Reacting to wonderful blog post from Lior Pachter

Saw this amazing blog post from Lior Pachter earlier today:

Unsolved Problems with the Common Core

In the post he presents many unsolved problems in math that could be used as a fun examples for kids in grade levels ranging from kindergarten to high school. It is a sensational read and I was happy to see lots of ideas that I’d not considered. As a way to show how unsolved problems can be used to talk about math with younger kids, I wanted show some projects that we’ve done that illustrate some of Pachter’s ideas:

Kindergarten: the 4 color theorem

Somehow or other we haven’t talked about the 4 color theorem, but I love the idea. We have two projects, though, that where coloring could come into play. One involves a neat problem about tiling octagons that Colorado math professor Richard Green wrote about on his blog:

Coloring sheets from Math Munch

Using a Richard Green post to talk geometry with my son

1st Grade – Ramsey theory

We’ve touched on Ramsey theory with several projects on Graham’s number. These projects require a little bit of understanding of how powers work, so maybe not 1st grade but still a fun topic:

The last 4 digits of Graham’s number

An attempt to explain Graham’s number to kids

2nd grade – Pascal’s triangle

I was not aware of the unsolved problem in Pachter’s post – what a cool problem! Pascal’s triangle is full of fun projects for kids. Here’s just a few of ours:

The hockey stick theorem and some fun geometry in Pascal’s triangle

Talking through Dan Anderson’s mod 2 Pascal’s Triangle

Pascal’s triangle and powers of 11


3rd grade – the Collatz conjecture!!

The Collatz conjecture is a great problem for kids to play around with. We’ve had some really fun conversations about it. Our most recent was inspired by a John Conway paper:

The Collatz conjecture and John Conway’s “amusical” variation

And here’s an old conversation with my younger son about the conjecture:

 

4th grade – The Goldbach Conjecture

We haven’t done any projects about the Goldbach conjecture, but we have done a bunch about primes and some basic number theory. Here’s a few including a neat game involving prime numbers (with the 1st one being inspired by a question from a 4th grade teacher):

Fun with abundant numbers

A review of Prime Climb by Math for Love

5th grade – Convex Polygons

Here Pachter presents another problem that I was not familiar with. While we haven’t discussed this problem in particular, I do think that a discussion of polygons can be really fun for kids. Our most recent polygon project was based on the new pentagon tiling pattern discovered last month:

Using Laura Taalman’s 3d printed pentagons to talk math with kids

and for graphing points in a plane, this was a really fun exercise:

Stuart Price and Joshua Bowman’s Pi-th roots of unity exercise

and Larry Guth’s “no rectangles” problem probably fits the bill for a project involving points in a plane:

Larry Guth’s “no rectangles” problem

6th grade – polygon nets

For me, the starting point with polygon nets was another Laura Taalman blog post:

3D printing and platonic solids

7th grade – an unsolved problem about runners on a track

The problem in Pachter’s blog here is another one that I’d not seen before and it is an amazing problem. We’ve never done anything like it. I’ve never seen anything like it πŸ™‚

8th grade – Beal’s conjecture

Beal’s conjecture is a great math problem for kids to play around with.

The idea Pachter is using to introduce Beal’s conjecture is exponents. I mentioned our Graham’s number projects above, but we’ve done a few others, too:

A few projects for kids from Richard Evan Schwartz’s “Really Big Numbers”

and then a neat project related to exponents inspired by a Steven Strogatz tweet:

The joy of x^x^x^x. . . . .

High school

Pachter’s first example about Gaussian integers is related to this project:

A really neat problem that Gauss solved

For the geometry example of the Euler Brick – this fun project inspired by Patrick Honner is related:

Mr. Honner’s 13-14-15 triangle and a surprising unsolved problem

Also it this paper was recently published claiming a solution to the perfect cuboid problem (I haven’t looked into it and do not know if the paper has been reviewed):

Has the perfect cuboid problem been solved?

For the function example he discusses Euler’s totient function. We’ve recently talked about a super fun problem posed on twitter by Christopher Long that sort relates to the totient function:

Talking through Christopher Long’s neat probability problem with kids part 1

Talking through Christopher Long’s neat probability problem with kids part 2

For the modelling example, I don’t have any similar project that we’ve done.

