematics, National Centre for Mathematics Education, Göteborg University, Göteborg, Sweden. Tout, D & Schmitt, MJ 2002, ‘The inclusion of numeracy in adult basic education’, in J Comings, B Garner & C Smith (eds), The annual review of adult learning and literacy: Volume 3, Jossey-Bass, San Francisco. VCAA 2005, Victorian essential learning standards, Victorian Curriculum and Assessment Authority, Melbourne. Walkerdine, V 1989, Counting girls out, Virago Education and University of London Institute of Education, London. Willis, S 1989, Real girls don’t do maths: Gender and the construction of privilege, Deakin University Press, Geelong. Willis, S (ed.) 1990, Being numerate: What counts?, Australian Council for Educational Research, Melbourne. Zaslavsky, C 1994a, ‘Africa counts’ and Ethnomathematics’, For the learning of mathematics, vol. 14, no. 2. Zaslavsky, C 1994b, Fear of math: How to get over it and get on with your life, Rutgers University Press, New Brunswick, NJ.

21


16/6/06 11:59:26 AM


Website http://www.vcaa.vic.edu.au/vcal/index.html Victorian Curriculum and Assessment Authority (VCAA) Victorian Certificate of Applied Learning website SUMMARY

• A response to dissatisfaction with what children are learning—or not learning—in mathematics classes, especially in the middle to latter years of secondary schooling, requires a different approach to the teaching of mathematics. It is needed if more students are to succeed in mathematics in these crucial years of schooling and become numerate adults. • Rich mathematics exists in many aspects of life including many everyday occupations and trades—this needs to be brought into the mathematics classroom and used as a stimulus and starting point for learning about and investigating mathematics. • Research from ethno-mathematics, functional mathematics, numeracy education and gender, backed up by the constructivist philosophy of knowledge and learning, argues for an approach that moves away from aspects of the traditional, transmission model of teaching and learning mathematics and that incorporates engaging in different ways with a wide range of mathematics content and skills that are embedded within social contexts and purposes. • Teaching mathematics based on applied or contextualised learning principles means that teachers teach mathematics from an actual or modelled situation or task in which the mathematics is embedded. Investigations and projects are used as vehicles for learning mathematics. • It is a way to achieve the aim of empowering as many students as possible to become more numerate citizens.

22


16/6/06 11:59:26 AM

CHAPTER

2

DESIGNING TEACHING PROGRAMS INTRODUCTION This chapter provides some ideas for and ways to plan and design programs based on an applied or contextualised learning approach to teaching mathematics. This includes brainstorming to identify themes, contexts and the starting questions/tasks; identifying the embedded mathematics that needs to be addressed; matching these against curriculum or standards; planning classroom activities and mathematics skills work; and incorporating assessment and reporting requirements. As well, there is some advice about what roles a teacher needs to take in the classroom to support students to learn mathematics this way, and what some of the skills are that students need to learn and be aware of.

HOW TO PLAN Teaching mathematics using an applied or investigative approach can be very demanding—certainly harder than teaching out of a textbook. It requires more planning and preparation if it is going to work successfully. The advance planning helps in terms of: • students having successful outcomes from their learning • teachers effectively managing what is happening in the classroom • teachers being able to successfully track and record the outcomes of their students • teachers being able to meet curriculum outcomes On the other hand, given the more open-ended approach to the learning, there may be unexpected and unplanned diversions—especially by more

23


16/6/06 11:59:26 AM


advanced and independent students. This is to be encouraged, provided it contains relevant and valuable knowledge and mathematical content. The teacher needs to decide how to handle these excursions: How long might it take—can the class and teacher afford the time? Is it valuable? Could it be delayed until later? Figure 2.1 maps out a process for negotiating and planning what to teach under an applied learning approach as described in Chapter 1. This is only a suggested process, and can be adapted to a particular curriculum or group of students. Appendix A provides a planning grid based on this process, which can be used to document a class investigation or project. 1. Negotiate a theme or context The first task is to decide what theme or context will be the topic for students to investigate and work on. It should be of interest to the whole group—it could be a current popular topic, something in the news, an issue of concern or interest to young people, or it could be related to some career or work interest. A list of possible generic themes or topics and related curriculum areas is included in Appendix B. This stage involves a number of processes and decisions. A starting point is identifying the students’ interests. Another important part of this is to discuss what topic you might choose with other teachers who teach the same group of students—their literacy or English teacher, their science teacher, and so on. This provides ideas of what the students might be working on in their other classes—if topics are chosen for mathematics that are compatible and related to what students are learning in their other subjects, this provides a more comprehensive and coordinated, cross-curricular approach to teaching the topic. This will not only benefit the students, but also the teacher. Teachers will know that students are learning about all aspects of the topic—the language and literacy aspects, along with other disciplines if relevant. It needs to also be remembered that the one topic or theme may not be of interest to all students in the group. It is important, therefore, that students are involved in the decision-making process if possible, so that they can feel that they own it in some way. Even if the final decision is to choose a subject that is not of interest to all students, then at least those students who didn’t choose the topic may feel that it was an area that other students did want to study and might therefore buy into the task instead of feeling no ownership or involvement at all.

24


16/6/06 11:59:27 AM

CHAPTER 2 Designing teaching programs

1. Negotiate a theme or context

Find a theme or context of interest to the group of students—current popular topic, issue of concern or interest, or work.

2. Brainstorm

Investigate what possible areas and topics exist that could arise out of the context/issue.

3. Identify a starting question/task

Decide on the actual starting point and initial question(s) to initiate the student investigation or task—often stage 2 will lead to too many options and pathways.

4. Identify the mathematics content

Investigate the mathematics that is embedded within the chosen topic and theme and the particular starting question. Refine the question/task if needed.

5. Map against the curriculum

Map the mathematics identified against any curriculum outcomes or descriptions that have to be met. Identify any gaps and refine the question/task if needed.

6. Identify the resources needed

Based on the above stages, identify the resources needed for the students to undertake the task/investigation and for teaching the mathematics content and skills.

7. Plan assessment

At the planning stage include decisions about how the students will be assessed—needs to be ongoing, formative assessment not just summative assessment at the end.

8. Do it—teach/ facilitate

Students undertake the investigation—the teacher is mainly a facilitator and resource person, but intervention to teach the mathematics and problem-solving skills is necessary.

Figure 2.1: A process for negotiating and planning what to teach under an applied learning approach

25


16/6/06 11:59:27 AM


There are several ways to involve students in the negotiation: talking and discussing with them, suggesting topics, brainstorming with them, working in small groups to develop ideas, asking them to write down ideas/topics on cards, and so on. It also needs to be noted that students may not even realise that they can study mathematics in this way—they may say ‘There’s no mathematics in sport’ and so on. The teacher may need to illustrate how this will work with some examples. In some cases, students might come up with topics that seem to have very little to do with maths—in this case the challenge is for the teacher to see what mathematics can be found that is embedded somewhere in the topic. An illustration of this was with a group of teachers teaching the Victorian Certificate of Applied Learning (VCAL) in a secondary college. The teachers decided that they would work on the same topics and themes across the curriculum, and negotiate this with the group of students. During this negotiation the students and teachers decided that a hot topic at that time for the group to work on was body piercing. Not surprisingly, the mathematics and numeracy teacher wondered how he might teach any mathematics involved with body piercing. He opened this up to his students, and eventually it was negotiated that the mathematics task was to be: ‘How many body piercings were possible on your body?’ Two weeks later they had completed the task in an engaging, hands-on and fun way, with students collaborating and deciding how they could solve the problem. The students made life-sized nets/maps of their bodies and calculated their surface areas and, by using agreed assumptions about how much area they needed for a body piercing, were able to calculate how many body piercings were possible. They successfully used measurement, space and shape, and number skills to undertake this investigation. It also needs to be acknowledged that negotiation is a two-way process— not just one-way from the students’ perspective. The teacher is there as an educational leader, and has a vital role to play with their knowledge and expertise about what investigations are achievable and manageable, and also what mathematics needs to be covered to meet curriculum guidelines. 2. Brainstorm A good way to investigate and document possible areas and topics that could arise out of the chosen context/issue is to brainstorm the topic. This usually shows a wide range of related areas that could be developed further from the central theme. Don’t worry at this stage about the mathematics content— finding what mathematics lies behind them comes later.

26


16/6/06 11:59:27 AM


What often happens is that the brainstorm will show up too many possible topics to investigate and work on—the challenge is to narrow them down to one or two main starting points or questions. This is the next stage. An example of a brainstorm diagram on the topic of cars is included in Appendix C. 3. Identify a starting question/task From the brainstorm diagram and/or lists from stage 2, there is the need to focus on one or two related areas that might be of most interest to students and also enable a good range of mathematics to be investigated. This stage involves making the decision on the actual starting point and initial question(s) that will initiate the student investigation or task. It is often required because the brainstorm stage leads to too many options and pathways, and because there needs to be a focus to the research and investigation that is to occur. An essential ingredient here, in order to meet many of the successful outcomes outlined in Chapter 1, is that the investigation and task enables students to follow their own particular approaches to the learning—using their own learning styles and problem-solving techniques, building on their own interests and skills, and using their existing connections with the topic. The best way of achieving this is to start with an open-ended question—one that connects to the chosen theme, but that enables students to go off in their own directions and use their own skills and knowledge. On the other hand, the question cannot be too open-ended, as this could lead to students doing completely different tasks and investigations, requiring quite different mathematical skills to be used and developed. So there is some need for the teacher to structure the question and task in such a way that most student investigations will require related and similar mathematics skills and understandings. In relation to the sample brainstorm provided in Appendix C about cars, some possible open-ended questions or tasks to pose to students could be: • If you were given $10 000 to buy a car, what car would you buy? Why? Describe the car you would buy and why you wanted it. • How would you pay for a car worth $12 000? (Compare at least three different ways.) • Choose a car you like and work out what its running costs would be. • Are young people more likely to die in car accidents than older drivers? • How does alcohol affect driving? • What does blood alcohol concentration (BAC) mean and how do different drinks affect BAC?

27


16/6/06 11:59:27 AM


• How do global positioning systems work in cars? In most cases, only one of these would be chosen for the class, although you may be able to combine two related questions. 4. Identify the mathematics content It is then important from a teaching perspective to be able to identify the mathematics that is embedded within the chosen topic and that lies behind the particular starting question. This requires further investigation and research into the theme and question, which can be done via a more detailed brainstorm or by listing the possible areas that arise out of the question. This requires the teacher to anticipate the possibilities and options that might arise, as a mechanism for planning what teaching activities and resources will be needed. It also provides the opportunity to refine the question/task if needed. As mentioned already, unexpected and unplanned side-trips can occur, and there is no way to predict these. The teacher needs to decide how to handle these excursions at the time. Appendix D provides an example of how this could look for one of the car questions above. 5. Map against the curriculum If there is the requirement to meet specific curriculum outcomes, this needs to be done next. Using the analysis of stage 4, map the mathematics identified against any curriculum learning outcomes or descriptions that have to be met. The major task is to identify any gaps and, if possible, refine the question/task if needed so that these gaps will be addressed and can be assessed. Sometimes, this is an easy thing to do, as it fits in naturally with what has already been decided, and it may simply require the teacher to pose an extra stimulus question to target students’ activities and work. However, if it does not fit in naturally to the investigation being planned, it is best to leave it aside and wait for a more natural topic to cover the mathematics skills. The teacher can write up the activities and questions to be posed against the learning outcomes, and this could easily incorporate the assessment aspects of the activity.

28


16/6/06 11:59:28 AM


6. Identify the resources needed The teacher then needs to identify the resources and equipment that the students will need to undertake the task/investigation and for teaching the related mathematics content and skills. These could include: • practical resources and equipment required to undertake the investigation (e.g. measuring tools, maps and plans, newspapers, magazines) • specialist facilities or rooms such as access to a kitchen if there is cooking involved, a workroom if building or making something. This could include resources outside the classroom or school (e.g. a sports ground or stadium, museum, workplace). • access to calculators, computers including specific software, and the internet (the section ‘Information and communication technology’ (ICT) also discusses how ICT may be used) • people resources—from within the school or from the community (e.g. a local car dealer to talk about buying a car) • worksheets and instructions for students that may need to be prepared, especially for less independent or less capable students who need more guidance and direction • supporting teaching materials for the teaching and practising of skills (e.g. games, activity sheets, textbooks) Appendix E includes a more detailed listing of possible resources that are relevant. 7. Plan assessment At the planning stage it is vital to make decisions about how the students will be assessed. Assessment needs to be ongoing and formative, as well as summative assessment at the end of the topic. In an applied investigative task, much of the learning and achievement of outcomes occurs during the process of undertaking the task, so it is important to be ready to record and document those achievements as they occur. Leaving it to a final assessment or evaluation activity may miss out on those outcomes and therefore not give credit for the students’ achievements. A number of assessment options and methods of recording student outcomes are suited to applied learning. These could include: • completion of a journal/diary • participation in group tasks • oral presentations

29


16/6/06 11:59:28 AM


• teacher checklists and observation sheets • brochures, posters and/or presentation software • actual products or reports from projects undertaken (this could include digital photos or videos of students working on the tasks) • student documentation and reports of planning, research and work When undertaking project-based activities, a good strategy for teachers to use is to develop an observation recording sheet in advance so that as a student is observed performing a skill the teacher can easily annotate the sheet to indicate this was achieved. For a comprehensive discussion on assessment in numeracy, including examples and templates for observations sheets, see Rethinking assessment (Marr, Helme & Tout, 2003). 8. Do it—teach/facilitate And finally, the purpose of all this planning—the students undertake the investigation. The teacher’s role is mainly as a facilitator and resource person. It is vital that the teacher monitors each student’s progress and actions, and intervenes when necessary to teach any identified mathematics and problemsolving skills that are necessary or missing for the task being tackled. Any mathematics or problem-solving skills that need to be explained and taught in order for the students to successfully achieve the learning and the outcomes expected can be introduced in a number of ways, depending on the skills and experience of the students. This could happen through: • whole-class activities and explanations prior or during the investigation as questions arise from the student’s work • small group activities based on explanations, worksheets or tasks provided by the teacher • individual skills and practice sessions—off worksheets, out of textbooks or on computers and/or the internet if activities are available

THE TEACHER ROLE In an applied learning or project-based mathematics classroom the teacher plays a crucial role. The students need to be supported and guided in order to learn the necessary skills to achieve the desired outcomes. Without this support and direction, the investigation and associated learning will not succeed. The teacher’s main role is as a facilitator—there to offer support and advice when needed, provide the necessary scaffolding and teach skills when necessary. It is vital that teachers remember to teach and instruct their

30


16/6/06 11:59:28 AM


students in any particular skill or piece of mathematical knowledge that is required for the task. The mathematical skills to develop in students In this style of teaching and learning, there are a range of different skills that students need to learn. Teachers need to develop students’ abilities to: • identify and recognise how mathematics can be used in real-life situations and contexts, enabling them to make connections between mathematics and the real world (a new skill to many students). This is vital to support students to be able to transfer their skills to new contexts. • undertake a range of mathematical operations, applications and processes including measuring, counting, estimating, calculating, drawing, modelling and discussing (the underlying mathematical skills and knowledge required to undertake the required investigations or tasks) • reflect and think about the mathematics they have been using: that is, interpret the results and outcomes of the investigation they have undertaken, including how appropriately and accurately the results fit the situation (another new skill to many students) • represent, communicate and discuss the results and outcomes of the use and application of mathematics in terms of their chosen investigation or project—this includes both written and oral skills and should incorporate the use of appropriate technologies Classroom activities Depending on the requirements of the task, the teacher will use the motivation of the investigation to have ready a number of classroom activities. These will need to include: • the work on the investigation/project itself—the initial and ongoing main aim of the activity • by choosing appropriate intervention times, the teaching and instruction of particular knowledge and skills—what is required to get the job done (e.g. if the task requires measurements to be made, then an understanding of measurements and the metric system is fundamental knowledge to be taught) • both whole group, small group and individual work on skills and practice on the underlying mathematics skills While the project or investigation is the starting point and the motivating factor, all these activities would occur in a normal classroom over a period of time. This is illustrated in Figure 2.2.

31


16/6/06 11:59:28 AM


context work projects/tasks

theory & knowledge

skills & practice group work

skills & practice individual work Figure 2.2: Classroom activities

Independent learners One important question about how a teacher might construct the investigation and the classroom activities relates to how independent and capable the students are. Some students may be quite dependent learners with limited independent learning skills and learning-to-learn skills. A teacher needs to take this in to account and help students move from being dependent to becoming more independent learners. What this means in the classroom is that lessons and activities need to progress from structured to less structured; from more directed and closed activities to more open ones; from modelled to less modelled; from supported activities to less supported. So the teacher role moves from being a supervisor to being more of a facilitator, scaffolding the learning for the students. Activities and tasks may progress from being provided as small chunks/tasks to being given as larger tasks/projects. Within the same classroom, some students may undertake the task based simply on the starting open-ended question, while others who are less independent or who are lacking specific content knowledge or problem-solving skills might be provided with structured questions, tasks and information that would enable them to still solve the problem. This supports all students—no matter what their level of skill—to be successfully involved in the same investigation and content. The more capable students may go way above and beyond what the teacher wanted or expected, but this is one of the positive benefits of teaching this way. Another way of working with such mixed ability groups in an applied learning classroom is to get students to work in small mixed ability groups,

32


16/6/06 11:59:29 AM


where the more capable students help the less capable students. Often this means that all students benefit: the students doing the teaching come to a better understanding of the skills or knowledge being imparted and the learner may understand the content better because it is being explained in the language of their peers. Managing student investigations When the range of different directions for student investigations on a topic or theme is considered, the tasks of resource provision and availability, classroom management, student assistance and supervision all become significant. Practically, the teacher will need to set guidelines and limits within which the student investigations will be conducted. Teachers can manage this process through an initial negotiation phase with students. A pro-forma that can be used in the negotiation, outlining directions, processes and intended outcomes of the investigation, is included as Appendix F. In completing the ‘contracts’, teachers and students will need to give consideration to factors such as: • the size and composition of student groups. Groups of 3 to 4 students per investigation will assist overall planning and coordination by teachers. • the overall directions and outcomes (including the final product) of the investigation. These should be complementary to the subject requirements and those of individual students; that is, the investigation will enable the students to demonstrate some of the required outcomes for the topic and support achievement of all outcomes over the term of the investigation. • a timeline and process for the investigation that gives consideration to the roles of the group members, any required resources and the nature of activities. Groups specify any requirements for research time, ‘field’ time, and so forth. • elaboration of required teaching of mathematical concepts and skills

I N FO R M AT I O N A N D CO M M U N I CAT I O N TECHNOLOGY ( ICT) Students should make use of information and communication technology, such as computer-based learning, multimedia and the internet where appropriate. The internet/world wide web will often be the source of useful and appropriate mathematical information and data that can be used as the basis of students’ research.

