Math in the alphabet

The children attended a well-run chess day camp this summer. Good people running things; a warm and welcoming atmosphere. Lots of varied activities to keep kids’ bodies engaged as well as their minds.

Sadly, this takes place on the complete opposite end of the Metro area from where we live. We had to drive all the way across St Paul, Minneapolis and deep into St Louis Park during rush hour. Ugh.

This led, one day, to my trying to find a topic of conversation to keep at least one of the children occupied while we drove home. I recount for you this conversation below.

Me: Tabitha. Can I ask you a question?

Tabitha (7 years old): Sure.

Me: What letter comes before I in the alphabet?

T: H. That was kind of an easy question.

I love that she has turned into a critic. If I am not challenging her, she calls me on it.

What she has not seemed to notice yet is that these questions she deems easy are just my openers for the good stuff.

Me: Yeah. Here’s a harder one. What letter comes two letters before S?

There is a fairly long pause here. This is a harder question because of how most of us know the alphabet—forwards. If we want to know what is 2 less than 71, it is not so hard to count backwards. We have lots of experience counting backwards. But we don’t have so much experience saying the alphabet backwards, so we need to make up a strategy.

T: Q and R.

Letter squares for q, r and s

Me: Q is two letters before S, yes. Now you ask me one.

T: What letter comes after Z?

Brilliant. What a great question. I wish I had thought of it myself.

Letter squares for w, x, y, z and three blank squares

Me: Oooooo. Good one. I say A. I say it starts over.

T: Nope.

Griffin has been listening in but not participating. He sees his chance to get in on the action.

Griffin (9 years old): Negative A.

Me: Wouldn’t that be what comes before A?

G: No. It comes after Z. It’s negative A.

T: Nope. Not that either.

Me: OK, then. I am stumped.

T: Nothing.

Me: Huh?

T: Nothing. No letter comes after Z.

So what do we learn?

This is a more sophisticated version of another mathy letters conversation I had with Tabitha a while back. Back then, we were trying to figure out which of two letters comes first in the alphabet. Here, we are more paying careful attention to precise placement (two letters before, not just before).

The other interesting thing going on is our three different ideas about what comes after the end.

My idea: After the end, we go back to the beginning, like the days of the week.

Tabitha’s idea: There is nothing after the end. It just ends.

Griffin’s idea: The end is like zero. When you get to the end, you repeat what you already had, only using negatives.

It is OK that we didn’t resolve who is right.

Starting the conversation

About a year ago, I started making a habit of having the kids ask me the next question. I highly recommend it.

You know how your children are always testing the limits of rules in everyday life? Like you say, “Do not touch” and they see how close they can get their finger to the forbidden object without actually touching it? That is normal and necessary behavior on the part of children.

They will do it in the world of ideas, too. Tabitha did not choose “What letter comes after Z” at random. She chose it because she knew it would be interesting to talk about. It probably would not have occurred to me to ask it. Our conversation was richer because she did.

 

Talking math with a word game [Product review]

Long ago, we were given the game “What’s Gnu?” as a gift.

whats.gnu

What better time to dig it out than in the waning days of summer leading up to the return of school? So Tabitha (six years old) and I did just that a couple weeks back.

Neither one of our children has been an early reader. They both love books. They are highly verbal with substantial vocabularies. And neither one has ever wanted to read aloud.

I am not worried about this.

But “What’s Gnu” is all about reading words aloud. This presented a problem.

So I got creative.

We played “War” with letters.

See that green mechanism to the right of the box in the picture? That is a letter dispenser. You move it back and forth, dispensing two letters at a time.

In our game, we took turns dispensing (it’s totally fun). Tabitha’s letter was on the left each time, mine was on the right. Whoever had the letter that came first in the alphabet won the round, taking both letters. Largest number of letters at the end of the game wins.

We discussed strategies for knowing who won each round. Tabitha described some version of each of the following strategies.

  1. Middle/end. Example: M and X. M is in the middle of the alphabet;  X is at the end. Therefore, M comes first.
  2. Recite alphabet from the beginning. This comes in two versions: (a) stop reciting at the first of the two letters, and (b) stop reciting at the second of the two letters. This one is useful for two letters that are in the same part of the alphabet. Example: H and N (both can be seen as in the middle of the alphabet).
  3. Recite alphabet from the letter you think is first. This is a more efficient version of strategy 2. Example: L and P.
  4. Adjacent. When two letters are next to each other in the alphabet, you can know right away. This may just be a very quick version of 3. Example: H and I.
  5. ABYZ. These letters are so close to the beginning (or end) of the alphabet that they MUST be first (or last), no matter what the other letter is.

There are relationships to something called subitizingwhich refers to knowing how many things there are without counting. You can recognize three objects, and probably also four, without counting or grouping them. But five objects you cannot; you probably group them as three and two without even noticing it. How psychologists measure this fact is super-interesting but not pertinent here.

Instead, notice that strategies 4 and 5 above are like that. Tabitha could recognize adjacent letters without thinking or reciting the alphabet. Reciting the alphabet is like counting. Similarly A, B, Y and Z she could compare to other letters without reciting.

But here’s the point. This counts as talking math.

We were comparing the order of things. Letters, like numbers, have an order. Anytime we are talking about how we know what order things come in, we are talking math.

I did mention that product reviews would not take the usual form on this blog, didn’t I?