Humanistic Mathematics Network Journal

Issue 9 Article 9

2-1-1994

Philosophy of Mathematics, MathematicsEducation, and Philosophy of MathematicsEducationYuxin ZhengNanjing University

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Recommended CitationZheng, Yuxin (1994) "Philosophy of Mathematics, Mathematics Education, and Philosophy of Mathematics Education," HumanisticMathematics Network Journal: Iss. 9, Article 9.Available at: http://scholarship.claremont.edu/hmnj/vol1/iss9/9

Philosophy of Mathematics, Mathematics Educationand Philosophy of Mathematics Education

Zheng Yuxin (Y. Zheng)Department of Philosophy

Nanling University. China (P. R. C.)

As a philosopher of mathematics. I have beenthinking about, or rather, worried about thefollow ing qu esti on: is there any importantrel ation ship between the philosophy ofmathematics and actual mathematical activities(including mathematical research, teaching andlearning)? Or, does the philosophy of mathematicshave any important infl uence on act ua lmathematical activities? I think the answeris 'yes';and I have also tried to do some things in thisdirection by working in the field of methodology of

While the advancement of hum ansociety, is the most importan texterna l impetu s, it is thetheoretical studies of mathematicseducation during the past decadewhich have laid the necessaryfoundation for the new reformmovement.

mathematics (cf. Y. Zheng, 1985, 1991a, 1991b).But it is only a personal opinion and has limitedinfluence, so when I came to the USA as a visitingscholar in 1991, this problem was still deeplyrooted in my mind. However, what I have learntin the field of mathematics education in the USA isreally a great pleasure for me, as it does showclearly that there is a close relationship between thephilosophy of mathematics, mathematics educationand mathematics as well: it is modem research inthe philosophy of mathematics which offers thenecessary ideological foundation for the newreform movement of mathematics education in theUSA, and then, in this way, the philosophy ofmathematics can exert a great influence on thefuture of mathematics,

The first and second parts of this paper will use themodem development of mathematics education inthe USA as a background to make an analysis of

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the great infl uence of the philosophy ofmathemati cs on mathematics education. The thirdsection discusses the problem of how to developthe subject 'philosophy of mathematics education ,which in fact can be regarded as an impetus frommathematic s education to the further developmentof philosophy of mathematics and philosophy ingeneral.

1. New Developments of MathematicsEducation

Mathematics education in the USA is nowundergoing a new reform movement, to whichEyerybody CouOls, published by the NationalResearch Council in 1990, gives the followingdescription:"Over the next cwo decades, the nation's schools,colleges, and universities will undergo majortransitions in mathematics programs-s-transitionsthat will involvefundamental changes in curricularcontent, in modes of instru ction, in teachereducation, in profe ssional dev elopment , inmethods of assessment, and in public attitudes,"(p.87)

While the advancement of human society, i.e. thetransition from the industrial age to the infonnationage, is the most important external impetus, it is thetheoretical studies of mathematics education duringthe past decade which have laid the necessaryfoundation for the new reform movement. In thissec tion, we ma ke a brief survey of the ncwtheoretical studies. They are chiefly: the emphasison probl em so lving , the psychology ofmathema tical learning, and the social-culturalapproach to mathematics education.

(1). The emphasis on problem solving

' Problem Solving' was the main slogan formathem atics educat ion during the eighties."Problem solving must be the focus of schoolmathematics" (NCTM, 1980, p. 2); and by

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'problem solving' . it means ' to use a variety ofmathematical knowledge and methods effectively tosolve nonroutine problems. including both actualproblems and those originated from mathematicsitself.

Putting forward the idea of focusing on problemsolving is a giant step for mathematics education.because the idea represents a great shift in theconception of mathematics education, i.e., the ideaitself is a direct negation of the traditionalconception of mathematics education, especially,the teaching method based on 'transmission ofinformation' and the trend of 'separating learningfrom application'. To explicate, the key points of'focusing on problem solving' are as follows:First. students should learn mathematics by theactivities of solving problems. That is, "'knowing'mathematics is 'doing' mathematics ....instructionshould persistently emphasize 'doing' rather than'knowing that' ... (NCfM, 1989, p. 7) Second. bysolving problems, especially those having actualmeaning. students can learn to value mathematics,and become more confident in their ownmathematical abili ty. Third, the final aim ofmathematics education should be to improvestudents' ability of problem solving, especiallyhelp them leam to think mathematically.

