What does it mean to know mathematics?

By William McCallum

A world where all learners know, use, and enjoy mathematics.

Perhaps the most mysterious verb in the IM vision—a world where all learners know, use, and enjoy mathematics—is the first one: know

Knowing mathematics means being able to “do the math”

While learning mathematics as an undergraduate math major, I often found myself in conversations with people about what it was exactly that I was doing. Often they seemed to see it as doing the same thing they understood from school, but just much more difficult, like adding up nine twelve-digit numbers or calculating the integral of a very complicated function. This view of mathematics as complicated procedures is out of fashion with educators these days, partly because it reduces mathematics to something that computers can do, but I suspect it is still a fairly common understanding among the general public. Someone who can add two-digit numbers fluently in their head or using a pencil and paper algorithm knows math. 

Although this point of view misses a lot, it captures something important. Some of the most beautiful products of mathematical thought are procedures that achieve powerful results seemingly without any need for conscious thought. The base-ten number system is one of the pinnacles of human ingenuity, enabling the reduction of complex calculations to simple iterative procedures that require only the knowledge of single digit sums and products. And although the utilitarian value of these procedures has been superseded by calculators, their value as ingenious tools remains. Humans love tools and enjoy using them. To this day I add up columns of numbers in my head, just because I can. Of course, calculators are fun tools to use as well and expand the repertoire of explorations that you can make with numbers.

Knowing math means seeing patterns

Another popular view of mathematics is that it is about seeing and understanding patterns. As an approximation, this isn’t bad; it captures a lot of what students learn in mathematics in K–12. An algorithm can be thought of as a sort of pattern. Understanding the steps in the standard algorithm for addition goes well beyond being able to “do the math.” Geometric patterns have a place in developing geometric understanding. 

Still, mathematics as patterns doesn’t quite capture everything. Is a geometric proof a pattern? How do patterns fit into my understanding of how to read a graph? If seeing patterns is understood more generally as seeing structure, then this view of mathematics becomes more accurate. To think of a proof, you often need to see some structure in a geometric figure; to read a graph you need to understand the structure of axes, scales, and coordinates. Two of my favorite practice standards—MP7, look for and make use of structure, and MP8, look for and express regularity in repeated reasoning—add a gloss to “seeing patterns” that is useful in understanding what it means to know mathematics. 

Knowing mathematics means interpreting and expressing mathematics

For me, this gets to the heart of the matter. Interpreting means, for example, reading a mathematical expression and being able to describe the calculation it represents. Expressing means being able to write an equation representing a word problem. Implicit in the words interpreting and expressing is the assumption that whatever you are interpreting or expressing has meaning. After all, we all understand in language arts that reading and writing does not mean reading and writing gobbledygook. And yet, for many students, that’s exactly how they see patterns of symbols they are required to read and write in the course of solving math problems. Faced with solving the equation (x – 2)(x – 9) = 0, the “do the math” approach might lead you to expand the expression on the left to put the equation in standard form and then factor it again or use the quadratic formula (I have seen this!). Whereas interpreting the equation as telling you that the product of two numbers is zero leads to a much quicker way to find the solutions. 

Implications for educational technology

The COVID-19 pandemic has increased interest in technology assisted instruction. Our understanding of what it means to know mathematics guides our desired outcomes from such instruction. There are plenty of technological solutions that can train students to do the math and graphical interfaces can often help students see patterns. But if the goal is to have students attach meaning to their work, then they need to be interacting with another sentient being, a teacher or a student. (Granted, there are some students who can learn math by interacting with their own brains, but such students are rare.) Until we have AIs capable of consciousness, educational technology will have to focus on facilitating discussions among students and between real-life teachers and students. I think this is an underexplored territory in educational technology; whereas there are quite advanced technologies for adaptive testing, handling mathematical input, and making sophisticated mathematical displays, technologies for handling the orchestration of productive mathematical discussions and the implementation of sophisticated instructional routines are underdeveloped.

There’s always a danger with technology that the piece of technology itself is at the center of the table, and everyone is trying to figure out what to do with it. Putting meaningful learning at the center of the table, and then asking what technology can do to facilitate it, is more likely to result in solutions that will truly help teachers and students adapt to the new world we find ourselves in.

Next Steps

Think about the technology you use in the classroom. How does it help students know the math?

William McCallum

Bill McCallum, founder of Illustrative Mathematics, is a University Distinguished Professor of Mathematics at the University of Arizona. He has worked in both mathematics research, in the area of number theory and arithmetical algebraic geometry, and mathematics education, writing textbooks and advising researchers and policy makers. He is a founding member of the Harvard Calculus Consortium and lead author of its college algebra and multivariable calculus texts. In 2009–2010 he was one of the lead writers for the Common Core State Standards in Mathematics. He holds a Ph. D. in Mathematics from Harvard University and a B.Sc. from the University of New South Wales.