What is the most valuable math skill for work and life?

The union of the mathematician with the poet, fervor with measure, passion with correctness, this surely is the ideal. William James

Throughout my working life, I have seen firsthand the critical role mental math can play in business success. I had written about this before and was delighted to find more evidence confirming my experience from an eminently more credentialed source, Dr Jo Boaler the Nominelli Olivier professor of mathematics education at Stanford University.

In her new book Math-ish, Professor Boaler makes a compelling case for an inclusive approach to learning math. The ‘ish’ part is that most valuable skill – our ability to judge whether a numerical answer or measurement is reasonable, to spot obvious mistakes, and to make timely corrections. She suggests it is a skill anyone can master with the right teaching and learning approach.

Suppose we begin all our calculations with a rough estimate of the answer. In that case, we will likely unlock our macro thinking and counterbalance our brain’s tendency to go into precision mode and lose sight of the bigger picture. This valuable interplay between macro and micro perspectives sharpens our thinking, helps us avoid significant mistakes, and occasionally allows us to hone in on deeper insights.

If only we learned math the right way.

Unfortunately, Western countries' math curricula and methods take the opposite approach and concentrate on solving for precise answers. As Professor Boaler points out, this seems to ignore the fact that most of the numerical applications we face daily involve ‘ish’ numbers. Such everyday calculations ask us to solve for expected temperatures, driving speeds and distances, food recipe portions, party costs, or the number of guests.

The way we learn math doesn’t add up. More effort must be made to connect real-world problems with math's capacity to help address them. Conrad Wolfram ’s 2010 Ted Talk makes a similar point, outlining four steps by which math can become part of our arsenal for quantitative thinking: Pose a useful question, abstract it into a computable form, do the computation, and then relate it back to the real world and answer the original question (what we often refer to in consulting as the “so what”).

Much of the value in learning comes not from getting the right solution but from becoming more efficient in solving problems.

How should we learn and teach math?

I was impressed by many of Professor Boaler’s ideas, but let me highlight three most critical takeaways.

1.???? Calculating the benefits of a growth mindset

We can achieve so much more when teachers and students jointly believe that anyone and everyone can learn math to the highest level if they put in the work. The best teachers set positive norms that encourage students to be curious and resilient. They emphasize that their math class is about learning and not about performing. They remind students that what will determine their success is the level of effort they put in. They explain that a depth of understanding is much more important than the speed at which they calculate, and they emphasize that a common sense for numbers comes from asking many questions (“Why does it make sense”?).

Boaler’s work underscores the value of struggle when paired with encouragement. One of my favorite aphorisms is, “Everything is hard before it is easy.” Learning comes from reflecting on difficulties and errors, not from our success. She recommends reinforcing the positive norm: “Mistakes are valuable – mistakes grow your brain.” Indeed, they do, and this is true also for adults, as more and more research on brain plasticity shows.? Conversely, when students are too shielded from struggle and constantly told how smart they are – sooner or later, they don’t build the growth mindset and grit to handle struggle (an issue Jonathan Haidt and Greg Lukianoff study in their book, The Coddling of the American Mind). In most schools, math tests stand out as the only subject where students are not shielded from negative feedback and lacking the right mindset, they start to detest math or assume it’s just not their subject.

The same applies to many professional environments. I mention often how wrong I have been to celebrate the motivational aspects of ‘insecure overachievers.’ While I still prefer some insecurity to overconfidence, the better way is a realization that progress toward excellence comes from struggle. We learn the most when we are at the edge of our understanding. Boaler even encourages teachers to ask their students to draw pictures of themselves standing next to a cliff edge. Likewise, when you experience such dizziness at work (aka imposter syndrome), it means you are about to undergo accelerated learning. Try to relish it.

Nonetheless, struggling and having a growth mindset is easier when you know others believe in your potential. In one experiment in the US, half the students received one extra sentence: “I am giving you this feedback because I believe in you” (the teachers did not know which students received them). That one sentence led to very different results for the group who got it – most markedly among people of color, as these students more often feel that teachers doubt their potential. In his book The Culture Code, Daniel Coyle similarly provided 19 words to make any feedback a lot more effective in professional environments: “I'm giving you these comments because I have very high expectations, and I know that you can reach them.”

