Counting

My five-year-old has been able to rattle off the numbers in order into the teens for a while, but when you ask her to count a set of objects, she can only get up to three consistently. “One, two, three–um–five?”

I have an idea of what might be going on in her head. It is pretty clear that she knows that four comes after three. I suspect that she has also figured out that when counting objects you are announcing at each step how many you have pointed to so far. When she points to the first object she knows that she has pointed to one object and announces “one.” Upon pointing to the second object she knows that she has pointed to two objects and announces “two.” When she points to the third things go similarly, but when she gets to the fourth she runs into trouble, because she doesn’t have any good idea how many have been pointed to.

I hear you saying “Whoa, if she knows that four comes after three then of course she knows that after pointing to one more than three things she has pointed to four things.” Not so, because she hasn’t learned that yet. Sure, you know what counting is all about, but kids don’t until they’ve been taught. What my daughter is doing when pointing to the first three objects is subitizing the number of objects pointed to so far. Subitizing is the act of being aware immediately of how many objects are in a collection without counting them. We humans can easily subitize collections of one, two, or three objects.

Up to this point my suspicion is merely that: a suspicion. My hypothesis has to be tested. We need to isolate the pieces of counting. The elements of counting are: 1) reciting the names of the numbers in order; 2) pointing to objects one-by-one; 3) doing 1 and 2 simultaneously; and 4) coordinating actions 1 and 2, saying exactly one number for each object pointed to. We isolate these elements in the case of my daughter by asking ourselves a series of questions:

First, can she recite the numbers aloud, in order consistently? Yes. She has no problem with this up to at least ten.

Second, can she point to objects one-by-one, in sequence? Yes. No problems there.

Third, can she point to objects one-by-one, in sequence while reciting the numbers aloud? Initially we aren’t sure, and we decide that we need to distance this skill from counting itself, so we ask ourselves:

a. Can she recite aloud a sequence of words other than the counting numbers? Sure, she knows the alphabet, as well as many songs.

b. Can she point to objects one-by-one, in sequence while reciting aloud a predefined sequence of words in order? We don’t know initially, but my wife tries it with her and she is successful (for instance, with the alphabet).

Fourth, can she coordinate the two actions above? That is, as she points to objects in sequence and recites words, can she match them up, pointing to exactly one object for each word she says? Again, we aren’t sure initially, but my wife tries it with her and she is in fact able to do it.

This last one is the complete skill of counting. Once she can do that she can count, right? So what’s tripping her up? Well, counting is two separate things: it’s a process as just described, and it’s a way to ascertain the number of objects in a collection. These are distinct facets of counting, but they are intimately connected. You do the process and the last number that you say is the number of objects in the collection. These two facets of counting are usually learned by children separately. It is my understanding that most children first learn the process of counting and only later learn that this process gives you information about how many things are in a collection. A young child who has “learned to count” may count the cookies on a plate “one, two three, four, five, six, seven,” but then when asked “how many are there?” reply with “I don’t know.”

I think that my five-year-old learned things opposite the usual order. I’m pretty sure that she has figured out that counting is all about determining the size of a collection, but initially she is unclear on how the process works. Only by divorcing the counting process from numbers were we able to make any progress. In the last several months she has progressed well, and is on her way to being a proficient counter.

See also: Wikipedia article on counting

The Math Games–One: Block Towers

We had some time today, so I suggested a little competition: try to build the tallest tower with our wooden blocks. My nine-year-old and I were the only ones to compete, and the nine-year-old got bored pretty quickly and essentially gave up. Below is his tower.

The rules that I laid out are as follows:
1) Build a tower by stacking the wooden blocks as you wish.
2) The tower is measured from base to tip. Your final score is calculated by taking the height of your tower in inches and subtracting an inch for each year of your age.
My son’s tower was 29.5 inches, so his calculated score was 19.5
My tower was 79.25 inches, so my calculated score was *****. Let’s just say that I won by a wide margin.

Of course the crux of designing these towers is managing your block resources. You want the base to be wide and sturdy enough to give good support to the tower, but you don’t want to waste too many blocks on any single layer.

