From 1989, the year of the publication by the National Council of Teachers of Mathematics in the United States (NCTM) of the Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989), until 2010, the year of the publication of the Common Core State Standards in Mathematics (CCSSM) (see McCallum, 2015), a debate about mathematics education raged in the United States. The debate, known as the math wars, was between a reform movement inspired by the NCTM standards, and a reaction to that movement which brought together under one umbrella a number of different groups, including parents puzzled by changes in their children’s curriculum, traditionalists worried about a decreased emphasis on basic skills, and mathematicians concerned about the correctness of school mathematics. The debate was political and not always informed by much evidence (Schoenfeld, 2004).

Neither camp was as homogeneous as their apparent unity at the height of the wars might suggest, as became apparent with the publication of the CCSSM, which drew support from both sides (and, to a lesser extent, opposition from both sides). My own career as a mathematician has spanned this period, from my involvement in the early 1990s in reforming the undergraduate calculus curriculum as a member of the Harvard Calculus Consortium to my role as one of the lead writers of the CCSSM. During that period, I have often wondered whether, underneath the irrationality and politics, there was a coherent duality which at least in part explained the difference between the two sides, and which also shows how they can come together, as they did with the writing of the CCSSM. This paper is an attempt to describe such a duality.

Views of Mathematics

I [want to] emphasize the practices, because from my point of view that’s where the content lives.

(Alan Schoenfeld, 3rd April, 2013)

at first, I thought no, that’s wrong, the practices live in the content standards, and then I realized we were both saying the same thing, namely that having this separate free-floating set of practices that are independent of the content is a bad idea. (William McCallum, 4th April, 2013)

The exchange above is from a meeting that Alan Schoenfeld and I attended at the Mathematical Sciences Research Institute in Berkeley, CA in 2013 (MSRI, 2013). The content and practices referred to are the Content Standards and Practice Standards in the Common Core State Standards in Mathematics (CCSSM "Common Core State Standards in Mathematics (CCSSM)", 2010, a collaborative effort of the fifty US states to write common standards. I will return to a discussion of this document later in this paper, but first I would like to use the exchange to lay out a dichotomy in views of mathematics.

Schoenfeld described a spectrum of views of mathematics:

At one end of the spectrum, mathematical knowledge is seen as a body of facts and procedures dealing with quantities, magnitudes, and forms, and relationships among them; knowing mathematics is seen as having “mastered” these facts and procedures. At the other end of the spectrum, mathematics is conceptualized as the “science of patterns,” an (almost) empirical discipline closely akin to the sciences in its emphasis on pattern-seeking on the basis of empirical evidence. (1992, p. 335)

A casual internet search on “mathematics as facts and procedures” does not find anybody advocating it as a complete definition, but finds many saying that mathematics is more than that. It is true, however, that this view of mathematics seems embedded in the culture of US classrooms. Stigler and Hiebert wrote:

In the United States, […] the level is less advanced and requires much less mathematical reasoning than in the other two countries [Germany and Japan]. Teachers present definitions of terms and demonstrate procedures for solving specific problems. Students are then asked to memorize the definitions and practice the procedures. (1999, p. 27; italics in original)

Despite efforts to reform this state of affairs going back to the 1989 NCTM standards, this culture remains prevalent today.

Schoenfeld associates one end of his spectrum, the view of mathematics as facts and procedures, with what he calls the content perspective:

A consequence of this perspective is that instruction has traditionally focused on the content aspect of knowledge. Traditionally one defines what students ought to know in terms of chunks of subject matter, and characterizes what a student knows in terms of the amount of content that has been “mastered.” […] From this perspective, “learning mathematics” is defined as mastering, in some coherent order, the set of facts and procedures that comprise the body of mathematics. The route to learning consists of delineating the desired subject matter content as clearly as possible, carving it into bite-sized pieces, and providing explicit instruction and practice on each of those pieces so that students master them. (1992, p. 342)

Note there are really two perspectives here, one on what mathematics is, and another on how it is learned. One could in principle hold the first and not the second. In contrast to the content perspective, and by preference, Schoenfeld proposed the process perspective. In writing about Everybody Counts, a report of the National Research Council (NRC, 1989), he claimed:

there is a major shift from the traditional focus on the content aspect of mathematics […] to the process aspects of mathematics—to what Everybody Counts calls doing mathematics. Indeed, content is mentioned only in passing, while modes of thought are specifically highlighted in the first page of the section. (1992, p. 343)

The process perspective has taken various forms over the years: the NCTM (2000) process standards, the focus on problem-solving as a core activity in reform curricula, and the practice standards of CCSSM "Common Core State Standards in Mathematics (CCSSM)" (2010). Again, one might hold a content perspective on what mathematics is and a process perspective on how it is learned; for example, problem-solving could be a way of learning facts and procedures. Schoenfeld (1992) described his own version of the process perspective as a view of mathematics as pattern-seeking.

