For now, this is a bit of a laundry list, without any special structure. If there is a theme, perhaps the main idea of the weekend was "games."

*English/Reading*

Over lunch on Saturday, we played a kind of improv game using sight word cards, these:

In our pack (from

FlashKids) there are 86 cards with a word on each side. I guess the theory of sight words is that these are common words a fluent reader should recognize on sight and read without sounding them out. We played a game where I would draw two cards and we would each take turns using those words in a sentence, flip the cards, repeat, then draw new ones.

We went through perhaps 10 rounds of cards (about 40 words, roughly 1/4 of the deck) in 10-15 minutes. The boys had no trouble with the words, so as a reading exercise this might be a bit too easy now, but they both were in the mood for this game and had a lot of fun. Pretty quickly, some themes emerged: Jin used his sentences to tell a story about the Boobaloo characters, while Jate turned all of his sentences into something about bottoms and poop.

We didn't really get up to much writing this weekend, but had some long reading sessions at bedtime. Jin has started reading to Jane at bedtime, though this weekend his rendition of

Spot's Treasure Hunt was a bit grudging and soon gave way to a simple race through lifting the flaps. Jate read us a Thomas story on his own, Jin read a couple of pages of

Asterix, and I filled in with

Alexander and the Terrible, Horrible, No Good, Very Bad Day and some sections from one of their children's encyclopedias. Strangely, the Alexander story needed to be jazzed up with some emotive reading (I'm great at imitating a petulant and whiny child, wonder why?) but they enjoy the encyclopedia entries straight.

*Math toys*

Mommy led two sessions from RightStart Math. Here's a sample page with the introduction of multiplication:

These are convenient books because they lay out lessons that we can essentially just open and go through with the kids. If something doesn't seem fun or is too easy, we either skip to the next activity or let them go through it very quickly. If she has time, Mommy will fill in the details of what she did for those sessions.

I think the abacus figured in, or at least it was something with which Jane starting playing, so here's a quick snap.

Both Jate and Jane spent some time working with the pattern blocks. I usually don't take pictures of them playing so we all can stay in the moment, so here is a non-action shot of the pattern blocks:

The pictures don't really stay the centerpiece of the activity, mostly we spend our time stacking the blocks, tesselating, comparing the blocks, making our own pictures, etc. You know: playing and investigating.

We played two other games over the weekend, both using a nice chinese checker set given to us as a leaving gift by some friends:

As a kid, I don't think I ever got the point of chinese checkers. To spill the secret, it it what you can see in the picture: yellow is about to make a big advance by jumping over the blue, over the next yellow, and over the green to move forward 6 spaces. Jin seemed to get the point quickly and seems about 50/50 playing against either Mommy or Daddy.

With the chinese checker set, tidy-up time also turns into a math game, one of you told us about this as a response to reading an earlier blog post: alternating turns, each player puts 1, 2, or 3 pieces in the box. The winner is the player that puts the last piece in the box. With a little prompting, the kids can figure out the strategy. Here's how I prompted Jin:

- When I got to my move with 4 remaining pieces, I asked what would happen if I took one piece? Two pieces? Three pieces? What does that all mean?
- Jin identifies that it means I had already lost when it was my turn with 4 pieces. What if it was his turn? Then he would be in a losing position.
- So, if it is Jin's turn with 5 pieces, how many should he take? What about 6? What about 7?
- Jin identifies that with 5, 6, or 7 on your turn, you have a winning strategy.
- What if it is my turn with 8 pieces?
- What about 9, 10, 11?
- What about 12?
- What about 13, 14, 15?
- What about 16?
- Have we noticed a pattern? At this point, he said that he could tell if someone was in a losing position by counting by 4s. Hmm, so this is where practice skip counting makes some future explorations easier?
- Play with 2 colors on the board (leading Jin to say, "how many are there? each color has 10, so that's twenty . . . Daddy, you go first.") We played several times with 20.
- Played with 30 and it was more difficult to figure out whether he wanted to go first or second. Skip counted by 4 up to 28, then he got it.
- Played with 10 and let him figure out whether he wants to be first or second and then play.

There are a bunch of extensions of this game. One that Jin suggested was instead of having a simple winner, maybe we should keep track of how many pieces each person collected and that would give them points. We discussed this a bit and then he realized that, alone, wouldn't be a fun game because everyone would just take 3 each turn. I said we could give the person who takes the last piece a bonus, but we didn't really have time to explore that, but here's where you could take it right away:

- how does the strategy change with a small bonus?
- what about a large bonus?
- Are there any critical values of the bonus that cause the strategy to shift?
- What if you change the game to try make the other player take the last piece (either the inverse of the basic game or the points game with a last-piece penalty, or the inversion of the points game where you want as few points as possible)?
- How does all of this work with a 3 player game or even more players?
- what other variants can you explore?

If I get around to it, I might make a pencilcode version of some of these. I have a basic AI from my work on Traffic Lights . . . which I still haven't gotten around to telling you about.

**Theoretical digression**

Multiplication happens to be a topic I've been puzzling over for a while. There is a well-established debate about the pros and cons of teaching multiplication as repeated addition, scaling, or something else. Here is part of the internet branch of this debate, a great post from

Keith Devlin.

I don't know if we are going to get it right, but my approach so far has been to try to present a range of different incarnations for each of the operations. Here is a nice

poster that claims to have 12 models for multiplication (I wish we had a copy of that poster!)

Even for addition, though, there isn't one single mathsematic concept that applies. For example, adding 2 apples to an existing pile of 3 apples gives you an ending pile of 5 apples, sure. But traveling 2 km and then going another 3 km is a bit different. Different again is modular arithmetic (or clock addition, for the under-7s).

Though I think Devlin's essay is helpful, I strongly disagree with his point that it is dangerous to change the rules on children (e.g,. "multiplication is repeated addition" at first, then later "multiplication is something else.") In math, we do this all the time and call it generalizing. For example, nearly everyone learns linear algebra hearing that a vector space is a real plane and then the concept is expanded. Perhaps this, then, is the real value embedded in how you teach multiplication: don't stress the association by saying "multiplication *is* repeated addition" but say "*one form *of multiplication is repeated addition" and ask if they can think of other forms. Talk about the relationship between them, use examples, draw pictures, stretch rubber bands, etc. What you really want is for them to appreciate that a concept in one form can be generalized to other areas and applications.