33


16/6/06 11:59:29 AM


It is important that teachers support students to be able to select and use appropriate technology to apply the mathematics efficiently to their tasks. These skills should include the ability to: • choose the appropriate technology for a given task or context • use a calculator for computation • produce tables of values, graphs, diagrams • organise and present information In Appendix E there is a listing of some of the relevant computer software that might be suitable. Reference Marr, B, Helme, S & Tout, D 2003, Rethinking assessment: Strategies for holistic adult numeracy assessment: A resource book for practitioners, policy-makers, researchers and assessors, Language Australia, Melbourne. SUMMARY

• In order to teach effectively using an applied learning approach, it is important to be prepared and to plan in advance. This includes brainstorming to identify themes and contexts and the starting questions/tasks; identifying the embedded mathematics skills; matching these against curriculum or standards; planning the supporting classroom activities and mathematics skills work; and also addressing any assessment and reporting requirements. • The teacher’s role is as a facilitator—there to offer guidance, support and advice, and provide the necessary scaffolding and teaching of skills when necessary. • Classroom activities will include the main investigation/project work itself, but supported by the teaching and instruction of particular knowledge and skills, and a combination of whole group, small group and individual work. • Information and communication technologies are important tools and media in applied learning.

34


16/6/06 11:59:29 AM

CHAPTER

3

MAKING MEANING INTRODUCTION A key skill in modern society relates to interpreting and reflecting on numerical and statistical information in various documents, texts and other media. This requires people to be able to view, read, understand and interpret such information no matter where or how it is presented. It is crucial in many parts of life—at home, at work, in the community and personally. Documents and texts these days can be in newspapers and magazines; on the internet; on CD-ROMs and DVDs; on TV; in information and leaflets from councils, banks, government agencies, utilities such as electricity, telephone companies and community organisations; in information printed on goods purchased from different sources and so on. To not understand the numerical and statistical information and explicit and implicit messages embedded within such texts prevents people from engaging with, and participating fully in, society. Teaching about reading, understanding and interpreting such numerical, quantitative and qualitative information is an important task that supports students to become numerate and empowered citizens. In some cases it may be critical information (e.g. about health or safety issues) or it may have financial consequences (e.g. about fines, overdue bills or gambling debts), or it simply may be a matter of enjoyment and engagement (e.g. reading a newspaper article of interest or about sports results). Types of mathematical information embedded in texts Most of the numerical and statistical information in such texts will involve these main types of mathematical information: • number • statistics and data analysis • chance

35


16/6/06 11:59:29 AM


• maps, plans and diagrams Some may require understanding of pattern, function and algebra. Mathematical information may be encoded in different ways such as: • a diagram or chart • graphs, tables and formulas • a map or plan of a real entity (e.g. of a city or a project plan) • various types of texts (such as newspaper articles or bank forms) Two different kinds of text may be encountered. The first involves numerical and statistical information represented in words or phrases that carry mathematical meaning. Examples are the use of number words (e.g. ‘three’ instead of ‘3’), basic mathematical terms (e.g. fraction, percent, average, proportion), or more complex phrases (e.g. ‘crime rate increased by almost a quarter’) which require interpretation. The second is where the information is expressed in notations or symbols (e.g. $3.27b, ±10, 24.5%, 450 m2), but is surrounded by text that, despite its non-mathematical nature, also has to be interpreted in order to be understood. Behind the scenes—reading, interpreting and more Not only is it important to be able to read and understand the numerical and statistical information in text. Readers may also need to make a judgment about how the information actually applies to the situation or context, or whether the data is being presented fairly or not. It can also incorporate a critical numeracy aspect, where the purpose of the task, the validity of the data or information presented, or the meaning and implications of the results need to be questioned. The implication for classroom practice is that mathematics or numeracy teachers need to be aware of how crucial the mathematical knowledge embedded within materials and information is, and how vital it is to understanding the messages and information. Therefore teachers need to use examples of such materials in their classrooms to demonstrate how the mathematics is embedded and make explicit to students what role the mathematics plays in first presenting the information, and then in being able to critically review and assess the information and any explicit or implicit meanings. In most instances the information will be quantitative and statistical and require mathematical knowledge related to number and data and statistics.

36


16/6/06 11:59:30 AM

CHAPTER 3 Making meaning

M AT H E M AT I C S A R E A S COV E R E D In Table 3.1, there is a list of possible topics related to reading and interpreting information that could be considered as areas for investigation, and therefore for teaching a range of mathematics skills. The area of interpreting and making meaning from numerical and statistical information included in texts can address many of the different types of mathematical knowledge and capabilities described in Chapter 1. In this area of making meaning of information, Category 6, Social empowerment through mathematics, is a primary purpose and outcome of analysing how information is used and embedded in different texts and materials. It is important to choose as examples texts and materials that are: • topical • of particular interest to the group of students • of general and popular interest There will always be current topics and issues that enable teachers to address the issue of making meaning of information, some of which are listed in Table 3.1. Combined with the knowledge about the different types of mathematical knowledge described in Table 3.2, and wanting to cover as many of these as possible, teachers will be able to use the planning strategies outlined in Chapter 2 to develop a number of learning activities and investigations that involve making meaning of information. The following examples illustrate how texts can be used to analyse the information and address the mathematical content. While the source of the material is from within a particular socio-cultural or political context, various situations of a like type can be readily identified in local materials of interest to the cohort of students. The suitability of material to be used and analysed goes hand in hand with local teacher knowledge—both of the local contexts but also of the student interests. Thus, while the sample materials in Figures 3.1 and 3.2 are drawn from an Australian context, the issues could be easily matched and used in other contexts. The issues of child drug use and gambling (Figure 3.1) are relevant to both developed and developing nations. Similarly in Figure 3.2, sports information, whether on the internet, or on TV or in newspapers, is of international interest in sports such as football (soccer), golf, netball, cricket and rugby, as well as national codes like AFL. Teachers need to use examples of texts such as these to demonstrate how mathematics is embedded and show students what role the mathematics plays in presenting the information and in helping them critically review and assess the information for any explicit or implicit meanings.

37


16/6/06 11:59:30 AM


Table 3.1 Topics related to reading and interpreting information

Topic and context

Areas of mathematics

Medicine and drugs • Medication labels and instructions • Articles and leaflets about drug usage, including alcohol and smoking • Websites about drug use

Number Statistics and data analysis Algebra

General news articles and information, including daily news, current affairs, etc. • Newspaper and magazine articles • Websites on the internet • TV shows including news and current affairs

Number Statistics and data analysis Chance Algebra

Music and entertainment • Newspaper and magazine articles • Radio shows and their websites • Music and band websites on the Internet • TV shows and their websites

Number Statistics and data analysis

Sports articles and information • Newspaper and magazine articles • On the internet • TV and radio shows

Number Statistics and data analysis Chance and probability Measurement

Advertising materials Number • For buying goods and services Statistics and data analysis • Best buys Algebra • Advertisements in newspapers and magazines Financial information • Brochures and leaflets from banks, utilities, government • Bills: utilities such as phones, electricity, etc. • Bank statements and details including via the Internet • Credit card payments and personal loans and finances

Number Statistics and data analysis Algebra

38


16/6/06 11:59:30 AM


Topic and context


Forms including tests • Learner drivers permit • Job application forms • Youth allowance application • Concession card application

Number Statistics and data analysis

Gambling • Popular forms of gambling: Tattslotto and the rest • Casinos and gambling: including financial implications for individuals and society • Sport related gambling—horse racing, other sports betting, etc.

Number Statistics and data analysis Chance and probability Algebra

Workplace documents and information • Employment forms • Standard operating procedures, instructions • Occupational health and safety guidelines • Safety warnings • Chemicals and dangerous goods • Quality control procedures and processes.

Number Statistics and data analysis Chance and probability Measurement Algebra

Home hazards and safety information • Leaflets on goods or their packaging e.g. electrical equipment • Chemicals and sprays


Statistical data and information Data published about national, regional and local data e.g.: • census data • data from research into car accidents, drug use, etc.

Number Statistics and data analysis Chance and probability Algebra

Weather information • Newspapers • the internet • TV or radio


39


16/6/06 11:59:30 AM


Yes. Reading, understanding and interpreting numerical and statistical information is a crucial and practical skill in many spheres of life—work, community and personally.

To be able to demonstrate useful mathematical and numeracy skills adequate for successful general employment and functioning in society.

To be able to solve practical problems with mathematics, especially industry and work centred problems.

To have an understanding and capabilities in advanced mathematics, with specialist knowledge beyond standard school mathematics (including advanced high school specialist study of mathematics to knowledge of university and research mathematics).

• Utilitarian knowledge

• Practical, workrelated knowledge

• Advanced specialist knowledge

Maybe yes, depending on whether the students are interested in examples and situations where higher level skills are required for interpreting and analysing the embedded maths. Examples could be found in higher level statistical analysis required, for example, in quality control processes using quality control charts based on upper and lower control units.

Yes. Would need to illustrate this through some vocational or workplace examples such as standard operating procedures, occupational health and safety instructions or quality control processes.

Covered here?


Table 3.2 Different types of mathematical knowledge and capabilities covered

MATHSWORKS FOR TEACHERS

Foundation Numeracy in Context

40

16/6/06 11:59:31 AM


Yes. Demonstrating how numerical and statistical information is a crucial and practical skill in many spheres of life and that there is vital mathematical knowledge and understanding behind this skill, should show students the significant role mathematics plays in culture and society in general.

To have an appreciation of mathematics as a discipline including its structure, subspecialties, the history of mathematics and the role of mathematics in culture and society in general.

To be confident in one’s personal knowledge of mathematics, to be able to see mathematical connections and solve mathematical problems, and to be able to acquire new knowledge and skills when needed.

To be empowered through knowledge of mathematics as a highly numerate critical citizen in society, able to use this knowledge in social and political realms of activity.

• Appreciation of mathematics

• Mathematical confidence

• Social empowerment through mathematics

(Source—columns 1 and 2: Ernest, 2004, 317)

Yes. This area of knowledge is almost the main purpose of studying this topic interpreting and reflecting on numerical and statistical information in documents and texts is because it supports students to become more numerate and empowered citizens.

Yes. Again, given the crucial and practical nature of understanding numerical and statistical information, working with students to realise this and make the connections between the mathematics and the different texts they will meet should provide a good basis for building their confidence and capacity.

Covered here?



41

16/6/06 11:59:31 AM


SOME EXAMPLES OF USING TEXTS Newspapers

Figure 3.1: Articles from a daily newspaper (reprinted with permission from The Herald and Weekly Times)

42


16/6/06 11:59:31 AM


Analysis of the texts in Figure 3.1 In the articles in Figure 3.1 the numbers embedded in the text contain the main message—if readers cannot read or understand the numbers and related terms, the text becomes meaningless. So, for example, readers need to understand the following mathematics: • understanding and comparison of whole numbers—place value including large numbers and abbreviations (e.g. $100m) • detailed understanding and comparison of decimal numbers and percentages • the importance of use of comparative words—‘jumped to’; ‘down from’; ‘more likely’; ‘topped’; ‘lowest’; ‘marginally more’ etc. In terms of critical numeracy and reading behind the mathematics and the numbers to look at how the information is being presented and what implicit or explicit messages are being given, questions that could be asked might be: • What message does the headline give? • Does the mathematics support this message? • Could there be a different message given based on the information? Questions such as these could be presented to students, and they could be asked to research the information and original source of the data to write up a different article with a different headline. The internet

Figure 3.2: Sports information on the internet (http://afl.com.au, reproduced with permission)

43


16/6/06 11:59:32 AM


Analysis of the text in Figure 3.2 The information and data available from this website is full of numerical and statistical information. It could be used as the basis for investigating and understanding the following mathematics: • understanding and comparison of whole numbers, decimal numbers and percentages • calculations with percentages • the data behind the league ladder and the individual teams and their players could lead to work including: plotting graphs; working out averages (e.g. mean and median kicks per game) Safety information

Figure 3.3: Chemicals at home

Analysis of the text in Figure 3.3 Common chemicals such as bleach are readily available in workplaces and homes and provide a way in to talking about ratios (and safety issues). This text therefore creates an opportunity to address this often difficult and misunderstood topic. It could be used as the basis for investigating and understanding the following mathematics: • understanding of metric volume (and common measures such as fractions of cups) • understanding the concept of and calculations with ratios and proportions

44


16/6/06 11:59:32 AM


Financial information

Figure 3.4: Phone bill extract (Copyright © Telstra, reproduced with permission)

Analysis of the text in Figure 3.4 In bills like this, the information and data available contain relevant and important financial information that is presented numerically and graphically. It could be used as the basis for investigating and understanding the following mathematics: • understanding and comparison of whole numbers and decimals • calculations with numbers • reading and interpreting data and graphs

45


16/6/06 11:59:33 AM


P OS S I B L E S TA R T I N G P O I N T S Below are some possible starting questions to pose to students for investigating the issues and the mathematics embedded within some of the above topics. Sports articles and information Topic: Researching your favourite sport Ask students to: • Find the results and the ladder or league table for their favourite sports competition or for a sport they are interested in investigating. Write up a report explaining and analysing how the ladder is worked out, showing the calculations for awarding points and how the teams are placed in order on the ladder, including their percentages. • Investigate and analyse the scoring system for their sport, and generalise the scoring into an algebraic sentence and equation (see Worksheet E in Appendix G). • Research and find statistics for one team in their chosen sport in at least two different games in the same competition and compare the two results. Represent them graphically and analyse them in terms of averages (mean and median). • Investigate where and how betting is now available for different sports. Write up a report showing the odds and betting system and how they work for at least one sport. Include an explanation on how odds in sport relate to how we calculate probability and chance in mathematics. Also note that Chapter 7 is all about using sport as a context for teaching mathematics. News articles and information Some classroom activities based on news articles (like those in Figure 3.1) would include: • Provide students with copies of the article with the crucial numbers whited out. Get them to work in small groups to try to estimate what the numbers should be. Each group reports back and the class discusses the results of everyone’s estimates. Then hand out the actual figures and discuss the differences and the reasons for these. • Get students to discuss what the messages are that the articles are trying to present. Are there particular messages that are being highlighted at the

46


16/6/06 11:59:34 AM


expense of the truth? Are their arguments supported by the statistics? Is there a bias in the reporting? • Find the original reports or data that the articles are based on and have students write their own newspaper-style article. They are to include at least one graph of the data in their article. Medicine and drugs Topic: Medicine labels and instructions Students could engage in the following activities: • Find out what tablets they and their family have around their home. Make up a spreadsheet that lists them all and details information in terms of their prescription rating, their weights per tablet, their ingredients, the frequency of recommended dosage, any age requirements, cost, etc. Identify which ones are the most dangerous and explain why. • Take a visit to a supermarket (or use the internet) to research the different types and brands of available over-the-counter headache medicines (e.g. paracetamol, neurophen). Compare and analyse the different medicines and brands in terms of their ingredients, recommended dosages and costs. Write a report describing their results. Topic: Articles and leaflets about drug usage • Have students use the internet to research drug usage by young people. They should include information provided by alcohol and drug organisations or see if any of the daily newspapers or TV news or current affairs shows have reported any information recently. (A good starting point is the Alcohol and Other Drugs Council of Australia (ADCA) at http://www.adca.org.au, which has good sources of information and articles, and links to many other related sites.) Students write a report showing the information they found, including some graphical representation of the statistics on drug usage by teenagers. • Get students to investigate how much money is estimated to be tied up with the illegal drug trade in Australia. They should include information provided by alcohol and drug organisations such as the ADCA, and then write a report showing the information they found, including some graphical representation of the information.

47


16/6/06 11:59:34 AM


RESOURCES Below are some specific resources that could be used to support the reading and understanding of mathematical information in texts. For more general resources about teaching mathematics in context, refer to the more comprehensive listing in Chapter 1. Books Erickson, T 1989, Get it together: Math problems for groups grades 4–12, Lawrence Hall of Science, University of California, Berkeley, CA. Flittman, E & Sharrock, H 2000, Foundation mathematics for the real world, Longman, Melbourne. Lawrence, G & Moule, J 1989–90, Mathematics in practice series (Accommodation, Health, Social issues, The consumer), Harcourt Brace Jovanovich, Sydney. Lowe, I 2001, Mathematics at work, (CD-ROM), AAMT, Adelaide. McLennan, W 1998, Statistics: A powerful edge, Australian Bureau of Statistics, Canberra. O’Connor, M & Gaton, B 2000, Foundation mathematics, Oxford University Press, Melbourne. Ritchhart, R 1994, Making numbers make sense: A sourcebook for developing numeracy, Addison-Wesley, Menlo Park, CA. Schmitt, MJ & Steinback, M (eds) 2004–2005, Math: EMPower series, Key Curriculum Press, Emeryville, CA. Thomson, S & Forster, I 1998, Trade and business mathematics 1, Addison Wesley Longman, Melbourne. Thomson, S & Forster, I 1999, Trade and business mathematics 2, Pearson Education, Melbourne. Vize, A 2005, Maths skills for living, Phoenix Education, Sydney. Vize, A 2005, Maths skills for working, Phoenix Education, Sydney. Weber, L 2003, The language of maths (secondary level), AEE Publishing, South Mission Beach, Queensland.

Online resources National statistics bodies and organisations usually offer a wide range of statistical information about a nation and its people. Such data can often be used as the basis for researching and analysing information that is encountered in articles. As well, they often offer specific resources, activities and support for schools. In Australia this is true of the Australian Bureau of Statistics (ABS). Their website has many such resources and information: http://www.abs.gov.au. Their Education Services section is available directly at: http://www.abs.gov.au/websitedbs/ d3310114.nsf/Home/Education%20Resources The Fedstats website at http://www.fedstats.gov is the gateway to statistics from over 100 US Federal agencies.

48


16/6/06 11:59:34 AM


CD-ROMs Hagston, J & Tout, D 2001, Brush up on your skills, TAFE Frontiers, Melbourne.

