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

Latin Squares, Squared Squares, and Legoed Squares

I introduced my kids to Latin Squares the other day. If you know Sudoku then you have seen examples of Latin squares. The idea is to fill in a grid of squares with colors or numbers or some other symbols, such that each symbol appears exactly once in each row and each column. (see also the Wikipedia Latin square entry). I handed them graph paper and let them loose. This is what my eight-year-old produced:
Note that she produced some Latin squares, but also explored other ideas.

I wasn’t sure initially how long my kids would be content to explore Latin squares, so I also planned to tell them about squared squares. A squared square is just a square made up of smaller squares (this is easy to accomplish). A perfect squared square is much more difficult, consisting of squares each of different sizes. For a long time it was thought that perfect squared squares didn’t exist, but they do! Here is the smallest (in some sense) one:
We got excited about it and decided to make one for our wall:

Next, the ten-year-old wanted to produce some squared squares of his own, but found the perfect ones difficult (they were thought to be completely non-existent for a long time after all). He settled for producing imperfect ones. After making an imperfect squared square on graph paper, he decided to reproduce it with Legos. Here is the result: This led us to considered perfect Legoed squares. That is to say, squares made up of some number of Lego pieces, each piece having a unique size. Here is one of our first examples: It’s a 3×3 made up of a 1×1, a 1×2, and a 2×3. Here’s another, somewhat larger: We found a bunch more (maybe ten or so total). I’m not sure that we got all of them. How many can you find?

Look for up-coming posts on: 1) further progress in building our marble computer, and 2) teaching my five-year-old to count (which required some deep thought about just what counting is).

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.

How to Eat an Equilateral Cheeseburger

My friend Sean worked out an elegant and practical solution to the Cheeseburger problem of my previous post.  I will try to summarize it here.

There are fifteen toppings.  To make things simpler we’ll represent them with the numbers one through fifteen.

1 = Mayo
2 = Relish
3 = Onions
4 = Lettuce
5 = Pickles
6 =Tomatoes
7 = Grilled Onions
8 = Grilled Mushrooms
9 = Ketchup
10 = Mustard
11 = Jalapeno Peppers
12 = Green Peppers
13 = A-1 Sauce
14 = Bar-B-Q Sauce
15 = Hot Sauce

Arrange these numbers around a circle at equal intervals, like the hours of a clock with fifteen rather than twelve hours.  Each cheeseburger can be represented on this circle as three points.  Think of these three points as the vertices (corners) of a triangle.  As has been shown in the comments of the previous post, there are 455 cheeseburgers total.  So the totality of cheeseburgers are represented by 455 triangles inscribed in the circle, which is a tangled, nasty mess (but not as big a mess as all 455 cheeseburgers sitting on your kitchen table).

Let’s get a handle on the mess.  We need some notation to make it easier to talk about these triangles.  If a triangle has vertices at points marked x, y and z, we’ll denote it by (x,y,z).  Some of the triangles will have the same shape as each other.  Others will be differently shaped.  For instance, (1,2,3) has the same shape as (2,3,4), but (1,2,4) has a different shape than (1,2,3).

In fact there are fifteen triangles with the same shape as (1,2,3) (including (1,2,3) itself).  They are: (1,2,3), (2,3,4), (3,4,5), (4,5,6), (5,6,7), (6,7,8), (7,8,9), (8,9,10), (9,10,11), (10,11,12), (11,12,13), (12,13,14), (13,14,15), (1,14,15) and (1,2,15).  Each of these triangles can be obtained from (1,2,3) by rotating it around the circle.  These fifteen triangles together form an equivalence class.

Likewise there are fifteen triangles with the same shape as (1,2,4).  Said another way (1,2,4) has fifteen triangles in its equivalence class (but we need to be careful here to emphasize that we are considering (1,2,4) and say (1,3,4) to have different shapes: they are mirror images of each other, but not rotations of each other).

In all, the 455 triangles fall into 31 equivalence classes.  30 of these classes have fifteen triangles.  The final equivalence class consists of five triangles.  More on that in a moment.

The strategy is this:  Pick an equivalence class.  Rotate through all of the cheeseburgers in that class.  Next move on to another equivalence class.

Let’s list the equivalence classes by writing down a single triangle in each class.  OK, I’m not actually going to write all of them out, but I will do the next best thing.  You’ll see.

