# Teach the Mathematics of Mathematicians

## Abstract

**:**

## 1. Statement of the Myth

## 2. The Community of Disciplines

Her juxtaposition of English and mathematics works well for me, as in the school curriculum these are the two disciplines that sit at the centre of the school program. English also has a strong generative component typically lacking in school mathematics. Students in English understand that they are expected to write creatively, but in math class they will claim that it is unfair to be asked to solve a problem they have not seen before. To discover why these two subjects differ in such a significant manner, I take a closer look at the nature of the high school mathematics curriculum.There are other indications that math is different from all other subjects. When we ask students what math is, they will typically give descriptions that are very different from those given by experts in the field. Students will typically say it is a subject of calculations, procedures, or rules. But when we ask mathematicians what math is, they will say it is the study of patterns that is an aesthetic, creative, and beautiful subject. Why are these descriptions so different? When we ask students of English literature what the subject is, they do not give descriptions that are markedly different from what professors of English literature would say.[3] (pp. 21–22)

## 3. The High School Math Curriculum

## 4. The Wisdom of the Past

Here Whitehead is clearly not denigrating algebra or the plays of Shakespeare, but he despairs of the narrow technical version that typically dominates the classroom. Whitehead certainly understands the critical role that technical mastery plays in learning and indeed in any creative enterprise, but it must be properly situated in what he calls the Rhythm of Education [9] (Chapter II). There he identifies three stages of learning: Romance, Precision and Generalization. To some extent, all our learning proceeds through these stages in order, such that, roughly speaking, the child is dominated by Romance, the youth by Precision, and the adult by Generalization. In practice, however, the stages cycle continuously like eddies in the fast-flowing stream of life (and indeed at different times we can all be children or adults).There is only one subject-matter for education, and that is Life in all its manifestations. Instead of this single unity, we offer children—Algebra, from which nothing follows; Geometry, from which nothing follows; Science, from which nothing follows; History, from which nothing follows; a Couple of Languages, never mastered; and lastly, most dreary of all, Literature, represented by plays of Shakespeare, with philological notes and short analyses of plot and character to be in substance committed to memory. Can such a list be said to represent Life as it is known in the midst of the living of it?.[9] (pp. 6–7)

Dewey’s main thesis is that the aesthetic experience is jointly constructed between painter and viewer, performer and audience, that both are called to be artists in a shared experience. For me this captures the essential character of the teacher–student relationship; I as teacher am the gardener, the student is the householder and we are working together to create beauty, and that means that we both have our hands in the earth. Seymour Papert warns that this cannot happen unless the activity brought into the classroom has meaning for the student:The word “aesthetic” refers, as we have already noted, to experience as appreciative, perceiving and enjoying. It denotes the consumer’s rather than the producer’s standpoint. It is Gusto, taste; and, as with cooking, overt skillful action is on the side of the cook who prepares, while taste is on the side of the consumer, as in gardening there is a distinction between the gardener who plants and tills and the householder who enjoys the finished product.[10] (p. 37)

In my own work I have little experience with art class, but I do have some experience with drama, and there I find a parallel between the response of the student and the teacher to the work being studied and constructed. I do not think it is unreasonable to strive for comparable harmony in the mathematics class.The important difference between the work of a child in an elementary mathematics class and that of a mathematician is not in the subject matter (old fashioned numbers versus groups or categories or whatever) but in the fact that the mathematician is creatively engaged in the pursuit of a personally meaningful project. In this respect a child’s work in an art class is often close to that of a grown-up artist.[11] (p. 249)

This certainly fits my own recent work, and I find myself paying considerable attention to the question of what structures are the right ones to bring into a class of students at various grade levels. Barabe and Proulx call Papert’s project-oriented approach a complete rebuild, “une reconstruction complète” of school mathematics [15] (p. 26), defining the mathematics curriculum itself not in terms of content but as the activity or experience of the students.This project-oriented approach contrasts with the problem approach of most mathematics teaching: a bad feature of the typical problem is that the child does not stay with it long enough to benefit much from success or from failure. Along with time-scale goes structure. A project is long enough to have recognizable phases—such as planning, choosing a strategy of attempting a very simple case first, finding the simple solution, debugging it and so on. And if the time scale is long enough, and the structures are clear enough, the child can develop a vocabulary for articulate discussion of the process of working towards his goals.[11] (p. 251)

## 5. A Complete Rebuild

This passage came my way at an important moment in my life, and supported my desire to move forward into the aesthetic. The cultural forces that Vickers is talking about are precisely those that have bent our school curricula to the grim task of preparing the students for a STEM future in a technologically-driven society. The irony is that employers who are hiring in the STEM disciplines are now more interested in the so-called complementary traits, the five C’s: creativity, critical thinking, collaboration, cooperation and care [1].the sad history of Western culture which, over the last two centuries, has so narrowed the concepts of both Science and Art as to leave them diminished and incommensurable rivals,—the one an island in the sea of knowledge not certified as science; the other an island in the sea of skill not certified as art… Moreover the two words “Ars” and “Scientiae” not only embraced virtually all skill and knowledge, but also overlapped each other’s territory without offense.[16] (p. 143)

Some students will be ready and eager to roll the task right up onto the stage of Precision; others who might not yet have the right analytical tools can still play and wonder on the stage of Romance. Indeed “wonder” is a magical word and Sinclair and Watson [18] have a marvellous book review in which they play with its two shades of meaning. In terms of the student’s future, it is worth observing that few of them will ever need to complete a square or simplify $\mathrm{log}\left(3y{e}^{2x}\right).$ However, they will often need to arrive at an understanding of the workings of a complex structure, perhaps every time they change jobs. If we can keep that firmly in mind in our assessment of the value of an activity, that might help to expand our viewpoint.Low floor, high ceiling tasks allow all students to access ideas and take them to very high levels. Fortunately, low floor, high ceiling tasks are also the most engaging and interesting math tasks with value beyond the fact that they work for students of different prior achievement levels…Such teaching, though demanding, is also extremely fulfilling for teachers, especially when they see students who lack confidence and were previously low-achieving take off and soar.[3] (p. 115)

## 6. Summing Up

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**An activity taken from the Ontario Grade 12 Advanced Functions curriculum [6] (pp. 97–98).

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Taylor, P.
Teach the Mathematics of Mathematicians. *Educ. Sci.* **2018**, *8*, 56.
https://doi.org/10.3390/educsci8020056

**AMA Style**

Taylor P.
Teach the Mathematics of Mathematicians. *Education Sciences*. 2018; 8(2):56.
https://doi.org/10.3390/educsci8020056

**Chicago/Turabian Style**

Taylor, Peter.
2018. "Teach the Mathematics of Mathematicians" *Education Sciences* 8, no. 2: 56.
https://doi.org/10.3390/educsci8020056