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

Expert Problem Solving

I've been reading Alan Shoenfeld's article Learning to Think Mathematically: Problem Solving, Metacognition, and Sense-Making in Mathematics, where he summarizes an interesting study about novice and expert problem-solvers.

Experts Don't Get Bogged Down

The study videotaped pairs of college students attempting novel problems and coded how much time they spent in different kinds of problem-solving activity. The following was typical of novice problem solvers:

"The students read the problem, quickly chose an approach to it, and pursued that approach. They kept working on it, despite clear evidence that they were not making progress, for the full twenty minutes allocated for the problem session. At the end of the twenty minutes they were asked how that approach would have helped them to solve the original problem. They couldn't say."

They did the same for pairs of math professors, and here is what they found:

From the article:

"The first thing to note is that the mathematician spent more than half of his allotted time trying to make sense of the problem. Rather than committing himself to any one particular direction, he did a significant amount of analyzing and (structured) exploring -- not spending time in unstructured exploration or moving into implementation until he was sure he was working in the right direction. Second, each of the small inverted triangles in Figure 4 represents an explicit comment on the state of his problem solution, for example "Hmm. I don't know exactly where to start here" (followed by two minutes of analyzing the problem) or "OK. All I need to be able to do is [a particular technique] and I'm done" (followed by the straightforward implementation of his problem solution) ....

... as he worked through the problem the mathematician generated enough potential wild goose chases to keep an army of problem solvers busy. But he didn't get deflected by them. By monitoring his solution with care -- pursuing interesting leads, and abandoning paths that didn't seem to bear fruit -- he managed to solve the problem, while the vast majority of students did not."

Teaching Self-Monitoring

A second study attempted to teach students this type of self-monitoring (though perhaps 'habituate' would be a better term). Shoenfeld had students spend approximately 1/3rd of their class time working on novel problems in groups of 3-4. His role was to be a roving consultant, and to press the students with the following 3 questions:

1. What (exactly) are you doing? (Can you describe it precisely?)
2. Why are you doing it? (How does it fit into the solution?)
3. How does it help you? (What will you do with the outcome when you obtain it?)

Initially, students can't answer his questions, because their way of working doesn't include thinking about them. Once they learn that he won't stop asking them, however, they begin to think about them ahead of time, before he asks. By the end of the semester, he reports, this type of thinking has become habitual.

Here is one activity graph for a pair of students who had completed this course.

He concludes:

"The point here is not that the students managed to solve the problem, for to a significant degree solving non-standard problems is a matter of luck and prior knowledge. The point is that, by virtue of good self-regulation, the students gave themselves the opportunity to solve the problem." [emphasis added]

This is like speed dating for math (if you'll indulge me the analogy for 2 paragraphs). Considering more candidate-strategies earlier gives you more chances of finding the right one for you. The difference is that speed dating prescribes a (short) fixed amount of time with each candidate. Problem solving would be more like speed dating if the speed-daters were left to decide for themselves how much time to spend with each person before moving to the next.

So how do you know when to move on? It's easy if you've run out of things to talk about, or if you don't know what to do next with your current strategy. But some people (and strategies) are seductive--it may take a long time to realize they're not taking you where you want to go. Savy and experienced daters, as well as mathematicians, learn to recognize signs of an interaction that seems promising or troubling. And this kind of subtle judgement is something that only comes with reflection on experience, I think.

What Now?

Thinking about this has suggested a few things I'm curious to try.

1). Brainstorm our options first. I might precede some problems with group- or class-wide brainstorming to get a short list of approaches that make sense to try.

2). Heuristics tell you what to try and what to look for. When I was teaching heuristics as skills, I focused on what each heuristic recommends that you try to do. Shoenfeld's questions make it obvious what this leaves out. Learning a problem-solving heuristic also means learning what to look for as you try, so you'll know whether the approach is working or not. It might also involve learning what you might do next with the kind of information you're likely to get. When first teaching students each heuristic through guided examples, I should make all three parts explicit--not just the first, as I had been doing.

3). Practice talking about all three parts. When a student was stuck, my first question was always "what could you try?" If they could give a coherent answer, I would often leave them to try. Instead, I should ask students all three questions: 1). What can you try? 2). What will tell you if it's working? (If you're looking for a pattern, can you quickly write a few types of pattern you might expect from prior experience?) 3). What then? (try to describe the pattern? connect its structure to the problem-statement? falsify it with test cases? represent it algebraically or graphically?).