Generally speaking, the idea that problem solvingmust be the focus of school mathematics is nowwidely accepted; and as this idea is directl yopposit e to the traditi onal conception ofmathematics education, it is said that ' solvingnonroutine problems is the central theme of thecurrent reform movement in school mathematics.'(T. Romberg, 1991, p. 9)

(2). The emphasis on th e psychology ofmathematical learning

The study of the psychology of mathematicallearning is itself a result of the further developmentof psychology: it has been beyond the level ofgeneral study and penetrated into special fields.Furthermore , where modern studies of thepsychology of mathematics learning are concerned,we should pay more attention to the cognitivescience approach to mathematics education and"theconstructivist view of mathematics learning".

To explicate. the basic position of cognitivepsychology is that the study of psychology shouldnot (as behaviorists suggest) be limited to 'vi sible

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behavior ' but penetrate into the inner infonnationprocessing of the mind, including the storage.retriev al , representation , development ofknowledge and so on. Also.the so-called'constructivist view' can be regarded as a mainconclusion of cognitive psychology: as far asmathematics learning is concerned. it asserts thatthe learning of mathematics is not a passivereception but a process of construction based onprevious experience and knowledge.

If the idea " focusing on problem-solving" is adirect negation of the traditional concept ofmathematics education, then the cognitive studiesof mathematics learning, especially theconstructivist view of mathematics learning. haveoffered further arguments for this fundamentaltransition from the microscopic view. And just for

we should take as a background thewhole culture of human society inthe study of mathematics education.This is to say, mathematicseducation should represent clearlythe features of the time.

this reason, the constructivist view on mathematicsleaming has recently attracted great attentions in thefield of mathematics education. For example. asR. Davis, C. Maher and N. Noddings say inConstructivist Views on the Teachin& and Leamineof Mathematics:

'The idea of "constructivismt-s-hardly mentioned afew years ago-nowadays attracts a lot ofattentionin the world of mathematics education. A greatmany people now think and write about it. andthepeople who do so do not agree with one another..Still, beneath the theoretical argumentation. thereis a substantial agreement about the nature 0/learners , the nature 0/ mathematics. and theappropriate form of pedagogy." (R. Davis, C.Maher and N. Noddings / 99/, p. / 87)

(3). The soclat-cultural approach tomathematics education

The first implication of the social-cultural study ofmathematics is that we should take as a backgroundthe whole culture of human society in the study of

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mathemat ics education . This is to say.mathematics education should represent clearly thefeatures of the time. In fact, it is exactly the mostimportant feature of the new reform movement ofmathematics edu cation in the USA: it is thetransition from the industrial soc iety to theinformati on society which offers the mostimportant impetus to the movement, and the finalaim of the movement is to create the kind ofmathematics education that not only meets the needof the time but also uses fully the new technology;in a word. we should crea te the mathematicseducation of the information age.

Secondly. the social-cult ural approach tomathematics education has also made clear thesocial nature of mathematics learning and teaching.Although the constructi on of mat hemati csknowl edge should be carried out rel ati velyindependently by all individuals, such activities arecanied on in some 'social environment' , and mustinclude th e proce sse s of ex pre ss ing ,communicating, comparing, criticizing, improvingand so on, so that it is in fac t a ' socialconstruction ' . Besides, the social nature ofmathematics teaching can be seen clearly by the fact

First, every mathematics teacher is(consciously or unconsciously)doing his work under the influenceof some conception of mathematicsand mathematics education, and thelatter are in fact manifestations ofthe social nature of mathematicseducation.

that the role played by teachers is ju st the' Intermediate ' between the whole system ofeducation and the objects of education. In otherwords, teachers' duty is to carry out the overall aimof mathematics education in facing the concretestudents and the concrete situation of teaching ingeneral.

