Monopoly arithmetic

who: J1
when: at bedtime (we were supposed to go to sleep, but stayed up to play)
where: bedroom

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We have recently had an outbreak of Monopoly.

Apologies for blurriness; my shutter speed isn't fast
enough to catch the lightening action of this game

J1 and I have been playing a lot of City Monopoly.  Basic play is similar to the classic version, but on every turn the player uses a little randomization device (not a dice!) to determine how many enhancements they can build on their properties (1-3 building blocks or a railroad). This device can also be used a minute timer for auctions and an hour timer for the overall game. I guess these extra functions justify why they didn't just use a tetrahedron dice.

Monopoly games have been known to last for a very long time, but we aren't allowed to keep a partially-completed game lying around the house.  Instead, we play for about 40 minutes and then tally up the assets to see who has won. One of J1's interesting observations: we usually end up with less money than when we started the game.

Where's the math?
There is a ton of thinly veiled arithmetic in this game as every action requires some type of calculation. This ranges from 2 dice addition when moving the pieces to simple multiplication when calculating the cost of building several blocks to 3 digit subtraction when making change.

There are slightly more subtle points around deciding where to build property enhancements based on which properties give a greater return on capital.  J1 is starting to build an intuitive sense and has made some good observations when comparing between properties.  The most sophisticated analysis he started was looking at the first ten squares and talking about which ones are "easy" to hit.  For example, the very first property can't be hit until players have gone all the way around the board and he was excited to realize that..

Where's the game?
Strategically, there doesn't seem much depth.  The only frequent choice on each turn is where to build the enhancements, including railroads.  Occasionally, there is also a choice about where to build a hazard or a bonus structure.

I said that we usually end up with less money than at the start.  Actually, we have played 10 times and always lost money.  Seems that the winning strategy is just to avoid taking action.  For us, that mostly means not building enhancements, except for the very rare times when they immediately increase the properties rental value as much as the enhancement cost.

Challenges (Programming class 6)

Who: Baan Pathomtham 5th grade class
Where: at home
When: break between terms

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This is a set of challenges meant to test what the kids have learned this term. Initially, I had planned to use these during our last session, but the schedule requires us to leave these for independent work at home during the break.

Instructions

Do as much as you can.
These are fun, but are supposed to be hard.
Work on the challenges in any order you want
Use your computer, old programs, and any references you want.
If you have a question, ask! If you are confused about something, ask!

Challenge 1:
Someone deleted a line of code in two of my programs, can you fix them?

A. Program http://jgplay.pencilcode.net/edit/class/testStar is supposed to make this star

What is missing? Copy the program and add the missing line.

B. Program http://jgplay.pencilcode.net/edit/class/testHi is supposed to write a little message

What is missing? Copy the program and add the missing line.

Challenge 2:
What does the following program draw?

Challenge 3:
Write a program that draws this spiral:

Extra Challenge:
Add the button and animation to the spiral in challenge 3.

Intro to Functions (Programming Lesson 5)

Who: Baan Pathomtham 5th grade class  + Pooh teaching solo (Josh was traveling)
Where: at school
When: 2 hours Monday morning

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Goals:

1) Review for loop

2) Introduction to writing functions

What we did today:

0) Discussing the homework.  They showed the car they modified last time

1) We started with a short program which shows a for loop and and if statement.  This program draws a 20 pointed star:

              speed 100

      pen red

      for x in [1..20]

         fd 80

         rt 100

         if x is 10

            pause 2

2) The children were then asked to create 20 pointed stars in 2 more colors.  Then I showed them how to use function so that they can call the function to draw stars of their color choices.

           

              speed 100

      star = (color) ->

         pen color

         for x in [1..20]

           fd 80

           rt 100

           if x is 10

             pause 2

      star red

3) Then we played with how to make command buttons.  Again, initial program supplied and then asked to extend the program to have more buttons.  I showed that button can do complicated things like drawing the 20 pointed stars as in previous example or even drawing a wheel of a car that was part of their homework last time.

