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Introduction to the Special Issue onPersonal Mathematical Knowledge inthe Work of Teaching/Introduction aunuméro spécial sur les connaissancesmathématiques personnelles dans letravail des enseignantsAnne Watson a & Helen Chick ba University of Oxford , Oxford , United Kingdomb University of Tasmania , Hobart , AustraliaPublished online: 31 May 2013.

To cite this article: Anne Watson & Helen Chick (2013) Introduction to the Special Issue onPersonal Mathematical Knowledge in the Work of Teaching/Introduction au numéro spécial sur lesconnaissances mathématiques personnelles dans le travail des enseignants, Canadian Journal ofScience, Mathematics and Technology Education, 13:2, 111-120, DOI: 10.1080/14926156.2013.784831

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CANADIAN JOURNAL OF SCIENCE, MATHEMATICSAND TECHNOLOGY EDUCATION, 13(2), 111–120, 2013Copyright C© OISEISSN: 1492-6156 print / 1942-4051 onlineDOI: 10.1080/14926156.2013.784831

Introduction to the Special Issue on Personal MathematicalKnowledge in the Work of Teaching

Anne was on a crowded bus looking over the shoulder of a young student who was trying to solvea geometry problem: to select, from several choices, the ratio of the radii of the inner and outercircles of a regular hexagon. The student was clearly stuck and going along irrelevant directions.When Anne asked, “How are you tackling this?” the young woman looked surprised and thentook out her earphones and said she had let the radius of the inner circle be 1. Anne said, “Goodchoice” (while thinking it would have been much more use to label the outer radius), and askedif that had been helpful. A discussion of possible approaches ensued, with Anne trying not togive any direct clues. One of the possible answers had

√3 in it and Anne asked if that suggested

anything. The student talked about angles of 60 degrees, and Anne said, “You have told meeverything you need to know to resolve this.” Before she left the bus, Anne said, “Whenever youlabel anything in a geometry diagram, choose a label that applies to as many aspects as possible.”

Observing Anne’s actions in this situation, it is clear that there are two personas acting inconcert, bringing to bear knowledge from a number of different areas. There is Anne-as-teacher,with a desire not to give answers or methods but “to be of some use”; indeed, the desire toact as a teacher overcame the norms of reticence on English buses. There is also Anne-as-mathematician, rapidly analyzing the problem, likely drawing on past knowledge of similarproblems, recognizing the greater optimality of choosing the outer radius, and appreciatingthe significance of

√3. Drawing on knowledge of geometry, of working mathematically, and of

pedagogical strategies for helping without being explicit, this composite Anne attempts to supportthe student in both solving the problem at hand and building strategies for working mathematicallyin future situations—although Anne’s departure from the bus prevents us knowing with whatsuccess.

A student of ours once described good teaching as “being alongside you during the methodinstead of only at the end.” To be an effective support “during the method” requires knowledge andexperience of doing mathematics and ways of articulating mathematical enquiry. Polya (1957)did this from his experience by naming a number of useful strategies for working mathematically.Although teachers could merely borrow his extensive list and offer it to students, the greaterthe distance between Polya’s words and teachers’ personal mathematical experience, the lesslikely they are to use his heuristics spontaneously and appropriately with students as featuresof doing mathematics in flow (Csikszentmihalyi, 1988). This distancing can be found in manymathematics curricula, too: for example, the term generalize in a curriculum is seldom applied todescribing mathematical structures as they arise in enquiry but often to the rather more specificcase of finding a functional formula for terms in a sequence.

Experiential knowledge of mathematical enquiry—evident in Anne’s interaction with thestudent on the bus despite the brevity of the encounter—is an aspect of being mathematical thatwe find absent from many descriptions of MKfT (mathematical knowledge for teaching), and yet

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it underpins the role of the teacher in enculturating mathematical ways of being. It is necessary, forexample, for teachers to make judgements of quality (Star & Stylianides, 2013) or to evaluate in-the-moment possibilities (Mason & Davis, 2013). It is a component of teaching and can contributeto the implicit provision of repeated mathematical experiences (see Watson & Harel, 2013) ratherthan depending on explicit repetitions that might be provided in a curriculum. It enables teachersto act on metamathematical issues, such as precision, ambiguity, reasoning, and aesthetics (Zazkis& Leikin, 2010). This is not merely about knowledge beyond the curriculum—since the relatedcontent might be well within the school curriculum—rather, it is also about the teacher’s personalfluency with the modes of mathematical enquiry.

