For Most Students, Competition Math is a Waste of Time
If you look at the kinds of math that most quantitative professionals use on a daily basis, competition math tricks don't show up anywhere. But what does show up everywhere is university-level math subjects.
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When a middle or high school teacher has a bright math student, and the teacher directs them towards competition math, it’s not because that’s the best option for the student. Rather, it’s the best option for the teacher. It gives the student something to do while creating minimal additional work for the teacher.
Competition math problems generally don’t require students to learn new fields of math. Rather, the difficulty comes from students needing to find clever tricks and insights to arrive at solutions using the mathematical tools that they have already learned.
A student can wrestle with a competition problem for long periods of time, and all the teacher needs to do is give a hint once in a while and check the student’s work once they claim to have solved the problem.
But if you look at the kinds of math that most quantitative professionals (like rocket scientists and AI developers) use on a daily basis, those competition math tricks don’t show up anywhere. But what does show up everywhere is university-level math subjects like multivariable calculus, linear algebra, differential equations, and (calculus-based) probability and statistics.
So, given that most students who enjoy math are going to end up applying math in some other field (as opposed to becoming mathematicians) – wouldn’t it be more efficient for them to get a broad view of math as early as possible so that they can sooner apply it to projects in their field(s) of interest?
The countering view is that “students should go ‘deep’ with the math that they’ve already learned – they’ll learn the other math subjects when they’re ready.” But, in practice, this is not true.
If students “learn the other math subjects when they’re ready,” then when is that? Is it when they complete a quantitative major during college? No – even most math majors only learn a tiny slice of all the math that’s out there.
(If you know someone who majored in a quantitative field, ask them if they took algebraic geometry, convex optimization, and control theory. Chances are, they haven’t taken any. On rare occasions, they may have taken one. These are just three out of hundreds of university-level math subjects.)
Do students “learn the other math subjects when they’re ready” after college, on the job? No – if you’re trying to solve cutting-edge problems that nobody has solved before, then there is no “known path” that can tell you what additional math you need. And to even realize that a field of math can help you solve your problem, you generally need to have learned a substantial amount of that field in the first place.
In practice, the only way for students to “learn the other math subjects when they’re ready” is to learn as much math as possible during school.
Follow-Up Questions
But doesn’t competition math help students develop general problem-solving skills?
In the cognitive science literature, there’s a lack of research evidence that people can actually increase their “raw” working memory capacity and generalization ability through training general problem-solving skills. There’s a mountain of evidence that you can increase the number of examples and problem-solving experiences in a student’s knowledge base, but a lack of evidence that you can increase the student’s ability to generalize from those examples. (For a brief overview, see Sweller, Clark, & Kirschner, 2010: Teaching General Problem-Solving Skills Is Not a Substitute for, or a Viable Addition to, Teaching Mathematics.)
However, there IS research evidence that you can effectively turn long-term memory into an extension of working memory if you acquire domain-specific foundational skills and develop them to the point of automaticity. And as you layer more advanced skills on top, those foundational skills naturally get compressed into more generalizable neural representations that can be applied more flexibly across different contexts.
The phenomenon of turning long-term memory into an extension of working memory was observed as early as 1899 by Bryan and Harter: “The learner must come to do with one stroke of attention what now requires half a dozen, and presently in one still more inclusive stroke, what now requires thirty-six. He must systematize the work to be done and must acquire a system of automatic habits corresponding to the system of tasks. When he has done this he is master of the situation in his [occupational or professional] field. … Finally, his whole array of habits is swiftly obedient to serve in the solution of new problems. Automatism is not genius, but it is the hands and feet of genius.”
For about a century, the evidence was gradually bolstered, and the 1980s marked a tipping point that led to an explosion of research interest and behavioral studies. For instance, Chase and Ericsson (1982) found that “rapid access to a sizeable set of knowledge structures that have been stored in directly retrievable locations in long-term memory … produce[s] an effective increase in the working memory capacity for that knowledge base.” And as explained by Unsworth & Engle (2005), ”..[I]ndividual differences in WM capacity occur in tasks requiring some form of control, with little difference appearing on tasks that required relatively automatic processing.”
Today, in addition to behavioral studies, this phenomenon of turning long-term memory into an extension of working memory can be physically observed in neuroimaging. At a physical level in the brain, automaticity involves developing strategic neural connections that reduce the amount of effort that the brain has to expend to activate patterns of neurons. This has been observed, for instance, by Shamloo & Helie (2016), who studied functional magnetic resonance imaging (fMRI) brain scans of participants performing tasks with and without automaticity and found that only subjects who had developed automaticity were able to perform tasks without disruption to background thought processes.
In summary, it appears that skill development all comes down to building domain-specific chunks in long-term memory that allow you to bring more information into working memory without actually increasing the amount of cognitive effort you have to put forth to rehearse that neural activation. In other words, the way you increase your ability to make mental leaps is not actually by jumping farther, but rather, by building bridges that reduce the distance you need to jump.
Now, I do expect that students who do well in competition math will tend to have some inherent cognitive “jumping” advantages (e.g., higher working memory capacity & generalization ability), and those cognitive advantages are advantageous in other fields. But I don’t think it’s the competition math training itself that confers the cognitive jumping advantage, just like it’s not basketball itself that confers a physical height advantage.
There are a lot of people who view competition math as some kind of all-encompassing holy grail of math learning, when really, it’s just another subject. Of course, competition math will force a consolidation of underlying skills, but so will layering on higher math skills that are more relevant to your area of interest.
I’m not saying competition math is zero ROI, I’m just saying that when you build a tower of skills, there’s a lot to be gained by building towards a direction that’s particularly relevant to your future. Like, sure, competition math will aid your abilities in higher math and to some extent translate to other quantitative problem solving domains. But you know what will aid your abilities MORE in higher math and translate MORE to other quantitative problem solving domains? Skilling up directly in higher math and those quantitative problem solving domains!
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