EL SCOUT ES PURO EN SUS PENSAMÍENTOS, PALABRAS, OBRAS Y EN EL USO DE TODOS SUS SENTIDOS.

Does the use of DGS enhance students’ geometric reasoning abilities?

In order to answer this question the study will reveal students’ geometric reasoning abilities before the introduction of DGS and then their abilities after the use of DGS.

5.1.1 Students’ geometric reasoning before introducing the software

Students’ reasoning ability is highly affected by the teacher’s teaching approach. Proof is taught in the participant class not because the teacher is convinced of its importance, but because it is required by the system. The teacher believes that “students’ brains are not ready yet”. This is why the teacher teaches proof as a procedure. When solving a problem involving proof, the teacher stresses on writing the given of the problem, drawing the figure, and writing what is required to prove as if it is a procedure. Afterwards, she explains the proof in a fragmented way; more specifically she explains the proof statement by statement. The stress is not on the understanding of the proof as a whole; however, the stress is that students understand each statement alone. She also stresses that for every statement a reason is presented. This reason should be a fact or a theorem. The study results show that this affected students’ perception of proof and the way they prove. In the paper-pencil quiz, students always start by copying the figure, what is required to prove, and then drawing the figure even before trying to understand the problem. However, 26.8% of the answers in paper- pencil quiz are fragmented proofs that include justifications whereby students write enough properties/ theorems needed to form a correct proof but either do not apply to the given problem, or are based on the required to prove, or invented by students. At other times, in 15% of the answers in paper-pencil quiz, students produced unrelated statements which together do not lead to a proof. These students aim at producing

fragmented statements that had a reason as a fact or a theorem. They do not care about whether the reasons apply to the problem or if they make sense.

The teacher assigns partial grades to writing the given and what is required to prove. This fact results in encouraging students to write these criteria even without understanding the problem. Also partial grades are assigned to correct statements, even when not related to the problem or when they do not lead to a proof. This is a second reason that makes students write fragmented proofs or unrelated statements.

The participant teacher does not explicitly teach ways to prove. To prove a conjecture, she asks students to make suggestions. A student who knows the answer gives it to the class. The other students adopt the proof without necessarily understanding how the student was able to get a correct proof. This may explain why some of the students are not able to produce proofs. In 13.2 % of the answers in paper- pencil quiz, no proof is included. Moreover, in 15.4% of the answers students were not able to conjecture.

Moreover, there is more stress on format of proof in the participant class than on proof itself, which results in students’ inclusion of details that do not add to the logic of the proof. 47% of the answers in paper-pencil quiz present such details. On the other hand, 71.3% of correct proofs in paper-pencil quiz were proofs that do not have enough details. In other words, students present details that do not add to the logic of the proof more than details that add to the logic of proof.

In addition, the way the teacher asks questions affects students’ ways of reasoning. For instance, the teacher asks students questions such as: “do they look

equal?” This is why students focus on the figure to make conjectures.

5.1.2 Students’ geometric reasoning after introducing the software

After students’ use of the dynamic geometry software, they had tools to understand the problem and the relations between the properties of the objects presented in the problem. As a consequence, the results of the analysis of students’ work on the

DGS worksheet reflect solutions for all the questions. Moreover, the category of unrelated statements is only 1.4% of the answers of the DGS worksheet. In addition, because they understand the problem, students gave a conjecture and a proof in 84.7% of the answers in the DGS worksheet. 83.3% of the answers presented are divided between correct proofs and fragmented proofs. This percentage represents proofs where students write the properties/ statements to prove the truth of the conjecture presented.

However, not all of the students in DGS worksheet were able to write correct deductive proofs because they only used Cabri during four sessions (two hours each). In particular, in 15.3 % of the answers on the DGS worksheet, students provided conjectures with no justification. Moreover, 26.4% of the answers were fragmented proofs, meaning that those students were able to write enough statements to form a correct proof but were not able to apply them to the given problem. Out of these 26.4%, 15.2 % of the answers were justified empirically. No fragmented invented proofs are found in the answers to the DGS worksheet.

In other words, students were able to better understand the problem and present a conjecture. DGS showed to be able to enhance their reasoning ability; however, students who were on low levels of reasoning were not able to prove deductively because they had many misconceptions related to the meaning of proof due to the way they were taught proof. They need to be introduced to proof in a different way and they need more time solving problems on DGS.