Using mathematical misconceptions in discussion
Often teachers encounter student misconceptions during class time. The urge to immediately correct the student or to ask another student for the correct answer is very strong, but we can miss opportunities to probe our students' thinking and to help them understand the mathematics if we do this. Use the prompts below to consider the variety of responses and questions you can provide to help class discussions grow. For each prompt....
 What questions would you ask the student to clarify what s/he is thinking?
 How might a typical student respond to your questions?
 What misconception are they likely working from?
 What sorts of questions and comments could you provide to help the student reconsider the problem as well as their answer?
 How might you engage other students?
 How do your responses compare with your mentor's?
Prompt #1
Students are working at their desks. The teacher is moving through the room observing how the students are progressing. S/he stops at one student's desk and, pointing at his work, asks:
T: Why do you say that 143 is prime?
S: Because 2, 3, 4, 5, 6, 7, 8, and 9 don't go into it.
Prompt #2
You ask Janelle to share her solution with the class at the overhead.
The problem was: Expand (x + y) 3. She provides the solution: x3 + y3 .
Prompt #3
While walking around the room checking student progress on the Pythagorean Theorem, you stop at a group and see this on their paper:
a2 + b2 = c2
92 + 122 = c2
so, c = 21
Prompt #4
At another group you see:
a2 + b2 = c2
92 + 122 = c2
so 225 = c
Teacher: Don't you need to take the square root?
Student: Oh, you mean divide by 2?