The last sentence in the second quotation above captures a danger of the process perspective: “content is mentioned only in passing”. The danger is that mathematics content is a backdrop to the action, a backdrop that can be inaccurate or forgotten.Footnote 1 For example, curricula written from the process perspective might be organized around large projects that pull different mathematical tools in at different times. Without careful planning there is the danger that mathematical dependencies get mixed up. Some curricula are organized around “big ideas,” lists of overarching themes that recur throughout the curriculum. This can work well if done judiciously; but some ideas in mathematics are not well-described as “big”: rather they are small but consequential. Completing the square is an example of such an idea (see McCallum, 2018).

Approaches from the process perspective—mathematics as pattern seeking, mathematics as problem-solving, big ideas—have in common what I call the sensemaking stance. In this stance, mathematics is a source of material for important processes such as problem solving and communication. It is an important stance, but it carries risks. If mathematics is about sense-making, the stuff being made sense of can be viewed as some sort of inert material lying around in the mathematical universe. Even when it is structured into “big ideas” between which connections are made, the whole thing can have the skeleton of a jellyfish.

I would like to propose a complementary stance, which carries its own benefits and risks.

The Making-Sense Stance

Where the sense-making stance sees a process of people making-sense of mathematics (or not), the making sense stance sees mathematics making sense to people (or not). These are not mutually exclusive stances; rather they are dual stances jointly observing the same thing. The making-sense stance is related to the content perspective described by Schoenfeld, without the unappetizing “carving it [content] into bite-sized pieces” (p. 342). It views content as something to be actively structured in such a way that it makes sense.

That structuring is constrained by the logic of mathematics. But logic by itself does not tell you how to make mathematics make sense, for various reasons. First, because time is one-dimensional, and sense-making happens over time, structuring mathematics to make sense involves arranging mathematical ideas into a coherent mathematical progression, and that can usually be done in more than one way. Second, there are genuine disagreements about the definition of key ideas in school mathematics (ratios, for example), and so there are different choices of internally consistent systems of definition. Third, attending to logical structure alone can lead to overly formal and elaborate structuring of mathematical ideas. Just as it is a risk of the sense-making stance that the mathematics gets ignored, it is a risk of the making-sense stance that the sense-maker gets ignored.

Student struggle is the nexus of debate between the two stances. It is possible for those who exclusively take the sense-making stance to confuse productive struggle with struggle resulting from an underlying illogical or contradictory presentation of ideas, the consequence of inattention to the making-sense stance. And it possible for those who exclusively take the making-sense stance to think that struggle can be avoided by ever clearer and ever more elaborate presentations of ideas.

The work entailed in the making-sense stance is mathematical work, so it is not surprising that much of the work of mathematicians in mathematics education falls under this heading. Wu (2015) has written about “textbook school mathematics” as a degraded subject that is not faithful to mathematics as it is understood by mathematicians. Howe and Epp (2008) have written about the mathematical ideas behind place value. Baldridge (2018) has constructed a vast edifice of grade-level-appropriate, internally consistent definitions of ideas that arise in school mathematics.

An important strand of research in mathematics education is composed of work where the two stances are taken simultaneously, often by pairs of mathematicians and education researchers. For example, Ball and Bass argue that:

Making mathematics reasonable is more than individual sense making […] making-sense refers to making mathematical ideas sensible, or perceptible, and allows for understanding based only on personal conviction. Reasoning, as we use it, comprises a set of practices and norms that are collective, not merely individual or idiosyncratic, and rooted in the discipline. (2003, p. 29)

Another example is the work of Izsák and Beckmann (2019), who propose a unified definition of multiplication that applies to the many situations modeled by multiplication. In their definition, a product is measured simultaneously by a base unit and by a group, which is itself measured by base units. Their work provides a nice example of co-ordinating the making-sense stance with the sense-making stance. On the one hand, their work is an attempt to make the diverse array of multiplication situations make sense through a unified definition. On the other, it recognizes the role of the sense-maker, the person who must make the choice of base unit and group in order to make sense of a multiplication situation.

We think that mathematics education as a field should seek more completely worked out coherent approaches to the [multiplicative conceptual field] based on consistency and logical interconnection. The absence of such articulation may be constraining our capacity to help students and teachers use prior knowledge and experience to effectively relate topics and construct interconnected bodies of knowledge. It is one thing to know that multiplication can be used to model a variety of situations and another to perceive a common underlying structure. (Iszák & Beckmann, 2019).

