Tricotomania and more

ALERT: If you have come directly to the page for this post (say, from a feed reader) you are going to get the whole thing. It includes solutions to the problem. I tried to put this after a break that you would have to click through to get but that only seems to work if you read it on the main page. Click here to go there now if you aren’t sure you want the solution.

Yesterday AnimalGirl came around to tackle this problem with Tigger, which I blogged about a couple of weeks ago. I thought I’d provide some detail of how they worked on it because I know some folks are interested. First go look at the problem. The girls read through the problem and then focused on the map. They weren’t sure where to start but one of them suggested working out what all the possibilities were and going from there. This was probably the best thing to do and they worked out what all the possibilities were pretty quickly and made a decision about where the princess should go. I thought that looked like a reasonable solution but something was bugging me that I couldn’t quite put my finger on. I know that there is a solution provided on the nRICH site, so I went and got that for them. (There is a link at the top of the page I linked above.) Their solution was somewhat different from what the girls had come up with so the 3 of us tried to work out how and why they differed. In the end, they agreed that the other one was probably a better solution (and thought they should change their view of which room the princess should go in).

I pointed out that the difference between the two rooms wasn’t that great and talked a bit about how probability problems often come up with answers that require some judgement. Whichever room you choose, there is still a pretty good chance that the poor peasant is going to get eaten by the lions. This problem didn’t take them very long but they enjoyed it and it did make them think.

For those who want to know what their solution was and how we differed from the published solution, I’ve got that after the break.
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One thing I love about blogging is how much I learn. I hope none of you were under the impression that I have everything figured out. I am a extrovert decision maker, which means that I need to talk about things with others in order to figure them out. And sometimes, just the word someone uses makes a whole bunch of stuff click.

“Probabilistic thinking” is a phrase Sarah used in her comment on my last post. (She also provided a link to a cool simulation so you might want to check that out.) I had been talking about how my general goal for math is to develop the skills needed to spend time working on tough problems and at least move towards a solution. But because the stuff we’ve been working on recently has been probability her response made me recognize a reasonable objective for the probability stuff: shifting to probabilistic thinking instead of “one right answer” thinking.

This ties in really well with the physics stuff I’d been reading because so much of physics relies on this way of looking at the world. As does so much of life, as I have also pointed out.

So now I have a clearer sense of what we’re doing and why. For now, anyway.

We’ve been working on more probability problems and this has forced me to recognize what the real goal of this approach to math is: learning to work for sustained periods on tough problems. The difficulties Tigger has faced in the past few days have not been about the math, per se. They seem to have arisen from the fact that the problem I have given her to work on is not one that she should be able to find “the answer” to in 5 minutes. And that even when she has been working for 20 minutes or more and doing good valuable work that contributes to finding the answer, I come an suggest other tacks she might take with the problem.

There has been some shouting, crying, and other frustrated behaviour. But we are making progress. We’ve talked about the importance of the process. About how math isn’t necessarily about solving easy problems in large numbers and getting all the answers right. We’ve talked about the ice-cream problem, how tough it was, the wrong alleys, and how we got to the answer. Also how good it felt when we figured out that formula after all those frustrating attempts.

Yesterday we said we’d put the problem away for the day and come back to it tomorrow. When she said “We’ll finish it tomorrow.”, I corrected her and pointed out that we might not finish it but we’d work on it some more.

Today, we worked on it some more. Together. She started falling into letting me do lots of the work and that led to some more serious discussion and frustration on my part. We talked it through a bit, went over the discussion of what the goals were, etc. We switched to playing the game the problem is based on. With dad instead of me.

Tomorrow, we’ll work on it some more but we might call it quits even if it isn’t “finished”.

Her friend is going to come over sometime next week to work on the Lady or the Lions. I am so glad she has a friend who thinks that coming over to do math together sounds like fun.

In the meantime we might work on the Birthday problem*. When I mentioned that just now, she said “But we know the answer.” and I reminded her that we don’t know why that is the answer. She thought for a moment and agreed that maybe it would be a good one to work on. Maybe we are making progress, slowly.