Finally, one more important implication of thesocial-cultural approach to mathematics educationis the imp ortance of conception both tomathematics teaching and learning: First, everymath emati cs teacher i s (consc ious ly orunconsciously) doing his work under the influence

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of some conception of mathematics andmathematics education, and the latter are in factmanifestations of the social nature of mathematicseducation. Secondly, as far as students areconcerned. the importance of conception lies in thefact that mathematics learning is a process in whichnot only mathematical knowledge is constructedbut also some conception, belief and attitude ofmathematics are formed, and the latter in tum willexert great influence on the learners' further studyof mathematics and even for their whole life (as apan of their whole ideology). For example, it isjust by such consideration that Curriculum andEvaluation Standards for School Mathemat ics ,which is one of the most important documentsshaping the new reform movement, lists ' learningto value mathematics' and 'becoming confident ofone's own ability' as the first two goals formathematics education.(pp. 5-6)

Obviously, if the psychology of mathematicslearning is the study on the microscopic level, thenthe social-cultural study belongs to the macroscopiclevel; and just as J. Kilpatrick points out in his AHistory of Research in Mathematics Education ,'Researchers were taking the social and culturaldimensions of mathematics educatio n moreseriously.' (1992, p. 30)

2. From the philosophy of mathematics tomathematics education

Research in the above three directions as a wholerepre sents a new conception of mathematicseducation, whose kernel is new ideas about thequestions 'what is mathematics' and 'what it meansto know mathematics' . At just these points, wecan see clearly the important influence exerted byphilosophy of mathematics on mathematicseducation.

To explicate. philosophy of mathematics had for along time been under the tradition of 'foundationalstudies '. The common position for all the mainschools in the study of mathematics foundations,i.e, logicism, intuitioni sm and formalism, tookmathematics as a body of mathematical knowledge,and it was hoped that, by the logical analysis of theinner structures of mathematical knowledge. theycould lay a finn foundation for mathematics so thatthe problem of the soundness of mathematics couldbe solved forever.

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The researchers of the above three schools hadproduced many important results. As far as theirfin al aims were conce rned, however, they allfailed , and as time passed , a big defici ency of thefound ation al studies has become clear, i.e, itdeviates tenibly from actual mathematical activities.So, after the period of the 'the golden age' (about1890-1940), the st udy of the philosophy ofmathematics stagnated.

In the sixties, mainly under the influence of thephilosophy of science , some new phenomenaappeared in the field of philosophy of mathematics,which in turn represented a transition of the basicpositions. The new position was that mathematicsshould be mainly regarded as creative activiti es ofhuman beings rather than a specific body of fixedmathematical knowl edge. Thus, in comparisonwith the traditional view of mathemati cs, the newconception-which may be called ' the human viewof mathematics'-contains the following changes:

First, the new view emphasizes the development ofmathematics: as creative activities of human beings,mathematics is not some thing static and fossilizedbut has been changing all the time and will keep onchanging in the fu ture. Particularly, as dailymath ematical activities are concerned, they arenecessarily complicated proce sses includin gconjectures, errors and tests.

Second, the development of mathematics is notonly a process of accumulation but also includesqualit ative ch anges. Th at is, there are alsorevolutions in mathematics.

Thi rd , the human view of ma thematics alsoconfirms that mathematics consists of meaningfulactivities, so that it should not be identified as themechanical manipulation of meaningless symbols.

As the human view of mathematics represents a bigtransition of basic idea s, it has also opened newdirections for the study of the philosophy ofmathematics.

For example, there is firs tly the social-culturalapproach to mathematics. To be con crete,mathema ticians in modern society are all workingin some social envi ronment, and therefore are, infact, members of 'mathemat ical communities'. Infact, the working aim for most mathematicians is toget mathematical sta te men ts which arerepre sentable by the language uniformly accepted

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by the community, and are resolutions to thoseproblems unifonnly regarded as important orsignificant by the community, and are based onarguments or methods uniformly accepted by thecommunity. (cf. P. Kitcher, 1984) In fact , such aprescriptive role of the mathematical communitybased on individual mathematicians is just theconcrete manifestation of what might be called'mathematical culture' (in the level of graduateschool and mathematics research).

