3For the Learning of Mathematics 28, 3 (November, 2008)FLM Publishing Association, Edmonton, Alberta, Canada

It has long been an aim of mine that school students shouldbe introduced to authentic mathematical activity such as ispractised by professional mathematicians, and those formsof exploration that contribute to the development of the sub-ject. Through this kind of activity, students get a sense ofmathematics as human invention, as certain habits of mind,that is more engaging and meaningful than learning a pro-cession of given facts, methods and question-types. Forexample, in one school well known to me, all mathematicsclassrooms display a text derived from Burton’s (2004)descriptions of professional mathematical behaviour:

Mathematicians …

• have imaginative ideas

• ask questions

• make mistakes and use them to learn new things

• are organised and systematic

• describe, explain and discuss their work

• look for patterns and connections

• keep going when it is difficult

Together we can learn to be mathematicians.

How far does it make sense to expect that school students canact like mathematicians? Before I start this exploration, letme make it clear that my own experience as a teacher con-vinces me that nearly all students can indeed act in the waysdescribed above, and can also generate mathematical ideas,identify and describe relationships between properties, con-struct effective algorithms. It is also my experience that thosewho have the opportunity to do this become better mathe-matical learners than those drilled in more traditionalmethods. This is now so well supported by research that ithardly needs stating. But to say that mathematical learningis better if learners ‘act like mathematicians’ is not to saythat it is possible to do this in institutional settings. In thisarticle I take ‘school mathematics’ to be the institutionaliza-tion of mathematical knowledge for novices, and it is therelationship between this and the discipline of mathematicsthat I am setting out to describe.

I shall claim that school mathematics is not, and perhapsnever can be, a subset of the recognised discipline of math-ematics, because it has different warrants, authorities, formsof reasoning, core activities, purposes and unifying con-cepts, and necessarily truncates mathematical activity inways that are different from those of the discipline. By ‘dis-

cipline of mathematics’ I mean the activities that advancemathematical knowledge: the forms of engagement, kinds ofquestions, and standards of argument that are accepted ascontributing to the conventional canon of pure or appliedmathematics. ‘School mathematics’ means the forms ofengagement in mathematics in formal teaching contexts forthe novitiate, including some undergraduates, or for thosewho do not even see themselves as novices but have mathe-matics thrust upon them.

In its worst form, school mathematics can be a form ofcognitive bullying that neither develops students’ naturalways of thinking in advantageous directions, nor leads obvi-ously towards competence in pure or applied mathematicsas practised by adult experts. The relationship of schoolmathematics to adult competence is similar to the relation-ship between doing military drill and military leadership;between being made to eat all your spinach and becoming achef; between being forced to practise scales and becominga pianist. There are some connections, but they are about hav-ing a focus on a narrow subset of semi-fluent expertise innegative social and emotional contexts, without full purpose,context and meaning. That some people become effectivemilitary leaders, beautiful pianists or inspiring cooks is inter-esting, but what is more interesting is the fact that mostpeople who go through these early experiences do notachieve these levels; instead, they merely follow orders, orhate green vegetables, or give up practising their instruments.

The image around which I hang this paper is the cafeteriaat the heart of the mathematics faculty at Cambridge. It is alarge room with coffee at one end, small tables covered withpapers and laptops, each surrounded by four chairs, students(mainly male), some undergraduate, some graduate, pureand applied, some alone, some in casual groups, some inself-organised study groups with their own internal disci-plines and plans, taking time, talking, arguing. There aremany furrowed brows, people leaning forward with armsand hands working away to express some thought; bodies,words and minds hauling together to communicate how theysee relationships and properties, and offering each other han-dles to work with abstract objects and multi-dimensionalextensions. No one is doing exercises or practising tech-niques; no one is interrupted by instructions or bells tellingthem to change to the next task; no one is taking their workto teachers to be marked. Every now and then there arewhoops of excitement or groans of frustrated realisation.

I recognise that behind such an apparently idyllic scenethere are a variety of pressures and frustrations that arisefrom institutional demands and the lived experience of

SCHOOL MATHEMATICS AS A SPECIALKIND OF MATHEMATICS

ANNE WATSON [1]

Fig

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mathematicians and mathematics students. One might alsosay that Cambridge is not typical, or question the dominanceof men and ask who is excluded, or simply laugh at howanyone could want to replicate such behaviour. Neverthe-less, this picture is of people doing mathematics together,in ways that fit the description ‘acting like mathematicians’.It is also possible to replicate much of this scene in schoolclassrooms. But even in these environments, where teach-ers may believe that their students are ‘working likemathematicians’, there are major significant differencesbetween school and ‘the discipline’.

Mathematical enquiryThe students in the cafeteria are in various stages of transi-tion between school maths and academic maths. With a fewexceptions they have rarely worked like this at school andonly a few may have been told explicitly that this is how theycould be working now. Mathematics as a discipline includesthe social and cultural characteristics that have contributed toits genesis. Most mathematical advancement does not arisein an isolated, independent way but is a product of its time,within the current paradigms, co-emergent with current tech-nological and economic needs and tools, co-emergent with thezeitgeist. Locally and globally, it is a product of social inter-action, even if that interaction does not take place until thereis a result (Burton, 2004; Davis & Hersh, 1981).

In school, things are generally different. I am not assumingany homogeneity about school practices – it is always possi-ble to read such assumptions and argue to exclude yourparticular patch of the world, or particular teachers, or a par-ticular curriculum. Instead, I focus on the mathematicalpractices in which, I presume, the students described aboveare engaged but which cannot be observed in the short-term,nor do they fit neatly into socio-cultural analyses of the learn-ing environment. For me, the starting point is what it meansto do mathematics and to be mathematically engaged.

In the discipline of mathematics, mathematics is the modeof intellectual enquiry, and effective methods of enquirybecome part of the discipline – so much so that mathemat-ics theses do not have chapters explaining methodology andmethods. This is not just about the availability of coffee,social groupings, the published discourse, or choicesbetween Maple and Mathematica, although these are cru-cial parts of the picture. ‘Doing mathematics’ ispredominantly about empirical exploration, logical deduc-tion, seeking variance and invariance, selecting or devisingrepresentations, exemplification, observing extreme cases,conjecturing, seeking relationships, verification, reification,formalisation, locating isomorphisms, reflecting on answersas raw material for further conjecture, comparing argumen-tations for accuracy, validity, insight, efficiency and power.It is also about reworking to find errors in technical accu-racy, and errors in argument, and looking actively forcounterexamples and refutations. Mathematics is about cre-ating methods of problem-presentation and solution forparticular purposes, tinkering between physical situationsand their models, and it also involves proving theorems. Ofcourse all these can be done in school, but it is the way inwhich these are coordinated and aligned that makes theoverall activity different.

Different kinds of mathematicsThe practices of mathematics just described are very farindeed from the concerns of psychologists who want to con-struct efficient methods of instruction, seeking for the fastestand most productive ways to teach students how to findanswers to broadly isomorphic problems. Research anddevelopment in psychology and neuroscience do not evenbegin to offer ways to induct students into the practices listedabove. My perspective on the discipline of mathematics isalso different from the concerns of the socio-cultural take onclassrooms and learners, which tells convincing and robuststories about the existence of differences in practice, and theprocess of development of identity in different cultural set-tings, but which cannot put detailed flesh on these stories interms of the development of specific mathematical ideas.

Of course we would expect to see different kinds of learn-ing and different classroom practices where there are differentviews of mathematics and different curriculum goals, and inparticular we would expect to see a political dimension tothis as countries connect the outcomes of education with theirfuture workforce. As Radford (2003) has said,

While the humanist view of mathematics emphasizesthe role this discipline plays in the development of log-ical thinking, abstraction, rigor and other highly prizedfaculties that have been the clear marks of men ofsophisticated spirit since the Enlightenment, the social-ist trend stressed the applicability of mathematics. Itsimportance was seen in terms of the utilitarian ability ofmathematics to master nature in the interests ofmankind. (p. 552)

If the curriculum takes a ‘humanist’ approach, rigorousargument and proof are taken to be important aspects of thesubject, yet many teachers treat ‘proof’ as a topic withinmathematics rather than as the way truth is examined andwarranted throughout the subject. They also take argumentas a specific focus for particular tasks or lessons. If this iscombined with fallibilist-inspired teaching styles such asinvestigative work, ‘What if…?’ questions, and algebraintroduced as an expression of generality, then in manyclassrooms two standards of argument are being offered.One is empirical, arising from specific cases, tables of val-ues, inductive construction of formulae and testing specialcases, while the other requires deductive reasoning and awillingness to engage in formal logical argument. The shiftlearners have to make between inductive to deductive argu-ment is hard and demands some ‘giving up’ of forms ofreasoning which serve students well elsewhere. For exam-ple, it is well documented that students who can produceexamples easily, in an inductive mood, find the productionand understanding of counter-examples, with their deductiverole, much harder (Zaslavsky & Ron, 1998). To use mathe-matical argument as the natural and normal method ofenquiry in mathematics it needs to be fully embedded inday-to-day lessons as part of classroom discourse. In real-ity, a curriculum written in accordance with what Radfordcalls ‘humanist’ principles is likely to be presented as ataught curriculum in which topics are presented in a devel-opmental order and few students will see the role for logicalthinking embedded in the development of mathematical

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understanding. If their own logical thinking does play a partin mathematics lessons, as it does with some teachers, thesehumanist curriculum aims might be realized in part.

Learners exposed to utilitarian mathematics have a dif-ferent experience compared to those taught with a moreabstract emphasis, but solving realistic and everyday prob-lems need not lead them to understand the role ofmathematics beyond providing ad hoc methods for realproblem-solving, or as a service subject that holds tools formoving forward in other domains.