This Special Issue arose out of a desire to examine the way in which knowledge aboutmathematics as a discipline impacts on the work of teaching. A teacher in Zazkis and Leikin(2010) says of his use of advanced mathematical knowledge (AMK):

I have found that I use AMK a lot in my teaching, from interjecting advanced or interesting “pieces”to full blown lessons based on what I have learned. I would say that rarely a day has gone when Ihave not used AMK. (p. 270)

Although many teachers in their study said similar things about their use of AMK, often theydid not provide specific examples. Zazkis and Leikin concluded their paper with a call forthe research community to pursue this, to find content-specific examples in which advancedmathematical knowledge contributes to teaching at the school level.

The articles in this collection go some way toward answering this call, by examining content-specific manifestations of mathematical modes of enquiry that arise from teachers’ personaladvanced mathematical experiences. They are largely reports of experience and hence the readershould treat them as phenomenological. This Special Issue advances from Even’s (2011) concep-tualization of the field—that is, the study of relations between teachers’ advanced mathematicalknowledge and their teaching—and also from Zazkis and Leikin’s (2010) aforementioned appealfor more knowledge of concept-specific connections between teachers’ advanced mathematicalknowledge (AMK) and their teaching. Our progress toward this is more than marginal, and weclaim that the articles indicate directions for future observations.

Even (2011) proposed three branches to the field of research into mathematical knowledge andits role in teaching. The first strand involves the conceptualization of teachers’ subject knowledge,identifying what is necessary and adequate. As we have already said, many such categorizationsdo not include the teacher’s fluency with mathematical modes of enquiry. Furthermore, much ofthe literature in this area relates to those intending to teach at elementary level without muchadvanced mathematics and on the kinds of courses they are offered. As yet, there is little researchfocusing on those who are teaching to the absolute limits of their own personal knowledge, or onthose with degree-level mathematics or beyond, and their application of AMK in teaching.

A second strand involves the empirical examination and testing of teachers’ knowledge—again,most often at the elementary level—showing that college-level mathematics on its own does notadequately support the knowledge needed to teach it well, thus giving rise to the question ofwhat else is required. These two strands combine to provide a plethora of studies about methodsof advancement of teachers’ knowledge for the mathematics in the school curriculum, the thirdstrand of research identified by Even (2011).

While it is well known that knowing advanced mathematics on its own is not an adequatepreparation for teaching mathematics, the converse question has not been posed extensively. As

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Even put it: “Is there a need for advanced mathematics studies in the professional education anddevelopment of secondary school mathematics teachers?” (2011, p. 942).

Clearly, many writers think there is a need, as do many course leaders who set up courses in,say, proof, number theory, problem solving, and so on for undergraduate prospective teachersand in-service professional learning programs (see, e.g., Rowland & Zazkis [2013], who reporton expecting prospective primary teachers to engage with number theory; and Barton & Paterson[2013], who report on several programs for teachers with advanced mathematics themes).

Toward answering Even’s (2011) question about need, we set up this Special Issue to openup for examination a fourth strand of research into the role of mathematical knowledge: that offinding out how teachers with advanced knowledge use it in their teaching. We look for evidencethat knowing significantly more than you are teaching and using the modes of enquiry of advancedmathematics enhances the teaching and learning of mathematics in profound ways.

