Learning basic arithmetic is a foundation of early math education. Algebra, trigonometry, and calculus are built on, among other things, the ability to quickly and easily solve math equations. Being able to solve math problems is also important for more general life purposes, like tipping, or paying taxes.
For that reason, students should be able to quickly solve simple math problems, but also be able to know the algorithm used to solve any kind of problem. As this implies, mathematical thinking is made up of many cognitive sub-processes and components.
In light of this complexity, how can we effectively teach math’s many aspects?
Multiplication, at least when I was a young nerd, was taught in two ways. For 1 × 1 through 12 × 12 we memorized multiplications tables. From what I gather on Youtube, similar efforts are ongoing today:
But we also learned how to multiply several-digit numbers algorithmically. Without a pen and paper, the multiplication algorithm actually relies on being able to multiply single digits with other single digits. The figure below, courtesy of Wikipedia, provides an example:
If a student is struggling with multiplication (or any aspect of math education, for that matter), it’s not always clear which aspect of a student is having difficulty with. Do they not know their multiplication tables? Or are they having trouble recalling the sub-products? Do they forget the steps of the algorithm?
Looking at accuracy or speed may not provide enough information—a wrong answer could be wrong for any of these reasons. Of course, asking students what they think they might be struggling with is important, but in general humans have pretty poor insight into their own cognitive processes.
To circumvent this issue, a recent study in the Psychonomic Society’s journal Cognitive, Affective, and Behavioral Neuroscience used a combined neuroimaging approach to observe 24 10-12 year-old children while they solved multiplication problems.
The children completed multiplication problems by writing the answers on a tablet while speed, accuracy, and brain activity were recorded. A schematic of the experimental design can be seen in the figure below, with a sample multiplication problem in panel a); the array of electrodes and fNIRS sensors in panel b); and a demonstration of the EEG-fNIRS setup in panel c):
The team of researchers led by Mojtaba Soltanlou measured children’s brain activity to see which brain regions and what brain dynamics were common when the multiplication involved two one-digit numbers (7 × 4) versus one two-digit and one one-digit number (16 × 5).
In adults, solving the more difficult multiplication problems results in more parietal lobe activation, which researchers interpret as mentally manipulating the numbers. It also resulted in more activation in the inferior frontal gyrus, which is interpreted as reflecting the higher working memory and attentional demands of the more difficult problems.
Would the same be true in children?
To address this question, Soltanlou and colleagues used two neuroimaging measures simultaneously—functional near-infrared spectroscopy (fNIRS) and electroencephelography (EEG). FNIRS uses infrared light to non-invasively measure, similar to functional magnetic resonance imaging (fMRI), the ratio of oxygenated to deoxygenated hemoglobin in the brain. This serves as a proxy for neural firing. EEG measures the sum of electrical activity from the scalp. The two make a dynamic pair, because fNIRS has high spatial resolution, while EEG has excellent temporal resolution.
The brain activation results from fNIRS can be seen in the next figure below. The findings from the adult studies were largely replicated with fNIRS—children recruit additional frontal regions during more difficult multiplication problems. Like adults, those problems require the additional recruitment of working memory and attentional resources. This can be seen most clearly in the Contrast brain maps below (which show differences in activation; two-digit minus one-digit), with increased activation in left inferior frontal gyrus.
The results from the EEG measures show evidence (by measuring the amount of desynchrony versus synchrony) that frontal regions were being more heavily recruited during the two-digit multiplication.
Based on previous research, this adds evidence to the authors’ contention that the more difficult math problems engaged increased attentional effort and executive control.
The behavioral results supported this story as well. In addition to getting more two-digit problems wrong, and taking longer to solve them, children were more likely to use a procedural strategy compared to a retrieval strategy (that is, children had to use an algorithm to solve the problem, rather than just knowing what the answer is). The children in this study also exhibited individual differences, which predicted their performance on and strategy in solving the multiplication problems. Children who scored higher on measures of spatial working memory were faster and more accurate on one-digit problems.
This study provides a template for research that uses multiple methodologies with a focus on educational applications. Multiple converging lines of evidence offer windows into training and, potentially, treatment. Although we are a long way from scanning children and diagnosing learning disabilities, using combined neuroimaging methods with behavior on learning outcomes can start to give us a richer picture of the neural landscape of the developing mind.
Reference for the article discussed in this post:
Soltanlou, M., Artemenko, C., Dresler, T., Haeussinger, F.B., Fallgatter, A.J., Ehlis, A-C., & Nuerk, H-C. (2017). Increased arithmetic complexity is associated with domain-general but not domain-specific magnitude processing in children: A simultaneous fNRIS-EEG study. Cognitive, Affective, & Behavioral Neuroscience. DOI: 10.3758/s13415-017-0508-x.