keith_devlin's picture
Mathematician; Executive Director, H-STAR Institute, Stanford; Author, Finding Fibonacci
Number Sense

When I graduated with a bachelors degree in mathematics from one of the most prestigious university mathematics programs in the world (Kings College London) in 1968, I had acquired a set of skills that guaranteed full employment, wherever I chose to go, for the then-foreseeable future—a state of affairs that had been in existence ever since modern mathematics began some three thousand years earlier. By the turn of the new Millennium, however, just over thirty years later, those skills were essentially worthless, having been very effectively outsourced to machines that did it faster and more reliably, and were made widely available with the onset of first desktop- and then cloud-computing. In a single lifetime, then, I experienced a dramatic change in the nature of mathematics and how it played a role in society.

The shift began with the introduction of the digital arithmetic calculator in the 1960s, which rendered obsolete the need for humans to master the ancient art of mental arithmetical calculation. Over the succeeding decades, the scope of algorithms developed to perform mathematical procedures steadily expanded, culminating in the creation of desktop and cloud-based mathematical computation systems that can execute pretty well any mathematical procedure, solving—accurately and in a fraction of a second—any mathematical problem formulated with sufficient precision (a bar that allows in all the exam questions I and any other math student faced throughout our entire school and university careers).

So what, then, remains in mathematics that people need to master? The answer is, the set of skills required to make effective use of those powerful new (procedural) mathematical tools we can access from our smartphone. Whereas it used to be the case that humans had to master the computational skills required to carry out various mathematical procedures (adding and multiplying numbers, inverting matrices, solving polynomial equations, differentiating analytic functions, solving differential equations, etc.), what is required today is a sufficiently deep understanding of all those procedures, and the underlying concepts they are built on, in order to know when, and how, to use those digitally-implemented tools effectively, productively, and safely.

The most basic of those new skills is number sense. (The other important one is mathematical thinking. But whereas the latter is important only for those going into STEM careers, number sense is a crucial 21st century life-skill for everyone.) Descriptions of the term “number sense” generally run along the lines of “fluidity and flexibility with numbers, a sense of what numbers mean, and an ability to use mental mathematics to negotiate the world and make comparisons.” The well-known mathematics educator Marilyn Burns, in her book, About Teaching Mathematics, describes students with a strong number sense like this: “[They] can think and reason flexibly with numbers, use numbers to solve problems, spot unreasonable answers, understand how numbers can be taken apart and put together in different ways, see connections among operations, figure mentally, and make reasonable estimates.”

In 1989, the US-based National Council of Teachers identified the following five components that characterize number sense: number meaning, number relationships, number magnitude, operations involving numbers and referents for number, and referents for numbers and quantities.

Though to outsiders, mathematics teaching designed to develop number sense can seem “fuzzy” and “imprecise,” it has been well demonstrated that children who do not acquire number sense early in their mathematics education struggle throughout their entire subsequent school and college years, and generally find themselves cut off from any career that requires some mathematical ability.

That outsiders’ misperception is understandable. Compared to the rigid, rule-based, right-or-wrong precision of the math taught in my schooldays, number sense (and mathematical thinking) do seem fuzzy and imprecise. But the fuzziness and imprecision is precisely why that is such an important aspect of mathematics in an era where the rule-based precise part is done by machines. The human brain compares miserably with the digital computer when it comes to performing rule-based procedures. But that human mind can bring something that computers cannot begin to do, and maybe never will: understanding. Desktop-computer and cloud-based mathematics systems provide useful tools to solve the mathematical aspects of real-world problems. But without a human in the driving seat, those tools are totally useless. And high among the “driving abilities” required to do that is number sense.

If you are a parent of a child in the K-12 system, there is today just one thing you should ensure your offspring has mastered in the math class by the time they graduate: number sense. Once they have that, any specific concept or procedure that you or they will find listed in the K-12 curriculum can be mastered quickly and easily as and when required. An analogous state of affairs arises at the college level, with the much broader mathematical thinking in place of number sense.