OUR ARTICLES

All six articles in this collection address fundamental questions about the nature of advancedknowledge and the nature of using that knowledge in teaching. The model of thinking aboutknowledge as a precursor to teaching seems very limited, as Chick and Stacey (2013), Wat-son and Harel (2013), and Rowland and Zazkis (2013) all report the emergence of advancedmathematical awareness during the flow of teaching—an observation that is theorized by Masonand Davis (2013). In Chick and Stacey’s (2013) cases, mathematical knowledge is drawn on torespond to unexpected events, but they highlight that necessary knowledge cannot be relied onto appear on demand even if the teacher has appropriate qualifications and a pedagogic approachto application of mathematics is necessary for fluent action. In Rowland and Zazkis (2013),advanced knowledge is seen as necessary to deal well with some incidents, and they furtherhighlight that personal mathematical knowledge can guide pedagogical implementation in bothsubstantive and syntactic ways. Syntactic here uses the meaning ascribed to Schwab (1978); thatis, fluency in the nature of subject-based enquiry, the acceptable warrants for action and veracityin the subject, and the transformations and representations involved in the subject. Indeed, syn-tactic knowledge—expertise in mathematical modes of enquiry—seems to be more important intheir examples than substantive higher mathematical knowledge. In Watson and Harel (2013),the teachers’ substantive knowledge flows throughout their teaching and is present as awarenessin the provision of repeated experience of both the content and the norms—what might be calledenculturation into particular mathematical viewpoints. This suggests the idea of syntactic fidelity,in which the activities and discussion in the classroom, as enacted by the teacher and developedin the students, reflect the nature of actual mathematical work. (This can be seen as an analogueof epistemic fidelity, as used by Meira [1998] to refer to the ways in which teaching materialscan clearly reflect the concepts they are intended to represent.) In Mason and Davis’s (2013)article it becomes clear that, although mathematics knowledge for teaching can be seen as a kindof applied mathematics (Bass, 2005; Chick & Stacey, 2013; Stacey, 2006), it flows both whenteachers are acting as mathematicians and also when they are acting as mathematics teachers—anintertwining illustrated in the anecdote with which we started this editorial.

The transformation of personal mathematical knowledge into teacher knowledge can takedifferent forms. Star and Stylianides (2013) show that transformations can focus on quality or

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on type of knowledge. Whereas type resonates to some extent, but not totally, with distinctionssuch as procedural/conceptual, quality relates more closely to our earlier comments on syntacticfidelity and to the examples throughout this Special Issue where teachers appear to be shapingstudents’ mathematical behaviors or mathematical habits of mind. For Star and Stylianides (2013),it is informed by the interplay of teachers’ mathematical knowledge and their knowledge of theeducation process. In Watson and Harel (2013), the mathematical knowledge of the teachersshapes the students’ educational experiences beyond individual lessons or lesson sequences andover time by providing purpose, need, development, and repetition of key experiences.

Much of the job that usually falls to editors—to demonstrate coherence among diversepapers—is done in the article by Mason and Davis (2013), who use features of their ownexperience to provide a bridge that arches between personal mathematical knowledge and in-the-moment mathematical behavior via fluency with the syntactics of mathematical structures andenquiry methods. Barton and Paterson (2013) provide several examples of teachers becomingarticulate and enthusiastic about that bridging process and the teachers’ reports of the value ofstudying higher mathematics while teaching that point to significant affective as well as syntacticand substantive benefits.

WHAT NEXT?

The work in these articles highlights the role of advanced mathematical knowledge in the workof teaching. The accounts contrast strongly with the self-report of two teachers in Zazkis andLeikin’s study (2010, p. 268):

I don’t think I use any AMK in my teaching. I would say that AMK is non-essential. (Nick-1)

I am able to teach secondary school mathematics without AMK. (Annie-1)

It may well be true that it is possible to teach without using content knowledge from advancedstudy, particularly if teaching focuses solely on procedures and routine questions. However, thiscollection shows that it may not be possible to teach well, and with syntactic fidelity, without thefluency of connection between the mathematical self and the teacher self. The questions that arisefrom this collection are rather different from the “What knowledge?” type. We are led to ask,How can those who have advanced knowledge be educated to transform the mathematician selfso that it also incorporates a mathematics teacher identity? Furthermore, how should advancedmathematical education be provided that influences not only mathematical knowledge-in-advance(about which much has been written, although mainly about elementary teaching) but alsomathematical knowledge-in-action?

Anne WatsonUniversity of OxfordOxford, United Kingdom

Helen ChickUniversity of TasmaniaHobart, Australia

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REFERENCES

Barton, B., & Paterson, J. (2013). Does mathematics enhance teaching? Does summer hiking tone winter thighs? CanadianJournal of Science, Mathematics and Technology Education, 13(2), 198–212.

Bass, H. (2005). Mathematics, mathematicians, and mathematics education. Bulletin of the American MathematicalSociety, 42, 417–430.

Chick, H., & Stacey, K. (2013). Teachers of mathematics as problem-solving applied mathematicians. Canadian Journalof Science, Mathematics and Technology Education, 13(2), 121–136.

Csikszentmihalyi, M. (1988). The flow experience and its significance for human psychology. In M. Csikszentmihalyi &I. Csikszentmihalyi (Eds.), Optimal experience: Psychological studies of flow in consciousness (pp. 15–35).Cambridge, England: University of Cambridge.

