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What Does Understanding Mathematics Mean for Teachers: Relationship as a Metaphor for Knowing

reviewed by Jason Samuels - November 25, 2012

coverTitle: What Does Understanding Mathematics Mean for Teachers: Relationship as a Metaphor for Knowing
Author(s): Yuichi Handa
Publisher: Routledge, New York
ISBN: 0415885973, Pages: 168, Year: 2011
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What does it mean to have a passion for mathematics, and how can a teacher achieve that state? These are the motivating questions of What Does Understanding Mathematics Mean for Teachers?: Relationship as a Metaphor for Knowing, by Yuichi Handa.

The book is framed through the lens of the author's journey in search of meaning in his teaching. Handa uses quotes and anecdotes to lay bare his own frustration. “I could not help at times but feel as if I were inflicting upon my students something that for the most part was unenjoyable for them... [I felt] uninvolved with mathematics... I felt like a fraud” (p. ix). Initially inspired by personal experience, reflections from both mathematicians and musicians on their relationship with their subject matter, and theoretical literature from philosophy and sociology, Handa developed a philosophy that would allow for the animation of mathematical study. In this book, he argues that personal meaning can be found in the content of study and instruction by developing a relationship with the material.

Most of the book is devoted to elaborating the characteristics of an individual's relationship with a field of study in general and with mathematics in particular. The framework is based on a triumvirate of conditions; when the first two states are achieved simultaneously, the third arises.

The first state he describes is grace, which Handa defines as an openness to being affected by a relationship (with mathematics). To reach this state, “learning and cognition require feeling...feeling requires letting go... letting go requires risk-taking... risk-taking requires courage” (p. 42). As a result of that courage, one can “become invested in the subject to the point where joy, frustration, and… emotions can arise in the interaction” (p. 43). Handa goes so far as to suggest that emotions may be the catalyst for remembering, rather than simple ancillary enhancers as they have often been treated. Aesthetic response is another form of being affected by mathematics. When a mathematician is engaged in inquiry, aesthetic considerations guide the selection of the next step of investigation. This operation is “not entirely rooted in logic, but in an aesthetic appreciation” (p. 47).

Handa also tweaks the typical definition of understanding by including a subjective component. He defines understanding as knowing combined with a ‘feeling of understanding,’ for which he uses the French word connaitre. Knowing without that subjective feeling is mere memorization, which he refers to as savoir. One of the central themes of the book is Handa’s conviction that this personalized aspect of education has been stripped away and should be reinstated.

Handa applies this savoir/connaitre dichotomy to good effect in considering the role of proofs. He gives the example of a teacher who can justify the steps of the proof that the Cantor set is without intervals, but does not viscerally believe it. Handa argues that this will happen when the proof does not deal with points of conflict. However, an individual may have a point of conflict at any step. Therefore, whether or not the proof is a proof-that-explains and convinces depends on the particular reader.

Handa’s second state is will, defined as the act of entering into the relationship (with mathematics). In the practice of mathematics, this often appears as the commitment to stay with a problem for continued inquiry, particularly through a phase of frustration, which ideally will eventually lead to understanding.

When a person has achieved both will and grace, he or she is in a state of interest, or being intertwined (with mathematics) which leads to the motivation to engage. As examples of the way in which interest may manifest, the author suggests the enjoyment of a challenge, the engagement of social connection, and curiosity as a force for further exploration. He provides lengthy excerpts from interviews with mathematics teachers that illustrate each.

In his last chapter, Handa explores how a relationship with mathematics can be cultivated in the classroom. This process should include 'doing' mathematics – producing instead of consuming —and is most completely realized if it includes the step of posing a question and then solving it. Handa explains that posing and then answering a question is more organic (i.e., personal) than simply answering someone else’s question. Moreover, whereas solving a given question is a process that terminates, including question-posing allows for a continued relationship, which in turn allows for variation, creativity, and infinite possibility. The author argues that most mathematicians engage in the cycle of problem posing and solving and appreciate its central importance to mathematics, but most teachers have no awareness or experience with it. He suggests that what is needed is a version of doing mathematics that is pedagogically relevant for teachers. He offers three classroom examples, one from infinite series (exploring 1-1+1-1+1-...), one from an arithmetic puzzle (arranging 5 digits into a maximal product), and one in algebra (multiple proofs that the square of the sum of the first n natural numbers equals the sum of their cubes).

The book concludes with a reflection on the author's evolution from disinterest to interest in mathematics and its teaching – or as he puts it, the replacement of savoir with connaitre. He emphasizes the crucial role a mentor played for him in this process and notes that a mentor should “present an embodied image for how to risk letting go into mathematical inquiry...and become overtaken by the excitement and enthrallment of such experience” (p. 130). In the classroom, a teacher can do this as well, or else recite “empty rationalizations for the study of the field” (p. 131).


This book has deep philosophical underpinnings, which result from “an ontological shift in the way [Handa] stood in relation to...mathematical activity” (p. xi). His goal is “reframing a person's meaning of mathematics less in terms of the knowledge that accrues and more in terms of the 'relationship' that evolves” (p. xiii). The style of the prose is also from the tradition of philosophical tracts. It is filled with abstraction such that there are often stretches in developing a concept that do not mention mathematics. There is extensive reference to other thinkers. Some of his central ideas are taken from the philosopher Buber, such as the central ideas of will and grace, as well as comparing the I-Thou relationship with the I-It relationship. In the entire book, there are very few pages that go without a block quote from another source; some pages have several such quotes. To investigate the aesthetic appreciation of beauty, the author makes appeals to the philosophers Kant and Schopenauer, and the writers Moritz and Joyce.

