What learning math sometimes looks like – when a problem suddenly makes sense

I really like this problem from the 1985 AJHSME:

Problem #20 from the 1985 AJHSME

Here’s the problem:

In a certain year, January had exactly four Tuesdays and four Saturdays. On what day did January $1$ fall that year?

This problem has always given my son a little trouble. Talking through it tonight he struggles a bit to get going, but slowly gets to the point where he’s drawn a calendar that represents the situation. It doesn’t look like a normal calendar, though.

Around 4:30 in the video he’s talking about why the picture looks wrong, but then comes a really cool idea to fix that problem – re-draw the calendar! From this new point of view the solution to the problem just falls off the page.

What my son went through talking this problem tonight is what learning math often looks like. From one point of view a problem can be nearly intractable and frustrating, but from another everything seems to make perfect sense.

My favorite math class

Saw a twitter hash tag #myfavoritemathclass going around a few days ago. I actually struggled to come up with just one to call my favorite, but it was really fun thinking through all of the classes. There were roughly four categories for me:

Junior High

Mrs. Whitney’s geometry class.

Up to that point in my math education everything was all about calculation. For me that was fine – I’m much more of a calculator by nature than anything else – but the new ideas about proof and constructions in geometry class were really interesting. The constructions, especially, I suppose and some of the problems that were not really accessible to me then stayed with me for a long time. I remember, for example, learning that $\cos(72^{o})$ = $(\sqrt{5} - 1) /4$ in high school and immediately heading off to construct a pentagon.

High School

This one is the only no-brainer – Mr. Waterman’s Enrichment Math class. This class was mainly for the math team at my high school and I took the class all three years I was in high school. Mr. Waterman also taught calculus, linear algebra, and differential equations, but it was enrichment math that really shaped how I think about math.

We learned all kinds of different math – basically whatever Mr. Waterman felt like talking about that day. Nothing was off limits – advanced geometry, number theory, fun algebra problems from old contests. Also super fun was having former students come back to give talks. After graduating I came back to give a talk to the class every year that Mr. Waterman was at Central.

College

There’s almost no way for me to pin it down to one class here – nearly every class had something amazing. The overall theme for me in college was learning that you could actually study math for math’s sake!

My introduction to complex analysis from freshman year convinced me to major in math. I remember nearly falling out of my chair seeing this integral on a homework assignment:

My analysis class out of Rudin was one of the biggest struggles that I’ve ever had in math. The professor even told me at the end of the semester that I didn’t have what it took to be a mathematician. Whatever, dude – I was just glad to have that class behind me ☺

The next analysis class was incredible – I don’t have the book anymore, but I think it was called “Measure Theory and Integration.” Who knows why, but Parseval’s theorem was really fascinating to me. I found a research job programming the data transmissions for a satellite telescope using ideas from Fourier analysis the summer after taking that class.

The most absurdly hard class for me in college was topology, which was by professor Munkres out of the topology book he wrote. At the start of the class he told us that he’d been teaching the class since the 60s and graded on a curve based on all of the students that he ever had. Given the size of the class, he estimated that there would be 2 A’s an 1 A-. There were 2 IMO gold medalists and 1 silver medalist in the class – ha ha. I didn’t really understand topology even remotely until grad school.

The class that was the most fun was Mike Artin’s Algebra course out of the now published Algebra book he was writing at the time. He had such a fun teaching style and was probably 2nd only to Mr. Waterman in his ability to communicate his love for the subject through his teaching. I was really happy to run into him at MIT’s graduation last spring and show him a fun project that we’d done based on a picture in his book.

A 3d project for kids and adults inspired by Kip Thorne

The class that just blew me away was Richard Stanley’s combinatorics class. The ideas that he showed us about generating functions were incredible – if there’s one thing that I wish I understood better now, it is the connection between algebraic expressions and counting. He had so many amazing examples – counting is really cool.

Graduate School –

From a “class” perspective, what I remember most about graduate school wasn’t a class per se, but a series of lectures that Cliff Taubes gave at Harvard. Two mathematical physicists – Nathan Seiberg and Ed Witten – announced some amazing breakthroughs applicable to geometry / topology / string theory in the mid 1990s and Taubes gave the “mathematical physics” to “math” translation.

