Understanding mathematical proof and the validity of proofs in elementary school

In the last three years of elementary school, pupils encounter substantiations or argumentations. These can be informal and based on arguments, which have or do not have mathematical validity, or even simpler formal ones. The basic role of mathematical proof in lessons is to encourage mathematical understanding, since a proof is most convincing, when it leads to understanding. An important mathematical activity is also the recognition of the validity of proofs, which includes the consideration, if an alleged proof is mathematically correct and if it includes a suitable logical mathematical reasoning. Research has shown that pupils do not feel the need for proving, they experience the proving procedure as hard and encounter most problems particularly with tasks, which require argumentations and proving. In the theoretical part of the master’s thesis, we discussed the concept of mathematical proof and described its role in mathematics and learning mathematics. We discussed the importance and incorporating proofs in lessons and presented the cognitive processes in proving from the perspective of the pupil’s development. We presented different functions of proof in mathematics and mathematics lessons, cognitive perspectives of proving and problems of pupils in learning proofs and proving. We also presented two classification of mathematical proofs: the classification according to Tall and the classification regarding the strictness of substantiation. In the empirical part of the master’s thesis, we used a quantitative and qualitative approach to identify the selected perspectives of proving in elementary school mathematics. Due to the influence of the textbook as a knowledge guide for pupils, we chose two classifications and used them to analyse different types of argumentations in the most widespread textbooks for mathematics in the last three years of elementary school. The analysis of the textbooks “Skrivnosti števil in oblik” has shown that according to the classification by Tall, the textbooks include mostly visual proofs, which means argumentations with figure and verbal support. According to the classification regarding the strictness of substantiation, in textbooks prevail generic proofs. The least common argumentations, according to the classification by Tall, are enactive proofs, and according to the classification regarding the strictness of substantiation, argumentations with authority. In terms of inclusion of argumentations in different topics, the most argumentations are included in the topic Geometry and Measuring. The research included a sample of 49 ninth graders from two different Slovenian elementary schools, out of which 27 were high achievers and 22 were low achievers. With a written examination, we analysed, if pupils accept and recognise the validity of an argumentation in a mathematical proof, meaning if pupils differentiate between a formal proof and informal argumentations. We also analysed, if they understand different types of argumentations according to the two chosen classifications. The data, obtained with the written examination, showed that pupils have problems to accept and recognise the validity of argumentations, since at the examination, a high proportion of pupils recognized the majority of types of argumentations as valid. According to the classification by Tall, the highest proportion of pupils recognised enactive argumentations as the easiest to understand, and argumentations with authority and argumentations with a figure according to the classification regarding the strictness of substantiation. As hardest to understand types, the highest proportion of pupils recognized the manipulative proof, according to the classification by Tall, and the formal proof, according to the classification regarding the strictness of substantiation. The statistical analysis of data showed that there are statistically significant differences between high and low achievers regarding the recognition and acceptance of validity and understanding of different types of argumentations in terms of the chosen classifications. We used a questionnaire to research the views of pupils on proofs and proving in mathematics lessons. A little more than half of the pupils find it hard to argument a certain mathematical statement, but at the same time they think they have enough knowledge to understand mathematical proofs. The highest proportion of pupils least agree with the statement that usually they find it hard to understand a proof presented by the teacher during lessons. A high proportion of pupils claim to know, what a proof in mathematics is and how it looks like, and know how to estimate, if a proof is correct or incorrect. The pupils expressed desires for learning proofs and proving in mathematics. More than a half of pupils would like that mathematical lessons would include more presentations, descriptions of proofs of mathematical statements, they would like to discuss about the correctness of mathematical statements and to solve proving tasks. A very high proportion of pupils want that the school textbook would include more proofs of mathematical statements and they think that proofs are interesting. A little less than a half of pupils agree that during lessons, the teacher tells them, why we bother to prove mathematical statements, and think that if the teacher writes a mathematical statement on the board during lessons, the statement is already proven. A high proportion of pupils also think that proofs are only for pupils with better grades and that, if a mathematical statement is written in the textbook, it does not require to be additionally proven. A high proportion of pupils think that they know the purpose of proving in mathematics and agree that a proof in mathematics is important, since it assures validity of the statement and explains, why a certain statement is correct. With the data, obtained with the questionnaire, we also analysed, if there are statistically important differences between high and low achievers regarding views on proofs and proving in mathematics lessons. The obtained data showed that most low achievers find it hard to argument a mathematical statement, usually find it difficult to understand a proof presented by the teacher during lessons, and find that they do not have enough knowledge to understand a proof of a mathematical statement. A significantly higher proportion of high achievers than low achievers agree that they know, what a proof in mathematics is and how it looks like, and that they usually can estimate, if a certain proof is correct or incorrect. Also, a significantly higher proportion of high achievers than low achievers expressed a desire for the school textbook and mathematics lessons to include more proofs of mathematical statements and more proving tasks. On the other hand, a significantly higher proportion of low achievers than high achievers agree that if a teacher during mathematics lessons writes a statement on the board or if the mathematical statement is written in the textbook, it does not require to be additionally proven and is therefore already proven. A higher proportion of high achievers than low achievers is aware of the purpose of proving in mathematics and the importance of a proof to assure the correctness of a claim. A higher proportion of low achievers than high achievers agree that proofs are not important for a good mathematics knowledge and that proofs are only for pupils with good grades. Both group of students expressed a similar level of agreement with the fact that during lessons, the teacher tells them, why we prove mathematical statements