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Learning to think mathematically: A reflectiveapproach for mature student‐teachersElizabeth Oldham aa Lecturer in Education , Trinity College , DublinPublished online: 18 Jul 2008.

To cite this article: Elizabeth Oldham (1998) Learning to think mathematically: A reflective approach for maturestudent‐teachers, Irish Educational Studies, 17:1, 194-207, DOI: 10.1080/0332331980170118

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194 Annual Conference, 1997

LEARNING TO THINK MATHEMATICALLY:A REFLECTIVE APPROACH FORMATURE STUDENT-TEACHERS

Elizabeth Oldham

Introduction

This paper is one contribution to a study of a course on mathematicalthinking and reflecting, given as part of a post-graduate diploma inprimary teaching in the Republic of Ireland. The theme of learning isaddressed in three ways. The students were learning aboutmathematics; they were learning about ways of learning mathematics;and teaching the course was also a learning exercise for the lecturer(the author). The course is examined with particular regard to itssuitability for mature students.

In the Republic, primary teachers usually qualify by taking aBachelor's degree in Education. Occasionally, however, "conversionprogrammes" are run, allowing graduates (in any subject) to enter theprimary teaching profession. The work described here was carried outin such a programme. It was addressed to a cohort of twenty-twostudents; some of them were recent graduates, but others werereturning to formal study after years in employment or in the home,and their levels of qualification in mathematics were very varied.Thus, a suitable course in mathematics appeared to be one dealing, notwith particular topics, but with mathematical thinking and the processof learning and doing mathematics. The theoretical background to thecourse is described in Section 1 of this paper. Section 2 offers anoverview of its aims and content, followed by an account of itsimplementation, focusing on the reflected experiences of the lecturer.The students' own reflections are analysed in Section 3. Conclusionsare drawn in the final section.

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Theoretical Background

The rationale for the course calls on several areas of research. Thechief ones are the nature of mathematics itself — in particular itsrelationship to teaching — and theories about learning the subject.

The nature of mathematics can be considered in terms Of Ernest'sthreefold categorisation of views of mathematics as platonist ("a bodyof knowledge, discovered rather than created"), instrumentalist ("a bagof tools") or problem-solving ("a continuously expanding field ofhuman creation") (Ernest 1985; Thompson 1992; Dossey 1992).Relevance to mathematics education can be found in claims thatteachers' views of their subject influence their teaching (Thompson1992; Kelly 1992). In particular, teachers who espouse a problem-solving view of mathematics are likely to emphasise mathematicalthinking and related activities. It should be noted that "problemsolving" can have different shades of meaning (Schoenfeld 1992): forinstance, it can be seen as a process of finding the correct solution to aclearly defined problem, or as a broader and more open-ended activityoften characterised in the English literature by the word"investigation." Similar ambiguities surround "mathematicalthinking." However, the latter may be taken to involve a cluster ofskills such as exploring, problem posing, hypothesising, abstracting,generalising, justifying and — eventually — proving. Increasedinterest in these areas in recent years is reflected in the prominence ofdiscussion of problem solving, especially in the United States in the1980s (for instance: Silver 1985; McLeod & Adams 1989), and by theinstitutionalisation of investigations in the English and Welsh NationalCurriculum (Department of Education and Science and the WelshOffice 1991).

A second strand of research considers how people learnmathematics. In recent years, "constructivism is certainly thedominant theory" (Lerman 1994, p.41). Particular attention has beenpaid to so-called radical constructivism (von Glasersfeld 1994;Ernest 1994); this leads to a paradigm in which learners build uptheir understanding and "ownership" of mathematics in their ownidiosyncratic ways, making mistakes but using them positively tocorrect inappropriately grasped concepts, and negotiating with peersto establish agreed meanings. A constructivist view of learning sits

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comfortably alongside a belief that mathematics is about solvingproblems. However, people with platonist or instrumentalist beliefsmay nonetheless espouse constructivist approaches to learning, seeingthem as appropriate ways for students to acquire relevant knowledgeand/or skills. A constructivist approach also seems particularlyappropriate for mature students. Research in this area is limited, butstudies in the field of adult learning suggest that females in particularbenefit from discussion, suitable activity methods, and reflection, all ofwhich are associated with constructivist practice; moreover, it appearsthat once adults have found their confidence, they can learn veryrapidly, notably, though not only, in classrooms displayingconstructivist principles (FitzSimons, Jungworth, Maass &Schloeglmann 1996). While a graduate class is not typical of groupsstudied under the heading "adult learning," the fact that some studentswere returning to college after many years suggests that these findingsmay have considerable relevance.

