Democracy and Mathematics Education: Rethinking School Math for Our Troubled Times, by KurtStemhagen and CatherineHenney, Routledge, 2021, 212 pp.

Reading Kurt Stemhagen and Catherine Henney's book on math education sparked some reminiscing on my experiences with mathematics inside and outside of school.1 I remembered taking advanced placement (AP) calculus in high school and successfully passing the exam while really only having the vaguest understanding of what I was doing and why — frankly, I was just manipulating symbols to get the “right answer.” On a beach vacation, many years after school math, an idle question came up about how far a person on the beach could see to the horizon. Using trigonometry and a few basic facts about the earth, I was able to stumble on the answer — about three miles. I remember being in awe that these symbols and tools could somehow relate to the world. Finally, I remembered figuring out how much tile I would need redo my kitchen floor, using math to solve a practical problem in my environment. In each of these memories, mathematics fit into my life in different ways. Stemhagen and Henney carefully consider this fit, this place of mathematics in human life, and use it to develop a powerful vision of mathematics education — school math with a human face. The book is structured around three different philosophies of mathematics: absolutist perspectives, constructivist perspectives, and evolutionary/pragmatic perspectives. The authors own view, Democratic Mathematics Education (DME), is most strongly related to the evolutionary view. The structure of each chapter is fair and judicious as it goes through the absolutist and constructivist perspectives. The authors even criticize their favored position, the pragmatic position, just as they criticize the other viewpoints. This lends a commendable air of honesty and intellectual seriousness to the book. I came away feeling like I had a basic grasp of the philosophy of mathematics — something I count as a small miracle, given my lack of past experience with the subject. The approachability of the text is, in fact, one of the book's many strengths: it admirably distills centuries of complex and seemingly esoteric ideas about mathematics in terms that are manageable to the average educated reader. In this way, the book is democratic in form as well as content. The absolutist sees math as a distinct form of knowledge leading to Truth. Absolutism comes in various flavors, including Platonism, formalism, and logicism. Platonism sees mathematical truths as “objects” existing in a world of rationality, indubitably real and in no need of empirical observation or confirmation. The problem of Platonism is how this mysterious, otherworldly existence of mathematical objects should be understood and how (short of positing a human spiritual preexistence) human beings have access to this dimension of reality. Trying to preserve the absolute certainty of mathematics without the Platonist baggage, the formalists respond by recasting mathematical truths as analytic truths, as necessarily true connections between sets of symbols, not as existing objects of contemplation. Truths are found in formalism by following proper rules and procedures; the truth is guaranteed by the system's internal consistency. While not stipulating the existence of mysterious mathematical objects, however, the formalist still struggles to appreciate and explain the practical applications of mathematics to the world that exists outside of its internally consistent systems. It also does not recognize a human element in mathematics. This way of thinking about math, Stemhagen and Henney argue, has influenced school math for the worse, making it seem disconnected and irrelevant to the everyday lives of students. In contrast to the absolutist approach, the constructivist approach focuses on the mind's activity in building mathematical knowledge. It is non-absolutist in outlook. The authors break constructivism down into two versions: psychological and social. For the psychological constructivist, mathematical knowledge is built by individuals as they engage with their unique experiences; mathematic truth is relative, in some sense, to these experiences. In the most radical versions of psychological constructivism, individuals are cut off from others and the world, casting doubt on the idea of objective mathematical knowledge. Social constructivism, for its part, emphasizes the reliance of mathematics on language and human practices and thereby reinstates the social. Here, math is emphasized as a product of social activity rather than a product of the solitary individual mind. Stemhagen and Henney reject both constructivist approaches, however, because they cannot account for the “remarkable stability of mathematical knowledge,” or if they do account for it, it is smuggled in through a backdoor of “inevitable structures, constraining language, or in general forms of rationality” (RSM, 77). The evolutionary/pragmatic perspective favored by the authors emphasizes the empirical dimensions of the development of mathematics. Math is linked to “changing social and empirical problems in our physical world” (RSM, 90). The point of mathematics is to improve human lives and solve the problems of human existence. Mathematics is a toolkit; the truths of math are dependent on the usefulness of the tool. Math is not valuable in itself but in how it enriches human life. In this sense, there is an historical contingency to mathematics that is often not recognized since the problems we confront are continually changing. Mathematics is not so much a body of knowledge but an activity filled with human intentions and purposes. A certain “objectivity” is reinserted in this way. Math is not an idiosyncratic human or cultural artifact without external standards of validity, but it is also not a rational object existing in a separate reality. Unlike an internally consistent symbolic system, it can be judged by how well it meets the specific problems it is intended to solve within its context of use. All of this builds toward the real contribution and achievement of the book, the development of Democratic Mathematics