- Children's Mathematics by Carpenter et al and Extending Children's Mathematics by Empson and Levi
- Kamii and Wu: I had assumed this was in reference to a joint work, but I couldn't find one. Here are links to their respective work: Wu's papers, and Kamii's)
- Fosnot et al: Young Mathematicians at Work series
- Ball/Bass/Hill: this is a link to Deborah Ball's work that should suffice
- Liping Ma's Knowing and Teaching Elementary Mathematics (not to be confused with this article by Askey)
- Nunes & Bryant. I'm not sure which paper/book of theirs is the top pick. Here are a couple: Children Doing Mathematics, Key understandings in Mathematics Learning (with Watson)
- The Common Core Standards themselves: math and English language
- NCTM Essential Understanding series
- Rational Number Project (fractions, multiple models)
- Christopher Danielson's own blogs: Talking Math With Kids and Overthinking My Teaching
- Mathminds blog
Sunday, January 25, 2015
CD's required reading
Christopher Danielson (@Trianglemancsd) started a twitter conversation listing essential reading for people who want to make pronouncements about mathematics education. Of course, this serves a dual use for anyone who wants to improve their own practice, in which spirit I offer this accumulate list and links:
Christopher 
Math on a plane
who: J1, J2, school friends
where: on a plane flying from BKK to Chiang Mai
when: super duper early in the morning (last week)
For our class trip to Chiang Mai, I prepared a bunch of games to play on the flight. Most of these come from Math4Love, though they can also be found elsewhere.
This was a real hit with J1 and J2. On the flight up, another school friend was (un?)lucky enough to sit next to us and I roped him into playing. I think he enjoyed it and, at least once, mentioned it to another classmate while we were in the north.
While this game mostly encourages deductive reasoning, I was surprised by some of the things I observed as we played:
Dots and Boxes and Hex were two other classics I thought we would play, but there was enough interest in the other games that we haven't gotten around to them yet.
Which Scripts
I thought this would be a fun activity from NRICH. So far, no real enthusiasm amongst my clan.
where: on a plane flying from BKK to Chiang Mai
when: super duper early in the morning (last week)
For our class trip to Chiang Mai, I prepared a bunch of games to play on the flight. Most of these come from Math4Love, though they can also be found elsewhere.
My prep
Pico Fermi BagelThis was a real hit with J1 and J2. On the flight up, another school friend was (un?)lucky enough to sit next to us and I roped him into playing. I think he enjoyed it and, at least once, mentioned it to another classmate while we were in the north.
While this game mostly encourages deductive reasoning, I was surprised by some of the things I observed as we played:
- The kids enjoyed this most as a collaborative puzzle and neither cared for nor needed a competitive twist to make it interesting.
- Being the puzzle master is already a fun challenge for the kids, especially the younger ones. You get to talk with them about whether they have chosen a number with distinct digits or not, answer questions about the definitions of Pico/Fermi/Bagel, and debate whether they are really sure when they assign points to a guessed number.
- M4L suggests 3 and 5 distinct digit games, but you should try other variations. J1 and J2 both asked questions about and made conjectures about 1, 2, and 6 digit games. "How many tries are required for a 1 digit game" is a low threshold question and a 2 digit game is a simpler level to explore the logical inferences available.
We only played this very briefly on the trip, but have had a longer discussion since returning. For what it is worth, J1 was immediately frustrated by the restriction on the hounds' movement and decided to create a super-hound that could also move backward. I think this game is worth another attempt, but probably more suited for younger kids.
Dots and Boxes and Hex were two other classics I thought we would play, but there was enough interest in the other games that we haven't gotten around to them yet.
Which Scripts
I thought this would be a fun activity from NRICH. So far, no real enthusiasm amongst my clan.
