I self discovered imaginary/complex numbers when I first got to high school. In a show your work setting. I used an x in a circle (IIRC) to represent a complex value and factored it back in to rational numbers to arrive at the correct answer. Even though the ultimate answer was correct and despite the fact that I had (not to toot my own horn or anything) made a mathematical and logical breakthrough - by myself - fundamental to advanced mathematics (this teacher even acknowledged that yet stated that it didn’t matter), I still got it wrong and was admonished for not doing it correctly - that was the last time I tried at school before dropping out.
Fresh in my memory when that happened (I’m still bitter about this one as I was just a wee lad at the time) was the time that I’d made a working wind-tunnel, in the sixth grade as a submission to the school science fair with a pulley system, scale tractor and trailer out of Lego and Balsa and a collection of air dams, wings, and other body modifications. Combined it demonstrated the effective drag coefficients of different types of aerodynamic body modifications, some I’d seem in use, others I’d devised myself, for tractor trailers (18’s), ultimately showing that a design I had come up with would produce the least drag and thus the greatest fuel efficiency for long haul truckers. I was kicked out (and the teacher actually threw my wind tunnel in the trash exploding the styrofoam it was made of) because the she insisted that it was obvious that my father had done this for me and that liars never prosper. I was new to the school, I’d been there maybe a month or two, so she really had no idea or basis for comparison of my relative scholastic capabilities. Never mind the fact that I was a latch-key kid at the time and never really interacted with either of my parents at that point in my life. What it taught me was that in America, you colour within the lines or they beat you into shape. I completely lost interest in school and eventually ended up in the adjunct schoolroom where the kid next to me had to wear a helmet lest he hurt himself. I started to try again when I got to high school, but that didn’t work out either.
Wow, that’s even worse than how they treated me (although the accusation of “your parents did this” sure sounds damn familiar). Glad you made it out of there!
Their explanation that “It matters for matrices” is a total cop out. They say:
The points didn’t come off because the answer was wrong. The points came off because the process the student was using won’t be helpful in the future.
Yes, sometime, more than a decade later (if ever) and in a completely different context, rows and columns will need to be distinguished. However, you need to understand numbers before you understand linear transformations. Commutativity of multiplication is part of that. Being able to flip a comparatively difficult problem 3+3+3+3+3 to the easier 5+5+5 is again part.
Okay, I do understand why the answer to 3x5 might be argued to be better or worse than the official one.
But the matrix ? Are they seriously claiming that rotating your head when looking at a bunch of bars may change their amount ? Isn’t that one of the earliest cognitive skills acquired by children ?
It’s not a matrix. It’s an array. Order matters in matrices, but do not matter when you’re multiplying two numbers using a rectangular grid for counting. The matrix explanation is a cop out. Especially since commutativity is important right now to the kid, and they’re teaching the kid that commutitativity doesn’t exist, while matrices won’t be taught for years. It’s basically a bunch of administrators trying to defend a section of the curriculum that if you follow to the letter of the book, is mathematically wrong.
yeah I can see where that could be quicker to actually calculate, but would be a pain in the ass to set up over the old school way. i would totes screw that lattice method up in all kinds of ways.
Yeah I would still mess that up though. While I did well in my college math courses that was all proofs and theory stuff. If I don’t have a calculator I will most definitely screw up the arithmetic somewhere along the line.