Education. According to DME, school math should be grounded in the context of student experience and the specific problems that students think are important. The mathematics classroom should be linked to human values and aims. It should not be compartmentalized or separated from other disciplines and subject matter. Furthermore, the aims of math education should be fully democratic, which means taking account of the link between individual and social interests. Democracy requires a broad view of how means connect with aims, and it should allow people to fully appreciate and participate in those aims, recreating those aims when necessary. These democratic ideals set the stage for the mathematics classroom. It should be collaborative, relevant to students, driven by their interests, and cross-disciplinary. Math should not be taught as “permanent, unchanging, and finished”; instead, democracy requires “live investigations of which the outcomes are not entirely predetermined.” Stemhagen and Henney write, “DME engages students in relevant, contextualized social inquiries designed to develop students' abilities to pose problems, to test solutions, to evaluate the work based in part on how well the work solves the initial problems. This work is intended to be done in concert with others and to foster agency” (RSM, 148). To me, this is all very welcome. DME seems like a wonderfully productive way of thinking about math education. Math should not be about learning to manipulate symbols to “solve” an equation. It is not a merely formal activity, as I experienced in my AP calculus class. It also is not about making conceptual advances in one's mind — an individual mental construction project. It is about solving important problems in one's environment. I can relate to the power of this approach to the extent that I have used math as a tool to solve problems I cared about (e.g., tiling my kitchen floor). Seeing math as a practical tool, DME can draw on the other philosophies of mathematics insofar as they help solve problems. Sometimes, for example, problem-solving is facilitated when we practice basic skills (as in traditional absolutist approaches to school math), and sometimes it will be helpful to focus on individual understanding (as in constructivist approaches). The key pragmatic questions, though, should always be the focus: How can math help me meet the challenges of my environment? What problems does it solve? The authors provide a few examples of DME (or DME-related practices) in action. In one case, borrowed from the literature, students are given a problem related to their lives: bus stops. The question is how far apart bus stops should be placed. Students determine the contours of the problem, thinking about whether the goal is to minimize walking distance or increase bus travel speed. They figure out the information they need and then make the appropriate calculations. This activity captures much of what is relevant in DME: student relevance and much deliberation about values and purposes. Another example involves weighted formulae. The students are asked to find a formula to evaluate NFL quarterbacks. They are presented examples that others have used to rank quarterbacks, and they uncover the contingent choices involved (e.g., how much should we value rushing ability over, say, leadership traits?). Students develop their own formula for quarterbacks, using their own values, and they eventually create weighting systems for other things they care about: song quality, taco trucks, and so forth. This is inspiring stuff and does seem to capture both mathematic rigor and democratic sensibilities. One way of thinking about DME, the authors say, is the kitchen metaphor. Borrowing from Reuben Hersh, they equate math to a restaurant.2 In a restaurant, all the pretty finished products are out in front, and all the busy messiness is in the back. School math is typically like the front of the restaurant, where everything appears tidy and settled. The mathematicians' math is in the back, where things get more tentative and messy. Our authors want kids to go to the back of the restaurant, where things are messy, tentative, and open to new possibilities. I think an even better metaphor to get at this idea would be the difference between a real chef and a cook who just follows recipes. You can follow a recipe without any understanding of cooking principles. In AP calculus, I was following a recipe with little understanding of what I was doing or why. The mathematicians' math is more like chefs cooking in the back, experimenting with different preparations, extending culinary concepts, and working without recipes because they understand the principles of cooking. Master chefs are judged on the food, not whether they follow a recipe. We want students, I would say, to be more like these master chefs than timid kitchen recipe-followers. DME is an approach to math, then, that is easy to get excited about. If this model ever comes to dominate our classrooms, we will be much better off as individuals and as societies. I would urge us, though, not to take a too truncated view of what “problems” are important and relevant for mathematics to “solve.” For centuries, mathematics has not just addressed practical economic problems, like building bridges or trading grain or tiling floors, but more existential and spiritual problems as well. Nel Noddings points to examples of how math has been used to think about the deepest questions of human existence. She points to Pascal's wager relating to Christian belief, mathematician Leonard Euler's nonsense proof for God's existence, and the gender differences in Edwin Abbot's mathematic fantasy, Flatland.3 Even beyond the application of math to solving existential problems, the very existence of something as “remarkably persistent” as math has itself inspired wonder and reverence. I experienced something of this wonder as my trigonometric equations seemed to work to calculate the distance to the horizon. The authors do note the “remarkable durability” of mathematical truths and argue that this is a difference in degree rather than in kind. I agree with this (I think?), but I do worry that the wonder of the durability