How much of the mathematics learnt in school is of use in the realworld? How can teachers teach mathematics so that students can makeuse of it? 10 Boaler has been finding out.

MAKING SCHOOL MATHEMATICS'REAL'

How do you teach mathematics which is of real useto students? In what sense can the mathematicsstudents learn at school be mathematics for the realworld? These questions prompted the research Iconducted in June 1991. My objective was to learnmore about the factors which influence students'ability to transfer their understanding of mathe-matics to the 'real world'. In my research Icompared the effectiveness of two different learn-ing environments: school A and school B.

School AIn this school the department works with thephilosophy that all tasks should integrate mathema-tical content and process.

The school was using the ATM SEG syllabus.Teachers do not use any published scheme ortextbook; instead students work through a selectionof open ended problems which enable them, withthe support of their teachers, to progress in differentdirections and at different rates.

Mathematical content is not taught to studentsin isolation, students work on extended problemsand encounter situations where mathematicalcontent needs to be applied.

School BIn this school the department focuses upon contentand process independently.

During lessons, students work through aselection of SMP 11-16 booklets; these are chosento suit individual students' needs. Investigations aregiven out for homework. Thus mathematicalcontent is encountered during lessons and mathe-matical processes during homework.

The two schools were chosen because of theirdifferent approaches to the teaching of process andcontent and because of their similarities in many

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other important respects. The schools are bothmixed comprehensives with an intake which issimilar in terms of socio-economic background.Both schools would be classified as having 'good',coherent mathematics departments with teacherswho are committed and have innovative ideas. Bothschools are convinced of the importance of learningmathematical processes and working investigatively.Both the classes I focused on were mixed ability.

The researchThe research took as its starting point theassumption that the way in which students learnmathematics influences the extent to which studentsare able to transfer the mathematics they learn to the'real world'. For the purposes of the research it wasalso assumed that if students could transfer theirmathematical understanding to tasks set in differentcontexts they would be more likely to be able totransfer their understanding to problems set in the'real world'. The research contrasted the effective-ness of the learning environments in the two schoolsby presenting students with questions set indifferent contexts and observing whether students'ability to transfer their mathematics was related tothe way they had learned mathematics.

Six questions were given to 50 Year 8 studentsfrom each of the two schools. The questions differedin the content they assessed and their use of contextsin assessing them.

The questions were adapted from questions inmathematics books and schemes in order that theywould be similar to those which are generally usedin mathematics classrooms. The questions weregiven out over two lessons and questions assessingsimilar areas of mathematical content were sepa-rated.

The three number questions (1, 2 and 3) allrequired students to put numbers into groups whichadded up to a given number.

MT141 DECEMBER 1992

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Question 3, a contextualised number questioncalled Fashion workshop, was adapted from aGAIM [2] activity.

Question 4 was the abstract fraction question.Questions 5 and 6, contextualised fraction

questions, were both adapted from questions inthe ATM book Working investigatively.

The questions were not given to students withthe belief that they were identical in all respects.They were given because I was interested to observewhether students used the same methods inquestions which assessed similar mathematicalcontent and whether their tendency to transfertheir methods was related to the way they hadlearned mathematics.

ResultsThere were two main findings of the research.

Firstly, the overall results for the two schoolswere not significantly different. To obtain overallresults I found the mean score for each question forall of the students in each of the schools.

Secondly, in school A, both the performance ofstudents and the processes they used were more or

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Question 2, a contextualised number questioncalled Cutting Wood, was adapted from a Mathe-matics focus worksheet [1].

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How did you decide who was best?

less independent of context in both the number andthe fraction questions. For students in school B,performance depended markedly on the contexts inwhich the questions were set. It is this finding whichI think is interesting and which I shall now describein more detail.

The questions were assessed against differentcriteria designed to reflect levels of difficulty andthese criteria were given a grade from 1 (the highest)to 4 (the lowest). An observation of grades enabled acomparison of the way that students had reacted todifferent contexts. For example, if a student hadattained 2, 2 and 2 for three questions, this wouldsuggest that the context had very little effect uponattainment; if a student attained 1, 2 and 3 thiswould suggest that the student's methods variedaccording to the context of the task.

