Academia.eduAcademia.edu

Outline

Title
Abstract
Figures
Introduction
Background
The Scope of the Paper
Method
Participants
Data and Procedure
Analysis
Limitations
Results
Discussion
References
All Topics
Education
Teacher Education and Professional Development

Preservice teachers’ knowledge of proof by mathematical induction

Profile image of Gabriel StylianidesGabriel Stylianides

2007, Journal of Mathematics Teacher Education

https://doi.org/10.1007/S10857-007-9034-Z
visibility

…

description

22 pages

link

1 file

Abstract

There is a growing effort to make proof central to all students' mathematical experiences across all grades. Success in this goal depends highly on teachers' knowledge of proof, but limited research has examined this knowledge. This paper contributes to this domain of research by investigating preservice elementary and secondary school mathematics teachers' knowledge of proof by mathematical induction. This research can inform the knowledge about preservice teachers that mathematics teacher educators need in order to effectively teach proof to preservice teachers. Our analysis is based on written responses of 95 participants to specially developed tasks and on semi-structured interviews with 11 of them. The findings show that preservice teachers from both groups have difficulties that center around: (1) the essence of the base step of the induction method; (2) the meaning associated with the inductive step in proving the implication P(k) ) P(k + 1) for an arbitrary k in the domain of discourse of P(n); and (3) the possibility of the truth set of a sentence in a statement proved by mathematical induction to include values outside its domain of discourse. The difficulties about the base and inductive steps are more salient among preservice elementary than secondary school teachers, but the difficulties about whether proofs by induction should be as encompassing as they could be are equally important for both groups. Implications for mathematics teacher education and future research are discussed in light of these findings.