Student worksheets Two sample worksheets that could be used to support students in undertaking two investigations based on the above outlines are included in Appendix G as Worksheets A and B. As explained in Chapter 2, any mathematics skills that arise out of the investigations, or are identified as required, will need to be explained and taught if the students are to achieve the learning and outcomes expected. Remember, these can be introduced in a number of ways, depending on students’ skills and experience. This could happen through: • whole-class activities and explanations before or during the investigation • small-group activities based on explanations, worksheets or tasks • individual skills and practice sessions References Australian Football League (AFL), Official site of the Australian Football League , accessed 24 August, 2005. Ernest, P 2004, ‘Relevance versus utility: Some ideas on what it means to know mathematics’, in B Clarke, DM Clarke, G Emanuelsson, B Johansson, DV Lambdin, FK Lester, A Wallby & K Wallby (eds), International perspectives on learning and teaching mathematics, National Centre for Mathematics Education, Göteborg University, Göteborg, Sweden. SUMMARY

• The ability to interpret and reflect on numerical and statistical information in documents and texts is crucial in many spheres of life—at home, at work, in the community and certainly personally. • Most of the numerical and statistical information behind reading and interpreting such texts and information will be related to number, statistics and data, chance and maps, plans and diagrams.

49


16/6/06 11:59:34 AM

CHAPTER

4

D R I V I N G A W AY INTRODUCTION The context of cars and driving is always seen as relevant and popular with teenage students who are getting to the age where they can learn to drive and are perhaps thinking about buying their own car. From a community perspective there is the issue of young people and safe driving, including saving young people’s lives or avoiding injury from car crashes. A number of areas are of interest and importance, including: the costs of purchasing, running and maintaining a car; safety and cars including accident rates, car safety, safe driving, alcohol and drugs; and using maps, street directories and satellite navigation systems in cars to get around. There is then the possibility of extending into other possible areas of interest such as the use of other forms of transport, including public transport, or working in the automotive industry, including retail sales, car mechanics and automotive engineering.

M AT H E M AT I C S A R E A S COV E R E D Table 4.1 lists some possible topics related to cars that could be considered as areas for investigation, and therefore for teaching a range of mathematics skills. This relates to the brainstorm discussed in Chapter 2 and provided as Appendix C. As Table 4.1 indicates, this one context of cars can be used to cover much of a traditional mathematics curriculum, as it is full of very different uses and applications of mathematics. Alone, it could take much of a year to cover in its entirety, and students would probably get bored by the topic and it could lose its motivating benefits. The issue, therefore, from a teaching point of view is to decide which of these possible topics are of most interest to the class being taught and/or to negotiate with the students which aspects or topics they

50


16/6/06 11:59:35 AM

CHAPTER 4 Driving away

Table 4.1 Car–related topics

Topic and context


Getting your licence • Road rules—speed limits, parking distances, directions, etc. • Costs

Number Measurement Space and shape Algebra

Buying a car • What car? Comparing characteristics of cars (e.g. fuel economy, safety features) • Costs—including on-road costs, registration, insurance • Paying for a car—different options

Number Measurement Statistics and data analysis Space and shape Algebra

Running a car • Running costs (fuel, repairs and servicing) • Registration and insurance

Number Measurement Algebra

Under the bonnet • Owner maintenance (oil, water, petrol, tyres, etc.) • Servicing (period/distance, costs) • Fixing your own car

Number Measurement

Accidents • The risks—statistics on driving factors (age, drinking, seat belts, types of cars, speed, locations of accidents, etc.) • Safe driving • Repairs and costs


Navigation—getting around • Using street directories and maps • Giving and following directions • Travel—distance, speed, time • Global positioning systems (GPS)


Drinking and driving • Rules and regulations and penalties • Alcohol content and drink driving • Quantities/bottle sizes/costs • Statistics on drink driving and accident rates


51


16/6/06 11:59:35 AM


would be most interested in investigating and learning about. As well, the teacher needs to decide which topics cover specific areas of the mathematics curriculum the best. The theme of cars and driving can address many of the different types of mathematical knowledge and capabilities described in Chapter 1 (see Table 4.2).

P OS S I B L E S TA R T I N G P O I N T S Based on two of the above topics, the following are some possible starting questions to pose to students for investigating the issue and the mathematics embedded within it. Buying a car Topic: What car? Comparing characteristics of cars (e.g. fuel economy, safety features) Ask students to: • Decide on a particular car they’d like to buy second hand (they should choose a fairly common brand of car). Search newspaper car advertisements or car sale sites on the internet and find out what choices they have. Analyse and report on what is available and which car they would buy. • Imagine that they have been given $10 000 to purchase a used car, and decide what car they would buy. Why? Describe the car they would buy and why they wanted it. • There are now some very fuel efficient cars available that use a combination of petrol and electric engines. Research one of these cars and do an analysis and comparison with a normal petrol engine car of a similar size. Consider aspects such as fuel efficiency, purchase cost, running costs and maintenance, insurance, etc. • Find out which is the safest small car (e.g. less than 2-litre engine) available on the commercial market. Explain why. • Analyse how cars depreciate in value. What cars retain their value better? Why does this happen? Illustrate this graphically by comparing a number of different cars. Topic: Costs—including on-road costs, registration, insurance • What are all the extra and hidden costs when you buy a car? How are these costs calculated? Is there a difference when you buy a new car compared to buying a used car?

52


16/6/06 11:59:35 AM

Associated capabilities


To be able to solve practical problems with mathematics, especially industry and work centred problems


Type of knowledge or capability


• Practical, workrelated knowledge



Possibly, yes, depending on whether specific examples and investigations could be found requiring higher level mathematical analysis, such as forensic analysis of the motion of cars involved in a crash.

Yes. Mathematics behind the use of cars in vocational or workplace situations can be important and there are the possible extensions into automotive industry work situations that could be of interest to some students, such as retail sales, car mechanics or automotive engineering.

Yes. Being able to drive cars and/or understand about how they work including use of other forms of transport is important in many spheres of life—work, community and personally.

Covered here?



53

16/6/06 11:59:35 AM


To be empowered through knowledge of Maybe, as awareness of some of these issues such mathematics as a highly numerate critical as accidents and drink driving can have a critical citizen in society, able to use this knowledge impact on students and their lives. in social and political realms of activity



Yes. Again, given the everyday, practical nature of the mathematics behind cars this could enable students to easily make the connections between mathematics and their everyday lives and hence build their confidence.



Yes. Although this topic of cars is a very personal and practical skill, it again may illustrate to students the vital role mathematics plays in everyday life.

Covered here?







54

16/6/06 11:59:36 AM


• Have students choose a car they’d like to buy, and compare at least three different insurance company policies and costs for insuring the car. Which policy would they take out? Why? Topic: Paying for a car—different options • Ask students how they would pay for a car worth $12 000? (Compare at least three different ways.) What would they pay in total for each way? • Have students compare two different options for paying off the same car by hire purchase through a finance company—and represent their results graphically. Navigation—getting around Topic: Using street directories and maps • Have students choose a capital city in a different state that they would like to visit. They should decide on how they would drive there and describe and draw a map of the route, showing distances travelled and the time they estimate it would take, how long they would stop for, including overnight stops, etc. Topic: Giving and following directions • Ask students to determine at least two alternative routes to drive from their school to the nearest hospital emergency department. They should write or record instructions for the driver to follow, and include a sketch map. Topic: Travel—distance, speed, time • Students estimate how long it would take them to drive from their school or home to the nearest major airport, draw a map of the route, and calculate the average speed. Topic: Global positioning systems (GPS) • Pose the question: How do global positioning systems work in cars?

55


16/6/06 11:59:36 AM


E X A M P L E O F M AT H E M AT I C S S K I L L S COV E R E D Example of the types of mathematics skills needed for some of these investigations are elaborated in Tables 4.3 and 4.4. As explained in Chapter 2, these mathematics skills will need to be explained and taught if the students are to achieve the learning and outcomes expected. Remember, these can be introduced in a number of ways, depending on the skills and experience of the students. This could happen through: • whole-class activities and explanations before or during the investigation • small-group activities based on explanations, worksheets or tasks • individual skills and practice sessions

56


16/6/06 11:59:36 AM


Table 4.3 Maths skills covered in ‘Buying a car’

Investigation Decide on a particular car you’d like to buy second hand (choose a fairly common brand of car). Search newspaper car advertisements or car sale sites on the internet and find out what choices you have. Analyse and report on what is available. Steps

Possible underpinning mathematics skills that need to be taught

Research car prices

Number • use place value concepts for whole numbers and decimals to interpret and compare written numbers • use common words for ordering and comparing numbers such as smaller, bigger, larger, less than, etc.

Record prices

Number • read and write whole numbers and decimals Statistics and data analysis • collect, sort and record data in a table

Graph prices

Statistics and data analysis • draw an appropriate graph for the data, labelling the graph and axes and marking in the scale

Analyse prices

Statistics and data analysis • interpret the meaning of graphs or tables • calculate with whole numbers and decimals • calculate with and interpret the meaning of mean, median and mode

Report on results

Statistics and data analysis • interpret and discuss the meaning of tables, graphs and accompanying text • use a range of descriptive language of graphs, tables and averages such as maximum, minimum, increasing, decreasing, constant, slope, average, above/below average, etc.

57


16/6/06 11:59:36 AM


Table 4.4 Maths skills covered in ‘Navigation—getting around’

Investigation Choose a capital city in a different state that you would like to visit. Decide on how you would drive there and describe and draw a sketch map of the route, showing distances travelled and the time you estimate it would take, how long you would stop for, including overnight stops, etc. (Note: this could be extended to include estimating the costs for the trip.) Steps

Possible underpinning mathematics skills that need to be taught

Read and interpret maps

Measurement and Shape and space • estimate length measurements • use appropriately the names and symbols of the units of length measurement e.g. centimetre (cm), millimetre (mm), kilometre (km) • convert within the metric system for length e.g. 3500 m is 3½ km or 3.5 km) • use maps or directories to describe or follow routes between locations • interpret key features of maps or directories • use map indexes and keys to locate particular places of interest • estimate and determine distances on maps and street directories using scales

Plan trip including Measurement and Shape and space distances, speed and • interpret and use the concepts of length and time • apply rates concepts such as speed to practical times calculations of time related to distances to be travelled • interpret and use terminology and symbols for distance, speed and rates, such as km/h, litres/100 km, $/l etc. Algebra • use simple formulas (speed in terms of distance and time) • substitute appropriately into formulas to find particular values • choose and perform arithmetic operations where appropriate

58


16/6/06 11:59:37 AM


Draw map

Measurement and Shape and space • draw sketch maps • use scales and directions on maps • choose appropriate measuring instruments and use them correctly to measure • read and interpret calibrations on rulers • use appropriately the names and symbols of the units of length measurement e.g. centimetre (cm), millimetre (mm), kilometre (km)

Estimate costs

Number • choose and perform arithmetic operations where appropriate • apply rates concepts such as speed, fuel consumption rates, price rates to calculate time and costs related to distances to be travelled • estimate and round off calculations

RESOURCES Below are some possible resources that could be used to support using cars as a topic for teaching mathematics in context. Books Dengate, B & Gill, K 1989, Mathematics for teenagers, Longman Cheshire, Melbourne. Flittman, E & Sharrock, H 2000, Foundation mathematics for the real world, Longman, Melbourne. Ford, K (ed.) 1985, Mathematics for living (series): Car mathematics, Travel mathematics, Holmes McDougall and Educational Supplies, Brookvale, NSW. Lawrence, G & Moule, J 1990, Mathematics in practice series: Transport, Harcourt Brace Jovanovich, Sydney. Lawrence, G & Moule, J 1990, Mathematics in practice series: Travel, Harcourt Brace Jovanovich, Sydney. Lowe, I 2001, Mathematics at work, (CD-ROM), AAMT, Adelaide Marr, B, Anderson, C & Tout, D, 1994, Numeracy on the line: Language based numeracy activities for adults, National Automotive Industry Training Board, Doncaster, Vic. Tout, D 2006, Car costs II, Council of Adult Education, Melbourne. Vize, A 2005, Maths skills for working, Phoenix Education, Sydney.

59


16/6/06 11:59:37 AM


Websites and other resources There are many state and national bodies and organisations that offer a wide range of information about cars and travel. Such information is very valuable as the basis for researching and analysing information about cars and driving. As well, they often offer specific resources, activities and support for schools and students. In Australia, there are state motoring organisations which provide a range of information about cars—including running costs, purchasing cars and safety issues. A starting point for all state organisations is the Australian Automobile Association (AAA) at: http://www.aaa.asn.au/

There are a number of specific sites related to car safety. These include: • ANCAP—the Australian New Car Assessment Program: http://www.aaa.asn.au/ancap.htm • How safe is your car: http://www.howsafeisyourcar.com.au/ • The European New Car Assessment Programme at: http://www.euroncap.com/

Student worksheets Included in Appendix G as Worksheets C and D are two sample worksheets that could be used to support students in undertaking two investigations based on the above outlines. Reference Ernest, P 2004, ‘Relevance versus utility: Some ideas on what it means to know mathematics’, in B Clarke, DM Clarke, G Emanuelsson, B Johansson, DV Lambdin, FK Lester, A Wallby & K Wallby (eds), International perspectives on learning and teaching mathematics, National Centre for Mathematics Education, Göteborg University, Göteborg, Sweden. SUMMARY

• The context of cars and driving is a relevant and popular topic with teenage students and can easily be used as a motivating and engaging area for investigation with students. • As it is rich in different uses and applications of mathematics, this context of cars and driving covers much of the mathematics curriculum.

60


16/6/06 11:59:37 AM

CHAPTER

5

MAKING ALGEBRA WORK INTRODUCTION One of the main areas of frustration and concern for a lot of secondary-school mathematics students is the study of algebra. Students often have a lot of difficulty understanding and using algebra—they don’t see its relevance, and are unable to come to grips with its foundations in generalisation, pattern and relationships and, ultimately, its abstraction (and beauty to a lot of teachers and students). Yet it forms the basis and subsequent content of much of the senior secondary school curriculum, and failure to learn how to conceptualise, use and manipulate algebra effectively is a barrier to many students’ future success and continuation with mathematics. Algebra has many aspects and benefits, and these need to be highlighted and used to motivate and engage students to learn about it. Algebra can be seen as a way of making and applying generalisations and developing generalisable expressions; it can be seen as connecting to work in relations and functions and about formalising patterns in number; and also as relational formulas in measurement. Teachers need to be able to see the usefulness of algebra and how it is used in this variety of ways, and to be able to demonstrate some everyday application and relevance which may help students to see that there is some way of connecting with algebra and seeing how and why it works the way it does. Teachers often say: ‘Learn this now: you will use it later in more advanced mathematics’ (Gough, 2005) to justify why students need to learn formal, abstract mathematics in secondary school—not a very compelling argument to the struggling mathematics student who has no intention of or interest in continuing to study formal mathematics. It is an example of the ‘just in case’, rather than the ‘just in time’ approach to education. Many students are studying formal mathematics ‘just in case’ they might need it later, whereas if teaching the more formal and abstract aspects of mathematics (in this case,

61


16/6/06 11:59:37 AM


algebra) can be connected to contexts, uses and applications from students’ current or potential future interests then the ‘just in time’ motivation may be able to be used to engage students with learning algebra. The aim of this chapter is to look at a few of the ways of giving meaning to and understanding algebra, and using of formulas. It suggests a way of developing and illustrating algebra through generalisations and abstractions of relevant, real-life situations, and shows how there is a need for formulas and algebra in real-life work-based situations. Hopefully, this is a way in which disengaged students can see a purpose for learning and using algebra and to therefore engage with it in their lives and work. As argued in Chapter 1, many occupations and trades use a wide range of mathematical skills every day. In many of these occupations there is the need to engage with the use of algebra and formulas.

A COMMON APPROACH TO TEACHING A LG E B R A AT S C H O O L What are the most common experiences of teaching and learning algebra at school, especially in the middle and upper years of secondary school? How is it introduced? How is it explained? How is it practised? Reflect on how algebra is taught. Think about these questions: • Was algebra made meaningful or relevant to the students? • What methods were used to teach it? • Were there any hands-on materials used, or was it predominantly taught by the teacher at the blackboard or out of textbooks? • How, or why, do teachers argue for the value and importance of learning algebra? • Was algebra related to everyday or real situations? • Did it cause students to feel worried or anxious about maths, or turn them off maths? See how responses to these questions relate to current practices and experiences in a school—is there a problem with how algebra is taught and with how students engage with it? Often algebra is seen and presented as a set of abstractions and rules that need to be followed, with page after page of repetitive practice exercises involving x’s and y’s. Many students never see the logic and patterns within these tasks—and rather than improving understanding, these approaches can lead to more confusion and disengagement (Hart, 1981, 212).

62


16/6/06 11:59:38 AM

CHAPTER 5 Making algebra work

INTRODUCING AND CONNECTING WITH ALGEBRA Algebra can be used to generalise situations where patterns or repeated calculations or situations occur, and while this is not what all of algebra is about, it is a very essential part of it, and may present a way in to motivating students to understand the workings and use of algebra. For example: • working out costs (e.g. how much you pay a service person, how much you pay for amounts of materials or goods) • working out football scores • working out how long it might take you to travel somewhere, and how much you might be paid if you get travel allowances • working out areas and volumes to calculate the amount of mulch, soil or sand needed for a job • converting between mass and volume if you know the relative density of the substance (e.g. asphalt) • calculating interest • generalising when calculating or estimating almost anything that has a regular pattern, cost, production or usage Often people don’t realise that they are using a form of algebra to solve a problem or work something out. This inability to see the relevance and use of algebra, even just in terms of its use in generalising, often stops students from engaging with mathematics in the middle years of schooling. However, algebra can be shown to be connected to and embedded in many areas of life, work and study. Types of relationships and terminology One of the benefits of putting algebra into context is that it then provides a vehicle for introducing and talking about the conventions, terminology and types of relationships that are used. For example, if the formula being discussed makes sense in the real world (as in the football example demonstrated later on), then terms and their meanings such as variable, dependent and independent variables, and constants can make sense to students—it often explains words and meanings that have previously seemed mysterious. They can make everyday sense and connect to student’s experiences and understandings. This approach enables students to realise that algebra can be useful—it isn’t just an abstract game that they need to understand for no particular reason. The challenge then is to extend that to explain how mathematics

63


16/6/06 11:59:38 AM


works as an independent and abstract system. How to teach the structure, conventions and manipulations of algebra is an important and vital aspect of teaching and learning mathematics, and there are now a wealth of resources and activities that support this (for example, see Lowe, 2005; Lowe et al., 1993–1994; Schmitt & Steinback, 2005 in the resources list at the end of this chapter). However, the aim behind this chapter is to show teachers how to engage students and to get them to see that there can be an underlying purpose, meaning and relevance to using algebra which can be a steppingstone to accepting and understanding the abstractions of mathematics.