           

            pen sienna

     button 'R', -> rt 10

     button 'F', -> fd 10

     button 'D', -> dot 'green'

4) Last we played with a program that takes user inputs.  It seems effective to have them play around with the program initially and then ask them to extend it in a certain way.  In this case, I ask them to extend the program to also take user input for the size of the polygon.  This one was a bit challenging for them and it was very gratifying to me to see them struggle with it before arriving at the answer.

    

            await read "Color?", defer color

     await read "Sides?", defer sides

     pen color

     for [1..sides]

        fd 30

        rt 360/sides

Homework

Draw a flower garden or a forest.  

How many animals (domino counting challenges)

who: J3, J2, and J1, but the real questions here are for older children
when: morning with J3, evening with the older ones
where: family room
what material did we use: a set of animal dominoes

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When J3 counts, she often says "eleven" after "six" instead of "seven." I set out to show her the difference between 7 and 11.

First, we've got 7 dominoes, I put the blank sides up so that the pictures wouldn't distract us from counting them all as a consistent collection:

Next to those, I arranged a group of 11 (again, all blank sides up):

By this point, she wasn't interested.  I think she found some take-apart cars that needed to be investigated. I, however, was interested in two follow-up questions:

1. how many dominoes are in this set?
2. how many different animal types are there on these dominoes?

The challenge of the first question is coming up with techniques that don't require counting the remaining dominoes.  I have one strategy in mind (other than pure guesstimating) that I think is too advanced for J2 and J1.

For the second question, I think even a direct count requires a thoughtful strategy (though some simple ones are available).  What direct counting strategies can you suggest?

Below, I'll show you how many dominoes are in the set, but I'm still only going to show you the blank faces.  Is that enough information to figure out how many animal types?

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I'll give you the easy way to figure out how many dominoes are in the set

Now, can you answer the second question: how many animal types are used in this domino set?

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Advanced challenge:
Do the same thing with a pack of Spot It cards.

Gratitude

A bit of a different post today.  Don't worry, the three little J's will be back in frame soon.

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Someone once said we only appreciate our parents when we have children of our own.*
Perhaps the same thing is true for teaching. Now that I'm trying my hand, I feel compelled to go back to some of my own influences and express my gratitude.

Before I name names, a disclaimer: these are not the only good teachers I had, nor the only ones who influenced me, nor the only ones I appreciate. What does distinguish them is that they each did or said something that I still think about nearly every day, decades later.


To

• Mrs Midgett, for a math problem that stumped me
• Mrs Cannon, for a puzzle that intrigued me
• Mrs Schenter, for critical edits on a book report
• Mrs Garing, for a book recommendation I really loved
• Mr McLean, for his delight at a proof with minimal steps
• Mr Fleischer, for demanding questions

Thank you very much!

The lesson I try to take into my own interactions with children are: guide them to rewarding experiences, let them struggle, push them to do more than they think they can and, most of all, remember that my little comment could be something they carry for years to come.

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*The parenting quote: the closest I could get to a source was 42 things that change. From my attempted google searches, it seems far more popular are complaints about how parents can mess up their kids. Yikes!

Loopy loops (programming lesson 4)

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:

One name in English

The first one done in Thai

Another one in Thai

This wasn't what I was expecting, but it works

What your child has learned

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. They each gave a surprise presentation (in English!) of their homework code.  Each explained a bit about how their program worked and then answered questions: which letter was most difficult, was there anything they would change/improve about their code?

2. More nested for loops. We played with their copies of this code (Nesting Catalog) to see (a) what happens if we move the fd 40 code between loops and (b) how to fix our code if we increase the second loop to run from 3 sides to 10 sides (or more).

3. We improved some code that Gan had shown to the class last time.  A friend had written this code (WoWCar) and it was just begging to be rewritten with a for loop.

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

Homework

This week, the challenge is to complete the car that the friend started.  Loops that draw the wheel will make it much easier to add another wheel (or maybe they are going to create a giant truck/lorry with 20 pairs of wheels?)

