Peer feedback provision as a learning opportunity

Providing feedback on the work of a peer can be beneficial for the PF providers, not only because they become actively involved in assessment, but also because they can engage in several high-order thinking processes including, making inferences, questioning, generating hypotheses, weighing alternatives, and evaluating different information (King, 2002; Topping & Ehly, 2001). A qualitative study by Nicol et al. (2014) revealed that during PF provision, students reported engaging in a critical evaluation of their peer’s and their own solution, experiencing self-reflection on their own learning, and generating explanations for their judgments. Another possible benefit of PF provision is being exposed to a wide range of solutions, which can enhance students’ understanding of the learning task at hand (Zerr & Zerr, 2011). Studies with undergraduate physics students already showed that their writing skills improved due to providing PF on lab reports (Cho & Cho, 2011; Cho & MacArthur, 2011). Furthermore, PF training was found to improve preservice teachers’ assessment skills (Sluijsmans et al., 2004). The development of assessment skills of some complex mathematical tasks such as proofs is essential for preservice teachers as they will need to teach and assess students’ performance on these tasks in their future classrooms. Indeed, proof validation is considered as an important part of proof comprehension (Selden & Selden, 2003), thus involving preservice mathematics teachers in PF is expected to be beneficial for their proof comprehension and proof validations skills, but only if the PF activities align with the mathematical norms against which the proofs are normally being judged. Furthermore, in order to develop beneficial PF training for preservice mathematics teachers, factors, and processes influencing PF provision on peer solutions to mathematical proofs should be examined.

3.1.1.1. Peer feedback provision and validating mathematical proofs

One of the key elements to scientific reasoning and argumentation in mathematics education is mathematical proofs (Heinze & Reiss, 2001). Several proof-related activities are essential for proof teaching and learning including, performing a proof correctly according to the mathematical norms of the teaching context (i.e., proof construction), understanding a proof from a textbook or from lecture notes (i.e., proof comprehension), and judging the correctness of proof performed by students or by an external source (i.e., proof validation) (Selden & Selden, 2015a). Proof validation is an important skill for preservice mathematics teachers because school mathematics curriculum is likely to depend on proofs and justifications (Selden & Selden, 2015b). Previous studies revealed that proofs are challenging to validate for high- school and university students (Inglis & Alcock, 2012; Reiss, Heinze, & Klieme, 2000; Selden & Selden, 2003). There is growing evidence that when undergraduate students are asked to validate proofs of different levels of correctness they do not reliably differentiate between the correct and erroneous proofs (e.g., Alcock & Weber, 2005; Inglis & Alcock, 2012; Selden & Selden, 2003). Nevertheless, a recent study revealed that students’ accuracy in proof validation depended on the type of error in the proof (Sommerhof, Ufer, & Kollar, 2016).

Geometry proofs are often used as initial context to teach scientific reasoning and argumentation in mathematics because they have a figure component that allows students to explore mathematical concepts visually and more easily by linking them to physical objects in the real world (Schoenfeld, 1986). Research with high school students (grades 7 and 13) showed that validating erroneous geometry proofs was more challenging to the students than validating correct proofs (e.g., Klieme, Reiss, & Heinze, 2003; Reiss et al., 2000). It is unclear, however, if preservice mathematics teachers also face the same challenge when validating geometry proofs.

demonstrated by the instructor without showing the reasoning processes involved in creating that proof (Reiss et al., 2008). Since students are presented with correct proofs most of the time, they are likely to approach them passively and barely try to engage with a proof actively or to validate and comprehend its components (Zerr & Zerr, 2011). It is, therefore, not surprising that students have difficulties validating proofs because this process requires a deep understanding of how the proof is constructed (Selden & Selden, 2015a). Proof comprehension research showed that making the proof instruction more active by using self-explanation training resulted in improved proof comprehension (see Hodds et al., 2014). One of the major differences between proof comprehension and proof validation, however, is that in the latter activity the correctness of the proof is not quite evident (Selden & Selden, 2015a).

Involving students in PF provision is another approach to active engagement. When providing PF on a proof constructed by a peer, the PF provider engages mainly in proof validation. Yet, the PF provider also requires some comprehension of the correct parts of the proof to be able to identify possible errors and to provide PF accordingly. Some students consider comprehending the proof as a necessity to be able to validate the proof (Selden & Selden, 2015b). In one qualitative study by Zerr and Zerr (2011), PF provision activities on erroneous proofs were implemented to teach proof validation within a proof-based mathematics course at university level. The researchers reported that their students were descriptively more successful at identifying correct parts in the correct peers’ solutions than being able to identify mistakes in the erroneous peers’ solutions. What remains unclear, however, is whether the correct and erroneous proofs are processed differently by students when they are providing PF on them and whether students would comprehend the validated proofs differently. A study by Inglis and Alcock (2012) showed that when validating proofs, undergraduates adopt different reading behavior as compared to experts. Undergraduates were found to focus on the “surface features” of the proof (i.e., formulae), whereas experts focus on the logical structure of the

proof. Nevertheless, no study yet to our knowledge has investigated whether preservice mathematics teachers cognitively process correct and erroneous peer solutions to geometry proofs differently when they provide PF on them and whether the quality of the peer solution influences preservice mathematics teachers’ comprehension of the proof.

The conflicting findings from research about learning from erroneous worked examples in mathematics education (e.g., Große & Renkl, 2007; Heemsoth & Heinze, 2014; Isotani, Adams, Mayer, Durkin, Rittle-Johnson, & McLearen, 2011; Tsovaltzi, Melis, McLaren, Meyer, Dietrich, & Goguadze, 2010), together with findings from proof validation studies (e.g., Inglis & Alcock, 2012; Reiss et al., 2000) suggest that preservice mathematics teachers are likely to comprehend the proof they are providing PF on when the peer solution is not an erroneous proof. Yet, this assumption should be tested empirically. When investigating the processing of geometry proofs during PF provision, the PF content cannot be ignored because it can indicate what the PF providers attend to while providing PF on the peer solution to the proof.