From Exploration to Proof Production

The impact of teachers' knowledge on the connection between technology supported exploration and mathematical proof

“…The study provides evidence about the difficulty of articulating proof and technology, in line with the difficulties addressed in the literature and related to mathematical proof (De Villiers, 1999;Hsieh et al, 2012), but it also offers evidence of the relevance of this articulation and of how the teacher's professional knowledge can impact the teacher's options. It would be interesting to deepen the articulation between proof and technology, paying attention to other types of tasks (beyond exploration tasks) and looking for understanding about how different roles of proof (beyond verification) are/can be addressed.…”

“…And this is because it is the teacher's knowledge of the students and their difficulties that seems to guide all the decisions. It is the teacher's knowledge of the type of task (as suggested by Rocha, 2020b) and the way in which the students approach them (often advancing and establishing conclusions based on a very small number of observations) that leads her to reinforce the importance of validating the conjectures, in line with the work of Hsieh et al (2012) (trying to make the students understand the relevance of thinking carefully, based on a set of cases, before formulating a conjecture; and transmitting the idea that a conjecture is something that seems to be true, but requiring a deeper analysis -the proof-before it is possible to be sure it is always true). And this is a decision that is based on the knowledge of the students, but also on what is the essence of Mathematics, as assumed by Blanton and Stylianou (2014), Dawkins and Weber (2017), Rocha (2019) and Schoenfeld (2009) (the teacher is aware about how the students can be convinced of the validity of a result based on the observation of some cases; but she also knows the relevance of proof in Mathematics).…”

“…The difficulty in getting students to understand the need for and importance of proof in Mathematics is, according to De Villiers (1999), well known to all secondary school teachers. This difficulty is accentuated when technology is involved because, according to Hsieh et al (2012), the dynamic character usually offered by it allows the carrying out of work of an experimental nature, which enhances the discovery of properties and the formulation of conjectures. Students can easily experiment and analyze various cases, reflecting on important mathematical ideas and, consequently, reaching a higher level of understanding (Goos & Bennison, 2008).…”

“…However, this possibility challenges the teachers' professional knowledge (Rocha, 2020b). The ease and speed with which it becomes possible to observe many cases of a given situation, brings conviction about the veracity of the formulated conjecture and fosters a feeling that nothing else is needed to be sure of it (Hsieh et al, 2012;Rocha, 2020b). Mathematical proof, assumed as the essence of Mathematics by several authors (Blanton & Stylianou, 2014;Dawkins & Weber, 2017;Rocha, 2019;Schoenfeld, 2009), thus tends to appear to the students as something dispensable (Hanna, 2001).…”

Using tasks to develop pre-service teachers’ knowledge for teaching mathematics with digital technology

“…However, depending on their mathematical experience (where mathematics is often validated externally, by the teacher, the textbook or even the parents (Tall et al 2012)), some students might tend to think of their conjectures as valid mathematical results. Therefore, the need to justify or even prove the conjectures might not be felt (Hsieh et al 2012). This means that, in certain circumstances, the DT can prevent students from learning about the central role of justification and proof in mathematics (Hanna 2001).…”