Figures (5)
Table 1 Possible student responses to the two multiple-choice parts of Task 2 and possible student difficulties in relation to focal idea 3 “ “ ™ OF _ . _— wee not include n = 4.° We deliberately set up Task 2 in this way in order to create a rich context for examining further students’ knowledge of proof by mathematical induction. In light of different possible responses to the two multiple-choice parts of Task 2 (Parts A and C), we could draw inferences about students’ knowledge. In this paper, we focus on four possible student responses (summarized in Table 1) that help us examine student difficulties related to focal idea 3.
Table 1 Possible student responses to the two multiple-choice parts of Task 2 and possible student difficulties in relation to focal idea 3 “ “ ™ OF _ . _— wee not include n = 4.° We deliberately set up Task 2 in this way in order to create a rich context for examining further students’ knowledge of proof by mathematical induction. In light of different possible responses to the two multiple-choice parts of Task 2 (Parts A and C), we could draw inferences about students’ knowledge. In this paper, we focus on four possible student responses (summarized in Table 1) that help us examine student difficulties related to focal idea 3.
Table 2 Percentages of student responses to Part A of Task 1 by major Note. The values represent percentages (rounded to the nearest integer) within major. MM19: Here we use the induction method but we don’t check for n = 1. When we check for n = | we see that the equation doesn’t hold. I (interviewers): Why is it necessary to check for n = 1? MM19: We are supposed to check for n = 1 because the induction method states that a sentence is true for all natural numbers, from 1 up to ... for all x. If we don’t check for n= | and we begin by setting n = k then we begin from something arbitrary that we don’t know whether it is true. Therefore, the induction hypothesis that the equation holds for n = k is false and we cannot proceed any further to prove the equation for n =k + 1. Had the equation been true for n = 1, one would be able to make the induction hypothesis and say that, if it is true for n = 1 and n =k, and is proved to be true for n = k + 1, then the equation is true in general.
Table 2 Percentages of student responses to Part A of Task 1 by major Note. The values represent percentages (rounded to the nearest integer) within major. MM19: Here we use the induction method but we don’t check for n = 1. When we check for n = | we see that the equation doesn’t hold. I (interviewers): Why is it necessary to check for n = 1? MM19: We are supposed to check for n = 1 because the induction method states that a sentence is true for all natural numbers, from 1 up to ... for all x. If we don’t check for n= | and we begin by setting n = k then we begin from something arbitrary that we don’t know whether it is true. Therefore, the induction hypothesis that the equation holds for n = k is false and we cannot proceed any further to prove the equation for n =k + 1. Had the equation been true for n = 1, one would be able to make the induction hypothesis and say that, if it is true for n = 1 and n =k, and is proved to be true for n = k + 1, then the equation is true in general.
Table 3 Percentages of student responses to Part A of Task 2 by major Note. The values represent percentages (rounded to the nearest integer) within major. This comment about the meaning of the word ‘always’ with regard to the domain of discourse of the statement in Task 2 is at the heart of focal idea 3 we aimed to investigate. The data suggest that many EMs considered that the proposed proof showed the statement to be true in some cases, because they thought that a valid proof should show that: (1) the sentence in the statement is true for all natural numbers, or (2) the sentence is true for all natural numbers that belong to its truth set (in this particular case, {n|n € N, n > 4}).
Table 3 Percentages of student responses to Part A of Task 2 by major Note. The values represent percentages (rounded to the nearest integer) within major. This comment about the meaning of the word ‘always’ with regard to the domain of discourse of the statement in Task 2 is at the heart of focal idea 3 we aimed to investigate. The data suggest that many EMs considered that the proposed proof showed the statement to be true in some cases, because they thought that a valid proof should show that: (1) the sentence in the statement is true for all natural numbers, or (2) the sentence is true for all natural numbers that belong to its truth set (in this particular case, {n|n € N, n > 4}).
Table 4 Percentages of correct student responses to Part C of Task 2 by major Stud depends on whether the proof estab consideration on the entire set of natural numbers rather than on its domain of discourse. Stud EM4 9 rejected the proof because discourse for which the inequality inva ent EM20 seems to believe that the validity of a proof by mathematical induction ishes the truth of the mathematical relationship under ents EM49 and EMS2 rejected the validity of the proposed proof for opposite reasons. he found a value for n (= 4) outside the domain of is satisfied. He seemed to believe that the proof is id because it is not as encompassing as it could be. EM52 failed to see that the inequality is satisfied for n = 4 and shows that the statement is always true.’ He therefore appears to think that a valid proof wou poss ibly all natural numbers. considered that this violated the assertion ‘the proof d show the truth of the inequality over a broader set than its domain of discourse, Wien eee Sis FS seek Peat A eds siete acs: Sees: See eeresimceee® weep ee ce
Table 4 Percentages of correct student responses to Part C of Task 2 by major Stud depends on whether the proof estab consideration on the entire set of natural numbers rather than on its domain of discourse. Stud EM4 9 rejected the proof because discourse for which the inequality inva ent EM20 seems to believe that the validity of a proof by mathematical induction ishes the truth of the mathematical relationship under ents EM49 and EMS2 rejected the validity of the proposed proof for opposite reasons. he found a value for n (= 4) outside the domain of is satisfied. He seemed to believe that the proof is id because it is not as encompassing as it could be. EM52 failed to see that the inequality is satisfied for n = 4 and shows that the statement is always true.’ He therefore appears to think that a valid proof wou poss ibly all natural numbers. considered that this violated the assertion ‘the proof d show the truth of the inequality over a broader set than its domain of discourse, Wien eee Sis FS seek Peat A eds siete acs: Sees: See eeresimceee® weep ee ce
ell ee A significant number of students from both majors, 38 EMs and 23 MMs, recognizec the validity of the presented proof, thus responding correctly to Part A of Task 2. From these students, only 15 EMs and 13 MMs responded correctly to all four special cases o Part C (Response Type 0), while three EMs and one MM said that the inequality is true fo all special cases (Response Type 1). The latter response type suggests a belief that th proof showed the truth of the inequality for values outside its domain of discourse (pos sibly all natural numbers). Almost all other students who responded correctly to Part A 0; Task 2 (18 EMs and eight MMs) said that the inequality is false for n = 3 and n = 4, anc true for the other two special cases (Response Type 2). This response type is most likely associated with the misconception that the truth set cannot include elements outside th« domain of discourse. euaaeseoeaes ON eee The remaining two response types are associated only with EMs. From the nine EMs who considered the presented proof as invalid, two said that the inequality is false for all special cases (Response Type 3). These students probably believed that, because the proof ‘failed’ to prove the statement, the truth set of the inequality is the empty set. Finally, from the 20 EMs who said that the presented proof shows that the statement is true in some cases, eight said that the inequality is false for n = 3 and true for the other three special cases (Response Type 4). These students most probably thought that the domain of dis- course of the inequality should coincide with its truth set.
ell ee A significant number of students from both majors, 38 EMs and 23 MMs, recognizec the validity of the presented proof, thus responding correctly to Part A of Task 2. From these students, only 15 EMs and 13 MMs responded correctly to all four special cases o Part C (Response Type 0), while three EMs and one MM said that the inequality is true fo all special cases (Response Type 1). The latter response type suggests a belief that th proof showed the truth of the inequality for values outside its domain of discourse (pos sibly all natural numbers). Almost all other students who responded correctly to Part A 0; Task 2 (18 EMs and eight MMs) said that the inequality is false for n = 3 and n = 4, anc true for the other two special cases (Response Type 2). This response type is most likely associated with the misconception that the truth set cannot include elements outside th« domain of discourse. euaaeseoeaes ON eee The remaining two response types are associated only with EMs. From the nine EMs who considered the presented proof as invalid, two said that the inequality is false for all special cases (Response Type 3). These students probably believed that, because the proof ‘failed’ to prove the statement, the truth set of the inequality is the empty set. Finally, from the 20 EMs who said that the presented proof shows that the statement is true in some cases, eight said that the inequality is false for n = 3 and true for the other three special cases (Response Type 4). These students most probably thought that the domain of dis- course of the inequality should coincide with its truth set.