M AT H E M AT I C S A R E A S COV E R E D In Table 5.1, there is a list of possible topics or work situations that are related to algebra and formulas that could be considered as areas for investigation. Table 5.1 Topics related to algebra

Topic


Australian Rules football and other sports

Number This is a situation where you can use algebra or a formula to work out something—in this case the score.

Service charges

Number Rates Algebra, formulas and relationships Extension—simultaneous equations and graphs

Asphalting and paving

Number Measurement Rates including speed, spray rates, etc. Algebra, formulas and relationships Areas and volumes, including triangles and circles Density (volume versus mass)

Catering

Number Measurement Rates

64


16/6/06 11:59:38 AM


Topic


Painting

Number Measurement Algebra, formulas and relationships Rates including area coverage Areas and volumes, including triangles and circles

Building

Number Measurement Algebra, formulas and relationships Rates Areas and volumes, including triangles and circles Density (volume vs. mass)

Gardening and horticulture

Number Measurement Rates Algebra, formulas and relationships Areas and volumes, including triangles and circles

Weather

Number Measurement Algebra, formulas and relationships Rates Probability/chance Statistics and data analysis Converting temperatures Chill factor

Travel

Number Measurement Algebra, formulas and relationships Speed vs. distance vs. time Rates Direct and inverse relationships

65


16/6/06 11:59:38 AM


Topic


Electro-chemical industries

Number Measurement Algebra, formulas and relationships for industry calculations (plus other areas such as matrices) Production rates Direct and inverse relationships

Plastics industry (e.g., film production, polymer beads and products)

Number Measurement Algebra, formulas and relationships Rates Formulas calculations for production

Finances: • Interest calculations • Taxation calculations • Travel allowance calculations

Number Measurement Algebra, formulas and relationships Use of formulas in spreadsheets Graphical analysis and comparisons

Introducing and teaching algebra in a context, and involving students actively in learning about algebra, can address many of the different types of mathematical knowledge and capabilities described in Chapter 1 (see Table 5.2). Apart from using the contexts and ideas listed in Table 5.1 and in the following samples of materials for demonstrating the use of algebra, it is important to follow this up by making algebra make sense through the use of hands-on materials. Refer to the Resources section on page 69 for suitable materials and websites for teaching algebra effectively.

EXTENSIONS The main aim behind this chapter is to show students that algebra can make sense and be useful, and hopefully show them some of the beginning procedures, conventions and manipulations of algebra. It mainly focuses on providing meaning to the development and use of linear equations and formulas and then substitution in those formulas. However, it is possible to extend this approach to demonstrate higher level applications and therefore

66


16/6/06 11:59:38 AM


Again, yes, as this topic of algebra and connecting it to real world applications can be the opening to seeing how advanced mathematics works and is used, and can easily be related to and extended to a wide range of more advanced mathematics.



Yes. Easily demonstrated by showing and using examples from vocational and workplace contexts. See Worksheet G in Appendix G.

• Practical, workTo be able to solve practical problems with related knowledge mathematics, especially industry and work centred problems

To be able to demonstrate useful Yes. Can be acquired through applying use of mathematical and numeracy skills adequate algebra to everyday contexts such as sport, for successful general employment and finances, gardening, etc. functioning in society


Covered here?




CHAPTER 5

Making algebra work

67

16/6/06 11:59:39 AM


To be empowered through knowledge of Yes. Access to the value and workings of algebra, mathematics as a highly numerate critical even to a limited extent, could be quite citizen in society, able to use this knowledge empowering to students. in social and political realms of activity



Yes. Given that algebra may have been a stumbling block and barrier to having confidence in mathematics, addressing it here in this practical way may well be an eye-opener and confidence booster.



Yes. As for the above type of knowledge, demonstrating how algebra works and its conventions and uses enables students to see and appreciate (maybe for the first time) how the abstract dimension of mathematics works. Could also get students to investigate some of the history of algebra and where it was derived and developed from.

Covered here?







68

16/6/06 11:59:39 AM


uses of algebra and algebraic and graphical analysis. Quite simple extensions to the examples demonstrated in this chapter provide insights and understanding for solving and transposing equations, simultaneous equations and graphical analysis. Use of graphical calculators could easily be incorporated into this. As well, many professions, industries and workplaces require higher level algebraic skills and analysis, some of which are included in the examples in Worksheet G (Appendix G), and some further searching will provide examples to demonstrate this.

RESOURCES Below are some possible resources that could be used to support teaching algebra in context and/or making algebra make more sense to more students. Books Flittman, E & Sharrock, H 2000, Foundation mathematics for the real world, Longman, Melbourne. Helme, S & Marr, B 1995 Some beginnings in algebra, Northern Metropolitan College of TAFE, Melbourne. Lowe, I 2001, Mathematics at work, (CD-ROM), AAMT, Adelaide. Lowe, I 2005, Active learning in number and algebra, Mathematical Association of Victoria, Melbourne. Lowe, I, Johnston, J, Kissane, B & Willis, S 1993–1994, Access to algebra: Books 1–4, Curriculum Corporation, Melbourne. O’Connor, M & Gaton, B 2000, Foundation mathematics, Oxford University Press, Melbourne. Schmitt, MJ & Steinback, M (eds) 2005, Seeking patterns, building rules (Math: EMPower series), Key Curriculum Press, Emeryville, CA. Thomson, S & Forster, I 1998, Trade and business mathematics 1, Addison Wesley Longman, Melbourne. Thomson, S & Forster, I 1999, Trade and business mathematics 2, Pearson Education, Melbourne. Tout, D 2001, Numeracy and mathematics: Health and fitness, TAFE Frontiers, Melbourne. Tout, D 2001, Numeracy and mathematics: Science and work, TAFE Frontiers, Melbourne.

Online resources The WWW references listed below are a few of the many now available and the list is not meant to be comprehensive. Searching the internet for a specific mathematics topic is sure to come up with a range of suitable websites.

69


16/6/06 11:59:39 AM


Maths300 is an Australian web-based service which aims to support teachers in the delivery of excellent mathematics education, and to resource members with extensive notes for at least 300 exemplary maths lessons (K–12): http://www.curriculum.edu.au/maths300/ Purplemath is a website which has been developed specifically to dispel the bad press usually given to the topic of algebra: http://www.purplemath.com The Knot a Braid of Links site aims to provide a guide to mathematics on the web. It holds lots of activities and you can contribute your own as well: http://west.camel.math.ca/Kabol/ The Math Archives holds a large number of links to various web-based resources on mathematics. Icons guide you to the levels of the activities and the types of web pages that are used (JAVA, animations, interactive etc.): http://archives.math.utk.edu/topics/ The Math Forum is an online math education community centre and is a site full of maths ideas and resources, including ‘Ask Dr. Math’, problem of the week, an online forum and a number of discussion groups: http://mathforum.org/ Coolmath.com is a website filled with activities and games for K-adult, which is a good interactive online website for practising math concepts: http://www.coolmath.com/

Curriculum authorities The website addresses of Australian state and territory curriculum and assessment authorities, boards and councils are provided in Appendix E. These include various teacher reference and support materials for curriculum and assessment and will include resources and support for teaching algebra in the middle and senior years of schooling. Student Worksheets Included as Worksheet E, Worksheet F and Worksheet G in Appendix G are some samples that could be used to support students to learn about how algebra works in realistic situations. Worksheet G lists a number of formulas used today in different workplaces. References Ernest, P 2004, ‘Relevance versus utility: Some ideas on what it means to know mathematics’, in B Clarke, DM Clarke, G Emanuelsson, B Johansson, DV Lambdin, FK Lester, A Wallby & K Wallby (eds), International perspectives on learning and teaching mathematics, National Centre for Mathematics Education, Göteborg University, Göteborg, Sweden. Gough, J 2005, Editorial, Vinculum, vol. 42, no. 2, June. Hart, KM (ed.) 1981, Children’s understanding of mathematics: 11–16, John Murray, London.

70


16/6/06 11:59:39 AM


SUMMARY

• Algebra is an area of frustration and concern for many secondary school mathematics students. Students often have a lot of difficulty understanding and using algebra—they don’t see its relevance or use, and are unable to come to grips with it. It needs to be taught effectively to all students. • Starting from real-life contexts and applications, algebra can be demonstrated to make sense and be useful, and this will hopefully motivate and engage students in learning about the procedures, conventions and manipulations of algebra.

71


16/6/06 11:59:40 AM

CHAPTER

6

C O O K I N G M AT H E M AT I C S U P INTRODUCTION The context of cooking and catering (and therefore eating) involves a lot of measurement skills. Extending this to hospitality and catering enables a much wider range of mathematical skills to be addressed. This can often relate to students’ interests in obtaining casual work in this area where opportunities exist as helping hands, shop assistants and waiters in bakeries, cafes and restaurants. In some cases, students may even be interested in pursuing a career in the hospitality industry and could be studying hospitality as part of vocational education and training (VET) subjects. The hospitality industry is one of the key and growing workplaces in today’s world of work and as such is a key context for the student cohort. Apart from that, there is personal skill development in terms of cooking for oneself, eating out, and health and nutrition. A number of areas are therefore of interest and importance, including: following recipes, measuring of ingredients and cooking food; catering for large numbers of people; budgeting for an event such as a party; costs of making your own food versus eating out or buying packaged food; nutritional aspects of foods; safe serving of alcohol; and safe handling of food.

M AT H E M AT I C S A R E A S COV E R E D Table 6.1 lists some possible topics related to cooking and catering that could be considered as areas for investigation, and therefore for teaching a range of mathematics skills. The theme of cooking and catering can address many of the different types of mathematical knowledge and capabilities described in Chapter 1 (see Table 6.2).

72


16/6/06 11:59:40 AM

CHAPTER 6 Cooking mathematics up

Table 6.1 Topics related to cooking and catering

Topic and context


Cooking • following recipes and measuring of ingredients • cooking food • calculations with time • reducing or extending recipes for more of fewer people

Number Measurement

Cooking and catering for a function • extending recipes for large groups of people • following recipes and measuring of ingredients • cooking food • calculations with time

Number Measurement

Budgeting • developing and costing for a function, including extending recipes for large groups of people • costs of ingredients • costs of venue, equipment and staffing

Number Measurement Statistics and data analysis Space and shape

Eating out • compare expense of eating out versus eating at home • costs of meals • the best deal—comparing restaurants • picking it up yourself—or home delivery • special deals—are they good value? • tipping • surveys of favourite foods and restaurants

Number Measurement Location, space and shape Statistics and data analysis

Drinks • mixing up drinks such as cordials or mixed drinks • alcohol and drinking • safe serving of alcohol

Number, including ratios Measurement Algebra

73


16/6/06 11:59:40 AM


Topic and context


Fresh or packaged? • cost and value of making your own versus packaged food • the benefits (and dangers) of making your own food • growing your own fruit and vegetables

Number, including ratios Measurement Algebra

Nutrition and health • ingredients of food • the food pyramid • diets • survey on eating habits • research into weight and obesity, including body mass index

Number, including ratios Measurement Algebra Statistics and data analysis

Safe handling of food • storage of food • handling and preparation of food

Number Measurement Statistics and data analysis

P OS S I B L E S TA R T I N G P O I N T S The following examples related to cooking and catering illustrate some possible starting questions to pose to students, based on some of the above topics and addressing a range of mathematical content. Remember that the decision about the actual topics and content of the material and questions to be posed and investigated require the input of local teacher knowledge—both of any local or national contexts, but also of the student interests. Cooking • In a small groups, students choose one of the recipes provided by the teacher that they would like to cook (and eat). They should write a shopping list for all the ingredients, purchase the ingredients, and prepare and cook (and eat) the food they chose. Was the food good to eat? If not, why not? How much did it cost per person? Do they think it was cheap or not? • Students choose their favourite recipe from home or a cookbook and bring it to class. They then write out the list of ingredients for making the recipe for a group of 20 people, and use the new list of ingredients to work out how

74


16/6/06 11:59:40 AM

To be able to solve practical problems Yes. Especially if this is connected to with mathematics, especially industry vocational or workplace situations such as and work centred problems catering or working in restaurants. To have an understanding and capabilities in advanced mathematics, with specialist knowledge beyond standard school mathematics (including advanced high school specialist study of mathematics to knowledge of university and research mathematics)

• Practical, work-related knowledge



Only maybe, depending on whether specific examples and investigations could be found requiring higher level mathematical analysis.

Yes. Mathematics related to cooking, and eating is a very practical skill.



Covered here?




CHAPTER 6

Cooking mathematics up

75

16/6/06 11:59:41 AM


To be empowered through knowledge of mathematics as a highly numerate critical citizen in society, able to use this knowledge in social and political realms of activity



Maybe, as this area is so down to earth and practical it may not lead to a lot of critical awareness or empowerment. But it is possible if connections are made with more crucial areas such as health and nutrition, and financial awareness of costs incurred, etc.

Yes. Again, given the everyday, practical nature of the mathematics behind cooking and catering this could enable students to easily make the connections between mathematics and their everyday lives and hence build their confidence.



Covered here?

Yes. Although this topic of cooking and catering is a very practical skill, it again may illustrate to students the vital role mathematics plays in culture and society in general.


• Appreciation of mathematics To have an appreciation of mathematics as a discipline including its structure, subspecialties, the history of mathematics and the role of mathematics in culture and society in general




76

16/6/06 11:59:41 AM


much it would cost them to purchase all the ingredients to make the food. How much would it cost per person? Cooking and catering for the ‘end of term lunch’ • The class is to cater and cook a meal for the whole group for the end of term lunch. Students are to work in small groups to prepare and cater for the meal, including drinks, at least three different types of food and some sweets or desserts. Once negotiated and agreed on what food is to be cooked, the students work out a timeline against tasks required to hold the function. Each group writes a shopping list for all the ingredients they need and estimates the cost of the ingredients. The total cost is then calculated and students share the costs. The ingredients are purchased using the money paid into a kitty, the meal is prepared and cooked and the event undertaken. Each group writes a report on the food, including any recipes and calculations they needed to undertake to purchase and make their part of the meal. Eating out • Students work in groups of 3 or 4, and choose a restaurant from a list supplied (collect takeaway menus from a range of local restaurants) and decide what they would buy if they only had $50 to spend and had to feed their group. What would they buy if they had $75 to spend? • Students investigate at least three different local pizza restaurants (or they could include fresh or frozen pizzas available from a local supermarket too) and work out who has the cheapest pizzas. They can decide on the criteria and then justify why they chose which restaurant. Variations could include: Which restaurant has the cheapest ‘special’ pizza working it out based on cents per square centimetre? Can you make a pizza for the same price as the cheapest one you have found? • Students investigate state or national ratings of restaurants (for example, in Australia the Age Good Food Guide, the Sydney Morning Herald Good Food Guide, or international equivalents such as the Michelin, GaultMilau, Zagat, AAA (American Automobile Association) guides, Fodor’s Restaurant Guide or even international internet sites such as http://www.cuisinenet. com). They could compare and analyse the criteria used and discuss the advantages and disadvantages of the different scoring systems. If they exist, they could compare how different guides rate the same restaurants (including graphing the results).

77


16/6/06 11:59:41 AM


Nutrition and health • Students investigate the nutritional content of common foods, using foods that list the nutritional content on the packaging. They could compare foods with a range of brands or types (e.g. breakfast cereals, milks, yoghurts) and analyse their nutritional content (e.g. fat, sugars, salt, fibre, etc.). Students write up a report of their finding including some graphs of their information. (Extension: compare the findings with how the different brands are advertised.) • Students undertake a survey on eating habits of students at their school. Write up a report, and include graphs and a statistical analysis. • Students analyse the food available from the school canteen. How healthy is it? How expensive or cheap is it? • Get students to research Australian or state statistics on weight and obesity, and compare different groups and/or different states and age groups, or compare Australian data with other overseas countries. This could include looking into the use of the body mass index graph and formula. Fresh or packaged? • Students undertake an investigation into the cost and value of making their own food versus buying commercially packaged food. Different groups could investigate different foodstuffs (e.g. pizzas, biscuits, muesli) and the class could then see if there were any differences. • How many oranges does it take to make a 2-litre container of orange juice? Which is cheaper? • Is it cheaper to grow your own vegetables?

E X A M P L E O F M AT H E M AT I C S S K I L L S COV E R E D Example of the types of mathematics skills needed for some of these investigations are elaborated in the Tables 6.3 and 6.4 below. As explained in Chapter 2, these mathematics skills will need to be explained and taught if the students are to achieve the learning and outcomes expected. Remember, these can be introduced in a number of ways, depending on the skills and experience of the students. This could happen through: • whole-class activities and explanations before or during the investigation • small-group activities based on explanations, worksheets or tasks • individual skills and practice sessions.

