Any suggestions for a math sequence that's advanced but not AOPS?

[Not_a_Number]

3 minutes ago, Clemsondana said:

But, I'm not a huge fan of discovery for complicated concepts.  Maybe it's from my teaching of biology, but...they gave Watson and Crick a Nobel for figuring out the structure of DNA, so it seems unlikely that vast swaths of high school students will discover it on their own.  I don't know that bioogy is unique in that...  🙂  

I basically teach entirely through discovery (I barely teach at all), but I scaffold it much, much more than AoPS. They don't spend enough time thinking about how kids actually think and building it up, I think.

I don't know that it would work for biology, lol, but I like how it's going for math. 

[Clemsondana]

17 minutes ago, Not_a_Number said:

I basically teach entirely through discovery (I barely teach at all), but I scaffold it much, much more than AoPS. They don't spend enough time thinking about how kids actually think and building it up, I think.

I don't know that it would work for biology, lol, but I like how it's going for math. 

I had a prof in college who taught part of metabolism regulation through something discovery-like - he would frequently say something like 'If you want to do X, what would be the easiest way to do that?' but it was with a class and if nobody could figure it out, he'd fill it in for us.  Similarly, I'll tell students about Chargaff and how they knew that the amounts of A and T were equal and G and C were equal and they can think through what that implies for structure.  But, realistically, those numbers were known for a long time and nobody figured out what it meant, and there are tons of examples like that, where the 'discovery' comes from somebody having the right insight into data that's been around for years, or decades.  I'm fine with asking students to make little steps, but generally don't find that it's worthwhile for most students to invest the time in literally figuring out every single thing on their own.  And, with older's dislike of inefficiency, I have to be careful because they don't like feeling like their time is being wasted.  But, yeah, the issue with AoPS seems to be that they want for kids to be willing to stare at things for a long time if they don't see it, and plenty of kids who can do challenging things don't consider 'pondering math' to be a pleasant way to spend their afternoons.  🙂  

[Not_a_Number]

1 hour ago, Clemsondana said:

I'm fine with asking students to make little steps, but generally don't find that it's worthwhile for most students to invest the time in literally figuring out every single thing on their own. 

I pretty much do let kids figure every little thing out, in some sense, although I suppose it's more like... I get them in a place where my observations won't feel like they make any less sense than their own. 

Maybe the point is that I don't ask kids to make leaps, and that requires a LOT of hands-on experience with how things work. But yes, I'll certainly show things... I just taught DD5 the commutative property of multiplication, for example, and I wasn't about to wait for her to discover it. But she was in a space where my explanation made perfect sense and got absorbed in one day. That's sort of what I go for. 

[UHP]

2 hours ago, Clemsondana said:

'If you want to do X, what would be the easiest way to do that?' but it was with a class and if nobody could figure it out, he'd fill it in for us.

Do you remember any examples? A year ago I read about the citric acid cycle and I remember that citrate or isocitrate or something like that inhibits an early step of glycolysis. I don't think I could have guessed from your profs prompt...

[Clemsondana]

Just now, UHP said:

Do you remember any examples? A year ago I read about the citric acid cycle and I remember that citrate or isocitrate or something like that inhibits an early step of glycolysis. I don't think I could have guessed from your profs prompt...

Funny enough, that's the example that I was thinking of.  Citrate regulated a lot of things, including the first step.  But, he'd talk about where you got buildup of things, and what steps were limiting, and then say 'So, what do you have an excess of?' and go from there.  Although my study partner and I joked with him that, when in doubt, the answer was citrate.  He'd also ask 'Where is the best place to regulate this pathway?', and the answer is usually at the beginning or at a step near where you make a product that can easily be used elsewhere.  Once you learn a few tricks, you can apply them in several places...at least for the basic parts of the pathway - I make no promises for the less-learned cycles!  

One example I give my students is how ATP is the same molecule as the A from A, T, G, and C for DNA  Why does that make sense?  In part, because if you have no energy (ATP), you can't get through the replication process.  There are other things that tie the paths together and it's obviously a lot more complicated, but that's one that students can wrap their heads around.  🙂  I talk about that with 'What do you need if you want to divide a cell?  Think about all of those microtubules being assembled and disassembled.  Enzymes are working. What does that take?  Energy!'...