Even, R. (2011). The relevance of advanced mathematics studies to expertise in secondary school mathematics teaching:Practitioners’ views. ZDM Mathematics Education, 43, 941–950.

Mason, J., & Davis, B. (2013). The importance of teachers’ mathematical awareness for in-the-moment pedagogy.Canadian Journal of Science, Mathematics and Technology Education, 13(2), 182–197.

Meira, L. (1998). Making sense of instructional devices: The emergence of transparency in mathematical activity. Journalfor Research in Mathematics Education, 29(2), 121–142.

Polya, G. (1957). How to solve it (2nd ed.). Princeton, NJ: Princeton University Press.Rowland, T., & Zazkis, R. (2013). Contingency in the mathematics classroom: Opportunities taken and opportunities

missed. Canadian Journal of Science, Mathematics and Technology Education, 13(2), 137–153.Schwab, J. J. (1978). Education and the structure of the disciplines. In I. Westbury & N. J. Wilkof (Eds.), Science,

curriculum and liberal education (pp. 229–272). Chicago, IL: University of Chicago Press.Stacey, K. (2006, December). What is mathematical thinking and why is it important? Paper presented at the

Tsukuba international conference 2007 “Innovative Teaching Mathematics through Lesson Study (II)” —Focusingon Mathematical Thinking, Tokyo & Sapporo, Japan. Retrieved from http://www.criced.tsukuba.ac.jp/math/apec/apec2007/paper pdf/Kaye%20Stacey.pdf

Star, J. R., & Stylianides, G. J. (2013). Procedural and conceptual knowledge: Exploring the gap between knowledgetype and knowledge quality. Canadian Journal of Science, Mathematics and Technology Education, 13(2), 169–181.

Watson, A., & Harel, G. (2013). The role of teachers’ knowledge of functions in their teaching: A conceptual approachwith illustrations from two cases. Canadian Journal of Science, Mathematics and Technology Education, 13(2),154–168.

Zazkis, R., & Leikin, R. (2010). Advanced mathematical knowledge in teaching practice: Perceptions of secondarymathematics teachers. Mathematical Thinking and Learning, 12(4), 263–281.

Introduction au numero special sur les connaissancesmathematiques personnelles dans le travail des enseignants

Anne se trouvait dans un autobus bonde et observait une jeune etudiante qui tentait de resoudreun probleme de geometrie consistant a determiner, parmi plusieurs possibilites, le rapport entrele rayon des cercles interieur et exterieur d’un hexagone regulier. De toute evidence, l’etudianteetait arrivee a une impasse et poursuivait des directions non pertinentes. Lorsque Anne lui ademande : « Comment abordes-tu ce probleme? », la jeune femme a d’abord semble surprise et,retirant ses ecouteurs, elle a dit qu’elle avait pose que le rayon du cercle interieur etait 1. « Bonchoix », dit Anne (meme si elle estimait qu’il aurait ete beaucoup plus utile de qualifier d’abordle rayon exterieur), et lui a demande si cette decision avait ete utile. Suivit une discussion surdifferentes possibilites d’approches, au cours de laquelle Anne evitait de donner a l’etudiante desindices trop directs. L’une des reponses possibles contenait

√3, et Anne a demande si cela lui

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suggerait quelque chose. L’etudiante a parle d’angles de 60 degres, ce a quoi Anne a repondu : «Tu m’as donne tous les elements necessaires pour resoudre ce probleme ». Avant de descendre dubus, Anne a ajoute : «Chaque fois que tu qualifies un element dans un diagramme geometrique,essaie de choisir une categorie qui s’applique au plus grand nombre d’aspects possibles ».