78


16/6/06 11:59:41 AM


Table 6.3 Maths skills covered in ‘End-of-term lunch’

Investigation The class is to cater and cook a meal for the whole group for the end of the term lunch. Students work in small groups to prepare and cater for the meal, including drinks, at least three different types of food and some sweets or desserts. Once negotiated and agreed on what food is to be cooked, the students work out a timeline against tasks required to hold the function. Each group writes a shopping list for all the ingredients they need and cost the ingredients. The total cost is then calculated and students share the costs. Using the money paid into the kitty, the ingredients are purchased and the meal is prepared and cooked and the event undertaken. Each group writes a report on the food including any recipes and calculations they needed to undertake to purchase and make their part of the meal. Steps

Possible mathematics skills

Decide on food for the meal

Number • use place value concepts for whole numbers and decimals to interpret and compare written numbers and metric measures • understand and estimate with common fractions and their use in practical contexts Measurement • identify common notations for metric measurement • demonstrate a sense of common and standard metric units

Establish Number ingredients required • use place value concepts for whole numbers and decimals to interpret and compare written numbers and metric measures • calculate and estimate with decimals and common fractions and their use in practical contexts Measurement • identify common notations for metric measurement • demonstrate a sense of common and standard metric units • calculate accurately and efficiently with decimals and fractions of metric measures

79


16/6/06 11:59:42 AM


Steps


Cost of meal

Number • use place value concepts for whole numbers and decimals to interpret and compare written numbers and metric measures • calculate and estimate with decimals and common fractions and their use in practical contexts • use number facts and rounding to give rough estimates of numerical calculations

Work out timeline and tasks

Measurement • use and calculate with time

Purchase ingredients

Number • understand, calculate and estimate with decimals and common fractions in practical contexts Measurement • estimate and interpret mass and volume measurements

Cook meal

Number • use place value concepts for whole numbers and decimals to interpret and compare written numbers and metric measures • understand, calculate and estimate with decimals and common fractions and their use in practical contexts Measurement • use of appropriate measuring instruments and the reading of digital and analogue scales and readings • estimate and interpret mass, volume and temperature measurements • identify common notations for metric measurement • convert for mass and volume • demonstrate a sense of common metric units • calculate accurately and efficiently with decimals and fractions of metric measures • use and calculate with time

80


16/6/06 11:59:42 AM


Table 6.4 Maths skills in ‘Nutritional content of food’

Investigation Investigate the nutritional content of common foods, using foods that have a listing of the nutritional content on the packaging. Compare foods with a range of brands or types (e.g. breakfast cereals, milks, yoghurts, etc) and analyse their nutritional content (e.g. fat, sugars, salt, fibre, etc.). Students write up a report of their finding comparing the different brands, including some graphs of their information. (Extension: compare the findings with how the different brands are advertised.) Steps


Research which foods to study and collect copies of the nutritional information

Number • use place value concepts for whole numbers and decimals to interpret and compare written numbers and metric measures • understand and estimate with common fractions, decimals and percentages and their use in practical contexts Measurement • identify notations for metric measurement

Record data

Number • read and write whole numbers, decimals, fractions and percentages Statistics and data analysis • collect, sort and record data in a table

Graph information

Statistics and data analysis • draw an appropriate graph for the data, labelling the graph and axes and marking in the scale

Analyse data

Statistics and data analysis • interpret the meaning of graphs or tables • calculate with whole numbers and decimals • calculate with and interpret the meaning of mean, median and mode

81


16/6/06 11:59:42 AM


Steps


Report on results

Statistics and data analysis • interpret and discuss the meaning of tables, graphs and accompanying text • use a range of descriptive language of graphs, tables and averages such as maximum, minimum, increasing, decreasing, constant, average, above/ below average

RESOURCES Below are some possible resources that could be used to support using cooking, eating and catering as topics for teaching mathematics in context. Books Dengate, B & Gill, K 1989, Mathematics for teenagers, Longman Cheshire, Melbourne. Flittman, E & Sharrock, H 2000, Foundation mathematics for the real world, Longman, Melbourne. Goddard, R & Regan, M 1995, The value of time: Numeracy for workers in manufacturing, Council of Adult Education, Melbourne. Lowe, I 2001, Mathematics at work, (CD-ROM), AAMT, Adelaide. O’Connor, M & Gaton, B 2000, Foundation mathematics, Oxford University Press, Melbourne. Ritchhart, R 1994, Making numbers make sense: A sourcebook for developing numeracy, Addison-Wesley, Menlo Park, CA. Schmitt, MJ & Steinback, M (eds) 2004–2005, Math: EMPower series, Key Curriculum Press, Emeryville, CA. Vize, A 2005, Maths skills for living, Phoenix Education, Sydney. Vize, A 2005, Maths skills for working, Phoenix Education, Sydney.

CD-ROMs Tout, D & Marr, B 1997, Measuring up: An interactive multimedia computer resource for numeracy learners, Protea Textware, Melbourne.

Online resources The WWW reference listed below is one known site that addresses cooking and mathematics, but there are probably others, so this is not meant to be

82


16/6/06 11:59:42 AM


comprehensive. Searching the internet for a specific mathematics topic is sure to come up with a range of suitable websites. Math in Daily Life: http://www.learner.org/exhibits/dailymath/

Toolboxes A wide range of online resources has been developed in Australia that are based on teaching and learning mathematics in a context—these have en called ‘toolboxes’. Some are generic, while others are specific to a particular industry or trade. http://flexiblelearning.net.au/toolbox/ To find out about all general toolboxes. http://toolbox.vetonline.vic.edu.au/519/entry_page.htm Where’s the party at?—a toolbox with a section specifically related to nutrition.

Student worksheets Included in Appendix G as Worksheets H and I are two samples that could be used to support students in undertaking two investigations based on the above outlines. Reference Ernest, P 2004, ‘Relevance versus utility: Some ideas on what it means to know mathematics’, in B Clarke, DM Clarke, G Emanuelsson, B Johansson, DV Lambdin, FK Lester, A Wallby & K Wallby (eds), International perspectives on learning and teaching mathematics, National Centre for Mathematics Education, Göteborg University, Göteborg, Sweden. SUMMARY

• The context of cooking, hospitality and catering is another area rich in mathematics that can motivate and engage learners. • Measurement skills form an underlying basis of the topic, alongside numerical and financial aspects related to costs and budgeting. Nutritional aspects of foods, safe serving of alcohol and safe handling of food are other areas that are both important and relevant, and can be the basis for learning and applying a range of mathematical skills.

83


16/6/06 11:59:43 AM

CHAPTER

7

A SPORTING OPPORTUNITY INTRODUCTION Sport is a major part of the lives of many young people and the communities in which they live. A 2003 survey by the Australian Sports Commission (Australian Sports Commission, 2004) found that over 90% of people aged 15–24 participated in at least one physical activity on a weekly basis. Participation levels for males and females were approximately equal. It is clear, then, that sport and related activities are a relevant and potentially engaging context for students to make connections between mathematics and their lives. All sports include an amazing array of scores, results, statistical data and analysis, which are extremely rich stimulus materials for working with students in mathematics. As well as the results and statistics, there are the shapes and dimensions of the playing fields, the rules for scoring (usually based on a formula) and elements of probability, which can be incorporated by looking at the chances of making finals and winning, and also betting on sports results. There is also the related issue of health, where investigations and research could include areas such as exercise and fitness, weight and diet, and the health and well-being of students and the adult population. Sports information, whether on the internet, on TV or in newspapers, is of local, state, national and international interest. It can include sports such as football (soccer), basketball, netball, cricket, hockey and rugby, as well as unique national codes of sport like Australian Rules football. As well, there are events such as the Commonwealth and Olympic games, World Championship events and so on, which can be highlights in a year’s sporting calendar and can be tapped into when appropriate and of interest to the cohort of students. There is likely to be a multitude of different sports and recreations which will have appeal and relevance to students. Swimming, for example, might only appeal specifically to a few students in a class—and even among that

84


16/6/06 11:59:43 AM

CHAPTER 7 A sporting opportunity

group there might be very different motivating factors. Some students might have particular interest in competitive swim training programs, others in swimming as part of triathlons and others again in relation to recreational surfing. Students with little interest in swimming themselves may be able to research and look at swimming results in popular events such as the Commonwealth and Olympic games. These different, but related, swimming contexts can be negotiated with the students taking into account issues of resource availability and classroom management. Clearly, in most circumstances teachers would not be able to adequately resource and manage the learning of a large number of students all following different avenues of investigation. Teachers need to set the parameters of study to be undertaken allowing for some flexibility in order to cater for particular student interests. In such an approach, teachers will also consider the curriculum requirements of the mathematics study and enable appropriate and suitable learning styles and approaches to be used by students. These negotiation and class management issues were addressed in Chapter 2, and the idea of developing contracts with students before beginning any investigations was suggested as a constructive way to handle these processes. As mentioned in earlier chapters, teachers need to use examples of different sports to model and demonstrate how mathematics is embedded and make explicit to students what roles mathematics has in playing, scoring and analysing sports.

M AT H E M AT I C S A R E A S COV E R E D Many areas of mathematics can be related to investigations involving sport in some way. The examples of topics shown in Table 7.1 should not limit student choices. Table 7.2 shows how the theme of sport can address many of the different types of mathematical knowledge and capabilities described in Chapter 1.

85


16/6/06 11:59:43 AM


Table 7.1 Topics using the context of sport

Topic and context


Student fitness and involvement in sport Investigations of the statistics of student participation in sports and their fitness such as: • participation rates (e.g. who plays what types of sports? How often? Compare to state or national data) • undertake tests of fitness and report and analyse, test and compare group of students’ performances on different events and compare with national or international records

Number Statistics and data analysis Measurement Algebra

Facts and figures on particular sports Number Investigations of the statistics of sports such as: Statistics and data analysis • game day statistics (e.g. points scored per Measurement player, game time, rebounds) • player statistics (e.g. age, size, games played) • match results (e.g. winning margins, highest scores, ladder positions, percentages) Playing the game Investigations into the playing conditions such as: • dimensions of the playing area and permitted variations duration of the game and stoppages • rules for scoring and winning operation of a finals series, including relative chances of winning from different ladder positions

Number Measurement Space and shape Algebra Statistics and data analysis including probability

Playing at a cost Number Investigations into the costs and rewards of Measurement participation in sports such as: Statistics and data analysis • cost of playing gear, equipment and facilities at community and salaries levels of elite sportsmen and women • cost of putting on a local or professional game • advertising and promotion of sports • financial value of an elite competition

86


16/6/06 11:59:43 AM


Topic and context


In the record books Investigations into record performances by teams and individuals such as: • world record performances (e.g. times, distances, winning margins) • comparisons of performances (e.g. How far down the track would I be as the winner crosses the line in the Olympic 100 m final?) • improvements in elite performances over time • comparisons of performances by men and women and relative improvements

Number Measurement Statistics and data analysis Algebra

Improving the game Investigations into changes that may enhance the sport such as: • changing the size of the scoring zone (e.g. widening a soccer goal, increasing the diameter of a basketball ring) • changing the size of playing equipment (e.g. the surface area of a table tennis bat, the width of doubles and triples on a dart board) • changing the rules of scoring (e.g. value of major and minor scores, penalties)

Number Measurement Algebra Space and shape

Making my own sport Investigations into creation of new games or variations of existing ones such as: • design of playing spaces and rules for playing and scoring • design and representation of tracks and courses around the local environment (e.g. orienteering, BMX track, skateboarding) • alternative finals series (e.g. how five teams might play off over different series lengths)

Number Measurement Space and shape Statistics and data analysis including probability

87


16/6/06 11:59:44 AM


Topic and context


A sporting chance Investigations into probabilities of sporting outcomes such as: • making of physical and/or computer based simulations of games • chances of winning games from various positions (e.g. a game of tennis from two sets down, a finals series from top ladder position) • chances of scoring from different field positions (e.g. kicking a goal from 30m, making a three point basketball shot, getting a bulls eye in darts from different distances) • legal betting operations including totaliser and government takes from betting pools

Number Measurement Space and shape Statistics and data analysis including probability

P OS S I B L E S TA R T I N G P O I N T S Based on some of the preceding topics, the following are some possible questions to pose to students for investigating mathematics in sport. Facts and figures on particular sports Students could engage in the following activities: • Find the results and the ladder for their favourite sports competition. Write up a report explaining and analysing how the ladder is worked out, showing the calculations for awarding points and how the teams are placed in order on the ladder, including their percentages. • Research and find statistics for one team in their favourite sport in at least two different games in the same competition and use graphs and averages (mean and median) to compare the two results. • Research and investigate characteristics of the best national players in their favourite sport. Research, analyse and present data on the physical attributes such as the height and weight of the players. See Example 1: Characteristics of the best players on page 92 for more details of this type of investigation. • Undertake a survey of students from the school and analyse how many participate in what types of sport, and how often they compete or participate. These results could be compared to state or national data. See Example 2: Student sport interests on page 92 for more details of this type of investigation.

88


16/6/06 11:59:44 AM


To be able to solve practical problems with mathematics, especially industry and work centred problems.

To have an understanding and capabilities in advanced mathematics, with specialist knowledge beyond standard school mathematics (including advanced high school specialist study of mathematics to knowledge of university and research mathematics).

To have an appreciation of mathematics as a discipline including its structure, subspecialisms, the history of mathematics and the role of mathematics in culture and society in general.

Practical, workrelated knowledge

Advanced specialist knowledge

Appreciation of mathematics

Yes. Recognition that all sports contain a wealth of opportunities for mathematical investigations will enhance appreciation of the role mathematics plays in culture and society in general.

Only maybe, depending on whether specific examples and investigations could be found requiring higher level mathematical analysis. Particular areas where this could occur will be in undertaking higher level statistical analysis of data and in investigations of probability and chance.

Possibly. Many of the investigations in Table 7.1 have clear links to problems encountered in trade and technical vocations. Also if students work in a sport or fitness related job, then this will add a stronger connection with this type of mathematical knowledge.

To be able to demonstrate useful Yes. Sporting participation and appreciation mathematical and numeracy skills adequate promotes healthy lifestyles and community for successful general employment and involvement. functioning in society.

Utilitarian knowledge

Covered here?




CHAPTER 7

A sporting opportunity

89

16/6/06 11:59:44 AM

To be confident in one’s personal knowledge of mathematics, to be able to see mathematical connections and solve mathematical problems, and to be able to acquire new knowledge and skills when needed.

To be empowered through knowledge of mathematics as a highly numerate critical citizen in society, able to use this knowledge in social and political realms of activity.

Social empowerment through mathematics


Mathematical confidence




Possibly. Through investigations students might become aware of the cultural and financial significance of sport to individuals and communities. Recognition of the winners and losers in sanctioned sports betting would be a significant outcome.

Yes. The use of personally recognised and understood sporting contexts provides for scaffolding of new skills and understandings and appreciation of links to other aspects of students’ lives.

Covered here?



90

16/6/06 11:59:44 AM


Playing the game Students could engage in the following activities: • Investigate and compare two or three similar sports (e.g. different football codes) in terms of the duration of the game, time for stoppages, the number of players, replacement strategies for injured or resting players, and infringement penalties. • Investigate and analyse the scoring system for their favourite sport, and generalise the scoring into an algebraic sentence and equation (see Worksheet E in Appendix G for a worksheet related to doing this). Write (and draw/illustrate) an explanation for the scoring in the game for someone who has never seen a game of their chosen sport before. • Investigate and analyse the operation of a finals series for their favourite sport or competition, including the relative chances of winning from different ladder positions using data from previous final series. • Investigate and analyse the dimensions of the playing area and permitted variations for their favourite sport. Draw a scale plan of a playing field/area for the sport accompanied by a written description. A sporting chance Students could engage in the following activities: • Investigate where and how betting is now available for different sports. Write up a report showing the odds and betting system and how they work for at least one sport. Include an explanation on how odds in sport relate to how we calculate probability and chance in mathematics. • Investigate and analyse the chances of scoring from different field positions in a particular sport (e.g. kicking a goal from 30m, making a three point basketball shot, getting a bulls eye in darts from different distances). See Example 3: A passing shot on page 92 for more details of this type of investigation. In the record books Students could engage in the following activities: • Investigate and analyse world record performances in a particular event or sport over the last ten years (e.g. times, distances, winning margins) and produce a report, including graphs analysing the changes over that period. • Undertake measurements of a student’s performances in a few athletic events and investigate and compare the student’s performances against either World, Olympic or Commonwealth games records. This could

91


16/6/06 11:59:45 AM


include, for example, comparing how far down the track the student would be compared to the winner in the World Championship or in the Olympic Games 100 m final, or how far is their throw in the shot put compared to the winner in the Olympic Games. Which event was the student the closest to the World or Olympic record? See Worksheet J in Appendix G for an example of a possible worksheet for an investigation like this.

E X A M P L ES O F N EG OT I AT E D I N V ES T I G AT I O N S Example 1: Characteristics of the best players Three students have a common interest in basketball. They suggest to the teacher that their investigation revolves around characteristics of the best national basketball players. They propose that they research and then analyse and present data on the physical attributes such as the height and weight of the players. The analysis of data will focus on statistics relating to central tendency and spread of data. The teacher suggests opportunities for the students to extend the investigation to include outcomes in Number (ratio and scale) and Measurement. The plan shown in Table 7.3 is agreed. Example 2: Student sport interests A group of students with a strong interest and skills in media production propose to create a short film of the sporting activities in which students are involved during school time. With no evidence of relevant mathematics outcomes in the proposal, the teacher suggests inclusions and directions to establish the plan, as shown in Table 7.4. Example 3: A passing shot Four students with a common interest in tennis form a group. They have previously completed investigations and class work in the areas of data analysis and probability, number, and algebra. They have done little in the areas of space and measurement, and it is agreed that these should form the foundation of the investigation. The plan as shown in Table 7.5 is developed.

92


16/6/06 11:59:45 AM


Table 7.3 Investigation: Superstar; Period: 4 weeks

General description of the investigation Course outcomes achieved We will investigate the height and weight of • Data presentation: the top 40 point scorers in the national frequency tables, competition. We will find the mean values and histograms the spread as summarised by the five number • Data analysis: central summary used to produce box plots. tendency (mean and median), spread (five Our final product will be a presentation to the number summary and class entitled ‘Can you be a basketball box plots) superstar?’ In the presentation of our results Measurement: making • we will include (a) a life-sized drawing of the and recording average superstar, and (b) a scaled drawing measurements, making showing the relative size of the ‘average full sized and scaled superstar’ and the average size of students in drawings our year level. • Number: converting measurements to scaled drawings Schedule Proposed times

Activities of group members

Resources required

Week 1

Research data on top 40 Internet access point scorers in national Measuring tape league Measure heights and weights of 40 students.

Week 2

Collected data analysed Possibly for central tendency and computer access spread for generation of Tables, graphs and data data displays summary statistics prepared

Mathematics required

Organising and recording data Random sample selection Data presentation as frequency tables and histograms Calculation of mean and median Calculation of five number summary and box plots

93


16/6/06 11:59:45 AM


Proposed times


Week 3

Skills work: application of developed skills to a range of tasks set by the teacher

Resources required


Application of skills in other contexts

Completion of all tables, graphs and calculations for the final presentation Week 4

Preparation and presentation of final report to class

Measuring equipment

Paper rolls for life sized Produce the life sized drawings of the average drawings superstar and the average student

Measurement and scale drawing Presentation and communication skills including graphical

Produce the scaled diagram showing the comparison between the average superstar and the average student Table 7.4 Investigation: Lunchtime, a snapshot; Period: 4 weeks

General description of the investigation We will make a video of the sporting activities that students are involved in on a typical lunchtime. We will survey students from each year level to get estimates of how many students are involved in different activities. We will include tables and graphs of the data in our final production. The video will be available for use by the school on open days.

Course outcomes achieved • Determination of sample size, and sampling processes. • Presentation of discrete data • Analysis of data • Application of technology in data analysis and presentation.

