Talking about Steven Strogatz’s “N dogs” question with kids

Earlier this week Steven Strogatz posted this question on twitter:

The problem received lots of attention (for a math problem on Twitter!) and I thought it would be fun to talk through the problem with the boys today. This type of problem is obviously beyond their ability to solve, but I thought they would have some interesting ideas about it anyway . . . and they did!

We started by just talking through the problem and getting a little clarity on what was going on with the dogs. One of the challenges in talking about a problem like this with kids is that elementary school kids don’t usually see problems involving motion, so it takes them a while to find the language and the ideas to describe what’s going on. They got there eventually, though, and by the end of this video the kids think that the dogs will move inside of the original polygon and eventually collide in the center.

 

With the introductions and initial thoughts out of the way, we tried to talk in more detail about how you would even approach solving this problem.

One idea my younger son had was to look at a triangle rather than trying to solve the problem about the n-gon. The interesting reason for this is that he thought maybe the dogs would run the same distance in all of the n-gon’s so why not look at a simple polygon first.

We tried to draw the shape that the dogs ran in the triangle. Their approach involved approximating the path with straight lines. Also, their drawing showed the idea of the dogs spiraling towards the center of the triangle.

At the end of this video my younger son has the idea to look at infinitely many dogs on a circle!

 

So the idea of looking at infinitely many dogs basically stopped me in my tracks. It seemed so fun to try to see what was going on in this situation. Here’s what they came up with:

 

Finally we moved to looking at one of the programs that Dan Anderson made for this problem. The program we looked at is from this tweet:

Sorry for the technical difficulties on my computer while we looked at Dan’s program. Despite Firefox crashing, the kids thought the curves were really cool:

 

So, despite being a pretty advanced problem, it is still a fun idea for kids to think through. From a “watching kids learn math” perspective, it is neat to hear the ideas that they have about how to approach this complicated problem, too.

Thanks to Steven Strogatz and Dan Anderson for sharing their ideas about this problem.

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What a kid learning math can look like

I first started paying attention to online math videos back in 2011 when, just by coincidence, several different friends pointed out Khan Academy to me and suggested that I should do something like it.

The idea was appealing in that I love talking about math, but essentially trying to duplicate what Khan Academy had done didn’t seem like that great of a pursuit. As I began to look around I saw lots of videos online with adults talking about math. It seemed to me that kids see adults talking about math all the time, but don’t really see kids talking about math nearly as much. I thought that maybe showing kids talking about math would be fun because:

(1) The ideas wouldn’t be prepared ahead of time and probably wouldn’t flow in a perfectly straight line like the “adults talking about math” videos often do,

(2) There would probably be many mistakes and false starts, so kids could see that math isn’t always a perfectly perfect process, and

(3) The ideas involved in solving a problem might be a little different or a little surprising compared to how an adult would approach the same problem.

Last night I had my younger son talk through problem #23 from the 2000 AMC 8. I chose this problem because my older son had struggled with it, but I thought that my younger son might have fun with it, too. His solution to this problem is has basically everything that I wanted to show about kids doing math.

The problem is here:

Problem #23 from the 2000 AMC 8

I’m sorry that the video is 7 1/2 minutes, but not all of the problems go super quickly. He has lots of ideas, has a few false starts, learns from those false starts, and in the end finds a clever solution to a really challenging problem. That’s what learning math looks like, and that’s what doing math looks like!

 

What learning math sometimes looks like -> averages

My older son struggled with a problem about averages from the 2000 AMC 8 this morning. Tonight we revisited it and it turned out to be an interesting example to work through with both kids.

Problem #23 from the 2000 AMC 8

Here’s the problem:

There is a list of seven numbers. The average of the first four numbers is 5, and the average of the last four numbers is 8. If the average of all seven numbers is 6 4/7, then the number common to both sets of four numbers is . . . . ?

My older son started with a nice picture of the situation, but then turns down a difficult path by assuming that the numbers that average to 5 are all 5’s and the numbers that average to 8 are all 8’s. After seeing that this approach is going to run into trouble he finds an different – and better – path to the solution.

 

After going through it with my older son, I thought that the problem would be accessible to my younger son, too, so we gave it a shot. He also started down the path of assuming all 5’s and all 8’s for those two parts of the problem. Although this approach is a tough way to tackle this problem, he stays with it until the end. There’s some great insights about arithmetic from him along the way.

 

So, a nice example of how a 4th and 6th grader approach a problem a bit differently. Hopefully a nice example of what learn math looks like sometimes, too.

Bob Lochel’s question

[only had 30 min to put this together before running out the door, sorry if it seems disjointed]

Saw this question from Bob Lochel today:

Without giving a direct answer to the question, let me share an experience I had with a different problem I saw on twitter last week. I wrote about the first part of it in the link below since I that part of the problem was great for students learning about probability and statistics:

A great introductory probability and stats problem I saw from Christopher Long

Here’s the next part of the story, and one that relates to Lochel’s question. As a reminder, here’s the question at hand:

Last year the Celtics finished with the 5th worst record in the NBA. Whether it’s two years from now or twenty years from now, what pick are the Celtics more likely to land first, the #1 pick or the #5 pick? Assume that the lottery odds and structure remain the same going forward. Further assume that the Celtics are likely to improve and every year the team will either:

Improve by 3 positions (e.g. they go from 5th worst to 8th worst) with a probability of 0.6
OR fall 2 positions (e.g. they go from 5th worst to 3rd worst) with a probability of 0.4.

On twitter, Ben Dilday and Christopher Long came up with a solution using Markov chains and absorbing states. The math behind that solution is something that I’ve seen but not something that I use in my work. BUT, it is clearly a great way to approach this problem and an important piece of math to know if you find yourself needing to solve problems like this one.

