Has any student ever gained real insight from the Pythagorean theorem diagram with the squares along each hypotenuse? I’m baffled by this every time I see it — the union of two squares is not a square! Is it supposed to be obvious to everyone at a glance that their areas add up in the right way?
It's reassuring to see that the facile animation purporting to clarify the Pythagorean Theorem has drawn this much attention. I concur with Dan that the idea that this animation is instructive is baffling. Having done my best to create a more enlightening animation, available at https://demonstrations.wolfram.com/EuclidsProofOfThePythagoreanTheorem/
I can say that it's not easy. Like Aman, I find the algebraically motivated proof dividing (a + b)^2 into 4 right triangles and either c^2 or a^2 and b^2 helpful, but it doesn't explain how the two smaller squares "fit" into the larger one. Student's suggested demonstration also offers a similar observational motivation. A nice aspect of Euclid's proof that compensates for its awkward nature is that it shows the fit. In specific, the altitude of the right triangle based in on the hypotenuse (and going to the right angle) splits the large square into two rectangles, one with area a^2 and the other with area b^2. The animation I created shows this in a way that makes sense to me. Others might prefer the animation created by Yasushi Iwasaki available at https://demonstrations.wolfram.com/AnIntuitiveProofOfThePythagoreanTheorem/
which creates the same intermediate parallelograms as my animation, but moves them in a different way.
In any case, SteveB's observation, "Because 'Is this thing my math teacher gave me actually correct?' is not a question very many students ask" is a cultural problem. Students ought to be asking their math teachers to demonstrate to the students' satisfaction the verity of claims made in the math classroom. We should seek to instill in students the sense that math is not an external authority limiting their actions, but rather a vehicle for exploring the universe around us in all its complexity. Mathematics ought to build a students sense of self confidence and self reliance. Not one of these ChatGPT animations strikes me as serving that purpose.
One friend told me his math teacher would show that diagram and then show how you could consistently cut up the two small squares and rearrange them to form the larger square, but I agree that it seems totally nonobvious to me. At the very least something like https://commons.wikimedia.org/wiki/File:Diagram_of_Pythagoras_Theorem.png gives some intuition behind the theorem.
Maybe the math teacher was trying to convince themselves that the Pythagorean theorem was correct? Because "Is this thing my math teacher gave me actually correct?" is not a question very many students ask.
Appreciate this as always. Just wanted to respond to this part of the deBoer quote:
"Many people are not academically equipped to get a college degree."
Speaking from my experience teaching at a community college:
- The number of people who have the academic and intellectual capability to succeed in college is much bigger than some people seem to think it is. Student ability is not the functional limitation!
- Given the choice, many students will choose to work on subjects that they find personally meaningful, whether or not they are directly connected with career preparation. Student interest is not the functional limitation!
- The struggles that students experience in college typically have much more to do with challenging life circumstances than academic capability.
- The nature of students' challenging life circumstances, the impact they have on students' learning, and many of the difficulties involved in providing effective support are all downstream of systemic inequities and political decisions about resource allocation.
- Even within those systemic problems, there are things we can do to make college work better for more students without compromising what college should be, and most of those things begin with building human connections.
It's a pretty good punchline for this hype'n'scam cycle for the Singularity-scented crowd, in their growing panic to actually make money with their spicy autocomplete, to just turn into another crappy edtech company making the same drag-the-slider widgets students have raced through for 25 years.
AI is being sold as an answer to a content-creation problem we mostly don't have. Don't I have enough worksheets, warmup questions, exit-slip questions, worked-out examples, instructional videos? Seems to me we're drowning in content, and AI's answer is "Here, let me help you make MORE!"
The real challenge is getting your students to engage with any of the content you have. To make some connection, to you, to the material, to other students. To which the AI boosters say, "Well, sure, but what if you had BETTER CONTENT!"
For the AI salesmen, better content must be the answer because content is what AI makes. Teaching must be a nail because we just spent a trillion dollars on a hammer factory.
I'm sure you know what you're talking about better than I do. Seems to me you're thinking about using technology to account for what and how you have always taught. It is just a matter of illustrating a concrete accounting for it. At best it is software engineering in service to instruction.
Why isn't education learning how to give students access to the power abstraction offers them in mathematics? Isn't freeing up resources to do this what technology is for?
I wish more administrators and policy wonks would read Freddie DeBoer's insights on education. I recently heard about Indiana's new graduation requirements, and I am heartened by the direction that state seems to be going.
Has any student ever gained real insight from the Pythagorean theorem diagram with the squares along each hypotenuse? I’m baffled by this every time I see it — the union of two squares is not a square! Is it supposed to be obvious to everyone at a glance that their areas add up in the right way?