Mathematical shifts of understanding in schoolmathematicsBut the choice between different possible ‘mathematics’ is,to my mind and in my experience, not the core problemabout mismatch between school mathematics and the disci-pline of mathematics, as I intend to show. I am going todescribe several shifts of understanding that have to be madefor learners to be successful within and beyond school math-ematics, whatever the dominant view of the discipline:

Additive to multiplicative reasoning

A shift from seeing additively to seeing multiplicatively isexpected to take place during late primary or early sec-ondary school. Not everyone makes this shift successfully,and multiplication seen as ‘repeated addition’ lingers as adominant image for many students. This is unhelpful forlearners who need to work with ratio, to express algebraicrelationships, to understand polynomials, to recognise anduse transformations and similarity, and in many other math-ematical and other contexts. Eventually another shift has tobe made to exponential reasoning.

Probabilistic reasoning

The concept of probability, understood mathematically,offers a different warrant for truth than is associated witheither deductive logic or induction from empirical evidence.To understand probability as a tool, or to see the world prob-abilistically, requires abstraction and imagination wellbeyond observable phenomena. Moreover, one cannotmerely follow algorithms to get answers except in very sim-ple contexts; often learners have to decide for themselveswhether events are independent, exclusive or not. The shifthere is from being told everything about a situation to hav-ing to identify characteristics and properties for oneself,before conceptually based action.

Integration

In the UK, integration used to be the first context in theschool curriculum where learners could not merely applymethods and be sure they would get some kind of answer.However students now are told not only what method to use,but also what substitutions to make if substitution is themethod examiners wish to see. The shift from being toldwhat methods and tools to use, to developing sensible selec-tion criteria for what is possible, has to be made to becomea mathematician.

Geometrical reasoning

Questions involving application of theorems can be avoided

in UK national tests at 16+ and students can still be awardedthe highest grades. Theorems and proofs of any kind, letalone geometrical contexts, do not play a part in higherschool examinations. In countries where reasoning fromaxioms still has a place in the school curriculum, these maybe taught mechanically, as a kind of memory or question-spotting activity rather than as a demonstration of deductivereasoning to explore phenomena and to establish a particularkind of truth. A shift from knowing what to look for, toselecting what to look at, then to deciding what to use andconstructing multi-stage arguments for oneself, has to bemade to become a mathematician.

Getting answers to gaining insight

A problem (in Polya’s sense) can be solved and nothing newbe learnt, even about problem-solving. A mathematician willusually have a purpose in mind when solving a problem, sothat the outcome is used to reflect on the context in whichthe problem arose, to decide if something unexpected hasarisen, to raise further questions, or in some other way toenrich or extend knowledge. The answer, if there is one, isnot the end of the process. A shift from getting answers togaining insight or constructing arguments has to be made. Innon-mathematical contexts, once the problem is solved thereis no motive for extending the work hypothetically.

Modelling

Mathematical modelling of realistic or artificial situationsis a feature of many mathematics curricula. While generali-sation might take place in order to create the model, andthis might be explored further to look for extreme cases orfurther variation, the modelling process itself does notrequire more abstraction or structural understanding than thesituation being modelled.

The shifts of perception, attention and engagement justdescribed do not take place without the learner being in anenvironment of tasks, language, representation, deliberatelydesigned to support those shifts – that is, a schooling environ-ment whose purpose is to ensure learners make these shifts.

The discipline of school mathematicsIt is not mere coincidence that these are all inherently ‘hard-to-teach’ aspects of mathematics. They all offerepistemological obstacles that require shifts of reasoning,or new ways to act with imagery, or encapsulation of previ-ous experience to be overcome.

Importantly, it is becoming more and more the case that,in an effort to ensure that more students can gain schoolmathematics qualifications, the difficult shifts that wouldhave to be made for school mathematics to be a subset of thediscipline of mathematics are being edited out of mathe-matics as a school subject, rather than edited in as thediscipline itself becomes more complex, more post-modernand less certain. I am talking here of all kinds of classrooms:reform and traditional; classrooms in which students areexpected to behave like little mathematicians and thosewhere they are expected to behave like acquiescent cogni-tive machines; classrooms in the developing world and thosein high-achieving countries. I am not arguing for or againstparticular kinds of school curriculum; I am saying that the

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task of schooling the mathematical mind to make the shiftsthat have to be made to become a mathematician is not, byand large, undertaken in schools.

In all the curriculum contexts I described above, and inmany others, there are features that are peculiar to schoolmathematics, and the way it is generally taught, which arenot part of the discipline as practised by adult experts. Forexample:

• there is a strong focus on answers and generalisa-tions rather than structural insight and abstraction;

• there is avoidance by teachers, tests, and curricula,of the need for uncertain choices;

• curricula seem to cling to topics and approachesthat can be represented experientially, diagrammat-ically or in concrete ways, rather than in abstractand imaginary ways;

• in school, inductive, empirical, and ad hoc reason-ing are privileged over deductive or probabilisticreasoning.

These characteristics, peculiar to school mathematics, canhinder the progress of students at university on pure andapplied courses, alongside inappropriate work habits, chal-lenges to identity, and unrealistic expectations of the subject(sometimes promulgated by popularisers). Mathematics asa discipline, by contrast to school mathematics, is concernedwith thought, structure, alternatives, abstract ideas, deduc-tive reasoning and an internal sense of validity and authority.It is also concerned with uncertainties about ways forward inits own realms of enquiry. To do maths includes holdingnagging questions in the mind while carrying on with life,and not expecting answers to be found, problems to besolved, within the confines of a particular room or timescale.The concerns of school mathematics pull learners in direc-tions that differ from these. The core activity in schoolmathematics is to learn to use mathematical tools and waysof working so that these can be used to learn more tools andways of working later on.

Whichever approach is taken by the curriculum, mostschool students are taught and examined on mathematics ofa kind that is done, both in the academy and in other work-places, by machines. They are taught not as an apprenticeshipto adult mathematics users, but as bottom-up preparationfor future mathematical activity that might one day be mean-ingful, either as an intellectual or an economic activity.Furthermore, they are taught this in regimented settings,with short time-scales, by teachers who themselves havelimited experience of the mathematical practices describedabove.

Those students who continue beyond school might bemotivated by a so-far satisfied need to have right answers, orby getting a kick from the resolution of puzzles, or by dis-covering special features which intrigue them, or byanticipating the delayed gratification of getting a furtherqualification, or of finally being able to study at the cuttingedge. They might also have what Krutetskii (1976) identi-fied as common features of mathematically gifted students:propensities to see the world and organize their mental activ-ity in certain mathematical ways. The list he identified has

little in common with the contents of formal taught lessonsand assessment regimes, and much to do with the list ofpractices above, and what Cuoco and his colleagues (1996)have called mathematical ’habits of mind’. Neither a cur-riculum based on rigour, historical genesis and conceptualdevelopment, nor a curriculum based on modelling andauthentic contextual questions, can achieve the educationof the mind required to engage consistently, habitually, withthe normal intellectual practices of mathematicians.

The roles of unifying conceptsA further striking difference is in the role of unifying con-cepts in mathematics as a discipline and in schoolmathematics. In mathematics as a discipline these orientatemuch of what is studied and researched. Such theories pro-vide opportunities for links and connections within thediscipline, new languages for discussing ideas, and newquestions to be explored. In school mathematics, however,the shaping of a curriculum with unifying concepts couldbe unhelpful. A unifying concept makes sense to mathe-maticians precisely because it offers unification ofpreviously disparate ideas of which they have a range ofexperience. For school students, to be told about a unifyingconcept before experiencing several widely differing exam-ples of it reduces the concept to ‘something else to be learnt’or ‘something else to be taught’ rather than an enablingencapsulation.

Linearity, for example, crops up throughout school andcould have been included in the ‘hard to teach’ list above,because it requires a shift from looking at functions as rep-resentations of relationships to comparing properties offunctions. However, this only makes sense when a learnerhas experienced several different linear and non-linear situ-ations so that there is a need for a word for those that behavein certain ways. Older mathematicians are supposed to havehad a range of experiences to draw on when they meet, forexample, vector spaces so that they have examples whichmight be activated by hearing definitions, theorems andquestions (Watson & Mason, 2002). To use the idea of a uni-fying concept successfully in school mathematics wouldrequire vertical coordination of teaching across years byteachers who have constructed a shared understanding ofhow that concept will be approached and learnt. In the dis-cipline of mathematics, unifying concepts arise through thepublished and broadcast insights of mathematicians drawingon the processes of historical cultural genesis – they movethe field on.

Constraints on teaching that alter the disciplineIt certainly is the role of school mathematics to provide arange of experiences of various kinds so that students under-stand the usefulness of mathematics, and can do variousnecessary calculations and estimations. It would also behelpful for students to understand the purpose and value ofpractice, and of basic standards of accuracy. School couldintroduce students to ways of working on mathematics, tothe kinds of questions mathematicians ask and the subjectmatter worked on, but would have to include in that someof the practices of number systems, algebraic manipulation,

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and ways of using diagrams, which are assumed within thediscipline. At the most extreme, schools use entirely differ-ent kinds of questions, enquiry, warrants and work habitsthan those of the discipline. At best, they introduce what is tocome, while moulding it to fit the institutional constraints,rather than to fit the development of mathematical ideas.

Many teachers and projects try to make their classroomsmore and more like mathematical workshops, but the schoolcontext overlays these with purposes that are not about thedevelopment of mathematics, but are more about learningwhat to do and how to be. In the best of these classrooms,teachers are explicit about appropriate forms of statementand discussion in ways that are not made explicit in mathe-matics faculties, but even the best teachers cannot be presentat every student’s side to be explicit about the many prac-tices of professional mathematics listed above. Mathematicsas a discipline includes discussion and critique of its ownmodes of enquiry, but does not include explicit teaching ofthese modes of enquiry, or practising core techniques, asschool mathematics has to do. Mathematicians decide forthemselves what they need to practise, and how to practiseit, and how to validate their techniques.

