I am creating a new category of posts about mathematical tasks aimed at developing teachers’ math knowledge for teaching. Most of the tasks I will present here have been used in studies about teaching and teacher learning. Mathematical knowledge for teaching was coined by J. Boaler based on what Shulman (1986) call pedagogical content knowledge (PCK) or subject-matter knowledge for teaching. I know this is a blog and not a discussion forum but with the comment section at the bottom of the post, there’s nothing that should prevent the readers from answering the questions and giving their thoughts about the task. Your thoughts and sharing will help enrich knowledge for teaching the math concepts involve in the task.

The following task was originally given to teachers to explore teachers beliefs to sufficiency of a visual argument.

###### The task:

Year 12 students, specializing in mathematics, were given the following question:

Examine whether the line y = 2 is tangent to the graph of the function , where

Two students responded as follows:

Student A: I will find the common point between the line and the graph and solving the system

The common point is A(0,2). The line is tangent of the graph at point A because they have only one common point (which is A).’

Student B: The line is not tangent to the graph because, even though they have one common

point, the line cuts across the graph, as we can see in the figure.

Questions:

a. In your view what is the aim of the above exercise? (Why would a teacher give the problem to students?)

b. How do you interpret the choices made by each of the students in their responses above?

c. What feedback would you give to each of the students above with regard to their response to the exercise?

Source: Teacher Beliefs and the Didactic Contract on Visualisation by Irene Biza, Elena Nardi, Theodossios Zachariades.

a) It looks like an exploration of student conceptions of tangency prior to the introduction of the derivative.

b) They each reflect a particular property of tangents to circles which does not match the general definition

c) I would suggest that Student A consider the case of y=3 and Student B think about the case of f(x)=x^3-x^2+2

Is that not a good point to introduce slopes and derivatives?