One of the things that surprised me when I studied math in college and grad school was just how much "fuzzy math" is out there being studied by mathematicians. Things like graph theory and game theory are their own branches of mathematics. You can talk with kids about counting up how many ways there are to partition a number (4 = 3+1 = 2+2 = 1+1+2=1+1+1+1) and it turns out that's an important fact in modern algebra. The Collatz Conjecture is something I've used with 5th graders, as we looked, in a fuzzy way, for patterns in this function, and it turns out that mathematicians are looking at it too. When I was in middle school, we played with "clock arithmetic" for fun, but it turns out that is the basis for all of group theory. And ironically, the Triangle Inequality is one of the foundational axioms of any metric. Without it, there would be no geometry, manifolds, or real analysis.
I think you name the most powerful aspect of these topics: they are great as low-floor activities. They bring in all students, since they are based on some sort of game or real-world experience. You don't need much background to understand them. They are also great because people naturally love looking for patterns, and they tend to lend themselves to that fairly quickly.
Maybe the most potent aspects of "fuzzy" math is that we just let it be what it is: exploration and learning for the joy of learning. Can you imagine giving students a test on how well they do a Sudoku? Calling home to parents to let them know that little Maya didn't study hard enough on her Sudoku homework, and now she is going to get a D? That would suck the joy right out.
I love your point about giving a test on Sudoku. The tone in how we approach, and motivate, and assess fuzzy math is different. And that's part of the point, the goal is different. And students absolutely love looking for patterns, as long as they manage to feel successful sometimes. Too much failure and it doesn't feel like looking for patterns at all.
Wow, that's such a cool insight! I love how you brought up pattern building with the Sierpinski triangle and number theory through Sudoku – it totally clicks. More Fuzzy Math logic can be used in teaching to build up a concept.
It's funny how my impressions of the triangle inequality theorem has been so different — I remember it being part of the curriculum when I was in high school in Wisconsin in the 90s, and then having to teach it for the SATs, and rather hating it. The very experience of something being mandatory (in this situation — I don't want to come across as a fundamentalist here) made bitter what could have been sweet.
A question for you — have you ever read "A Mathematician's Lament" (the essay, not the book) by Paul Lockhart? It takes a sort of fundamentalist (!) stand on this very issue. That might a fun thing to get some education writers together to argue over.
I think you're actually pointing to something really important when you describe your experience of the triangle inequality theorem -- the experience of learning fuzzy math can differ dramatically depending on the context and the individual. My goal isn't for every student to love every topic, that's not reasonable. But each fuzzy math topic is a kindof roll of the dice, and maybe that will be the thing that ignites the passion of a few students.
I have read the Lament. I have complicated feelings on it. To start, I think Lockhart doesn't understand how much practice is necessary to become proficient at his level. (Maybe he's just a really smart guy and doesn't need that much practice? But most students do.) And I think he underestimates how accomplished students can feel by figuring something out and getting proficient at it. I recently watched a student beaming with pride when he finally got his head around a set of integer subtraction problems. That's a cool experience for that kid, but it took a lot of practice and wouldn't appeal to Lockhart.
While I don't think I agree with Lockhart on how math education should be structured, I do think he points to something that should exist in every math program. For me it's a ∃ argument and not a ∀ argument. Students should have chances to explore and see the open and creative side of math. We should do our best to give them those experiences. But that's not what all math education should look like. In a great math program there exists opportunities to be creative and explore, but it's not for all lessons, the lesson should be structure as a creative exploration.
I, too, am a moderate on Lockhart — although perhaps I'm enough toward his side that this would be a fun thing for us to fight about on a podcast or something! Actually, have you ever gotten to experience a Julia Robison Math Festival? For me, they're a WONDERFUL way to bring this "fuzzy math" into students' lives at the highest levels. I used to volunteer in them when I lived in Seattle; they changed some of my experience of what math education could look like.
I'd love to see something like that in action. I guess my take right now is that I maybe believe is those ideas could play a larger role in practice if they were executed very very well...but that's hard to do and not something I've ever seen. Feels pretty far away from my day-to-day reality.
Interesting! Good to know, though I will say that I'm skeptical that example will land to convince 7th graders the triangle inequality theorem is worth learning. And that's fine, I think we can harness curiosity and visual puzzles to motivate this topic. I'm sure most math topics have some application somewhere, I guess it's a question of where they find their greatest use to figure out which category a topic falls in.
