# Kindergarten mathematics

This is a call for help. My son’s kindergarten teacher has invited parents to come in and talk about their careers. I’d like to go in and talk about math. I’d like to have some interactive hands-on mathematics activities for the kids to do. I also want them to be activities outside the typical kindergarten curriculum. [Update: my math lesson has already happened. Read about it here.]

Do you have any suggestions? If so, please leave them in the comments below.

I have no expertise in childhood development, but here are some of the facts I’ve observed based on the abilities of my son and his friends.

• Most of the kids are 5 years old (a couple are 6).
• They can count, but they don’t necessarily know any arithmetic (although some do).
• They know the alphabet, but they can’t read or write except maybe the most basic words (such a their names).
• They can’t draw very well (eg. straight lines, circles, etc.).
• They can uses scissors, tape, glue, staplers, etc., but their accuracy is not great.
• They can do some simple logical reasoning.
• They have short attention spans.

Here are some ideas that I came up with. (Some of these were suggested by my colleagues and my followers on Twitter.) This list is the result of a brainstorming exercise, so I know that some ideas are half-formed and some are too advanced for this age group, but I still kept them on the list.

Paper folding, cutting, taping

Bubble activities

Geometry

Pattern recognition

• Teach them the game Set
• List three things in a sequence, ask for the fourth (for example, a picture of a triangle, a square, and a pentagon)
• What do these have in common?

Drawing and coloring

• Simple bridges of Königsberg/graph tracing problems
• 4-color theorem
• Coloring patterns in square, triangular, hexagonal graph paper

Counting

• Permutations (using a tree): We have 3 pairs of shoes, 4 shirts/dresses, and 3 hats. How many outfits are possible?
• Rock-scissors-paper tournament

Numbers

Stick puzzles

Knot theory (some of these are definitely too advanced)

• Have everyone stand in a circle with hands thrust toward the center of the circle. Have the children grab random hands. The result is a giant human knot or link. Have them unknot themselves by taking turns letting go, changing a crossing, and grabbing hold of their partner’s hand.
• Take a long string and tie the ends around Alice’s wrists. Bob’s hands are tied together in a similar way, except his string passes through the loop made by Alice’s arms and her string. Can they become disentangled without pulling the looped string off their own hands?
• Alice holds a long unknotted string with one free end in each hand. Can she hand the string to Bob (one end to one hand, the other end to his other hand) so that when he receives it it is knotted?
• Charley is wearing a big, baggy t-shirt. He clasps his hands in front of him. Can Alice and Bob take off and manipulate the shirt so that it goes back on Charley inside out without Charley unclasping his hands?
• Bob is wearing a big, baggy t-shirt. He stands face-to-face with Alice and holds her hands to form a circle. Can Charley take the shirt off of Bob and put it on Alice without them letting go of their hands?
• Tie three strings to a chair. Braid them together in any way (no knots though!) so that the left strand ends in the left position, the middle one in the middle, and the right-most one on the right. Tape the free ends together. Figure out how to unbraid it without untaping the ends. (It is always possible.)

Play dough/clay

• Turn a coffee cup into a donut without breaking a loop

1. Most of these look too hard to me. In the paper cutting section, the mysterious paper flap looks best, especially if you can start it with a flourish, so it does seem mysterious to them. (So much of the world is still amazing at 5. Will this be any more amazing than things they see every day?)

I think Set is too hard. But Blink is great. (It might not wow a grownup as much, but it’s a great game.) With slightly older kids, after they’ve played Set for a while, making their own deck of Set cards is fun.

The Kaplans (hemathcircle.org/thefounders.php) like to draw a number line from 0 to 1, and ask kids if there’s anything in between. Their method depends on not telling, just asking good questions. I haven’t yet gotten good at it, being used to lecturing in college math classes. But I’ve seen great things happen between them and young kids. Their book, Out of the Labyrinth, describes their work in math circles.

An easy topic to get the kids discussing is Big Numbers. No lecture, just questions, is easier for me with this one. Start with “What’s the biggest number?” and just see where it takes you.

I like your coffee cup into donut one, but I think that one will likely be too hard also.

If you submit this to the math teachers at play blog carnival, you might get tons of response.

1. Thanks for all of your comments, Sue. I figured that some (a lot) of these would be too difficult for them to do alone, but maybe I could do some for them and have other things for them to do themselves.

I’ve had good luck playing Set with my son, but I always removed a trait (for example, green cards only).

I especially like your idea of just starting a conversation with them and seeing where it goes!

2. Jason Dyer says:

Here’s an actual session with a mathematician that supposedly worked well.

http://toomai.wordpress.com/2008/07/06/sequences-and-creative-math-for-kindergartners/

1. Thanks, Jason. That’s fantastic. I think I’ll use that.

3. Cathy says:

There are lots of areas of discrete mathematics that are adaptable to Kindergarten, I think. This is a book I read about it some time ago that has lots of interesting ideas.
http://www.amazon.com/Discrete-Mathematics-Across-Curriculum-K-12/dp/0873533054
When my younger son was that age, he particularly liked finding Euler circuits in graphs. He was also into doing arithmetic mod 2 (even and odd numbers!) For awhile he had himself convinced that there must be more even than odd numbers, because there are more ways to “make” even numbers (i.e., by multiplying them together). You can also investigate lots of ideas about divisibility and modular arithmetic by looking at the kinds of patterns that you can make by threading colored beads on a string.

