If you are connected with the world of K-12 mathematics education, it’s highly unlikely that a day will go by without you uttering, writing, hearing, or reading the term “number sense”. In contrast everyone else on the planet would be hard pressed to describe what it is. Though entering the term into Google will return close to 38 million hits, it has yet to enter the world’s collective consciousness. Stanford mathematician Keith Devlin explains what it is.
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, I experienced first-hand a dramatic change in the nature of mathematics and how it played a role in society.
The shift began with the introduction of the electronic 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 packages such as Mathematica and cloud-based systems such as Wolfram Alpha 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.
In many ways, that change in the society-required focus of mathematics education echoed a change in the nature of mathematics as an intellectual discipline that had taken placed in the late nineteenth century. Beginning in Germany (in particular, the small university town of Goettingen), the primary focus of mathematics shifted dramatically at that time from executing procedures to solve problems, to analyzing and understanding properties of, and relationships between, abstract mathematical concepts.
The most basic of today’s new mathematical 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 2007 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 (in the nature and to the degree required, given that procedural execution can be done by machine) quickly and easily as and when required. An analogous state of affairs arises at the college level, with the much broader notion of mathematical thinking in place of number sense.
Make no mistake about it, acquiring that modern-day mathematical skillset definitely requires spending time carrying out the various procedures. Your child or children will still spend time “doing math” in the way you remember. But whereas the focus used to be on mastering the skills with the goal of carrying out the procedures accurately — something that, thanks to the learning capacity of the human brain, could be achieved without deep, conceptual understanding — the focus today is on that conceptual understanding. That is a very different goal, and quite frankly a much more difficult one to reach.
This, by the way, is the new state of affairs that the mathematics Common Core was created to address. Outsiders, including politicians in search of populist issues to incite voters and others with an axe to grind, have derided caricatured portrayals of this important new educational goal, by describing it as “woolly” and “fuzzy”. But I disposed of that uninformed red herring already. The fact is, number sense is (rightly, and importantly) the primary focus of 21st Century K-12 mathematics education that millions of children around the world are receiving today. Children who are not getting such an education are going to be severely handicapped in the world they are being educated to inhabit.
In a follow-up article, I look at some of the research behind the move away from computational mastery of procedures to the development of number sense in the K-12 education world, and say something about how it can best be taught.
This article is an expanded version of an essay published on Edge.org on January 1, 2017.