My solution was far less sophisticated and much more brute force – no Markov chains, just a simple random walk that started over after every time the Celtics got the 1st or 5th draft pick.

After 10 billion steps in this walk I found 929,797 times when the Celtics received the 5th pick before the 1st pick, and 865,011 times when the reverse was true. So, roughly 48.2% of the time the Celtics got the 1st pick first in my model, and 51.8% of the time they got the 5th pick first. Almost exactly matching Dilday’s and Long’s numbers.

Dilday and Long have nothing to learn from my solution. In fact, Lochel’s question features an almost perfect description of my solution -> correct but wildly inefficient.

I, however, do have a lot to learn from their solution, and have learned from it already. Learning from better solutions (how ever you want to define “better”) is an incredibly important part of development in math. While being able to get to the answer is nice, growing the number of tools in your mathematical toolkit is even nicer!

Michael Pershan’s geometry problem part 2 + an extension

[note – sorry, no editing whatsoever on this one, had to run out the door for the evening . . . ]

Yesterday we talked through a geometry problem that Michael Person posted on twitter:

Michael Pershan’s geometry problem

Today he posted another one:

My older son had an after school activity today, so I talked through this problem with my younger son first. He looked at it in two different ways.

First he looked at the entire rectangle and subtracted away the areas that were not shaded. The arithmetic gave him a little difficulty, but he was able to work out the area:

Next I challenged him to find an alternate approach. This time he thought about shifting the top triangle to the left one unit. This approach is a nice little challenge for a younger kid.

When my older son got home I’d planned on going through a problem from an old AMC 8 that gave him a little trouble this morning (partially because of time). I hadn’t looked at the problem, though, and when I did look at it I got a nice surprise – it was a problem we could approach “in your head” just like the two problem from Pershan.

Here’s the problem on Art of Problem Solving’s site:

Problem 25 from the 1999 AMC 8

and here’s my older son talking through the problem this afternoon:

Even if was just a coincidence, I’m happy to see how the ideas you use to talk about beginning problems are also really useful in problems that seem much more complicated. Math is like that – the basic ideas can be really (and surprisingly) powerful.

Michael Pershan’s Geoemtry problem

Yesterday Michael Pershan posted this geometry question:

The question interested me for two reasons. First it is always neat to hear how kids think out loud. The challenge if this problem is to solve it in your head. Second, by coincidence, we’d just talked about Pick’s theorem last week, and this would be a good chance to review the idea (even though the number of grid points is pretty large).

Here’s my older son’s initial reaction – he sees two different ways to approach the problem:

Next I asked him to approach the problem using the ideas in Pick’s theorem. The interesting thing to me here was how he counting the various grid points, and the little bit of difficulty that he had counting these points made me happy that we looked at the problem this way.

Next up, my younger son. We had to run to an even program that he’s in on Monday’s so I only looked at the problem one way with him. Interestingly, though, his first idea was to approach the problem via Pick’s theorem, though we ended up talking about a more traditional geometric approach.

So, a fun question from Michael Pershan. It is always nice to hear the ideas that are happening in their head rather than just crunching out the numbers on paper.

When is proof appropriate?

Got this question on twitter last night:

It was a nicely timed question because I had just talked through two problems that had given the boys trouble earlier in the day. Each of the problems essentially asked the kids to find the largest / smallest solution to a specific problem. Both times the boys were able to find a single solution, but weren’t really sure if the solution they had found was the largest / smallest.

I think that Cathy O’Neil’s piece about what math teaches you applies to the situation my kids were struggling with yesterday:

Mathematicians know how to admit they’re wrong

This piece helped me understand one way that mathematicians see the world that is different than the way most non-math people see the world. In fact, the idea she lays out here is probably one of the most important things that I’ve come to understand in the last couple of years:

To be a really good mathematician you need to be a skeptic and to walk around with a metaphorical gun to shoot holes in other people’s arguments. Every time you hear a reasoned explanation, you look for the cases it doesn’t cover or the assumptions it’s making.

And you do the same thing with your own proofs to help yourself realize your mistakes before looking like a fool. Because putting out a proof of something is tantamount to asking for other people to shoot holes in your argument.

I obviously don’t know the right age to introduce formal proof, but what O’Neil explains in the next paragraph is one of the beginning features of proof / mathematics that I’m trying to work on a little bit with my kids:

For that reason, every proof that one of these young kids offers up is an act of courage. They don’t know exactly how to explain their thinking, nor do they yet know exactly how to shoot holes in arguments, including their own. It’s an exercise in being wrong and admitting it. They are being trained to get shot down, to admit their mistake, and then immediately get back up again with better reasoning. The goal is to get so good at being wrong that it doesn’t hurt, that it’s not taken personally, and that it’s even fun to be wrong and to improve your argument.

So, with that introduction, here are the two problems.

First up my older son talking through Problem 23 from the 1997 AJHME

This problem asks you to find the largest numbers which has two specific properties. It is hard for him to see that any number than 17 meets the requirements and it takes a while, in O’Neil’s language, for him to see how to shoot holes in the 17 argument.

Next is my younger son talking through a problem from an old MOEMs test. This problem asks for the lowest possible value of a high score. Working during the day he found several different possible values of the high score, but struggled to find a way to determine the lowest value. Working through the problem in the evening, he finds a really nice way to explain the problem:

So, I don’t have the slightest idea what the right age to introduce proofs to kids, though I’m pretty sure it is older than 4th and 6th grade! I do, however, really like the ideas about proof from Cathy O’Neil’s post I linked above. Working on explanations and working on a little bit of skepticism about their own work seems like a nice way to help them start down the path to understanding mathematical proofs.