Coherence

Coherence is the sine qua non of the making-sense stance. Schmidt et al. (2005) talk about coherence of standards:

We define content standards […] to be coherent if they are articulated over time as a sequence of topics and performances consistent with the logical and, if appropriate, hierarchical nature of the disciplinary content from which the subject-matter derives. […] This implies that, for a set of content standards to ‘to be coherent’, they must evolve from particulars […] to deeper structures. (p. 528)

This definition was elaborated by Cuoco and McCallum (2017) to include coherence of curriculum and coherence of practice. Izsák and Beckmann (2019) argue for a coherent view of multiplication in mathematics education research. Attempts to bring coherence to school topics also underlie the work of mathematicians mentioned above.

Coherence was a guiding principle in the writing of the Common Core State Standards in Mathematics (CCSSM) (McCallum, 2015) in 2009–2010. An important precursor was the report in 2008 of the National Mathematics Advisory Panel, which laid out the following principles:

A focused, coherent progression of mathematics learning, with an emphasis on proficiency with key topics, should become the norm in elementary and middle school mathematics curricula. (NMAP, p. xvi)

By the term ‘coherent’, the Panel means that the curriculum is marked by effective, logical progressions from earlier, less sophisticated topics into later, more sophisticated ones.

Standards have an inherent tendency to interfere with focus and coherence, in that they attempt to reduce a subject to a list, Schoenfeld’s “bite-sized pieces”. The pieces can lose connection with each other, breaking coherence, and there is a danger that everybody’s favorite pieces get added to the list, breaking focus. Maintaining focus in CCSSM was a matter of resisting temptation. Maintaining coherence was a matter of building structures that transcended the bulleted list. See Daro et al. (2012), Zimba (2014) and McCallum (2015) for more detail on the process.

One important way of maintaining coherence was to build the standards on progressions: narrative descriptions of how the mathematical ideas in a particular domain evolve over a sequence of grades (CCSSWT, 2018). These were the first documents produced in the writing of the standards. For example, there was a progression for Number and Operations in Base Ten (NBT) in grades K–5, which told the story of that domain over the grades. Different progressions were tied together by cross-domain connections. For example, it makes sense that the place in the NBT progression where students learn about multiplication should come in the same grade where the geometry progression talks about area of rectangles. These connections tied the different stories together into a coherent whole.

A particularly knotty area in mathematics curriculum is the progression from fractions to ratios to proportional relationships. Part of the problem is the result of a confusion in everyday usage, at least in the English language. In common language, the fraction, the quotient a ÷ b, and the ratio a: b, seem to be different manifestations of a single fused notion. Here, for example are the mathematical definitions of fraction, quotient, and ratio from Merriam-Webster on-line:Footnote 2

fraction: a numerical representation (such as, or 3.234) indicating the quotient of two numbers;

quotient: (1) the number resulting from the division of one number by another (2) the numerical ratio usually multiplied by 100 between a test score and a standard value;

ratio: (1) the indicated quotient of two mathematical expression (2) the relationship in quantity, amount, or size between two or more things.

The first definition says that a fraction is a quotient; the second says that a quotient is a ratio; the third one says that a ratio is a quotient. Thus, it would appear that these words all mean the same thing. The definitions are not wrong as descriptions of how people use the words. For example, people say things like, “mix the flour and the water in a ratio of 3/4,” confusing ratios with fractions.

From the point of view of the sense-making stance, this fusion of language is out there in the mathematical world, and we must help students make sense of it. From the point of view of the making-sense stance, we might make some choices about separating and defining terms and ordering them in a coherent progression. In CCSSM, the following choices were made.

  1. 1.

    A fraction a/b is the number on the number line that you get to by dividing the interval from 0 to 1 into b equal parts and putting a of those parts together end-to-end. It is a single number, even though you need a pair of numbers to locate it.

  2. 2.

    It can be shown using the definition that a/b is the quotient a ÷ b, the number that gives a when multiplied by b. (This is what Iszák & Beckmann, 2019, call the Fundamental Theorem of Fractions.)

  3. 3.

    A ratio is a pair of quantities; equivalent ratios are obtained by multiplying each quantity by the same scale factor.

  4. 4.

    A proportional relationship is a set of equivalent ratios. One quantity y is proportional to another quantity x if there is a constant of proportionality k such that y = kx.

    Note that there is a clear distinction between fractions (single numbers) and ratios (pairs of numbers).

This is not the only way of developing a coherent progression of ideas in this domain. Zalman Usiskin (private communication) prefers to start with (2) and define a/b as the quotient a ÷ b, which is assumed to exist. One could then use the Fundamental Theorem of Fractions to show (1).

There is no a priori mathematical way of deciding between these approaches. Each depends on certain assumptions and primitive notions. But each approach is an example of the structuring and pruning required to make the mathematical ideas make sense; an example of the making-sense stance.