gets lost a bit in the evolutionary perspective. Many of the truths discovered millennia ago are still truths today. In some cases, these truths were discovered independently across different cultures and different time periods.4 Bernoulli numbers, for example, were independently discovered in Switzerland and Japan in the seventeenth century. Pappus–Guldin theorems were independently discovered by Pappus around 300 AD and by Guldin in the seventeenth century. The Pythagorean Theorem was discovered a thousand years before Pythagoras under much different historical circumstances. Mathematics does seem to transcend specific cultural contexts and problem situations. Emphasizing math largely as a tool to enrich our lives should not mean that we entirely overlook the ways that math is different from other endeavors, at least by degree. We should not simply ignore this difference in the education context, but should, at least sometimes, try to figure out what it means. I do not think, of course, that Stemhagen and Henney want to ignore the difference, to be sure. But there is no mention of exploring the meaning of this difference in the book. This durability of mathematical truth has played a role in helping some people in their quest for meaning. Seeing math as a simple tool to solve everyday problems, or overemphasizing this as the singular role of mathematics, might leave less room for the sense of awe and wonder that comes from the beauty of ordered, interconnected, surprisingly extensive systems. Yes, I say, let us think about the ways math can solve practical problems, but let us not forget that the larger problem of human meaning, and related spiritual problems, are among the most important. This deep question of meaning does not require learning math “for its own sake” — but it does require pausing to appreciate the aesthetic and even spiritual dimensions of math. One ambiguity I noted in the book is about the exact place of democracy in mathematics. Throughout much of the book, the focus is on how math is applied. Democracy enters mathematics through mathematical applications to real-world problems. People must decide how to use it by designating which problems are important to solve, which things are counted, and how things are weighted. Students are using mathematical concepts, in social groups, to solve important personal problems. Without a doubt, there is plenty of room here for messy contestation and interpretation in a mathematics classroom, even about the question of whether a question is a mathematical one at all. In this way, democracy has a legitimate and important role to play in mathematics, a role that should be emphasized and celebrated. But, even in the authors' example of the weighted formula, students are creating formulas democratically while also using accepted toolkits of multiplication, subtraction, division, ratios, and addition — the “stuff” of math. Once the parameters of the problem are set, democracy seems to end, and mathematical tools are employed. At that point, there becomes a “correct answer.” If students conclude, after all the interpretive work is done, that 1 + 1 = 3, then it seems we should turn to the “remarkably durable” side of math to correct them rather than to the democratic side of math, which demands openness. In some instances, the authors seem to say that democracy constructs the tools themselves, the stuff of math, rather than guiding the application of the tools, implying that some sort of social, democratic process creates the mathematic truth, not simply the parameters of its application. For example, the authors write: “12 divided by 3 is equal to 4 not because it matches some fact in an unknowable realm … [but rather] because if the student is in a group of three and has to disperse 12 objects, each student should receive 4” (RSM, 167). This use of democracy in math seems much less grounded. To me, it seems we should not say that 12 ÷ 3 = 4 is true because it solves the problem; rather, it solves the social problem because it is true. This scheme of distribution favoring equality of distribution is indeed a choice that should be open to democratic debate. But, once equality in distribution is decided, then the math dictates the answer. Once a student wants to distribute the objects “equally,” no amount of debate or discussion will change the right answer. This presents an important question for the authors: How democratic are we willing to be about the “stuff” of mathematics? One interesting feature of the book is the overlap between DME and Critical Mathematics Education (CME), or mathematics with an explicit focus on social justice. The authors are quite sympathetic to CME and its political aims but also express concerns. DME is about students exercising agency in choosing their own problems. In CME, the teachers often seem more directive, choosing problems for the students. One example from the book is a teacher asking students to calculate the cost of the B-2 bomber and how this represents “a welfare state for the rich” (RSM, 179). While an important problem, to be sure, it is unlikely that such a problem arose from students lived experiences. Both the problem itself and the interpretation of the experience are presented to students rather than formulated by them. CME is also faulted for leaving the “stuff” of math “as it is,” while DME is open to questioning even math itself, as we have seen. The authors argue that the purposes of democracy — and social justice — might be better served by allowing students more agency. The authors endorse an indirect approach to social justice, using DME to construct agency and virtues that will affect their sense of social justice. I remain confused about how much DME should present the “stuff” of mathematics as open to democratic debate. The authors are right, however, to be wary of the overly doctrinaire nature of CME, which flirts with being a “banking approach” to education rather than a problem-posing approach. The authors' commendable emphasis on student agency made me realize, though, that the book actually presents few examples in which the students mathematical and problems. 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