First we have twelve equivalence classes with triangles of the form (1,2,n), where n=3,4,5,…,14.  We don’t allow n=15 (i.e. (1,2,15) because that is the same class as (1,2,3).  That gives us 180 cheeseburgers.

Then there are nine classes of the form (1,3,n), where n=5,6,7,…,13. (n=15 would give the same class as (1,2,4) and n=14 the same class as (1,3,5)).  That’s another 135 cheeseburgers.

Next we get six classes of the form (1,4,n), with n=7,8,9,…,12.  Another 90 cheeseburgers.

Similarly there are three classes of the form (1,5,n), with n=9,10,11.  45 more cheeseburgers.

Finally, we have our one strange class with only five cheeseburgers in it.  This class is represented by (1,6,11).   Let’s list all of the triangles in this class: (1,6,11), (2,7,12), (3,8,13), (4,9,14), (5,10,15).  These are the five equilateral triangles in the set.  Just for fun let’s list the five cheeseburgers that these triangles represent.

The equilateral cheeseburgers are:

Mayo, Tomatoes, Jalapeno Peppers

Relish, Grilled Onions, Green Peppers

Onions, Grilled Mushrooms, A-1 Sauce

Lettuce, Ketchup, Bar-B-Q Sauce

Pickles, Mustard, Hot Sauce

So go into a burger joint.  Ask for an equilateral cheeseburger.  See what kind of look you get.  Then just tell them you want mayo, tomatoes and jalapeno peppers on it.

OK, seriously, back to the math for a minute.  Sean also realised that you can rotate through the classes such that no two burgers in a row have any of the same toppings.  Great!  That’s what I was asking for in the first place.  Let’s use our first equivalence class as an example.  We start with the triangle (1,2,3).  Now we can rotate one slot by adding one to each number in this triple to get (2,3,4).  But that’s not what we want: we would have two burgers in a row where only one topping changed.  So instead let’s rotate three slots so that none of the toppings are the same twice in a row.  We get (1,2,3) then (4,5,6).  Keep going to get: (7,8,9), (10,11,12), (13,14,15).  Now when we do the next addition: (13+3,14+3,15+3) we have to be careful to do clock arithmetic on a 15-hour clock.  That is, counting up one from 15 gives one, rather than 16 (we mathematicians call this modular arithmetic).  So (13+3,14+3,15+3)=(1,2,3).  That’s a problem because we’ve already visited the triangle (1,2,3), but there are still ten triangles that we haven’t visited in this class.  Our problem was moving three slots at a time.  15 is divisible by 3.  In fact, what we should do is pick a number that is relatively prime to 15, which means a number that is not divisible by three or five (the prime factors of 15).  Four will work.  So start with (1,2,3).  Add four to each slot: (5,6,7).  Add four again: (9,10,11).  And again: (13,14,15).  Again: (2,3,4).  Continue in this manner.  Here is the list we get for this whole equivalence class:

(1,2,3)

(5,6,7)

(9,10,11)

(13,14,15)

(2,3,4)

(6,7,8)

(10,11,12)

(14,15,1)

(3,4,5)

(7,8,9)

(11,12,13)

(15,1,2)

(4,5,6)

(8,9,10)

(12,13,14)

and we’re done with this class.  We will now move on to the next class starting with (1,2,4).  For this class rotating four slots will also work:

(1,2,4)

(5,6,8)

(9,10,12)

(13,14,1)

(2,3,5)

(6,7,9)

(10,11,13)

(14,15,2)

(3,4,6)

(7,8,10)

(11,12,14)

(15,1,3)

(4,5,7)

(8,9,11)

(12,13,15)

Done with that class.  The next one in line starts with (1,2,5).  Now if we rotate by four we will get overlap between successive triangles.  Instead, we could choose to rotate by two:

(1,2,5)

(3,4,7)

(5,6,9)

(7,8,11)

(9,10,13)

(11,12,15)

(13,14,2)

(15,1,4)

(2,3,6)

(4,5,8)

(6,7,10)

(8,9,12)

(10,11,14)

(12,13,1)

(14,15,3)

And we’re done with that class.  And so the process continues through all 455 cheeseburgers.  So the amount we rotate depends on the equivalence class we’re working in, but we can always pick a rotation that is both relatively prime to 15 and doesn’t result in any overlap between consecutive triangles.  In fact, it turns out that we can always choose our rotations from the set {1,2,4}.