One of the things that surprised me when I studied math in college and grad school was just how much "fuzzy math" is out there being studied by mathematicians. Things like graph theory and game theory are their own branches of mathematics. You can talk with kids about counting up how many ways there are to partition a number (4 = 3+1 = 2+2 = 1+1+2=1+1+1+1) and it turns out that's an important fact in modern algebra. The Collatz Conjecture is something I've used with 5th graders, as we looked, in a fuzzy way, for patterns in this function, and it turns out that mathematicians are looking at it too. When I was in middle school, we played with "clock arithmetic" for fun, but it turns out that is the basis for all of group theory. And ironically, the Triangle Inequality is one of the foundational axioms of any metric. Without it, there would be no geometry, manifolds, or real analysis.
I think you name the most powerful aspect of these topics: they are great as low-floor activities. They bring in all students, since they are based on some sort of game or real-world experience. You don't need much background to understand them. They are also great because people naturally love looking for patterns, and they tend to lend themselves to that fairly quickly.
Maybe the most potent aspects of "fuzzy" math is that we just let it be what it is: exploration and learning for the joy of learning. Can you imagine giving students a test on how well they do a Sudoku? Calling home to parents to let them know that little Maya didn't study hard enough on her Sudoku homework, and now she is going to get a D? That would suck the joy right out.
I love your point about giving a test on Sudoku. The tone in how we approach, and motivate, and assess fuzzy math is different. And that's part of the point, the goal is different. And students absolutely love looking for patterns, as long as they manage to feel successful sometimes. Too much failure and it doesn't feel like looking for patterns at all.
Wow, that's such a cool insight! I love how you brought up pattern building with the Sierpinski triangle and number theory through Sudoku – it totally clicks. More Fuzzy Math logic can be used in teaching to build up a concept.
It's funny how my impressions of the triangle inequality theorem has been so different — I remember it being part of the curriculum when I was in high school in Wisconsin in the 90s, and then having to teach it for the SATs, and rather hating it. The very experience of something being mandatory (in this situation — I don't want to come across as a fundamentalist here) made bitter what could have been sweet.
A question for you — have you ever read "A Mathematician's Lament" (the essay, not the book) by Paul Lockhart? It takes a sort of fundamentalist (!) stand on this very issue. That might a fun thing to get some education writers together to argue over.
I think you're actually pointing to something really important when you describe your experience of the triangle inequality theorem -- the experience of learning fuzzy math can differ dramatically depending on the context and the individual. My goal isn't for every student to love every topic, that's not reasonable. But each fuzzy math topic is a kindof roll of the dice, and maybe that will be the thing that ignites the passion of a few students.
I have read the Lament. I have complicated feelings on it. To start, I think Lockhart doesn't understand how much practice is necessary to become proficient at his level. (Maybe he's just a really smart guy and doesn't need that much practice? But most students do.) And I think he underestimates how accomplished students can feel by figuring something out and getting proficient at it. I recently watched a student beaming with pride when he finally got his head around a set of integer subtraction problems. That's a cool experience for that kid, but it took a lot of practice and wouldn't appeal to Lockhart.
While I don't think I agree with Lockhart on how math education should be structured, I do think he points to something that should exist in every math program. For me it's a ∃ argument and not a ∀ argument. Students should have chances to explore and see the open and creative side of math. We should do our best to give them those experiences. But that's not what all math education should look like. In a great math program there exists opportunities to be creative and explore, but it's not for all lessons, the lesson should be structure as a creative exploration.
I, too, am a moderate on Lockhart — although perhaps I'm enough toward his side that this would be a fun thing for us to fight about on a podcast or something! Actually, have you ever gotten to experience a Julia Robison Math Festival? For me, they're a WONDERFUL way to bring this "fuzzy math" into students' lives at the highest levels. I used to volunteer in them when I lived in Seattle; they changed some of my experience of what math education could look like.
I'd love to see something like that in action. I guess my take right now is that I maybe believe is those ideas could play a larger role in practice if they were executed very very well...but that's hard to do and not something I've ever seen. Feels pretty far away from my day-to-day reality.
Here's an example of how the triangle inequality theorem helped untangle a complicated protein folding problem: https://youtu.be/P_fHJIYENdI?si=D-Ny0S-j99HkSRfW&t=995
Interesting! Good to know, though I will say that I'm skeptical that example will land to convince 7th graders the triangle inequality theorem is worth learning. And that's fine, I think we can harness curiosity and visual puzzles to motivate this topic. I'm sure most math topics have some application somewhere, I guess it's a question of where they find their greatest use to figure out which category a topic falls in.