# Results

The results from the first phase of the study, the questionnaire, support previous findings that many teachers have difficulties with deciding what a valid mathematical argument is. When asked to comment on a student’s false conjecture about a connection between the perimeter and area of a rectangle, 26 of the participants provided an acceptable answer and one did not respond to the item. 30 participants accepted the student’s claim without questioning its empirical base.

Fifty-four participants responded to another questionnaire task in which they were to explain why the sum of two odd numbers is even. Twelve provided an acceptable or almost acceptable answer, some of them phrasing it with little use of mathematical symbolism (e.g. “an odd number has a remainder of 1 if divided by 2. Two odd numbers with remainder 1 that are added have the remainder 2 —> no remainder”). Among the unacceptable answers, there are 11 empirical arguments (e.g. “One can see this, if you just try a sufficient number of times”; “1 + 1 = 2, 3 + 1 = 4, 1 + 5 = 6, 1 + 7 = 8, . . .”).

More surprisingly, there is little connection between the teachers’ affective relation with mathematics and their assessment of their own mathematical qualifications on the one hand, and their proficiency with R&P on the other (Larsen et al., 2018). There are 46 research participants, who consider themselves highly qualified in mathematics and/or who are fond of the subject. Only 18 of these participants provide an (almost) acceptable response to the perimeter-and-area item, and 8 do so to the item on the sum of two odd numbers.

In the observations from the college classroom in the last part of the second phase of the study, the teachers face considerable problems arguing how or why the video clips they selected from their practicum are related to R&P. Four of the groups do not provide a coherent explanation for why they selected the episode, and three of the other groups have selected their clips for reasons that are unrelated to mathematical reasoning (e.g. the teachers’ supervisor chose the video clip or the technical quality of the recording was good). The last group claims that their clip is on reasoning, but it shows students making number stories for tasks on fractions.

There may, of course, be many reasons why the prospective teachers brought video clips back from their practicum with little or no connection to R&P. For instance, they may have focused on more general educational problems, or their students may have found R&P difficult and for that reason refrained from engaging in these processes. Flowever, there is nothing in what the prospective teachers said or did when back at the college that indicates that any of this was the case, and they in no way suggested that they were not satisfied with their selection of the clips as a reasonable response to the task that had been set before their practicum. We therefore suggest that their selection is based on insufficient prior experiences with R&P.

Looking at the clips themselves, rather than at the teachers’ reasons for selecting them, three have no connection to mathematical reasoning (e.g. the teacher presents the solution to a procedural task on the board). Other episodes have some potential for student involvement in R&P, but the teachers do not emphasize aspects of R&P in the discussion with their students in the classroom.

Learning to teach to reason 55

In one episode with some potential for student involvement in R&P, middle school students are to find the point equidistant from the vertices of a triangle. In the video clip, a school mentor, the teacher normally teaching the class, unintentionally shows the students an incorrect procedure for constructing perpendicular bisectors. The students use the incorrect procedure, but having measured the distances on their drawings, they realize that something is wrong. They then shift their attention to the question of how to draw a perpendicular bisector, but they pay no attention to how and why it may help them solve the initial problem.

In their discussion of the video clip at the college, the teachers focus on what they describe as lack of conceptual understanding on the part of the students and on what they appear to consider general drawbacks of whole-class instruction. There is no discussion of if and how the task may become a point of departure for the students’ exploration of the problem or for formulating and justifying a conjecture about the properties of perpendicular bisectors.

Another episode with some potential for involving the students in the phases of the R&P cycle concerns the pattern in the number of squares in a sequence of figures. The teachers introduce the task and show the two first figures in the sequence on the board (Figure 4.1). The students, who are in Grade 6, use small cookies to represent the squares. The video clip shows two students, who have 33 cookies. They have written “7” and “y = 4x + 5” and made the drawing in Figure 4.2.

FIGURE 4.1 The two first figures in the sequence as shown on the board.

FIGURE 4.2 The group’s drawing of their cookies.

The communication between the teacher and the students in the video clip is as follows:

TEACHER: The equation, you are working with, what is x in that equation? I can see you have written 7, but what is the x in your figure? If you look at the cookies, what is the x? . . .

ANNA: If we count. . . . Those five cookies, you don’t count them [points at the five cookies in the middle]. You only count these [points at the “arms”], don’t you?

TEACHER: The figure that was on the board, how about it?

ANNA: Do you count these [the cookies in the middle] and add them to the “arms”?

TEACHER: In the beginning there were some cookies, right? What do you think are going to be the “arms”?

ANNA: How do you make the “arms”? Do you take that cookie [points at one of the cookies next to the one in the middle]? I think you count how long your “arms” are.

TEACHER: Is this cookie a part of the “arm”? [Points at one of the cookies next to the one in the middle.]

ANNA: Yes, I think so, right?

TEACHER: How can we characterize the “arms”, if we want this cookie to be included |points at the cookie next to the one in the middle]?

ANNA: Then these are the “arms” [points at a row of cookies, leaving out only the one in the middle], the rest of the cookies belong to the “arms” [moves the cookies around]. But this is not right!

BENNY: I think there are too many cookies, right? [Counts the cookies in the “arms”.]

TEACHER: It looks as if you have counted them perfectly . . .

The students have trouble linking the equation with the geometric representation they made themselves, and as the teacher joins them, the emphasis of the discussion is on the meaning of x and the length of the arms of the cross. These difficulties appear also in other groups, who have written the same equation.

Back at the college, the teachers’ discussion of the episode tends to revolve around general pedagogical issues. Major concerns are student motivation; how it may become a successful experience for the students, “success” understood in terms of the students being comfortable and having a good time; and how much time students should spend on working on their own before receiving help from the teacher. The discussion does not focus on the potential of the episode as an opportunity for the students’ learning of R&P.

Issues related to the contents of the episode do come up in the discussion. The teachers focus primarily on the students’ problems with linking the sequence of figures to the symbolic representation in the equation, and on their difficulties deciding on the value of the constant, b, in the equation у = mx + b:

TEACHER (from the group that selected the episode): They [the students who responded to the task] begin by showing the 5 cookies in the middle. And that’s what confuses the students. After that they write у = Ax + 5 instead of 4x + 1, and where the first cookie in the arm is included. And that’s their problem. . . .

TEACHER (from the group that selected the episode): [Tjhey have no problem making the additions. . . and finding rules or patterns. The problem appears when they are to make the equation. And that’s what we want them to make - an equation. And here they also have some problems with symbolic understanding.

Another teacher suggests that there may be some potential in discussing how the number of squares in each figure may be calculated from the number of squares in the previous one:

TEACHER (not from the group that selected the episode): It would be nice with a table and count down to one [that is, to the first figure in the sequence]. And then they have a function that says that four will be added every time.

The teachers do not discuss if the equation could be considered a conjecture that could be tested and revised or verified. In this sense, mathematical justifications do not become an issue.