Si on observe le comportement d’Anne dans cette situation, on s’apercoit que deux personnesentrent en jeu et agissent de concert, se fondant sur des connaissances d’origines differentes. Il ya d’une part Anne l’enseignante, qui veut eviter de donner des reponses ou des methodes, maisqui veut « etre utile »; en effet, son desir d’enseigner est ici plus fort que la reticence typiquedes passagers d’un autobus anglais. D’autre part il y a aussi Anne la mathematicienne, qui atout de suite analyse le probleme, en se fondant probablement sur ses connaissances passeesde problemes semblables, ce qui lui a permis de reconnaıtre la priorite theorique du rayonexterieur, et d’apprecier l’importance de

√3. Grace a ses connaissances en geometrie, a sa pensee

mathematique et a ses strategies pedagogiques visant a donner de bonnes pistes sans toutefoisetre trop explicite, cette Anne composite a tente de soutenir l’etudiante a la fois pour la resolutiondu probleme et pour l’elaboration de strategies mathematiques susceptibles d’etre appliquees ad’autres problemes futurs (meme si l’histoire ne dit pas avec quel degre de succes puisque Anneest descendue de l’autobus).

L’un de nos etudiants decrit un bon enseignant comme quelqu’un qui « est a tes cotes pen-dant toute la methode et non pas seulement a la fin ». Pour etre un soutien efficace « pen-dant toute la methode », il faut avoir les connaissances necessaires et l’experience du travailmathematique, et savoir expliciter la demarche de l’investigation mathematique. C’est ce quefait Polya (1957) a partir de son experience lorsqu’il donne quantite de strategies utiles dans letravail mathematique. S’il est vrai que les enseignants pourraient se contenter d’emprunter cetteliste pour la proposer en vrac a leurs etudiants, il est egalement vrai que plus la distance estgrande entre les mots de Polya et l’experience mathematique personnelle des enseignants, moinsil est probable que ceux-ci se serviront spontanement de son heuristique de facon appropriee avecleurs etudiants dans le cadre de leur enseignement (Csikszentmihalyi, 1988). Cette distance semanifeste aussi dans de nombreux curriculums de mathematiques : par exemple, dans les cur-riculums le terme « generaliser » est rarement utilise pour decrire les structures mathematiquesau fur et a mesure qu’elles se presentent au cours de la demarche d’investigation, mais plussouvent lorsqu’il s’agit simplement de trouver une formule fonctionnelle pour les termes d’unesequence.

L’experience pratique de l’investigation mathematique—manifeste dans l’exemple del’interaction entre Anne et l’etudiante dans l’autobus (malgre la courte duree de cetteinteraction)—est un aspect du travail mathematique qui est absent de bien des descriptionsdes connaissances mathematiques necessaires a l’enseignement, et pourtant il est a la base durole de l’enseignant, qui doit eduquer aux activites mathematiques. Il est necessaire, par exem-ple, que les enseignants emettent des jugements de qualite (Star & Stylianides, 2013) et qu’ilsevaluent les possibilites qui se presentent spontanement en classe (Mason & Davis, 2013). Celafait partie de l’enseignement et contribue a fournir de nombreuses experiences mathematiquesrepetees (voir Watson & Harel, 2013), evitant ainsi de dependre des repetitions explicites sus-ceptibles de figurer dans le curriculum. Cela donne aux enseignants l’occasion d’agir sur desquestions de metamathematiques, par exemple la precision, l’ambiguıte, le raisonnement etl’esthetique (Zazkis & Leikin, 2010). Il ne s’agit pas simplement de connaissances qui vontau-dela du curriculum—car ces contenus connexes peuvent tres bien figurer dans les curriculums

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scolaires—, il s’agit aussi de l’aisance personnelle des enseignants en matiere d’investigationmathematique.

L’idee de ce numero special est nee d’un desir d’analyser la facon dont les connaissances surles mathematiques comme discipline influencent le travail pedagogique. Un enseignant cite parZazkis et Leikin (2010) raconte comment il utilise ses connaissances mathematiques avancees :

J’ai constate que j’utilise beaucoup mes connaissances en mathematiques avancees dans mon en-seignement : parfois j’insere un « commentaire » avance ou interessant, parfois je donne des leconscompletes basees sur ce que je sais. Je dirais que je passe rarement une journee sans me servir desmathematiques avancees. (p. 270)

De nombreux enseignants cites dans cette etude font des affirmations du meme type sur leur utili-sation de connaissances mathematiques avancees, mais peu d’entre eux fournissent des exemplesprecis. Zazkis et Leikin concluent leur article par un appel a la communaute scientifique afinqu’elle poursuive cette avenue de recherche, pour trouver des exemples de contenus specifiquesou les connaissances mathematiques avancees contribuent a l’enseignement au niveau scolaire.