The following links will help with the geometric commands:

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.

  
Some great things they did
As a reminder, this is just a couple of highlights from the class.  Overall, the four are really engaging well and taking advantage of the time to explore and test the programming system (including some little boy/non-traditional variable names).

Titus surprised me today when he was working on the problem of extending the shapes catalog to more sided shapes.  He found a way to shrink the shapes so that they would remain in a grid pattern (at the same time reducing the step between each shape.)  I always love it when someone has an approach that I didn't consider.

Boongie was fun to watch during the nested for loop exploration, particularly as he worked on the problem of getting all of his shapes to appear in the preview panel.

What I learned
The kids just need a little encouragement and a small hint to get going on an exploration.  With a light start, they can be off and running on their own steam.

I think they need a bit of encouragement to ask more questions.  Perhaps the language is an issue (English vs Thai)?

Tools for 2 year olds

Who: J3
When: early afternoon
Where: all over the house
What did we use: assorted play and real tools

Today, J3 was engaged in a serious construction project.  Putting together and taking apart this airplane:

Ours is fun, but the nuts and bolts don't levitate like this

One main step, left out of the manufacturers instructions, was to check whether there were any hidden bolts or screws in my head that could be loosened with the toy power drill. She kept mentioning various pieces that were coming out, but I hope we got them all back in.

Frankly, I love toys like this.  Obviously, there is a lot of counting along the way (do we have all the parts, do we have the right amount of each part), often there is matching (nuts and bolts, usually), and a lot of shapes to discuss and compare.  When we play together, I ask them to describe how things fit together, to encourage them to visualize the connections and to plan how they are going to sequence the construction.

Then, we moved on to the real tool box.

We did two main activities with the real tools. First, J3 found a small flashlight and explored the size of the lit spot she could make.  She had a lot of fun testing her theories about how to make it larger or smaller.  I tried asking if she thought the light would be smaller on her than on my because I'm bigger, but she'd already had enough experience shining it on large walls to realize that didn't matter.

Our second activity was an exploration of the wrenches (few) and screwdrivers (many) that we've collected.  She had near perfect results separating flat and Phillips heads:

That's quite a relief, given how prominently this task figures in so many standardized tests these days!
We arranged them by size, with some interesting discussion about the short 1/4 flat head:

Tool on the bottom: smaller because it is shorter or larger because it is fatter?

A question I posed that was lot on J3 (for now) and I'll try again with the older ones: if that screwdriver is 1/4 made in the USA, where was the other 3/4 made? Also, the longer one says "1/8 made in the USA." Why did they make different amounts of each tool in different places?

I'm afraid, but fully willing to admit, that this behavior shows that a I have a full-blown case of dad humor.

Worksheets (aka how I learned to stop worrying and love the bomb)

who: J1 and J2
when: after bath, before bedtime
where: in the bedroom
what: worksheets (gasp!)

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So...Whoever loves worksheets, raise your hands!

You can't see, but my hands are firmly down at the keyboard. In other words, not raised. I'll tell you first why I don't like worksheets and then why we were using them last night.

The bad
My first issue with worksheets is that they (usually) don't facilitate understanding. Take this typical example:

Tiny pic, to contain the evil.
Also, not a worksheet we used!

From these exercises, we don't learn anything about what subtraction is (or isn't), we don't learn anything about how to apply it, we don't connect any additional language with the operation.

The ugly
One danger is that worksheet activities can define "doing math" for kids. They seem so official, so that must be what math is all about, right?  In the usual process, these will be done (alone, silently) given to the teacher, marked up in red pen, and then returned with right and wrong clearly labelled.  This isn't the message I want for my kids.

The good
Having all that written out explicitly, it is pretty clear that each of the weaknesses with worksheets can be addressed.

