Synchronicity

I first heard the word synchronicity when I was sitting in an airport, waiting for a plane, working on this math problem (for fun) and over-hearing a man’s cell phone conversation. I thought that it was a good word for this phenomenon. Here it is:

Have you noticed that 3 times 7 is 21 and if you expand 7 base 3 it is 21. That is:

This seems like an interesting coincidence. Could it be that this happens for any other numbers? Maybe it happens infinitely often, maybe never. Asking such questions is where the adventure begins. Of course we are implicitly working base 10 here. That is, in the first line the string of numerals “21″ is interpreted as a base 10 number. In the second one it is (explicitly) interpreted base 3. You could allow 10 to be replace by other bases also.

If we restrict our attention to base 10 (in the first line), I claim that, by exhaustive computer search, there are only five other instances of this phenomenon. They are:

and

and

and

and

and

Math with scissors

The other day I visited my son’s class and did the following math demonstration.

Take a strip of paper and draw a line down the middle. Our line is red.

Tape one end to the other, but put a “half twist” in it, just like shown here. This is a Moebius strip. It only has one side to it. A pretty cool object in and of itself.

Now for some real fun. Take a pair of scissors and cut the Moebius strip in half along the red line–But before you do that try to guess what you will end up with!!! I’m not going to show you. You have to try it yourself. Just make sure that you only cut along the red line!

There is more that we can do. Let’s make a + out of paper with two red lines drawn as shown here.

Now tape two opposite ends to each other with no twists. Like this.

Tape the other two ends to each other, but put in a half twist. What you have here is actually a Klein bottle (whatever that is) with a hole in it.

Now cut along the two red lines. Make sure not to make any other cuts except along the lines.

Keep cutting…

Keep cutting.

In the end what do you get?

(Make a guess first.)

(!)

One more: Make a star shape just like this. Make sure that all of the strips are the same length and that the angles between the strips are the same. Draw three red lines. One down each of the strips.

Here is what it should look like.

Now tape two opposite ends together with no twist.

Take another two ends…

…and tape them, with no twist, just like this.

Now for the tricky part. You have to do this part right to get nice shapes in the end. But, don’t worry too much. If you do it “wrong” you’ll just get something other than what we got. Flip your paper over. Tape the last two opposite ends with a full twist to the left. Just like shown here.

Here is another view. Now cut along the red lines and see what you get. Bailey’s class loved it.

Multiplication tables

After reading Lockhart’s Lament (see yesterday’s post) you may wonder what my experience with math in school was and how I got to be a mathematician. Here are a few, rather disjointed vignetes :

Once (upon a time) in 1st grade my teacher was teaching us about multiplication. She said that she had a little secret that she knew: when you multiply a number by one you get the same number that you started out with! So there you go, we should have no trouble multiplying by one now. I thought that it was great that she had discovered this fact and I wanted to get in on the act also. I had noticed a pattern myself: when you multiply by two you get double the number that you started out with! Now there was a fact that my teacher hadn’t yet discovered! So I went to her and told her this little secret that I had learned. I thought that surely she would gather the class and tell them that: hey look, here is another little trick! She never did. I wasn’t crushed, but I was puzzled. After all, at the time at least, my secret seemed just as good as hers. I mention this for two reasons:
1) Because of the incident I know that I had a natural, intuitive understanding of what “doubling” meant before learning about multiplying by 2, and, at the time that I was first learning multiplication, these two things seemed like distinct and separate operations. (If I didn’t remember this incident, I wouldn’t remember how I thought about doubling and multiplying by 2 at that age).
2) Also at that age I thought of math as a process of discovery that everyone was allowd to participate in (which it is!). The system hadn’t yet drummed that idea out of me. I guess that the system never really did drum it out of me completely.

My second memory comes from first grade also, I’m pretty sure, but it could have been second grade. We were learning how to add multi-digit numbers and we were supposed to be working on a sheet where we calculated dozens of such sums. I thought it was cool that the technique we were using would let you add two numbers no matter how big they were! So I flipped my sheet over, wrote down two really big numbers, and started adding them. My teacher caught me before I had gotten very far and made me turn my sheet back over and work on what I was supposed to be doing.
Retrospective comments:
1) Why did she do that? Wouldn’t I have gotten just as much practice from working out my own sum? Yes! And I would have been having more fun. Again, I was participating in discovery for myself. But, of course, it would have been harder for her to check my work for accuracy…
2) I was having some fun with the playground in my mind! These big numbers had nothing to do with any real-life application. Kids just like big numbers! In fact kids love big numbers. My niece really likes the number 1,082. It was a kid who made up the words googol and googolplex. Why don’t we let kids play with numbers in a math class?

The next scene is from fourth grade. My teacher called two students to the front of the room. (I think that she had arranged this with them ahead of time). On the board she had a problem for each of them. One was to calculate $\latex 7+7+7$. The other was to calculate . They set about calculating. Well, I should say that the first student set about calculating . The second student simply wrote down . When they had both finished, our teacher said “See, multiplication is just a fast way of adding.” I was appalled. I thought about what would have happened had it been me up there. I didn’t have memorized, so I would have had to calculate . It’s the exact same problem! There is no easier way (that I was aware of at the time) to calculate other than calculating . Of course, if you already have the answer memorized, then of course you are going to be able to do it quickly. You just write it down! I don’t know if it was this demonstration, or something else (or just the fact that I am bad at memorizing facts) that turned me off to memorizing my times tables. In any case, I remember staying after school (or maybe it was staying in from recess) in order to learn my times tables with the other slow kids.
My comments here are:
1) Looking back I realize that the teacher could have had a much better demonstration if she had given something like to the first kid and to the second, or, even better, for the first and for the second, then we would have seen the power of this cool little algorithm for multiplying. You know, the one that you learned–just do it a digit at a time.
2) I don’t think (and I could be wrong on this) that multiplying is just a quick way to add. It certainly can be a quick way to add, but I think that going from adding to multiplying is a pretty big leap. Multiplying is a whole other animal. It is related to adding, and it interacts with adding in interesting ways, but I think that it is more than just fast adding. (Like I said I could be wrong. This is a philosophical issue, as much as a mathematical one.)

That’s enough for now, but look for more such vignettes in later posts.

Math class is stupid and boring

…but it doesn’t have to be. Scroll down to my last thesis update. See those nice pictures? Do you like them? Are they pretty?  That’s what math is all about: neat ideas, pretty pictures.

I just read a really good article. Here is an excerpt:

…if I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done— I simply wouldn’t have the imagination to come up with the kind of senseless, soulcrushing ideas that constitute contemporary mathematics education.

Check out the article. I really liked it:

Lockhart’s Lament

To be fair to Keith Devlin I should say that it was from his article that I found Lockhart’s Lament:

Devlin’s Angle

Thesis Update

I have turned in a draft of my thesis to my committee. I’m waiting for feed-back from them. Here are some pictures that I produced for it:

Thesis Update

Arlynda has been very good the last few days about taking care of everything so that I could work on my thesis. So I have made a lot of progress. Today I took my office’s laptop to Panera bread and worked for most of the day. I think I have basically all of the content and most of the formatting.
Still to do: 1) give to my adviser and get feedback 2) draw my figures and insert and label them.

Crunch Time!

I realized today that I have 6 weeks to get my thesis done. (!) I hit the panic button and worked on it all day instead of doing something fun with my kids, which is my usual Saturday activity. I now have a draft that is (…uh…almost) ready to be handed over to my adviser for criticism.