As people began to understand the Seiberg-Witten framework, long standing problems in math were falling like dominoes. There was so much energy in the room for Taubes’s lectures. I’d never seen anything like it in math. I really got caught up in the excitement and dove into Freed and Uhlenbeck’s “Instatons and 4 Manifolds” and Lawson and Michelson’s “Spin Geometry.” It was a fun time to be hanging around math ☺

Sorry I couldn’t pick just one class!

Anna Weltman’s Loop-de-Loops

Saw this tweet when I got up this morning:

Enter my fall math contest and win a class set of @AnnaWeltman's awesome book, This is not a Math Book. http://t.co/WgMQmxnYtH

— Dan Meyer (@ddmeyer) September 29, 2015

My older son had to leave a little early for school today, so I decided to use some of the extra time I had with my younger so to play around with the loop-de-loops.

I printed out the two sheets instruction page and student work page linked in Dan Meyer’s post and tried out the activity.

Here’s the introduction and an attempt to replicate the picture on the instruction page just to make sure we knew what we were doing. I published these videos in hd, but it still may be difficult to read the black writing against the red background – my camera skills just aren’t that great.

and here’s the attempt to draw a 6-3-5 loop-de-loop:

As I was writing this up, Dan Anderson tweeted about a Scratch version of Weltman’s activity – looks really neat!

Programming and @AnnaWeltman ’s This Is Not A Math Book and @ddmeyer ’s Contest http://t.co/rOoLsygboN Enjoy! pic.twitter.com/cBaOQaC9q4

— Dan Anderson (@dandersod) September 29, 2015

Finally, as I mentioned in the last video, on other maybe interesting extension of this activity is combining it with some of the “fold and cut” projects we’ve been doing recently (though I have absolutely no idea how difficult this extension would be):

Fold and Cut Activity #3

So, a really fun little project this morning with the loop-de-loops. Glad I happened to see it in my twitter feed this morning 🙂

Fold and cut part 3

We decided to do one more fold and cut project tonight. Our first two are here:

Fold and Cut project #1

Fold and cut project #2

Tonight we decided to try to fold and cut some of the letters that Katie Steckles demonstrated at the end of the Numberphile video. My older son chose the letter ‘B’ and my younger son chose the letter ‘L’. We didn’t quite get the ‘L’ right, but overall both letters were a great challenge.

Here’s my older son and ‘B’:

and here here’s my younger son with ‘L’:

Feels like we could do projects like these for a long time! For some more amazing one cut shapes, check out Erik Demaine’s MoMath lecture (with the one cut shapes starting around 2:30 in the video below): (h/t to Patrick Honner for this amazing lecture by Demaine, btw)

The fold and cut theorem is awesome

Yesterday we saw this incredible Numberphile video:

We did a really fun project this morning based on the video, too:

Our One Cut Project

There was much more to do, though – this activity has so many different possibilities for kids! One thing that was on my mind all day was shapes with holes. While my younger son was taking a bath, I decided to try out a simple square in a square shape with my older son:

When my younger son got out of the tub he gave it a try. The cool thing was that he did it a slightly different way!

Just to emphasize that comment near the end: “hey cool . . . the fold and cut theorem is awesome!”

Can’t wait to try more shapes!

Our one cut project

Saw this amazing video on the “one cut” problem from Numberphile and Katie Steckles yesterday:

After watching the video, my younger son spent some time trying to make various shapes last night:

That gave me the idea to try out the idea for our little Family Math project today.

The first shape we tried was also the first example in the video – a square:

Next I drew a rectangle to see if the kids could apply some of the same ideas to fold the paper and cut out the rectangle with one cut. There’s a lot of nice mathy discussion from the kids trying to figure out the fold here:

For the next challenge – an equilateral triangle. I thought that a general triangle would be too difficult for a 5 minute video project, but the equilateral triangle worked really well. Again, there’s some great math conversations:

Finally, I had them follow the folds that Steckles made in the video to produce a star. This was a challenge for them, but we got pretty close. Not bad for our first try 🙂

So, a super fun project. Once again Numberphile has produced an incredible video you can use to talk math with kids.

A problem that’s always bugged me a little

Note – just to be clear from the start – this post is only half serious, and I understand that I’m playing a little fast and loose with a few things – especially the definition of a limit – but it is true that the concept I’ve outlined below has always bugged me.

I started reading (actually listening to) Eugenia Cheng’s How to Bake Pi today. It is a fun book so far, and certainly made working out at the Harvard stadium this morning much more enjoyable than usual.

