Learning math is often about learning specific math topics. But it can also be useful to step back and take a higher-level view of math learning. Keith Devlin, professor of mathematics at Stanford University and creator of the popular Introduction to Mathematical Thinking course on Coursera, says modern students of math have to master two types of thinking:

- In K-12, the goal is to develop strong
*number sense*. - In college, those who continue their studies in STEM need to learn
*mathematical thinking*.

## Number Sense

In a 2017 article, Devlin uses the definition of number sense from *About Teaching Mathematics* by Marilyn Burns, who writes that number sense is the ability to:

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.

Rather than K-12 math, let’s apply this definition to technical work. Imagine that you’re a software developer working on a high-traffic website. A best practice when developing online systems is to collect telemetry and use it to inform feature design. How would each of the components of number sense apply to this scenario?

**Think and reason flexibly with numbers**

It’s possible to think and reason flexibly about software design *without* using numbers. You can collect verbatim feedback from customers and use that to decide what change to make to a product. But as the number of users gets larger, it becomes more difficult to take action on individual pieces of feedback. Instead, you have to aggregate feedback and look for trends. That requires making judgments about data, like the percentage of users who abandon an online shopping cart. The telemetry system gives you the data, but humans have to decide what data to collect and how to interpret it.

**Use numbers to solve problems**

After you collect and interpret the data, you have to do something with the results. For example, you could plan to change your design in a way that makes the checkout process easier for users. To keep using number sense in this phase, you can use techniques like A/B testing to make sure your design changes are moving your metrics in the right direction.

**Spot unreasonable answers**

Telemetry can give you data, but it can’t tell you if the data is realistic. For telemetry to be useful, humans have to look at it and perform a sanity check. If 97.2% of carts are abandoned, but your company is still profitable, there may be something wrong with the telemetry.

**Understand how numbers can be taken apart and put together in different ways**

In the shopping cart scenario, one way to take numbers apart and put them together is to evaluate how one metric affects another metric. For example, maybe the total cost of the items in the cart correlates with the chance of cart abandonment. That might lead you to present a reward, like a coupon, for carts over a certain size.

**See connections among operations**

In K-12 math, connections among operations are things like the connection between division and subtraction, or logarithms and exponentiation. In the shopping cart scenario, good number sense might mean good intuition about how a change to the system might affect the telemetry data.

**Figure mentally**

Data science tools are powerful. They let you experiment with your source data and find hidden insights. But if you’re brainstorming with team members on the whiteboard, switching to a computer can break the flow of discussion. This is when mental calculation comes in handy. If you can remember and manipulate data in your head, you can bring it to any discussion, whether you’re in a conference room or brainstorming while you’re out for a run.

**Make reasonable estimates**

In interviews for technical positions, companies have been known to ask questions, like “How many piano tuners are there in Chicago?” The goal of these so-called *fermi problems* or *estimation problems* is to test a candidate’s number sense. Companies are looking for candidates who know that a large metropolitan area in the US has about 10 million people, not 1 million or 100 million, and who can combine this estimate with other estimate to find an answer without getting bogged down in precise calculation.

## Mathematical Thinking

In more advanced K-12 math classes, the focus shifts from manipulating numbers to manipulating symbols that represent numbers, or manipulating mathematical structures like sets and functions. But even at this level of abstraction, students still learn *procedures* for finding results. For example, there are procedures for factoring various kinds of equations.

Keith Devlin defines mathematical thinking skill as the ability to solve a mathematical problem even when you don’t have a procedure to use. In other words, you just have to think it through.

In his number sense article, Devlin points out that services like Wolfram Alpha can almost instantly solve “any mathematical problem formulated with sufficient precision.” Textbook problems fall into this category if they don’t involve writing proofs or other creative thinking.

What’s the point of teaching people to carry out procedures that computers can do faster and more accurately? One reason is that computers, as the saying goes, can only do what you tell them to do. It takes a human to make sure the computers are being told to do the right things and “spot unreasonable answers” that the computer might come up with. Doing that requires knowledge of how mathematical procedures work. So the focus of mathematical education should be on understanding procedures, not carrying them out manually.

## Learning Number Sense and Mathematical Thinking

To improve your number sense and get better at mathematical thinking, approach mathematical subjects with those skills in mind. For number sense, the main goal is to do your own calculation and manipulation of numbers, rather than relying on computers. When an opportunity comes up in work, study, or everyday life, start with mental calculations before verifying them electronically if necessary.

Since mathematical thinking is about solving problems without procedures, look for opportunities to practice solving problems that don’t lend themselves to step-by-step solutions. Brilliant takes that approach more than Khan Academy does, but any math problem can work if you approach it from first principles rather than by looking for a cookbook solution.

(Image credit: Karola Riegler)

*I’m writing about discrete math and competitive programming this year. For an introduction, see A Project for 2019. To read the whole series, see my Discrete Math category page.*