Another relevant source is Schön's work on the reflectivepractitioner (Schön 1983). The main "practitioner" was the lecturer,the course offered her a chance to carry out action research anddevelop her teaching. In addition, she could model the role of areflective practitioner for the students and help them enhance theirreflective skills.

Altogether, therefore, arguments for a course on mathematicalthinking can be made on three grounds: consonance with views onthe nature of mathematics, possible effectiveness as a way of learningmathematics, and importance in allowing students to reflect. In anIrish context the work is timely. Relevant research has beenemerging (Kelly 1992; Kelly & Oldham 1992; McNamara 1994;McNally 1995), and the climate is appropriate for furtherdevelopment. In particular, the forthcoming primary mathematicscurriculum is emphasising both problem-solving and constructivism(National Council for Curriculum and Assessment 1997); so teachereducation courses need to respond appropriately.

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Course Design and Implementation: the Lecturer's Story

An outline of the course is displayed in the Appendix; fuller detailsare given in a companion paper (Oldham 1997). Some points maymerit elucidation. It was felt that the students might not readilyappreciate the relevance of "mathematical thinking." Hence, theaims were formulated so as to emphasise students' development asprimary teachers, and the content section on "Changes in the primarymathematics curriculum" started from their likely perceived needs (tobe able to teach "new" aspects of the curriculum). This section wasdesigned to be taught relatively conventionally, but availing ofconcrete materials and discussion. Hopefully, the ground would thusbe prepared for the less familiar investigational work involved in"Learning, doing and thinking about mathematics." For the latter,students would be given chances to address various mathematicalpuzzles and problems — drawn chiefly from a collection used in anearlier Irish project (McNamara 1994) — covering different topicsand catering for differing levels of expertise. Emphasis would be puton group work; on the skills of exploring, problem posing,hypothesising, abstracting, generalising, justifying and proving, aslisted above; and on recording the various processes (both abortiveand successful) as well as the products of the investigations. Asregards assessment, an important element would be writing aboutmathematics, so the students would be asked to compile portfolioscontaining records of their investigations and their reflections duringthe course (see Appendix).

Moving from intention to implementation, it is appropriate forthe author, acting as a reflective practitioner, to highlight criticalincidents that seemed to shape the development of the course. Whilethe selection recorded here owes something to hindsight, they aredescribed on the basis of observations and reflections made at thetime. Again, a fuller account is given in the companion paper(Oldham 1997).

The first two critical incidents concerned setting the tone for theentire unit. Particular care was taken in formulating the initiallecture, ample time being given for discussion and negotiation ofmeaning (cf. Moreira 1995). This may have been important inestablishing a climate in which students could reflect and then make

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decisions for themselves. The second critical incident occurred in thefirst lecture on probability. The students were asked to list words thatreferred to "likelihood;" the words were written on sheets of paper,and then were ordered from least to most likely (Nugent & Waugh1991). The students participated eagerly, producing a long list ofwords and working together to arrange the sheets of paper on thelecture room floor. A collaborative, appropriately cróss-curricularand slightly zany tone was set, in which mathematics appeared as anentertaining activity involving more than "doing sums."

The one-week interval between lectures allowed for a good dealof forgetting. It was therefore decided to end the section onprobability with the consolidation exercise that formed portfolio item4(a) (see Appendix). However, almost all the students — intenselypreoccupied at that time with preparations for their impendingteaching practice — forgot that they had portfolio items to completeon the day in question, and, uniquely, attendance was very poor. Thisepisode, with its aftermath, provides the third critical incident. Thelecturer chose to react lightheartedly, using the incident to illustratethat learners under pressure often fail to absorb information, andasking for the work to be completed after the end of teaching practiceif necessary. A different approach could have destroyed the relaxedatmosphere that had been fostered in the lectures, and hence couldhave militated against openness to learning in the following crucialsection of the course.