Furthermore, as mathematical researches are socialactivities, we can therefore study the impetus andlaws for the development of mathematics from ahigher level. This is to say, we can transcend anindividual's work and take the whole humansociety as a background for the study of thehistorical development of mathematics. (cf. R.Wilder, 1981 ) Obviously. such studies denote thatphilosophy of mathematics has extended from dailymathematical activities to macroscopic studies.

Also, from the microscopic view, mathematicalactivities are all mental processes. In particular, thecreation of all mathematical concepts is a process ofconstruction. To be concrete, mathematical entitiesare not objects existing in the empirical world butcreati ons of abstraction. Furthermore, in strictresearch , no matter whether the entities concernedhave or do not have empirical backgrounds, wecannot rely on intuition but on deduction from thecorresponding definitions. Therefore, the processof mathematical abstraction is, in fact, an activity of

as creative activities of humanbeings, mathematics is notsomething static and fossilized buthas been changing all the time andwill keep on changing in the future.

construction. That is, mathematical entities areconstructed by the corresponding definitions(including explicit and implicit definitions), andonly by those processes of "logical construction"can the corresponding mathematical entities betransferred from ' the inner creations of the mind' to'the ou ter independent existence'. (cf Y. Zheng,1991 b.) Furthermore, because mathematicalentities are not objects in the empirical world, thestudy of mathematical entities must include a

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,

process of 're-creation' (in comparison with ' theprimary creation' ). That is, people must actuallyconstruct the corresponding mathematical entities inthe mind, so that what had been 'objectified ' withthe aid of language can be transferred back into'inner elements of the mind'.

Putting together the above discussion of themodem developments of mathematics educationand of the philosophy of mathematics, we can seeclearly that it is modem research in the philosophyof mathematics which has offered the importantideological foundation for the new reformmovement of mathematics education in the USA.For example, the emphasis on problem solving isobviously a necessary consequenc e of the humanview of mathematics. In fact, an important starting

the distinct feat ure of the 'newmath' was that littl e attention waspaid to the actual cognitiveprocesses, of how hum an beingsthink about mathematics.

point of the new reform movement of mathematicseducation is ju st the recognition that schoolmathematics under the old tradition is not 'realmathematics', and the idea of 'foc using on problemsolving' in turn is to put students in the samesituation as mathematicians. T. Romberg says onthis point:

'For over two thousand years, mathematics hasbeen viewed as a body of infallible truth farremoved f rom the affairs and values of humanity.These views are being challenged by a growingnumber of philosophers of mathematics....Such adynamic view of ma themat ics has powerfu leducational consequences. The aims of teachingmathematics need to include the empowerment 0/learners to create their own mathematicalknowledge; ....When mathematics is seen in thisway, it needs to be studied in living contexts thatare meaningfu l and relevant to the learners,including their languages, cultures, and everydaylives, as well as their schoolbased experiences ,'(T. Romberg, 1992, p . 751)

Secondly, altho ugh 'constructiv ist ' is a newterminology in the world of mathematics education,

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it is quite familiar to philosophers of mathematics.Therefore, the 'rise' of the constructivist view onmathematical leaming and teaching can be regardedas an extension or transition from philosophy ofmathematics to mathematics education. Whatshould also be noted is that, mathematics educatorshave found important illuminations for instructionfrom modern studies of the philosophy ofmathematics and philosophy of science in general.For example, based on the discussion aboutscie ntific revolutions, especially about thetransition of 'paradigm' in philosophy of science,some mathematics educators suggest that forming'conceptual conflict' is a requisite and efficient wayfor promoting students' mathematical thoughts,especially for the correction of their wrong ideas.

Finally, the social cultural approach to mathematicseducation obviously corre sponds directly to thesocial cul tural studies of mathematics . Forexample, in correspondence with the concept of'mathematical community', mathematics educatorshave introduced the concept of ' mathematicsedu cati on com munity', whic h consis ts ofmathematics teachers, mathematics educationresearchers, directors for mathematics teacher' straining, supervisors for mathematics curriculum,makers of policies for mathematics education,designers of mathematics examinations and so on,and the main feature of a mathematics educationcommunity is also that all its members share(consciously or unconsciously) somewhat the sameconception of mathematics education.

