Archived Information

State of the Art: Mathematics - July 1993

image omitted Students learn mathematics best when they construct their own mathematical understanding.

Ms. M.: "The African Elephant ate 37 peanuts. The Indian elephant ate 43 peanuts. How many fewer peanuts did the African elephant eat than the Indian Elephant?"

Ms. M.: "Got it? How many fewer did the African elephant eat..?"

Ubank: "Six."

Ms. M.: "Does everyone agree with that? . . . How did you figure it out, Ubank?"

Ubank: "Well, I had 43 here (pushing out 4 stacks of ten cubes and 3 additional cubes joined together), and I had 37 here (pushing out 3 stacks of ten cubes and a stack of 7). I put 30 on top of these 30. I took 3, and I put them here. There were 4 left, so I took 4 off, and there were 6 left. . ."

Ms. M.: "Did he do it a good way? . . . Did anyone do it a different way?"

Marci: "I took 37, and I needed 43. So I counted up 3 more. That was 40. Then I took 3 more to 43."

Ms. M.: "Good. Does her way work well? . . . It sure does. Did anybody do it differently?" 

                                (Carpenter and Fennema 1992, 462-463)
Students who construct their own mathematical understanding transform their mathematical potential. It takes courage to begin using the "constructivist" approach in the classroom, but the rewards can be great. Teachers often start with an experiment--a somewhat ill-defined but interesting mathematical problem or application for students to solve. They resist pleas to solve the problem for their students. They often find that their best students resist the change in teaching and learning at first--after all, the best students have succeeded in the old mode, even if they found the mathematics boring. The teachers give the experiment time to succeed.

One of the most difficult shifts for teachers is to relinquish their role of keeper of "the right answer." As students grapple with constructing their own knowledge, they may ask questions that the teacher cannot answer. They may go down mathematical paths that their teacher had not trod. They may devise algorithms that are unknown to their teacher. Teachers too need to construct their own mathematical and pedagogical knowledge. As teachers become learners they model the mathematical behavior they expect of their students.

Teachers must assume a new role if students are to construct their own mathematical understanding. Rather than just being the information givers--pouring mathematical knowledge into the student's head--teachers must provide stimulating mathematical problem situations that encourage mathematical learning. Students must change from being passive recipients to becoming active seekers of knowledge. Students must also learn to verify their own mathematical knowledge.
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[Teachers need to listen to students and to incorporate into their instruction what they learn from listening.] [Table of Contents] [Students need to learn more and different types of mathematics.]