Here are some more shots of my tower:

Evening of Math VI

It was late and rushed tonight, but I did manage to do some math with the kids. Actually, A was throwing fits, so I put her to bed. I had B working on the fact that if you have a connected graph on 5 vertices then there is a connected subgraph on 4. He worked out this conjecture himself, but doesn’t have any justification for it. I’m thinking he should develop the idea of a spanning tree, but I’m not sure how to nudge him in that direction. I guess I’ll have to think up an intermediate problem.

A Morning of Math

I’ve been sick and had to stay home from work the last several days. This morning I told my kids that they could choose to put away their clothes or do math. They chose math. I set both B and A working on different problems in our partitions of graphs research project. B had previously shown that 2+2 is a Q1 partition with the claw being it’s Q1 graph. I asked him to decide whether any of the other partitions of 4:
4
3+1
2+1+1
1+1+1+1
could be Q1. He started with 1+1+1+1. He said this partition you can always make with any 4-vertex graph, so it is not Q1. Next he tackled 2+1+1. He gave me an algorithm to make this partition from any (connected [all of our graphs start out connected so far. I think next time I might introduce the idea of a graph not being connected and still being considered one graph]) 4-vertex graph. Here is the algorithm: choose two adjacent vertices. Do not cut the edge between them, but cut all other edges in the graph. This gives the partition 2+1+1. Next he dealt with the partition 4. If the graph is connected, then of course 4 is a partition you can make trivially (again we’ll have to come back to this once we talk about starting with disconnected graphs). 3+1 he doesn’t believe can be Q1. He thinks that it may also be a partition allowed by any connected 4-vertex graph. He is going to think about why, and we’ll return to it at our next session.

A had started analyzing what she called the ice cream cone. I had her redraw it for me today. I think that it had more vertices before, but today it ended up being a 4-cycle (the square). We checked one by one that you can make any partition of 4 with this graph. Next I had her think about the 5-cycle, then 6 and 7 and finally 8. After doing 7 she was pretty sure that you can always make all partitions of n with an n-cycle. She hasn’t given me any justification, but I think she sees what’s going on. Next session I think I will have her think about what partitions can be made with stars.

The Chicken’s Claw

I had my fifth evening of math with my kids.  I got a box of 100 washers from the hardware store and some yarn.  I had the kids draw some graphs and then make some graphs with the washers as nodes and the yarn as the edges.  I let them make some of thier own, then I had them make these:

The square:

o-o
| |
o-o

and the claw:

  o
  |
  o
 / \
o   o

and asked them if they were the same (“What do you mean, Dad? Of course they’re not the same.”)

Then I had them make the pentagon and the pentagram. I asked if they were the same (again “of course not”). So I took two copies of the square. “Are they the same?” “Of course.” “Well then, why aren’t the square and the claw the same?” With time they answered that one had a node that was hooked to three other nodes rather than just two. Next they spent some time trying to get the pentagon to look like the pentagram. I suggested they try to work with the pentagram itself. B picked it up and it immediately fell apart into the pentagon.

Next I showed how I could cut two edges of the pentagon to get two connected graphs, one with two nodes and the other with three, thus making the partition of 5: 3+2. I then asked how many partitions of 4 they could make with the claw. They discovered that the only one that they couldn’t make was 2+2. I asked this question.

Q: How many partitions of 4 can’t you make with the claw?

The answer is 1, so we call the claw a Q1 graph.

A couple nights later I had them look for more Q1 graphs. B tried one that he called the chicken claw:

    o
    |
    o
    |
    o
   / \
  o   o
 /     \
o       o

He quickly figured out that this is not a Q1 graph. In the mean time A worked on what she called the ice cream cone:

o-o-o
|  /
o o
|/
o

But she wasn’t able to finish her analysis of it inthe time we had.