In school the truncation of mathematical activity takesplace when enough has been learnt for now, or at the end ofa lesson, or when the curriculum requires moving on, or asyllabus has been ‘covered’, or students leave school. Inmaths the truncation of enquiry happens when a problem hasbeen solved, a proof has been accepted, a model has beenproduced, or everyone is tired of the enquiry – but all ofthese can be the starting points for new enquiry. Of coursetemporary truncation takes place because of teaching,administrative and family commitments, but the mathemati-cian is not expected simultaneously to worry about academicproblems in geography, language, science, citizenship aswell as mathematics.

Another aspect of school mathematics that differs fromthe discipline of mathematics is the nature of authority.Freudenthal (1973), Vergnaud (1997) and others haveemphasised that mathematics includes its own methods ofvalidation, and a good teacher can enable students to usethe structures of mathematics to verify their own work andideas. In the education system as a whole, however, valida-tion and authority are highly structured through textbooks,examinations, and curriculum systems rather than by mathe-maticians working as a community. Formal axiomatic proofis inaccessible for most school-age children, although manychildren of all ages can develop robust arguments withappropriate scaffolding. Even so, for most of what is taughtin school the correct application of procedures and check-ing answers using inverse procedures, or checkingreasonableness, are available as warrants for truth, and theprevailing warrants in classrooms are usually teacher’sapproval, or checking answers.

ConclusionIn conclusion, therefore, I find myself claiming that schoolmathematics is necessarily not a subset of the discipline ofmathematics, whatever the nature of mathematics beingtaught, whatever the way it is taught. Essentially, this is forthe following reasons:

Schools have to prepare students for future study and/oremployment and this requires them to be explicit about waysto develop recall, fluency, accuracy, and ways of working;these constitute the major part of the goals of school mathe-matics.

Mathematics requires shifts away from the everydaythinking of mathematics lessons to special forms of mathe-matical thinking. Someone has to do this work, whichrequires knowledgeable teachers in school and university.We have to recognise that many schoolteachers of mathe-matics do not have personal experience of what it means todo mathematics over time, exploring questions that haveintellectual purpose, not pedagogic purpose.

Limited time slots, curricular pressures, and assessmentregimes constrain or prevent the development of the kinds ofquestions and ways of working which characterise the dis-cipline.

Authority in school mathematics lies with teachers, text-books and assessment regimes, not with mathematicalargument.

It is not only ways of working and goals that are differentbetween school and disciplinary maths; it is the way thatthese shape the available forms of mathematical enquiry thatmakes school mathematics a different discipline, with itsown rules, purposes, authorities and warrants.

Note[1] The substance of this paper was presented originally at a working groupon “Disciplinary Mathematics and School Mathematics” at the Sympo-sium on the Occasion of the 100th Anniversary of ICMI (InternationalCommission on Mathematical Instruction), Rome, March 5–8, 2008.

ReferencesBurton, L. (2004) Mathematicians as enquirers: learning about learning

mathematics, Dordrecht, NL, Kluwer.Cuoco, A., Goldenberg, P. and Mark, J. (1996) ‘Habits of mind: an orga-

nizing principle for mathematics curricula’, Journal of MathematicalBehavior 15, 375–402.

Davis, P. and Hersh, R. (1981) The mathematical experience, Brighton, UK,Harvester.

Freudenthal, H. (1973) Mathematics as an educational task, Dordrecht, NL,Kluwer.

Krutetskii, V. A. (1976, translated by J. Teller; edited by J. Kilpatrick andI. Wirszup), The psychology of mathematical abilities in school children,Chicago, IL, University of Chicago Press.

Radford, L. (2003) ‘From truth to efficiency: comments on some aspectsof the development of mathematics education’, Canadian Journal of Sci-ence, Mathematics and Technology Education / Revue canadienne del’enseignement des sciences, des mathématiques et des technologies,4(4), 551–556.

Vergnaud, G. (1997) ‘The nature of mathematical concepts’, in Nunes, T.and Bryant, P. (eds.) Learning and teaching mathematics: an interna-tional perspective, London, UK, Psychology Press.

Watson, A. and Mason, J. (2002) ‘Extending example spaces as a learn-ing/teaching strategy’ in Cockburn, A. and Nardi, E. (eds.), Proceedingsof the 26th Annual Conference of the International Group for the Psy-chology of Mathematics Education, Norwich, UK, University ofNorwich, 4, 377–384.

Zaslavsky, O. and Ron, G. (1998) ‘Students’ understandings of the role ofcounter-examples’, in Olivier, A. and Newstead, K. (eds.), Proceedingsof the 22nd Conference of the International Group for the Psychologyof Mathematics Education, Stellenbosch, South Africa, University ofStellenbosch, 4, 225–233.

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I concur with Watson in referring to mathematics as an activ-ity and a way of engagement, rather than as a body ofknowledge. Her main claim is that “school mathematics isnecessarily not a subset of the discipline of mathematics”, asthere are “major significant differences”.

Focusing on these differences, a superficial reading mightprompt one to interpret her claim to mean that school math-ematics and disciplinary mathematics are disjoint sets.However, a more attentive reading will acknowledge claimssuch as “it is also possible to replicate much of this scene inschool classrooms” and “Of course all these can be done inschool” – where the phrase “all these” refers to activitiesinvolving mathematicians advancing disciplinary mathe-matics. Thus, while I accept the “not a subset” claim, I preferto consider intersecting rather than disjoint sets, and thisintersection is of my interest here.

I recognize, considering historical trends and cultural set-tings, that school mathematics and disciplinary mathematics,as well as the intersection thereof, are constantly changing.However, my central claim is that teacher education plays apivotal role in determining the nature and the size of thisintersection, in any context. That is, guiding prospectiveteachers through the experience of “working and thinkinglike a mathematician” may eventually result in instillingthese ways of working in students and thus create a largerintersection between approaches practiced in teaching andlearning school mathematics and approaches employed indeveloping disciplinary mathematics. While Watson recog-nizes teachers’ experience and motivation as one of theobstacles, I suggest that this also is a key to the solution.

In describing school mathematics, Watson mentions “twostandards of argument” and acknowledges the demandingshift from inductive to deductive argument. However, ratherthan presenting this as a dichotomy and a need for a shift,school mathematics may focus on considering inductiveargument as an introduction to a deductive one. That is,empirical exploration of special cases is conducted not todraw conclusions, but to conjecture and find out what isthere to prove. I exemplify this approach in what follows.

Example: Exploring tridians and conjecturingSimilarly to a median, we define a tridian as a segment thatconnects a vertex of a triangle to a point on the oppositeside that is marking 1/3 of the side’s length. As shown inFigure 1, BX, BY, AW, AZ, CS and CT are tridians in a tri-angle ABC.

Inspired by Brown and Walter’s (1983) “what if not”

strategy, one of the tasks I present to prospective secondaryteachers is as follows:

Tridians generate many interesting relationships at var-ious points of intersection and of various areas they cut.Compare for example the areas of triangles ABC,AZW, IJH and BIZ. With the help of the Geometer’sSketchpad, make several conjectures about tridians in atriangle. Formulate and prove at least 2 of your con-jectures.

What if not? What if not a triangle? What if not tridi-ans? What if … Explore. Present at least one variationon a theme of tridians in a triangle. Prove at least someof your observations.

There are many possible relationships. While some are quiteobvious and easy to prove from the definition of a tridian,such as the ratio of areas ABX:ABC = 1:3, others are ratherchallenging, such as the ratio of areas IJH:ABC = 1:7. [1]

What’s in a task?The tridians task exemplifies several important features of“thinking like a mathematician”. First, there is an invitationto observe relationships, to notice, to wonder and then to for-mulate and test conjectures. The software helps in exploringand verifying conjectures: preservation of relationship when“dragging”, while not a proof, is a very good indicator of arelationship to be proven. This activity is similar to the workof mathematicians in constructive mathematics: the com-puter is used not to prove theorems but to find out what isthere that may be proven (Borwein, Beiley & Girgensohn,2004). Of note, while computer feedback presents wonder-ful encouragement for testing conjectures, it is not anecessary component in this kind of activity.

Second, there are choices of engagement and of levels ofdifficulty. Though the task directs learners to one particulartriangle, different individuals will observe different proper-ties and will wonder about and justify different patterns andrelationships. Working with students at different levels ofmathematical sophistication, these choices, while limited,are extremely important – after all, mathematicians chose for

Responses to Watson

Looking for a possible intersection

RINA ZAZKIS

Figure 1: Tridians in a triangle

themselves what would be their next investigation or theirnext proof.

And third, there is an incentive in the task – that wouldhopefully develop into a habit – not to leave the problemwhen it is solved, but to continue with explorations. Oncethe initial question is answered, the task is not left to rest.Answers generate further questions that involve either vari-ation or generalization of the previous ones, allowing newobservations and conjectures to emerge.

ConclusionThe task discussed above presents an opportunity for manyactivities – such as empirical exploration, logical deduc-tion, seeking variance and invariance, conjecturing, seekingrelationships, verification or proof – that Watson attributes todisciplinary mathematics. Similar engagements do notrequire one to be working at the frontiers of the discipline;they can be presented at any level of difficulty in helpingstudents develop mathematical habits-of-mind. “Workinglike a mathematician” experiences in teacher education openthe gate for similar experiences for students and increasethe intersection between school mathematics and discipli-nary mathematics.

Note[1] When one proves this conjecture using linear algebra, vectors or affinecoordinates, the challenge is to prove it using only methods of Euclidiangeometry. I thank Guershon Harel for presenting me with this challenge.

ReferencesBorwein, J., Beiley, D. and Girgensohn, R. (2004) Experimentation in

mathematics: computational paths to discovery, Natick, MA, A. K.Peters.

Brown, S. and Walter, M. (1983) The art of problem posing, Philadelphia,PA, Franklin Institute Press.

