Doodle games

Who: J1 and J2
What: paper, pencil, crayons or colored pencils
When: while waiting at doctors' offices, violin lessons, during meetings
Where: any place with a table and some writing tools
Why: fun designs, stimulate geometric thinking (maybe)

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Take a moment and watch this fun video from Vi Hart: http://www.youtube.com/watch?v=heKK95DAKms

I'll forgive you if you just wander off and watch a bunch of her other videos.  She is great and I'm sure to include more links to her work.

For a while, I've played these doodle games with the older children at times when we were waiting around. At long last, I've managed to remember to take some pictures to complete this (very short) little post.

Counting with 6 hands

Please read for the *questions* below as I'd love to have your thoughts in the comments.

Who: Baan Pathomtham 1st grade class (J1's class)
Where: at school
When: 2 hours Tuesday morning

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We (P and J0) got a chance to spend the morning talking about subtraction with J1 and his classmates. It was an opportunity to see some differences between talking math one-on-one (or one-on-two) and a larger group.  Here are a couple of tidbits from the discussion.

Practice with poker chips
Of course all children need to be familiar with the standard gambling implements: dice, cards, and poker chips. The first two are already well known, so we did an activity with poker chips this time. How would you count all the chips in the picture?  Well, what if "you" were actually a group of 3 first graders?

Here's the strategy one group implemented, spontaneously, as far as I could tell:
(1) divide the chips into equal piles for each child
(2) count the remainder in the center of the pile (in this case, one chip, so the count was trivial and done without an explicit effort)
(3) take turns putting one new chip into the pile
(4) all count together as the new chips are added

Actually, this was their second strategy. At first, it was a free-for-all with all three trying to count all the chips and messing up each others division between counted and uncounted chips. 

Note: counting the chips was just accidental to the activity we were doing, so this shared counting strategy was just a cool thing we noticed along the way. If it had been more central, I would have talked with them about the equal piles at the start (which links with multiplication) and why there was a remainder (which links to the division algorithm).

Enthusiasm
Most importantly, the kids were all really excited and enjoyed working on math. They liked asking mathematical questions about a picture we presented, had fun doing calculations, trying new modeling tasks, and playing the mathematical game.

This confirms, once again, that enthusiasm and curiousity are things we (usually) kill during the educational process.  Not at our school!

Explaining
Given their enthusiasm and apparent facility with the calculations, I was surprised that they struggled to explain their calculating strategies. I can't tell if this is a language issue, if the calculations they were asked to describe are so ingrained that they don't consciously think about them, or if they don't really understand what they are doing.

My key take-away: I will focus a lot more of my discussion time on getting the J's to talk about how they calculated something, see if they can draw a picture, and see if they can explain using a concrete object.

Extensions
As preparation, P and I talked about 3 models of subtraction: taking away, differences, and counting back. P made two comments:
(1) Word problems are harder than straight calculations (said while we were discussing what types of problems to use to have the kids investigate the three models)
(2) "Counting back is such a waste, I always knew the answer through another method and had to artificially demonstrate counting back."

Word problems seemed, to me, the natural way to motivate using a particular model for subtraction. For three quick examples:

• You started with 5 cookies and ate 3, how many are left? This is taking away, obviously.
• Don has 27 poker chips and Tanya has 13. Who has more and how many more do they have? Differences, naturally.
• Walking along a straight line, you go forward 6 meters and then back 2 meters, how far are you from your starting point? Counting back suits this one.

*Question* is this the wrong way to use alternative models? Is it necessary to force them to use "unnatural" models to demonstrate proficiency (for example, using take-away to resolve the differences question)? Does this create difficulties for problems involving alternative missing values in the same types of questions (i.e., you started with 10 cakes and now have 3, how many did you give away?)

Counting back is the same as the movement model, which I called "forward movement" in my post on addition models. This model leads nicely and really easily to emphasizing the role of 0, negative numbers, and subtraction of negatives. Looking a bit farther ahead, it links with vector addition by just extending our operation to more dimensions. Taken along another path (ha, the puns!) it can be used for modular arithmetic (replace directed movement on a straight line with directed movement on a circle).

Triangles (programming lesson 3)

Who: Baan Pathomtham 5th grade class (all present)
Where: at school
When: 2 hours Monday morning

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Animations

Again, starting with some code that the kids wrote:

A golden star for their first for loop

More complex 3 levels of nested for loops

What your child has learned

This is just a quick list; you can ask them to show you and, if anything seems unclear, I've got some links that can help or just send me a Line message.

0. Reminder: You can see code that the students are writing through this directory. My own folder with a lot of examples is here: jgplay.pencilcode.net/edit and the class folder is here jgplay.pencilcode.net/class/edit.

1. Basics of for loops: this is a way they can run commands repeatedly without having to write repeated lines of code. There are some spacing and punctuation issues to get this right, so encourage them to use the blocks to get things lined up correctly.

2. How to use the command line to run single lines of code. Easier for them to show you than for me to explain it.  This is something they can use to test an idea or in debugging.