94


16/6/06 11:59:45 AM


Schedule Proposed times


Resources required

Week 1

Develop questionnaire to survey students

Computer access How to select a random sample and Camera for conduct a survey lunchtime use

Determine the exact type and nature of data required Get permission and conduct survey for each year level Start producing tables and graphs


How to make frequency tables and graphs Use a spreadsheet to make tables and generate graphs

Begin scripting the final video Get some sports footage at lunchtimes Week 2

Complete tables and graphs

Computer access Appropriate ways of displaying Camera for different forms of Use the collected data to lunchtime use data get overall numbers playing each sport Using sample data to make inferences Make some conclusions based on the data

Week 3


Application of skills in other contexts

Ensure that all material (tables, graphs, footage, script outline) is prepared

95


16/6/06 11:59:46 AM


Proposed times


Week 4

Complete final video

Resources required

Data projection and editing Project the tables and graphs onto a screen for facilities filming


Presentation and communication skills including graphical

Final editing with inclusion of tables and graphs, titles, and soundtrack Table 7.5 Investigation: A passing shot; Period: 4 weeks

Course outcomes achieved • Measurement of length and area • Approximations and error analysis • Scale drawing We will each produce a short report with scaled • Use of dynamic geometry software diagrams to show the relative difficulty of hitting winning shots. General description of the investigation The investigation will have two parts. The first will look at the dimensions of a tennis court. In the second, we will use technology to investigate the angles that players use to hit winning shots.

Schedule Proposed times


Resources required


Week 1

Measurement of the Measuring tapes Reading of scales to school tennis courts and or wheels required accuracy comparison with Access to courts Error analysis regulation size Calculation of Determination of the perimeter and area amount of tape required to mark a court Possible extension: Find a different sized court that uses the same amount of tape

96


16/6/06 11:59:46 AM


Proposed times


Resources required


Week 2

Production of scaled Computer access Ratio and scale diagrams of courts using and dynamic Use of dynamic the computer geometry geometry software software Investigation of angles Angle available to players in measurement different parts of the court

Week 3


Week 4

Complete final computer presentation with scale drawing and results of different angles of shot and showing the relative difficulty of hitting winning shots

Application of skills in other contexts Computer access and dynamic geometry software

Presentation and communication skills including graphical

Data projection

Set up software for presentation via data projector

RESOURCES Below are some possible resources that could be used to support using sport as a topic for teaching mathematics in context. Books Dengate, B & Gill, K 1989, Mathematics for teenagers, Longman, Melbourne. Flittman, E & Sharrock, H 2000, Foundation mathematics for the real world, Longman, Melbourne. Lowe, I 2001, Mathematics at work, (CD-ROM), AAMT, Adelaide. Perso, T (ed.) 1994, Mathematics for living: Mathematics in sport (rev. edn), The Mathematical Association of Western Australia, Perth. Vize, A 2005, Maths skills for working, Phoenix Education, Sydney.

97


16/6/06 11:59:46 AM


Online resources Most major sports have comprehensive websites with up-to-date information, latest scores, ladders, player profiles and match statistics. Below are some specific ones which may be useful. The Australian Sports Commission has general information about different sports: http://www.ausport.gov.au The Australian Institute of Sport also has general information about different sports: http://www.ais.org.au The International Olympic Federation: http://www.olympic.org The Commonwealth Games Federations: http://www.thecgf.com The International Association of Athletics Federations: http://www.iaaf.org

Student worksheets Included in Appendix G are Worksheet J, which shows an example of how to support students in undertaking work and investigations based on sport, and Worksheet K, which is an example of a cooperative logic problem based on a sports scenario. References Australian Sports Commission 2004, Participation in exercise, recreation and sport: Annual report 2003. Viewed at (http://www.ausport.gov.au/scorsresearch/ ERASS2003/findingsERASS2003.pdf) Ernest, P 2004, ‘Relevance versus utility: Some ideas on what it means to know mathematics’, in B Clarke, DM Clarke, G Emanuelsson, B Johansson, DV Lambdin, FK Lester, A Wallby & K Wallby (eds), International perspectives on learning and teaching mathematics, National Centre for Mathematics Education, Göteborg University, Göteborg, Sweden. SUMMARY

• Sport and related activities such as fitness are a relevant and potentially engaging context for students to make connections between mathematics and their lives. • There is a wealth of mathematics embedded in sport, especially numerical information and data and statistics. But this is easily extended to include measurement and space and shape.

98


16/6/06 11:59:46 AM

CHAPTER

8

CURRICULUM CONNECTIONS Different school systems and educational jurisdictions have particular features in their senior secondary mathematics curricula that have been developed over decades, and even centuries in some cases, to meet the historical and contemporary educational needs of their cultures and societies. When these curricula are reviewed, it is often the case that this includes a process of benchmarking with respect to corresponding curricula in other systems and jurisdictions. This may be in a local, county, state, national or international context. Over the past few decades, particularly in conjunction with renewed interest in comparative international assessments (such as TIMSS and PISA, OECD), curriculum benchmarking has been employed extensively by educational authorities and ministries. Such benchmarking reveals much that is common in curriculum design and purpose in senior secondary mathematics courses around the world. Some key design constructs that are used to characterise the nature of senior secondary mathematics courses are: • content (areas of study, topics, strands) • aspects of working mathematically (concepts, skills and processes, numerical, graphical, analytical, problem-solving, modelling, investigative, computation and proof) • the use of technology, and when it is permitted, required or restricted (calculators, spreadsheets, statistical software, dynamic geometry software, computer algebra systems) • the nature of related assessments (examinations, school based and the relationship between these) • the relationship between the final year subjects and previous years in terms of the acquisition of important mathematical background (assumed knowledge and skills, competencies, prerequisites and the like) • the amount and nature of prescribed material within the course (completely prescribed, unitised, modularised, core plus options)

99


16/6/06 11:59:47 AM


• the amount of in-class (prescribed) and out-of-class (advised) time that students are expected to spend on completion of the course In broad terms, it is possible to characterise four main sorts of senior secondary mathematics courses. Type 1 Courses designed to consolidate and develop the foundation and numeracy skills of students with respect to the practical application of mathematics in the lives, in their community, or related to employment or other areas of study. These often have a thematic basis for course implementation. Type 2 Courses designed to provide a general mathematical background for students proceeding to employment or further study with a numerical emphasis, and likely to draw strongly on data analysis and discrete mathematics. Such courses typically do not contain any calculus material, or only basic calculus material, related to the application of average and instantaneous rates of change. They may include, for example, statistics and data analysis, businessrelated mathematics, linear programming, network theory, sequences, series and difference equations, practical applications of matrices and the like. Type 3 Courses designed to provide a sound foundation in function, coordinate geometry, algebra, calculus and possibly probability with an analytical emphasis. These courses develop mathematical content to support further studies in mathematics, the sciences and sometimes economics. Type 4 Courses designed to provide an advanced or specialist background in mathematics. These courses have a strong analytical emphasis and often incorporate a focus on mathematical proof. They typically include complex numbers, vectors, theoretical applications of matrices (for example transformations of the plane), higher level calculus (integration techniques, differential equations), kinematics and dynamics. In many cases Type 4 courses assume that students have previous or concurrent enrolment in a Type 3 course, or subsume them.

100


16/6/06 11:59:47 AM

CHAPTER 8 Curriculum connections

Teaching mathematics in context and courses The courses that could utilise the approaches and ideas in this book are mainly Type 1 or Type 2. These courses are often offered at the penultimate year of senior secondary mathematics schooling (typically Year 11), where studying a mathematics course is either compulsory or has a high rate of student optional selection. The approaches and ideas are also part of several curricula in the final year of senior secondary schooling (typically Year 12). The following mapping refers to both of these contexts. It should also be noted that many of the ideas and approaches used in Foundation Numeracy in Context will be applicable to middle years mathematics courses too—typically Years 7 to 10. Table 8.1 provides a mapping in terms of curriculum connections between the chapters of this book, Type 1 and/or Type 2 courses, and those courses currently offered in various Australian states and territories. As this book is a teacher resource, these connections are with respect to the usefulness of material from the chapters in terms of mathematical background or relevance, rather than direct mapping to curriculum content, or syllabi, in a particular state or territory. Table 8.2 provides a mapping in terms of curriculum connections between the chapters of this book, Type 1 and/or Type 2 courses, and some of the courses currently offered in two English-speaking jurisdictions from around the world. In other key English-speaking countries such as the United States and Canada, for example, courses of this type relate to state or provincial based courses, and their curriculum authorities would need to be consulted in order to align the book to their curricula. Adult Education curricula Foundation Numeracy in Context is suitable for a wide range of numeracy and mathematics courses in adult education. In Australia this would be at Certificate Levels I or II, in numeracy and mathematics modules in courses such as the Certificates in General Education for Adults (CGEA), the Certificate in Adult Foundation Education (CAFE), Certificates in Vocational Access, and Introductory Vocational Education (IVEC).

101


16/6/06 11:59:47 AM


Table 8.1 Curriculum connections for senior secondary final years’ mathematics courses in Australia

State or territory

Type of course

Relevant chapters

Victoria

1: VCE Foundation Mathematics and/or VCE General Mathematics (Year 11)

all

1: VCAL Numeracy (Year 11 or Year 12)

all

2: VCE Further Mathematics (Year 12)

3, 4, 5 and 7

1/2: HSC General Mathematics (Year 11 and Year 12)

all

1: HSC Mathematics Life Skills (Year 11 and Year 12)

all

New South Wales

Queensland

1/2: Mathematics A (Year 11 and Year 12) all

South Australia/ Northern Territory

1/2: Stage 2 Mathematical Applications (Year 12)

all

1: Stage 1 Mathematics (some topics) (Year 11)

all

1/2: Discrete Mathematics (Year 12)

3, 4, 5 and 7

1: Mathematics in Practice (Year 11)

all

1: Vocational Mathematics (Year 11)

all

1/2: Mathematics Applied (Year 12)

3, 4, 5 and 7

1: Mathematics for Living, Mathematics at Work, Mathematics after College (Year 11)

all

1/2: Mathematical Applications (Year 11 and Year 12)

3, 4, 5 and 7

1: Trade and Business Mathematics (Year 11)

all

Western Australia

Tasmania

ACT

102


16/6/06 11:59:47 AM

CHAPTER 8 Curriculum connections

Table 8.2 Curriculum connections for senior secondary final years’ mathematics courses in some jurisdictions around the world

Country

Type of course

Relevant chapters

International Baccalaureate Organisation (IBO)

1/2: Mathematical studies SL

all

UK

1/2: GCSE level Mathematics

all

1: Free-standing mathematics qualifications

all

References The following are the website addresses of Australian state and territory curriculum and assessment authorities, boards and councils. These include various teacher reference and support materials for curriculum and assessment. The Senior Secondary Assessment Board of South Australia (SSABSA) http://www.ssabsa.sa.edu.au/ The Victorian Curriculum and Assessment Authority (VCAA) http://www.vcaa.vic.edu.au/ The Tasmanian Qualifications Authority (TQA) http://www.tqa.tas.gov.au/ The Queensland Studies Authority (QSA) http://www.qsa.qld.edu.au/ The Board of Studies New South Wales (BOS) http://www.boardofstudies.nsw.edu.au/ The Australian Capital Territory Board of Senior Secondary Studies (ACTBSSS) http://www.decs.act.gov.au/bsss/welcome.htm/ The Curriculum Council Western Australia http://www.curriculum.wa.edu.au/ The following are the website addresses of various international and overseas curriculum and assessment authorities, boards, councils and organisations: International Baccalaureate Organisation (IBO) http://www.ibo.org/ibo/index.cfm/ Qualifications and Curriculum Authority (QCA) UK http://www.qca.org.uk/ OECD Program for International Student Assessment (PISA) http://www.pisa.oecd.org/

103


16/6/06 11:59:48 AM


Trends in International Mathematics and Science Study (TIMSS) http://nces.ed.gov/timss/ The international Adult Literacy and Lifeskills (ALL) Survey which includes a specific numeracy component http://nces.ed.gov/surveys/all/

104


16/6/06 11:59:48 AM

APPENDIX A Planning grid Task/investigation/project: __________________________________________ Initial question and description/purpose

Curriculum learning outcome(s) covered

Mathematics skills and supporting activities

105


16/6/06 11:59:48 AM


Resources

Assessment strategies and records

106


16/6/06 11:59:48 AM

APPENDIX B Possible contex ts

Context/topic

Area of mathematics

Sport • scoring • ladders • payments to players • reading information/data • fitness

Number Algebra Data Measurement

Food and cooking • health • cooking and recipes • eating out

Number Measurement Space and shape

Medication and health • medicines—dosages etc • fitness/illnesses • diet

Number Measurement Data

Drinking • alcohol content and drink driving • quantities/bottle sizes/costs

Number Measurement Data Space and shape

Gambling • chance in different games • winnings/losses • impact on individuals and the community • data on participation

Algebra Number Data

Getting around • street directories • maps • giving and following directions • GPS

Space and shape Measurement

107


16/6/06 11:59:48 AM


Context/topic

Area of mathematics

Signs and symbols • road signs • safety signs • logos

Space and shape

Clothes • designing and making clothes – design and measurement aspects • sizes of clothes • costs of buying clothes (versus making your own)

Space and shape Measurement Number

Music, art and craft, hobbies, etc. • designing and making things (clothes, jewellery, woodwork, recording music, etc.) • selling items that you make • popularity of music – the ‘top 40’ charts, etc. • surveys on music likes and dislikes • costs of producing your own CDs • designing CD covers

Space and shape Measurement Number Data

Reading and understanding newspaper articles that incorporate numbers/data/graphs

Data Number

Shopping • costs • packaging • weights

Number Measurement Space and shape

Budgeting • living • parties • holidays

Number Measurement

Finances • banking • credit cards • loans (e.g. buying a car)

Data Number Algebra

108


16/6/06 11:59:48 AM

APPENDIX B Possible contexts

Context/topic

Area of mathematics

Plans and construction • building • gardens • parks • sports fields, etc.

Space and shape Number Measurement Algebra

Holidays • planning (where, when, etc.) • travel distances and times • fuel economy and costs • budgets (accommodation, fares, meals, etc.)

Space and shape Number Measurement Algebra

Bills • phones • utilities (gas, electricity, water) • credit cards

Number Algebra Data

Elections and voting

Number Data

Cars • getting your licence • buying a car • running a car • under the bonnet • accidents • travel—distance, speed, time • navigation—getting around

Number Measurement Data Space and shape Algebra

Eating out

Number Measurement

Work-related contexts • quality control info/charts • orders/stock control/recording information • following job sheets and instructions • costs/money • measurement • Occupational health and safety • parcels—weights and charges • delivery routes etc.

Number Measurement Data Space and shape Algebra

109


16/6/06 11:59:49 AM

APPENDIX C Sample brainstorm

110


16/6/06 11:59:49 AM

APPENDIX D Mat hemat ics skills example

Starting question If you were given $10 000 to buy a second-hand car, what car would you buy? Why? Mathematics skills Possible mathematics skills that could be covered or encouraged would include: • understanding numbers and money – reading and writing numbers and money, comparing, calculating • understanding measurements related to cars (e.g. capacity of engines, distance travelled) • understanding rates related to cars (e.g. fuel economy, speed) Refining the question You may decide to specify the requirements of the task a bit more, especially if you think students need more direction and guidance. Preparing worksheets can also support this: If you were given $10 000 to buy a second-hand car, what car would you buy? Compare at least three different cars and describe which of these cars you would buy and why.

Integrating with other skills If integrating the teaching with other areas such as information communications technologies (ICT) and literacy, skills that could be covered include: • researching information (in newspapers, magazines, and on the internet) • reading advertisements including terminology and abbreviations • using ICT to collect, document and analyse data and costs • writing and presenting reports, including using ICT – PowerPoint, digital stories using video or photos, etc.

111


16/6/06 11:59:49 AM

APPENDIX E Resources

Real-life resources In an applied learning approach, real-life resources, especially materials from students’ own life experiences and interests should be used as much as possible. These could include: • newspapers, magazines or advertisements • actual items and products such as containers of drinks, food, and so on • internet sites and materials • videos of TV shows • information leaflets and materials, e.g. about mobile phones, allowances, concession cards • shopping and advertising materials • maps, plans, street directories Information and communication technology Students should have access to: • calculators, including graphics calculators where appropriate • computers preferably with internet access Internet access can be used for a range of purposes, including: • searching for information and texts • finding answers to specific questions • using educational software and teaching sites Appropriate software should be available, such as: • MS Excel or other spreadsheet packages (including the drawing and graphing package) • MS Word or other word processing package (including the drawing and graphing package) • MS PowerPoint or other software presentation packages • Educational software.

112


16/6/06 11:59:49 AM


Hands-on materials There is a wide range of hands-on materials that are useful for teaching, especially for teaching the underlying mathematical skills. This would include materials such as: • dice, including 10-sided dice • place value materials such as MAB blocks, straws • shapes and containers such as packaging, bottles, tins • domestic measuring equipment such as scales, cups, builders’ and dressmakers’ tapes, etc. • fraction materials such as fraction circles • rulers, paper, card, scissors, glue, etc. • clocks, watches, calendars Print-based resources The references listed below include some possible teacher and learner resources. It is not meant to be a definitive list. Some particular resources are also listed in each chapter that are relevant to the context being discussed. Dengate, B & Gill, K 1989, Mathematics for teenagers, Longman Cheshire, Melbourne. Downie, D, Slesnick, T, Stenmark, JK 1981, Math for girls and other problem solvers, Lawrence Hall of Science, University of California, Berkeley, CA. Erickson, T 1989, Get it together: Math problems for groups grades 4–12, Lawrence Hall of Science, University of California, Berkely, CA. Flittman, E & Sharrock, H 2000, Foundation mathematics for the real world, Longman, Melbourne. Ford, K (ed.) 1985, Mathematics for living (series): Car mathematics, Mathematics around the home, Outdoor mathematics, Rural mathematics, Travel mathematics, Holmes McDougall and Educational Supplies, Brookvale, NSW. Goddard, R, Marr, B & Martin, J 1991, Strength in numbers: A resource book for teaching adult numeracy, Holmesglen College of TAFE, Holmesglen, Vic. Goddard, R & Regan, M 1995, The value of time: Numeracy for workers in manufacturing, Council of Adult Education, Melbourne. Kimball, DB 1990, Math for the real world: Book one, New Reader Press, Syracuse, NY. Kimball, DB 1990, Math for the real world: Book two, New Readers Press, Syracuse, NY. Lawrence, G & Moule, J 1989–90, Mathematics in practice series (Accommodation, Design, Health, Social issues, The consumer, Travel, Transport), Harcourt Brace Jovanovich, Sydney. Lowe, I 2001, Mathematics at work, (CD-ROM), AAMT, Adelaide. Lowe, I 2005, Active learning series (Measurement, Chance and data; Number and algebra and Space) Mathematical Association of Victoria, Melbourne.

