Estilo reflexivo

2.2.3.1. Geometric construction tasks

The tasks presented preservice mathematics teachers with a set of geometric objects (e.g., a line, a circle, and an angle) and asked to construct – with ruler and compass – a specified object (e.g., a tangent to the circle that would intersect the line in a congruent angle). The task requirements were to (a) perform the construction, (b) describe their construction step by step, and (c) provide reasoning to show why their construction yields the specified object. When the construction task was introduced to the participants, they also received a basic geometric knowledge sheet that described basic definitions / rules and acceptable construction rules according to the mathematical norms of the teacher-training program. Fictional (peer) solutions were created for two construction tasks, one for the pretest and one for the posttest, based on pilot studies. Participants were instructed to provide written PF to a fictional peer about each part of the solution. All of the participants received the same peer solution. The fictional peer solutions contained (a) some correct steps, but partly followed an incorrect strategy, (b) correct descriptions of some but not all steps, (c) correct as well as incorrect reasoning steps, and (d)

vague language in some parts of the solution. The graphical construction in the peer solution matched the description but was not performed with complete accuracy.

2.2.3.2. Training

All participants received PF provision training which consisted of two stages. In the first stage, two instructional sessions were held each lasting for 45 minutes. In the first instructional session, the notion of PF was discussed with the preservice mathematics teachers. The participants shared their thoughts about PF, its benefits, how it should look like, and their insecurities regarding PF. Then the feedback levels (task, process, self-regulation, and self) were introduced and discussed with the preservice mathematics teachers. At that point, the participants also received PF provision prompts accompanied with a task-specific evaluation rubric. In the remaining part of the first session and the second session, the preservice mathematics teachers were involved in several individual and group activities to understand the different levels of feedback better. They had to (a) identify each feedback level in written PF comments, (b) transform one feedback level to a higher level, and (c) work in groups to provide written PF on a solution and share and discuss their PF with the rest of the class. In the second stage of the PF training, which also involved two sessions, each participant received a fictional peer solution and practiced providing written PF on that solution with the help of the instructional scaffolds (i.e., prompts, evaluation rubric and worked example).

2.2.3.3. Instructional scaffolds

Participants received several instructional scaffolds that we introduced at different points during the PF training. In the first stage of the training, the preservice mathematics teachers received PF provision prompts and an evaluation rubric. These scaffolds were used in the instructional practice activities. In the second stage of the training, a worked example of each geometric construction task was provided in combination with the PF provision prompts and the evaluation rubric.

Peer feedback provision prompts

A visual organizer (developed by Gan, 2011) with progressive prompts reflecting different levels of feedback according to Hattie and Timperley’s (2007) model was used (see Hattie & Gan, 2011 for the visual organizer). We extended Gan’s visual organizer with additional prompts mostly knowledge integration/self-reflection prompts (adopted from Chen, Wei, Wua, & Uden, 2009; King, 2002; Nückles, Hübner, & Renkl, 2009) (see Table 2). Most of the added prompts were at the self-regulation level because it is assumed that the PF provider benefits more from providing knowledge integration/self-reflective questions as s/he needs to think deeply about the learning task (King, 2002).

Evaluation rubric

The evaluation rubric consisted of a set of criteria that could be used in combination with the PF provision prompts to judge the peer solution, and produce written PF. More specifically, the preservice mathematics teachers had to judge: (a) the construction of the geometric object, (b) the description of the construction and (c) the reasoning provided to prove the construction true.

Worked example

All participants received a standard worked example of the geometric construction task they had to provide written PF on in the second stage of the PF training (i.e., in the practice sessions). Since the purpose of this study was to improve preservice mathematics teachers’ PF skills, and not their domain knowledge, we provided them with the worked example. This was done to ensure that those with low domain knowledge can still practice providing written PF at the higher levels (i.e., process and self-regulation) and not spend most of the sessions’ time trying to understand the task. The geometric construction tasks used in this study always had more than one solution approach, and the solution proposed in the worked example was

different from the fictional peer solution that the preservice mathematics teachers had to provide PF on.

Table 2

Prompts added to Hattie and Gan (2011) feedback levels visual organizer and their sources

Prompts Peer feedback

level

Source

Which parts were not sufficiently clarified? Task Nückles, Hübner & Renkl (2009) How can you best explain…? _{Self-regulation}

How would you use…. to…..?

Self-regulation King (2002) What would happen if you….?

How does….tie to what we learned before? What conclusions can you draw about…? Explain why…..

Explain how….

How are …and … similar? How are… and ….different? What can you infer from….?

Self-regulation Chen, Wu & Uden (2009) What can you think of from….?