What Is Mathematical Thinking?
Date: 03/08/2005 at 09:57:40 From: Penny Subject: Can all thinking be considered mathematical Can all thinking be considered mathematical? For example, I have found that there is no such thing as mathematical thinking because you are essentially just thinking about mathematics, not the entity of mathematical thinking--as this does not exist. That being said, can we reverse the tables and say that all thinking is mathematical? I'm trying to get my head around the entity of mathematical thinking not really meaning anything but thinking about mathematics. There is no such thing as thinking mathematically. As my opening title affirms, this question explores what it means to think mathematically. In addition, these questions will ask the question if you are unsure of what mathematical thinking is, how will you ensure that your students are meeting the goals of the mathematics classroom (i.e. engaging in rich mathematical thinking)? Below, I have attempted to explain the position that Andrea Pitt, the author of ‘Mathematical Thinking?’ has taken while also researching that of the National Council of Teachers of Mathematics (NCTM) on their respective views of this matter. When reading Pitt’s article it was clearly evident that she feels there is no such thing as mathematical thinking. In fact, she believes that people use the entity too loosely, when what they truly mean is that they are thinking about mathematics and not thinking mathematically, because this does not exist. (We will explore how she is able to think about mathematics with her students near the end of this paper as we first need to understand the concept of what mathematical thinking is truly about). This had me wondering what I believe, and whether the NCTM feels the same way as Pitt. After all, is this really a fictitious statement? Is there truly no such thing as mathematical thinking? I turned to the NCTM documents to help justify Pitt’s allegations. When searching for a definition of mathematical thinking from NCTM, I found inconclusive, indirect statements of what it means. Nowhere was I able to find a true definition of what the NCTM believes that mathematical thinking means. Based on Principles and Standards for School Mathematics, published by the NCTM, mathematical thinking involves connections in order to construct some sort of mathematical understanding. Learning to think mathematically comes about when you have a mathematical standpoint and apply it to a given situation. It also means developing mathematical capabilities using the tools necessary in the mathematics classroom. They also state that problem solving requires thinking about mathematics as you need basic knowledge of mathematics to solve problems. From the NCTM findings they use words such as reasoning, problem solving, communicating, and connections when referring to mathematical thinking. These words are all encompassing proving the definition to be very omnipresent. These words do not signify a true meaning of mathematical thinking, only what should be incorporated in that line of thought. To go along with Pitt’s view, the NCTM’s outlook of mathematical thinking is so broad that the definition could be applied to all other subjects in school and in life in general. For instance, Pitt states in her article that some mathematicians, such as Dick Tahta, define mathematical teaching as "...going beyond the information given or trying to find a pattern..." but can you not apply this statement to physics, or chemistry, or any other subject matter for that matter? The NCTM derived a list of methods for encouraging mathematical thinking through such techniques as asking non-leading questions, paraphrasing student’s answers to aid him/her at thinking about what was just said, using wait time to allow sufficient time for the students to think through and explain their reasoning. The NCTM believes that these procedures can help students to think mathematically. In comparison, Pitt alludes that these procedures are methods of thinking about mathematics, but that does not necessarily mean that you are thinking mathematically. With that being said, Pitt has come to the conclusion that there is no such thing as thinking mathematically and encourages her students to think about mathematics in her classroom through such techniques as those the NCTM has listed in the above paragraph. In particular, the NCTM believes in the use of reflective questions as these questions can help pin point areas of the lesson that are important. Using these questions forces the students to explain math, not to just answer the problems at hand without thinking about them. In this way, it is encouraging all students, at any level, to think about the mathematics, which is the aim of Pitt’s classroom. As a result, effective discourse is the basis for promoting students to think about mathematics. One concluding statement that Pitt has disagreed with is that we could say all thinking is mathematical; instead of the reverse, having no such thing as mathematical thinking. She believes that mathematical thinking is the same sort of thinking for all subject areas and that it is too broad to be solely included in the mathematics domain, but she does not believe that all thinking is mathematical. I am not sure if I believe this or not...
Date: 03/08/2005 at 14:00:06 From: Doctor Edwin Subject: Re: Can all thinking be considered mathematical Hi, Penny. Nice question. You've obviously been thinking about it (mathematically or otherwise) for some time. I will throw in something to suggest that there is in fact something you might call mathematical thinking--a well-known example from cognitive science. If I say to you, "All Kpelle men are rice farmers. Steve is not a rice farmer. Is Steve Kpelle?" You will instantly answer, "No, because otherwise he would be a rice farmer." But when the question was asked of Kpelle rice farmers, a variety of answers came out, mostly along the lines of "How should I know? I've never met him." Some answers were more interesting, like, "If you have never even heard of someone, you can't answer questions about what they do for a living," or, "With a name like Steve, probably not." This phenomenon is not confined to the Kpelle. The experiment has been repeated in many places where schooling is rare. Check out this answer: "Everyone in the village who owns a house pays a house tax. Alfonse doesn't pay a house tax. Does Alfonse own a house?" "I would say that if he can't afford to pay the house tax, he probably can't afford to own a house." Some people were able to answer these questions the way we expect, and some were not. As I alluded to above, the difference was schooling. As little as three months (!) of schooling would grant the ability to answer these kinds of questions correctly. I think that the essence of mathematical thinking (okay, maybe just AN essence) is the ability to lift the abstract structure of a situation away from the specifics and answer it based on that alone. I don't have to figure out two people times three meals, because I already know what two times three is. There is a formal reasoning (that is, reasoning about the form) that is apparently inculcated in schools, among other places. What were they being taught during those three months? Certainly not logic puzzles, syllogisms, or what have you. Probably some mathematics, probably some writing. Whatever it was, the teachers were certainly teaching something else in addition to whatever they thought they were teaching. I would suggest that it was a type of mathematical reasoning. It may be hard for us to see it as separate from other types of reasoning. I am reminded of the saying, "I don't know who discovered water, but I'm sure it wasn't a fish." If mathematical reasoning is everywhere we look, and especially if it's part of what we look WITH, it may be hard to notice. I'm going to leave this question in the "incoming questions" bin, in the hope that other doctors will have more to add. - Doctor Edwin, The Math Forum http://mathforum.org/dr.math/