4. Cathy says:

Since you are a topologist, there is probably some fun activity that you can do with minimal surfaces. I can vaguely remember an activity when I was in kindy (I think) where we dipped some bits of wire twisted into loops into some sort of viscous red stuff that would then set, forming a minimal surface.

1. Thanks for the book recommendation. I’ll have to track that down.

I’m interested in the red viscous stuff! This is essentially what I wanted to do with the bubble examples. If you put enough glycerine in the bubble mix, they are pretty resilient.

1. Cathy says:

I’m sorry that I can’t be more specific, but it was over 30 years ago. It set really hard (like paint) after a few minutes, but was slightly translucent, so our creations really were quite beautiful. Maybe an art supply shop or a hobby shop would have some ideas about what you could use.

5. Sara says:

There are some terrific trade books you can read and use to start a mathematical discussion- I’m happy to share titles if you like the idea. Authors to consider: Steve Jenkins, David Schwartz, Virginia Walton Pilegard, Angeline Lopresti, Cindy Neuschwander – many others but those are a few to get you thinking.

It’s also very interesting to ask Kindergarten students to walk along a number line (a knotted piece of rope). Start at 2 and walk 3 steps. Start at 3 and walk 2 steps backwards. What if you walked 4 steps backwards? Where would you be? It makes the number line concrete and starts them thinking about how numbers relate to one another.

1. Thanks for the book and activity suggestions. I’ll have to look for books by the authors you suggest!

6. Scott says:

Their scissor work is going to be very imprecise, judging by my high schoolers, haha.

Set is going to be too hard I think.

I think Jason Dyer’s link is a good one. (I initially came here to post it)

How about paper folding (how many times can you fold it in half?) I’d also envision posting up the sequence of folds of a square piece of paper. 1, 1/2, 1/4, 1/8, etc are all visually shown.

The Berkeley Math Circle runs an elementary program, but again I think this is pretty advanced: http://mathcircle.berkeley.edu/index.php?options=bmc|bmc_elementary|BMC%20Elementary

1. Thanks for the link and the suggestions. As I wrote above, the full version of Set would definitely be too difficult, but I think removing a trait brings it into reach by kids this age. I’m always surprised by how well kids do with this game.

7. Scott says:

here’s one of the Berkeley Math Circle activites for elementary school: http://mathcircle.berkeley.edu/archivedocs/2009_2010/lectures/0910lecturespdf/lesson3_report.pdf

8. misha says:

Try the Dirac belt trick or the coffee cup trick related to it.
http://gregegan.customer.netspace.net.au/APPLETS/21/21.html

1. Misha. Thanks, yes, that’s a great one. Maybe I’d have them use a sippy cup just in case…!

9. toomai says:

Someone already pointed you to my post about number sequences which really did go well. Here is another session I did with my son’s class with cutting moebius bands and generalization. It wasn’t so much hands on for the kids since I did all of the actual cutting, but I had them guess what we would end up with and they loved it:

http://toomai.wordpress.com/2008/03/30/math-with-scissors/

What I love best about kindergartners is that they haven’t learned to be afraid of math yet.

My feelings on the game Set: There would be some kids who would get it right off and some for whom it would be completely opaque. I play it with my wife 8-year-old and 6-year-old (with the full deck). The standings usually come out like this:
1)wife
2)6-year-old
3)me
4)8-year-old

1. I really liked the sequence suggestion. I think there is a very good chance that I’ll use it on Wednesday when I visit the class. The Mobius band example was my very first thought when I was asked to meet with them. I was a little concerned because it might be difficult the get them involved (I’m not sure they’re dextrous enough to carefully cut around the loops)—but maybe I could do it and have them guess the outcome, like you said. Also, as one previous commenter mentioned, I think they are young enough that they don’t really know what is surprising. To older kids it is weird to have a 1-sided surface or one in which it can be cut down the middle and not become disconnected. But I was a little afraid that it would be lost on the kindergartners. Maybe doing several different examples with different outcomes would be fun for them.

I’m looking forward to exploring the rest of your blog. Thanks!

10. toomai says:

The way I approached it was I first showed them a single strip of paper. I asked them what shape it was (a rectangle was their answer). I taped the two ends together without a twist. Now what shape? (a circle). I cut it in half, but asking them first what I would get (two circles). So they saw that I got two circles. No surprises yet. Next I took another strip, put a half-twist in it and taped the ends together. What shape is it (no one knew). I explained that this is called a Moebius strip. What do I get when I cut it in half? (two moebius strips, they naturally assumed). I did the cutting, and if things work out just right the strip with a full twist that you get naturally falls into a heart shape. This was shocking to them! For the other two shapes I cut up they were making all kinds of guesses as to what the outcome would be (e.g. a sock, a snowman).