There is a video of Prof. Georg Polya, giving a lecture to a group of students. Polya was a mathematics professor at Stanford, known best perhaps for his seminal work on problem solving called “How To Solve It”. In the 60’s, the Mathematical Association of America filmed a lecture of his on the role that guessing plays in mathematics.
It is worthwhile watching the video (linked to above). Polya is very charming and gives a good lecture. Unfortunately, the lecture tends to get misinterpreted.
In the past, I have offered criticism of the trend in math ed to give students a type of problem called the “growing tile” problem. Students as young as fifth graders are given these type of problems in which they attempt to find the pattern that describes the sequence of a growing number of tiles or dots. These types of problems are given supposedly to develop the “habit of mind” to see a pattern and then to represent it mathematically.
Such problems are reliant on intuition — i.e., the student must be able to recognize a mathematical pattern using inductive reasoning. There is nothing wrong with inductive reasoning, or guessing what a pattern is. In fact, as Polya points out, it is an essential part of solving problems in mathematics. He poses a problem in his lecture and asks his students to guess what the next number (in this case, the number of spaces created by a specific number of planes). He then, at about 25:50 in the video, asks his students if the guesses they have generated prove anything. The answer from the students, and echoed by Polya, is “No.” He goes on to say “Don’t believe in all your guesses.” And in fact, the problem he poses in the video does not conform to the pattern of guesses that seems so apparent.
This is the part that sometimes gets left out of the “habit of mind” and “growing tiles” approach that is popular in various math programs (CPM being one of many) and advocated by Jo Boaler and others. While some problems conform to the inductive conjectures that students produce, other problems such as Polya’s do not. For example, this problem, taken from Moise-Downs “Geometry” textbook, is from a set of problems to reinforce the idea that intuition often does not coincide with deductive reasoning.
In the above problem, although 32 is a natural guess from observing the pattern, it is not correct. There will never be 32 regions. In general, there will be 31 regions formed. If the six points are equally spaced around the circle there will be 30 regions formed.
Presenting growing tile problems in early grades is more of an IQ test than fostering any kind of mathematical thinking. But more important than that is the unintended habit of mind that springs from such inductive type reasoning. Specifically, students learn the habit of jumping to conclusions based on inductive responses. This develops a habit of mind in which once a person thinks they have the pattern, then there is nothing further to be done. Such thinking becomes a hindrance later when working on more complex problems.
Polya’s lecture has become a kind of poster child for this type of “math is about patterns” type of thinking. Polya says guessing and patterns are important for sure; he devotes a whole lecture to this topic. But he also says you ultimately have to prove your guess. That’s the part that doesn’t get played up too much by the “math is all about patterns” crowd.
6 thoughts on “Guesses, Induction and Math is All About Patterns, Dept.”
I have mainly come across such problems as part of an approach that is meant to result in an understanding of (linear) functions. I don’t like them used in this way because they are discrete rather than continuous and the y-intercept becomes confusing. I much prefer the example of a taxi fare because although not strictly continuous, it is closer to what I want the students to understand. The problems don’t seem to represent much that is useful as standalones.
I remember a colleague who told me about the event that made up his mind for him to pursue the study of mathematics: He was sitting in a math class and the teacher presented something, and the students were to say what they thought the actual situation was. Then a piece of mathematics was done, deductively, and it turned out that everyone’s guess was wrong. He suddenly realised that mathematics is a form of “sight” that allows you to see the hidden world of truth that is inaccessible to the intuition — and continually fools it. And he realised that he wanted to develop that kind of “seeing”. Indeed, I think this is correct; mathematics is like a 6th sense. Like intuition, it operates deep in the mind rather than via a physical organ like sight, taste or hearing. But unlike intuition, when properly used, it provides the closest thing we can get to unassailable verity concerning the world to which it applies.
“That’s the part that doesn’t get played up too much by the “math is all about patterns” crowd.”
True, but vastly understated. I would replace “too much” by “AT ALL”. In fact, many of the proponents of the approach can’t even do a good mathematical induction proof of the obvious much less one of these, I’d like to see Jo Boaler attempt this “regions” one, maybe even a nonlinear but trivial one would be enlightening.
For all the genuflecting to the memory of George Polya, he was only talking about students with special mathematical ability. A good historical clue was his refusal to participate in the seminal meeting (or anything thereafter) for the New Math that was held at Stanford itself and there was plenty of badgering to try to get him to change his mind. He was convinced that an ordinary classroom of math students in the hands of (“under the guidance of” for ideological purity) an ordinary math teacher would lead to failure. Exactly what happened where it was initiated at all (at lots of them it was not, including my Proviso East High School near Chicago where I was teaching at the time). It was a great opportunity for a few bright students with a strong teacher but doomed to fail across the spectrum.
Ben Blum-Smith assembled a list of these “patterns that break” several years ago. Useful for helping students see the value in deduction over induction.
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I see no transference of inductive practice to other problems in a general sense. I’ve had many teachers who tried to lead students down the inductive (and deductive) primrose path with clues. The goal was to produce some sort of light bulb or discovery effect for a general concept. It rarely worked. Besides, each example in a new area showed no improved deductive skill. Success was more associated with developed content knowledge and skills. And as an in-class group process, it’s more of a failure because most students go along for the ride. If a few students discover something, then that makes the teacher feel good about the process. The other students are left with engagement (or not) and direct teaching by other students.
In detailed preparation for tests like the SAT I and II and the AMC, the goal is not to develop some sort of generalized inductive or deductive reasoning skill. The goal is to practice everything possible – to do all of the historical problems possible. It’s not induction or deduction that gets results, but practice and remembering the problems and tricks. You will more easily see that hidden radius or remember the special form or use of a logarithm with practice on real problems (cough – homework set). You will never get there using induction or deduction if your starting point is 5 steps away. I have a few colleagues who like to show-off some sort of deductive light bulb genius to the naïve when discussing a new problem. I know that it’s simply because they’ve seen it (or something close) before. Dick Feynman liked to play that game.
Starting at the top with deductive reasoning depends on content knowledge and skills, but most K-12 pedagogues like to use a top-down process as the path to justify and master content and skills. That doesn’t happen, so the parents of the best students have to do the job at home and with tutors. Starting at the bottom by ensuring content knowledge and skills (for all) does not preclude projects or developing other generalized problem solving skills, like my favorite one – let the math do the thinking for you.
Of course, all of this has little to do with giving students what they really need – a curriculum that ensures basic skills on a grade-by-grade basis (you know, the skills portion of the balance they love so much) and one that leads to a proper algebra I class in 8th grade. There can be no “deep” understanding without mastery of skills. No amount of inductive or deductive practice and engagement will fix that.
Reblogged this on The Echo Chamber.