One of her statements that got me thinking was the idea that nearly everyone hits a wall in mathematical abstraction at some point. I may not have Cheng’s exact words exactly right, but the idea was pretty interesting to me. It made me wonder about some fairly abstract ideas that I’d never completely reconciled in my mind.

What came to mind is strange property of the harmonic series (1 + 1/2 + 1/3 + 1/4 + 1/5 + . . .). Nearly everyone I’ve shared this idea with thinks I’ve completely lost my mind, so with that warning (and with the extra note at the start!), I guess, here we go 🙂

A standard proof that the harmonic series diverges goes something like this:

Step 0: 1 is bigger than 1/2
Step 1: 1/2 + 1/3 is bigger than 1/2
Step 2: 1/4 + 1/5 + 1/6 + 1/7 is bigger than 1/2
Step 3: 1/8 + 1/9 + 1/10 + 1/11 + 1/12 + 1/13 + 1/14 + 1/15 is bigger than 1/2

The reason that the sum in step 3 is larger than 1/2 is that all 8 terms are larger than 1/16, so the sum is greater than 8/16 which is 1/2.

Step n: $1/2^n + 1 / (2^n + 1) + \ldots + 1/(2^{n + 1} - 1)$ is greater than 1/2

The reason that this sum is greater than 1/2 is more or less the same reason given for step 3. Each of the $2^n$ terms is larger than $1/(2^{n+1})$ so the sum is larger than $2^n / 2^{n+1}$ which is 1/2.

At each step m, you have a sum with $2^m$ terms which is greater than 1/2, and that tells you that the finite sum $1 + 1/2 + 1/3 + \ldots 1/2^m$ is greater than $m/2$ and so as you let m approach infinity, the sum also goes to infinity.

Totally seperately, when you learn about set theory you learn about some of Cantor’s theorems about power sets. The “power set” of a set is the set of all subsets of the set, including the null set and the original set itself. The notation that I learned for the power set of a set $S$ was $2^S$ . That notation, I assume, comes from the fact that for a finite set $S$ with $n$ elements, the number of elements of the power set of $S$ is $2^n$ .

Cantor proved that the cardinality of the power set of a set $S$ is always greater than the cardinality of the set itself. In particular the cardinality of the set of positive integers – the “smallest” infinity and usually denoted by $\aleph_0$ , is larger than $\aleph_o$ . In fact, the cardinality of the power set of the set of positive integers is equal to the cardinality of the real numbers. If we call the cardinality of the real numbers $R$ , what I’ve written above can be summarized by saying:

$R = 2^{\aleph_0} > \aleph_0$

Loosely speaking, that equations tells you that there are “more” real numbers than there are integers.

Ok, so back to the standard proof that the harmonic series diverges. The way that I’ve constructed the proof above, at every step m we show that the sum of $2^{m+1}$ terms starting at $1/2^m$ is greater than 1/2.

So, as $m$ goes to infinity, the number of terms we add in each additional step – namely $2^{m+1}$ terms – also gets *really* large. In fact, so large that it is actually a different infinity! Right?? That means that the “last” sum that is greater than 1/2 in the proof above – namely the step that “finally” gets you to infinity requires adding up more terms than there are integers!

A little crazy, I know, but at least I felt less crazy about being puzzled by this idea after learning that $1 + 2 + 3 + \ldots = -1/12$ 🙂

Basic stats: A math club homework problem and a problem from the 1989 AIME

My older son had this interesting basic statistics problem as part of his math club homework:

For a list of 8 positive integers, the mean, median, unique mode and range are 8. What is the greatest integer that could be in this set?

My older son and I talked through this problem yesterday as part of our short, morning math talks: MathyMath15. It is such a nice problem, though, that I wanted to turn into a project.

Part of the reason was the similarity to this problem from the 1989 AIME:

Problem 11 from the 1989 AIME

Here’s the slightly simplified version of that problem I used in our project today:

A sample of 121 integers is given, each between 1 and 1000 inclusive, with repetitions allowed. The sample has a unique mode (most frequent value). Let $D$ be the difference between the mode and the arithmetic mean of the sample. What is the largest possible value of $D$?

I really like both of these problem for getting kids to think about different ways numbers can be combined to produce similar (basic) statistical results.

The 20-ish minute conversation we has about these two problems is below. I let my younger son tackle the first problem mostly on his own since my older son had already worked through it yesterday. There are several fun “aha” moments in the conversation:

Part 1: Introducing the first problem (including defining some of the words) and my younger son’s initial thoughts. His first thought uses all 8’s:

Part 2: My younger son tries a few more ideas based on noticing that the sum of the numbers needs to stay at 64. He finds a solution that uses 12 as the largest number.