Until students start to get a feel for how to answer the questions, this will probably have to be on a small-group or individual basis. Once students gain experience, I might stop all problem-solving work at 5 minutes, then at 10 minutes, then at 15, and have students write concise answers to these questions (no more than 1 minute each question).

4). Practice forced switching. Knowing how to make a mental bookmark of your current approach is immensely useful. It's required for high-level task-switching, but also to simply listen to and understand someone else's ideas. Being forced out of one's comfort zone is also sometimes required to experience the benefits of a new approach. Students might work for 5 minutes down one path, summarize their progress, then work down another path from scratch. This would be good practice for #3, and also let them know what it feels like to mentally start over, which is something that happens all the time in problem solving.

Those "getting to know you" handouts

Every teacher I know gives a "getting to know you" questionnaire at the beginning of the year, including me! Cleaning up all my papers in June, I'm always surprised at just how much students really told me about themselves in that first week, and how I wished I'd used that information better somehow. What that in mind, here are a few of my ideas about What to ask and How to use the answers.

1. Everyone's got something

For the past 3 years I've asked students the following unexceptional questions:

a). Describe yourself in 3 words: ___________, ___________, ____________
b). Describe school in 3 words: ___________, ___________, _____________
c). What's something you're really good at?
d). What's something you wish you were really good at?
e). What do you want to do "when you grow up?" (or what do you definitely NOT want to do?).

I've often been surprised by their interesting answers. "I can swim a technically perfect butteryfly", "I wish I were better at controlling my will", "I want to be a physical anthropologist (!)", "I know how to breed giant tortoises".

I make an overhead for each question with an anonymous jumble of answers. Even though most juniors and seniors recognize each other by the time I get them, they don't really know each other. I find that sharing the responses helps creates a really positive atmosphere of excitement and curiosity about each other. Also, the parents at back to school night love seeing the answers as well.

2. All I had to do was ask

One year on the back of my expectations form I also asked:

a). Which of these expectations do you think you'll have the most trouble with?
b). What can I do to help you succeed?

Common responses include "always doing the homework", "not being tardy", but also things like "I have trouble asking for help when I need it" and "I have trouble working in groups". These provide great heads' up for all kinds of things. The answers to part (b) are typically less useful, but sometimes can be a real help. Every now and then you'll get a student with some really bad habits who actually asks you to be really tough on them, which is a great point to come back to with them later.

3. Matchmaking

Something I haven't tried but would like to is asking the students something like this:

(a) Circle things you're pretty good at
(b) Underline things you still struggle with

The list could include specific math topics, more general skills, and "soft skills" like working with others, explaining your thinking verbally or in writing, asking good questions, etc.

Aside from providing a very rough pre-assessment of knowledge, this information could be used for creating working pairs, or surreptitiously choosing warm-up problems to boost the confidence/status of particular individuals.

Another nice question for designing working groups might be to ask about working style. Something like:

(c) Rate how much each statement applies to you:
I like to work by myself.
I like to explain my thinking out loud.
I like to have someone explain their thinking to me.
I like to try things myself, but have someone close-bye to help if I get stuck.
etc.

Clearly there's a lot of ways to get at this, and I don't particularly like this version, but that's the general idea.

4. Stealing ideas through the students

One year I asked:

(a) What are things that teachers n your past have done that really helped you learn?

I've gotten all kinds of great suggestions from this one. The opposite question of what to avoid is usually less helpful as students say things like "don't be boring".

Something that has worked for me, however, is asking students at the end of the first week

(b) What about the class is "working for you" (helping you learn)
(c) What about the class isn't working for you? What could I change?

Last year, students told me that I should: "slow down", "do more examples more slowly", and "give us more chances to do examples at the board." Obviously not all suggestions will be consistent with your goals or values, but the students are telling you something that needs paying attention to. Maybe their suggestions reveal that they don't understand what you expect of them in the same way you do.

On my end-of-the-year suggestion form for students, I noticed that some of their suggestions were the same as the ones they gave me in that 1st week (ouch!). I tried, but clearly it wasn't enough. Next year I'm thinking of re-phrasing some student suggestions I want to work at and having them all rate my success every few days so I can tell if I'm getting better.

5. Actually doing something

All that said, the main challenge I have isn't asking the questions but actually doing something with the data. Without a specific plan (typically including dates) it's too easy to file the answers away until June cleaning. Two other practical issues are the sheer time involved in compiling some of the data, and finding a useful and accessible format to put it in for when I need it. That's all a little much for the denouement of a post, so I'll leave those details until next year when I actually try to implement some of these again.