Related papers

Mathematical Induction in School: An Illusion of Rigor?
Francis Lowenthal

School Science and Mathematics, 1992

View PDFchevron_right
Secondary-level Student Teachers' Conceptions of Mathematical Proof
Dr. Thomas Varghese MSc, MSc, MEd, PhD

Recent reforms in mathematics education have led to an increased emphasis on proof and reasoning in mathematics curricula. The National Council of Teachers of Mathematics highlights the important role that teachers' knowledge and beliefs play in shaping students' understanding of mathematics, their confidence in and outlook on mathematics education, and their ability to use math to solve fundamental problems. It is crucial that teachers, especially the uninitiated, understand on a deep level the mathematical concepts that they are expected to teach to adolescents. Thus, it becomes critical for teacher educators to assess the understanding and abilities of student teachers in constructing mathematical proof. The analysis in this study is based on three factors: 1) meaning of proof, 2) ideas about teaching methods on proof, and 3) ideas about the use-fullness of proof in a mathematics classroom. An analysis of the data collected from this study indicates that current student teachers' conceptions of mathematical proof are limited. The uneasiness expressed by student teachers about mathematical proof may suggest an examination of students' experiences with the mathematical proof in both secondary and post secondary classrooms.

View PDFchevron_right
The Views of Elementary Mathematics Education Preservice Teachers on Proving
Ebru Aylar, Çiğdem Alkaş, Belma TÜRKER BİBER

This study has been prepared with the purpose to get the views of senior class Elementary Education Mathematics preservice teachers on proving. Data have been obtained via surveys and interviews carried out with 104 preservice teachers. According to the findings, although preservice teachers have positive views about using proving in mathematics teaching, it is seen that their experiences related to proving is limited to courses and they think proving is a work done only for the exams. Furthermore, they have expressed in the interviews that proving is difficult for them, and because of this reason they prefer memorizing instead of learning.

View PDFchevron_right
Quantitative Analysis of Pre-service Elementary Mathematics Teachers’ Opinions about Doing Mathematical Proof
Halil Eksi

Meaning and importance of proof in mathematics and education increases gradually. Therefore levels of doing proof, proof-related opinions and perceptions of the teachers and pre-service teachers who will train the students in future are of importance. Accordingly, this study aims to determine the proof-related opinions of pre-service elementary mathematics teachers who still study at different grade levels. In line with this purpose, a questionnaire developed under the title “Questionnaire for Constructing Mathematical Proof” was used to determine the pre-service teachers’ opinions about proof. The questionnaire comprises 27 items based on 5 point Likert-type. In the study, developmental research method was conducted and the questionnaire was applied to 187 pre-service elementary mathematics teachers from different grade levels. As a result of the study, it was revealed that pre-service teachers have positive views about proof. Also, the study revealed that confidence of pre-service teachers in proving is lower than mental process, self-assessment and belief, and attitude factors.