Mason, J., Burton, J. and Stacey, K. (1982) Thinking mathematically, Read-ing, MA, Addison-Wesley.

Is all course-based mathematicsspecial?

DAVID W. HENDERSON

Wow! Watson set off contradictory reactions in me: I findmyself at the same time both strongly agreeing and stronglydisagreeing with her arguments. I do not mean agreeing withsome of her arguments and disagreeing with others; rather Iam in the position of strongly agreeing and strongly dis-agreeing with her overall argument. I recognize this as apotentially very creative place that I find myself in; so let metry to examine where I am standing.

Here is what I find myself strongly agreeing and stronglydisagreeing with: Watson describes what she means by the“discipline of mathematics” and what she means by “schoolmathematics”. She then argues that school mathematics is,for reasons she delineates, “necessarily not a subset of thediscipline of mathematics”. After internalizing my conflict,I find that I have trouble with the large (I think I would even

say ‘huge’) gap between ‘school mathematics’ and the ‘dis-cipline of mathematics’. In particular, there is ‘college/university mathematics’ that engages mathematics majorsbut also many other student who take mathematics courses.I think that Watson’s description of ‘school mathematics’ as“the forms of engagement in mathematics in formal teachingcontexts for the novitiate … or for those who do not even seethemselves as novices but have mathematics thrust uponthem” is a description that also applies (at least in USA uni-versities) to almost all undergraduate courses and graduatecourses up to the level of the graduate comprehensive exams(in the USA most of the first-year and some second-yeargraduate courses are geared to these exams) – I shall callthese pre-comprehensives courses.

For example, the students in these graduate first-yearcourses may not have “simultaneously to worry about acad-emic problems in geography, language, science, citizenshipas well as mathematics”; but they do have simultaneouslyto worry about academic problems in algebra, analysis,topology, and, maybe, logic, probability, or geometry. Inundergraduate years students normally take at least fourcourses at a time, at least one (usually two) of which is notmathematics.

Watson states in her first page: “let me make it clear thatmy own experience as a teacher convinces me that nearlyall students can indeed act in the ways described [in Burton’s‘professional mathematical behavior’]” and then describesin what ways this does not in practice generally happen inschool courses. I would make the same statement about pre-comprehensives courses. Note that in her Cambridgemathematics cafeteria there is no mention of courses – and,if they are taking courses, are these courses different in kindfrom their school courses? Do not many (most?) pre-com-prehensives courses have “authority” that “lies withteachers, textbooks and assessment regimes, not with math-ematical argument”. If not, why have comprehensiveexams? Maybe Cambridge is very different; but I suspectthat, even in Cambridge, most students who study mathe-matics are not mathematics majors and are not among thosethat Watson observed in the cafeteria.

I agree with Watson’s descriptions of the difficult shiftsin students that are necessary for them to participate in thediscipline of mathematics. Of course, some mathematics stu-dents make these shifts and become mathematicians; but, isthere a “typical” college or university where this happens incourses because of “an environment of tasks, language, rep-resentation, deliberately designed to support those shifts,that is – a schooling environment whose purpose is to ensurelearners make these shifts” or is the most common occur-rence that the student (at all pre-comprehensives levels)somehow makes this shift outside of courses? – one reasonthat I have often heard cited by mathematicians is the inspi-ration from a special teacher (school or university) – I do notrecall any mathematician saying that their inspiration camefrom a specific course.

So what are my conclusions? I look at Watson’s state-ment: “But to say that mathematical learning is better iflearners ‘act like mathematicians’ is not to say that it is pos-sible to do this in institutional settings”. However, Watsonhas described in other publications examples of school class-

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rooms that go in this direction; and I suspect that we allknow, at the school and undergraduate levels, examples ofspecial courses that accomplish this to a greater or lesserdegree. So, I am left with the question that Watson’s articledoes not resolve: Would it be possible for typical pre-com-prehensives mathematics courses to contain “anenvironment of tasks, language, representation, deliberatelydesigned to support those shifts – that is, a schooling envi-ronment whose purpose is to ensure learners make theseshifts”? I think the answer is ‘yes’ – it is possible; but I agreethat it does not, at present, happen in the typical pre-com-prehensives course. Some of the reasons that it does nothappen are the limitations delineated by Watson; but, Ibelieve, that these limitations apply at all pre-comprehen-sives levels, though less at the university than in schools.

Maintaining the mathematicalintegrity of school curricula: the challenge

GUERSHON HAREL

In essence, Anne Watson’s paper can be interpreted in termsof and organized around four fundamental questions. Thefirst three are: (1) What is mathematics? (2) Do currentschool curricula maintain the mathematical integrity of theircontent? and (3) Can, in principle, school curricula main-tain the mathematical integrity of their content? Watson’sjustifications as to why her answer to the third question isnegative raises a fourth question: (4) What are the currentbelief systems and social and institutional constraints thatmake school mathematics not mathematical, or, using Wat-son’s words, “not a subset of the discipline of mathematics”?

Regarding the first question, let me begin with thepremise that there is one and only one mathematics. In thisrespect – and despite my inference that Watson shares thispremise – the sentence “school mathematics is not a subsetof the discipline of mathematics” is logically inconsistent.This premise is critical because it entails that curricularobjectives of school mathematics must be formulated interms of the elements that define mathematics. Elsewhere, Idefine mathematics as the discipline consisting of all theways of understanding and ways of thinking that have beeninstitutionalized by communities of mathematiciansthroughout history. The terms “ways of understanding” and“ways of thinking” have technical meanings, but for our pur-pose here, they can be thought of as two categories ofknowledge, the first refers to “subject matter,” such as par-ticular definitions, theorems, and proofs, and the second to“conceptual tools,” such as heuristics and beliefs about whatconstitutes proof in mathematics (Harel, 2008). Watsonoffers many examples of mathematical ways of thinking,and she convincingly argues that these have nothing or lit-tle to do with the ways of thinking targeted and promotedby schools. There are two significant aspects to herapproach. First, collectively, her list of examples typifies,

rather than just exemplifies, mathematical practice. Second,for Watson these ways of thinking are not a priori constructsbut products of human activities – they are what mathemati-cians do to learn, engage in, and produce mathematics.

An implication of the first aspect is that mathematicalways of thinking should be identified and studied byresearchers and, accordingly, targeted as instructional objec-tives by curriculum developers and teachers. The secondaspect is even more critical: it implies that learning – as ahuman activity – necessarily involves the construction ofimperfect and even erroneous ways of understanding anddeficient, or even faulty, ways of thinking. The history ofmathematics is rich of cases that attest to this fact.

The shortcomings of current curricula and instruction thatWatson points to cannot be justified on the basis of thisnature of the learning process. Rather – and this is the thrustof Watson’s argument – they are rooted in the character andstructure of our educational systems, particularly theirshared meanings of knowledge, learning, and teaching andtheir shared perspectives on the social, cultural, behavioral,and emotional factors involved in the learning and teachingof mathematics. These meanings and perspectives dictate theresponsibility and school-related behaviors of students,teachers, school administrators, and parents. They entail thekinds of tools for measuring what students know and setsexpectations for what they ought to know. And they orientthe educational systems as to what, why, and how to carryout school-related actions.

Rather than calling for actions to study and accordinglyreform our educational systems, Watson concludes that“school mathematics is necessarily not a subset of the dis-cipline of mathematics, whatever the nature of mathematicsbeing taught, whatever the way it is taught” (emphasisadded). Research findings unequivocally support Watson’sargument that current school curricula do not maintain themathematical integrity of their content. On the other hand,her conclusion that school mathematics cannot, in princi-ple, be mathematics is unfounded and surprising. It issurprising because her paper is a guide as to the nature of thereform needed. Specifically, Watson lists four reasons forher conclusion, each of which is a reason for change ratherthan for acceptance of impossibility of change. The compe-tencies for authentic mathematical practice that Watson listsat the end of her paper as her first reason are required fromany professional mathematician. The question for researchin mathematics education is how to help students developthese competencies not through authoritative need butthrough intellectual need. The second reason about teachers’knowledge is not a justification for impossibility of changebut rather a justification that change is imperative. The thirdreason – about time constraints, curricular pressures, andassessment regimes constraints – is a call for a fundamentalreform of perspective; that, for example, teaching should beabout the quality of the ideas students uncover rather thanthe quantity of the material being covered in a lesson. Thefourth, and last, reason that “authority in school mathemat-ics lies with teachers, textbooks and assessment regimes, notwith mathematical argument,” is an acknowledgement that adash of authority is not completely harmful and perhapsunavoidable, but it is essential that teachers recognize that

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students must not be helpless without an authority at handand that students should never regard a mathematical justifi-cation of a result as valueless and unnecessary.

ReferenceHarel, G. (2008) ‘What is mathematics? A pedagogical answer to a philo-

sophical question’, in Gold, R. B. and Simons, R. (eds.), Current issuesin the philosophy of mathematics from the perspective of mathemati-cians, Washington, DC, Mathematical Association of America, pp.265–290.

School and academic mathematics

ALF COLES, NICK PEATFIELD

(Alf Coles is Head of Mathematics at Kingsfield School, UK,where Nick Peatfield spent four months on teaching practice.)

Alf: You have been a professional mathematician and arenow near the end of a one-year teacher education course.From these experiences, where would you locate the differ-ences between mathematics as an academic discipline andschool mathematics?

Nick: Although I would agree with Watson’s conclusionsabout the differences between academic mathematics andthe mathematics of many schools, at Kingsfield, I have seenmost of them contradicted. Here the main goal and empha-sis, throughout the mathematics curriculum, is on thelearners being encouraged to behave as mathematicians.Still, it is the case that at school the students aren’t pushingthe boundaries of knowledge, and so are correct when theytend to assume that the teacher “knows the answer”.