The grand winner
So, even though Pico/Fermi/Bagel was loved, I was surprised by the activity that really captivated J2 on this trip: calculating decimal expansions of fractions!Wednesday, January 14, 2015
Simple splits: sharing money
Who: J1 and J2
When: just after dinner
Where: dining room floor
Taking the skytrain (BTS) yesterday, I ended up with interesting change: 48 Baht comprised of four 10s, one 5, and three 1s. Why is this interesting change, you ask? While 48 has a lot of factors, this set of coins makes it impossible to evenly divide into any smaller amount!
This gave me an idea for a sharing discussion with the kids. Here's the intro to our conversation:
When: just after dinner
Where: dining room floor
Taking the skytrain (BTS) yesterday, I ended up with interesting change: 48 Baht comprised of four 10s, one 5, and three 1s. Why is this interesting change, you ask? While 48 has a lot of factors, this set of coins makes it impossible to evenly divide into any smaller amount!
 |
| Just chop it right here! |
This gave me an idea for a sharing discussion with the kids. Here's the intro to our conversation:
- J0: Hey, I just realized I have 48 baht
- J1: Can I have it?
- J0: Not yet, I want to ask you a couple of questions. If you were going to split it equally with J2, how much would you each have?
- J1: 24. Can I have it now?
- J0: (I write down 48 split for J1 and J2 means 24 for each). What if you were going to split it with your sister, too?
- J1: still 24
- J0: Oh, I mean you split with J2 and J3. All three of you get the same amount
- J1: (losing interest, the coins don't seem closer to his grasp) Uhh, I don't know. Let's do something else.
- (proceeds to wander around the room for a bit, does some other activity for a while. I have a guess he is thinking about the question and avoid pressing)
- J1: 16, we each get 16 baht!
- J0: Good! How did you figure it out?
- J1: I divided 48 by 3. Can I have the money now?
- J0: Yes. Here are the coins. Can you show me how you would split it like we said? Show me how to share it with J2.
- split them as close as possible and randomly decide who gets which pile
- split them as close as possible and then J1 gets more because he is older
- J1 splits them as close as possible, gets more and gives the rest to J2. This is considered fair because every child ends up with more money than they had at the start, so they should be happy. This is a version of the ultimatum game and I was really surprised that J1 came up with this reasoning on his own.
- Split evenly what we can and give the rest back to daddy.
- Ask for change for the 5 baht coin and then split evenly.
- Just cut some of the coins in half (physically cut them)
- Buy something with the money that we both want and can split evenly (ice cream, yogurt drinks, etc)
- Split 25/23 and J1 gives J2 something of value to balance
Determining value
Point 7 led to a mini-conversation about how much value the extra item should have: 2 baht or 1 baht. J1 gave a bunch of examples (used toys, some services) and asked me if they had the right value, and I explained it would really depend on whether they both agreed because there wasn't a separate way to determine the value.
The conversation propagates
J1 had so much fun with this conversation that he then got J2: "hey, i want to show you something. How can you split 48 baht into two?" He didn't completely recreate the discussion, but the two got a lot of the same ideas out together and it was great fun to watch.Important Lessons
- When they don't seem to be focusing on a question or challenge, (sometimes) they still are thinking about it. Let them have space and don't force it.
- Even mundane items and observations can be gateways to deep ideas
Monday, January 12, 2015
Calendar Tricks and Break the Bank (math games class 8)
Who: Baan Pathomtham 1st and 2nd grade classes
Where: In school
When: after science and before lunch
As our warm-up today, we played calendar games (one for the 1st grade and 2 for the second grade).
Without telling me their choice, the children select a 2x2 square with 4 numbers and add them all together. They tell me their sum and then I 'magically' know which square they chose and write it down.
For example, they might say 44 and then I write down:
Homework: to think about how I know which numbers they chose.
They should play with some examples and look for patterns. In particular, what is the smallest sum they can make? What is the largest sum? Do either of them depend on the month? In a given month, if they form as many sums as possible, do they notice something about all of the sums?
