What does it mean to “do mathematics” and how do we get our students to truly “do mathematics?” Is doing mathematics all about following a well-known procedure that leads to the right answer? Or is doing mathematics all about makings sense of big ideas that can be used to solve problems? Teaching and learning mathematics for understanding is undoubtedly the goal of most mathematics educators. Many teachers chose to become mathematics educators because they felt that they could help students succeed in learning mathematics. It is implied that this learning would be with understanding. Unfortunately, learning mathematics without understanding has long been a common outcome of school mathematics instruction.
Paul Lockhart, author of “A Mathematician’s Lament”, asserts that “mathematics is about problems, and problems must be made the focus of a student’s mathematical life. Painfully and creatively frustrating as it may be, students and their teachers should at all times be engaged in the process of doing mathematics — having ideas, not having ideas, discovering patterns, making conjectures, constructing examples and counterexamples, devising arguments, and critiquing each other’s work.”
Often, however, students view mathematics as being about remembering facts and following procedures that make no sense. “Good” students remember what to do and when to do it and the “bad” students do not. Does this make the “good” students good mathematicians or good at trivial pursuit? For example, “the area of a triangle is equal to one-half its base times its height.” Lockhart shares that “students are asked to memorize this formula and then “apply” it over and over in the “exercises.” Gone is the thrill, the joy, even the pain and frustration of the creative act of deriving and making sense of this idea!” Rather, we give students with problems that usually have 2 numbers given. The formula (one-half times base times height) also has 2 places to plug in these numbers — how convenient! Students can do an entire page of homework problems related to area of triangles without actually knowing anything about the area of a triangle! I paint the worst case scenario here but unfortunately, this is the status quo in many cases.
Often, we fall into the pattern of teaching in the same ways that we were taught. We ask if students have questions on last night’s homework. We show them how to do the next procedure. We assign the next batch of homework. This has not worked, generally, to create more enthusiastic and proficient learners. This has not worked to develop students who are persistent problem solvers and able to reason mathematically.
We would love to engage in a discussion here on this blog about the status of mathematics education in America. What do you think of this posting? Is it accurate? What can we do to overcome this traditional view/beliefs/status of mathematics education?
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