This is more of a general question, but topic-relevant: Have you tried exploring what Claude or ChatGPT can create by way of age-appropriate math lessons? Both have made a serious effort in targeting STEM education and, in a matter of minutes, can create interactive lessons by guiding the educator through a series of questions to specify the learning objective. It's really worth exploring. I would imagine that this will also be very helpful for homeschoolers.
And it actually has some practical use too! For example, calculating cube roots in your head might seem pretty esoteric, but it's actually what you need to figure out how large of a box you would need to hold a stack of books.
I think my son would have loved this. Possibly too much. He actually skipped out on his *gifted* math class because he wanted to *win* the math competition in his "ordinary" fourth-grade class. Awkward. Perhaps just as well as he's going into music composition...
P.S. I *really* want to talk to you about Philosophy of Math before your next post...
A particular kind of mental math that I think comes in extra handy is the set of divisibility rules. If you know the divisibility rules, it's much easier to do prime factorization. And if you are comfortable with prime factorization, a variety of things from finding GCFs and LCMs to counting the number of factors a number has become obvious. Plus it's the fundamental theorem of arithmetic so it's weird that it's not typically given more attention!
I've noticed that Beast Academy/Art of Problem Solving helps kids keep using mental math by designing problems to almost entirely avoid the use of calculators. I think some AoPS problems specifically instruct the student to use a calculator for a specific thing, but unless otherwise stated everything is designed to be done without a calculator. I actually have no idea how common this is among math curricula, but it's a huge departure from the way I learned math in the 90s. This means lots of practice with simplifying radicals and prime factorization and exponent rules resulting in wielding them confidently. At first I wasn't sure about this approach because the problems are written very carefully to make things cancel and whatnot. In the wild math obviously won't always be so courteous, and it can lead to kids spending more time looking for the easy way to do the problem than it would take them to just do it a mildly tedious way. But I've come to believe it's on balance great, not just because of the skill practice, but because of the sense that the problems you're given are thoughtfully woven puzzles. They're evidence that someone who made that puzzle cared about math, and cared about math learners. And I think we need way more of that.
In ways I like this idea, but in truth I would rather give kids more time to work on puzzles than to memorize interesting mental math. My professional life in science (PhD in physics and worked in industrial research in varying fields including optics, hard drives, and bioinformatics) has made me see that the most valuable skill I learned in grad school was the willingness to stick with a problem and know that with enough effort, I could at least make progress on understanding it. I feel like kids can't get enough practice on struggling through problems and practices like the one in this post would cannibalize time better spent on just exploring math and getting to "ah-ha" moments. That said, I feel like my 3 boys are very good at mental math from working through beast academy, but when asked, they feel like math is their weakest subject and it tends to fall into the bucket of least liked activities. Perhaps, had I given them an impressive "superpower", they might view themselves differently. I'm not quite convinced enough to implement this, though =).
This is more of a general question, but topic-relevant: Have you tried exploring what Claude or ChatGPT can create by way of age-appropriate math lessons? Both have made a serious effort in targeting STEM education and, in a matter of minutes, can create interactive lessons by guiding the educator through a series of questions to specify the learning objective. It's really worth exploring. I would imagine that this will also be very helpful for homeschoolers.
You'll probably mention this in the next post, but I think the best use of this kind of mental math is as a foundation for quick estimation.
That's also something that can grow with students as they learn more math, so it can provide an enduring thread throughout their math education.
Feynman had a lot of fun with these kinds of problems: https://math.stackexchange.com/questions/4837744/how-did-feynman-produce-solutions-in-under-a-minute
And it actually has some practical use too! For example, calculating cube roots in your head might seem pretty esoteric, but it's actually what you need to figure out how large of a box you would need to hold a stack of books.
I think my son would have loved this. Possibly too much. He actually skipped out on his *gifted* math class because he wanted to *win* the math competition in his "ordinary" fourth-grade class. Awkward. Perhaps just as well as he's going into music composition...
P.S. I *really* want to talk to you about Philosophy of Math before your next post...
A particular kind of mental math that I think comes in extra handy is the set of divisibility rules. If you know the divisibility rules, it's much easier to do prime factorization. And if you are comfortable with prime factorization, a variety of things from finding GCFs and LCMs to counting the number of factors a number has become obvious. Plus it's the fundamental theorem of arithmetic so it's weird that it's not typically given more attention!
I've noticed that Beast Academy/Art of Problem Solving helps kids keep using mental math by designing problems to almost entirely avoid the use of calculators. I think some AoPS problems specifically instruct the student to use a calculator for a specific thing, but unless otherwise stated everything is designed to be done without a calculator. I actually have no idea how common this is among math curricula, but it's a huge departure from the way I learned math in the 90s. This means lots of practice with simplifying radicals and prime factorization and exponent rules resulting in wielding them confidently. At first I wasn't sure about this approach because the problems are written very carefully to make things cancel and whatnot. In the wild math obviously won't always be so courteous, and it can lead to kids spending more time looking for the easy way to do the problem than it would take them to just do it a mildly tedious way. But I've come to believe it's on balance great, not just because of the skill practice, but because of the sense that the problems you're given are thoughtfully woven puzzles. They're evidence that someone who made that puzzle cared about math, and cared about math learners. And I think we need way more of that.
In ways I like this idea, but in truth I would rather give kids more time to work on puzzles than to memorize interesting mental math. My professional life in science (PhD in physics and worked in industrial research in varying fields including optics, hard drives, and bioinformatics) has made me see that the most valuable skill I learned in grad school was the willingness to stick with a problem and know that with enough effort, I could at least make progress on understanding it. I feel like kids can't get enough practice on struggling through problems and practices like the one in this post would cannibalize time better spent on just exploring math and getting to "ah-ha" moments. That said, I feel like my 3 boys are very good at mental math from working through beast academy, but when asked, they feel like math is their weakest subject and it tends to fall into the bucket of least liked activities. Perhaps, had I given them an impressive "superpower", they might view themselves differently. I'm not quite convinced enough to implement this, though =).