* How many people do you need to have in a room for the probability of 2 of them having the same birthday to be 1 in 2? The answer is 23.

That ice-cream problem worked so well that I thought I should keep going with that kind of approach. We haven’t covered much probability so I thought that I would continue in that vein.

This morning we worked a bit with dice talking about the chances of rolling a particular number, rolling it 3 times in a row, rolling it once in 3 rolls, etc. We went on to talk about rolling 2 dice and the probability of different sums, something that is important strategically in some games (including Settlers of Catan, which marks the probability on the number tiles).

But this was not grabbing Tigger’s interest as much as the ice-cream problem did. So I’ve been looking for some other problems. And then I remembered to check my own links page. Duh. The NRICH site is a treasure trove of interesting problems. I did a search on probablity and then looked at a bunch of the different detailed topics. I narrowed each set of results to Key Stage 2 & 3 + Problems. You could narrow it differently depending on your kids. (Steph, you might want to look at Key Stage 1 for your younger ones.) There is an explanation of the Key Stage thing somewhere on that site for those not familiar with the UK system.

The Lady or the Lion looks like something that might appeal to Tigger. I’ve printed it along with couple of other things.

There is even an article on the history of probability.

I’m also thinking that we could do worse than just go to that site and try one of the monthly problems 3 times a week or something. Rotate between topics. Whatever. Hmmm…

If anyone else has some cool problems they want to tell me about, shout out in the comments. I’m not buying any books though.

Taking more tips from Steph, we had some fun with math yesterday. I figured it would be more fun with an extra kid so we invited a friend over. The girls had a great time. Honest.

As I have recently discovered math is really all about patterns. Figuring them out is what mathematicians do. And what I’ve learned at the Living Math forum recently is that a good thing to do in the middle school years is a lot of problem solving. Real problem solving. Stuff that might involve really puzzling things out for a long time. So the 31 Flavors problem that Steph described seemed like an ideal activity. Our local ice-cream shop isn’t that same big multi-national chain though and they have 48 flavours!

I had explained the basic problem to Tigger the other day when I suggested inviting AnimalGirl over to work on it with us (with a visit to the ice-cream shop of course). So she’d been pondering the smaller numbers for a few days. Just after lunch she got out some graph paper to work on and the two girls sat and worked through some of the problem. I let them get started and was cleaning up from lunch and making a coffee behind them. At one point I suggested a table and got them started. They worked through the table to 10 flavours but were somewhat baffled about the pattern. They were trying some different things to try to work it out.

Once my coffee was ready, I joined them at the table and we worked on it together. I noticed that one of their early attempts (failed) might actually lead us in the right direction. They had put the numbers 1 to 5 across the top of a table and down the side but then filled in the whole box. I said that I thought they might be on the right track there and set out another grid. Then I filled in the diagonal because that was all the cones that were 2 scoops of the same flavour. We used their other knowledge of the options to fill in the appropriate part of the grid.  …

I’ll leave you with that because we found that once you got to that point, the visual gave us big clues as to how to figure out the formula we needed. Our first attempt wasn’t quite right but since we knew the right answer, we could work out what we needed to do to get there, figure out why that made sense, and then test the formula on a couple of other numbers. I let them use the calculator to do multiplication with bigger numbers.

We all were really pleased when we figured it out. And that math history came back when Tigger said “The Greeks were right. Everything is about geometry.”

While I finished my coffee they found out what time the bus was going to pass and got ready. We headed down to the ice cream shop, chose ice-cream and counted how many flavours. After we ate our ice cream we got the calculator out and figured out how many choices there are if you have 48 flavours. And since the woman behind the counter said that they make 100 flavours but only have 48 out at a time, we also worked out how many choices there would be if all 100 were available. Lots of fun.

And that last bit led to a discussion of the difference between plugging numbers into a formula and doing math. I emphasized that the math part is figuring out what the formula is. This is a great problem. I highly recommend it.

