The Joys and Dangers of Rock Groups - I

In Strogatz' 2nd column, he discusses how thinking of numbers (positive integers, anyway) as groups of rocks can make what seems boring more approachable and interesting.

For example, you might notice that you can make different kinds of shapes with your rocks, like squares and triangles.

(The 4 means that it's a square with 4 rocks on a side. The 3 means it's a triangle with 3 rocks on a side).

But only certain numbers of rocks can make certain shapes.

It's easy to get curious. The triangular numbers (in red) go 3, 6, 10, 15. What is it about these numbers that lets you make triangles out of them? And what about 4, 9, 16 lets you make squares? If you think about it, the number 1 is sort of like the smallest square AND the smallest triangle. Are there any other numbers that you could make both a square and a triangle out of?

Using rock piles, you can fiddle with these questions and discover lots of cool number patterns visually. For example, you might notice that you can build a triangular number by adding increasing integers starting at one.

Think about that. All triangles are increasing sums.

You may also notice that you can get a square number by adding two triangular numbers.

In other words...

Triangle_5 + Triangle_4 = Square_5

And each triangle is just an increasing sum. SO...

Triangle_5 + Triangle_4 = Square_5

(1 + 2 + 3 + 4 + 5) + (1 + 2 + 3 + 4) = 5*5

And it's a pretty sure bet that the pattern will continue...

(1 + 2 + 3 + 4 + 5 + 6) + (1 + 2 + 3 + 4 + 5) = 6*6

This kind of thing can be useful as well. There's a legend about the mathematician and physicist Carl Gauss. When he was in 2nd grade, as the story goes, he was acting up and his teacher told him to find the sum of all the numbers from 1 to 100. Instead of adding them, as was expected, he took advantage of a pattern and solved the problem in under a minute.

Maybe you can see it too if we look at what that sum would look like as a group of rocks.

One pattern you might notice is that the sum looks like half of a square.

As you can see, it's actually a little more than half of 10^2, because there's all those half-circles that cross the half-way line. There are 10 half circles there, so that's really 5 whole circles.

That means that....

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 10*10 + 5

That pattern makes it easy to do the same thing for the sum of 1 to 100. By the same reasoning, the answer should be 100*100 + 50.

Or you might see a different kind of pattern. Imagine taking the lower triangle here...

And moving it to create a rectangle.

This shows that the same sum is really 11 * 5!

Pretty clever! So sometimes playing around with number patterns turns out to be useful. In fact, that could be the subtitle for The Story Of Math. Time and time again, mathematicians who enjoy playing with patterns discover something interesting...and then 100 or 200 years later, the pattern turns out to be incredibly useful for something.

This visual way of thinking about numbers can also help you avoid a mistake that almost every high school student makes at one time or another.

If you had to make a guess about what (6 + 5)^2 is, what would it be?

A pretty reasonable guess might be 6^2 + 5^2. But is it true? Don't reach for your calculator!

It's very easy to check if you just imagine what shapes the numbers make.

Nope! Clearly the 6^2 and 5^2 only make up a part of (6 + 5)^2

How much more is needed to fill in the whole square? It looks like the two faint rectangles are each 6*5. So, we have that....

Big Square = Blue Square + Red Square + Two Rectangles

(6 + 5)^2 = 6^2 + 5^2 + 2*(6*5)

When I was in school, the way you figured out (a + b)^2 was by using an acronym: FOIL. It stands for First-Outer-Inner-Last, which is supposed to describe how you mechanically multiply the numbers to get the answer. It is also notoriously difficult for some students.

Area models not only make it easier, but also make it obvious WHY they work.

SO, for me, the key idea here is: Having a physical or visual representation for math concepts makes them easier to understand, get curious about, and play with. These patterns are really numerical patterns. The visual model just helped us see them more clearly.

There's nothing really magic about rock groups. If you decide to leave groups of rocks behind and just think about the area of a square, you can get even more surprising results.

In this picture, an infinite number of squares and rectangles representing successively smaller fractions can all fit inside a square with an area of 1. It's kind of surprising that an infinite sum of numbers could have a finite total, but there it is!

Using visual models and metaphors is not without its dangers, however. We'll look at a few of those next time...

