The Necessity of Grinding Through Concrete Examples Before Jumping Up a Level of Abstraction
If you go directly to the most abstract ideas then you're basically like a kid who reads a book of famous quotes about life and thinks they understand everything about life by way of those quotes.
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Many learners fail to understand that grinding through concrete examples imbues you with intuition that you will not get if you jump directly to studying the most abstract ideas.
If you go directly to the most abstract ideas then you’re basically like a kid who reads a book of famous quotes about life and thinks they understand everything about life by way of those quotes.
The way you come to understand life is not by just reading quotes. You have to actually accumulate lots of life experiences.
And you might think you understand the quotes when you’re young, but after you accumulate more life experience, you realize that you really had only the most naive, surface-level understanding of the quotes back then, and you really had no idea what the hell you were talking about.
It’s the same way in math. In general, the purpose and power of an abstract idea is that it compresses a zoo of concrete examples. But if you haven’t built up that zoo of concrete examples then you miss out on that power.
If you shy away from grinding some messy math then you will never truly know what the hell you’re talking about.
Skipping the concrete examples is a one-way ticket to existential crisis.
If you’ve lived and breathed concrete examples, they’ll get compressed into tangible, meaningful abstractions that inject you with a dose of vitality every time you work with them – but if you haven’t, then the abstractions will feel dull and lifeless, and you’ll constantly wonder what’s the point of pushing meaningless abstractions around in arbitrary patterns of allowed manipulations.
For instance, a company’s balance sheet can tell an incredibly interesting story if you have visceral experience with success and failure in business – but if you don’t, then analyzing financials will make you feel like a robot checking whether numbers match semi-arbitrary conditions for being “good” or “bad”.
Grinding the concrete examples is NOT about turning yourself into a robot and shielding you from intellectual awakening.
It’s the opposite.
It’s about equipping you with invigorating experiences that can live on through the abstractions, empowering you to actually know what the hell you’re talking about.
Follow-Up Questions
Is it really necessary to learn matrix multiplications by hand? They are just tedious and that’s it.
Good luck teaching a student that matrix multiplication is non-commutative, that an m-by-n matrix times an n-by-k matrix produces an m-by-k matrix, that a matrix has no inverse if a row or column is a linear combo of others, etc., without grinding through concrete examples.
How do you incorporate concrete examples in proof-based courses like Real Analysis and Abstract Algebra? Isn’t the whole point to be abstract, not concrete?
Concrete examples have a place in all levels of math, even in proof-based courses like real analysis and abstract algebra. Unfortunately, they are often left out for no reason other than the instructor’s laziness, causing students to needlessly struggle. A snippet from The Pedagogically Optimal Way to Learn Math:
- Just to name one example: last year, I tutored a student who was taking analysis at an elite university, and each problem set consisted of those "think really hard for a long period of time" problems. Things were going the way of a train wreck: despite her best efforts, she was spinning her wheels on these problems and making very little progress. Not only was she unable to solve the problems, but also, she was not noticeably improving any supporting knowledge by trying and failing to solve them.
What I ended up doing was engaging her in deliberate practice on all of the component knowledge that was being pulled together in each problem. Something like this (the following is non-exhaustive):- Deliberate Practice on Definitions: I give you a mathematical object and you tell me whether it meets the definition (and why or why not). Repeat over and over again increasing in difficulty. Okay, now suppose we remove some criterion from the definition. What's an object that didn't meet the original definition but does now after dropping that criterion? Repeat over and over dropping different criteria.
- Deliberate Practice on Theorems: I give you a scenario and you tell me whether it meets the assumptions of the theorem. If so, you tell me specifically what else you know is true about the scenario, according to the theorem. Repeat over and over again increasing in difficulty. Okay, if this is a one-way implication, tell me some scenarios where the converse does not hold.
As soon as we took a step back from the homework problems and started doing that, she started making actual progress on her supporting knowledge. After enough cycles of deliberate practice, she'd re-attempt the homework problems, often solving them completely or at least getting a lot further before starting to spin her wheels again.
The result: her exam performance skyrocketed and she ended up finishing the course with an A.
What could have happened: without this deliberate practice intervention, she would have gotten a low grade in the course and possibly even dropped out of the major entirely.
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