Notice that my first 15 cheeseburgers will be isoceles.  Then I will be eating scalene burgers for quite some time.  Not until the very end do I get to the equilateral cheeseburgers.  I have decided that I am going to try to do two per month.  That way it will take me just under 19 years to finish.  Read Mayo, Relish Onions: A Cheeseburger Review for an account of the first trial of this experiment.  Bon Appetit!

Latest picks from the library

Here are the two books that I grabbed recently from the library:

How Math Works: 100 Ways Parents and Kids Can Share the Wonders of Mathematics, by Carol Vorderman.  Lots of activities here, but not all of them are all that mathematical and not all of them look all that fun, but I think there are some gems.  One that looks interesting is on page 167 of the edition that I have.  The point of the activity os to make a solid out of modeling clay that will be able to plug a circular hole, a triangular hole, and a square hole.  To quoate the text: “The experiment encourages you to think about non-regular shapes.”

On Beyond a Million: An Amazing Math Journey, by David M. Schwartz.  this is a quick picture book about big numbers and counting by powers of ten.  The highlight is the speech bubbles of the child characters in the books.  They ask lots of questions, not all of which are completely answered…

Go Figure

I’ve only looked through this book a little so far, but it looks awesome.  My 7 and 5 year old have looked at it and had fun and I learned several math facts I didn’t know from it today.  I highly recomend it for math lovers and mathophobes alike.

Origami II

In answer to yesterday’s post:

Yes!  It is possible

I didn’t beleive it at first either!  But I have done some experiments.  Try making a simple curved fold yourself.  I’ve found that it is pretty easy to make a curved fold with a strip of paper such as a receipt.

Check out this article for more explanation and instructions for some curved origami that you can do at home.

Origami

We all know that you can fold a piece of paper along a straight line.

Here is a question to ponder:

Can you fold a piece of paper along a curve that isn’t straight without crumpling the paper?

Thoughts, theorems, expiriments, counter-examples, and half-baked ideas are all welcome.  I’ve been thinking about this question for a while and done some experiments, but I don’t think I have a difinitive answer.

Knot Tying Schema

I’ve noticed that whenever my two-year-old wants to tie a knot, she takes the two strands that she’s interested in and repeatedly twists them around each other.  Of course this doesn’t produce any kind of knot at all (or I should say it only produces the unknot).  But this twisting appears to be the only basic move that she knows (and she learned it herself) for producing knots.

It made me wonder what the basic moves are that we use to tie knots.  Every knot theorist knows that there are three basic moves for transforming one picture of a mathematical knot into another (possibly quite different looking) picture of the same knot.  These moves are called the Reidemeister moves.  Any two pictures of the same knot can be made to look like each other using just these moves (and “plane isotopies”).  Similarly it seems to me that mathematically you really only need two moves to tie any knot, or what I mean to say is to go from a picture of a straight string to a picture of any knot.  Here is an illustration of those two moves:

Of course after doing several of these moves, you will want to glue the two ends of the string together if you wnat to get a picture of a mathematical knot.

When people tie knots, it does seem like there is some finite set of moves they use, but the catalogue of moves seems to be bigger than these two moves I illustrated above.  For instance there are moves like “wrap one end around a loop” or “put one end through a hole” or “follow one strand through a whole series of moves.”  Of course each of these can be broken up into a sequence of the two basic moves, but it is not always useful to break things down into the simplest possible moves in practice.  It makes me wonder what the catalogue of moves is that people use.

A couple more sentences.

Sentence A: Sentence B has 1 1, 3 2s, 2 3s, 3 4s, 2 5s, 3  6s, 1 7, and 1 8.

Sentence B: Sentences A and B together have 5 1s, 6 2s, 6 3s, 4 4s, 3 5s, 4 6s, 2 7s, and 2 8s.

Sentence C: Sentence D has 5 1s, 2 2s, 1 3, 3 4s, 1 5, 1 6, and 1 7.

Sentence D: Sentence E has 4 1s, 4 2s, 1 3, 1 4, 2 5s, 1 6, and 1 7.

Sentence E: Sentence C has 5 1s, 2 2s, 2 3s, 1 4, 2 5s, 1 6, and 1 7.