113


16/6/06 11:59:50 AM


Marr, B, Anderson, C & Tout, D 1994, Numeracy on the line: Language based numeracy activities for adults, National Automotive Industry Training Board, Victoria. Marr, B & Helme, S 1987, Mathematics: A new beginning, State Training Board, Victoria. Marr, B, Helme, S & Tout, D 2003, Rethinking assessment, Language Australia, Melbourne. O’Connor, M & Gaton, B 2000, Foundation mathematics, Oxford University Press, Melbourne. Ritchhart, R 1994, Making numbers make sense: A sourcebook for developing numeracy, Addison-Wesley, Menlo Park, CA. Schmitt, MJ & Steinback, M (eds) 2004–2005, Math: EMPower series, Key Curriculum Press, Emeryville, CA. Thomson, S & Forster, I 1998, Trade and business mathematics 1, Addison Wesley Longman, Melbourne. Thomson, S & Forster, I 1999, Trade and business mathematics 2, Pearson Education, Melbourne. Vize, A 2005, Maths skills for living, Phoenix Education, Sydney. Vize, A 2005, Maths skills for working, Phoenix Education, Sydney. Weber, L 2003, The language of maths (secondary level), AEE Publishing, South Mission Beach, QLD.

CD ROMs Tout, D & Marr, B 1997, Measuring up: An interactive multimedia computer resource for numeracy learners, Protea Textware, Melbourne. Hagston, J & Tout, D 2001, Brush up on your skills, TAFE Frontiers, Victoria.

Online resources The references listed below are a few of the many now available and the list is not meant to be comprehensive. Searching the internet for a specific mathematics topic is sure to come up with a range of suitable websites. Math in Daily Life: http://www.learner.org/exhibits/dailymath/ Maths300 is an Australian web-based service which aims to support teachers in the delivery of excellent mathematics education. It aims to resource members with extensive notes for, at least, 300 exemplary maths lessons (K-12): http://www. curriculum.edu.au/maths300/ Online data in mathematics, science and technology education contains links to sites that hold recent data in mathematics, science and technology education. These include Olympics statistics, weather data and US Census data: http://www. mste.uiuc.edu/data/data.html The Knot a Braid of Links site aims to provide a guide to mathematics on the web. It holds lots of activities and you can contribute your own as well: http://west. camel.math.ca/Kabol/

114


16/6/06 11:59:50 AM


The Math Archives holds a large number of links to various web-based resources on mathematics. Icons guide you to the levels of the activities and the types of web pages that are used (JAVA, animations, interactive etc.): http://archives.math.utk. edu/topics/ The Math Forum is an online math education community centre and is a site full of maths ideas and resources, including ‘Ask Dr. Math’, problem of the week, an online forum and a number of discussion groups: http://mathforum.org/

Toolboxes A wide range of online resources has been developed in Australia that are based on teaching and learning mathematics in a context – these have been called ‘toolboxes’. Some are generic, whilst others are specific to a particular industry or trade. To find out about all toolboxes: http://flexiblelearning.net. au/toolbox/ But below are some specific examples. Basic Skills in the Cybercentre: http://www.flexiblelearning.net.au/toolbox/ demosites/series4/424/index.htm Where’s the party at?: http://toolbox.vetonline.vic.edu.au/519/entry_page.htm Your Online Learning Assistant (YOLA): http://www.flexiblelearning.net.au/ products/yola.htm

Curriculum authorities The following are the website addresses of Australian state and territory curriculum and assessment authorities, boards and councils. These include various teacher reference and support materials for curriculum and assessment and will include resources and support for teaching in the middle and senior years of schooling: The Senior Secondary Assessment Board of South Australia (SSABSA): http://www. ssabsa.sa.edu.au/ The Victorian Curriculum and Assessment Authority (VCAA): http://www.vcaa.vic. edu.au/ The Tasmanian Qualifications Authority (TQA): http://www.tqa.tas.gov.au/ The Queensland Studies Authority (QSA): http://www.qsa.qld.edu.au/ The Board of Studies New South Wales (BOS): http://www.boardofstudies.nsw.edu. au/ The Australian Capital Territory Board of Senior Secondary Studies (ACTBSSS): http://www.decs.act.gov.au/bsss/welcome.htm The Curriculum Council Western Australia: http://www.curriculum.wa.edu.au/

115


16/6/06 11:59:50 AM


The following are the website addresses of various international and overseas curriculum and assessment authorities, boards, councils and organisations: College Board US Advanced Placement (AP) Calculus: http://www.collegeboard.com/ student/testing/ap/sub_calab.html?calcab International Baccalaureate Organisation (IBO): http://www.ibo.org/ibo/index.cfm Qualifications and Curriculum Authority (QCA) UK: http://www.qca.org.uk/

116


16/6/06 11:59:50 AM

APPENDIX F Contract for the outline of the invest igat ion

Title of investigation: ____________________________________________________ T

General description of the investigation (including final product).

Course outcomes achieved

Proposed times

Resources required



117


16/6/06 11:59:51 AM

APPENDIX G Sample worksheet s

WORKSH EET A : DRUG USAG E BY YOU NG PEOPLE

Investigation Use the internet and newspapers to research about drug usage by young people. Write a brief report showing the information you found, including some graphical representation of the statistics on drug usage by young people.

Possible resources •

•

Information provided by alcohol and drug organisations (a good starting point is the Alcohol and Other Drugs Council of Australia (ADCA) at http://www.adca.org.au, which is a good source of information and articles, and links to many other related sites). Articles in any of the daily newspapers. You can search on their websites to find any that may have been published recently.

Tasks and questions 1. Research and collect relevant data Make sure you identify, after your initial researching, what data and information is manageable for you to study. For all of Australia or just your state? For what age groups? Don’t try to cover too much. You may need to restrict what you study—for example, a particular type of drug— or you could group them together in some way if that makes it easier.

2. Possible extension: conduct local survey To compare the data you found for Australia or your state or locally, you could conduct a local survey to do a comparison for your community. You will need to seek permission to do this from your teacher and school, and any relevant authorities, and make sure that they approve the survey and its questions before you start to ask people to complete it. You will need to make sure you can guarantee anonymity and confidentiality.

118


16/6/06 11:59:51 AM

APPENDIX G Sample worksheets

3. Record data collected or found Put the information about drug usage that you found into a table, and any other information you found that is important. (You could use a spreadsheet in Excel or another software package to record this information.)

4. Graph information Plot your information onto a graph or graphs. (You could use the graphs software in MS Excel or MS Word or another software package to plot the graph.)

5. Analyse data • •

What did you find out? How did the drug usage vary? In what ways?

6. Report on results • • •

Write up a report of your research, including any graphs of your main findings. What did you find out? Did anything surprise you? What is the main message you would want to highlight in your report?

Note: Remember to ask your teacher for support and explanations if you need help with understanding and using any particular mathematics skills to complete your project.

119


16/6/06 11:59:51 AM


WORKSHEET B: GAMBLING AND BETTING

Investigation Investigate at least one instance of gambling or betting. You could choose to compare games of chance or analyse how betting on sports works. Write up a report showing the odds and the betting or gambling system and how they work. Include an explanation of how odds in the sport or the game relate to how we calculate probability and chance in mathematics.

Tasks and questions 1. Research different games or sports betting Identify which games of chance or which sport you will study. Games of chance could include Tattslotto, Super 66 and their equivalents; or games played at casinos like Two-up, Blackjack and Roulette. Most popular sports now accept betting—so you could look at how betting works on a sport you are interested in—but you could just analyse the more traditional sports such as betting on horse racing. Research your chosen games or sports and the betting systems (the internet might be the quickest, but your school or local library may well have books that explain the systems) to find out how they work.

2. Possible extensions: conduct a local survey or invite a speaker to talk about gambling As part of your research you could conduct a local survey to see how many people (or students at your school) gamble or bet and what they bet or gamble on. You will need to seek permission to do this from your teacher and school, and make sure that they approve the survey and its questions before you start to ask people to complete it. Or you could contact a local or state-based organisation that helps people with gambling problems and ask for a representative to come and talk to your class about the problems with gambling in our society.

3. Look at the chance and probability involved in the game or betting Analyse your game or betting scheme in terms of: • the rules of the game and how you can win • the probability of winning (or losing) • the system used for paying you your winnings, including how much the ‘bank’ or dealer keeps

4. Report on results • • •

Write up a report of your research. What did you find out? Did anything surprise you? What is the main message you would want to highlight in your report?

Note: Remember to ask your teacher for support and explanations if you need help with understanding and using any particular mathematics skills in completing your project.

120


16/6/06 11:59:51 AM


WORKSHEET C: BUYING A CAR

Investigation Decide on a particular car you’d like to buy second hand (choose a fairly common brand of car). Search newspaper car advertisements or car sale sites on the internet and find out what choices you have. Analyse and report on what is available.

Tasks and questions 1. Decide what car would you like to buy Brand: Model: Year or years made: Size of engine:

2. Search advertisements for the car at used car yards You could use car sections out of the daily newspapers (often the Friday or Saturday papers have lots) or you could search on the internet for cars on sale). Or you could visit some car yards. • Where did you find the car for sale? • How many did you find for sale? • Did you need to change which car you wanted because there were not many available for sale? If so, what did you change?

3. Record your results Put the costs of the cars you found for sale into a table (and any other information you found that is important). (You could use a spreadsheet in Excel or another software package to record this information.)

121


16/6/06 11:59:51 AM


If you found lots of cars, you could group them together if that makes it easier.

4. Graph the prices Plot your information onto a graph or graphs. (You could use the graphs software in MS Excel or MS Word or another software package to plot the graph.)

5. Analyse the prices • • • •

What did you find out? What was the most common price for the car you chose? What was the average price? How much did the prices vary by?

6. Report on your results • •

What did you find out? Did anything surprise you? Out of the cars you found, which car would you buy and why?


122


16/6/06 11:59:52 AM


WO R KS H EE T D : N AV I G AT I O N — G E T T I N G A RO U N D

Investigation Choose a capital city in a different state that you would like to visit. Decide on how you would drive there and describe and draw a sketch map of the route, showing distances travelled and the time you estimate it would take, how long you would stop for, including overnight stops, etc.

Tasks and questions 1. Which capital city in a different state would like to visit? Visit a tourist bureau or use the internet to find out information about the city. • Which city do you want to drive to? Why? • What do you want to see while you are there or on the way? • About how far is it from where you live?

2. Investigate about getting there Find a map or maps that show your routes from home to the city. Then use all the information you have collected to consider these questions: • What maps are you using? • What is the scale of the map?

123


16/6/06 11:59:52 AM


• • •

Which route will you take to get there? Will you drive back the same way? If not, which way will you come back? What is the total distance you will need to drive? Do you want to stop and see places on the way? What are they? Where are they?

3. Plan your trip Use all the information you have found in order to plan the trip. You could write up your plan using computer software—for example, in Word, Excel or PowerPoint. Write up a detailed plan for the trip. Include information about: • road numbers and names • key towns you drive through • which towns you will stay at overnight and how long you will stay in each place • distances between the key towns • estimated travel times between the key towns

4. Make a sketch map Draw up a rough map for the trip to show your friends or family. Include basic information on the map about key towns and distances.

5. Costing your trip (optional) Use a spreadsheet to estimate the total costs for the trip. Include how you calculated the different costs. Include information about: • petrol costs • accommodation • food • costs of places you might like to visit • other costs—like souvenirs, presents Note: Remember to ask your teacher for support and explanations if you need help with understanding and using any particular mathematics skills in completing your project.

124


16/6/06 11:59:52 AM


WORKSHEET E: ALGEBRA AND AUSTRALIAN RULES FOOTBALL In Australian Rules football, you get: • 6 points for each goal you kick • 1 point for each behind you kick The team with the highest number of total points wins the game. This is a situation where you can use algebra to show how you can work out something—in this case the score. Most children and adults who know about Australian Rules football can work this out in their heads, without realising they are really using a form of algebra.

How does this work? Here are some scores for a football game between two teams, Essendon and Brisbane: Team

Goals

Behinds

Points

Essendon

11

16

82

Brisbane

12

10

82

Because each goal is worth 6 points and each behind is worth just 1 point, to work out the total number of points you need to multiply each goal by 6, and then add on the number of behinds. So for Essendon you have 11 goals, worth 11 times 6 points, which is 66 points. Then you need to add that to the number of behinds (16). So altogether Essendon’s score of 11 goals and 16 behinds is worth 66 plus 16 points, which gives 82 points. With Brisbane it will be 12 times 6 points for the 12 goals plus 10 for the number of behinds. So the number of points will equal 12 times 6 which equals 72, plus 10 more which gives us a total of 82 points. In this case, the game would be a draw because the two teams had the same total number of points.

So what has this got to do with algebra? No matter what each team kicks, we have a scheme that enables us to work out their total score. There is therefore a general method that applies in any situation. This is a generalisation that a formula in algebra does for us—it is a way of representing and writing down what we need to do. Let’s have a look at this with these examples. In the last calculation above for Brisbane, we have written a mathematics sentence: The number of points will equal 12 times 6 which equals 72, plus 10 more which gives us a total of 82 points. If we write all the maths words using their symbols (that is, times is a x, and plus is a + and equals is =, then we have something looking like: Number of points = 12 x 6 + 10 = 72 + 10 = 82 points.

125


16/6/06 11:59:53 AM


However, this is just for this one score—we can generalise this even more for any number of goals and behinds: Number of points = number of goals x 6 + number of behinds. This is actually an algebraic sentence or a formula. In mathematics we are often lazy and use shorthand ways of writing things down. We use a single letter to represent what we are talking about. Here we can use a letter to stand for the number of points (let’s say p), a letter for the number of goals (let’s say g), and finally a letter for the number of behinds (let’s say b). Then we use the letters instead of the words in the sentence. This would give us: Number of points = number of goals × 6 + number of behinds p= g ×6+ b p=g×6+b Now that’s more like algebra we’ve seen at school. It’s a formula for working out the number of points a team scores in Aussie Rules football. We use italics for the pronumerals to avoid confusion.

One more step We can go a few more steps based on short-cuts in how we write down algebra and formulas in mathematics. A particular short-cut is for when we multiply a number by a letter, like in g × 6 above. We shorten this further to be simply 6g—it still means 6 lots of g or 6 multiplied by g. For multiplication we normally put the number in front of the letter. So the formula we got above can be made very short by writing: p=g×6+b p = 6g + b

Substitution Here are some other football scores. Team

Goals

Behinds

Carlton

9

15

Adelaide

15

8

Fremantle

10

12

Sydney

15

8

Points

To work out the score for Carlton, who kicked 9 goals (so g = 9) and 15 behinds (so b = 15), we need to put 9 and 15 into the formula (this is called substitution): p = 6g + 6 p = 6 x 9 +15 p = 54 + 15 = 69 points Number of points scored by Carlton was 69.

126


16/6/06 11:59:53 AM


Can you use the formula to work out the number of points scored by the other three teams? Team

Goals

Behinds

Points

Carlton

9

15

69

Adelaide

15

8

Fremantle

10

12

Sydney

15

8

If you follow other games such as like rugby or basketball, you will find that there are similar scoring schemes where you can work out the scores using algebra. Try and do it for another sport.

Extension You could go further with using this formula for football scores. Think about how you might work out these questions: a If a team scored 100 points and kicked 14 goals, how many behinds did they kick? b If a team scored 100 points and kicked 10 behinds, how many goals did they kick? c How many ways could a team score 100 points if they kicked at least 12 goals? d Investigate the different scoring system used in the pre-season AFL (Australian Football League) competition, and write a formula for it.

Conventions and rules We saw some rules and short cuts that we use in mathematics. Here they are written out again, with a couple more as well. Operation or words

Symbol/letter

Any letter can be used (here we used g, b and p). The value being used (like goals, behinds and When we use a letter like this to stand for something it is called a pronumeral. points) They represent variables—because they can vary from one situation (game) to another. Add

+

Subtract

–

Multiply

× [or * on some calculators or computers] or no symbol or sign at all in algebra

Divide

÷ or ⁄ or — like a fraction

Equals

=

127


16/6/06 11:59:53 AM


WO R KS H EE T F: FO R M U L A S AT WO R K

1. Service charges and rates Leak & Family Plumbers charge a $50 attendance fee for coming to your home to fix a leak, and then charge $80 per hour for the time the plumber spends fixing it. How can we work out how much they will charge? In this case, the charge will be the number of hours the plumber comes for multiplied by his rate of $80 per hour, then we need to add on the flat rate of $50 he charges for coming to your house. This is another situation where you can use algebra to help work it out. So if he came for 1 hour, he will charge just 1 lot of $80 plus the $50 attendance fee, giving a charge of $130. If he came for 2 hours it would be 2 times the $80 plus the $50, which gives $160 plus $50, which is $210. This still works for times of less than an hour, provided we do the time as a fraction of an hour. For example, a call of half an hour would be a half of $80 plus the $50, which would give you $40 plus $50 equals $90. Now, let’s see about trying to write this using algebra. As a sentence it would be something like: Charge is number of hours multiplied by $80 plus the $50 fee Using letters: let’s say C = charge, h = number of hours, so we can write this formula as: C = h × $80 + $50, which can be shortened to: C = 80h + 50

Extension You can do some more work based on this example of paying for a plumber’s time. Investigate these situations. a How many hours of Leak & Family Plumbers time could you get for $100? For $200? b Draw a graph of the cost versus the time for Leak & Family Plumbers to work at your house. Draw it for up to 8 hours work. Use it to work out the times for the above question. Relate this to algebraic graphs—what does the y-intercept mean in this example? What does the gradient mean? (Ask your teacher for help if you don’t understand what these are or how to work them out.) c Another plumbing company, Tip Top Taps P/L, charges an $80 attendance fee for coming to your home to fix a leak, and then charges $50 per hour. Compare the two companies, Leak & Family Plumbers and Tip Top Taps P/L, by drawing graphs of both their costs. Who would you use when and why? Use the graph to work out how many hours of service would make their costs the same? Can you work this out algebraically? This can be solved on graphing calculators too. (Hint: this is called simultaneous equations).