Alf: I’m interested in the issue of the teacher knowingthe answer, and the students, in your words, who ‘tend toassume’ this. One of the things I have become aware of inmy own teaching practice is when I ask questions that arebasically “guess what’s in my head”; questions; and I havebeen trying to avoid themse as much as possible. I like toplace my attention in the learning and sense making of thestudents. When I am most “in flow” as a teacher it really isthe case that “the answer” is not in my head – not because Idon’t know the answer, but because I am not engaging thataspect of myself. I am genuinely interested in what studentssay or do in mathematical contexts, and enjoy being exposedto perspectives I had never considered. There is an inherentambiguity, for me, in mathematics – answers are always pro-visional on assumptions, and part of my work with our firstyear classes (aged 11) is getting them to become aware oftheir assumptions, or the conditions under which their gen-eralizations do and do not apply. A counter-example does notso much prove a conjecture wrong as lead to a modification(to borrow John Mason’s language).

Nick: Your description of ambiguity and becoming awareof assumptions chimes with what I would call, in a univer-sity context, a good mathematical conversation. Onemathematician (often more of an expert in the particulartopic under discussion) will have a conjecture, or methodof attacking a problem, which both then try to test, and thenamend. ‘Being exposed to perspectives [they] had never

considered,’ which the other mathematician can provide,will often result in a breakthrough of some kind. ‘Answersalways being provisional on assumptions’ is important here,as it can be hidden assumptions in the “expert” view whoseremoval can be key in a solution. The idea of counter-exam-ples being openings rather than endings is also important –they often lead to more interesting questions and almostcertainly to a deeper understanding.

What I have seen at Kingsfield makes me think there is nonecessity for school mathematics to be anything distinct,for example if students are actively pushing the boundariesof their own knowledge. It feels like you have set up struc-tures in the department with the aim of making this as closean approximation to “the real thing” as is possible; any dif-ferences lie in the short-comings of the approximation (andapply just as much, if not more, to most undergraduatecourses). Of course, as teachers you frame the context forany enquiry, but that is not so different from university. Thecontext there is set by the lecturer, or even sometimes at PhDlevel by the supervisor. Arguably, in almost any mathemati-cal question the context has been set by previousmathematicians (the one who first asked the question, andothers who have thought about it). It is often breaking outof this context and re-framing the problem that might lead toits solution.

Alf: In talking about Kingsfield, I am wary of Watson’scomment that ‘it is always possible to … argue to excludeyour particular patch of the world’. I read her, in essence,as saying that ‘whatever the way it is taught’ the phase oflearning mathematics is different from the phase of mas-tery. I am reminded of Varela (1999), in the context of ethics,discussesing a mode of reflection on life that ‘takes the mid-dle way between spontaneity and rational calculation’ (p.31), whereby ‘after acting spontaneously’ we ‘reconstructthe intelligent awareness that justifies the action … as a step-ping-stone for continued learning’ (p. 32). Varela states‘even the beginner can use this sort of deliberate analysis toacquire sufficient intelligent awareness to bypass deliberate-ness altogether and become an expert’ (p. 32). I have alwaystakening this quotation to imply that learning and mastery donot have to be distinct phases, and that I can use myself insuch a way to learn through mastery from the beginning. Itry to work with my students so that they use such ‘deliber-ate analysis’ in learning mathematics.

This point takes me to the issue of the role of practice inlearning mathematics. In Watson’s image of the cafeteria atthe mathematics faculty in Cambridge, ‘[N]o one is doingexercises or practiscing techniques’. She also states ‘Math-ematics as a discipline … does not include … practising coretechniques’. Would you say that in your post-doc research,when you were presumably pushing boundaries of knowl-edge presumably, that practice and drill played any part inwhat you did? (If so, would you consciously choose to dothis? I am reminded of my brother who is a professionalmusician and yet continues to practice his scales.)

Nick: I definitely did “drill” myself in various differenttechniques, even as a post-doc, and I believe that other math-ematicians also do this, especially when embarking on a newproblem, in an area slightly unfamiliar to them. And it cer-tainly was a conscious choice. When exploring a new

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structure in mathematics, one has to play with it. You posea lot of questions, and then try to solve them. Asking andworking on many similar problems can lead to conjecturesand more questions. When you think you have a method thatwill solve the problem you are really interested in, it isimportant to practise it a lot so that you are familiar withhow it works and can be applied in different contexts. I thinkthat this is probably very similar to what mathematics teach-ers are trying to get their students to do in many schools.

ReferenceVarela, F. (1999) Ethical know-how: action, wisdom and cognition, Stan-

ford, CA, Stanford University Press.

There can be, should be, andsometimes is a connection

VICKI ZACK

Watson opens by asserting “that school mathematics is not,and perhaps never can be, a subset of the recognised disci-pline of mathematics”, and re-emphasizes the point in herconcluding section by omitting the ‘perhaps’ when statingthat school mathematics “necessarily is not a subset of thediscipline of mathematics”. I differ, and insist that, whenconditions are propitious, school mathematics can be – andin some cases is – a foundation for a mathematician’s work.There must be a connection.

I do agree with much of what Watson says in regard tothe differences between what happens in schools and what isentailed in the work of mathematicians. However, the foun-dation mathematicians’ work is often laid in school, if thesettings are such as the school Watson mentions at the outset(i.e., the one using the marvelous reference to Burton’swork). Was Watson herself a teacher at that school, so “well-known to her”? I think also of the progressive school(Phoenix Park) in which Boaler (1997) did her study andthe rich questions those students tackled.

I was inspired by the Phoenix Park students’ work. I wasalso distressed by Boaler’s postscript that the school boarddecided to shift back to a traditional curriculum for PhoenixPark. The authentic mathematical activity that Watsondescribes and the project-based curriculum that Boalerdescribed represent exactly what we should strive for. Hencemy disappointment when Watson seems to be saying that theworlds of schools and of mathematicians are separate. Howcan that be? It is for the most part in challenging and creativeclassroom settings that students can apprentice.

Watson distinguishes between situations in which chil-dren work by rote and ones in which the focus is makingmeaning. Her life’s work has been around the latter. How-ever, I do not, even when it is done for the sake of making apoint, wish to hear of the two lumped together (“whatever thenature of the mathematics, whatever the way it is taught”).My own school experience leads me to decry situations inwhich children learn by rote. I was one of a multitude whowas “cognitively bullied”. As I came to know about the

beauty of mathematics as a mature adult, my anger grew atbeing cheated. No one offered me a glimpse of the creativityof mathematics when I was young – in elementary school orin high school or out of school. I spent half a lifetime feel-ing inferior about my mathematics ability. We have to lookcritically at what is not working in traditional classrooms.

To that end, Lockhart (2008) looks at the situation inschools, deploring the fact that what could be a “rich …adventure of the imagination has been reduced to a sterile setof “facts” to be memorised and procedures to be followed”(p. 5). He describes mathematics as “the art of explanation”:

If you deny the students the opportunity to engage inthis activity – to pose their own problems, make theirown conjectures and discoveries, to be wrong, to becreatively frustrated, to have an inspiration, and to cob-ble together their own explanations and proofs – youdeny them mathematics itself. (p. 5)

The work in which Lockhart, an elementary and secondaryteacher, engages with the students is nothing less than a full-fledged mathematical endeavor.

I returned to the classroom in 1989, at age 43, to teach fulltime and do research, looking at what 5th-grade childrencould do with non-routine problems. It was then that I beganto perceive the complexity and beauty of mathematics (cf.Zack & Reid, 2003, 2004). I have gloried in the children’s,and my own, investigations and discoveries. They were ableto do work as mathematicians do: discovering patterns, mak-ing conjectures, constructing counterexamples, andpresenting powerful arguments. The work done by adults ina community of mathematicians is a mature version of thesekinds of endeavors.

The joy that I continue to feel in regard to the children’sendeavors and accomplishments in mathematics has led meto think about the joy I feel as a learner and as a teacher. Ihave always looked for ideas, books, art that would touchme, make me laugh, make me cry, put me in awe of theirpower. In my work with students I have always shared myexcitement about diverse topics – but have encouraged themto pursue answers to questions that they found fascinating.That was true for me in regard to literature and social studiesprior to 1980 and, although a bit intimidating, was as trueabout mathematics when I came to be intrigued by it. I knowthat the children will far outdistance me, both as 10-year-olds, and later on.

It might seem trite to speak about instilling a lifelong loveof learning and of seeing the beauty in various disciplines –literature, mathematics, and history – but that has been mygoal. I am the lover of literature who can write critiques butwill never write a literary work worthy of publication. But Iknow that some of my students will. Our investigations ofquestions in mathematics will inspire some of those studentsto do something of significance in the mathematics. A num-ber of them will go on to do great things. What Watsondescribes as the discipline of mathematics has its roots inschool-based endeavors.

ReferencesBoaler, J. (1997) Experiencing school mathematics: teaching styles, sex and

setting, Buckingham, UK, Open University Press.

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Lockhart, P. (2008). Lockhart’s lament, available (last accessed, 2008August 05) at: http://www.maa.org/devlin/devlin_03_08.html

Zack, V. and Reid, D. (2003). Good-enough understanding: theorizingabout the learning of complex ideas (pt. 1), For the Learning of Mathe-matics 23(3), 43–50.

Zack, V. and Reid, D. (2004) Good-enough understanding: theorizing aboutthe learning of complex ideas (pt. 2), For the Learning of Mathematics24(1), 25–28.

