One dogma that Anglophone maths teacher all have to profess is their belief in conceptual understanding. All maths teachers, including those from East Asia, want children to understand mathematics, but those most under the influence of American academics take this a step further. American maths teachers tend to be more fundamentalist, insisting that conceptual understanding must come before procedural knowledge.
The story goes something like this: Large numbers of children are put-off mathematics at school because they are led to believe that it is just about manipulating meaningless symbols, instead of seeing the beauty and relevance of mathematics. We therefore need to relentlessly push the practical relevance of mathematics and we need to develop students’ conceptual understanding so that they realise that maths is not just a set of rules to memorise.
According to this account, knowledge and conceptual understanding are qualitatively different. I reject this idea. Understanding is important but it is made up of layers of knowledge. It is not spooky, mysterious and somehow set apart from, or in opposition to, knowledge. There is, in fact, evidence that students gain understanding by learning procedural knowledge and vice versa.
The conceptual understanding story resonates so strongly with researchers that they tend to see everything through this lens. For instance, the fact that Chinese students tend to outperform American students on international assessments could be down to a range of factors. For instance, international assessments only tend to sample from the big Chinese cities and so comparing these results with the whole of the U.S. is unfair. There are also cultural differences between the two countries, with the Confucian culture of China often seen as valuing hard academic slog. Moreover, if you are relatively less wealthy, you are likely to see maths as a possible route to a better career and persist with it more.
Nevertheless, the bizarre idea has spread that the reason East Asian students perform better at mathematics than American students is because they are taught to have better conceptual understanding. This notion persists in the teeth of evidence that East Asian teachers use fewer of the strategies that Western academics associate with teaching for understanding. Such is the power of confirmation bias.
A new paper has tested this theory in a rather odd way. Researchers asked students in America and China a series of meaningless questions about fractions. Firstly, they were asked to suggest fractions to place in the empty boxes either side of an operation e.g:
Next, given a pair of fractions, or a mix of whole numbers and fractions, they were asked to suggest an operation e.g:
Finally, they were given problems and asked to match the operator with the problem. For instance, they might be asked to guess which problem is an addition in the following pair of examples:
If you think that these are odd questions to ask students then I am inclined to agree with you. The researchers asked them because of the way that fractions operations work. To add or subtract fractions, we need to write them with a common denominator i.e. they have to have the same number on the bottom. Even questions that start with different numbers have to pass through this stage. So researchers wondered whether students would have learnt a spurious association between fractions that have a common denominator and addition/subtraction and fractions that have a different denominator and multiplication/division.
The authors had a number of ideas about what might happen and, specifically, the possible differences between Chinese students, with their supposedly superior conceptual understanding, and American students. As the authors suggest:
“One reason to suspect that strong conceptual knowledge of mathematics prevents learning of spurious associations is that children who understand the conceptual bases of correct solution procedures do not need to rely on mathematically irrelevant associations to select solution procedures.”
What were the results? Both Chinese and American students made the same spurious associations. There is no evidence from this experiment to support an argument about differences in conceptual understanding. Funnily enough, students from across the world are very similar.
I think that a mistake researchers make in this kind of study is to assume that subjects, even when in possession of conceptual understanding, will use it. In a recent post, I invoked Daniel Kahneman’s idea of instinctual and quick ‘System 1’ thinking versus the slower, logical ‘System 2’ thinking. I suggested that sometimes students may make a mistake, even if they understand it is a mistake and why it is a mistake, because they are using System 1 thinking instead of System 2 thinking. I suggested that the drain on resources that a complex maths problem imposes could lead to the use of System 1 thinking for solving sub-components of that problem.
I cannot think of anything more likely to induce System 1 thinking than the fractions tasks outlined above. There is no way of reasoning them out using System 2 thinking and so students will reach for any associations they can think of and the fact that they have often seen addition problems in the past where the denominators are the same is the only thing they can think of that seems relevant.
What is more, the fact that Chinese students seem to make the same System 1 associations as American students does not seem to be something that gets in the way of them being better at maths. The true believers in conceptual understanding will need to go looking for a new target.
Filling the pail 
Agree with your analysis here. In fact, Dan Willingham argues that while understanding the deep structures of a discipline such as mathematics is an important goal of education, “if students fall short of this, it certainly doesn’t mean that they have acquired mere rote knowledge and are little better than parrots.” (Daniel T. Willingham, “Inflexible Knowledge: The First Step to Expertise,” American Educator 26, no. 4 7 (2002): 31–33, 48–49.)
Rather, they are making the small steps necessary to develop better mathematical thinking. Simply put, no one leaps directly from novice to expert.
In an odd way this supports our very premise that a great deal of practice and memorization will lead to higher order thinking…something that many Asian countries practice at the primary level and something that is receding dramatically in most Western nations. My eldest is currently enjoying this semester abroad in Japan and she is constantly amazed at how well her classmates have their math facts so well memorized. They don’t even know what calculators are; they have never seen them. In Grade 11. These kids calculate their answers faster than how my daughter can plug the question into her own calculator. This has led to many interesting and thoughtful discussions in their math class, exploring multiple concepts – all led by the knowledgeable teacher at the front of the class – encouraging students to explore their thoughts and answers in a respectful fashion. They are LIGHT years ahead of our students over here in Canada.
Simply concluding that Asian students have better “conceptual understanding” completely misses the journey, and understanding about how they got there. As Barry has already suggested…making the necessary steps to greater understanding needs to be acknowledged. Too bad the researchers didn’t examine that as part of their research.
Many thanks for this, Tara. I’ve long suspected that automatic recall of number bonds is taken for granted in the Far East, but I’ve yet to come across any research on the subject. Anecdotal evidence is a lot better than no evidence at all!
I like the idea of system 1 and system 2 thinking. I’d like to stress that both need to be trained, and that the two modes of thinking can amplify each other.
To give an example: when teaching integration, both the substitution rule and partial integration are procedural — prime examples of system 2 thinking. However, knowing which technique to use is often an intuitive system 1 choice. When my students ask how they can know which method to use, I emphasize that they need, through a lot of practice, to develop a gut feeling of roughly what the result of each integration technique would be, and whether this would result in a solvable (or simpler) problem.
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As an author of the article being criticized, I think that Mr. Ashman missed the point of the study. The tasks that Mr. Ashman describe assess whether children have learned the associations between the numbers and operations that are present in textbooks in both the US and China; the data indicate that children in both countries have learned those associations. The point of the study is to demonstrate that despite having these mathematically irrelevant associations, it’s still possible for children to generate high levels of accuracy in fraction arithmetic. Our argument is that the Chinese children’s superior conceptual understanding of the underlying mathematics shields them against the influence of the spurious associations. Phrasing it in terms of Type 2 thinking protecting children against Type 1 thinking in a context where the Type 2 thinking is needed is a fine way to think of the relation between the two; we in fact cite Kahneman’s distinction as a way of explaining our ideas in a recent article. Thus, I’m unsure what Mr. Ashman’s problem with our article is.
Hi Robert. I don’t think I am particularly critical of the study. I think it is an odd thing to measure but the results are quite interesting. Otherwise I wouldn’t have written about them. The bulk of my criticism is for those who set up dichotomies between the allegedly rote learning of procedures and conceptual understanding.