The disconnect between the ubiquity of probability in our culture, especially in relation to health and illness, and the poverty of understanding of this area of mathematics strikes me as highly problematic. If one goal of eduction is to prepare our children (and ourselves) to participate in the world, then a solid understanding of probability should definitely be included.

But, as my opening statement suggests, while the world offers many opportunities to delve into the topic in relevant ways, it offers few guides to understanding. There is a lot of misinterpretation of statistics out there that a good education should equip us to recognize and counter.

Take, for example, the relationship between a particular illness and various factors that seem to be correlated with the incidence of this illness. If we exercise regularly, eat well, scorn smoking, and drink moderately should we feel cheated if we develop heart disease? Does the fact that we might die of heart disease anyway mean that none of these behaviours are worthwhile? Or, in the famous example, if one might smoke all ones life and still live to 97, does that mean that the relationship between smoking and lung cancer is all smoke and mirrors?

No. Most of this health information, and there is lots of it out there, is based on epidemiological studies. Epidemiology is the study of population health. It uses statistics to calculate the correlation between various diseases and various possible contributing factors. It does not determine cause. It determines probabilities, which can also be called risks. Just as individuals do win the lottery, despite the long odds, some smokers will live to 100. And some people who “do everything right” will get cancer.

What most of this health reporting is telling us is that there are correlations, sometimes strong correlations, between certain behaviours or characteristics and certain illnesses. Correlation is not cause. Human bodies are incredibly complex. Despite the enormous advances in scientific knowledge in the past 100 years or so, we still have very little idea of how that complexity works. Because of that, we know little of cause.

We do know that viruses cause the flu. And we know how viruses are transmitted. We only know in the most general terms why some people exposed to a virus will get sick and others will not, and why some will get sicker than others. We know that those things are correlated with factors like age, general state of health, diet, etc. And we can calculate the relative risks. But we don’t know exactly why.

So much of the current health debates seem to misrepresent these basic facts. There is a tendency to try to allocate blame to individuals for their illnesses. There is also a tendency to think that we should be able to eradicate illness. If only we all ate properly, no one would have heart attacks. The fact is that the probability you will die sometime is 100%. The part that is difficult to predict is when and how.

For those who want to learn more about probability, I recommend Chapter 2 of Nathalie Angier’s The Canon or Chapter 7 of Keith Devlin’s The Language of Mathematics. The archived radio programme on the history of probability from the BBC’s In Our Time series, is also informative. Devlin’s chapter also has a lot of history in it. There is also good list of readers for various ages at Living Math.

Willa has sent me off to read more interesting things. In addition to the points she drew out of this post at Rational Mathematics Education, I wanted to highlight what seems to me a very sound argument for the negative political consequences of the dominant mode of mathematics education.

The piece starts with a long, and very interesting, quote from Fred Goodman, in which he elaborates on the importance of games (as distinct from puzzles) in mathematics education. It is from this that Willa quoted and pondered. I note particularly his statement:

As the world moves closer and closer to a world where Gods collide and their followers depend with greater and greater certainty on the correctness of their God’s solution, we need to look more closely at the relations that might exist between games, Gods and grades. If learning is conceived primarily as a matter of finding the one correct answer according to the teacher who already knows the answer, and students’ sense of worth is tied to their ability to discover, understand and accept that correct answer, we may be encouraging, even in our secular schools, a tendency towards sectarian thinking.

As I have been reading about mathematics and physics, I am struck by the sense of uncertainty, of working towards better knowledge but of never having the “right” answer. Indeed in First You Build a Cloud, there is a whole chapter on Right and Wrong, which explains the ways that physicists see these questions. And they are very different from dogmatic approaches. She quotes physicist David Bohm:

The notion of absolute truth is shown to be in poor correspondence with the actual development of science. … Scientific truths are better regarded as relationships holding in some limited domain.

I got the same impression from the Keith Devlin book that I reviewed a few weeks ago.