In addition to the above discussion, what should benoted is that we can analyze the relationshipbetween the philosophy of mathematics andmathematics education in a more general sense.For example, the tradition al conception ofmathematics education reflects to a great extent the'absolute view of mathematics' (we should also seethe influence of mechanism here). Besides, it isthe foundational study mentioned above that offersthe necessary ideological foundation for the 'newmath movement' , which was seen throughout allthe western countries during the sixties . In fact,the distinct feature of the 'new math' was theemphasis on the logical structures of mathematicalknowledge and little attention was paid to the actualcognitive processes, of how human beings thinkabout mathematics. We can See here very clearlythe influence of the ' foundadonists' . The Frenchmathematician R. Thorn, while commenting on'new math' . clearly raised the following question:

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"'Modern" Mathemat ics: an Educat ional andPhi losophical Error?' And as an answer, he says ,for example,

'Set theory....is the essential litany intoned bythose who advocate the so -called modernmath ematics. Some affirm that the use of settheory permits the entire renovati on of mathematicsteach ing and that, thanks to th is change, theaverage student will be ab le to achieve mastery ofthe curriculum. Needless to say, this is pureillu sion .. .Everything con sidered , the excessiveoptimism bred by the use of set theory symbol hasits roots in a philosophical error.' (Thorn, 1971, p.75)

To summarize, we should definitely confirm thatthere is an important rel ationship betwee n thephilosoph y of mathematics and ma them aticseducation; and then, via mathematics education,philosophy of mathemati cs will exert greatinfluence on the future of mathematics.

3. Towards a philosophy o f m athematicseducation

The above discussion shows clearly the importantrel a tion ship be tween the p hilo sophy ofmathematics and mathematics education; however,at the same tim e we should not identify thephil osophy of mathematics with the theoreticalfoundation of mat hem atics educat ion. In otherwords, mathematics ed ucation should have its ownrelatively independent theoretical foundation.

In fact , every subject has its ow n history duringwhich it has formed its special field , problems andtheories. With thi s view, we ca n see clearly thedifferences between the philosophy of mathematicsand the theoretical fou nda tion of math ematicseducation:

On the one hand , the philosophy of mathematics,as philosophical ana lysis of mathem atics, has itsspecial problems. In fact , in comparison withthose problems mentioned above, the onto logy andepistemology of mathematics are of a more basicnat ure. Th e ontology of mathematics ca n bedescribed as: do mathematical en tities have anindependent existence? If the answer is ' yes' , thenwhat kind of existence is it; if the answer is ' no',what is the meaning of mathematics? On the otherhand, the ep istemology of mathemat ics is focusing

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on whether mathematical statements are a priori orempirical. The fac t that the ontology andepistemology o f math emati cs have occupiedimportant po sitions in the philosophy ofmathematics is a natural result of the speciality ofmathematics, especially i ts abs tractness (thespecia lity of mathema tics lies not only in thecontents of mathematical abstraction, but also in itsdegree and method. cf. Y. Zheng, 1991). And justfor this reason, although there have been some newdirec tions in the fiel d of the philosophy ofmathemati cs since the six ties, any sy stematictheory in the philosophy of mathematics still has togive definite answers (or detailed analysis) to theontology and epi stemology of mathematics. Infact, ju st as P. Benacerraf points out in his paperMathem atical Touh, the difficulty in the study ofphilosophy of mathematics just lies in 'the dilenunaof the onto logy and epis temology of mathematics',This is to say, those theories which are satisfactoryin the ontology always have serious deficits in theepi stemo logy ; while the others which aresatisfactory in epistemology always have defici ts inthe ontolog y. However, all the se discussions donot seem to have important implications formathematics education.( P. Benacerraf 1983)

On the other hand, the theoretical foundation ofmathem atic s education obviously should incl udethe following contents:

(1 ) The View of Mathematics. This is theanswer to the question ' what is mathematics'. Itsho uld incl ude no t on ly ana lys is about there lation ship be tween objective mathematicalknowled ge and the creative activities of the human,but also an explication of the subject (and nature)of mathematics, For example, according to themodern view, mathematics should be defined as't he science of patterns' (cf L. Steen, 1988 and Y.Zheng, 1991 ), and this definition seems to be aconfirma tion of the duality of mathematics, i.e., itis both descriptiv e and prescriptive.