Teacher as model learner

HILARY POVEY

In her paper, Watson reminds us of the distinction between“school maths” and the activities of mathematicians and shewrites compellingly and convincingly about the latter. Shedescribes in detail and with care the processes in whichmathematicians engage, the processes that define what it isto do mathematics. She has had as an aim for a long time that“school students should be introduced to authentic mathe-matical activity such as is practised by professionalmathematicians”; but, she fears, “school mathematics is nec-essarily not a subset of the discipline of mathematics” (myemphasis). I share Watson’s concern that the difficult shiftsof reasoning demanded by mathematics are being “editedout” of the mathematics experienced in school and appreci-ate that the alternative is rarely seen. However, despite thefact that she argues forcefully and accurately that deep insti-tutional structures militate against (unfortunately, veryeffectively) students “acting like mathematicians” inschools, I remain, finally, unconvinced of this necessity.

She has called up a vivid image of undergraduate and grad-uate students of mathematics working mathematicallytogether in their Cambridge coffee room. Watson is right that,for these highly privileged and highly successful students ofmathematics, there is a pretty good chance that no-one explic-itly told them to behave like this; but this does not seem tome to cause a fundamental breach between their learning anddoing mathematics and the learning and doing mathematicsthat might happen in school classrooms if we had the will.Watson does not, but I do, see them practising techniques – fora well defined and self-chosen purpose, of course; I see themeven, in the same spirit, doing exercises or practise problems;and certainly, I see them wandering over from time to time,and very pleased when the opportunity presents itself, tocheck out results, ideas, muddled thinking and so on with theirteachers who have also popped in for a coffee and a bun.

I agree with Watson that, even in the mathematics class-rooms in school that take most seriously the injunction“Together we can learn to be mathematicians”, other thingswill happen than doing mathematics. But so will they in theCambridge coffee room. People will gossip, will drink cof-fee, will have to leave to take their laundry out of the washingmachine and so on. Also, more importantly, they will alsostruggle to understand the already existing mathematics ofothers; will practise routines, perhaps to gain deeper algo-rithmic understanding, perhaps to become more proficientand fluent; will welcome instruction; will sometimes just

accept closure on uncompleted problems; will accept, attimes, that a model “works” without fully understanding whyand will not pursue the matter further; will use inductive rea-soning, pattern spotting, experimental results, sometimeseven as worthwhile ends in themselves. And, whilst Watsonis right to draw attention to the fact that a vital and significantunderlying purpose of their activity is to sustain and developmathematical habits of mind, those undergraduate and grad-uate students in the coffee room will sometimes experiencetheir studies as ‘a “performance” route’ (Mann, 2003, p. 20)to success and will, from time to time, have their mathema-tising undermined by high stakes external assessment.

There is one aspect of the injunction to which Watsonrefers at the beginning of her paper that I should like toexplore a little further: the suggestion is that together weshould learn to be mathematicians. In a personal communi-cation, one teacher (Corinne Angier) who attempts to modelher practice on this wrote, “I was thinking about the studentsas being apprenticed as learners of maths and of me being amodel learner not a teacher”. Watson draws attention to thenature of authority that is appealed to in school classrooms;as Alrø and Skovsmose have it (1996, p. 4), the teacher, thetextbook and the answer book make up a united authoritythere which needs no specification or justification. But it ispossible for classrooms, even school classrooms, to be dif-ferent from this – to have school students, occasionally,successfully challenging what a teacher, a textbook, ananswer book says. This is going to happen in a classroomwhere there is a strong sense of the learner and the teacher,from time to time, being engaged in the same, not a differ-ent, activity. In the teaching of art, the model ofapprenticeship resonates much more closely than is usual inmathematics classrooms. Why might this be so? I see tworeasons that are relevant here. First, whilst the students are,of course, practising techniques, doing exercises, acquiringknowledge of the art already completed by others, there isalso an expectation that they are throughout, from time totime, themselves working as artists. Second, almostuniquely among subject teachers (in England, at least), artteachers will themselves be practitioners of the subject theyare teaching. They will be artists, and will describe them-selves as such, as well as being teachers of art: indeed, theywill often practise their craft side by side with their studentsin the same classroom. Some mathematics teachers, albeit afew, also behave like this; that suggests to me that the factthat the mathematics of school classrooms is “coordinatedand aligned” differently from the discipline of mathematicsis a contingent rather than a necessary phenomenon.

One cannot think of what happens in mathematics class-rooms in school as being only modelled on whatmathematicians do. Watson is right to say that there are alter-native (and some probably necessary and even desirable)purposes to which attention is paid. But, and I am aware ofthe risk that I am being utopian, she has not successfullyshown me that “learning to be a mathematician” cannot bethe organising principle for teaching mathematics in school.

ReferencesAlrø, H. and Skovsmose, O. (1996) ‘On the right track’, For the Learning

of Mathematics 16(1), 2–40.

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Mann, C. (2003) Indicators of academic performance, available (lastaccessed 2003 December 17) at: htp://www.admin.cam.ac.uk/reporter/2003-03/weekly/5913/6.html

Mathématiques scolaires etMathématiques

VIVIANE DURAND-GUERRIER

Dans son article, Anne Watson écrit: « Authority in schoolmathematics lies with teachers, textbooks and assessmentregimes, not with mathematical argument ». Ceci conduit àse poser la question de ce qu’est un argument mathématique.

En accord avec Barallobres (2007), je soutiens que la vali-dation intellectuelle (mathématique) ne se limite pas à ladémonstration, mais engage « ce processus complexed’allers-retours, d’intuitions, de doutes et de méfiances quicaractérisent le développement de la pensée mathématique »(op.cité, p. 41). Dans la perspective du modèle théorique ini-tié par Tarski (1969), ceci relève de l’articulation entre le« domaine de réalité » (lieu de l’action des sujets avec lesobjets) et la théorie mathématique émergente. Ceci renvoieégalement à la dialectique action/formulation/validation quiest au cœur de la Théorie des Situations Didactiques(Brousseau, 1997, 1998).

Prenons l’exemple classique de la situation d’agrandisse-ment du Puzzle dans l’ouvrage cité en référence. Les élèvesde la fin de l’école primaire (10–11 ans) sont invités àagrandir le puzzle de sorte qu’une pièce qui mesure 4 cm surle modèle devra mesurer 7 cm sur le Puzzle agrandi. Chaqueélève d’une équipe de 4 ou 5 agrandit sa pièce, avant dereconstituer le Puzzle. La mise en défaut de la procédureadditive se fait par l’impossibilité de reconstituer le puzzleavec les pièces obtenues. Ce sont ici les rétroactions dumilieu qui vont permettre aux élèves de remettre en causela procédure utilisée, ce qui permettra au professeur derelancer la recherche, et de faire, peut-être, émerger uneprocédure de type multiplicatif.

Dans le modèle proposé par Brousseau, le travail desélèves se fait dans un va-et-vient entre les actions sur lesobjets et la formulation de conjectures. À ce stade du cur-riculum, le théorème de Thalès ou l’homothétie ne sont pasencore disponibles. Les élèves peuvent alors chercher dif-férentes manières de « passer de 4 à 7 » et les tester enréalisant un nouvel agrandissement. Si le résultat obtenu estprobant, on peut voir apparaître une sorte d’ « axiomatisationlocale » qui consiste à poser comme « axiome » un énoncéqui, réinterprété dans la situation, donne le résultat attendu.[1] Ce que le professeur ou les manuels doivent alors garan-tir, c’est que cette « axiomatique locale » peut s’insérer dansles théories mathématiques de référence dans lesquelles lasituation s’inscrit. Nous avons également rencontré ce typed’élaboration « d’axiomatique locale » chez des enseignantsnon experts en mathématiques à qui nous avions proposéde déterminer tous les polyèdres réguliers convexes, en met-tant à disposition, outre le matériel classique de géométrie,des pièces en plastiques permettant de faire et défaire facile-

ment des assemblages (Dias & Durand-Guerrier, 2005). Un« axiome local » est apparu très rapidement : « on peut con-struire un polyèdre régulier pour chaque polygone régulier ».À l’issue d’un important travail sur les objets et après denombreux échanges contradictoires, les membres du groupesont finalement tombés d’accord pour poser que « si lasomme des angles au sommet est égale à 360°, alors on nepeut pas faire un angle (sous-entendu, un angle trièdre) », cequi détruit l’ « axiome local » précédent. Cette même situa-tion proposée à des étudiants mathématiciens de Masterpremière année (4e année d’université) donne des résultatssimilaires, illustrant le fait que les cours classiques degéométrie ne permettent pas d’expliciter ce type de résultat.

Dans ces deux exemples, le travail proposé aux élèves ouaux professeurs en formation nous ramène à la question desarguments mathématiques. De fait, une part importante del’activité du mathématicien est d’élaborer des nouveauxobjets et des axiomatiques permettant de rendre compte dephénomènes observés, que ce soit à l’intérieur des mathé-matiques elles-mêmes, ou dans d’autres domaines del’activité humaine. Or, selon Chevallard, « dans une mathé-matisation complète, on s’assure qu’un fait est vrai sur unebase expérimentale, puis on s’assure que dans la théorie qu’ona bâtie par ailleurs, on peut déduire le fait en question. »(Chevallard, 2004, p.35), ce qui correspond à une dialec-tique de la vérité (dans les domaines d’interprétation) et dela validité (dans les théories) (Durand-Guerrier, 2008). Lesdeux exemples précédents et ce qui est développé dans Bar-allobres (op. cité) montrent selon moi que l’on pourraitconsidérer qu’un processus analogue de mathématisation està l’œuvre dans l’apprentissage des mathématiques. J’a-jouterais en outre que ceci vaut dès le plus jeune âge : il suffiten effet de penser à l’apprentissage du nombre commemesure de la taille des collections discrètes chez les jeunesenfants, notion abstraite s’il en est qui, quelques années plustard, sera devenu suffisamment familière pour les élèves pourpouvoir soutenir la construction de l’algèbre élémentaire.

Note[1] Ici : « Pour agrandir, il faut multiplier toutes les dimensions (longueurs)par un même nombre ».

RéférencesBarallobres, G. (2007) ‘Introduction à l’algèbre par la généralisation : prob-

lèmes didactiques soulevés’, For the Learning of Mathematics 27(1),39–44.