The following table shows the number ofstudents in school A attaining each of the gradesin the abstract number question and the contextquestion 'fashion workshop'.

Fashion Workshop

An observation of the answers given by studentsin school A revealed that they nearly always usedintuitive methods even when faced with stereo-typical 'text book' style questions such as theabstract fraction question. Examples of thesemethods are shown below.

The performance of students' in school Abecomes particularly interesting when comparedwith students' performance in school B.

In school B, student performance was markedlydissimilar across all task contexts. There was no, orvery little, relationship between performance on anypair of questions set in different contexts. Thefollowing table shows the results for students inschool B on the abstract and fashion workshopnumber questions and illustrates a pattern betweenthe two schools which was repeated for each pair ofquestions.

Fashion Workshop

The following extract demonstrates the sort ofdisparity of performance across questions whichwas common to students in school B. The studentattained a grade 1 in the context fraction question onpenalties and a grade 3 in the context question onfertilizers (because the student simply consideredthe numerators in deciding whether 14/45 plants intub A was better than 9/30 in tub B).

Who is better at talcing penalties?

Many students in school B attained a 2 on theabstract fraction question and a 3 or 4 on thepenalties and fertilizer questions. This meant that

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their answers to the abstract question revealed arealisation that the fractions represented propor-tional amounts but in the context question studentsmerely added up the number of goals scored ornumber of plants grown. Eight students in school Battained a 1 on the abstract fraction question,successfully using school-taught algorithms tocompare the fractions. In the context questionsonly one of these students even attempted to makeuse of the algorithm with which they were familiar.The context fraction questions demanded more ofthe students by presenting three sets of results tocompare each time and this may have inhibited theunderstanding of some students. The answers givendid however show that the students could add theresults to obtain one 'fraction' but could not thenregard the fraction of plants or goals as somethingthat could be compared in the same way as anabstract x/y. The following example demonstrates acommon pattern of performance in school B whenstudents attempted to recall the fraction algorithmincorrectly, demonstrating a confusion and lack ofunderstanding of the algorithm.

It is not possible to conclude from the resultswhy context may be affecting procedure but thevariance in results from school B on both numberand fraction questions and with the insertion orvariation of context, does suggest that procedurewas, to some extent, determined by task contexts.The final example shows the work of a student inschool B whose methods and strategies in thefashion question were determined by the illustra-tion in the corner. This suggests that, not only maythe task context determine performance, but also thetask illustrations!

Analysis of resultsI have attempted in this article to illustrate brieflysome of the findings of the research study and havenot been able to report fully upon all of the reasonsfor drawing the conclusions which I am about todraw. Reading and marking the six answers given byeach of a hundred students enabled me to noticesome clear patterns between the schools. In schoolA students generally seemed to be unaffected bycontext and demonstrated a tendency to challengeand explore the mathematics of each task whateverthe context. The answers given by students inschool B suggested a strong tendency for the

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students to react to superficial cues and applymethods on the basis of these cues. They seemedto be attempting to recall the 'correct' school-learned method and used cues such as the context,illustrations and layout as prompts for the recall ofthis method.

The response of students in school B wasconsistent with previous research findings on theeffect of context. Students in school B were notunusual in their response; students in school Awere. Burton [3] and others have suggested that thecommon response to 'superficial' cues shown bystudents in school B is indicative of their learningmathematical methods without developing genuinemathematical understanding. Thus, the isolated useof SMP booklets in school B may have ensured onlythat students were well practised and competentwith the various techniques and procedures offeredin the booklets; because techniques were not used ina realistic problem or activity which students hadmade their own, any genuine understanding of themeaning of these techniques, their derivation, theirunderlying processes or their applicability in the'real world' was missing. This is not to suggest thatschool B is in any way remiss in its approach to theteaching of mathematics. Schools in Great Britainhave, for many years, been driven by therequirement to prepare students for exams at 16;exams which require the recall of content facts andmethods. The syllabus that school A used enabledthe school to reject this narrow focus and the resultsof this small-scale research have suggested that theschool's subsequent approach to the teaching ofprocess and content may also enable students totransfer their 'school mathematics' to the 'realworld'.