From Fish to Pre-School

Since Steve Strogatz is writing a series on math, I thought I would create a sort of supplement to his with bits and pieces I've been collecting over the past year. Steve's first entry, "From Fish to Infinity" starts with pre-school. I will start with the number sense that pre-schoolers share with fish (and pigeons, and dolphins, and lions and many other animals) and how humans are uniquely able to build on those capacities to use numbers as tools.

* * *

The innate number sense we share with much of the animal kingdom boils down to two capacities: We can represent small numbers exactly, and we can represent larger numbers approximately. (The following largely follows Feigenson, L. Dehaene, S. and Spelke, E. (2004))

Small Numbers, Exactly

A 1994 experiment with wild lions illustrates the first of these numerical capacities. In the experiment (McComb, Packer & Pusey 1994), researchers used hidden speakers in the African jungle to play between 1 and 5 lion calls. If a group of female lions hears fewer calls than the number in their group, they will go and explore. If they hear more calls than the number of their group, they will leave, presumably to avoid being outnumbered. To do this, lions must be able to represent and compare the number of sounds the hear with the number of lions in their group. Clearly it's important that they be exact; being off by even one could be a life-or-death mistake!

Consider another experiment (Feigenson, L. et al. 2002), demonstrating this ability in rhesus monkeys and human infants. In the experiment, 10- and 12-month-old infants watched an experimenter hide crackers, one-by-one, in two buckets. The infants were then allowed to choose a bucket. Infants spontaneously chose the bucket with more crackers as long as there are 3 or fewer crackers in each bucket. However, when the experimenter puts, for example, 2 crackers in one bucket and 4 in the other, the infants choose randomly. They also choose randomly for 3 vs. 4 crackers, 3 vs. 6 crackers, and even 1 vs. 4 crackers. The same experiment performed on (adult) rhesus monkeys yielded similar results. Like the lions, infants and monkeys were able to represent small numbers. This experiment also highlights the limit of this capacity.

We adults are also familiar with our limited ability to discriminate all but the smallest numbers. If we see a grouping of 2 or 3 things, we can quickly, confidently, and correctly tell you how many there are. This ability is called subitizing. When there are more than about 5 or 6 items, however, we begin to make mistakes, become less confident, and eventually must resort to explicit counting.

The experiments sketched here barely scratch the surface of existing research, and you can undoubtedly think of alternative interpretations for their results. Infant and animal research is notoriously difficult, and I would encourage the curious to pursue the links in the references section further for more detail about the experimental designs. However, for the time being, let's look at our second numerical capacity...

Larger Numbers, Approximately

Immediately intuiting small numbers is only one part of our innate number sense. Here's a quick demonstration of the other. Could 43 + 17 be 500? You don't need to think much to answer. Like our ability to subitize, humans can make judgments like this quickly, confidently, and correctly. However, as the numbers all become larger, or the magnitudes closer together, we take longer, become less certain, and make more mistakes. Could 43 + 17 be 59? This judgment is far less intuitive.

We already know that infants can't represent more than about 3 or 4 things. Yet six-month-old infants are able to tell the difference between much larger pairs of numbers. For example, they can discriminate between 8 vs. 16 dots and 16 vs. 32 dots. For these experiments (Xu, F. and Spelke, E.S., 2000), it isn't the exact numbers of dots that matter, but the ratio of dots to be distinguished--how "close" the two numbers are. While six-month-olds are able to tell the difference between numbers in a 1:2 ratio, they fail with a 2:3 ratio (as when, for example, they are presented with 8 vs. 12 or 16 vs. 24 dots). All of these experiments control for non-numerical cues that you might expect someone to use such as size of the display, density of items, and so forth. Adults, by the way, can discriminate ratios as small as 7:8. This suggests that infants' do have some way of thinking about larger numbers, but that it's in terms of their approximate magnitude.

Trained rats can also represent approximate numerical magnitudes. In one experiment (reviewed in Dehaene, S., 1997), rats were rewarded with food whenever they pressed a lever in their cage a specific number of times, and then switched to press a different lever. For larger numbers, such as 41 presses, the actual number of times the rats pressed before switching formed a perfect bell curve around 41. Since individuals don't press at the same speeds, the experimenters were able to show that they could not have been using elapsed time to help them. It seems that they had to have some notion of the number 41, but that it was only an approximate one.

