What does it mean to be good at math? There are students who were good at math before they hit algebra, and then struggled. There are students who were good at math, but just weren’t great at the proofs of geometry class. There are kids who were good at math until they hit calculus. There are kids who are good at math, but just can’t do word problems. There are kids who are good at math, but keep making slopping mistakes. There are kids who are good at math so long as they have already learned how to solve that kind of problem, but particularly struggle when faced with novel problems.
So, what does it mean to be good at math?
Can a student be really good at math if they struggle with algebra, proofs, calculus, word problems, novel problem and make sloppy arithmetic mistakes? Clearly not. These things are all aspects of mathematics. The best math students excel at all of them and the worst excel at none. But most students are better at some and worse at others.
When we have large constructs (e.g., math), but students different in which parts they are better at and which parts they are worse, it is multi-dimensional. Math is not one thing; it is not unidimensional.
English Language Arts is not just one thing, either. Reading is not just one thing, and neither is writing. One can be a good speller, but have poor command of conventions of formal grammar. One can write good sentences, but struggle with developing a single cohesive paragraph. One can struggle to but together a cohesive piece that organizes and ideas and supports for it. And quite differently, one can write imaginatively—a certain kind of creativity. One might be good at writing evocative descriptions, or real-seeming characters. One might imagine interesting plots, or write realistic dialogue. Reading also has many components that different readers are better or worse at.
Not only can people differ in which dimensions of a larger construct they are good at, the kinds of lessons and practice that might help to improve offer differ from dimension to dimension even within a construct. Learning to be a better speller is a very different process than learning to write real-seeming characters. Learning to be more careful with arithmetic is different than learning to solve word problems.
The thing is, it’s not just that mathematics is multi-dimensional. Even arithmetic is itself multi-dimensional. Even multiplication is multi-dimensional. Even single-digit multiplication is multi-dimensional. When someone learns their multiplication tables, they can be better at some parts of it than others. 2’s, 5’s and 10’s are easy. The others…well, there are tricks and there is memorization. If we all focused on 8’s first, we might know them better than 6’s, but we tend to focus on 6’s before 8’s. Eventually, however, when we are past those learning stages, we process all of that complexity more automatically and the dimensionality of multiplications tables reduces. It might even become unidimensional, differing by our level of command with the individual differences that we had when we were first learning them. Some people know them all and the ones who don’t tend to make the same mistakes. That is, once it is safe to assume that we obtained the level of proficiency with single-digit arithmetic that we are going to obtain, it is unidimensional—but that is past the point when it is a skill worth measuring.
So, some people remain better at algebra, while others might remain better at the reasoning skills of proofs, and others better at the diligent care of avoiding slopping mistakes. Similarly, some writers are better at dialogue, others at character and others at plot. Moreover, science, social studies, foreign language, psychology, each sport and most everything is actually multi-dimensional.
Even sprinting—running a footrace—is multi-dimensional. Track and & field coaches talk about the biomechanics of i) the start, ii), acceleration, iii) drive and iv) deceleration—though some think there are more and some think there are fewer dimensions. Thinking through this example puts a lie to the idea that unidimensionality can be meaningfully built of a constant combination of separate components. Different sprint distances (e.g., 10m, 40m, 100m, 200m) each constitute a different ratio of these different components, and the is not an absolute or definitive reference for what ratio represents sprinting. It is always an arbitrary decision which one to favor.
So, from an educational measurement perspective what is unidimensionality? If we care at all about the substance and what we are measuring, then unidimensionality is an arbitrary fiction created to serve some convenience—and perhaps never even able to serve that convenience well.