One of the people I have learned the most from online is Grant Sanderson, the man behind the math explainer channel 3Blue1Brown where he uses an inquiry-based learning approach to teach math. He was one of the first people that truly made me understand the importance of visualization and spatialization for understanding math.
In the first video of one of Sanderson’s most popular series, “the essence of calculus”, he mentions that:
“my goal is for you to come away feeling like you could have invented calculus yourself”
This is obviously a great philosophy to have, and if you feel this after watching the series it is clear that you have understood the key concepts in some depth. That being said, while I learned a lot from the series, I did not come away “feeling like [I] could have invented calculus myself” at all.
Grant’s videos differ heavily from traditional teaching resources in many positive ways. Rather than presenting the abstract idea and then giving examples, Sanderson will provide a specific problem, question or inquiry and then take you through a step-by-step solution that can be generalised to other more complex examples. Also at the beginning of the first video in the series, he quotes David Hilbert:
“The art of doing mathematics is finding that special case that contains all the germs of generality”
This is an approach that many (including me) consider better for providing a deeper understanding to the reader than simply presenting the abstract idea as something to memorise.
The problem is, that during the video, the “special case that contains all the germs of generality” is simply given to the learner through a series of very specific questions which you would be extremely unlikely to just “stumble accross”. We are not “finding that special case” at all. You could even say that according to David Hilbert, we are missing out on the “art of doing mathematics”.
Of course, after the video I understood the concepts better, but I was no closer to understanding how I could have “invented calculus myself”.
The human brain does not achieve such amazing feats as inventing calculus through sheer luck of stumbling across the right ideas. Instead, it works via pattern recognition. As humans, we recognise patterns in our environment and create abstractions from them. This is how motivation should be provided as it appeals to the nature of our brains, and does not introduce the chasm of luck in discovering new ideas.