Les articles de ce numero fournissent des elements de reponse a cet appel, en analysantles manifestations des demarches d’investigation mathematique qui naissent de l’experiencepersonnelle des mathematiques avancees chez les enseignants. Il s’agit principalement de comptesrendus d’experiences, c’est pourquoi il convient de les traiter comme phenomenologiques. Cenumero special part de la conceptualisation d’Even (2011)—c’est-a-dire l’etude des rapportsentre les connaissances mathematiques avancees des enseignants et leur enseignement—et ausside l’appel lance par Zazkis et Leikin’s (2010) mentionne plus haut. Nous estimons que notrecontribution a ces objectif n’est pas indifferente, et nous affirmons que les articles publies iciindiquent les directions a prendre dans les observations futures.

Even (2011) propose de diviser en trois branches le champ de recherche sur le role quejouent les connaissances mathematiques dans l’enseignement. La premiere se penche sur laconceptualisation des connaissances des enseignants sur leur sujet, pour determiner ce qui estnecessaire et adequat. Comme nous l’avons deja dit, souvent ces categorisations ne tiennent pascompte de l’aisance de l’enseignant en matiere de demarche d’investigation mathematique. Deplus, une bonne partie de la litterature sur la question porte sur les enseignants qui se destinentau niveau elementaire (sans mathematiques avancees) et sur le type de cours qu’on leur propose.A ce jour, il y a peu d’etudes centrees sur ceux qui poussent leur enseignement jusqu’aux limitesde leurs connaissances personnelles, ni sur ceux qui sont de niveau universitaire et qui appliquentleurs connaissances avancees a l’enseignement.

La deuxieme branche porte sur l’analyse empirique et l’evaluation des connaissances desenseignants—encore une fois surtout au niveau elementaire—et montrent que les mathematiquesde niveau preuniversitaire ne suffisent pas a elles seules a fournir le niveau de connaissancesnecessaires pour les enseigner de facon adequate, ce qui souleve la question de ce qu’il faudraity ajouter. Ensemble, ces deux branches fournissent de nombreuses recherches sur les methodesvisant a perfectionner les connaissances des enseignants sur les mathematiques qui figurent dansle curriculum, soit la troisieme avenue de recherche identifiee par Even (2011).

S’il est vrai que la connaissance des mathematiques avancees ne constitue pas a elle seule unepreparation suffisante pour l’enseignement des mathematiques, la question inverse n’a jamaisete examinee en details. Comme le demande Even : « La recherche en mathematiques avancees

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est-elle necessaire pour la formation professionnelle et le perfectionnement des enseignants desmathematiques au secondaire ? » (2011, p. 942).

Il est clair que de nombreux chercheurs estiment cette recherche necessaire, tout comme lefont de nombreux professeurs qui etablissent des cours sur, par exemple, les preuves, la theoriedes nombres, la resolution de problemes, etc. dans les cours de premier cycle destines aux futursenseignants et dans les cours de perfectionnement professionnel (voir par exemple Rowland &Zazkis [2013], qui presentent le cas de futurs enseignants de niveau elementaire qui s’attaquenta la theorie des nombres, et Barton & Paterson [2013], qui presentent plusieurs programmes deformation des maıtres incluant des themes de mathematiques avancees).

Sur la question de necessite soulevee par Even (2011), nous avons mis sur pied ce numerospecial dans le but d’ouvrir une quatrieme avenue de recherche sur le role des connaissancesmathematiques, soit celle de determiner comment les enseignants se servent de leurs connais-sances avancees dans leur enseignement. Nous cherchons des preuves que le fait d’en savoir beau-coup plus que ce qu’on enseigne, et d’utiliser les demarches d’investigation des mathematiquesavancees, ameliore fondamentalement l’enseignement et l’apprentissage des mathematiques.

