Math is interesting in that we routinely task young children with learning very sophisticated concepts. We now teach kids in third grade things that took mathematicians centuries to figure out. My research revolves around gaining better insight into how people understand abstract math patterns and concepts, and why some people are better at these skills than others. Hopefully, by better understanding math cognition, teachers will be able to develop better instruction and curricula.
Using Math Understanding as an Indicator
There are many ways in which simple math abilities can be an indicator of foundational skills and future performance in mathematics. It’s a question I’m very interested in: Why are some people good at recognizing math patterns and some aren’t?
For example, if a person is bad at grasping negative numbers, it might be due to something so basic as not having a sense of the size of numbers or the ability to conceptualize them. There are two general intuitive notions when it comes to negative numbers – the first is seeing them as laid out on a number line and the second is imagining them as “positive” and “negative” particles, with one cancelling out the other. In our experiments, we find out which way people are understanding numbers, and how their understanding changes over development.
This seems counterintuitive; you would think a person’s ACT score would be determined by the more advanced concepts taught in high school, like trigonometry. But our research indicates that standardized test performance can be predicted from how well (or how poorly) people do on very simple number tests. This suggests that these foundational skills are incredibly important. We may be using very simple exercises in these experiments, the kind even an elementary-age student could do. For example, one task might be identifying whether there are an odd or even number of objects laid out on a table. However, each set might be arranged in a different pattern, for example, paired up in groups arranged in a row. It turns out that some people are much better at picking up on the math patterns in these simple tasks.
The people that are more sensitive to mathematical structure are the ones that generally earn higher ACT scores. Think of it like music: musicians are can hear the structure of music and different tones, while other people struggle. It’s the same with math for some people; they pick up on math patterns early on, and this advantage might pay off later in their schooling.
Adding it Up
The results of our research offer a path to better understanding. We still don’t know exactly why basic numerical and arithmetic skills are an indicator of a person’s standardized test performance. But our findings (along with those of other researchers) provide a clearer picture about what it means to understand mathematics. A future goal of our work is to provide more effective instruction for students, and we think we’ve found a path towards improvement. More generally, if we can identify the things which kids who are bad at math do differently on an individual level, then we can figure out what’s different about the way they think. A student who is lower in a more general ability such as executive function and is also bad at math might indicate a relationship between the two otherwise seemingly unrelated aspects of thinking. That insight could lead to a targeted intervention for the student to improve their executive function, with the goal of also hopefully improving their math skills.
In the Cognitive Architecture Lab, my team and I are engaged in a range of research projects related to mathematical thinking and understanding, problem solving, and other complex forms of cognition. We also build computational models that embody the explanations we’re trying to offer and test those models in experiments. The reason our work is different than most cognitive science research – and why we’re housed in the College of Education and Human Development – is because we are ultimately interested in developing theories that will inform the work of our colleagues in education and help them teach math more effectively in the classroom.
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