For the geometry example, we did a different unsolved geometry problem tweeted out by Steven Strogatz. This is another really neat 3d printing project, too:

3D Printing and “Rollers”

Finally, for statistics and probability, though not unsolved problems, we’ve done some fun projects such as the Christopher Long-inspired project above. I also think that geometric probability is a neat subject that most people don’t get to see:

I can’t wait to talk about geometric probability with my kids

and for a more stats-related project, I’ll pick one from my own working life:

Perfect Brackets

So . . . I loved seeing Pachter’s blog post this morning and love the idea of using unsolved math problems to inspire kids of all ages. Can’t wait to try out some of the new ideas that I learned from this post!

“Simplify don’t complify”

My older son had trouble with a practice problem from his math club this morning. Part of the problem can from trying to slog through lots of arithmetic.

We went through the problem this morning using his approach – we were running a little bit low on time so we did some (though not all) of the calculations with a calculator. Part of the point of what I wanted to show him this morning was that all of this arithmetic did indeed make the problem a little confusing:

 

When he got home from school we talked through the problem using a slightly different approach. This approach involved trying to do as little calculating as possible:

 

I love my son’s old math motto – “simplify don’t complify”. He came up with this motto when he was 5 or 6 πŸ™‚ This is a nice example of a problem where getting caught up in the calculations makes the problem much more difficult and confusing than it needs to be.

Michael Pershan’s factoring question

Saw this question from Michael Pershan on Twitter today:

I guessed that my kids would multiply out and then pick out 2 as an alternate factor, but there’s really only one way to know for sure what they’d do . . .

For each kid, I asked them to solve the problem and then asked for a second solution that they could get using a different approach. Here’s what they did:

 

 

 

 

So, there’s 4 mostly different ways that a kid might approach the problem. I’m actually pretty happy that their ideas really didn’t overlap that much πŸ™‚

Counting digits

My younger son had a problem from his math club that was pretty fun:

For today’s project, we revisited that problem, made up our own version, and then finally solved one of the problems posted in response to the original problem:

Here’s my younger son’s solution to the problem I posted on twitter:

Next, I asked my older son to make up a similar problem and solve it. His problem was – if you write the integers from 1 to 805, what is the 805th digit that you write. Great! Here’s his solution to that problem:

Finally, we moved on to solving the 1,000,000th digit problem. This one took a little longer simply because the numbers involved were larger, but we managed to find our way to the solution of this one, too.

One bit that took a little time was the subtraction problem 1,000,000 – 488,889. The boys chose to evaluate this subtraction problem in sort of a strange way – which is totally fine – one of the nice things about these Family Math projects is sometimes getting in a little bit of arithmetic practice πŸ™‚

So, a neat problem from my son’s math club gave us a chance to talk a little bit about counting techniques, arithmetic, and even a little bit about problem solving. Definitely made for a fun morning!

A fun physics game

Saw this tweet from John Golden yesterday:

Here’s a direct link so you can play the game yourself:

Play Potato Here

I spent the morning watching my kids play the game and even published and uploaded the videos of the project. Watching them just now, though, I’ve decided to not include them because they ultimate give away how to play the game. Since one of the super fun things about this game is actually figuring out how to play (there are no instructions), I felt it was better to not show the game play.

Anyway, a large part of the fun this morning was watching and listening to them discuss the game and (what they thought were) the rules. Definitely good math and problem solving related conversations.

A really fun game to check out with kids.

Moving on from trial and error

My son wanted to talk through an interesting problem from his math club homework this morning:

Mary notices that the number of miles on her car’s odometer is (i) a multiple of 1,000 and (ii) is also a perfect cube. She does some calculations and determines that she has to travel another 4,921 miles until the odometer will be at a perfect cube again. What mileage is her odometer showing now?

My son solved the problem through trial and error, so we spent a little time today talking about an alternate way to solve the problem. I’m sad that the conversation is a little rushed, but that happens sometimes.

Here’s how we got started with the conversation today. He introduces his trial and error method and we talk a little bit about last digits of perfect cubes just to clarify a small point in his thinking.

So, with the little last digit point behind us, we returned to talking about the problem. He notices that 1,000 can’t be the answer because the next cube is 1,331 and that’s not 4,921 away from 1,000. He also thinks that he can eliminate 8,000, but isn’t sure.

His idea about 8,000 inspired a short conversation about how you could walk through a list more efficiently than checking every number.

