Addition, it's just making a bigger collection, right?

Who: J1 (also guest appearance from J2)
When: at dinner
What materials did we use: our mouths (just talkin') and then a bunch of miscellaneous items later

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J1: [Friend X] isn't good at multiplication.
M: That's not surprising. Has he started learning multiplication in school?
J1: Actually, he's not so good at addition either. [Friends Y and Z] also don't really understand addition.
D: Great, so maybe you can help them. Is there a picture you can draw that shows how you think about addition? For example, 4+3?
J2: (looks at his fingers for a moment) that's 7! (exclamation, not factorial)
D: So what does that mean, when you add?
J1: (making a big gesture with his arms) Gathering things together, collecting them.
D: Are there other models of addition that you know? Other things you do where you need to add?
J2: counting, then counting some more
. . . tbc

So, how many models of addition have your kids seen? How many are there? Let me know what you think in the comments.

Repeated Counting

For a hobbyist counter, like J3, this activity is an excuse to count up to 20.  Twenty? Well
5 fingers + 5 almonds + 5 fingers + 5 almonds = 10 fingers + 10 almonds = 20 etwas.

My argument is that the fingers and almonds aren't really being combined, they are all serving as objects for counting, then counting some more.

Grouping Together
On the 4 different stems, we see 1 longan, 2 longan, 3 longan, and 4 longan. How many ลําไย are there all together?

I'm calling this distinct from repeated counting because the 10 ลําไย do make a coherent collection all together in a fruit bowl, while their separation into 4 subsets probably won't be relevant for their future destiny and, in fact wasn't relevant to why they came to our table (they were all bundled together in a cluster).

Forward Movement
You've seen this game before: roll the dice and move your fierce dinosaur closer to the finish. A number line gives you a similar model, with the benefit of fractional steps, but I liked the fact that the motion is only sequential by convention here.



Combining Mass


A lucky one as the measurement error doesn't destroy our perfect whole number addition here.

Combining Volume
Yes, it was necessary to pour out 1/4 cup + 1/2 cup = 3/4 cup of chocolate milk in the course of this investigation!



As with the mass model, note that this is imprecise and the experimenter must decide from the context what gap can be tolerated between the theoretical and empirical results. This is especially critical for valued substances like chocolate milk!

(Musical) Time
One quarter note plus four eighth notes get played for a time lasting one measure, in 3/4 time.  Or, if you want to be measure independent, four eighth notes together last the same time one half note.

Silly
This is a bluebird of happiness. It is not a model of addition . . .  or is it?

How many cows?

Who: J3 (2 yr old)
When: at breakfast and again at lunch
Where: the dining table
What did we use: a cup with a repeated picture on it

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Look closely at the cow (cows?) on the cup below. I've taken the picture from 4 orientations, so you can see all around the cup. How many cows are there?

I discussed this with J3 the other day.  She pointed to the cow and counted, then I turned the cup until she saw another cow, pointed at it and counted.  We kept doing this until she had counted up to 5.  I asked her to try counting them again, and again I stopped turning when we got to 5. I asked how many cows are on the cup and she said 5. We did it again, but this time I kept turning until she counted 6.  We did it again up to 10. Finally, I kept turning until her counting got very unorthodox (mid-teens) and with large skips and also going back to smaller numbers.

Later in the day, I put a piece of tape next to one of the cows (this is actually when I took the pictures above). I repeated the same turn-and-count with J3 (I think up to 10).  Then I drew her attention to the tape and repeated the counting.  Same result.

As I typed this, she looked over my shoulder, pointed to the pictures and proceeded to count up to 5 cows.

We do a lot of counting objects, moving them between piles, associating them with fingers, comparing which is larger, so I already knew that her one-to-one association of objects and the count is shaky. Still, I was surprised that she was willing to accept an arbitrary number of cows on the cup.

Challenge
Did you notice the white plus signs on the cup? How many are there?

If you didn't notice them, perhaps this was an invisible gorilla experience.

If you did notice and also counted them, particularly if you distinguished large and small ones for a more detailed analysis, then you can be an honorary member of our family!

Turtle Houses (Programming Course Lesson 1)

Who: Baan Pathomtham 5th grade class (minus one student, plus one friend)
Where: at school
When: 2 hours in the morning

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We had our first lesson today and it was a lot of fun.  Here's a quick synopsis:

Robot Turtles
We started with the basics of robot turtles.  Everyone got a pack of command cards (forward, left turn, right turn) and then I executed the moves as they played. We cycled through a couple of times with them giving commands in turn, then each student took a turn as the moving turtle. By chance (?) each turn ended when we crashed the turtle into something and I exclaimed "Crash! Next player."

