What does cognitive science say about whether to learn mathematics top-down from research papers?

Question

There is currently a popular question on math.SE on whether it is effective to learn math top-down.

By top-down I mean finding a paper that interests you which is obviously way over your head, then at a snail's pace, looking up definitions and learning just what you need and occasionally proving basic results. Eventually you'll get there but is this a bad idea? Is learning each required math area by textbook the better way?

I thought that cognitive science might have something to contribute to this discussion. Specifically:

• What would cognitive science have to say about the pros and cons of learning from the ground up versus taking a difficult mathematical text and following up on the details?
• What decision rules would cognitive science have for deciding whether to adopt a given a approach?
• Has any empirical research explicitly dealt with this question?

Answer 1

This is a part answer to this question, specifically if research has been done for this problem. Also, some perspectives of a Maths and Physics teacher of over a decade experience. Somewhat ironically, I am going to start by making a parallel to a topic that is arguably an equivalent in the Sciences: Physics - as in my experience, I have seen similarities in terms of top=down vs bottom-up learning.

According to "Learning to think like a Physicist: A review of research based instructional strategies" (Van Heuvelen, 1991), many studies then have found that conventional teaching methods (bottom-up in a lot of places), in their use of "primitive formula-centred problem solving" fail to properly develop topic and subject related reasoning and problem solving.

The point here is that enquiry based problem solving, which can be considered as 'top-down' allows the student to learn the topics and concepts within a context, in doing so, they see how various topics link together, that may not be so apparent when working with formula-centred methods. A caveat here, it is absolutely critical that the students have some background knowledge and skills that could be used in their enquiry, these would potentially be enhanced through a top-down type of enquiry.

This is emphasised further in the chapter "Research on Teaching Mathematics: The Unsolved Problem of Teachers' Mathematical Knowledge" (Ball, 2001), who asset that the most common encounter with mathematics is a set of rules to be memorised, equivalent to the 'primitive formula-centred problem solving' model described in the previous article, resulting in the subject not only being misunderstood, but underappreciated. This is an advantage of a top-down approach, according to the article (and my own experience), where if the student starts with a query (from an article or unfamiliar question), then it turns into a mathematical challenge, rather than a pedagogical one.

Finally, a key point from the chapter "Learning to think mathematically: Problem solving, metacognition, and sense-making in Mathematics" (Schoenfeld, 1992), make several key points outlined above, and:

Present problem situations that closely resemble real situations in their richness and complexity so that the experience that students gain in the Learning to think mathematically, classroom will be transferable.

This includes taking an unfamiliar problem, and using skills and knowledge that they have to disassemble the problem to understand its components. Essentially, a great skill can be developed in the top-down approach - how to break down a big unfamiliar problem into its more familiar ingredients.

Hope this helps.