That section, dealing explicitly with mathematical thinking,began when the students returned from their teaching practice. It wasduring the vital early stage of introducing the class to aninvestigational approach that the fourth critical incident occurred.One of the students had spent time teaching in England, where shehad encountered and used the approach; and at this stage sherevealed her expertise to the lecturer and her classmates, giving anaccount of her experience. Importantly, she offered support for thewhole idea of investigative learning, as well as evidence that theapproach was feasible in the world outside college lecture rooms.The students then worked, generally in groups, on variousinvestigations of their choice. A final critical incident showed theextent to which they accepted and became attuned to the approach.The lecturer had introduced the class to a particularly famous puzzle;

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she decided to follow up with a "lecture" on the mathematicsinvolved, giving the students a short break from investigations.However, the change of tactics was not made sufficiently clear.When she circulated a handout intended only for brief use and futurereference, the students promptly formed groups and launched intoinvestigative work, effectively ignoring her well-meaning attempts toprovide them with information.

Perhaps the course tailed off somewhat towards the end, asanother session of teaching practice approached. Overall, however, ithad provided the lecturer with the learning experience she required,and also with much enjoyment. Whether it had done the same for thestudents is examined below.

The Outcomes: in the Words of the Students

The first two aspects of learning cited in the Introduction can now beconsidered. Data are drawn from the students' portfolios (seeAppendix); the students themselves are referred to by pseudonyms ornicknames, in general ones of their own choice. As indicated earlier,there were twenty-two students with varied mathematicalbackgrounds. All were female; the mean age of the group wasaround thirty, with a standard deviation of eight years.

The students' experiences of learning about mathematics weredescribed in the companion paper (Oldham 1997), and findings aresummarised here. The paper focused on the change in students'reported views of the subject — in terms of the categories describedin Section 1 — over the period of the course. There was a rise insupport for the problem-solving view and a corresponding fall for theinstrumental one. Perhaps some students gave the types of answersthey felt were required; but the refreshing enthusiasm of many ofthe responses suggests that, despite the short timescale, the coursehad made a genuine impact The mood is captured by two comments,respectively from Hannah (one of the more mature students) andMim (one of the youngest): "It was exciting to find that mathsweren't just 'sums'! ... It was absolutely thrilling to discover thepattern [in one of the investigations]," and "it has given me a newway of looking at Maths."

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The present paper deals mainly with learning about variousways of learning mathematics. It examines three aspects inparticular use of concrete materials and activity methods; workingin groups; and verbalising mathematics (both orally and in writingup results). These aspects were chosen with a view to their probablerelevance for adult learners, as outlined in Section 1, but also becausethey emerged spontaneously from the data.

The first aspect is the use of concrete materials and activitymethods. This was addressed explicitly in one of the questionnaires(item 4(b)), and several students made comments — generallypositive — elsewhere. A typical remark is that of Dulcie: "I did findthese helpful, it made the new concepts more real to me and a loteasier." Even Sara, who noted that "I usually prefer to reasonsomething out in my head," found concrete materials "somewhathelpful." Sinéad commented that "[if] these aren't available ... (atleast initially)... I let the problem go over my head," adding: "Eventhe use of paper and pen can help me to visualise the problem". Herphrase "at least initially" is noteworthy; the views of students whoappreciated the approach, but felt the need to complement or move onfrom it, are of special interest because of the abstraction andgeneralisation involved in mathematical thinking. Anne, amathematics graduate, said that "concrete materials and activitymethods are extremely helpful and probably necessary," but "I needto be able to work it out in my head in order to be satisfied that Iunderstand what I am doing." Sophie, pragmatically, "would like tohave gone on to get and apply [fonnulae] so that I wouldn't have toalways work out all the possibilities." Laura's comment is a classicin the field, as it suggests a real readiness for abstraction: "I find atthis point that I am eager to learn some true formulae."

The second aspect, working in groups, also appeared popular.Eighteen of the students made positive comments in the relevant partsof their final reflections, while only six made negative ones. Typicalpositive views are encapsulated in Máire's remark: "I personallyfound it very helpful to work in a group and enjoyed sharing anddiscussing possible methods. It is nice to start an investigation with afew angles [from] which to approach it." For some, group work wasa confidence booster (see Oldham 1997); Sarah remarked that "eventhough I often felt less able[,] adequate and intelligent I saw that I too

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had something to offer and thereby gradually gained in confidence."The mixed-ability issue was addressed differently by Boris: "I find ithelpful where the partner is about at the same level as [1 am], not ifthey're way behind and certainly not if they're way ahead. Idealpairing results in 'peer scaffolding' (Vygotsky)." Interestingly, the"expert," Niamh, felt under pressure in a group situation because hercolleagues had high expectations of her. Other negative comments inthe final questionnaires referred to difficulties when the thoughts ofgroup members were going in different directions.

