I know this is going to be super controversial, but I genuinely find illustrations of mathematical concepts below a minimum threshold of complexity totally useless and frequently detrimental.
Below a certain level of complexity the human brain is much faster and efficient operating on abstract symbols, like 'x' and 'y'. You can solve equations and figure things out in a fraction of the time it takes you to visualize bananas, goats, coins, bread, milk, etc.
Visualizations have a role in developing intuitions about complex structures, such as what the a matrix does to a vector or what cosine similarity means, and so on.
But in recent years, everyone and the next man has suddenly assumed that visualizing the number 1 or 2 in terms of every day objects somehow helps learning. It doesn't.
Everyone is different! I personally find examples and visuals a very important part of teaching.
> But in recent years
Just to expand on this a bit: I have been teaching this way since at least 2016, when I published a book on algorithms called Grokking Algorithms. It is an illustrated guide to algorithms. If you didn't like this post, I imagine you won't like the book either :)
I think the milk and bread is just a helpful real world example of how an object might contain two number that need to be solved for simultaneously (carbs and protein). It's more of a why than a how.
Below a certain level of complexity the human brain is much faster and efficient operating on abstract symbols, like 'x' and 'y'. You can solve equations and figure things out in a fraction of the time it takes you to visualize bananas, goats, coins, bread, milk, etc.
Visualizations have a role in developing intuitions about complex structures, such as what the a matrix does to a vector or what cosine similarity means, and so on.
But in recent years, everyone and the next man has suddenly assumed that visualizing the number 1 or 2 in terms of every day objects somehow helps learning. It doesn't.