by jerkstate | 2025-04-20 14:34:46+00:00
I guess these workbooks usually come in two different "shapes" - one, a guided workbook with a high-level goal that combines several concepts, and another would be a practice worksheet where we do a bunch of exercise of the same algorithm (say, long division, calculating summations, or matrix multiplication) over and over. For the "workbook" pattern, we first discuss with the LLM the final goal (e.g. a scratch program that can calculate a rocket's position using the rocket equation). Then we flesh out the steps towards the goal - is it reasonable to add the math for air resistance? air resistance that decreases with altitude? gravity turn? how do we integrate the velocity and position for each frame? How can we relate the integration by step size to the underlying integral, by showing that the result gets more precise (but slower to calculate) the smaller your delta-t is? Then, produce a Scratch code sample that implements the velocity and position calculation. Of course, there are things subtly wrong with the code sample (usually, if the math formulas are well-known, they're correct), which requires debugging - just another type of problem-solving.

The second shape, worksheets, is a lot more straightforward. Just define the type of problem you want to practice and have chatgpt make a bunch of problems. Then switch to one of the newer reasoning models and have it work the problems, and refine to get rid of any bogey problems (for example, for polynomial exercise, you could tell it to make sure the roots are integers)

The worksheets are more "hands off" - I run them through the algorithm once and check their work once or twice and then let them do the rest. The important thing is that the worksheets are connected to their high-level goal, and they understand that in order to solve the big, hairy problem that they're interested in, they need to build up certain specific skills.

Usually the worksheet goal is a pretty substantial conceptual stretch for my kids so they need to go through a series of fundamental worksheets. But the great thing about the LLM is, you can just tell it you're having a problem understanding some concept and to help build the scaffolding by listing all of the required skills to understand a concept, and picking the ones that needs improvement the most and practicing them.

My approach draws a little from "The MathAcademy Way" - https://www.justinmath.com/files/the-math-academy-way.pdf but instead of building fundamentals evenly in all topics before advancing (like expanding a sphere), we look only at the scaffolding required to support some higher-level goal - it's sort of like the masters/PhD process but guided through existing human knowledge: https://www.openculture.com/2017/06/the-illustrated-guide-to... . As a side note, I think it's really fun to include the history (mathematicians who contributed to the ideas) as well as the notation (using the greek letters, explaining why it's common to use them). When the kids notice the names like Pythagoras, Newton, and Euler reappearing frequently, and get a sense of the time scale these discoveries happened on, they treat the current state - and their ability to go learn thousands of years of math in months - with more reverence.