Group work tends to involve discussion and thus, moregenerally, verbalising: the third aspect to be considered. Discussionwas addressed in portfolio item 4(b), and received a predominantlygood press. The remark of the student nicknamed "Summer"provides an overview: "lively discussions usually occurred during theclass and while initially adding to some people's confusion ... moreoften meant that their worries were sorted out and other people'sthoughts reinforced." However, other aspects of verbalising were notalways viewed so positively. Particularly as regards recording workin progress, comments were far more mixed; in the final reflections,eighteen students gave positive and fourteen negative remarks.Sadhbh's views were typical; she "found it helpful to record thematerial especially when I wanted to go back over something later,"but "at the time ... found it difficult to remember to write downeverything." Some had no reservations: Niamh, from her experiencein the classroom, described recording as "crucial;" and Sarahreported progress, as with group work: "I found it very helpful andnot at all annoying.... I felt that I had actually achieved something."However, Sophie — again pragmatic — perhaps best encapsulatedthe feelings of many of her classmates when she said that "It'shelpful to have a recording to see the steps that you took but I foundthe level of recording expected of us to be tedious."

Altogether, the quotations indicate that the students had indeedlearned about ways of learning mathematics, and had reflected ontheir learning. An appropriate summary is offered by Maire, whosaid that she felt "a lot more confident" that she could tackle aninvestigation, "especially if I can work with a group, where it maytake just one comment from someone else to spark you off."

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Conclusion

The main focus of this paper has been on the learning that can bebrought about when students — especially mature students —undertake a constructivist-oriented course in mathematical thinkingand reflecting. Research was quoted in Section 1, suggesting thatadult females in particular would benefit from discussion, suitableactivity methods, and reflection, and that they would learn fast oncetheir confidence increased, especially in a constructivist setting. Thestudents' reactions, recorded in Section 3, show the extent to whichthis was true for the given group. The course appeared to play to thestrengths of mature students in other ways also. It allowed them touse their life experience; for instance, Niamh's background inteaching has already been mentioned, and Jane could see differentlearning styles exemplified in her own young children. Perhapsmature students are also particularly well able to pinpoint reasons fortheir likes and dislikes; several illustrations involving the formerhave been given earlier, and an instance of the latter is offered byHilary's reflection on learning probability, which she "did not findtoo interesting ... I suppose.... because I don't really care if a coinlands head or tail first." However, the lecturer learned enoughoverall to be hopeful that such a course could be developed so as tohelp all students, including younger ones, to reflect. The last wordin this respect can be given to Laura: "I feel that the reflection on mymathematical experience has been very valuable. I have been able toidentify my strengths and weaknesses.... I would not say that myunderstanding of maths has changed greatly but my understanding ofmyself and maths has !"

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REFERENCES

Dossey, J. A. (1992)The Nature of Mathematics: its Role and Influence, in D. A.Grouws (ed.) Handbook of Research on Mathematics Teachingand Learning: A Project of the National Council of Teachers ofMathematics. New York: MacMillan, pp.39-48.

England and WalesDepartment of Education and Science and the Welsh Office(1991) Mathematics in the National Curriculum. London: HerMajesty's Stationery Office.

Ernest, P. (1985)The Philosophy of Mathematics and Mathematics Education, inInternational Journal for Mathematical Education in Scienceand Technology, Vol.16, No.5, pp.603-612.

Ernest, P. (ed.) (1994)Constructing Mathematical Knowledge: Epistemology andMathematical Education, Studies in Mathematics EducationSeries. London: Falmer Press.

FitzSimons, G.E., Jungworth, H., Maass, J. and Schloeglmann, W.(1996)Adults in Mathematics (Adult Numeracy), in A. Bishop, K.Clement, C. Keitel, J. Kilpatrick and C. Laborde (eds.)International Handbook of Mathematics Education, Dordrecht,Kluwer, pp.755-784.

Glasersfeld, E. von (1995)Radical Constructivism: A Way of Knowing and Learning,Studies in Mathematics Education Series. London: Falmer Press.

Kelly, L. (1992)Perceptions of Mathematics and Mathematics Teaching: a CaseStudy of the Views and Practices of Four Irish PrimaryTeachers, Master in Education thesis, University of Dublin,Trinity College.

