Sense-Making and Making Sense

WILLIAM MCCALLUM

In the early part of this millennium, when the math wars were raging, I gave some testimony to the National Academies panel that was working on the report Adding it Up. Somewhat flippantly I said that which side of the math wars you were on was determined by which you were paying lip service to, the mathematics or the students. I was recently invited to give a plenary address at the ICMI Study 24 in Japan on school mathematics curriculum where I decided to expand on this remark, because I think it is worth going beyond the flippancy to map out an important duality of perspectives in mathematics education. What follows is an edited summary of what I said in that address.

In my address I talked about two different stances towards mathematics education: the sense-making stance and the making-sense stance. The first manifests itself in concerns about mathematical processes and practices such as pattern seeking, problem-solving, reasoning, 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 propose a complementary stance, the making-sense stance, which carries its own benefits and risks. 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 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 the 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 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 take the making-sense stance to think that struggle can be avoided by ever clearer and ever more elaborate presentations of ideas.

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 a , 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 online:

Fraction: A numerical representation (such as 3/4, 5/8, 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 one 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. These 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 .”

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 writing the Common Core State Standards in Mathematics we made the following choices:

(1) A fraction a as 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) 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 Sybilla Beckman and Andrew Isz´ak call the Fundamental Theorem of Fractions.)

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

(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 has told me that he prefers to start with (2) and define a/b as the quotient a ÷ b, which 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. One might take the point of view that the distinction between the sense-making stance and the making-sense stance is artificial or unnecessary. 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. However, this coordination of the two stances does not always happen. Rather than provide examples, I invite the reader to think of their own examples where one stance or the other has become dominant. This has been particularly a danger in my own work in the policy domain. I hope that spelling out the two stances will contribute to productive dialog in mathematics education, 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.

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