Goldenberg, the author of Rational Mathematics Education, goes on the connect Goldman’s long discussion to the broader issue of democracy:

The mentality that has been used to teach mathematics to the masses in this country (and in many others) has for far too long been grounded in authoritarianism. It cannot be a coincidence that progressive-minded reformers continue to call for approaches to classroom teaching that are more student-centered and which stress communication of mathematical ideas, offering sound reasoning for mathematical answers and procedures, while anti-reformers decry this as “time-wasting,” “fuzzy,” and somehow too “touchy-feely” to matter.

and later

my concern here is for the way that subjects are taught and what the political lessons are that aren’t explicitly stated or acknowledged. And those lessons are fundamentally anti- and undemocratic. The focus upon single right answers that are arrived at by (generally) one approved method speaks volumes towards the underlying values of the teacher, the school, the district, right on up through the state and federal governments. The job of students becomes not learning and thinking, but anticipating what teachers expect exactly as they expect it: no less, and generally no more. And therein lie a host of tragedies, even were there not the anti-democratic issues to consider.

I cannot do justice to the argument with excerpts. The whole piece, basically a long quote from Goodman followed by further discussion by Goldberg, is excellent and raises many important points. As many homeschoolers already know, the idea that learning comes in neatly divided boxes labeled “mathematics”, “civics”, “language”, etc is a fiction. What this article nicely points out is that it is a dangerous fiction.

In the frontispiece of First You Build A Cloud, I find this quote:

Newton himself, as well as those … who attacked him … would have all alike been amazed at the more recent contention that natural science has nothing to do with “values,” that it can and should itself remain “value-free,” and that those seeking a direction for human life have nothing to learn from our best knowledge of the nature of things. Even a little science … is a thing of infinite promise for human values. (John Herman Randall, Jr., Newton’s Philosophy of Nature)

Whatever our values, we need to be aware of how the methods we use to teach are promoting or undermining them.

That article by Lockhart (referred to in a previous post of mine) was linked from Keith Devlin’s page at the MAA site. I ended up surfing through his previous articles. There is a good pairing on what mathematics education should be, not at the elite level (preparing future mathematicians) but the general level (ensuring that we all have basic literacy). The main outline can be found here, but it refers to an earlier article here. That article led me to Devlin’s course at Stanford (outline and philosophy are both worth reading and can be downloaded from that link). Which, in turn, led me to his book, The Language of Mathematics. As luck would have it, it was in my public library.

As you can see it has taken me a couple of months to read it but I finished it last night. It was hard going sometimes and I can’t claim to have understood everything, but I think that I now have a better sense of what mathematics is about than I did before. And some areas were truly fascinating and really advanced my own knowledge.

A couple of summers ago, I did some work on the nature of research in the humanities disciplines. It was very interesting. And I have felt for a long time, though without a lot of knowledge, that mathematics is really more like the humanities than like the sciences. Although mathematics is very useful to scientists, mathematicians don’t really approach the world in the same way that scientists do. They are more like philosophers. Reading The Language of Mathematics confirmed this view for me. Mathematicians are a lot like philosophers. Reality does not concern them very much. Abstraction is very important to them. And finding abstract patterns really excites them. Mathematicians put a lot of value on elegance. And simplicity.

An illustration of the importance of beauty and aesthetics to mathematicians can be found in chapter 3 of Devlin’s book. Don’t worry if you don’t quite understand what he is talking about her, I’m not sure I do completely either. My point in quoting it is to highlight the use of aesthetics as an argument for the acceptance of something.

With the gradual increase in the use of complex numbers spurred on by the obvious power of the fundamental theorem of algebra and the elegance of Euler’s formula, complex numbers began their path toward acceptance as bona fide numbers. That finally occurred in the middle of the nineteenth century, when Cauchy and others started to extend the methods of the differential and integral calculus to include the complex numbers. Their theory of differentiation and integration of complex functions turned out to be so elegant — far more than in the real-number case — that on aesthetic grounds alone, it was, finally, impossible to resist any longer the admission of the complex numbers as fully paid-up members of the mathematical club. Provided it is correct, mathematicians never turn their backs on beautiful mathematics, even if it flies in the face of all their past experience. (page 135)

Throughout, Devlin is quite clear that mathematicians are not directly concerned with reality. In the last chapter he spells this out quite clearly when talking about the nature of light.