1. They don't facilitate understanding.
   Two possible fixes here.  First, some worksheets give nice pictures or diagrams that can aid understanding. In the concrete-pictorial-abstract schema, when you are on the picture stage, it is actually a benefit for children to get their own paper with some pictures to investigate.
   The other fix for this is maybe you aren't working on understanding and you need the worksheet for something else. It could be a diagnostic tool too see what the student doesn't understand (facilitated by discussion, see my comments below on grading). Or, it could be pure practice (see the practice point).
    
2. Done alone
   If that's your concern, then have them done together, in pairs or small groups. That also addresses the language issue by encouraging the kids to talk about what they are doing.
 
3. Graded with right and wrong
   At home we, would go over the problems together and talk about them. Sometimes we would take one they did correctly, ask if they are sure of the answer and ask how they got it. Other times, we'll focus on one that wasn't right and ask the same questions. How did you get this answer? Can you draw a picture? Can you use some objects to show me? This is the best way to tease out differences of interpretation, misconceptions, alternative approaches mis-executed, or just a mistake.

Practice
All of those rehabilitation steps aside, at long last I've concluded that there is a role for repetition and practice for our young mathematicians.  However, I think the point at which you can take out a pure practice worksheet in good conscience is when . . . the kid asks for it.

How did J1 ask? Playing Munchkin, actually.

J1 wanted to be faster at calculating so he could easily see whether adding a wandering ancient plutonium dragon to my current combat would let them win. And winning at Munchkin, like all worthwhile things, takes some practice!

Munchkin(s)

who: J1 (and occasionally J2)
when: many times this weekend, whenever we have some free time
where: wherever it is safe from J3's disruption
what: another game

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Your friend is level six, armed with a grand collection of magic items (bonuses shown below) and has just started fighting the Stoned Golem. You could add a wandering Squidzilla to the battle and are trying to decide whether you need to add extra intelligence (+5 for the monsters' side).  Time to do some calculating . . .

We just started playing Munchkin together and J1 really loves it.  As you can see from the cards, it is a bit silly, fairly fast to play, and strategically simple. Also, unsurprisingly, the build-as-you-play characters really appeal to J1's 7 year old sense of gaming and achievement. I say "unsurprising" as this was part of the idea for the game, though in that context it was formulated as a parody take on the unsophisticated attitude of some players in role-playing games.

What do they learn?
First, many of the rules of the game are written on the cards themselves.  Look at the picture.  That wizard card is full of text explaining the strengths (most of them) and weaknesses (some of them) of being a wizard. J1 is at the level that he can read all the cards, but it takes some effort and is good practice for him.

Second, the basic game mechanic is really about repeated addition.  Every battle consists of simply comparing your combat strength to your opponents combat strength.  That means you start with a base level and add a collection of bonus or penalty modifiers.  As a result, most of the additions are +1, +2, +3, but powerful characters (like above) inspire a search for more efficient strategies than just adding every card individually.  That means grouping, skip counting, and multiplying.

In addition, there are opportunities to add wandering monsters to a battle and to double a monster, or fighting character. For a final extra set of calculations, the players can sell their treasure (value noted on the bottom of each card) for levels that determine the game winner.

On a more sophisticated level, they start to detect (more subtle) patterns.  For example, what are the weaknesses to playing as an elf?  They aren't listed on the elf cards, but you recognize them over time when you see that many of the monsters get an automatic bonus when they fight an elf.  Similarly, the listed benefits of playing as a dwarf seem minor, but then there are some powerful magical weapons and armour that are only available for dwarves.

Is it fun, as a game?
Yes, but . . .
One key element of the game is backstabbing your friends (the other players). Sometimes you help them, sometimes you harass them, sometimes you do both at the same time! However, like we experienced with Settlers of Catan, this type of game isn't for J2 (yet?) He was turned off the first time he thought he was going to defeat a monster and then J1 added an extra monster to the battle.

For more sophisticated players, there is probably a sweet spot for maximum enjoyment.  People who have some experience with D&D will enjoy the little puns and jokes more.  However, the game play is really simple, randomness is high and strategic choices are limited.

If you are in Bangkok, you are welcome to come play a round with us!

