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).