Looking back at my own education, I realize I never really understood what grades meant. In school I always had straight A’s in math - through my ‘math/physics magnet middle/high school’ and the early university years. Then, near the end of my undergraduate studies and into graduate level, the abstraction level finally outran me. What once felt easy became a fog of symbols and proofs I could barely follow. My grades dropped from A’s to barely passing.
Now my son is in middle school, facing math tests that seem simple compared with what I used to do, and yet he usualy brings home B’s and C’s. At first I was worried he was struggling. But after thinking about how math actually builds from one layer to the next, I began to see grades differently…
Looking back at my own education, I realize I never really understood what grades meant. In school I always had straight A’s in math - through my ‘math/physics magnet middle/high school’ and the early university years. Then, near the end of my undergraduate studies and into graduate level, the abstraction level finally outran me. What once felt easy became a fog of symbols and proofs I could barely follow. My grades dropped from A’s to barely passing.
Now my son is in middle school, facing math tests that seem simple compared with what I used to do, and yet he usualy brings home B’s and C’s. At first I was worried he was struggling. But after thinking about how math actually builds from one layer to the next, I began to see grades differently. Math is cumulative: arithmetic => fractions => basic algebra => trigonometry => calculus => abstract algebra and other abstract branches. To pass any stage, even with a C, a student must already handle the lower levels - fractions, negative numbers, operations, with near-A competence. In that sense, every “passing” grade is built on a pyramid of earlier mastery. A C in algebra implies solid command of arithmetic; otherwise the algebra would collapse. And mathematics is abstract all the way down. Even counting natural numbers is an act of abstraction - something crows and primates can do. As we climb the ladder, the required abstraction doesn’t suddenly appear; it grows, from trivial symbolic mapping to ever more demanding mental models. And eventually each person reaches their ceiling - the point where abstraction exceeds what their working memory and reasoning capacity can handle.
This leads to what I see as a two-axis model of learning math. One axis is vertical, measuring mastery of the current/latest subject - the top of the stack of skills built layer by layer. The other is horizontal, measuring the mind’s abstraction capacity - how far one can extend that structure before it becomes too complex to hold. Grades only measure the first axis. The second one, which defines how far a person can ultimately go, remains totally invisible. Piaget hinted at this when describing the “formal operational” stage — the onset of abstract reasoning — but reality is more continuous. The ability to handle higher abstraction isn’t a stage that suddenly starts; it’s a spectrum that stretches through life, unevenly across individuals.
And here lies the problem: all education systems remain centred on test scores that capture only the vertical axis. It rewards fluency and accuracy but ignores the horizontal reach of abstraction that truly defines mathematical potential. Until education finds a way to see and nurture that dimension, grades will continue to tell only half the story, and I would argue not the most important one.