3. How do direct a named object. Using this program, they got to test their knowledge of the command line and movement commands by shifting around the named triangles: jgplay.pencilcode.net/edit/class/shapes.

Remember this reference: http://guide.pencilcode.net/home/

Homework

This week, the challenge is to draw their name in Thai and English.  These links are really useful if they need help understanding the turning and arc commands lt (left turn) and rt (right turn)

http://pencilcode.net/material/measuring.pdf
http://pencilcode.net/material/arcs.pdf
http://david.pencilcode.net/home/explainer/turns
http://david.pencilcode.net/home/explainer/curves

Please look through these with your children.

Here is a version of my name in Thai:

  
Some great things they did
I'm going to start flagging things the students showed in class that are really good and we should encourage.
For loops: when asked to match code with output, Win made some good observations that helped him form a hypothesis without tracing through every line of the code.  In the golden star code, he saw that the for loop would run 5 iterations, so he was looking for a picture where 5 was a prominent feature. In addition, he saw that the pen was going to be gold, so he could match up the color to add confidence to his prediction. Over time, as they encounter more complex code, they will need to use features like this to get a sense of the program, even if there are commands they don't understand.

At the end of the class, Kan started showing programs his brother had written.  Really fantastic to see that Kan enjoyed what we were doing enough to share with his family, that he had learned enough to help get someone started.  This also got the whole class excited about exploring code written by other users. We want to encourage them to learn from other examples they see, which means:

• searching for examples
• thinking about what other programs do
• testing them
• changing them and seeing the effect

What I learned
Make the tasks concrete and explicit, at least for one example, before letting them pursue a more general/less structured investigation.

Have a back-up for when the network is down.

Fair sharing Exploration (warm-up)

who: J1 and J2
when: over a course of weeks (this is a plan, not a historical record)
what material: the objects to be shared

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This post: http://letsplaymath.net/2014/08/13/fractions-15-110-180-1/#more-28158 gave me the inspiration to create an extended plan to explore ideas around fair sharing. The idea got a further boost from our warm-up discussion about breaking swords and cutting cakes last weekend.

In our house, issues of fairness lurk just below the surface of almost every interaction between the children. Actually, that's not accurate, since fairness is often a visible dark cloud hanging over the proceedings.
Part of the idea for this exploration is to harness their strong feelings on this topic to examine:
- fractions (naturally)
- approximations
- competing theories of fairness/equality
- their own intuition and biases around what is fair and why something should (or shouldn't, or doesn't have to be) fairly distributed, including concepts of ownership ("that is mine!"), earned privileges ("he got X because he did Y"), and private valuations ("you like X more than Y, but she likes Y more than X").

The basic idea is to present different types of sharing problems as thought experiments: talk through or play-act the scenarios, do some analysis of different sharing tactics (maybe using manipulatives, diagrams, etc) and later come back to these in live examples.

Examples of different classes of sharing problems I see:
- cakes/pies: things that can easily be cut into small fractions
- sausages, or apples: something that can be cut reasonably accurately into moderate fractions (maybe down to 1/8th)
- ice cream, or soup/rice/etc: something that needs to be weighed or volume measured
- KEX cookies/small candies: something that can, at best, be cut in half, maybe not cut at all.
- ballons/babies/bicycles/scooters: items that are indivisible objects
- balls/group toys: items that increase in value as more people play (up to a point)

some types of questions are:
1) technically, what tactics can be used to divide, what are the pros and cons
2) strategically, how can you get buy-in that your approach to divide is fair?

Let me know if you have any tips or thoughts on how this will all turn out.

Mathematical distraction (argh!)

who: J0
when: J3's naptime
what material: scale, bottle, water, formula

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I looked down at my hands and saw a partially prepared bottle of baby formula.  Hmm, did I add the right amount of each ingredient?!

Stepping back, how I had I gotten into this (minor) predicament? By day-dreaming about some other math (which I hope to post in the near future), I had managed to mess up this simple recipe:
1. add 4 ounces of water to the bottle
2. add 2 scoops of formula powder to the bottle
3. close and shake the bottle
4. dispense to the consumer

My solution: use a little more (very simple) math to figure out what I had done.

I got out my trusty scale and made the following measurements:
(1) empty bottle: 60 grams
(2) 2 scoops of formula: 20 grams (pictured below is one scoop, I had tared out the lid and the scooper)
(3) 4 ounces of water is approximately 120 mL/approximately 120 grams
(4) my partially made bottle: 155 grams

Hmm, 155 grams? My target mass was supposed to be 200 grams. If I had forgotten one scoop of powder, it should have been 190 grams. So...

Looking more carefully, I had only put in about 3 ounces of water (mass about 90 grams).  I concluded that I needed an extra 30-40 gr of water and another scoop of powder.

What lesson did I learn?
Maybe I need to be cautious thinking about engaging math problems when I'm driving?