Math with Bananas

Tonight was my fourth evening of math with my kids.  It was rather frustrating, but turned out good in the end.  I pulled out the number line again, which had markings for 0, 1 and 2, as well as 1/2, 1/3, 1/4, etc. through 1/17 and then an annotation saying: “infinitely many more like these.”  I said “the zero has lots of friends, but what about the one?”  Mostly they grumped around and doodled for a while.  Then they kind of stared at me blankly as if expecting me to answer my own question.  They weren’t talking to each other at all, so I left the room and told them to work it out on their own.  B called me back a while later saying that 1 had some more friends now.  I came back into the room and he showed me that he had written 1/2, 1/3, 1/4, etc. through 1/17 between the 1 and 2.  I prompted him to tell me what the difference between the 1/2 between the 1 and 2; and the one half between the 0 and 1 was.  He said that they were the same, except that they were in different places.  I asked him what he meant and he continued to insist that they were the exact same number, but that they show up in those two places.  Next he told me that the 1/2 between the 0 and 1 is supposed to mean half of zero, while the 1/2 between 1 and 2 is half of one.  Hmmm…something was going wrong here.  I asked what half of something means.  B said that it means you split a number up.  A basically wasn’t participating.  I was starting to get frustrated, so I took down a bunch of bananas.  I asked them to show me one banana.  Now A perked up and B shut off.  A held up one banana.  I told here to put it in its spot on the number line.  She did.  Then I asked them to show me two bananas.  A picked up two bananas, and I prompted her to put it in its spot.  Then I asked B if she was right that this was two bananas.  He said that he didn’t know (!).  Obviously he was frustrated.  He knows what two bananas is.  Finally he agreed that A was right with her two bananas.  Next I asked for zero bananas.  They both understood that with no problem.  “Now, how many bananas do we put by this 1/2?” I asked.  B claimed that we should put zero there.  He was still thinking of it as half of zero.  I’m beginning to understand that he was confused with the concept of half as a number (an element of the real numbers) and half as a scalar multiplier of the integers.  (“half of zero is zero.  Half of one is, well it’s half of one.  Half of two is one,..etc., but what is one half?  It’s a different kind of object…right?”  that’s what I imagine was going on in his brain, but I’m not sure.)  Finally I got out a knife: “Show me half a banana.”  He was starting to get it now.  He cut a banana in half.  There was lots more frustration before the night was out, but eventually we got 5/3, “one and a half,” 4/3, and 2/3 on the number line in their correct places (!) and ended up with a bunch of cut up bananas.  Smoothies for breakfast!

Ok, tonight was not tons of fun for any of us.  We’ll give the number line a break for a while.  It’s tough for me to work with the kids on something where I know the answer and they don’t.  I need to get better at that.  But next week I plan on moving them closer to doing some math research where I don’t know the answers.

An Evening of Math III

I’ve been reading Out of the Labyrinth: Setting Mathematics Free, by Robert and Ellen Kaplan, who are the founders of “the Math Circle.”  As I understand it a math circle is where you get a bunch of people together, ask a couple of provocative questions to get them thinking, and then get out of the way and let them create mathematics.

I decided that I would try something akin to their math circle with my kids.  I settled on using the opening example from the book:  draw a number line with 0 and 1 labeled, ask if there are any other numbers in there, and step back to see what numbers the kids can find.  The idea is that they start marking fractions, get comfortable with those and eventually come up with an irrational number.

A was easily distracted tonight.  She kept wanting to put in numbers beyond 1.  B put in 1/2, then 1/4 in approximately hte right places.  Later he moved them so that he could fit more numbers in.  He ended up with 1/2, 1/3, 1/4, etc. down to 1/13 spaced approximately evenly.  A added 1/14, 1/15, 1/16.  I asked how many more there were.  ” I can’t say how many.  Lots more.  Infinity many.”  Is what B replied.  He ended up writing “infinitely many numbers like these” between 1/16 and 0.  We also ended up with 1/1 somewhere between 1 and 1/2.  For now I’m letting that persist until they figure out that its the same thing as 1.  A wanted to put in 0/2, but B told here that that was the same thing as 0.

A also said that one thousand two hundred thirty four was her favorite number and wrote it out: 1234, with a box around it.

Next week we’ll pull the number line out again and since 0 has so many friends next to it, I’ll ask whether 1 has any.