Handa writes with great passion and caring. His dedication and concern for personal involvement with mathematics are apparent, and often contagious. When he presents quotes from interviews he conducted, the accompanying discussion clearly demonstrates his empathy for his subjects’ emotional highs and lows.

The prose abounds with metaphors, which the author uses to convey his personal, subjective sentiments at different junctures of his journey. Limitations are a “stiff suit.” He writes that, for one student, mathematics is “emotionless, [and] in turn would rebuff her feelings” (p. 78). There is also extensive use of anecdote, often pulling the book far afield. The author tells the story of a friend who is a bird-watcher in order to point out the intimate relationship developed with a familiar bird. The prose is full of repetition. Often, an idea is presented and defined, and then it is then referred to later and defined again - several times. Chapter sections begin with summaries of previous points. The height of this occurs in the final chapter when, in support of a point, the author block quotes himself from earlier in the book.

The author uses linguistic analysis very frequently to make points and develop arguments. For example, in English, 'to know' serves many purposes, but the author makes a finer distinction by invoking the French verbs 'savoir' (to know) and 'connaitre' (to be familiar with), as it is a transition from the former to the latter which he seeks to achieve. Further, he offers three other equivalent pairs representing this distinction. For many words, the author presents the Latin derivation. For example, he deconstructs “interest” to its Latin roots, inter and esse, defines them, and in many cases used “inter-esse” in his discussion. When the author offers a list of equivalent words for 'will' - commitment, exertion, discipline, effort, intention, persistence, affectivity - the linguistic analysis is repetitive and at its worst.

The aspirations of the book are delineated, not only by what is in the book, but also by what is not in the book. Many elements of Handa's relationship framework are very similar to major areas of mathematics education research that pass without mention. For example, the author's emphasis on the personal nature of learning bears a lot in common with constructivism, the theory of learning which states that each learner builds knowledge through their own thoughts and actions, and that knowledge construction is personal and different for each learner based on his or her own faculties. This theory emphasizes process instead of product, problem-solving, and social interaction (Brewer & Daane, 2002). Despite all these commonalities, that theory is not mentioned. Because of the centrality of constructivism in the field of education, it might have been helpful to make this connection in support of his thesis. Handa raises the notion that the learner must examine his or her own process in order to best learn from problem solving, which is known in the literature as metacognition or self-regulation. The idea has been extensively investigated in articles and books (e.g., Schoenfeld, 1987), but that work is not referenced here.

The final chapter offers examples of implementing this philosophy in the classroom. It offers the reader a chance to see a manifestation of the most useful aspect of Handa’s philosophy, the notion that teachers should ‘do mathematics.’ The three examples are promising, but feel isolated and leave us wanting more. The biggest obstacle to doing mathematics is probably not a lack of desire, but rather possible productive starting points, and finding them is not trivial. Handa’s ideas make a tremendously interesting philosophy. They would have been more persuasive if, in addition to linguistics and philosophy, he had more thoroughly included the relevant work in the field that dovetails with his ideas. Teachers might also be more absorbed by a collection of examples or a road map for how to implement Handa’s theory of relationship in their own lives and classrooms.


Brewer, J. & Daane (2002). Translating Constructivist Theory into Practice in Primary-Grade Mathematics. Education, 123(2), 416-426.

Schoenfeld, A. (1987). What's all the fuss about metacognition? In A. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 189-215). Hillsdale, NJ: Lawrence Erlbaum Associates.

Cite This Article as: Teachers College Record, Date Published: November 25, 2012
https://www.tcrecord.org ID Number: 16944, Date Accessed: 5/24/2022 4:53:01 AM

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About the Author
  • Jason Samuels
    City University of New York
    E-mail Author
    JASON SAMUELS is an Assistant Professor of Mathematics at the City University of New York. He received his Ph.D. in Mathematics Education from Columbia University. His main area of research is innovation and assessment in calculus instruction. He is currently writing a calculus textbook based on his approach. He also does research on effective instructional strategies in e-learning.

    Samuels, J. (2012) The Effectiveness of Local Linearity as a Cognitive Root for the Derivative in a Redesigned First-Semester Calculus Course. In S. Brown, S. Larsen, K. Marrongelle, and M. Oehrtman (Eds.) Proceedings of the 15th Annual Conference on Research in Undergraduate Mathematics Education (CRUME), Portland, OR: The Special Interest Groups of the Mathematical Association of America on Research in Undergraduate Mathematics Education, p155-161.

    Samuels, J. (2012) The Use of Technology and Visualization in Calculus Instruction. Saarbrucken, Germany: Lambert Academic Publishing.

    Samuels, J. (2011) The relationship between learner characteristics and learning outcomes in a revised first-semester calculus course. In Wiest, L & Lamberg, T (Eds.) (2011) Proceedings of the 33rd Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Reno, NV: University of Nevada, Reno, p666-674.

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