(2) Th e Analysi s of the Nature orMathematical Learning. Differing from thestudy of epistemology in the ph ilosophy ofmathemat ics, the final ai m of the analysis ofmathemat ical learni ng is not to get a definiteconclusion abou t the a priori and empirical natureof mathematical statements but rather to study theac tual information process of the mind andexplic at e it s implicat ions for mathematicsedu cation . Therefore, the key question here is

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whether mathematics learning is a process of' transmission of information' centering on teachersor an activity of di scovery (re-creation) bystudents. Besides, from the social-cultural view,there is also the question whether mathem aticseducation is an isolated activity or an organic panof the whole cultural system of the human.

(3) The Aim of Mathematics Education. Asa conscious act ivit y of hum ans, mathematicseducation has its definite aim which should reflectthe features of the time, i.e., it should meet theneeds of the time and reflect the advance of scienceand technology. Particularly, we should analyzecarefully the great influ ence on math ematicseduc ation exerted by the transition from theindustrial age to the information age and the rapiddevel opment of compu ter technology. Forexample, as the information age is in some sense'the age of mathematizing' , the development of thesociety has made a higher standard for everystudent an historical necessity for mathematicseducation (cf. NRC. 1990. 1991). In addition, therapid development of computer technology has notonly offered efficient tools but also opened a newprospect for mathematics education. For example,with the help of computers, people can really befreed from the influence of the traditionalconception of mathem atics education thatemphasizes ver y routine skills , and thenconcentrate on the promotion of the students'ability in problem-solving.

By the above discussion, we can now see clearlythat there are both some important relationship anddifferences between the philosophy of mathematicsand the theoretical foundati on of mathematicseducation. What is more , it is obvious that weshould also differentiate naive conceptions ofmathematics education from systematic theories.Therefore, there is a deep need to introduce theconcept 'philosophy of mathematics education' .To be explicit, philosophy of mathematicseducat ion co nsists mainly of the followingcontents: the view of mathematics, the analysis ofthe nature of mathematics learning (and teaching),and the aim of mathematics education; and as awhole it forms the theoretical foundation formathematics education.

To make things clearer, we are going to make abrief introduction and co mment on the mostpopular view of mathematics education in China.According to this view, the theory of mathematicseducation mainly consists of the followin g threepans: the theory of mathematics curriculum, thetheory of mathematic s teaching, and the theory ofmathematics learning. 'The theoretical foundationsof the theory of mathematics education include thefoll owing subjects: d ialectica l materiali sm(phil osoph y) , ma them atic s , educ ation ,psychology, logic, and com puter science: (CaoCal-han. 1989. p. 9 ) So. the basic framework ofthis theory of mathematics education is as follows:

philosophy

I1he his tory 0 f meUlOOOlDtY 0 computerma1hematical Ima1hematics I mathematics leducation I Ipsycholoty I llotic science

I

1he theory ofmathematic" education

the theory of the theory of 1he theory ofmathematics rnatherna.tics mathematicscurriculum telll:hing le8.l'Ilinl:

Figure 1

t

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Philos ophy of mathematics education

Itheory of mathematics education

II I I I I I

1he 1heoryof 1he 1heoryo the 1heory of the 1heory of 1he 1heory of 1he 1heoryofmathematic, mathematic:s ma1hematic, problem evaluation of lIechniq.u incumculum 1Ie aJ:hiIlg learning .olving maths . edu malhs. edu.

Figure 2

However. we now know clearly that the analysisof the theoretical fo undations of mathematicseducation should no t be limited to listi ng all therelevan t subjects; instead, we should se t up its owntheoretica l foundation. i. e.• the philosophy ofmathematics education . Therefore, we are, in fac t,introd ucing the fo llowin g new th eoretica lframework for ma the ma tics educa tion ( in wh ichwe have al so made so me imp ro vement andextension of th e conte nts of th e theory ofmathematics education; however, it goes beyondthe topic of thi s paper), as shown above.