Date: 03/08/2005 at 14:23:32 From: Doctor Ian Subject: Re: Can all thinking be considered mathematical Hi Penny, >For example, I have found that there is no such thing as >mathematical thinking because you are essentially just thinking >about mathematics, not the entity of mathematical thinking--as this >does not exist. I would disagree. I think there is very much a kind of thinking-- which can be used to think _about_ anything at all--that is "mathematical" in nature. And it differs radically from other kinds of thinking. >That being said, can we reverse the tables and say >that all thinking is mathematical? The world would be a very different place if that were true. >Trying to get my head around the entity of mathematical thinking not >really meaning anything but thinking about mathematics. There is no >such thing as thinking mathematically. See http://mathforum.org/library/drmath/view/52370.html >When reading Pitt’s article it was clearly evident that she feels >there is no such thing as mathematical thinking. Maybe she's just never experienced it. I might make the same kind of claim about religious ecstasy, because I've never experienced that. But a lot of others, who _have_ experienced it, would disagree. >When searching for a definition of mathematical thinking from NCTM, >I found inconclusive, indirect statements of what ‘it’ means. No >where was I able to find a true definition of what the NCTM believes >that mathematical thinking means. Perhaps their purpose for using the term is something other than providing a clear definition. In fact, perhaps providing a clear definition would actually run _counter_ to their purpose. >Based on Principles and Standards for School Mathematics, published >by the NCTM, mathematical thinking involves connections in order to >construct some sort of mathematical understanding. Learning to >think mathematically comes about when you have a mathematical >standpoint and apply it to a given situation. It also means >developing mathematical capabilities using the tools necessary in >the mathematics classroom. They also state that problem solving >requires thinking about mathematics as you need basic knowledge of >mathematics to solve problems. That sounds pretty muddled, doesn't it? It also sounds untrue. People solve problems all the time without using mathematics. ("The Smiths are coming over for dinner tomorrow. What should we serve? Mrs. Smith is allergic to seafood, and last time they were here, Johnny wouldn't touch the chicken we served...") Having said that, given any particular problem, there is definitely a mathematical way to approach it, which doesn't necessarily mean reducing it to sets of equations, or even using what we normally think of as "mathematics" to solve it. The mathematical approach, as illustrated in the joke I cited earlier, is to reduce the problem before you to one or more simpler problems, and to keep doing that, over and over, until you end up with a set of trivial problems, i.e., problem whose answers you already know because you've already solved them in the past. Contrast this with the many other ways that you might go about trying to solve a problem, e.g., looking up the answer, asking someone else to solve it for you, trial and error, guessing, prayer, solving a different problem instead, and so on. These are _not_ mathematical ways of approaching the problem. We joke about this all the time in my family, because I'm usually looking for the "mathematician's solution" in any given situation. For example, say we need to go pick up the dry cleaning, and we also need to go drop off a movie that we rented. My wife would get out a map and try to find a route _from_ the dry cleaner's _to_ the movie place; but I would drive to the first one, then drive back home, because now the second problem (getting to the movie place) is one that I already know how to solve. It's more wasteful in terms of time and fuel; but it's more economical in terms of allowing me to use my limited cognitive resources for things I find more interesting than figuring out how to get from point B to point C. And it is, I believe, a distinct way of looking at the world. There's another very nice story about Enrico Fermi, which I think illustrates the difference between mathematical thinking and other kinds of thinking. One evening Fermi was watching his wife knit, and suddenly he got up and went to his study. An hour later, he came back and announced that, topologically, there were only two ways to tie the kinds of knots she was doing, and showed her the other would work. "I know, dear," she said. "The other one is called perling." Now, what makes his approach mathematical isn't that he _used_ "mathematics", i.e., topology, to investigate the problem. What made his approach mathematical is that it occurred to him in the first place that there might be some way to reduce what his wife was doing to some axiomatic description, and then use that description to consider _all_ the ways that it might be done, all at once, instead of one at a time. _That_ is mathematical thinking, and it's not really very much like any other kind of thinking at all: http://mathforum.org/library/drmath/view/52350.html So to sum up, the way it looks to me is that "mathematical thinking" is primarily a very sophisticated kind of laziness, which can be learned. When confronted with a problem, the mathematician tries to avoid having to do any more work than absolutely necessary by going in one of two directions: down (breaking the problem into smaller problems that have already been solved), or up (abstracting the problem so that he can solve not just the one in front of him, but a whole bunch of others that might arise in the future). Note that to learn this kind of thinking, you don't necessarily have to work with what we commonly think of as "mathematics". But it's a lot easier, because mathematics provides an imaginary world in which it's possible to control, with great precision, the amount of complexity that a particular problem involves. By abstracting out the millions of details that arise in any _real_ situation, we can give students the luxury of focusing on the task at hand, i.e., learning to transform problems into other problems, and to look for ways to reduce sets of individual rules or formulas to more general ones, e.g., http://mathforum.org/library/drmath/view/54685.html Now, whether this kind of thinking is being taught in classrooms is another story. I think it's not, largely because nearly every math teacher I've ever explained this view to has responded in the same way: first, by saying "That makes perfect sense"; and second, by asking "Why didn't anyone ever explain that to me before?" I hope this helps. Write back if you'd like to talk more about this, or anything else. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
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