Rather than having him multiply out a bunch of numbers, I decided to show him an algebraic solution to the problem. This is where I really wish we weren’t running low on time – I would have liked to have talked through the idea here in much more depth.

So, definitely not our best day in terms of talking about ideas, but I did want to help him see how to move beyond trial and error solutions.

Revisiting lines

My older son is participating in an amazing program called Idea Math and I wouldn’t be able to say enough great things about this program if I tried. Last weekend I had the pleasure of listening to the academic director – Dr. Zuming Feng – lay out his philosophy for the program. It was wonderful!

You can see him talking a bit in the beginning of this old video about the 2006 US IMO team:

The program intentionally does not give out much homework, maybe just an hour or so per week. Probably less, actually, though we are obviously brand new to it, so I don’t really know the exact number. Today my son started in on one of the homework problems about lines.

It has been a long time since we talked about lines, so I was really happy to see that he’s studying lines again. The first part of the problem involves drawing some lines given the equation of those lines and then estimating the coordinates of the intersection points:

The follow up question asked him to find the solution to the two equations using algebra. He seems to be fairly naturally inclined towards calculation, so this part was fairly straightforward for him. I’m excited to talk about the final part of this problem tonight – describe what happens geometrically when you add these to equations for a line together?

A problem that led to an interesting math conversation this morning

My younger son’s math club had a homework problem that boiled down to this idea:

Your backyard is 600 square yards. You want to pay someone to mow the yard for you, but want to pay less than 5 cents per square yard. What is the maximum amount of money you can pay?

It was interesting to me how both of my kids interpreted this problem differently than I did.

Harvard’s Gender Inclusivity in Mathematics Talk

[sorry for the total lack of editing on this one – had an hard time stop today for work]

Last night I attended a talk about Gender Inclusivity in Mathematics at Harvard. This talk was the first of several talks planned for this year – more information on the series is here:

Harvard’s Gender Inclusivity in Mathematics Speaker Series

The speakers from last night were Cathy O’Neil and Moon Duchin. I’ve followed O’Neil for a long time online through her blog posts at mathbabe.org and her participation on the Slate Money podcast. Actually, I found out about the event last night through one of her posts:

Bloggy young nerd women

I’d not come across Duchin, who is a math professor at Tufts, previously, but **holy crap** is that my loss. She had a fascinating perspective on how to approach your own math education as an undergraduate student and as a graduate student, too. It made me wish her talk had somehow been teleported back to the fall of 1989 when I started college.

As a funny (to me at least!) aside, Duchin mentioned that she has a book on the history of mathematics coming out soon. I forgot to ask for more info about that book at the end of the talk last night, but looking later for more information online I found out that she grew up on Stamford, CT. So, after completing our move from Stamford to Lexington just last week, my first bit of fun cultural activity in the Boston area was attending a talk given by folks who grew up in Lexington and Stamford. Some sort of weird birthday paradox happening there πŸ™‚

Three things from the talk last night that are resonating with me today:

(1) The ideas from O’Neil and Duchin on advice they’d give to undergraduates.

Duchin started off by describing how she approached her undergraduate education. She said that she took many graduate level classes in various different departments, but that what really stuck with her 15 or so years later were the ideas that she came across in the more general undergraduate classes. One bit of advice from her was to resist the urge to accelerate as fast as possible in college. She mentioned that even today she’s taking an undergraduate class (in philosophy, I think) to continue to broaden her education.

O’Neil offered the idea that if you were planning on going to graduate school, taking some graduate courses in that field was probably a good idea. Duchin agreed and also mentioned that some research work she did in math as an undergrad helped her prepare for graduate school.

(2) O’Neil’s idea that math training helps you be better at admitting when you are wrong

This is a powerful idea that I’ve actually only heard from her. She’s also written about that idea here:

Mathematicians know how to admit they’re wrong

The passage that struck me reading this piece back in 2012 was:

Not every person gets trained in being wrong and admitting it. I’d wager that most people in the world, for most of their professional lives, are trained to do the opposite in the face of being wrong: namely, to wriggle out of it or deflect criticism. Most disciplines spend more time arguing they’re right, or at least not as wrong, or at least they have different mistakes, than other related fields. In math, you can at the most argue that what you’re doing is more interesting or somehow more important than some other field.