Fina lly , wha t should be emp hasized is that ,al though there is already some preliminary work inthis direction (c.f., P. Ernest, 1990), philosophyof mathematics education is still a new field waitingfor fu rth er s tud ies. We can see by the abovediscussion that the fou nd ing of a systematic theoryof philosop hy of mathe matics edu cation need scooperation between philosophers and educators .Actually, the most importan t thing is to introspectone's own co ncept of mathemati cs education, so asto tran sfer fro m the old, back ward co nce ption tothe advanced a nd sc ien ti fic conceptio n ofmathem atic s edu ca tio n. In fact, just as EverybodyCounts, which is another important document forthe new reform of mathematics education in theUSA, po ints out, the followi ng transition s ' willdom inate the process of c hange during theremainder of this century' :

Transition 1: The focus of school mathematics isshifting f rom a dua lism mission-minimalmathematics f or Ihe majority, advanced

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mathematics f or a few-i-to a singular focus on asignifi cant common core of mathematics for allstudents,

Transition 2: The teaching of mathematics isshifting f rom an authoritarian model based on"transmission 0/ knowledge" 10 a student centeredpractice featuring "stimulation oflearning."

Transition 4: The teaching of Mathematics isshifting f rom preoccupation with inculcatingroutine skills 10 developing broad basedmathematical power. (p . 81-82)

These transitions, of course. can not be carried outspontaneous ly in practice; just the opposite,' naive ness ' in philosophy (one frequent fonn of'naiveness ' is the ignoring of philosophy) alwaysleads people to become slaves of some 'modem'.but a t the sa me time 'bad' philosophy. Forexample, what is called 'the radical constructivistview ', which seems to be a 'modern fashion' in theworld of mathema tics education in the USA, is, infact, a revision of intuitionism in the philosophy ofmathem atics. And as intu itionism necessarily leadsto mathematical mysticism' and 'mathematicalsolipsism ' by its den ial of the representability andobjecti vity of mathematics, this philosophical viewhas already been widely criticized. Obviously, itshows more clearly the importance of the study ofphilosop hy of mathematics education; and it in turncan also be regarded as an impetus for mathematicseducat io n to th e further development of thephilosophy of mat hemati c s and philosophy ingeneral.

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Kircher, P., 1984, The Nature of MathematicalKnQwledge. Oxford Univer. Press;

National Council of Teachers of Mathematics,19 80, An Ag enda for Acti on:RecQmmendations for School Mathematicsof the 1980.;

Referenc es

Nati onal Council of Teachers of Mathematics,1989, Curri culum a nd Ev aluationStandards for School Mathematics.

National Council of Teachers of Math ematics,1991, Professional Standards fQrTeachingMathematics:

40 HMN Journal #9

Wilder , R.,1 981, Mathem at ics as a CylturalSystem, Pergamon Press;

Zheng, Y.,1985, Introduction to Methodology ofMat he ma tics (in Chinese) , Zhej iangEduca tional Publishing House, P.R.C.;

Zheng , Y. & Xi• • J.•1986. Western Philoso phy ofMath em at ic s (in Chines e), People ' spublishing House. P.R.C.;

Zhen g, Y.,1990, New Theories in Philosophy ofM ath em atic s (in Chinese), JiangsuEducationa l Publi shing House, P.R.C.

Zheng, Y., 1991a, Methodology of Mathematics(in Chi nese) , Gu an gxi EducationalPubli shing House, P.R.C.;

Zheng, Y., 1991b, Phjlosqpby of Mathematics inChina, PhUosop bia Mathematica, Vol. 6,No. 2. 174-199.

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HMN Journal #9

Poems by Lee Goldstein

Plighted Symbolism

Through the credential nonverbalityAre theorems of that that is of the not benamed abstraction,And where the verbality, thence the symbolism seems may be to willing,While the masters of the symbolismKnow that unwillingness and yet can act them;For the locus of what is not benamedBodes mathematical,And such constructivism is believedIn this very alterity.

1993

The Imagination

The infiniteth inbeing of desire expressed objectively ,For instance, 'the set of all sets which do not include themselves',Implies an ineluctable phenomenonThat precludes mental escape,Unless there is admitted the glamourous searchOf the not at the object,Bur of a living, instead, past the paradoxes implicit in desired (or undesired) objectsWhere truthful objectedness arises, identically,Only upon a nonce imagination of the "things ideal."

1993

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