Flags (programming lesson 2)

Who: Baan Pathomtham 5th grade class (all present)
Where: at school
When: 2 hours in the morning

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Apologies that this post is a little bit of a muddle. I am including material for the school teachers and other parents so that  you know what we are doing and can help your child if they want. For the more general audience, I want to explain what we did and share my thoughts about what I learned and teaching more generally.

Animations

Let's start with the really fun bit: some code that the kids wrote:

A Thai flag

Colorful yin-yang (inspired by the Korean flag)

What your child has learned

This is just a quick list; you can ask them to show you and, if anything seems unclear, I've got some links that can help or just send me a Line message.

0. You can see code that the students are writing through this directory. My own folder with a lot of examples is here: jgplay.pencilcode.net/edit and the class folder is here jgplay.pencilcode.net/class/edit.
1. Basics of setting up an account, logging in to our accounts, opening a new file for programming, saving our work, and changing the name of the file. All of these are summarized quickly on this page: Making a Pencilcode Account.

2. How to test the program and run it in full screen mode, how to use blocks to build a program and how to toggle between code and blocks.  Play around clicking on the basic screen and these are all pretty easy to see.

3. Basics of moving the turtle (fd, rt, lt, jump) and drawing (pen, dot, pen path, fill). This handout has enough to get started and is what we worked on for the first lesson.

Important there is a lot of helpful material here: http://guide.pencilcode.net/home/. Feel free to make use of any of this, even code from which I have drawn the in-class activities and homework assignments.

Homework
The challenge this week is to replicate this turtle drawing (the last figure on the hand-out from last week):

What are they really learning?
This course isn't really about learning how to make a little turtle move and maybe it isn't even about programming. Here are things we did and what I wanted to show them.

To start the lesson, we looked at the programs each student wrote to draw a Thai flag. I've given one example above, but suffice it to say that all five of the programs were different! I was a little surprised we got so much diversity from this simple flag, but that was good for the kids to see: there isn't a single right answer.

Some students were only partially successful.  That's also fine! We got to talk about where they were stuck and learn another pencilcode command (pen path . . . fill <color>), which we later spent most of the session investigating.

I showed them my flag programs: South Korea (more complex shapes and animation) and Thai (simple animation and music). Was I just showing off to the beginners? No, letting them know that there isn't a limit to what they could build (it isn't just moving a little turtle around).

The main meat of the day, though, was spent playing with filled paths, drawing arcs, and drawing simple shapes. This was really a geometry lesson and I was delighted to hear them asking each other: "how many degrees? how many sides, how long should this be relative to that?" They also did a lot of experimenting with new commands, trying different angles and sizes to get things to fit together. Overall, a lot of great math.

And. . . once again we saw that simple things can be attacked from many different directions.  For the task of drawing a simple equilateral triangle, the kids came up with 3 different approaches (though they couldn't get all of them to be equilateral).

What I learned
First message I got was to ratchet down my expectations for how quickly we can go through the programming material.  My intention had been to introduce for-loops today (and that was even a step back from my original plan to go through for-loops and nested loops!) We need give them a chance to really get used to the basic operations and elements around programming.

The second lesson is to make the math part more explicit. Inherent in all their drawing programs will be distances, angles, circles, arithmetic, and some algebra. For students this age, I need to respect that these are also important concepts and big challenges.

Fair Sharing (warm-up)

who: J1
where: at Kuu, our regular lunch destination at the mall
when: while waiting for ice cream
what did we use: chopsticks

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As we enter this snippet of conversation, J1 was describing/acting out some martial fantasy scene with different elements battling and weapons getting destroyed.

J1: What if I broke his sword into 11 pieces?
J0: 11 equal pieces?
J1: yes, 11ths.  Then he would cry 11 times.
J0: I wonder, is it easy to cut something into 11 equal pieces?
J1: no, hard to get them all the same size.
J0: What is easier, cutting into halves or thirds?
J1: (thinks for a bit) cutting into half is easy. (He then makes a swipping motion with his chopstick and a blade-swooshing sounds.) If you cut it into thirds, you do chop (one smaller slice), chop (another slice about 120 degrees of the previous one), and then you really need to put your back into it (as he makes a third slice toward himself).
J0: (laughing) What did you say?
J1: (laughing, then repeats the last cutting motion) then you really need to put your back into it.
J0: Wow, you were dividing a round cake.  I was just thinking of sticks.  How many cuts do you need for those?

We proceeded to have some discussion of how many cuts were needed, some contemplation of why a it takes n radius cuts to divide the cake into n pieces (for n>2), and a hypothesis about why halves are the easiest to divide (because you only have to compare two pieces and make one cut).

One further comment was worth flagging, related to my n>2 provision above:
J1: hmm, thirds take 3 cuts, but halves only take one cut?

As this was in the middle of something else he was describing, I didn't follow that branch of the conversation, but may return to it later.

*Apology* sorry I didn't include any pictures on this post. I'll see if we can draw a picture of the enemy with  a sword broken into 11ths.