In what sense is this rapidly moving entity a wave? Strictly speaking, what the mathematics gives you is simply a mathematical function — a solution to Maxwell’s equations. It is, however, the same kind of function that arises when you study, say, wave motion in a gas or a liquid. Thus, it makes perfect mathematical sense to refer to it as a wave. But remember, when we are working with Maxwell’s equations, we are working in a Galilean mathematical world of our own creation. The relationships between the different mathematical entities in our equations will (if we have set things up correctly) correspond extremely well to the corresponding features of the real-world phenomenon we are trying to study. Thus, our mathematics will give us what might turn out to be an extremely useful description — but it will not provide us with a true explanation. (Devlin, page 311-312; emphasis mine)

In some ways this is quite freeing. What we are being introduced to in this book is a new way of thinking; a way of viewing the world through mathematicians eyes. At times this kind of makes your head hurt a bit. And if you really wanted to understand everything he was saying, you would get frustrated very quickly. But if you are willing to let some of it flow over you a bit in an attempt to grasp something of that way of seeing, then it is well worth a read. I suspect that if one was really interested, this book would take several readings. A bit like philosophy really.

Indeed, as noted above, I discovered The Language of Mathematics when investigating Keith Devlin and what he did. He uses this book as the core text for a course he teaches. And it might be the kind of thing that is better taken slowly, over 3 months, pondering each chapter and engaging in actual mathematical work as a means to develop understanding. I’m not sure how easy that would be to do without the guidance of a professor, the structure provided by a course, and the opportunity for discussion in a seminar. But I’m sure it would be worthwhile.

However, for those of us who do not have the time or energy right now to take on that kid of commitment, The Language of Mathematics is still useful book. Individual chapters would be worth reading to get a sense of the bigger picture in some specific field of mathematics. The Prologue gives an overview of the discipline. Chapter 1 “Why Numbers Count”, opens our minds to new ways of thinking of numbers and what we can do with them. Chapter 2 ” Patterns of the Mind” introduces us to mathematical logic and proof, the “discipline” of mathematics not in the sense of “a branch of knowledge” but in the sense of “the practice of training people to obey rules or a code of behaviour” (OED). (These two senses of the term are not unrelated.)

Chapter 3 “Mathematics of Motion” gave me a much better sense of the calculus and what it is trying to do. It is interesting that this topic comes so early in the book and perhaps that has had some influence on my plan to introduce it earlier in Tigger’s education than is usually the case in the school scope and sequence. It is followed by “Mathematics gets into Shape” (Chapter 4) a fascinating discussion of geometry that begins with Euclid and moves on to demonstrate why Euclidean geometry, though terribly useful in everyday life, is not actually the geometry of the “real world”. The 3 angles of a triangle only add up to precisely 180 degrees in the imaginary Euclidean plane. We live on a sphere. For most everyday purposes, even of scientists, the piece of the sphere we live on is so close to a plane that we can ignore this fact and work as if it were in fact a plane. But if we were flying an airplane, the difference makes a difference.

Now if you are the kind of person who likes things to be True and can’t see the point of studying anything that is only approaching the truth or approximating the truth, this might be terribly disturbing. For someone like me who has long embraced a sort of epistemological uncertainty, this just brought mathematics and the sciences into the same set of discussions that the humanities and social sciences have been grappling with in a different room. That’s a lot of big words, sorry. “Epistemology” is just the study of how we know what we know. So the central issue here is not “is there a Truth?” (big T) but “can we know it?”. I have become less concerned with the Truth, and more at ease with approaching it, approximating it, and generally learning. This book takes very much that approach, as the quotes above demonstrate.

