The Real World
The Real World
The term "the real world" bothers me, especially when it is used in discussions about teaching mathematics. It suggests to me that there is some other, more authentic place I ought to be rather than the place I am, especially if that place is the mathematics classroom.
The term often appears in discussions of problem solving. A case in point is NCTM Standard 1: Mathematics as Problem Solving. I have no quibble with the Standard; it's the statements made in the accompanying rationale for the Standard that bother me. Statements like: "[Children] should encounter problems that arise from both real-world and mathematical context."; " ...a balance should be struck between problems that apply mathematics to the real world and problems that arise from the investigation of mathematical ideas."; " ...problem situations, which for younger students necessarily arise from the real world, now often spring from within mathematics itself."
These statements imply to me that there is the real world and then there is something else called mathematics. So where does that leave me, someone who has been involved with this thing called mathematics for the past half-century? Have I been marginalized? Shipped to an alien planet? Am I drifting around in dreamland? If mathematics is not part of the real world, then where is it? What other world is there? Where in the world am I?
I can understand how this otherworldly view of mathematics arises. Many would agree with Churchill who, in his autobiography My Early Life, described his experience of school mathematics as being in "an Alice-in-Wonderland world" inhabited by all sort of strange creatures like quadratic equations and sines and cosines which, when he finally completed his math requirements, "passes away like the phantasmagoria of a fevered dream" never to be encountered again. But even Churchill, as mathophobic as he was, caught a glimpse of the mathematical world that was hidden from him during most of his schooling. "A much respected Harrow master," he tells us, "convinced me that Mathematics was not a hopeless bog of nonsense, and that there were meanings and rhythms behind the comical hieroglyphics."
If mathematics seems remote to students, it's not the fault of mathematics, but rather of the way it's taught. I suspect that those who wrote the Standards would agree. But I don't think the situation is improved by suggesting that mathematics is unreal and by embroidering its occurrences outside the classroom.
Most of the problems I've seen that carry a "real-world" label are contrived for the classroom and don't really reflect what goes on elsewhere. Take, for example, the "real-world problem situation" cited in the NCTM Standards: "In a two-player game, one point is awarded at each toss of a fair coin. The player to first attain n points wins a pizza. Players A and B commence play; however, the game is interrupted at a point at which A and B have unequal scores. How should the pizza be divided fairly?" Now, just where in the world did that scenario take place?
It's not that the problem is uninteresting or not instructive. Problems on the division of gambling stakes have attracted attention for a long time. But why give the impression that anybody out there is making repeated tosses of a coin to determine who wins a pizza? Our students know that doesn't happen and to pretend that it does, besides being untruthful, feeds the very notion that one is trying to counter, namely, that school mathematics is out of touch with reality.
Also, I don't want to give the impression to my students that mathematics is not a legitimate, fascinating and accessible part of the world in its own right—as real as any other part of the world. It does have connections to many other parts of the world and it is true that exploring these connections can be a useful and informative learning experience.
I suggest we abandon our attempts to contrive "real-world" problems for the classroom and concentrate on presenting "real" problems. By that, I mean, problems that are instructive and interesting; problems that students will put energy into investigating. If they connect to other parts of the world, well and good. If they don't, that's okay too. If it helps to clarify them, cast them in non-mathematical language and relate them to students' past experiences. But, don't pretend they replicate situations one is likely to encounter outside the school world.
Very little of what goes on in the mathematics classroom is ever going to be encountered per se in the average person's life. That's not the point. The point is to develop one's mathematical competence and confidence so that whenever one does encounter a mathematical situation later in life, whatever it may be, one is willing to tackle it and has some background and knows some strategies for doing that.
The world of mathematics need not be an "Alice-in-Wonderland world" nor a morass of "comical hieroglyphics." There are "meanings and rhythms" in the world of mathematics and we can unfold those to our students. Mathematics need not be divorced from the stuff of life. One can use common experiences to lend meaning to mathematical discourse. I can, for example, talk about arithmetic progressions in terms of stair steps. But I do this not because carpenters knowingly apply a knowledge of arithmetic progressions when building a flight of stairs, but because students' familiarity with them provides useful images that help them grasp the concepts being introduced.
There is only one world, and mathematics—and the mathematics classroom—is a very real part of that world. A part of the world in which many of us spend a good deal of our time and most of our energy. Let's not deny its existence.