Metacognitive Strategies for Learning Math

24 Jul 2024

Metacognitive learning strategies have made me a better mathematician and a better educator.

Metacognition means “thinking about thinking.”

I’d like to discuss six metacognitive strategies that I find invaluable for learning new math.

Explain-Alouds

Once you’ve grasped a new concept, imagine you’re teaching it to a student. Explain it clearly and loudly (yes, really!) using a pencil and paper or whiteboard.

By vocalizing your thought processes and distilling them into a simple explanation, you become more aware of their strengths and weaknesses.

Don’t shy away from important technical details; the goal is to simplify, not oversimplify.

Keep revising your explanation until you’re happy.

Q&A

What questions would your student ask? Can you answer them? More importantly, can you answer them well?

Don’t skip a question just because the answer seems obvious. Often, the most obvious questions are the best.

Why is (-1) x (-1) = 1? Can you explain it? If not, learn why it’s true and how to explain it.

If your answers aren’t great, go back to the drawing board and improve them. Then, return to your student with a better explanation.

Remember, “If you can’t explain it simply, you don’t understand it well enough.”

Retrieval Practice

Retrieval practice means recalling information from memory.

Struggling to remember key ideas or formulas is vital for transferring important information to long-term memory and ensuring it stays there. This is the so-called testing effect.

Numerous studies have also shown that retrieval practice significantly enhances learning compared to strategies such as repeated reading.

While studying, maximize the testing effect by recalling as much information as possible from memory! Test yourself regularly.

If you constantly refer to notes, you’re doing yourself a disservice, probably without realizing it. Refer to notes only when you’re completely stuck!

Identity Structure in Mathematical Formulas

“Rote” is a dirty word in educational circles. However, to become a successful mathematician, some rote learning is unavoidable.

The challenge is knowing what information to rote-memorize and which to internalize using an alternative (metacognitive) strategy.

Students often rote-memorize mathematical formulas. But wherever possible, structure should always be identified in a mathematical formula. Identifying structure means breaking a formula down into its component parts.

Not only does this provide a deeper understanding, but you’re also more likely to remember it! It’s a win-win.

Some math formulas are opaque and have no obvious structure. Trigonometric identities and standard derivatives/integrals are results where instant recall is required and, therefore, should be memorized.

Derive Key Results

Few things are more satisfying in math than deriving an important result for yourself.

My approach to this is similar to that of worked examples: read how it’s done, then put it aside and try it yourself. Repeat this until you get it right.

Invoke spaced repetition to make sure your new knowledge sticks. In other words, reconstruct your derivation a day or so later, then maybe a week later again.

If you’re feeling brave and have some spare time, try to derive the same result using an alternative method.

Diagrams

Draw diagrams and flow charts to help form mental models that make it easier to transfer information to long-term memory.

Flow charts help break down complex processes into individual components and can be invaluable for decision-making.

Visuals also help distribute the cognitive load of the task between two learning channels (visual and written), thus helping to avoid overload.

By integrating these strategies into your studying routine, you’ll develop a deeper understanding of the material, enjoy studying more, and ultimately enhance your mathematical skills.