The idea of not bothered by being wrong has helped me tremendously in my job. It is funny how often in business people try to get a leg up – or just outright try to bully you – by telling anyone who will listen that you are wrong. But training in math teaches you to think about ideas from many, many different angles, and search and search for ways that ideas can be wrong. I think that O’Neil is right that this sort of training is unusual. But the more comfortable you are in searching for ways that you might be wrong, the more likely you’ll be to find the right (or at least a pretty goo) answer, I think.

The idea is pretty similar to the philosophy from this old article “Winning thee Loser’s Game” that success comes from avoiding mistakes rather than from successfully swinging for the fences:

Winning the Loser’s Game

(2) Mindset

I know that “growth mindset” probably long since crossed over into being a cliche, but that doesn’t mean it is a bad idea.

There was a fairly long discussion about the idea of what happens when you encounter a difficult problem. Two possible choices are:

(A) Hey, I can’t figure this out right now, but if I work hard I’ll be able to, and

(B) I can’t figure this out right now, so I must not be very smart.

The discussion last night, obviously, was that (A) is a better way to approach math (and the world of learning in general) than (B).

My first encounter with this idea was on Josh Waitzkin’s book The Art of Learning, though there’s plenty of other places to read about the idea. A suggestion from the audience was that girls are often taught that (B) and that boys are often taught (A). All I can say is that I certainly do try to teach my own kids (who are boys) that (A) is is a good way to approach learning.

Sort of on the same topic, O’Neil has written a lot about how math contests can lead to ideas about not being good in math, or about ranking. For example:

Math contests kind of suck

Jordan Ellenberg, who (among other things) was one of the top US math contest kids my year in high school, also writes about how he used to think that the top math contest kids would become the top mathematicians in his book How not to be Wrong and how he’s moved way past that idea now.

Some concern was expressed last night that Harvard’s undergrad math program can seem like a program where only the stars get attention. Having not been in Harvard’s math program, I don’t know, but I hope the ideas that O’Neil and Duchin talked about in relation to mindset help students a little with the ideas about approaching their own math education and also with the ideas that success in math is just for super stars. Hopefully Harvard’s math department is aware that people feel this way about their program and can take steps to change the source of those feelings.

Finally, one other thing that’s been on my mind that O’Neil and Duchin talked about is what they called the paradox of prizes for women. They gave a a pretty compelling argument that special prizes only for women can be damaging because when there is an open prize and a prize for women, men win the open prize and women win the other prize. Their argument extended beyond prizes, too, and applied to areas like hiring when special money in universities is set aside for hiring female faculty members.

It made me wonder if the two awards I’m sponsoring for ultimate could have unintended negative consequences:

The Michelle Ng Inspiration Award

The True Veteran Award

I going to have to seek some more advice about this issue before the end of the year.

So, definitely a fun night last night. It was nice to see 100-ish people there for the talk. Hopefully this new series of talks will generate some good ideas both for the students in Harvard’s math department and hopefully some meaningful changes, too. Maybe some of those ideas and changes will extend beyond Harvard yard!

What learning math sometimes looks like – a distance and time problem

Last week the math club at my older son’s school got started. The first day the kids received a handout with a few hundred practice problems and we’ve been working through them slowly – usually 5 problems per day.

One of the problems today gave my son a tremendous amount of difficulty. I was surprised – not by the fact that it was difficult for him – but rather the extent of the difficulty. Reading through the questions ahead of time, I’d thought all 5 problems were roughly the same level. It turned out that 4 were about the same level and this one was essentially 4 times as hard.

Even when we went to talk about it on camera I thought it would take 5 minutes – it took 20. I present that conversation below without much comment. Maybe not my best work helping him out either, by the way, but the 20 minute conversation below is what learning math sometimes looks like. Not a straight line, but zig-zaggy struggle.

Here’s the introduction to the problem and the first steps toward the solution, as well as a discussion about a few things that confused him the first time through the problem:

 

At the end of the last video my son was close to writing down some equations that would help solve the problem. Here he does write down those equations, but, unluckily, with one little error that we’ll straighten out in a bit.

 

At the end of the last video my son noticed that there must be a mistake in his equations – here we search for that mistake. Part of the difficulty in finding the mistake comes from translating the ideas from the problem into math.

 

In this last part we get to the solution of the problem – and the best thing is that after our long talk here things do seem to start making sense to him. That was nice to see.