One point we did discuss was whether the 4 numbers are related. Because weeks are organized in rows, we know that any square has the following pattern:
Parents: using concrete examples, ask your children to explain this pattern. Why are the numbers on the lower row 7 greater than the upper row?
You now calculate $14 \times 8 - 7\times 15$, but don't tell your child what you are doing. They will tell you the answer.
Homework: Again, try to figure out why this works. Play with patterns, look at parts of the calculation. Have fun practicing multiplication and think about different ways to do the calculations.
Example: Nikki's idea for 7x15
During the Determined Determinants trick, Nikki needed to calculate 7x15. She got several pencils out of her pencil case and said:
Example: I know I've busted!
Playing Break the Bank, Tanya exclaimed: "Oh no, I know I've already busted!" Looking at her grid, she had five numbers filled:
Jin and Ji Ping were next to her, so I asked the there of them: "Are you sure Tanya has busted already? How do you know?" Part way through the discussion, one of them said: "no, this just adds up to 95, so it isn't busted." They thought for a bit more and then realized that the next digit would add at least 10, so, busted indeed.
Where: In school
When: after science and before lunch
As our warm-up today, we played calendar games (one for the 1st grade and 2 for the second grade).
I can guess your numbers
Source of this game is Calendar Puzzles which I read about in Denise Gaskin's monthly newsletter.
Choose one month on the calendar. In my examples, I'll use January this year:
| Sunday | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday |
| 1 | 2 | 3 | ||||
| 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 11 | 12 | 13 | 14 | 15 | 16 | 17 |
| 18 | 19 | 20 | 21 | 22 | 23 | 24 |
| 25 | 26 | 27 | 28 | 29 | 30 | 31 |
Without telling me their choice, the children select a 2x2 square with 4 numbers and add them all together. They tell me their sum and then I 'magically' know which square they chose and write it down.
For example, they might say 44 and then I write down:
| 7 | 8 |
| 14 | 15 |
Homework: to think about how I know which numbers they chose.
They should play with some examples and look for patterns. In particular, what is the smallest sum they can make? What is the largest sum? Do either of them depend on the month? In a given month, if they form as many sums as possible, do they notice something about all of the sums?
One point we did discuss was whether the 4 numbers are related. Because weeks are organized in rows, we know that any square has the following pattern:
| N | N+1 |
| N+7 | N+8 |
Parents: using concrete examples, ask your children to explain this pattern. Why are the numbers on the lower row 7 greater than the upper row?
Determined Determinant
We only showed this next trick in 2nd grade. I'm not going to go through all the details since the kids wanted an opportunity to try to amaze their parents. Here's what you'll be asked to do:- As before, choose a month on the calendar
- Without telling your child what you've selected, choose a 2x2 square with 4 adjacent numbers
- Now, multiply the bottom left by the top right
- Multiply the top left by the bottom right
- Subtract your second product from the first
- Don't tell your result, but tell your child that you are ready
- They will tell you the answer you got
| 7 | 8 |
| 14 | 15 |
You now calculate $14 \times 8 - 7\times 15$, but don't tell your child what you are doing. They will tell you the answer.
Homework: Again, try to figure out why this works. Play with patterns, look at parts of the calculation. Have fun practicing multiplication and think about different ways to do the calculations.
New Game: Break the Bank
For the first time, we are playing with dice. This game comes from math 4 love.- Roll your die and enter the number in your grid
- Repeat 6 times for the 2-digit game, 9 times for the 3 digit game
- Add up your 3 numbers
- If your sum is 100 or greater (for the 2 digit game), you get 100 points. If it is 1000 or greater for the 3 digit game, you get 1000 points
- If your sum is lower than the threshold, you calculate your points as Threshold - sum
- Player with the least points wins
Observations
The children really enjoyed these tricks and games. As we were playing with them, there were many opportunities for interesting little discussions and ideas from the children. As a result, these activities were much more engaging and stimulating than we might have guessed from the written explanation on the page.Example: Nikki's idea for 7x15
During the Determined Determinants trick, Nikki needed to calculate 7x15. She got several pencils out of her pencil case and said:
- here are 7 pencils
- this pencil is 15
- two of them are 30
- Grouping them in pairs, I can skip count by 3, so 30, 60, 90
- I have one pencil left over, so I need to add another 15
- Together, I get 105
Example: I know I've busted!
