Math Without Numbers, or Words: A Case Study of Individualized Learning

How Language Data Solved a Math Problem

A Case Study on Individualized Learning

At the end of first grade, the student who was destined to become Cajal’s first pupil was behind in his addition facts. By the end of second grade, he was behind in his addition and his subtraction facts—and had become so convinced that he was “no good at math” that it had become a significant trigger for his school-based anxiety.

And yet, at home his favorite pursuit was to drum, and over the course of a year he had become an extraordinary, self-taught drummer who improvised with highly complex rhythms and shifting time signatures. When he got access to a synthesizer for the first time, he immediately sat down and composed a song over five minutes in length that was so perfectly divided into segments that we were able to chart it out on a graph. Along the way, he had intuitively leveraged intervals well over an octave, perfectly nailing consonant and dissonant combinations to suit his purposes. It was hard to believe this kid could really be “no good at math.”

At first, it seemed there was nothing for it but to soldier on. Perhaps, we thought, we were just seeing the results of what had become significant anxiety, sending him into a state of freeze-fight-flight that is hardly conducive to computational success. Day after day, we dutifully pulled out cheerful yellow ‘one’ cubes, ‘ten’ rods and ‘hundred’ blocks and showed him how they could be regrouped into tens to make additional “easier.” And every day those cubes, rods and blocks would end up on the floor—and his conviction that he could not do math was only stronger than it had been the day before. “I just can’t,” he would say, his face unmistakably full of self-loathing.

But when we dug more deeply into the significant quantities of data we had on Kaido’s neuropsychological, auditory processing, visual processing and language profiles, a different picture began to emerge. Integrating data points across his complex and multi-variate profile, we saw that he was challenged by expressive language, the visual processing of columns and symbol relationships, and his output organization deficit made the task of breaking orally-presented directions into discrete steps, prioritizing and motor planning based on them inordinately difficult. Thus, we set for ourselves a mandate, to teach him math without using words, symbols or column alignments.

At first, this task seemed utterly impossible: math curriculum, so far as our certified teacher had ever seen it done before, consisted of a linear progression of bundled computational, numeric and linguistic concepts. She wondered how it was possible that she had taught math for 17 years in a public school, without ever having been taught to understand the granular cognitive skills engaged in the process—and thus which aspects of math were to blame when she could easily see a student struggle.

Going back to his individualized profile, we zeroed in on the areas that had been identified as unusually strong, and the key fact that across music and athletics, we saw that he could teach himself to do pretty much anything with his body—a handy “workaround” to find outlets for his drive to learn while avoiding his own challenges with processing external instruction. Thus, we shifted from teaching him math, to making the tools available for him to discover math himself, by presenting colored, multi-length rods without using the words that sound like math.

“How many pinks could you trade in for this purple?”

“Can you hand me 17? 23? How many different combinations can you make?”

“Can you give me four rods of the same color that are the same as this combination here?”

Suddenly, he was handing us the answers literally faster than we could check his work. Combinations of 23 came flying up out of the box of rods—not in ten rods but in deftly-assembled combinations of threes and fours that were far from intuitive to the rest of us. All those weeks of hitting our head against the concept of regrouping into tens suddenly made sense: unlike us, his brain doesn’t rely on base 10 and is just as comfortable in base 3 or 4—so our insistence on regrouping into 10’s had seemed to him like a confusing and idiosyncratic exercise. It turns out he’s just so good at math that the parts of his brain that we use to describe math can’t keep up, so traditional ways of teaching it to him simply hadn’t made sense.

This discovery transformed his sense of self—and the way we teach. By removing what was for him a numeric language and symbol relationship problem, he was able to access his high math aptitude for the first time, and began flying through new mathematical concepts at a rate he never thought possible. Three weeks later, the child who had never shown any interest in rulers was cutting a pipe into precision lengths and made his own pan pipe —without using a ruler at all. An entire unit on measuring angles, the difference between acute and obtuse angles and how two angles together add up to 90 degrees was covered in one very enjoyable morning kicking a soccer ball into a goal. Using string and a protractor, he started at ninety degrees and decreased by ten degrees with each attempt, measuring each time, while viscerally experiencing how much harder that goal was becoming no matter where he stood on that string.

Meanwhile, leveraging well-established occupational therapy techniques, we added movement-based games like bouncing a basketball or running an obstacle course while skip-counting, to improve his facility to recall the math language he was missing. Rather than drill a sheet of 15 math facts every day, we endeavor to keep up with his appetite for absorbing new mathematical concepts through their real-world, tangible applications, while strategically targeting the language he’s missing to express them—all in a way that feels like fun. And “I can’t do math” changed to “Wow - I’m really good at this!” What had started as his biggest academic anxiety had turned into his superpower.

This child’s math transformation was dramatic—but it wasn’t happenstance. By breaking academic tasks down into very granular, component cognitive tasks we can identify and ‘quarantine’ the cause of the problem, and then give those problem skills the focused attention they deserve. We describe our data-driven process for individualizing instruction here. And, when we integrate motor-sensory input in the learning process, we increase the neural networks involved so that ‘fun’ becomes doubly effective, not only reducing anxieties but leading children to retain the concepts far longer and without any of the drill.

This is but one example of how taking the time to do a deep dive into a child’s profile both within and across functional areas can transform their learning outcomes. The power and the promise of today’s science is in our ability to methodically and analytically determine how we can best help each child experience their own gifts. When we instead impose a ‘one-size-fits-all’ approach, or cycle through one experiment after another until we hit on something that fits, we burn through their confidence, their trust and their sense of self-worth. Before we had the science, that was the best we could do. Now that we have the science, it’s up to us to use it.