Abstract

This study reports on the result of the analysis of some graduate mathematics students of University of Science and Technology of Southern Philippines and Mindanao State University-Iligan Institute of Technology in writing or constructing mathematical proofs. The study used the quantitative and qualitative method of research. Results revealed that the teachers who finished their undergraduate degree in pure mathematics perform very well in writing mathematical proofs and females are likely to find difficulty in doing proof than males. Poor experience in writing proof in high school, and college, poor foundation in logic , being non-pure mathematic major and poor understanding of mathematical concepts are some of the causes of difficulties in writing mathematical proofs.

1. Introduction and Theoretical Framework

“Quad Erat Demonstradum” (abbreviated Q.E.D.) are the three words in Latin that traditionally mark the end of the mechanical proofs. These words simply means “which was to be shown”. Proving sometimes is a matter of convincing one’s peer that has indeed shown no more no less. In almost all areas of mathematics construction of proofs of mathematical statement is considered essential and inevitable task. In fact, before a mathematical statement can ever be labelled a lemma, a theorem or a corollary, its proofs have already been constructed or established. Many mathematicians and mathematics educators said that the traditional view to the mathematical proof concept is that it is a formal and logical line of reasoning that begins with the set of axioms and moves through logical steps to a conclusion. It is pointed out that one of the purposes of proving a theorem or a mathematical assertion is to establish its mathematical certainty. A proof simply confirm truth for a mathematician or mathematics educator the way experiment or observation does a natural scientist. However many mathematics educators and some mathematicians believe that proofs are much more than this. Mathematics educators and mathematicians further believe that establishing the truth of the statement in only one of the many reasons for constructing a proof. By looking at a proof a reader can understand why a certain statement is true. De Villiers noted that proof can also be used to organize previously disparate result into unified one ¹. By working continuously on a system deductively, one may actually discover that some argument may be fallacious, circular and incomplete. Despite all these, purposes of a proof laid by mathematicians and mathematics educators, still others such as Hanna ² and Hersh ³ argued that explanation should be the primary purpose of proof in the mathematic classroom.

There is a reality of awareness of the importance of proof and proving in mathematics, however there is a hindrance since most students have difficulties in proof and writing mathematical proofs. Sadly this deficiency is not only evident in the performance and proving ability of undergraduate students but also of the graduate mathematics students. The researchers who are graduate school professors can attest of such difficulties, hence there is need to investigate the causes to remedy the situation.

Difficulties of students with proof and in writing mathematical proofs are believed to stem from variety of factors. Some of these factors may be attributed to the individual and knowledge earlier acquired. One of the cognitive development theories of Piaget stated that human beings are distinguished from animals through his ability to do “abstract symbolic reasoning” ⁴. He identified four stages and the formal operational stage is of special interest in this study. It is theorized that intelligence of an individual is demonstrated through logical use of symbols to relate to abstract concept and he suggested that the development is the end point in goal.

Vygotsky on the other hand believed that the development is a process that should be analyzed instead of product to be obtained ⁵. He said that the development begins birth and continues until death is too complex to be defined by stages. He further say that cognitive development was dependent on social interaction which he called it Zone of Proximal Development (ZPD). In these phenomena the development is through a problem solving task under the guidance of an adult of a more capable peer. A student can perform a task under adult guidance or a peer that cannot be achieved alone. Epp added that at any given point in the learning process, the insights and intuitions of the learner previously developed provides a basis that help to accomplish the next stage of instruction, hence the teacher has a great role in this regard ⁶.

In addition, there are form of categories to attain success in mathematics like resources, heuristics, control and beliefs ⁷. These categories are needed in the task of constructing mathematical proofs. It was also noted that poor conceptual understanding and ineffective proof strategies are some reasons for difficulties in constructing mathematical proofs ⁸ and believed that there is need for a focus of mathematical reasoning starting with logic to build up confidence ⁴, determining truth or falsity of statement and experience on the use of logical terms.

This study seeks to determine some factors that contributed to the difficulty in writing mathematical proofs of graduate mathematics students of MSU Iligan Institute of Technology and the University of Science and Technology of Southern Philippines.

