Download Errors in the classroom: social, sociomathematical, and didactical

IMAGES OF FUNCTIONS DEFINED IN PIECES: THE CASE
OF ‘NON-INFLECTION’-INFLECTION POINTS
Regina Ovodenko*and Pessia Tsamir**
*Centre for Educational Technology (CET)
**Tel Aviv University
Teachers are continuously encouraged to conduct class discussions where students
present their solutions, raise assumptions, and evaluate each others’ suggestions (e.g.,
NCTM, 2000). Clearly, the ability to conduct such lessons is dependent on the
teachers’ knowledge regarding the validity of students’ suggested ideas (SMK), as
well as on the teachers’ ability to pose challenging follow-up problems (PCK).
Research findings, however, indicate that prospective teachers’ SMK and PCK
related to various mathematical topics are not always satisfactory and hence, there is
a call to promote this knowledge. Here we focus on an activity aimed at promoting
prospective teachers’ SMK and PCK regarding functions-in-pieces. Our questions
were: (1) what are the prospective teachers’ concept images of functions-in-pieces
with reference to the notions “continuity”, “differentiability”, “extreme points” and
“inflection points”? (2) What follow-up tasks may promote prospective teachers’
related knowledge? and (3) Which of these tasks do the prospective teachers regard
as “good tasks” and why?
We investigated 23 prospective secondary school mathematics teachers’ conceptions
at Tel Aviv University. They were asked to solve the task:
Look at the function f ( x)  tgx, for x  0 and answer the questions:
2
A(0, 0)  f(x)
x , for x  0
1. In your opinion, is the function f(x) continuous at A? Explain.
2. In your opinion, is the function f(x) differentiable at A? Explain.
3. In your opinion, is A an extreme point? Explain.
4. In your opinion, is A an inflection point? Explain.
and to suggest some follow-up questions. The participants’ solutions reflected their
views of functions-in-pieces, and their grasp of the notions “continuity”,
“differentiability”, “extreme points” and “inflection points”. For example,
prospective teachers claimed that “the function has a ‘non-inflection’, inflection point
at x = 0, because on the one hand it changes from concave down to concave up, and
on the other hand it’s not differentiable at zero”. This led to a discussion about
different mathematical definitions to a concept and to the examination of the
equivalency of these definitions. A number of follow-up tasks were suggested and
discussed from a mathematical and from a didactical point of view. This type of
activities (e.g., Tsamir & Ovodenko, 2005) seem to be valuable in teacher education.
References
National Council of Teachers of Mathematics (2000). Principles and standards for school
mathematics, National Council of Teachers of Mathematics, Reston, VA.
Tsamir, P., & Ovodenko, R. (2005). “Erroneous tasks”: prospective teachers’ solutions
and didactical views. Paper presented at PME 29, Melbourne, Australia.