A sense of numbers

Ramanujan and John Nash definitely had it. Your neighbourhood grocer has it. And some kids in my son’s elementary school have it. But I’m guessing a lot of us don’t.

I am talking about a “feel for numbers”.

And it’s more than just knowing our multiplication tables. Or the number of zeroes in a million or the number of bytes in a gigabyte.

I’m talking about feeling the passage of time. How many of us can estimate accurately the time needed to read out a 1500-word speech to our teams? 1 minute? 5 minutes? More?

I’m talking about doing the math in our heads quickly. During your company’s onboarding exercise, each of the 30 new employees needs to undergo an 8-minute medical test and we need to wrap up the whole exercise in an hour, how many doctors should we arrange? (Quickly now, please, before the elevator stops!)

I’m saying, it’s a rare person who can sense the extremities of data, if informed that the mean is 5 and standard deviation is 0.8.

I’m saying, not many of us can look at a largish crowd and estimate accurately how many people there are. Two thousand and ten thousand will look the same to us (both are “lots of people”). Ditto with terabytes and petabytes…both look like “more memory than I’ll ever need”, before we’re proved wrong!

This feel for numbers is about speed, acceptable accuracy and familiarity with numbers. It’s about the numbers talking as much to your gut as to your brain.

Why aren’t we all good at this?

It’s a gaggle of reasons.

Some things are intangible, hence tough to grasp. Time is one example. Another example is human-invented metrics like average, standard deviation or correlation coefficient, which can’t be “seen” in nature but are essentially computations.

Other things are tangible, but weren’t taught to us correctly during early years. Examples are distances, areas, weights, fractions, densities, volumes and counts. Instead of “feeling” or “seeing” them, we experienced them as numbers written on paper. That robbed them of the multi-sensory experiences they could have been.

Our understanding of large numbers suffers because we “see but don’t count” when we encounter them. We’ve all seen “hundreds” of people at a busy railway station or rock concert, “thousands” of stars in the night sky and “millions” of grains of sand on a beach. But this kind of “seeing” is fleeting and passive. We don’t spending time and effort in actually estimating how many we saw. As a result, we will be fairly accurate when estimating the number of biscuits in a small box, but our answers will be way off when counting larger numbers.

We aren’t quick at mental calculations (unlike the grocer) because we don’t do it often enough, waiting for the answer to come from the calculator or the other person we’re talking to. Practice matters here.

What can be done?

I read somewhere recently, “The best time to plant a tree was 10 years ago. The second best time is now”. It’s the same with developing an intuitive sense of numbers.

Early education does much to make or mar our understanding of all things quantitative.
If you have a small child, let them “sense” the numbers before you prod them to “read” numbers. At Swadhaa, the Waldorf Kindergarten my kids attend, primary schoolers learn about distance in a very “feeling-sensing” way. They first construct their own metre-scale out of a strip of wood, getting a sense of millimetres and centimetres. Then, they step out of school and walk a kilometer, measuring the road a thousand times with the metre scale.
More examples…
To learn about weights, they sit on one side of a huge beam balance while the teacher sits on the other.
To “feel” multiplication, they use a number wheel, (see picture, credit Jennifer Compton) doing as much work with their hands and eyes, as with their brains. And they play games, where they rhythmically clap or stomp their feet in sets of 3, then 6 and so on. It’s almost as if the “body” learns the math before the “mind” does it.


Such learning is deeper… haven’t we often noticed that we forget names and numbers and facts within a short time, but our body easily recalls how to swim or ride a bike after a gap of years.

But if you (just like me) missed the bus at that age, what are our choices now, as adults?

For one, immersion and effortful engagement. We live in a sea of numbers, and if you take the time, it’s possible to dance with these numbers. At the airport, guess how many people are there in the boarding queue, then actually count them and compare the results… you’ll find that you’re getting better with time. During the office commute, estimate time and distance, and then cross check using your watch and the car’s odometer. In the kitchen, when the cake recipe asks you to measure out “115 grams of butter”, take a moment to weigh it (feel it) in your palm.

Some other things aren’t amenable to such casual rumination. So, a more “classroom style” approach helps.

To develop an understanding of percentages (fractions), I sometimes try this exercise in a workshop. Take a sheet of paper, tear it into half. Tear one part into half again. Repeat. Just the act of handling and watching ever smaller pieces of paper helps get a feel of fractions. We can see “just how small is one sixteenth, or 6.25%”.

To develop a sense of data distributions, I have learners imagine birds sitting on a stick. Then, the “average” is the fulcrum, the point where the stick can be balanced on a finger. The “median” bird has an equal number of neighbours on its left and right. Given 2 such sticks with birds sitting on each, the stick with lesser “standard deviation” is the one where the birds crowd around the center. Later, learners double check what their intuition says, by doing the math with real numbers.

To help develop an ability of quick estimation, exercises like this help: 43 people per batch * 3.5 hours per person * 19 batches = How many hours in all? It sometimes devolves to a problem of which 2 numbers to multiply first, and where can a rounding-off be done. There are several ways… my mental chatter was “approximately 40*4*20, but a little less”. I arrived at 3000 hours, an acceptably accurate in some situations (The exact answer is 2859.5 hours)

The practical question

All this is fine, if being good with numbers is intrinsically valuable for you (Csikszentmihalyi’s concept of Flow comes to mind)

But, if it’s a means to an end, is the effort worth it? Could having a way with numbers confer an edge? Could it help you get ahead in life?

My assertion is this: an understanding of time, physical space, magnitudes and proportions underlies many areas of decision making. With an intuitive grasp of these, we’re faster and more likely to make the right decisions. Without it, we’ll still manage, but only after much delay and struggle with the minutiae of spreadsheet models.

If you’re an account manager / salesperson on the move, and good with numbers you’d spot opportunities faster, and act quickly without losing momentum. You could do the mental math quickly and make a decision, rather than say “Umm… Could you please email this to me. I’ll look at the numbers more closely, and I’ll get back to you”. In deal making, seizing the moment is key.

If your work involves planning and estimation, the quality of your decisions would go up, if you had a feel for “large” numbers. Example: How much circulation space should you plan for, if you’re expecting footfalls of 10,000 to 15,000 people during a 3-day symposium? If this boggles your mind, you’re losing time and you may make arrangements that are either inadequate or too extravagant.

If you’re a data scientist, you could take a look at what the computer spat out and say “Wait a minute, this doesn’t seem right. Let’s inspect the input parameters again”. It’s like being able to smell a rat before you see it.

My perspective and yours

My point of view is that of someone who likes numbers… I compulsively count, measure and estimate (which does get in the way of simply savouring the moment, I admit). So, for me, number crunching doesn’t have to be “profitable”. But I’m curious about what others think.

If you’re reading this article, I’d like to know your perspective. Is quickness with numbers all that useful in your work? Which kind of number-work do you find most challenging? Does any of you specialise in numbers which are either very large or very small?

I’m all ears.

Photo credit: Heather Zabriskie via Unsplash and Jennifer Compton via Etsy