2. Methodology

The study used quantitative and qualitative method of research with triangulation and with stratified random sampling procedure to get the sample of the population. It used proportional allocation to determine the size the sample. There were 40 students as respondents who have not completed their respective program of study. The questionnaire was designed to gain overview of the background information about students’ attitude towards proof writing, previews experience of proof writing, methods of writing proof, understanding of mathematical concepts and control of resources. The proving task underwent validation and reliability coefficient was determined. The respondents were required to accomplish the questionnaire and answer the proof writing task. The data gathered were tabulated and the proof writing task done by respondents were assessed using rubrics. The data was analyzed using non parametric tested at 0.05 level of significance.

3. Results and Discussions

The table shown below gives the results of the analysis of the data gathered from the respondents answer on the questionnaire given to them using Chi-square with probability value of 0.05 level of significance.

As shown in Table 1, the Chi-square value for the association between the respondents proving ability and the ranges of age is 1.436 with a p-value of 0.310. This implies that there is no significant association of respondents’ proving ability with their corresponding ages. This findings did not support the assertion that all individuals will automatically move to the next Piagetian cognitive stage as the biological nature of man.

In terms of gender and proving ability the analysis yielded a Chi-square value of 7.016 for the association of respondents’ proving ability and gender with a p-value of 0.017 less than the critical value at 0.05 level of significance. This implies that males have significantly high ability in doing proving task than their female counterpart. This findings support the many previous studies that as males become more mature they become more logical and better in their critical thinking ability, particularly in doing a proving task.

With regard the analysis of the undergraduate degree of specialization and proving ability, the result of the analysis of association between the two variables is 15.824 with a p-value of 0.000. This implies a high degree of association signifying that those who finished pure mathematics degrees were able to present a more elegant and logical proof in the given task compared to those non pure mathematics graduates. This implies further that when students are trained to do proof writing , they tend to do better in doing a proving task and develop their logical and critical thinking ability. Results from interview from non-pure mathematics graduates also revealed that they did not experience mathematics proof writing in high school and in college.

As regards respondents’ feeling towards doing proving task and their ability to do proof writing, the analysis yielded a Chi-square value of 1.200 with a p-value of 0.333, which implies that their feeling is not associated with their proving ability. This means feeling toward a given task do not influence their logical reasoning and critical thinking ability especially doing proof writing.

Furthermore, the same table shows the analysis on the association of the respondents’ proving ability and previous instruction on proof writing. It yielded a Chi-square value of 8.864 with a p-value of 0.006 less than the critical value at 0.05 level of significance. This implies that respondents‘ proving ability has a significantly high association with their previous instruction in lower years of mathematics courses. This result further implies that teachers’ role in training students to do proof writing is very vital. The interview further revealed that their proving task is not only deficient their secondary mathematics instruction but also in college.

For conceptual understanding and logic as independent variable, in association with respondents proving ability , the analysis yielded a Chi-square value of 6.593 with a p-value of 0.017 which is significant at 0.05 level. This implies that the respondents who are classified as good in conceptual understanding and logic have s significantly high proving ability. This means that good conceptual understanding and logical thinking are important ingredients in proving a mathematical task. The interview also revealed that the respondents with low proving ability do not possess a profound conceptual understanding of basic mathematics, set theory, number theory, logic and methods of proof writing. This result also agrees with Weber ⁸, who said that poor conceptual understanding can cause ineffective writing strategies and difficulty in proof writing

On the variable, use of resources which refers to the ability to decide when and what resources and strategies to use, when doing mathematical proof and the association to the proving ability, the Chi-square yielded a value of 5.293 with a p-value of 0.041 less than the critical value of 0.05 level of significance. This implies that there is a significant association of proving ability and use of resources. This implies further that when respondents possess the ability to control the resources and strategies, they are likely to exhibit an elegant, logical and correct proof of a given proving task. Interview further revealed that faulty use of logic is due to lack of ability to decide on what and when to use the resources and technique appropriately on the given task. This findings support that the resources can boast success constructing mathematical proofs ⁷.

4. Conclusions and Recommendations

Based on the analysis and findings of the data collected the researchers conclude that the respondent gender, their undergraduate degree finished, experience in proof writing, conceptual understanding of mathematical terms, logical line of reasoning and control of resources are the factors that have caused difficulties in doing proof writing tasks while age and feelings are not contributory to their difficulties. It is recommended that teachers need to inculcate the importance of proof writing, give emphasis on conceptual understanding, provide good and sufficient background on fundamental of logic and different methods and strategies of doing mathematical proofs.

References