Part 3: Here the three of us work together to find the largest integer that can be in the set:

Part 4: Now we move on to the problem from the 1989 AIME. There are more numbers to keep track of, and the arithmetic is a little more difficult, but the ideas we need to solve this problem are pretty much exactly the same as the first one. Both kids have several really nice ideas about how to tackle this problem over the next two videos:

Part 5: While skipping over the arithmetic, we walk down the path to the solution of the problem.

So, a fun project connecting some ideas from one math club problem to an old AIME problem. AIME problems obviously aren’t usually going to be great problems for kids, but I thought this would be an interesting exception. It is pretty neat to me that more or less the same ideas solve both problems 🙂

The Common Core check

Several friends have written to ask me about the Common Core check story as well as the follow up post by Hemant Mehta:

The Dad Who Wrote a Check Using “Common Core” Math Doesn’t Know What He’s Talking About

The short answer is this – math education debates exhaust me . . . .

Here’s a longer answer.

We taught the kids at home until this year, and now they are attending the local public schools in 4th and 6th grade. Other than making sure that they’ve completed their homework, I don’t pay that much attention to the math the kids are doing in school. Instead I try to find fun supplemental activities to do with them such as our “Family Math” projects on this blog.

I’m sure, though, that there will be some point where they ask me about their math homework and I will be in the position of not really understanding what they are being asked to do. At that point I’ll put Tom Lehrer’s “New Math” on the playlist and they hopefully take the approach that Mehta did by spending a few minutes on Google trying to see what’s going on.

For the specific example that caused all of the fuss here – “ten frame” cards – I’ve never heard of them before reading Mehta’s piece. Thanks to Mehta I get the general idea and understand why someone might want to spend a little time talking about numbers with kids using the idea. The general idea isn’t something that I think is new at all, btw.

My Montessori pre-school in Omaha (so, 1975 ish) had a big collection of red and blue rods that were used to represent numbers. Maybe they were unique to my school, or maybe lots of schools had them – I honestly have no idea – but I loved playing with them. Sister Lorraine used them to teach me math all the way up to division. Maybe my parents were so frustrated by this approach to learning about arithmetic that sent checks to the school using blue and red rods, I’m not sure, but for me they were a fun way to think about numbers. Also, those rods from 40 years ago don’t really seem that different at all from the “ten frame” cards being discussed this week.

While teaching the kids at home I used several different geometric ideas to help the boys think about and learn about numbers. Three examples come to mind as I’m writing:

(1) For learning place value, we spent a lot of time talking about numbers in different bases and made a “binary adding machine” out of duplo blocks. This was a really fun activity that I probably spent two weeks on with both kids:

(2) When we learning about dividing fractions, we also turned to numbers represented by blocks to see what was going on:

(3) Lastly, I used a representation of numbers in rectangles to show why a negative number times a negative number is a positive number:

So, representing numbers is ways that don’t seem like numbers doesn’t bother me that much. You never know what is going to grab a kid’s attention – those red and blue rods from pre-school sure grabbed my attention. 40 years later I’m still happy to use non-number representations of numbers if it helps kids learn math.

Why I like using old math contest problems

Math contests inspire a wide range of reactions in the math world. Many people do not like them. I’ll take Cathy O’Neil’s post Math contests kind of suck as a nice example from that camp.

On the other end of the spectrum there’s the super-star status that some of the top competitors achieve. A good representative article here is Zach Wener-Fligner’s piece: The Last Contest: Hanging With the Big Dogs at the 2013 Putnam Math Competition

I grew up in math in the world of math contests, so I’m probably pretty biased toward liking them. The reason that I like using them with my kids has nothing to do with competition, though. I like them because the wide variety of questions and topics on the contests help me see where they have misconceptions.

With no school today I had the boys spend a little bit of time on some problem from the 2014 AMC 8:

The 2014 AMC 8 hosted at Art of Problem Solving

My younger son had some questions on problem #18 and that led to a fun conversation about probability:

My older son wanted to talk about question #20. Talking through this problem led to a nice conversation about approximating $\pi$ . It has been really interesting to me to watch his ideas about approximating numbers develop.

So, a fun morning talking through a few old AMC problems. I’m happy that these problems are available online now – they are a great resource.