Emprical Argumet or Proof? Can 7th Grade Students' Awareness of the Difference Between Them be İmproved?

Author(s):

Ebru Aylar(presenting / submitting)

Conference:

ECER 2014

Network:

21. Emerging Researchers' Group (for presentation at Emerging Researchers' Conference)

Format:

Paper

Session Information

ERG SES D 14, Education and Teachers' Practice

Paper Session

Time:

2014-09-01

13:30-15:00

Room:

FPCEUP - 254

Chair:

Fiona Chambers

Contribution

Mathematical proof is important in mathematics teaching in terms of the comprehension of mathematical knowledge. Thus, proof has critical value in the teaching process in terms of the prevention of memorization in mathematics, the construction of conceptual knowledge, and the realization of meaningful learning. Again, the tendency to consider proof as only a subject requiring advanced level mathematical knowledge is continuing. It would not be a mistake to say that report of “The Principles and Standards for School Mathematics” published by NCTM in 2000 created an important breaking point for this tendency. In this report, NCTM discussed proof as an important component of mathematics teaching for every age group and has led to interest and discussions being directed to this area. NCTM does not consider proof as a special activity of certain subjects of the curriculum conducted at certain times. Proof and reasoning must be a part of the process of teaching a lesson no matter what the subject is (NCTM, 2000).

“Reasoning and proof”, which NCTM dealt with as a process standard, is an important method of the comprehension of mathematical content and knowledge. NCTM mentions of the importance of comprehending proof in the understanding mathematics. Contrary to this, the most recent primary school and middle school curriculum attempting to largely include the process and content standards of NCTM in its content is observed to not emphasizing proof at the same degree.

In Turkey, with the transition to the practice of 12 years of compulsory education, curriculums were updated in 2013. Together with this correction, when curriculums are examined, it can be observed that proof is not included in the primary school and middle school curriculum as a concept. In the curriculum, proof is dealt with in the curriculums of the 9^th and 11^th grades as mathematical skills and competencies that are aimed to be developed.

In parallel with the increase of teaching of proof in high school and advanced levels of education, a large proportion of studies conducted regarding proof discuss the teaching of proof in primary and middle school and some studies can even state that proof in school mathematics is suitable for students in the advanced secondary education level and middle school students do not understand and do formal proof (Bell, 1976; Fischbein, 1982; Knuth, 2002). Contrary to these discussions, recently there is an increase in studies advocating that proof teaching can start in the early age group starting from preschool education (Ball et al., 2002; Cyr, 2011; Stylianides, 2007). In addition to this, most of the research shows that students who are attaining at all grade of school levels, tend to accept emprical arguments as proofs of mathematical generalizations (Healy & Hoyles, 2000; Goetting, 1995; Stylianides & Stylianides, 2009).

Studies conducted in this area in our country are relatively limited. The title of “the relation of early age period and proof” for students in Turkey is an unknown title. Can middle school students prove? Can they understand the difference between emprical argument and proof?

For the purpose of finding some answers to the problems above to some extent, the concept of proof could be acquired by 7^th grade students was aimed in this study. First of all in the study, applications that 7^th grade students can perform proof were focused on and after these performed applications, their perceptions and skills towards proof were attempted to be determined with the test and interview conducted subsequently.

Method

This study was designed as action research, which is one of the qualitative research approaches. As specified by Yıldırım and Şimşek (2005), this study can be considered under the title of "the trial of a new approach". Students of the 7th grade were applied proof teaching, which is an extracurricular title, and a new approach was trialed and it was aimed that obtained findings were presented to educators as information for the development of teaching practices and curriculums. In the data collection and analysis process of the action research, various techniques and methods can be used. Qualitative and quantitative data have been utilized together in this study. The data collected after the application has been subject to a descriptive analysis. The study was conducted with a total of 54 7th grade students. Exam 1 aiming to determine the perceptions of students on proof, exam 2 and 3 aiming to determine the skills of students on proving, a readiness test, and interview forms prepared as data collection instruments under the scope of this study. Each exams has 4 questions. First of all, a 14 week lesson plan was prepared for the application process prepared for developing the proof skills of students. In the application process commencing in November, 2012, the readiness test consisting of four questions was applied to both classes. Afterwards, a 14 week lesson application was performed in classes with 1 hour a week. In this process, the subjects of numbers, sequences, unity, pairs, and divisibility were focused on and examples of proof regarding these subjects were provided in the classroom. A discussion was made on the difference between mathematical verification by using emprical argument and proof over the examples discussed in the classroom for 14 weeks. Following the application that lasted for 14 weeks, 3 exams were applied to the students. After that, 16 students were selected in a manner covering the diversity of the answer provided by students of both classes and semi-structured in-depth interviews were conducted with these students. In these interviews, students were requested to explain and justify each answer and with the selected response, attempts were made to obtain more detailed information on their perception towards proof by questioning whether or not they reached a generalization.

Expected Outcomes

Data obtained under the scope of the study were analyzed in this paper on the basis of questions in the test. The data of the exams were examined and the data was encoded according to the responses of the students and presented in the form of a table. This data was detailed and supported with findings from the interviews. At the end of this study conducted, an increase was observed in the awareness of students on the difference between verification with an example (emprical argument) and proof. When the findings from the readiness test and the test after the application are compared, it was observed that students tended to prefer answers containing deductive reasoning. Contrary to this, an important proportion of the students continued to have the idea that the trial of a few conditions would be adequate for proof. Students with this idea have difficulties in understanding and applying algebraic expressions. As a result, they prefer verifications with examples for proof. Contrary to this, these students were included in the interviews conducted after the test. In the interviews conducted with them, it was observed that students could perceive the difference between verification with examples and proof sometimes with difficulty and long discussions and sometimes with ease with support and guidance of the investigator.

References

Ball, D.L.; Hoyles, C.; Jahnke, H.N.; Movshovitz-Hadar, N. (2002). The teaching of proof, Proceedings of the International Congress of Mathematicians (Ed. L.I.Tatsien), Vol. III, Higher Education Press, Beijing, 907–920. Bell, A. W. (1976). A Study Of Pupıls' Proof-Explanatıons In Mathematıcal Sıtuatıons, Educational Studies in Mathematics, 7, 23-40. Cyr, S. (2011). Development of beginning skills in proving and proof writing by elementary school students, Proceedings of the Seventh Congress of the European Society for Research in Mathematics Education, University of Rzeszów, Poland. Fischbein, E. (1982). Intuition and Proof. For The Learning of Mathematics, 3 (2), 9-18. Goetting, M. (1995). The college students ' understanding of mathematical proof. Unpublished doctoral dissertation. University of Maryland, College Park. Healy, L., & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 31 (4), 396-428. Knuth, E. J. (2002). Teachers’ conceptions of proof in the context of secondary school mathematics. Journal of Mathematics Teachers Education, 5, 61 – 88. NCTM (National Council of Teachers of Mathematics), (2000). Principles and standards for school mathematics, www.nctm.org Stylianides, A. J. (2007), Proof and Proving in School Mathematics, Journal for Research in Mathematics Education, 38 (3), 289-321. Stylianides, G. J., & Stylianides, A. J. (2009). Facilitating the transition from empirical arguments to proof. Journal for Research in Mathematics Education, 40 (3). Yıldırım, A.; Şimşek, H. (2005). Sosyal Bilimlerde Nitel Araştırma Yöntemleri (5. Baskı), Ankara: Seçkin Yayınevi.

Author Information

Ebru Aylar (presenting / submitting)

Ankara University

Departmant of Secondary School Science and Math. Education

Ankara

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