128


16/6/06 11:59:53 AM


2. Travel costs You have a sales job where you need to travel a lot. Your employer pays you $50 a day food and meal allowance when you travel out of town. As well, you get paid a car allowance of 60 cents per kilometre. • How much would you get paid for one day’s work if you travelled a total of 300 km in that day? • How much allowance would you receive if you did 500 km over two days? • Write a formula that will work out your allowance for any number of days and any number of kilometres? a How much would you get paid for one day’s work if you travelled a total of 300 km in that day? Travel allowance would equal the food and meal allowance ($50) plus 300 km at 60 cents per km. This would equal $50 plus 300 # $0.60 = $50 + $180 = $230 b

How much allowance would you receive if you did 500 km over two days? Travel allowance would equal the food and meal allowance for two days (2 # $50) plus 500 km at 60 cents per km. This would equal 2 # $50 plus 500 # $0.60 = $100 + $300 = $400

c

Write a formula that will work out your allowance for any number of days and any number of kilometres. Travel allowance would equal the food and meal allowance for the number of days plus the number of km at 60 cents per km. So travel allowance = $50 # the number of days + $0.60 # the number of kilometres. If we call TA the travel allowance, and d the number of days and k the number of kilometres, then we can generalise this to: TA = $50 # d + $0.60 # k Writing this mathematically we would get: TA = 50d + 0.6k dollars

3. Examples of using a formula—substitution The aim in using a formula is to work out answers for any situation. We do this by putting in the figures into the formula to work out the answers. We should also check that the answer we get matches what we estimate or expect the answer to be.

129


16/6/06 11:59:54 AM


Try these yourself: a b c d e f g

What would be the score for an Australian Rules football team that kicked 8 goals and 13 behinds? What would be the score for an Australian Rules football team that kicked 20 goals and 21 behinds? What would Leak & Family Plumbers charge for a three quarter hour visit? What would Leak & Family Plumbers charge for a two and a half hour visit? Use the formula for your travel allowance, TA = 50d + 0.6k , to work out how much you would get paid for 2 days travel where you travelled 350 km? Use the formula for your travel allowance to work out how much you would get paid for 4 days travel where you travelled 650 km? The formula for the area of a rectangle is:

length, l width, w

h

Area = length # width A = l#w

What is the area of the floor of a rectangular shaped room 3.5 m x 4 m? As with area, there is a formula that we use to work out the volume of a boxshaped object (called in maths, a rectangular prism). If the sides of such a shape were called length (l), width (w) and either height (h) or depth (d):

length, l

height, h or depth, d

width, w

Then the formula for working out the volume of a box shape is: volume = length # width # height V = l#w#h What is the volume of a swimming pool 25 m # 12 m # 1.2 m ?

4. One for you—hiring a stall to sell some craft Amanda decides to sell her photos by hiring a stall at the next school market. The charge to hire the stall is $100 for the day. She decides to sell her photos for $25 each. a Write a formula to show how much money Amanda makes from the market, showing a connection between the amount of money she takes home and the number of photos she sells.

130


16/6/06 11:59:54 AM


b c d e

How much money does Amanda make if she sells 10 photos? Show how you do this using your formula. How much money does Amanda make is she sells 24 photos? How many photos does she need to sell to break even on the day—that is cover her hire of the stall? How many photos does Amanda need to sell to make $200?

Extension—graph of this formula You can do some more work based on this example of selling photos. Ask your teacher for help if you need to. Draw a graph of your formula showing how much money Amanda makes versus the number of photos she sells. a Is there anything unusual about this graph? [Hint—something about when she makes a loss on the day.] b What does the y-intercept mean in this case? c What does the x-intercept mean in this case? d What does the gradient mean?

Rates We use rates a lot—in one example here it was how much the plumber charges per hour (dollars per hour), but it can be: • how much we might pay for food (meat, vegetables, etc)—dollars per kilogram • how fast we drive a car—kilometres per hour • how much you get paid for work—dollars per hour • your heart rate—beats per minute • how fast an engine turns—revolutions per minute • how much a machine in a factory produces—metres per minute, kilograms per hour, square metres per kilogram Rates are a way of describing one quantity (e.g. pay in dollars) in relation to another quantity (e.g. time in hours). It tells you how many of the first quantity you get for each of the second quantity—so for the plumber’s pay rate it is how many dollars you pay them for each hour they work. We use the word ‘per’ for rates (e.g. dollars per hour, kilometres per hour). It means ‘for each’ or ‘out of’. We use the symbol ‘/’ to abbreviate this. Here are some common rates and how we write them: • dollars per kilogram = $/kg • kilometres per hour = km/h (sometimes kph) • dollars per hour = $/h • revolutions per minute = rev/min (sometimes rpm) We often are given rates and need to calculate with them, as in the plumber example we are working on. To work out a total (like how much you pay the plumber, or pay for the meat at the butchers) you need to multiply the rate by the quantity you are using or buying.

131


16/6/06 11:59:54 AM


WORKSHEET G : SOME EX AMPLES FROM THE WORKPLACE Below are some examples of formulas used in some different everyday situations, jobs and workplaces. These do not include more mathematically based occupations such as engineering, surveying, architecture, and so on. It is surprising how many occupations require the use and application of formulas.

Areas and volumes Common to many occupations such as painting, landscaping, asphalting and paving is the need to use formulas for calculate areas and volumes for different shapes. Here are some:

Rectangle Area = length # width A = L#W

W L

Triangle Area of right-angled triangle = 21 # length # height A=

1 2

H

#L#H

L

Circle Area of circle = p # radius # radius A = p#r#r A = p # r2 A = pr 2

r

Box shape (rectangular prism) Volume = length # width # height (or depth) V = l#w#h

h l

w

Quality control In many industries, from shop floor workers through to supervisors and management, there is an expectation that employees will be responsible for some form of involvement and participation in the quality control processes of the workplace. This may be about keeping tabs and records on quality of the materials being processed and produced (in terms of their sizes and tolerances, or their imperfections) or in terms of accident rates, days of absence, performance (yield and production rates, meeting time schedules, etc.) or a range of other quality factors. This requires not only an understanding of measurement, numbers and rates, but also of statistics and data and its analysis. This requires the understanding of a number of statistical formulae, some of which are below.

132


16/6/06 11:59:55 AM


Mean (average)

/x

x= n

Standard deviation s=

/]x - x g2 n-1

Asphalting and paving Calculating the number of tonnes of asphalt The relative density used as the industry constant for asphalt is 2.4 tonnes per cubic metre. In order to work out the number of tonnes of asphalt, there are two calculations that need to be made: Step 1: Calculate the volume of the hole or patch in cubic metres Step 2: Calculate the mass of asphalt in tonnes using the formula: T = V # 2.4

Estimating quantities To estimate the quantity of asphalt needed for a job: Step 1: Work out the number of square metres per tonne for that job. The formula is: Square metres per tonne = 400 ' T Step 2: Estimate the tonnes of asphalt by dividing the area you have by number of square metres per tonne (the answer you get for Step 1): Tonnes of asphalt = area of job ÷ square metres per tonne

Weather—converting temperatures and chill factor Fahrenheit to Celsius degrees temperature 5 C = ]F - 32g 9

New Bureau of Meteorology apparent temperature formula AT = Ta + 0.348e - 0.7ws +

0.70Q

]ws + 10g

- 4.25

Where: Ta = dry bulb temperature (°C) e = water vapour pressure (hPa) ws = wind speed (m/s) at an elevation of 10 metres Q = net radiation absorbed per unit area of body surface (w/m2)

133


16/6/06 11:59:55 AM


Music, sound production and telecommunications Again, a wide range of formulas are used in working in these areas. Some examples:

Relative power gain Ap =

Signal to noise ratio

Po Pi

S SNR = 10 log10 c m dB N

Amplitude modulation AM waveform: e AM ]t g = Vc ^1 + m sin wm t h sin wc t

Travel—speed Constant speed s=

Constant acceleration d = ut + 21 at 2

d t

Electronics industry Ohm’s law

Resistance (series)

V = I#R V I= R V R= I

R]totalg = R1 + R2 + R3 + g Resistance (parallel) 1 1 1 1 = + + R]totalg R1 R2 R3

Meat industry The formula to calculate the percentage of total solids lost: P=

100 # S ]1.1 # D g - B + B D-S

where: P = percentage of total solids lost S = percentage of solids in the stick water D = percentage of solids in the dewatered blood B = percentage of solids in the raw blood

134


16/6/06 11:59:56 AM


Health and fitness Body mass index (BMI) Metric:

Imperial:

W BMI = 2 H

BMI =

where: BMI = body mass index W = weight in kilograms H = height in metres

where: BMI = body mass index W = weight in pounds H = height in inches

W # 700 H2

Financial Compound interest A = P]1 + r gn

135


16/6/06 11:59:56 AM


W O R K S H E E T H : E N D - O F -Y E A R M E A L

Investigation Your class is to cater and cook a meal for the whole group for an end-of-year meal. The class is to plan, organise, cater and prepare the food. You need to work with a small group of classmates to prepare and cater for some part of the meal. The meal should include drinks, at least three different types of food and some desserts. The tasks you need to complete are set out below.

Tasks and questions 1. Decide on food for the meal • • •

Negotiate with the rest of the class and the teacher to agree on what food is to be cooked. Write out the menu for the day. What part of the meal is your group responsible for?

2. Establish the ingredients required • • •

Find and write out a recipe for what you are to prepare. How many does your recipe serve? Work out how many times you need to multiply your recipe by to feed the whole class. How much is required to feed everyone? Write a shopping list for all the ingredients you need

3. Work out the cost of meal • •

Go to a supermarket or use a supermarket site on the internet to cost the ingredients. Use the rest of the class’s costs to work out the total cost of the meal and work out the cost per student.

4. Work out timeline and tasks • •

With your group work out what the tasks are you will need to do in order to have the food ready for the time you want to all eat. Make a timeline - list the tasks against the days and times so you can meet the deadline. Remember to take into account time to buy the ingredients, storage, preparation and any cooking time.

5. Buy ingredients •

With your share of the class money, go with your group and buy the ingredients and food you need.

136


16/6/06 11:59:56 AM


6. Cook meal • •

Now prepare and cook your part of the meal. Enjoy the food!


137


16/6/06 11:59:57 AM


WORKSHEET I : NUTRITIONAL CONTENT OF FOOD

Investigation Investigate the nutritional content of common foods, using foods that list the nutritional content on their packaging. Compare foods with a range of brands or types. Good examples include breakfast cereals, biscuits, milks and yoghurts. Once chosen, analyse their nutritional content (e.g. fat, sugars, salt, fibre). Write up a report of your findings comparing the different brands, including some graphs of the information.

Nutrition information servings per package – 27 average serving size – 30 g (1 metric cup) average quantity per serving

average quantity per serve with 1 cup skim milk 2

average quantity per 100 g

Energy

475 kJ

669 kJ

1582 kJ

Protein

2.3 g

7.0 g

7.8 g

< 0.1 g < 0.1 g

0.2 g 0.1 g

0.2 g < 0.1 g

Carbohydrate – Total – Sugars

25.1 g 2.4 g

31.6 g 8.8 g

83.6 g 7.9 g

Dietary fibre

0.8 g

0.8 g

2.6 g

246 mg

303 mg

820 mg

28 mg

234 mg

93 mg

Fat – Total – Saturated

Sodium Potassium

Tasks and questions 1. Research which foods to study and collect copies of the nutritional information Identify which type of food you want to investigate, and collect copies of the nutritional information from the packaging. This is usually in a table that lists the different nutritional content such as fat, sugar, salt, fibre, etc. There may be too many ingredients for you to investigate effectively, so focus on a few that you are interested in researching and comparing.

138


16/6/06 11:59:57 AM


2. Record data Put the information about the nutritional content of each brand into a table. (You could use a spreadsheet in Excel or another software package to record this information.)

3. Graph information Plot your information onto a graph or graphs. (You could use the graphs software in MS Excel or MS Word or another software package to plot the graph.)

4. Analyse data • • •

If appropriate you could calculate averages of some of the figures. Analyse the graphs and the data. How did the nutritional ingredients vary from brand to brand? In what ways?

5. Report on results • Write up a report of your research, including any graphs of your data. • What did you find out? Did anything surprise you? • What is the main message you would want to highlight in your report? Extension: compare your findings with how the different brands are advertised. Is there a connection or relationship? Note: Remember to ask your teacher for support and explanations if you need help with understanding and using any particular mathematics skills in completing your project.

139


16/6/06 11:59:57 AM


WORKSHEET J : HOW CLOSE TO THE MARK?

Investigation Undertake measurements of your own or another student’s performances in three different athletic events and investigate and compare the performances against either World, Olympic or Commonwealth Games records. • In which event was the student the closest to the record? By how much? Depending on access to an athletics track or not, possible events to consider would include: • one or two running events the 100, 200, 400, 800 or 1500 metre races • a throwing event such as the shot putt or discus throw • high jump

Tasks and questions • • • • • •

•

What events are you going to compare? Who is going to be the student (or students) to try each event? Which records are you going to compare them with? World, Olympic or Commonwealth records? Record the times or distances for each event. You could average the times or distances out over three attempts or take the best of the three attempts. Use the internet to find the records for the three events you have chosen. Do your analysis of the student results versus the records. Include analysis of the following: – What is the difference in performance? In time or distance, and as a percentage. – Draw a graph or a diagram that illustrates the comparison in performances. In which event was the student the closest to the record? By how much?

140


16/6/06 11:59:58 AM


WO R KS H EE T K : CO O P ER AT I V E LO G I C

2004 Olympic Games 4 x 100 metres final Cooperative logic problems are an excellent way to encourage students to think mathematically, to problem solve, and to share their mathematical knowledge and language. And they have fun doing it. You need to photocopy each of the pieces below, preferably onto coloured card and cut them out and store each set in an envelope or bag. You need to have enough sets for each group of students. Students work in small groups of 4 to 6 to jointly solve the problem they are given. You need to explain that the aim is to solve the problem by working together cooperatively. Each student is to read out their clue to the group and they then discuss what that means and use the other cards to find a solution that satisfies everyone’s clues.

Instructions for students • • • •

Empty the contents of your envelope onto the table. Place the Question card and the names of the countries face-up on the table. Share out the clue cards so that everyone has at least one clue card. Each person now takes it in turn to read out their clue to the rest of the group and group members work together to find an answer that they all agree with.

Answers The answers to the cooperative logic question are: 1. Great Britain 2. USA 3. Nigeria 5. Poland 6. Australia 7. Trinidad and Tobago

4. Japan 8. Brazil

141


16/6/06 11:59:58 AM


✂

Question: In what order did the teams finish in the final of the Men’s 4 x 100 m athletics relay at the 2004 Olympic Games?

✂

Brazil

Nigeria

Trinidad & Tobago

Japan

USA

Great Britain

Poland

Australia

142


16/6/06 11:59:58 AM


✂

The last four teams were Australia, Brazil, Poland and Trinidad & Tobago, but not necessarily in that order

Japan had a time of 38.49 seconds but did not win a medal

Brazil had a time of 38.67 seconds

The USA was just one of a second behind Great Britain

1 100 th

Australia with a time of 38.56 seconds finished 4 places behind the USA

Great Britain had a time of 38.07 seconds

Trinidad & Tobago was 0.04 seconds behind Australia

Nigeria had a time of 38.23 seconds

143


16/6/06 11:59:58 AM

NOTES


16/6/06 11:59:59 AM

NOTES


16/6/06 11:59:59 AM

NOTES


16/6/06 11:59:59 AM

NOTES


16/6/06 11:59:59 AM

NOTES


16/6/06 11:59:59 AM

NOTES


16/6/06 12:00:00 PM

NOTES


16/6/06 12:00:00 PM

NOTES


16/6/06 12:00:00 PM

NOTES


16/6/06 12:00:00 PM

acr001_fn02

11/5/06

11:33 AM

Page 1

MathsWorks for Teachers Series editor David Leigh-Lancaster

Foundation Numeracy in Context David Tout and Gary Motteram Foundation Numeracy in Context describes an approach to teaching mathematics based on applied and contextual learning principles. This means that the teaching and learning of mathematics proceeds from a contextual, task-based and investigative point of view—where the mathematics involved is developed from a modelled situation or practical task. Practical investigations and projects are principle vehicles for student learning in such an approach. This text is written for teachers working with students who have become disengaged from learning mathematics during the middle to latter years of secondary schooling, and will likely have had limited success with mathematics. The approach used will be helpful for teachers of students who need a practical rather than formal mathematical background for their everyday life skills and further education, training or career aspirations. The text illustrates how this approach works through some sample contexts such as cars and driving, sport, cooking and catering, and draws together mathematics from the areas of number, measurement, space, data and statistics, and algebra.

Series overview

MathsWorks for Teachers has been developed to provide a coherent and contemporary framework for conceptualising and implementing aspects of middle and senior mathematics curricula. Titles in the series are: Functional Equations David Leigh-Lancaster Contemporary Calculus Michael Evans Matrices Pam Norton Foundation Numeracy in Context David Tout & Gary Motteram Data Analysis Applications Kay Lipson Complex Numbers and Vectors Les Evans

Foundation Numeracy in Context

Issues in Teaching Numeracy in Primary Schools

School Numeracy Practices

Numeracy for Teaching

Roman Portraits in Context (Image & Context)

Truth in Context

Truth in Context - Introduction

Truth in Context Preface

Calvin in Context

Avicenna’s Metaphysics in Context

Bioethics in Social Context

Transitions in Context

Writing (English in Context)

James Joyce in Context

Foundation

Foundation

Foundation

Foundation

Foundation

Foundation

Foundation

Context

Calvin in Context

Restorative Justice in Context

Metaphor in Context

Roman Portraits in Context

Responsibility in Context: Perspectives

The Mind in Context

Locke (Philosophers in Context)

Operations Management in Context

Foundation Numeracy in Context

Issues in Teaching Numeracy in Primary Schools

School Numeracy Practices

Numeracy for Teaching

Roman Portraits in Context (Image & Context)

Truth in Context

Truth in Context - Introduction

Truth in Context Preface

Calvin in Context

Avicenna’s Metaphysics in Context

Bioethics in Social Context

Transitions in Context

Writing (English in Context)

James Joyce in Context

Foundation

Foundation

Foundation

Foundation

Foundation

Foundation

Foundation

Context

Calvin in Context

Restorative Justice in Context

Metaphor in Context

Roman Portraits in Context

Responsibility in Context: Perspectives

The Mind in Context

Locke (Philosophers in Context)

Operations Management in Context

Recommend Documents