Brousseau, G. (1997) Theory of didactical situations in mathematics, Dor-drecht, NL, Kluwer.

Brousseau, G. (1998) Théorie des situations didactiques, Grenoble, FR,La Pensée Sauvage.

Chevallard Y. (2004) ‘Pour une nouvelle épistémologie scolaire’, CahiersPédagogiques 427, 34–36.

Dias T. et Durand-Guerrier V. (2005) ‘Expérimenter pour apprendre enmathématiques’, Repères-IREM 60, 61–78.

Durand-Guerrier, V. (2008) ‘Truth versus validity in proving in mathemat-ics‘, Zentralblatt für Didaktik der Mathematik 40(3), 373–384.

Tarski, A. (1969) Introduction à la logique, Paris-Louvain, FR, Gauthier-Villard.

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The purpose of having mathematics in schools tellswhat school mathematicsshould be

ROMULO LINS

The core of Watson’s argument is, I think, that the purposeof school mathematics is quite different from that of disci-plinary mathematics. All the other differences somehowderive from this. For instance, it could be the case that thepurpose of school mathematics is to teach (or induce) pupils(all) to act as mathematicians; even in this case, there wouldbe a crucial missing component, namely that in disciplinarymathematics one is operating on the boundaries of theknown, while in (such) school mathematics pupils wouldbe only acting as if they were there. This is the reason whya Lakatosian (fallibilist) ‘foundation’ for mathematics edu-cation does not make sense.

On the other hand, is it a fair (or even interesting) goal formathematics education in schools that all pupils come to act asmathematicians? Perhaps it would be fair to aim at enablingall pupils to act as mathematicians whenever they wanted to.But that does not seem reasonable, and the reason for this is inthe huge difficulty in getting mathematicians not to act asmathematicians when talking about mathematics (forinstance, when discussing school mathematics and the learn-ing of mathematics). It seems the drive is too strong for themto keep acting as mathematicians. In other words, maybe asone truly crosses the border and begins acting as a mathe-matician, it is too late to come back, even if eventually.

Watson correctly states that the “discipline of mathemat-ics has different warrants for truth, different forms ofreasoning, different core activities, different purposes” inrelation to school mathematics, and it seems wise to extendthis observation to everything ‘mathematical’ normal peoplealso do outside school. And in doing so we are naturally ledto the question ‘why is mathematics in school?’ While Wat-son spends quite a while on the question “what is‘mathematics’ in school?” it is less visible a concern with“why is ‘mathematics’ in school?” If mathematics as a dis-cipline disappeared altogether at the disciplinary (research)level (production of new mathematics) would school mathe-matics also disappear? Not necessarily. The other wayaround? Yes, possibly in a couple of generations.

Watson also says that, in her experience, “those who havethe opportunity to do this [act like mathematicians] becomebetter mathematical learners than those drilled in more tradi-tional methods” – but that could well be because thetraditional approach is (with respect to any subject) boring tomost (but not all) people, not because of any peculiarity ofmathematics. In school I loved to read and write, but hatedformal grammar; I loved to play (real) football, but hateddrilling passes or even tactics. Maybe normal people only liketo (drill-like) practice the things they enjoy enough as to wantto become a pro in them. And that choice sounds perfectly

healthy. That points to a fundamental and irreconcilable dif-ference between school and disciplinary mathematics:people who do the latter do it voluntarily. And I sense thatfits quite well with the fact, mentioned above, that these peo-ple also do it for real.

Let me push the analogy further. In Brazil, youngsters whogo to football clubs to be trained act exactly like the pros:they drill passes, shooting, tactics, carrying the ball and soon. And not only they do not complain of this traditionaldrilling, they will actually tell their peers who only play forfun that these are not doing it the right way (if they everbother to comment, much as the mathematician does not usu-ally comment with non-mathematicians on what he does).And, quite naturally in this case, in the discipline the waysof acting are transparent, but in school mathematics theyare a teaching goal, as Watson and others correctly point out.

It could be that “the institutionalization of mathematicalknowledge for novices … is not, and perhaps never can be,a subset of the recognised discipline of mathematics” notjust because of different warrants for truth but, ultimately,because we are looking at an institutionalization for all andthat means that school mathematics should not be an incu-bator of future mathematicians, as much as physicaleducation in school should not be (although it similarlycould) be an incubator of Olympic athletes.

Of course, I agree that the public image of mathematics is,in too many countries, quite negative, and there is plenty ofroom for improvement in that area. But surely that is notgoing to change simply by making people eat all their maths.

Watson’s argument sufficiently supports the point shewants to make. If anything, I would just add that perhaps asit stands the effort to understand the relationship betweenschool and disciplinary mathematics is better framed as astruggle between mathematics educators and mathemati-cians, possibly representing professional interestsideologically dressed as ‘truths’.

And maybe, paradoxically, that struggle means that therelationship will not be best understood by remaining onlywithin our specialist fields.

What’s so great about doingmathematics like a mathematician?

HEATHER MENDICK

Watson argues that school mathematics should try to pro-duce people who act like mathematicians. And that this is notpossible because of the different conditions in which thepractices of school and disciplinary mathematics are embed-ded. I agree with the second part but not the first part of herargument. I start with the part where we agree.

Watson mentions several reasons why there will always bea mismatch between school and disciplinary mathematics:

• schoolteachers do not have the experience of doingmathematics over time;

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• schools need to prepare students for future studyand/or employment;

• and, within schools, authority lies with the text-books and timetable and curriculum andassessment systems constrain what can be done.

Except for the constraints of assessment and curriculum,these were not the first reasons that came to mind. Instead,I came up with the following list:

• school students do not get paid for doing mathe-matics;

• they do not apply for opportunities to do it;

• and, even when they have an identity that isinvested in being good at it, mathematics neverdefines them in the way that one’s employmentdoes as an adult.

The differences between these two lists are indicative of dif-ferent emphases and approaches. If I had more space I mighthave chosen to dissect these differences in detail and to arguethat, while socio-cultural approaches may not be able to put“detailed flesh on [their] stories [of identity development] interms of the development of specific mathematical ideas”,sociological approaches can do this. Hopefully some of thiswill come through as I talk through our main point of dis-agreement over whether school mathematics should try toproduce people who act like mathematicians.

About her image of the Cambridge cafeteria, Watsonwrites: “One might also say that Cambridge is not typical, orquestion the dominance of men and ask who is excluded, orsimply laugh at how anyone could want to replicate suchbehaviour”. In this way Anne tantalisingly offers, and thenforecloses, four interesting points of departure. I will take-upone of them: Why would anyone want to replicate thisbehaviour? Should disciplinary mathematics be our modelfor school mathematics? Katrina Miller (personal communi-cation) has suggested that school science should be taughtby sociologists so that we no longer attempt to separate sci-ence from what it does in the world. Perhaps we should wantat least some people to do mathematics like sociologistsrather than like mathematicians – or like historians or geog-raphers …. This questioning of mathematicians as our modelfor school mathematics is coming from a range of directions,including the mathematical literacy curriculum in SouthAfrica [1] and Appelbaum’s (2007) suggestion that mathe-matics teachers be ‘professional amateurs’ using the popularculture practices of young people and other school subjectsas models for their practice.

We could take this argument further and ask why profes-sional mathematicians would want to replicate thisbehaviour? In fact, many do not. They feel excluded by it(another of Watson’s foreclosed points of departure) and findother ways through, as far as the discipline allows, as Hen-rion’s (1997) biographical studies of female mathematiciansshow. However, Watson seems to give a universal status tothe ways of being a mathematician, for example, throughdiscussing shifts that “ha[ve] to be made to become a math-ematician”, without acknowledging either the cultural and

historical specificity of these shifts or the variation in prac-tices among contemporary mathematicians. Taking asideways step into the popular, this part of her argumentreminds me of The Rules (Fein & Schneider, 2000) – a self-help book which tells women how to act in ways that get aman to propose. These rules include: never accept a Satur-day date after Wednesday, never call a man and never goout without make-up. It appears to be very successful but atthe expense of leaving in place the status quo of sexual dou-ble standards. We need to teach people both to bemathematicians (the rules of the game) and to teach them tobe critical of the practices of mathematicians and of mathe-matics (to want to change the rules).

Note[1] Graven, M. and Venkat, H. (2008) Mathematical literacy: issues forengagement from the South African experience of curriculum implementa-tion, presented at the International Congress of Mathematics Education,Monterrey, MX, 2008 July 6–13.

ReferencesAppelbaum, P. (2007) ‘Afterword: bootleg mathematics’, in de Freitas, E.

& Nolan, K. (eds.), Opening the research text: critical insights andin(ter)ventions into mathematics education, New York, NY, Springer,pp. 221–248.

Fein, E. and Schneider, S. (2000) The complete book of rules: time-testedsecrets for capturing the heart of Mr Right, London, UK, Thorsons.

Henrion, C. (1997) Women in mathematics: the addition of difference,Bloomington, IN, Indiana University Press.

Reflections on mathematics aspracticed in schools and inacademia

UBIRATAN D’AMBROSIO

In asking whether the bases of School mathematics and Aca-demic mathematics are the same, Watson touches a sort oftaboo. Are good mathematics students potentially goodmathematicians? The main questions are “Why is mathe-matics a school subject?” and “What justifies the inclusionof mathematics in the curriculum?”

Watson’s reasons for distinguishing between Schoolmathematics and good Academic mathematics are signifi-cant and well justified. They also point to some interestingideas about ways of doing mathematics, the nature of math-ematics as a discipline, and of its organization as a field ofknowledge.

It is important to understand why societies, all over theworld, maintain some form of schooling. I see two mainreasons:

• to prepare new generations for citizenship, and

• to enhance creativity.