We were having trouble with the network connection, so we then played three rounds of robot turtle on the board:

1. no obstacles, each player plays one card at a time and I moved the turtles
2. Add walls that they need to negotiate, each player plays one card at a time
3. With walls, each player plays 3 cards at a time for a batch move.

This was all easy, but seemed engaging.

In the future, I will plan robot turtle activities either as a warm-up or when we have to take a break from the computers (or network is giving us trouble).

Pencilcode Intro
I showed them how to create accounts and open their first program.  Then, we worked through the first two activities on this worksheet: http://pencilcode.net/material/lesson1-handout.pdf

Periodically, I would stop them to point out actions or utilities:
- the run triangle to play their program
- how to save
- run in full screen
- open a new program file
- shift to block mode and shift back to code mode
- move through folders to explore other users' code

They did a good job adding a door to the house from the worksheet and immediately started experimenting with other things they could draw.  Here's an example:

I think it worked really well to show them the blocks after they had written a decent program first. They were able to drag in new commands to see what effect it had and those effects were often surprising, especially when they put a new line in the middle of their existing program.

Finally, I explained their homework and gave them some time to work on the challenges, think about them, and ask questions.

Homework
This week, the kids have 3 homework activities and 2 open activities:

1. Show their parents how to draw a Thai flag using pencilcode
2. Figure out the code to draw each of the designs on the left (from the first "chapter" of the pencilcode book)
3.  Draw what they think each of code blocks on the right do and then test it (this is from the second "chapter" of the pencilcode book)
4. Look through other programs and highlight ones they find interesting.  I'm not expecting anything concrete on this, so it is a half assignment.
5. Think about something they would like to program as a project.  Again, want them to start thinking, but I'm not looking for anything concrete yet.

What did I learn?

1. Important to have an offline, back-up activity when the network connection isn't performing
2. Most of my priors were confirmed: the kids were eager to experiment, they benefited from a little guidance and some prompting to try particular things, they were good at explaining to each other how to do something, and the intro activities were really easy to blend into a fun lesson.

Matific Games (quick review)

Who: J2 (with some help from J1 at times)
Where: dining table
When: after lunch (was everyone finished eating? Not clear.)
What we used: https://www.matific.com/us/en on our laptop

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J2 saw I had an open website and started playing some of the games: Bees and Flowers, Spy-a-Meerkat, Monster Shop I, and Tile-a-Shape. For a free (it seems) site, I was impressed: the animation was cute and the activities were engaging, even if these four were a bit easy for J2.

Two points of discussion:

• Spy-a-Meerkat: J2 was fine up to 5, but then had difficulty when there were 6 or more.  J1 had no trouble and I should have had him talk about his strategy for figuring out the answer
• Since all of the questions were pretty easy, the two together didn't get any wrong, until J1 said: "I wonder what will happen if we get one wrong. Let's try it." I love that.

They got stuck finally on the last shape tiling.  See if you can figure out the answer.  You can only translate and rotate the purple shapes, no scaling:

21 with uno cards

Who: J1 and J2
When: after bath and before bedtime
Where: bedroom
What material did we use: pack of UNO cards

I'm working through the games and activities listed on this page of ideas for J1's school. Tonight, three of us played a version of 21 with the UNO cards.  I considered removing the non-numerical cards, but instead made up the following rules:

• everyone is dealt 7 cards at the start, you draw a card every time you play one, so should stay with a 7 card hand
• numerical cards played are all added, up to 21, with the last player able to play collecting the stack
• reverse and skip cards can be played at any time (perhaps I should have restricted it to play on a card of the same color?)
• wild automatically take the stack to 21 if the player can say what value is required
• draw 2 and draw 4 lead the next player to draw extra cards (so it helps an opponent)

Overall, I think the game was fairly enjoyable.  The most interesting point was asking J2 the value of a wild card he had played onto a stack with value 5 and he replied "5+5+5+1." He did this again several times, showing a couple of things about his thinking: first, he wasn't really subtracting, he was filling in a gap, he is comfortable with some number bonds (often his first step is to get to 20 or 10, and he likes adding by 5s if there is room.

Extension discussions
For several reasons, we just played the game and didn't talk a lot about the structure.  Here are two areas of discussion we could have pursued:

1. Are there strategic plays? Given the numbers available are 0 through 9, how many turns are required before we get to 21? Generally, if you have a choice of two cards, should you play the smaller or larger one?
2. Hey, this is a game I just made up.  What do you think about the rules? What is fun, what's not? What else could we try?