2.???? Developing numbers sense

When you practice math with your kids, it may be the first time you’ll discover that they think about simple math calculations differently. Some young kids may quickly see that 19+6 is the same as 20+5; others may count up from 19 six times. When asked to do 38x5, some may convert this to 19 times 10. Others may do 30x5 and add to it 8x5 or do 40x5 and subtract 2x5, and so on.

Boaler points out that it comes as a big surprise to people to learn that there can be so many ways to approach the same numerical problem. Accordingly, much of the value in learning comes not from getting the right solution but from becoming more efficient in solving problems.

Cognitive development is not a matter of genetics, nor should it be a matter of happenstance. Stanford’s Cognitive Neuroscientist Bruce McCandliss has been researching how our brains get rewired for new human abilities to emerge. This is the act of intentionally focusing the mind’s attention by priming it with auditory and visual exercises to activate different brain responses before engaging in a new learning situation. He found that the right stimulus significantly improved learning effectiveness. McCandliss describes this as “capturing lightning in the bottle” and suddenly being able to investigate the lightning much more intensely.

Math teaching which helps students to learn through visual patterns and use them to build mental representation, leads to dramatically better results and everlasting number sense. McCandliss and his colleagues expanded the idea of subitizing (this is the ability to know how many objects we are seeing without counting them - such as with dots on a die) into groupitizing – taking on a more complex set of numbers and being able to chunk and cluster them to parse and add numbers. When assessing kids between the 3rd and 8th grades, the found that the extent to which students were good at groupitizing was a major predictor of math achievements years later. This factor also dramatically reduced the (otherwise heavy) influence of household income in predicting their success. This is just one of several examples of visual and physical models Boaler suggests can have a huge impact on numbers sense.

3.???? Becoming critical thinkers

“Ours is not to reason?why;?just invert and multiply!”

Most students are taught to simply invert and multiply when learning how to divide by a fraction. This robotic approach may help students pass exams, but it often has little lasting learning value. Enduring value comes from reasoning, and exploring the relationship between numbers is foundational for critical thinking.

When asked to divide 1 by ?, kids can rapidly invert it to 1 multiplied by 4/3. Alternatively, they can evaluate how many times three quarters would fit into a space of 1. Similarly, dividing 3/4 by 2/3 may be very hard to visualize. But when you first seek a common denominator, it becomes dividing 9/12 by 8/12 and you can now easily estimate how many times 8/12 fits into 9/12.

Even when kids are taught how to make sense of numbers and think conceptually, rules and standard algorithms can interfere with cognitive development. This occurs when sense-making disappears and is replaced with rule-following. Researchers compared groups that first received five lessons of traditional instruction (focusing on formulas) and then three lessons of conceptual instruction (where students were trying to devise ways to calculate area problems) and groups that received only the three conceptual lessons. In subsequent assessments, the students who learned a lot less overall but had only (the same amount of) conceptual lessons did much better. The first group’s learners developed fixed ideas – the rules interfered with their problem-solving. Teachers often say they don’t have time for conceptual teaching and focus almost exclusively on rule teaching and practicing. This evidence shows that it’s possible to improve both the speed and the quality of teaching.

In a recent McKinsey Global Institute report about the future of work we analyzed which skills will be more or less in demand by 2030. Creativity, critical thinking and decision making, advanced communication and negotiation skills are expected to rise in importance alongside other technological, social, and emotional skills. The skills most in decline are basic quantitative skills. Math as a means without its end will become increasingly meaningless.

We can all play a role in our communities and families to encourage the big shifts that will convince every child that they can thrive with math and that depth of understanding, visualizing, and conceptualizing are a lot more important than the speed of calculating.

Every one of us can develop the math-ish capabilities that matter the most today and will matter even more in the years to come.