These reasons are two sides of the same coin. One, the con-servative facet, draws on established practices and values;the other calls for innovation, for the new. How does Schoolmathematics fit in these two objectives?

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Let me be very clear about my position. In what followsI am considering School mathematics and Academic math-ematics as practiced, since late-18th century, by the“included” population. I refer to the world that, since then,has come under western influence (or dominance). Myreflections do not refer to other periods, or to “excluded”populations and to other cultures, or to different or “alter-native” school systems and academic mathematics (whichmay be considered within the general framework of Ethno-mathematics).

I see School mathematics and Academic mathematics ashaving different methods and, above all, different objectives.They may have a few very basic elements in common, butnot all of these are essential and might be overcome. In Wat-son’s words, “most school students are taught and examinedon mathematics of a kind that is done, both in the academyand in other workplaces, by machines.”

The usual practice of School mathematics is, essentially,catechetic. More emphasis is given to the conservative orcitizenship facet. Students are required to accept and to per-form in a valid way. According to Watson “validation andauthority are highly structured through textbooks, examina-tions, and curriculum systems rather than by mathematiciansworking as a community”.

We have to recognize that some Academic mathematics ispracticed in very much the same way, with strict limitations.This is what Pierre Samuel (2002) called the style “âne quitrotte”, and brings to my memory what J.D. Salinger (1961)says in Franny and Zooey: “they’re not real poets. They arejust people that write poems that get published and antholo-gized all over the place, but they’re not poets”. Indeed, wehave a very large academic population recognized as math-ematicians, whose contributions are hundreds of printedpages (discussing details of sameness) – read by practicallyno one and soon forgotten. These writings serve the pur-poses of earning degrees and securing promotions. Suchactivity is accepted by the academy as a whole, with theargument – which I believe valid – of the necessity of a crit-ical mass of workers. It is, metaphorically, the concept ofcitizenship in the academic world.

Something similar occurs in School mathematics. Lowachievers are part of the system, and considerable resourcesare aimed at forcing them to achieve a mediocre passingstatus. Some are low achievers in mathematics, but brightin other subjects. Regrettably, mathematics educators havegiven practically no attention to low achievers, which canseriously damage their creative capacities. I will not pursuethis argument in this short commentary.

I feel it important to address creative mathematics –which is, essentially, open-ended. The paragraph in whichWatson describes the cafeteria of the Cambridge mathemat-ics faculty is a very good one. Such a scene is typical wherecreative mathematics is going on. Sometimes a seminar ini-tiates with a challenging question. After three or four hoursof discussion, the question remains unanswered. These arevery creative hours! We all know that after working forweeks on a mathematical question, often the only resultsare scribbled and discarded pages and a still-unansweredquestion. Normally, creative mathematicians “adjust” thequestions, modify the hypotheses, create new situations, and

reach new results. This is a major characteristic of the hum-bleness that is intrinsic to creative mathematics. It has muchto do with the ethical facet of mathematics and with theintrinsic character of mathematics as the essential languageof Science, which led Wigner (1960) to refer to the “unrea-sonable effectiveness of mathematics”. He describesmathematical creation in a most interesting way:

The great mathematician fully, almost ruthlessly,exploits the domain of permissible reasoning and skirtsthe impermissible. That his recklessness does not leadhim into a morass of contradictions is a miracle in itself:certainly it is hard to believe that our reasoning powerwas brought, by Darwin’s process of natural selection,to the perfection which it seems to possess. (p. 2)

As Watson discusses and exemplifies, neither this open-ended style nor the essentiality of mathematics to Scienceand the ethics associated with it belong to current Schoolmathematics. Although School mathematics does not aimto prepare future mathematicians, an appreciation and thepossibility of acquiring the essence of mathematics reason-ing should be achieved – for example, through exposure toabstract thinking and symbolic reasoning, or through prac-tices leading to inquiries about major issues. These highercapacities of the human mind are, in a sense, discouragedin School mathematics, which emphasizes an answer orresult to a question. Elsewhere I have discussed a proposalof curriculum aiming at the development of these highercapacities of the human mind in our civilization dominatedby technology (D’Ambrosio, 1999).

Watson provocatively, although briefly, raises the issueof “authority”. This point may lead to a very interesting dis-cussion, absolutely pertinent to this “era of standardizedtesting”. This opportune article gives also some hints onwhich contents might be brought into School mathematics.

ReferencesD’Ambrosio, U. (1999) ‘Literacy, matheracy and technoracy: a trivium for

today’, Mathematical Thinking and Learning 1(2), 131–153.Salinger, J. D. (1961) Franny and Zooey. Boston, MA, Little, Brown and

Company.Samuel, P. (2002) ‘Résultats élémentaires sur certaines équations dio-

phantiennes’, Journal de théorie des nombres de Bordeaux 14(2),629–646.

Wigner, E. (1960) ‘The unreasonable effectiveness of mathematics in thenatural sciences’, Communications in Pure and Applied Mathematics13(1), 1–14.

Just Yoking? on the Conjunctionof ‘School’ and ‘Mathematics’

WILLIAM HIGGINSON

Almost exactly 40 years ago the distinguished literary criticNorthrop Frye (1968) addressed a national meeting of Cana-dian educators on the theme of “The Social Importance ofLiterature”. Frye concluded his address with the followingobservation:

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The study of literature, as I define it, is not a panacea; itis not a cure; it does not solve social problems. What itdoes is to base education on the sense of a participat-ing community which is constantly in process andconstantly engaged in criticizing its own assumptionsand clarifying the vision of what it might and could be.The teaching of literature in that sense, and in that con-text, seems to me to be one of the central activities ofall teachers and educators in their continuous fight forthe sanity of mankind. (p. 23)

This view of Frye’s with its early emphasis on the educa-tional importance of participating communities criticizingand clarifying their assumptions and visions has alwaysseemed to me to resonate strongly with the FLM ‘statementof purpose/epigraph’ found on the inside front cover ofevery issue of this journal since it was first drafted by thefounding editor more than a quarter of a century ago. Onereason for raising the ‘journal aims’ at this point is becauseI think (high praise indeed) that Watson’s contribution isvery much the sort that David Wheeler would have beenpleased to include in this journal when he was editor. Muchmore than most mathematics education papers, even in thislocation, it “stimulates reflection”, “promotes study of prac-tices and theories”, and reveals that “the learning andteaching of mathematics are complex enterprises aboutwhich much remains to be revealed and understood”. Beforeturning to some specific points I would like to make anobservation about the ‘solicited comment’ approach. In myview the nature of the FLM community makes this style ofdialogue a desirable and worthwhile one. Whether or not itis effective for individual readers may vary, but it is worthnoting that there are journals in some academic areas thatmake this a major feature of their standard publishing model.One of the most prominent of these ‘Open Peer Commen-tary’ journals, Behavioral and Brain Sciences, on occasionco-publishes as many as twenty-five reactions to a givenlead article. It is perhaps a model that the editors and advi-sory board might consider extending to some degree infuture issues.

From the comments made above it will be clear that I findmuch of merit in this paper. Watson’s ‘feel’ for school math-ematics curricula, classrooms, children and teachers isnuanced and deep. Her ideas on necessary ‘shifts of under-standing’ are provocative and interesting. Is it really the casethat “most school students are taught and examined on math-ematics of a kind that is done … by machines”? Does‘bullying’ parse so neatly that we can really have somethingcalled “cognitive bullying”? All good food for thought.

Her conclusion that school mathematics is necessarilynot ‘junior’ mathematics, namely a subset of the disciplineseems uncontroversial to me. The conjoining of concepts(think a * b, for some entities a and b, and operator * : con-sider ‘turtle’ and ‘geometry’; ‘water’ and ‘board’, ‘Holmes’and ‘Fortnum’) is far from a straightforward process, as 20th-century scholars in computer science, botany, zoology,linguistics, philosophy, and mathematics have found. Bringtogether (yoke, merge, unite, concatenate, …) an institution(technology?) and a body of knowledge (discipline) and theresult is influenced by both but isomorphic to neither. Muchof the paper documents the very strong influence of theschool side of the family tree of school mathematics in theform of resulting homogenization and routinization.

If I had to quibble (it comes with the commentator terri-tory) it would be mainly with the paper’s portrayal of themathematical side of the dyad. In particular, the image of theCambridge cafeteria/conjecture-palace doesn’t work for meas an ideal for authentic mathematical activity. Yes, the con-versations are animated, the cerebral leaps impressive, butthis is still (unless it has changed from my last visit) largelya group of students mostly working on assigned questionswith most of the time and configuration constraints thatcharacterize school mathematics. In general, I see the disci-pline as being much more fragmented (fractalized?) than issuggested here. While there are family similarities (à laWittgenstein) between and among, for example, ‘experi-mental’, ‘applied’ and ‘recreational’ mathematics, theirdifferences are significant. While the author never says soexplicitly, my sense is that when she says mathematics, shemeans research in pure mathematics.

While one might feel that a case could be made for thisarea of endeavour as a pearl of human achievement, it is lessclear that its methods and procedures should form a centralpart of the ‘common curriculum’ for all learners. In con-cluding, let me go back to Frye’s view, noted above, andsuggest two things. First, that if one were to replace ‘litera-ture’ by ‘mathematics’ in the statement, its validity would beweakened very little. Second, that in today’s world it is notjust mankind’s “sanity” that is in doubt. Mathematics hasclearly contributed much more than literature to the cre-ation of today’s global problématique. Does this not implythe need for a common mathematical curriculum more con-nected to issues of human survival?

ReferenceFrye, N. (1968) ‘The social importance of literature’, Educational Courier

39 (10/11), 19–23.

Group Photos

Ten people in a team line up in two ranks for a group photograph. They are all of differentheights, and each person in the back row stands directly behind a person in the front row. Inhow many ways can they be arranged so that every person in the back row can look over thecorresponding person in the front row? (unknown origin; selected by Leo Rogers)