DIY: Madlibs

who: J1 and J2
where: lounging around the house
when: 10 minutes before violin lessons
what: a language game
*warning* this episode contains some little boy humor that not everyone will find appropriate.

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A picture of something that is not yet, but will eventually become,
an example of my sons' current favorite double duty noun/verb.

Mad Libs - the background
I enjoyed MadLibs as a kid, so I introduced them to the family. Instead of taking the easy route (buying some) or the easier route (downloading from the web) or the easiest route (getting an app), I had the great idea to make my own!  The idea was to take basic sentences, whatever their original context, take out some of the key words, then ask for substitutes from the same parts of speech.

Of course, this also required teaching a little grammar along the way:
Nouns: person, place or thing (also animal or idea, according to wikipedia).
Verbs: action words (also occurrence or state of being, again, wiki)
Adjectives: words that describe nouns (ha, I got this one!)

I decided these three were enough to get us started.

Our Libs
The first try, this came from the literary classic, Spot's Treasure Hunt:

Daddy has planned a thin event for Mommy.  Run and jump with the pillows in every temple to help Mommy and her playdough search for hidden earth.

If you guessed that the yellow highlight is for the words (or phrases) we added, then give yourself a gold star.

The next one, direct from the back of Origami Paper Planes:

The poop and snails are happy to run you on board. The fleet consists of 126 testicles and other mad kidneys.

Finally, from the front page of the BBC news webpage:

Feces are running near the south-eastern Ukrainian city of London.  Earlier, the blue and sad presidents shook as the saliva was pooping.

You can see there is a bit of a theme to the word selection...

Did they love it?
After our third round, J1 asked suspiciously: "Daddy, is this the game?"

P thought that was hilarious.

Oh well.

the measure is 27

Who: J3 (also something for 13+)
Where: at home on the reception floor
When: just after breakfast

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A quick picture to show some standard activities in our home.

When in doubt (i.e., too tired to think creatively), I reach for the trio blocks and polydrons and start putting them together. Inevitably, the children will join and take over the activity. We had our tape measure lying around, so J3 started measuring our creations and Ms Rabbit.  After putting the tape measure up to something and looking carefully at the numbers, she would proclaim: "27." She did this several times, each time announcing the same length: 27.

I guess this is similar to her lack of 1-1 correspondence when she's counting: just a developmental step she hasn't yet taken.

A further exploration
Did you notice the star-shaped polydron construction?  It is a cube with the faces replaced with square pyramids. Though it is pretty obvious, I was delighted when we realized that square faces in our constructions could be replaced with 4 triangles arranged as a square pyramid and equilateral triangles could be replaced by 3 sides of a tetrahedron.  Here's an NRICH exploration I found when trying to determine the name of our construction (the cube with pyramids instead of faces).

Subitizing and Frustration

Who: J2
Where: at home on the dining room floor
When: just before dinner

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Last night, we practiced counting without counting. Blocked from J2's view, I put xiangqi (chinese chess) pieces on the ground, asked him how many and gave him 3 seconds to see, then covered them again.

I started with the 3 black pieces, then did 4 red (the top two quadrants below). Three seconds was really too long as he was able to count them, so this level was easy.

Next, I combined the two previous groups (bottom left quadrant).

He answered 8, quite confidently.  When I removed the cover and counted them, he was frustrated. Instead of persisting with more examples,  I asked him to make a challenge for me.

Smiling mischievously, he asked if I was ready and then uncovered the configuration in the bottom right. Well, of course, I couldn't count them all and I couldn't work out any subgroups that would help me count clusters.  I said "I'm not sure, but I think there are 30."

Here's my thought process:
- from a visual estimation, I thought there were more than 20
- I guessed that he had used all the pieces and, as with western chess, there are 32 pieces in a full set
- Because he had been frustrated on his last round, I didn't want to get the answer right. I wanted to get close and then have a discussion about getting it a bit wrong.

J2 proceeded to count them and there were 30! Oh no! He was stunned, not the intended outcome.

Turns out that little J3 had stolen two of the pieces earlier in the evening

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).