Getting from "fish-fish-fish-fish-fish-fish" to "6 fishes"

Strogatz' post includes an excellent clip from Sesame Street which he describes as "the clearest and funniest explanation of what they [numbers] are and why we need them." In it, one of the characters learns that the number 6 is a more reliable way to communicate a dinner order than saying "fish" six times. But how do children make the conceptual leap illustrated by the clip? How do they go beyond the numerical abilities they share with other animals to harness numbers in a more useful way?

No one yet has a full answer to these questions, but language and writing both appear to play an important role. Unlike the character in Sesame street, children aren't suddenly able to use the number system as adults do: they reliably go through several intermediate learning stages (detailed in B.W. Sarnecka, S. Carey, 2008).

Children spend countless hours learning and practicing counting. But children who can count to 10 aren't immediately able to use this skill to help them, for example, put six fish on a plate.

Initially, children who have learned to recite their numbers haven't actually learned the meaning of any of the number terms. They will count successfully, but when asked to "Give three cookies to the bear", they will grab any number of cookies at random. They will do this even when asked to count the cookies out. For this reason, children at this stage have been termed "grabbers". They can perceptually distinguish between 1, 2, 3, and 4 items; they know the word "three" comes between "two" and "four"; but they don't know what the word "three" actually means.

Over several months, children successfully learn to give 2 items, then 3, then 4. At no time do they use counting to help them--they still grab, but they grab the correct number. And at each stage, a request to give a larger number results in the random grab. At this stage, they are learning that the word "three" names their perceptual experience of 3 things. But they still don't know how counting works.

Then a revolution happens. Children suddenly become able to give any number of items up to their counting limit. When this happens, they graduate from "grabbers" to "counters". Indeed, whereas they used to grab to give 3 items, or 4 items, they now dutifully count the items out. At this stage, children have learned the key principal of counting: The number of items that a number names is determined by its position in the counting sequence.

Only this insight can get a child from "fish-fish-fish-fish-fish-fish" to "six fishes" with the aid of counting. Even if they can't perceptually distinguish 6 things from 5 things or 7 things, language has given them the tools to conceptually distinguish between them. They know that the numbers occur in a sequence, that each number-word names a specific amount, and that each successive word in the sequence adds one to that amount.

The Language and Culture of Numbers

As the last section suggested, number words provide a crucial cognitive scaffold for learning to count, and thereby to conceptualize larger numbers exactly. They act as anchors in a sea of thought to which meanings can be attached, organized, and preserved.

Though I can't seem to locate the references right now, a number of experiments have been performed with speakers from an isolated Amazon tribe whose language contains no exact number words. Their only number words translate roughly to "one-ish" and "two-ish", or perhaps better, "less" and "more". In these experiments, adult speakers performed equivalently on numerical tasks with children prior to learning language.

For me, the Amazonian culture brings out the question of what features of a culture create a need for counting and naming exact numbers. It's easy to imagine why hunter-gatherer cultures might not need to distinguish 57 fro 58, or even 19 from 20. Though it's harder, I can also imagine that, for smaller collections, naming the individual items could suffice, obviating the need for specific numbers like "three".

I know there is research on these topics, but unfortunately I haven't been able to find it (and, sadly, without a university connection I can't read online journal databases!). If anyone knows any general references, I would love to follow up on this strand.

References:

Feigenson, L. Dehaene, S. and Spelke, E. (2004) Core Systems of Number. Trends in Cognitive Science Vol. 8 No. 7. 308-314

B.W. Sarnecka, S. Carey (2008) How counting represents number: What children must learn and how they learn it. Cognition 108, 662–674

Dehaene, S. (1997), The number sense: How the mind creates mathematics, New York: Oxford University Press, ISBN 0195132408 (The precise is freely available online).

Feigenson, L. et al. (2002) The representations underlying infants’ choice of more: object-files versus analog magnitudes. Psychol. Sci. 13, 150–156

Xu, F. and Spelke, E.S. (2000) Large number discrimination in 6-month old infants. Cognition 74, B1–B11