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ConclusionsI would suggest that activities need to help studentsto make links between the process and contentdomains. Students may deepen their awareness ofprocesses during investigations, but if this activity isseparate from the use of mathematical content inbooks and cards, then the development of genuineunderstanding is inhibited. Students will not behelped to transfer 'school mathematics' by thepresentation of contexts which are intended torepresent reality, or by being told by teachers thatlinks exist. Students need to work with problemsthat expose the links; they need to negotiate, tochallenge, to use their own experience and values, toform their own problems and solutions and, in thisway, to develop understanding. Faced with a 'real'problem in the 'real world' students have to weighup a complexity of variables; ignore some, controlothers, apply their own awareness of real worldobjects, develop their own meaning and chooseappropriate mathematical strategies. In order to dothis successfully students either need a considerableconfidence in simultaneously abstracting andgeneralising from their separate experiences ofmathematical process and content, or they need tohave encountered problems in school which are asopen, as complex and as close as possible to theirview of reality. When students really are encouragedto tackle problems and to investigate the dimensionsof a task, without searching for cues which may hintat the correctly learned procedure, then mathe-matics will begin to become more meaningful.

Einstein is often quoted as saying that:"As far as the laws of math refer to reality, they

are not certain; and asfar as they are certain, they donot refer to reality" [4, page 5]

I do not think that many students in this countrywould agree with these sentiments. Mathematics iswidely thought of as a subject which is certain andobjective yet possibly the most important schoolsubject with respect to real world demand.Unfortunately, when students are faced withproblems in the real world they find that thecertainty which they have learned is not applic-able. These sorts of discoveries often lead adults tobelieve that they are mathematical failures and thatmathematics is extremely difficult. I do not thinkthat these perceptions are caused by a misunder-standing of what is learned in the mathematicsclassroom but by students learning techniqueswithout really knowing what they mean, whenthey can be used, how they may be derived andwhere they fit into a more global mathematicalpicture.

I'm sure that there is no simple solution to theproblem of transfer across contexts, or across the'school-real' divide but I think that the problemswill be reduced if mathematics has more individual

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meaning for students. Over the last five or ten yearsdifferent researchers across the world, withcompletely different initial concerns, have beendrawing the same conclusions. They suggest thatthere is a need to acknowledge that mathematics isculturally based, that it must connect with astudent's individual construction of understandingand it must acknowledge the social environment inwhich students are forming their understanding. Ifmathematics is to appear more real to students itshould also offer a balanced perspective and Ibelieve that this requires the integration ofprocesses with the content to which they areapplied.

More research is needed into the way thatstudents can be helped to apply their mathematicsand I believe that this should aim to determine theeffectiveness of the few mathematical environmentswhich challenge the assumptions of SMP and otherwidely used schemes. The research which Iconducted suggests that when students are encour-aged to learn mathematics through the integrateddevelopment of process and content then they reallybegin to understand mathematics and becomecapable of using this mathematics in differentsituations. Further research will be needed to findout whether this is a valid assertion and, if it is, howthis learning may be more accessible to a largernumber of students.

ReferencesCentre for Innovation in Mathematics Teaching(1989) Mathematics focus, School of Exeter,University of Exeter

2 GAIM (Graded Assessment in Mathematics) Team(1988) GAIM Development Pack, Macmillan:London

3 L Burton (1980) The teaching of mathematics toyoung children using a problem solving approach,Educational studies in mathematics pp 43-58

4 M Fasheh (1982) Mathematics, culture and authorityFor the learning of mathematics 3(2) pp 2-9

Jo Boaler is a research officer at King's College, London.This research is to be published shortly in the journalEducational studies in mathematics.

Could you fill this gap?When an issue of Mathematics Teaching isbeing put together there are often awkwardlittle gaps to fill - like this one.We would welcome contributions from MTreaders to fill the gaps. A gap-filler could be amathematical problem, a poem, a cartoon, ananecdote, or a favourite quote. And there areno doubt many other possibilities. Please sendyour ideas to the ATM Office.Editors

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