Automaticity in math: getting kids to stop solving problems with inefficient methods
Tags: Staffroom

Many early grade math programs teach students to solve addition and subtraction problems by representing quantities with a physical symbol, like lines. To add, kids simply draw lines for both numbers and count them up. To subtract, kids draw lines for the starting quantity (the “minuend”), then cross out the subtracted quantity and count what’s left.

Photo by Crissy Jarvis on Unsplash

By using lines for subtraction like this, what kids are actually doing is solving by “counting back” because they can’t compute yet — they haven’t memorised that 8–5 = 3, so to simplify the expression, they physically cross out 5 of the 8 lines and find that only 3 lines remain, so the answer must be 3.

This is all fine, and even necessary, in the beginning stages of numeracy development, but it certainly can’t go on forever. As soon as digits get higher than 20, the strategy becomes painfully inefficient, and as soon as the problems require regrouping, it becomes much more difficult to apply. At some point, kids have to stop using it, or they won’t really learn to add or subtract.

In fact, proctors for the Early Grade Mathematics Assessment are required to prompt students to use another strategy if they attempt to solve problems with “inefficient methods” like tick marks, and record it if they don’t, because arriving at a correct answer that way isn’t really the same thing as doing the math.

We recently discovered that relying on the line method for too long is getting in the way of achievement for some of our kids. Results from a recent study from Oxford Policy Management showed that while 91% of second graders at Bridge schools could solve at least 3 out of 4 simple subtraction problems, only 44% could solve 3 out of 4 problems that involved regrouping with 2 or 3 digits — like 44–27. This particular struggle is not unique, given that Bridge students in this study performed on par with students in other Lagos private schools, and better than public schools. Some observation notes from the study’s author revealed that kids were attempting to solve problems like that by drawing sticks, like this:

Solving maths problems with ‘sticks’.

The kids sampled in the Lagos study are not alone. An internal study of our students in the Liberian Education Advancement Program ([LEAP] formerly Partnership Schools for Liberia) revealed a similar challenge in performance on EGMA’s Subtraction 2 subtask, which included problems like 25–9 and 53–19.

Both of these data points have made it obvious that there is a skill gap between counting lines and true automaticity in computation that our program hasn’t yet addressed effectively. For our instructional design team, the question is how to systematically and methodically bridge that gap. Putting a problem on the board one day out of the blue and telling kids they can’t draw lines doesn’t mean that they suddenly know how to compute. Even many more reps of problem after problem won’t change the fact that kids still don’t know that 9–7 = 2 without counting it first.

Kids have to know that 9–7 = 2.

And that 9–2 = 7.

And because of that, they have to know that 2 + 7 = 9, as does 7 + 2.

In other words, the way to span the gulf between counting and computation is to teach kids that they don’t need to count as soon as they know the relationships between numbers — or “fact families” — cold. One program we’ve learned a lot from, Connecting Math Concepts, extends the definition of fact families beyond digits 1–10. If kids are fluent in the relationships between 9, 4 and 13, for example, solving a problem like 213–84, even with the regrouping, can become a natural extension.

Our program, we concluded, needs to teach fact families better. This might not seem like a revolution, but for a team dedicated to educating as many of the world’s kids as efficiently as possible, realisations like this make an outsized difference, especially relative to the size of the fix. The intervention itself amounts to a simple drill a day — carefully planned and sequenced — just five minutes of connecting three numbers with plus and minus signs until counting up and back is a thing of the past. No need for a teacher retraining, no tech or new books, not even flashcards. Just a blackboard, pencils, and a few sets of numbers. And no more lines.