An Evening of Math II

Tonight I gave the kids the choice of doing more partitions, or something else. They chose something else. In the book Math Tricks, Puzzles & Games by Raymond Blum I found this problem:

TUNNELS

Try to connect each rectangle with the triangle that has the same number.  Lines cannot cross or go outside the diagram.

When I showed the problem to the kids, A was very upset and said that she didn’t know how to do it because she had never learned.  She threw a fit and started drawing a picture instead of working on it.  B immediatly started working on it.  He worked for about 20 minutes, drawing lines, erasing, drawing more lines.  He eventually got frustrated, saying that it was impossible.  In spite of A’s protests I brought her over next to B and had him explain what he had tried, and why it was impossible.  B connected rectangle 1 to triangle 1, then connected rectangle 2 to triangle 2, but at that point rectangle 3 was completely cut off from triangle 3.  Then A took a pencil and a copy of the puzzle and without pausing connected rectangle 1 to triangle 1 and rectangle 3 to triangle 3.  Just as B was saying that she wouldn’t be able to connect the last pair, she did!

Getting A to look at a problem is 95% of the battle.  I reminded her that it was B’s work on the problem that helped her to her solution.

An Evening of Math I

My wife has taken on the challenge of homeschooling our children this year. My main participation in this is a weekly math session with the kids in the evening on any subject of my choosing!

Tonight was our first session. I decided to do partitions with them. I am priming them to be able to help me with my research on Q1 graphs.

We pulled out cubical blocks and I told the kids to make partitions with them. The hardest part about this is keeping my mouth shut and staying out of their way.

B invented Ferrers diagrams. Meanwhile I set A to work on making all partitions of the small numbers. She found 1 partition of 1, 2 partitions of 2 and 3 partitions of 3. She started working on 4 and found 4 partitions. Then B chimed in with a 5th partition of 4. This upset A and she refused to accept it as a partition, because it didn’t follow the pattern that she had seen. She stormed off, but came back and was ready to accept the 5th partition of 4.

I tried to get them thinking about how we could know that we had got all of them. They haven’t come up with anything along those lines yet. B (of his own volition) started working on an algorithm to generate all of the partitions of a given number. The algorithm needs work, so far only generating the n partitions of n: (1,1,…,1), (2,1,1,..,1), (3,1,1,..,1),…,(n-1,1), (n). He was also working on an algorithm for getting partitions of n+1 from partitions of n. I think that one was pretty incomplete too.

Their minds were still going, but it was getting late, so I sent them to bed. It was a good evening of math.

Math Card Games

By Joan A. Cotter, Ph.D. Published by Activities for Learning, Inc.  A link to their site about this product can be found here: Math Card Games: 300 Games for Learning and Enjoying Math.  To see a price list, click on the picture at the bottom of the page.

These games are designed to assist in the memorization of mathematical facts and the reinforcement of concepts by practice, but in a way that makes the players think a little more than doing a worksheet, and is a lot more fun and interactive.  The games tend to remove many of the boring aspects of learning elementary level math.

Games cover the following subjects: Numeration, Addition, Clocks, Multiplication, Money, Subtraction, Division, and Fractions.

Many of the games are memory games, with a twist.  The twist is where the genius comes in.  The game that I played today with my oldest was designed to assist him in learning the dreaded times tables.  In this game, number cards with multiples of chosen numbers on them are used, and the twist is that instead of getting matches, you have to find the cards in increasing order.  If you are looking for multiples of 3, and turn over 6 first, you can’t take it, you have to find the 3 first.  The great part about it is that the after you find the 3-card, your mind turns to finding the 6-card, after that you are looking for the 9-card, etc.  Your mind begins to anticipate what the next number is as you play.  After playing the game multiple times, you become familiar with the number sequences.

Many of the games are modified versions of common games, such as fish and rummy, so it makes it easy for the child to play them.  The rest have pretty simple rules that the child can easily understand.  Several of the games can be played with multiple players, or as solitaire games.

This book and the accompanying cards have been well worth the money, and we’ve have 270+ more games to go in the book.

Reviewed by Arlynda