Conclusions about the game
J1 was generally pretty fast with all of the little calculations required, so it felt like playing a game rather than drilling arithmetic. Based on this test run, I'd say it is a decent game for J1's class, or a similar group of students.

I really liked the rule about wild cards and will use that again when we play next. However, J1 didn't like the treatment of draw 2 and draw 4; he suggested that the player should get to draw the extra cards for them self.

Skip and reverse also seem like duds. As noted above, I may restrict their use to when they can be played on a like color. Another idea is to assign them a numerical value (10 and -5 both appeal to me). Another possibility is that they reset the pile value to 0.

Just play UNO? 

On the other hand, playing with the UNO cards made them feel like we should just be playing UNO.  We ended up playing two rounds of 21 and then one game of UNO.  It has become accepted that J2 will always win UNO when we play as a family. Of course, this isn't likely to be true over time, but tonight was no exception. In fact, his last card was a lucky play: after a string of reds, J1 played a red 5 allowing J2 to drop his last card, a yellow 5!

Prizes

Major international mathematics prizes were awarded yesterday at the International Congress of Mathematicians. For the Fields Medalists and Nevanlinna Prize winner, Quanta magazine has really nicely written profiles.

I just want to offer a personal note of congratulations to Manjul Bhargava (Fields Medal) and Jacob Lurie (Breakthrough Prize).  We were undergraduates together and it was a lot of fun to work on mathematics together so long ago!

Since this post doesn't have any actual math in it, here's an animation to make up for it:

Currency conversion and 1001 nights

who: J1 (and a bit of J2)
where: bedroom
when: bedtime (particularly after lights out)
what materials: internet connection (not really necessary)

This is another Talking Math With Your Kids -style conversation.

• J1: Daddy, what's 100 pounds (British pounds) in Baht (Thai currency)?

some discussion about which way he wanted to convert as I hadn't really been listening

• D: Well, there are about 50 baht per pound.
• J1: So, I need 50 groups of .  . .can you use the computer to calculate it?
• D: Yes, I can, but we don't need to.  First, though, we need to figure out what calculation to do. You said you have 100 pounds and there are about 50 baht per pound

some mumbling, not really getting anywhere; I'm tempted to comment and guide, but hold back.

• J1: I have 100 groups of 50.  What's that?
• D: What about 10 pounds in baht?
• J1: (pause) that's 500
• D: how did you calculate that?

I was assuming some strategy for directly calculating 10 x 50, either just adding a zero or building from 10 x 5 (which he would calculate as 10-20-30-40-50), or 50-100-150-200-250-300-350-400-450-500.

• J1: well, I know 20 pounds is 1000 baht.  Then 10 is half of 20 and 500 is half of 1000.
• D: Interesting.  How many 20s are in 100?
• J1: (thinking) 5000 baht in 1000 pounds!

some conversation about what  you can buy with 5000 baht.

• D: what about 15,000 baht.  How many 5,000s are in 15,000?
• J1: 3, so 300 pounds.  Wow, that's a lot

Why did I find this so interesting?
First, I was really surprised by the strategy to calculate 10 x 50.  This reinforces the magical phrase "how did you think of that?" Sometimes my own preferred approach seems so obvious that I feel there won't be anything interesting gained by hearing the child's approach and, in this case, asking the question was just because I wanted to build a good habit.  I was so surprised by the answer that I forgot to tell him about an alternative strategy.

Second, there had been several other times earlier in the day when I tried to lead him into a mathematical conversation and he wasn't taking the bait. I guess I should relax and see where chances arise instead of controlling it.

Ok, but 1001 Arabian Nights?
We've started reading the Project Gutenberg version of 1001 nights. Nice mathematical title, no? Well, tonight we started The Story of the Husband and the Parrot. What you need to know is:

I am reading the story of Sheherezade to J1 and J2.  In this story . . .
. . . Sheherezade is telling King Shahriar The story of the Fisherman, in which . . .
. . .the fisherman is telling a genie The Story of the Greek King and the Physician Douban, in which . . .
. . .the King tells his Vizir about a story told by another vizir to King Sinbad, in which . . .
. . . we get The story of the Husband and the Parrot (which involves the Parrot telling the Husband a short story).

Counted generously, that's 6 stories-within-a-story.  And now I've told you, so that's 7 layers.