NOS ARTICLES

Les six articles de ce numero abordent des questions fondamentales sur la nature des connais-sances avancees, et sur l’utilisation de ces connaissances dans l’enseignement. Le modele depensee qui veut que les connaissances soient le precurseur de l’enseignement semble plutotlimite, comme l’indiquent Chick et Stacey (2013), Watson et Harel (2013), ainsi que Rowland etZazkis (2013), lorsqu’ils discutent de l’emergence d’une conscience des mathematiques avanceespendant l’activite d’enseignement—une observation theorisee par Mason et Davis (2013). Dansle cas de Chick et Stacey (2013), les connaissances mathematiques sont une ressource permettantde reagir aux imprevus, mais elles soulignent le fait que les connaissances necessaires ne sontpas toujours disponibles sur demande, meme si l’enseignant a les qualifications appropriees,et qu’une approche pedagogique pour l’application des mathematiques est necessaire pour agirsur le moment. Pour Rowland et Zazkis (2013), les connaissances avancees sont vues commenecessaires pour bien reagir a certains incidents, et les auteurs soulignent que les connaissancesmathematiques personnelles peuvent guider l’application pedagogique aussi bien de facon sub-stantive que syntactique. Le mot « syntactique » est utilise ici au sens que lui donne Schwab(1978), et renvoie a l’aisance de l’enseignant en matiere d’investigation, de justifications pouragir, de veracite, de transformations et de representations du sujet. En effet, il semble que lesconnaissances syntactiques—ou l’experience de l’investigation mathematique—soit plus impor-tante dans leurs exemples que les connaissances mathematiques vraiment plus avancees. ChezWatson et Harel (2013), les connaissances des enseignants transparaissent dans tout leur en-seignement et se manifestent comme conscience au cours de l’experience repetee des contenus etdes normes—ce qu’on pourrait appeler la diffusion de points de vue mathematiques particuliers.Ceci renvoie a l’idee de fidelite syntactique, selon laquelle les activites et les discussions enclasse, telles que proposees par l’enseignant et developpees chez les etudiants, refletent la naturedu travail mathematique reel. (Il s’agit d’une idee analogue a celle de fidelite epistemique, telleque l’utilise Meira [1998] pour parler des facons dont les manuels pedagogiques peuvent refleterclairement les concepts qu’ils sont censes representer). Dans l’article de Mason et Davis (2013)

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il apparaıt evident que, meme si les connaissances mathematiques necessaires pour enseignerpeuvent etre vues comme une forme de mathematiques appliquees (Bass, 2005; Chick & Stacey,2013; Stacey, 2006), elles entrent en jeu aussi bien lorsque les enseignants agissent en tant quemathematiciens que lorsqu’ils agissent comme enseignants de mathematiques—un double rolequ’illustre bien l’anecdote avec laquelle nous avons commence cet editorial.

La transformation des connaissances mathematiques personnelles en connaissancespedagogiques peut prendre differentes formes. Star et Stylianides (2013) montrent que les trans-formations peuvent etre centrees sur la qualite ou sur le type de connaissances. Bien que le mot« type » soit lie en partie, mais non completement, aux distinctions entre le procedural et leconceptuel, le terme de « qualite » est plus intimement lie a nos commentaires precedents surla fidelite syntactique, ainsi qu’aux exemples qu’on trouve tout au long de ce numero lorsqueles enseignants visent a former le comportement mathematique des etudiants ou leurs habitudesde pensee mathematique. Pour Star et Stylianides (2013), cette qualite depend entre autres del’interaction entre les connaissances mathematiques des enseignants et leur connaissance des pro-cessus educatifs. Chez Watson et Harel (2013), les connaissances mathematiques des enseignantsinfluencent et faconnent l’experience formative des etudiants au-dela des cours individuels ou dessequences de cours, et a long terme en fournissent les objectifs, les besoins, le developpement etla repetition d’experiences cles.

Une bonne part du travail qui revient normalement aux redacteurs—soit celui de mettre enevidence la coherence entre les differents articles—est realisee ici par l’article de Mason et Davis(2013), qui se servent d’elements tires de leur propre experience pour jeter un pont entre les con-naissances mathematiques personnelles et les activites mathematiques sur le moment, par le biaisde la syntactique des structures mathematiques et des methodes d’investigation. Barton et Paterson(2013) fournissent pour leur part plusieurs exemples d’enseignants qui s’enthousiasment pour ceprocessus, et les comptes rendus des enseignants sur la valeur de l’etude des mathematiques plusavancees pendant la pratique de l’enseignement indiquent qu’elle apporte des benefices affectifset syntactiques substantiels.

ET MAINTENANT?