Back to the chapters… “The Mathematics of Beauty” (Chapter 5) deals with symmetry, tiling the plane, sphere packing and related issues, introducing the mathematical concept of the “group”. Chapter 6, “What Happens when Mathematics Gets into Position”, introduces topology beginning with the very useful, but not at all to scale, London Underground Map and leading on to the concept of “networks” and some very funky ideas about n-dimensional universes. (I did warn you that mathematicians are not concerned with reality.) “How Mathematicians Figure the Odds” (Chapter 7) is obviously about statistics but introduces some interesting connections among gambling, insurance, and global finance.

Devlin closes with a discussion of light and the universe, “Uncovering the Hidden Patterns of the Universe”, that includes a very simple explanation of Einstein’s theories of relativity that makes perfect sense. (Or maybe it only makes perfect sense once you have read the whole book and come to terms with epistemological uncertainty.) Approaching the problem of the nature of light from a different angle, Einstein developed his theory of special relativity

Building on Lorentz’s theory, Einstein went one significant step further. He abandoned the idea of a stationary ether altogether, and simply declared that all motion is relative. According to Einstein, there is no preferred frame of reference.

The theory of general relativity is an extension of this. I think now is a good time to go back and look at the Einstein exhibition the American Museum of Natural History put together (which I saw when it toured here). As I recall there are some very good illustrations of this principle.

Of course that gets mathematicians back into n-dimensional universes, and the shape of them. That whole curvature of space-time notion is a bit weird from our place on a sphere that looks very like a plane most of the time. But at the end of a book which has thoroughly demolished any idea you might have had that mathematics is some sensible discipline about the real world, it is quite fascinating. Which is the point, I think. Utility, especially immediate utility, limits the pursuit of knowledge so much. Many of the weirder things that Devlin talks about do eventually have some utility but sometimes it takes hundreds of years and several mathematicians to develop these ideas to a point where they can relate to some problem in the “real world”. That isn’t what motivates them to keep working on the ideas though. Mostly, mathematicians keep working on these problems because they are fascinating.

And this brings me back to where I started, wondering about how we teach mathematics. Reading Devlin has opened my mind to a different way of seeing the whole subject. And has confirmed that the objective of mathematics teaching should be to fascinate our students. We must also discipline them: train them to obey the rules or code of behaviour of mathematics as a branch of knowledge. But part of that is just giving them the tools to understand what mathematicians are doing.

And thus even if you can’t face reading The Language of Mathematics, I would strongly urge you to read Devlin’s thoughts on the goals of a mathematics education. Many of them seem perfectly in line with the kinds of things that homeschoolers discuss. In fact, his first goal, the one he says is least common, seems to be a primary goal of the Living Math crowd.

By the way, we should be teaching our children that the rules and codes of behaviour are different in different disciplines. That’s what distinguishes the disciplines as branches of knowledge. The point of learning how to write an elegant mathematical proof is not that this is a “transferable skill” but that this is the way that mathematicians go about demonstrating the truth of something. Although some of the logical principles are also used in other disicplines, practitioners of those disciplines will use them differently, being concerned with a different kind of elegance, perhaps.

On that recurring theme, someone on the LivingMath Forum posted a link to a talk by Ron Eglash on fractals in the design of African villages and other aspects of African cultures. Very interesting. I might have to go find some of his books. There is an article available online about the origins of binary code in African patterns, something he talks a bit about in the video.

He also has a website with tools to create designs using African fractals. And one specifically designed for kids as well as a set of tools specifically designed to relate to specific cultures. Hmmm. Lots of possibilities.

This post is basically a place-holder for me. I came across a recommendation for Calculus without Tears. It looks like it might be a good resource for us. Tigger is looking forward to the Calculus chapter in Challenge Math and likes the idea that her dad never learned it. And this book is aimed at kids her age, with an approach that is related to physics. I like Maria’s reviews and she was positive about it. So this might be a good buy given that we are planning on using physics to do math in the fall.