Fidelity

Another principle of the making-sense stance is fidelity. I define fidelity as, “the extent to which a curriculum, or a collection of curriculum materials, faithfully presents the underlying mathematical concept as it is situated in the discipline of mathematics” (McCallum, 2019, p. 80). I go on to say that, “mathematical fidelity is not the same as mathematical formality; a mathematical concept can be presented in a way that is appropriate for the age of the students, while still being presented with fidelity” (p. 80).

Examples of lack of fidelity abound on the internet. Consider, for example, this representation found at LumenFootnote 3 (Fig. 33.1)

Fig. 33.1
A model diagram of the fruit halving function h, which takes apple fruit as input and outputs half of the fruit.

Fruit-halving function (this shows a function that takes a fruit as input and releases half the fruit as output)

Full size image

The image would seem to violate the condition that a function have one output for each input, since an apple has two halves. Or, if we take the caption to mean that the machine is throwing away one of the halves, there is still the question of which half. A function does not randomly choose outputs from two possible choices.

Fidelity is to some degree a matter of taste. Consider, for example, the distinction between order of operations – the set of rules for how to read arithmetic expressions, such as giving precedence to multiplication over addition – and the properties of operations – the set of rules governing how operations work, such as the distributive property. In school mathematics these topics are often given equal salience. However, most mathematicians would regard the first as merely convention and the second as fundamental law. The order of operations could be changed; there is nothing mathematically wrong with saying that addition takes precedence over multiplication, in which case the distributive property would be written a·b + c = (a·b) + (a·c). But the distributive property itself is fundamental, and has the same meaning no matter how it is notated. Although it would not be mathematically incorrect in a curriculum to present order of operations and properties of operations in a flat list with the same degree of emphasis, it would be a little tone-deaf.

This subjective aspect of fidelity means that there can be reasonable disagreements about it. A making-sense stance takes seriously the task of discussing those disagreements with evidence from the professional norms of the discipline.

Concluding Thoughts

I have spent most of this paper describing properties and examples of the making-sense stance: the properties of coherence and fidelity, the example of ratios and fractions. However, a complete view of mathematics and learning takes both stances at the same time, with a sort of binocular vision that sees the full dimensionality of the domain. An example of this is Arcavi’s (Arcavi, 1994) article on symbol sense, which shifts beautifully back and forth between the two stances.

In wondering about how the duality between the two stances relates to the math wars, I am drawn to the observation that some participants in that debate may have made unwarranted assumptions about what each stance implied on the other side. It was sometimes assumed that a proponent of mathematical correctness – the making-sense stance – would also be in favor of instructional practices that come under the heading of “stand and deliver”: the teacher standing at the front of the room, explaining a concept or demonstrating the procedure for solving a certain type of problem, and then asking the students to mimic the procedure with a set of similar problems. It was also sometimes assumed that a proponent of the sense-making stance would embrace an arrangement of mathematics by extra-mathematical organizing principles, such as large real-world projects, or nebulous big ideas.

However, we have known better for a long time. For example, in Hiebert et al. (1996), they describe a reform approach to instruction based on the principle that, “students should be allowed and encouraged to problematize what they study, to define problems that elicit their curiosities and sense-making skills” (p. 12), a principle which falls squarely in the sense-making camp. However, they illustrate this approach with an account of a second-grade class where students work working on the most traditional of word problems: “find the difference in the height of two children, Jorge and Paulo, who were 62 inches tall and 37 inches tall, respectively” (p. 13). Thus, the sense-making stance is applied to an ostensibly traditional organization of material. In the other direction, the freely available curriculum (Illustrative Mathematics, 2017)Footnote 4 takes the making-sense approach to ratios and proportional relationships prescribed by CCSSM, organized into lesson plans that support problem-based instruction as described in Hiebert and colleagues (1996).

However, this co-ordination of the two stances does not always happen. This is partly because there is a political aspect to the division, as illustrated by the math wars in the United States (Schoenfeld, 2004). The Common Core is an existence proof of the possibility of overcoming these political differences. Part of that success was the result of the usual grind of diplomatic work; a lot of listening and trying to find third ways, while at the same time insisting on principles of mathematical coherence and pedagogical appropriateness. It is difficult to draw a general lesson there, apart from the lesson that if you keep trying at something you occasionally succeed.

But there is one lesson worth pointing out, about the power of the word “common”, meaning shared. In the end, the fact that almost fifty states agreed on the same set of standards was at least as powerful as the quality of those standards. Having a shared set of standards means being able to share curriculum, teaching strategies, and resources across state lines. And, although a country with a centralized education system has already solved that problem, I think the idea of shared understanding also has the power to bring together the two sides in whatever version of the math wars might be happening in that country, or indeed in the international community.

I hope that spelling out the two stances will contribute to productive dialog in mathematics education, such as the one that started this article, allowing for conscious recognition of the stance one or one’s interlocutor is taking and for acknowledgement of the value of adding the dual stance.