It's reassuring to see that the facile animation purporting to clarify the Pythagorean Theorem has drawn this much attention. I concur with Dan that the idea that this animation is instructive is baffling. Having done my best to create a more enlightening animation, available at https://demonstrations.wolfram.com/EuclidsProofOfThePythagoreanTheorem/
I can say that it's not easy. Like Aman, I find the algebraically motivated proof dividing (a + b)^2 into 4 right triangles and either c^2 or a^2 and b^2 helpful, but it doesn't explain how the two smaller squares "fit" into the larger one. Student's suggested demonstration also offers a similar observational motivation. A nice aspect of Euclid's proof that compensates for its awkward nature is that it shows the fit. In specific, the altitude of the right triangle based in on the hypotenuse (and going to the right angle) splits the large square into two rectangles, one with area a^2 and the other with area b^2. The animation I created shows this in a way that makes sense to me. Others might prefer the animation created by Yasushi Iwasaki available at https://demonstrations.wolfram.com/AnIntuitiveProofOfThePythagoreanTheorem/
which creates the same intermediate parallelograms as my animation, but moves them in a different way.
In any case, SteveB's observation, "Because 'Is this thing my math teacher gave me actually correct?' is not a question very many students ask" is a cultural problem. Students ought to be asking their math teachers to demonstrate to the students' satisfaction the verity of claims made in the math classroom. We should seek to instill in students the sense that math is not an external authority limiting their actions, but rather a vehicle for exploring the universe around us in all its complexity. Mathematics ought to build a students sense of self confidence and self reliance. Not one of these ChatGPT animations strikes me as serving that purpose.
There’s a lovely physical demonstration of the Pythagorean Theorem using the three squares and water: https://www.youtube.com/watch?v=CAkMUdeB06o
I think it’s the best version of intuitive proof for the theorem I’ve seen.
Yes, that's lovely.
One friend told me his math teacher would show that diagram and then show how you could consistently cut up the two small squares and rearrange them to form the larger square, but I agree that it seems totally nonobvious to me. At the very least something like https://commons.wikimedia.org/wiki/File:Diagram_of_Pythagoras_Theorem.png gives some intuition behind the theorem.
Maybe the math teacher was trying to convince themselves that the Pythagorean theorem was correct? Because "Is this thing my math teacher gave me actually correct?" is not a question very many students ask.
Appreciate this as always. Just wanted to respond to this part of the deBoer quote:
"Many people are not academically equipped to get a college degree."
Speaking from my experience teaching at a community college:
- The number of people who have the academic and intellectual capability to succeed in college is much bigger than some people seem to think it is. Student ability is not the functional limitation!
- Given the choice, many students will choose to work on subjects that they find personally meaningful, whether or not they are directly connected with career preparation. Student interest is not the functional limitation!
- The struggles that students experience in college typically have much more to do with challenging life circumstances than academic capability.
- The nature of students' challenging life circumstances, the impact they have on students' learning, and many of the difficulties involved in providing effective support are all downstream of systemic inequities and political decisions about resource allocation.
- Even within those systemic problems, there are things we can do to make college work better for more students without compromising what college should be, and most of those things begin with building human connections.
Thanks for saying all this, that quote grated on me too.
It's a pretty good punchline for this hype'n'scam cycle for the Singularity-scented crowd, in their growing panic to actually make money with their spicy autocomplete, to just turn into another crappy edtech company making the same drag-the-slider widgets students have raced through for 25 years.
AI is being sold as an answer to a content-creation problem we mostly don't have. Don't I have enough worksheets, warmup questions, exit-slip questions, worked-out examples, instructional videos? Seems to me we're drowning in content, and AI's answer is "Here, let me help you make MORE!"
The real challenge is getting your students to engage with any of the content you have. To make some connection, to you, to the material, to other students. To which the AI boosters say, "Well, sure, but what if you had BETTER CONTENT!"
For the AI salesmen, better content must be the answer because content is what AI makes. Teaching must be a nail because we just spent a trillion dollars on a hammer factory.
I'm sure you know what you're talking about better than I do. Seems to me you're thinking about using technology to account for what and how you have always taught. It is just a matter of illustrating a concrete accounting for it. At best it is software engineering in service to instruction.
Why isn't education learning how to give students access to the power abstraction offers them in mathematics? Isn't freeing up resources to do this what technology is for?
I wish more administrators and policy wonks would read Freddie DeBoer's insights on education. I recently heard about Indiana's new graduation requirements, and I am heartened by the direction that state seems to be going.