Playing Break the Bank, Tanya exclaimed: "Oh no, I know I've already busted!" Looking at her grid, she had five numbers filled:
| 6 | 4 |
| 2 | 5 |
| 6 |
Jin and Ji Ping were next to her, so I asked the there of them: "Are you sure Tanya has busted already? How do you know?" Part way through the discussion, one of them said: "no, this just adds up to 95, so it isn't busted." They thought for a bit more and then realized that the next digit would add at least 10, so, busted indeed.
Shooting to the moon (observational botany 4)
who: J2 and J1
where: dining room floor
when: Sunday morning
Time for another update on our plants.
Orange Seeds
The bad news is that our orange seeds aren't making any progress. Unknown to me, J1 had been putting the seeds out in the direct sun for several hours each day. This has possibly cooked the seeds. It is possible they weren't fertile in the first place. Of course, it is also possible they will still manage to sprout. We haven't tossed them out yet.
Avocado
The avocado has been a clear success. Since the last post, we hit the stage where the sprout was 6 inches long. This was the point our original instructions suggested we cut back to 3 inches. We were wondering why? After some searching on the internet, we concluded that most growers don't prune back at this stage and the purpose of doing so is to encourage a shorter, more bushy plant suitable for growing indoors. J1 and J2 were both keen to get the plant to grow as tall as possible (100 meters!) so we decided not to prune.
Some quick observations:
Root observation
We spent part of the morning sketching the roots. J1 was initially resistant, suggesting we should just work through technology ("why don't we just take a picture?") and that he had already drawn a picture of plant roots ("I already did that for school!") I explained that I was planning to draw the roots so I could practice close observation and asked if I could use his microscope. Telling him why I was going to do it seemed a very effective rhetorical strategy.
Here are our drawings:
Avocado drama of the week
Our little baby is going touring this week! Well, actually just to school for one day, then back home the next. Will it return intact?
where: dining room floor
when: Sunday morning
Time for another update on our plants.
Orange Seeds
The bad news is that our orange seeds aren't making any progress. Unknown to me, J1 had been putting the seeds out in the direct sun for several hours each day. This has possibly cooked the seeds. It is possible they weren't fertile in the first place. Of course, it is also possible they will still manage to sprout. We haven't tossed them out yet.
Avocado
The avocado has been a clear success. Since the last post, we hit the stage where the sprout was 6 inches long. This was the point our original instructions suggested we cut back to 3 inches. We were wondering why? After some searching on the internet, we concluded that most growers don't prune back at this stage and the purpose of doing so is to encourage a shorter, more bushy plant suitable for growing indoors. J1 and J2 were both keen to get the plant to grow as tall as possible (100 meters!) so we decided not to prune.
Some quick observations:
- J1: it is 25 cm tall
- J2: it has tertiary roots!
- J1 and J2: is that white stuff on the roots mold?
Root observation
We spent part of the morning sketching the roots. J1 was initially resistant, suggesting we should just work through technology ("why don't we just take a picture?") and that he had already drawn a picture of plant roots ("I already did that for school!") I explained that I was planning to draw the roots so I could practice close observation and asked if I could use his microscope. Telling him why I was going to do it seemed a very effective rhetorical strategy.
Here are our drawings:
 |
| AR1 by J1 |
 |
| AR1 By J2 (right) and J0 (left) |
Avocado drama of the week
Our little baby is going touring this week! Well, actually just to school for one day, then back home the next. Will it return intact?
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