Le travail presente dans ces articles souligne le role des connaissances mathematiques avanceesdans la pratique de l’enseignement. Les comptes rendus sont en net contraste avec les rapportsfournis par deux enseignants dans l’etude de Zazkis et Leikin (2010, p. 268):

Je crois que je ne me sers jamais de mes connaissances en mathematiques avancees dans monenseignement. Je dirais que ces connaissances ne sont pas essentielles. (Nick-1)

Je suis en mesure d’enseigner les mathematiques au secondaire sans me servir de mes connaissancesen mathematiques avancees. (Annie-1)

Il est probablement possible d’enseigner les mathematiques sans utiliser ses connaissances decontenus provenant des etudes avancees, en particulier si l’enseignement est centre sur les ques-tions et les processus de routine. Toutefois, cette serie d’articles montre qu’il n’est peut-etre paspossible de bien enseigner, en respectant la fidelite syntactique, sans qu’il y ait communicationfluide entre le soi mathematicien et le soi enseignant. Les questions que souleve ce numero sont

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bien differentes de celles du type « Quelles connaissances ? ». On se demande plutot commentceux qui ont des connaissances avancees peuvent etre eduques a transformer leur identite demathematiciens de facon a ce qu’elle integre aussi une identite de pedagogue. D’autre part,comment peut-on fournir une formation en mathematiques avancees susceptible d’influencer nonseulement les connaissances mathematiques « donnees a l’avance » (au sujet desquelles on abeaucoup ecrit, bien que surtout en rapport avec l’enseignement au niveau elementaire), maisaussi les connaissances mathematiques « en action » ?

Anne WatsonUniversite d’OxfordOxford, Royaume-Uni

Helen ChickUniversite de TasmanieHobart, Australie

REFERENCES

Barton, B., & Paterson, J. (2013). Does mathematics enhance teaching? Does summer hiking tone winter thighs? CanadianJournal of Science, Mathematics and Technology Education, 13(2), 198–212.

Bass, H. (2005). Mathematics, mathematicians, and mathematics education. Bulletin of the American MathematicsSociety, 42, 417–430.

Chick, H., & Stacey, K. (2013). Teachers of mathematics as problem-solving applied mathematicians. Canadian Journalof Science, Mathematics and Technology Education, 13(2), 121–136.

Csikszentmihalyi, M. (1988). The flow experience and its significance for human psychology. In M. Csikszentmihalyi &I. Csikszentmihalyi (Eds.), Optimal experience: Psychological studies of flow in consciousness (pp. 15–35).Cambridge, England: University of Cambridge.

Even, R. (2011). The relevance of advanced mathematics studies to expertise in secondary school mathematics teaching:Practitioners’ views. ZDM Mathematiques Education, 43, 941–950.

Mason, J., & Davis, B. (2013). The importance of teachers’ mathematics awareness for in-the-moment pedagogy.Canadian Journal of Science, Mathematics and Technology Education, 13(2), 182–197.

Meira, L. (1998). Making sense of instructional devices: The emergence of transparency in mathematics activity. Journalfor Research in Mathematics Education, 29(2), 121–142.

Polya, G. (1957). How to solve it (2nd ed.). Princeton, NJ: Princeton University Press.Rowland, T., & Zazkis, R. (2013). Contingency in the mathematics classroom: Opportunities taken and opportunities

missed. Canadian Journal of Science, Mathematics and Technology Education, 13(2), 137–153.Schwab, J. J. (1978). Education and the structure of the disciplines. In I. Westbury & N. J. Wilkof (Eds.), Science,

curriculum and liberal education (pp. 229–272). Chicago, IL: University of Chicago Press.Stacey, K. (2006, December). What is mathematics thinking and why is it important? Paper presented at the

Tsukuba international conference 2007 “Innovative Teaching Mathematics through Lesson Study (II)” —Focusingon Mathematics Thinking, Tokyo & Sapporo, Japan. Retrieved from http://www.criced.tsukuba.ac.jp/math/apec/apec2007/paper pdf/Kaye%20Stacey.pdf

Star, J. R., & Stylianides, G. J. (2013). Procedural and conceptual knowledge: Exploring the gap between knowledgetype and knowledge quality. Canadian Journal of Science, Mathematics and Technology Education, 13(2), 169–181.

Watson, A., & Harel, G. (2013). The role of teachers’ knowledge of functions in their teaching: A conceptual approachwith illustrations from two cases. Canadian Journal of Science, Mathematics and Technology Education, 13(2),154–168.

Zazkis, R., & Leikin, R. (2010). Advanced mathematics knowledge in teaching practice: Perceptions of secondarymathematics teachers. Mathematics Thinking and Learning, 12(4), 263–281.

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