Kelly, L. and Oldham, E. (1992)Images of Mathematics among Teachers: Perceptions, Beliefsand Attitudes of Primary Teachers and Student-Teachers in theRepublic of Ireland, paper presented at the Eighth InternationalCongress on Mathematical Education, Québec, August 1992.

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Lerman, S. (1994)Articulating Theories of Mathematics Learning, in P. Ernest(ed.) Constructing Mathematical Knowledge: Epistemology andMathematical Education, Studies in Mathematics EducationSeries. London: Falmer Press, pp.41-49.

McLeod, Douglas B. and Adams, V. M. (1989)Affect and Mathematical Problem Solving: A New Perspective.New York: Springer-Verlag.

McNally, S. (1995)When the Problem is not the Question and the Solution is notthe Answer Effective Mathematical Instruction in PrimarySchool Classrooms, Master in Education Thesis, University ofDublin, Trinity College.

McNamara, A. (1994)Investigational Mathematics in Transition Year: A Pilot Project,Master in Education Thesis, University of Dublin, TrinityCollege.

Moreira, Cândida Queiroz (1995)Bridging Two Worlds, paper presented at the 20th AnnualConference of the Association for Teacher Education in Europe,Oslo, September 1995.

National Council for Curriculum and Assessment (1997)Curriculum for Primary Schools: Mathematics — Draft.Dublin: National Council for Curriculum and Assessment.

Oldham, E. (1997)Encountering Mathematical Thinking: Reflections andReactions of a Group of Student-Teachers, paper presentedat the 21st Annual Conference of the Association for TeacherEducation in Europe, Glasgow, September 1996.

Schoenfeld, A. (1992)Learning to Think Mathematically: Problem Solving,Metacognition, and Sense Making in Mathematics, in D. A.Grouws (ed.) Handbook of Research on Mathematics Teachingand Learning: A Project of the National Council of Teachers ofMathematics. New York: MacMillan Publishing Company,pp.334-370.

Schön, D. (1983)The Reflective Practitioner. New York: Basic Books.

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Silver, E. A. (ed.) (1985)Teaching and Learning Mathematical Problem Solving:Multiple Research Perspectives. Hillsdale, New Jersey:Lawrence Erlbaum Associates.

Thompson, A. (1992)Teachers' Beliefs and Conceptions: a Synthesis of theResearch, in D. A. Grouws (ed.) Handbook of Research onMathematics Teaching and Learning: A Project of the NationalCouncil of Teachers of Mathematics. New York: Macmillan,pp.127-146.

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APPENDIX

COURSE OUTLINE

Aims

The course aimed to:

help students reflect on their mathematical experience;enhance their understanding of mathematics and theirappreciation of the variety and power inherent in the subject;give them practice in communicating (talking and writing)about mathematics;hence, prepare them to teach the subject in primary school withpurpose, insight and vision.

Course content

1. Mathematics and myself- introduction (lecture 1);- mathematical autobiography (lecture 2).

2. Changes in the primary mathematics curriculum:- probability (lectures 3-8);- a problem-solving approach (lectures 9-17 — see below).

3. Learning, doing and thinking about mathematics:

- investigations (lectures 9-17).

4. Conclusions (lecture 18).

Method of working

The early stages of the course were "taught". In the later stages,topics for investigation were selected by individuals or groups (frombooklets and handouts provided) so as to match their interest andmathematical level.

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Assessment

This was based on portfolios of work done during the year, chiefly inclass. Portfolio items (with times of collection) were as follows.

1. An initial questionnaire gathering data on the students'backgrounds and views of mathematics (first lecture).

2. (a) An attitude questionnaire;(b) an autobiographical essay(second lecture).

3. Comments on the experience of learning probability (near theend of the unit on probability).

4. (a) A consolidation exercise in probability (students didexamples and recorded whether they achieved correctanswers and understood the process);

(b) a questionnaire, based largely on the responses in item(3), but collecting feedback systematically;

(intended to be completed in the final lecture on probability;some came in later).

5. Investigation^ work, together with comments on the students'reaction to it (from five weeks into the section on mathematicalthinking; after the first occasion, students submitted their workfor their portfolios when they chose).

6. A final questionnaire, based on responses to item (5) inter alia,gathering reflections on the course as a whole, and repeatingearlier items on views of mathematics and attitudes tomathematics (completed after the final lecture).

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