From this HuffPost article, written by a mathematician talking about the shift from procedural math knowledge (doing problems) to what we really need to know about math these days:
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.
He goes on to name this understanding as number sense:
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.”
It comes down to deep understanding of the “why” and making connections. So tell me again why a lot of the math education I’ve seen in the past few years is still all about making sure students know how to do procedures without the understanding or connections between them attached?
We can do better.