Higher Math Textbooks and Classes are Typically Not Aligned with the Cognitive Science of Learning

Research indicates the best way to improve your problem-solving ability in any domain is simply by acquiring more foundational skills in that domain. 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. Yet, higher math textbooks & courses seem to focus on trying to train jumping distance instead of bridge-building.

Why do so many people struggle with higher math?

One reason is that higher math textbooks & classes are typically not aligned with (and are often in direct opposition to) decades of research into the cognitive science of learning.

Not saying that higher math would be “easy” if taught properly. Just that many more people would be able to learn it, than are currently able to learn it.

Higher math is heavily g-loaded, which creates a cognitive barrier for many students. The goal of guided/scaffolded instruction is to help boost students over that barrier.

Of course, the amount of work it takes to create a textbook explodes with the level of guidance/scaffolding, so in practice there’s a limit to the amount of boosting that is feasible, especially if the textbook is written entirely by a single author.

But most textbooks don’t even come close to the theoretical limit for a single author, much less the theoretical limit for a team of content writers.

Anyone who’s ever tried to learn math from a textbook has run into the following frustrating situation: a worked example demonstrates a special case, but a practice problem requires a logical leap that wasn’t explicitly covered.

Amusingly, many people think the solution to this is “abandon worked examples and focus exclusively on trying to teach general problem-solving skills,” which doesn’t really work.

In the cognitive science literature, there’s a mountain of evidence that you can increase the number of examples & 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)

In other words, research indicates the best way to improve your problem-solving ability in any domain is simply by acquiring more foundational skills in that domain.

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.

Yet, higher math textbooks & courses seem to focus on trying to train jumping distance instead of bridge-building – especially once you get into serious math-major courses like Real Analysis and Abstract Algebra.

What actually works in practice is simply creating more worked examples, organizing them well, and giving students practice with problems like each worked example before moving them onto the next worked example covering a slightly more challenging case.

You can get really, really far with this approach, but most educational resources shy away from it or give up really early because it’s so much damn work! ;)