It’s been a good long while since this particular project occupied my attention and focus. However, I’m currently motivated to finish it — or at least finish the nine pages that I already have outlined and planned. There are six more pages that are unplanned except for the margins, which means that I have a total of fifteen pages left to write, and maybe a card or panel to put in the pocket of the book, an afterword of sorts to explain the project a little better than I did at the beginning.
What project am I talking about? This one, the geometry book that I began a long time ago practically in a galaxy far, far away. In fact, from the earlier entries from 2013, I can tell that I was already about sixteen pages into it. Now, I’m thirty-seven pages into it, and I have fifteen left. I’m almost the opposite point in this project as I was four years ago. Funny how these things circle around, right?
Of course today is the day that I made a mistake. I drew out the process of comparing 1:√2, and didn’t discover my error (on the right-hand page) until I had already inked the diagram and written the explanatory text. Always check your work in geometry before you render it in pen!
No matter. I had the room to be able to describe the process incorrectly, add in A WARNING IN CAPITALS AND RED, and then offer the correction. Typical medieval manuscript at this point, really — sometimes errors creep in, and the lowly scribe has to figure out how to offer the correction clearly and legibly in less space. I managed.
Why did I return to it, though? Well, first, I’m trying to clear my desk of unfinished projects. This one has been a big one, and it’s been on my mind to complete for a while. But for another, I recently took up the opportunities and challenges of tutoring again. And I’m tutoring a few young people in geometry. So this project is serving to lubricate and rub the rust off of my geometry skills. Even so, I’m finding that the knowledge of actual geometric proofs isn’t quite as useful as one might imagine.
A lot of the work that students do in geometry class these days appears to be algebra. There will be one diagram (with a note beside it to say, not to scale or not rendered accurately), and then a lot of algebraic notation, and the student is expected to work without a ruler and compassed just their brain power and maybe a calculator, to solve the problem.
I don’t understand.
Are we teaching geometry, or geometric algebra? It looks like the latter, rather than the former. And I understand that teaching actual geometry is challenging, and that it involves looking at a lot of diagrams and working out a lot of constructions by hand… but heck, that’s what we do as human beings. Isn’t it?
But that’s (more or less) true — we use our hands to instruct our brains, and vice-versa. How do we actually learn geometry if we’re not using the tools that geometry has used for thousands of years (or reasonable electronic replacements, though I’d argue that such tools are not as good as actually using hands to manipulate a compass)?
In any case, here’s a place where abstraction often gets the best of us. I think it’s time to bring back some actual geometry to the classroom, and not simply ask students to do it algebraically. This is a set of skills that belongs in our students’ hands, and not just in their heads.