View PDFchevron_right
Developing Pre-service Mathematics Teachers’ Pedagogical Content Knowledge of Proof Schemes: An Intervention Study
Fikret Cihan

2019

The purpose of this paper is to examine the effects of an intervention study which aimed at improving pre-service mathematics teachers’ pedagogical content knowledge (PCK) of proof schemes. Sowder and Harel’s (1998) framework of proof schemes constitutes the conceptual framework of this paper. To explore twenty-two pre-service teachers’ PCK, we designed a survey using a scenario. The quantitative findings of the study revealed that the intervention had a meaningful and large effect on participants’ PCK of proof schemes. The qualitative findings of the study indicated that participants had difficulties identifying especially symbolic, example-based, transformational and axiomatic proof schemes prior to intervention, but they had overcome these difficulties after the intervention.

View PDFchevron_right
Preservice Mathematics Teachers’ Determination Skills of the Proof Techniques: The Case of Integers
Muhammet Doruk

International Journal of Education in Mathematics, Science and Technology, 2019

The aim of the study is to reveal the preservice mathematics teachers’ ability to determine the techniques of proofs on integers. A qualitative case study approach was adopted in this study. The participants of the study consisted of 172 preservice teachers enrolled in an elementary mathematics teaching program in their second and third years at a state university in Turkey. The data of the study were obtained from the Proof Techniques Determination Form (PTDF) which consists of six proofs on integers proven by different techniques and semi-structured interviews with five preservice teachers who were successful in different achievement levels of PTDF. The preservice teachers were asked to determine the proof techniques presented to them and express their warrants. The proof techniques used in PTDF are; direct proof, proof by induction, proof by contradiction, proof by contraposition, proof by counterexample and proof by confliction. At the end of the study, the preservice teachers w...

View PDFchevron_right
Preparing prospective secondary teachers to teach mathematical reasoning and proof: the case of the role of examples in proving
Orly Buchbinder

ZDM – Mathematics Education

Mathematics teacher education programs in the United States are charged with preparing prospective secondary teachers (PSTs) to teach reasoning and proving across grade levels and mathematical topics. Although most programs require a course on proof, PSTs often perceive it as disconnected from their future classroom practice. Our design research project developed a capstone course Mathematical Reasoning and Proving for Secondary Teachers and systematically studied its effect on PSTs’ content and pedagogical knowledge specific to proof. This paper focuses on one course module—Quantification and the Role of Examples in Proving, a topic which poses persistent difficulties to students and teachers alike. The analysis suggests that after the course, PSTs’ content and pedagogical knowledge of the role of examples in proving increased. We provide evidence from multiple data sources: pre-and post-questionnaires, PSTs’ responses to the in-class activities, their lesson plans, reflections on ...

View PDFchevron_right
The case of mathematical proof in lower secondary school :Knowledge and competencies of pre-service teachers
jill brown

2008

A case study of the preparedness of pre-service mathematics teachers to teach proof at lower secondary is reported. Data collected by questionnaire and problem-centred interviews were subjected to in-depth qualitative content analysis. Knowledge and competencies demonstrated by two future teachers are examined. Both displayed high affinity with proving and a formalist image of mathematics but their inferred abilities to convey this affinity to students at this point in their careers differed. Implications for teacher education and university mathematics are outlined.

View PDFchevron_right
Mathematical Induction, A Pedagogical Discussion
Paul Ernest

This paper discusses the nature of mathematical induction, and analyses it mathematically and psychologically, including behavioural and conceptual dimensions and prerequisites, and common errors and misconceptions. It analyses the treatment of MI in a selection of school and college texts (the selection is rather dated now). Overall a multifaceted pedagogical discussion of MI.

View PDFchevron_right

References (28)

  1. Brousseau, G., & Otte, M. (1991). The fragility of knowledge. In A. J. Bishop, S. Mellin-Olsen, & J. van Dormolen (Eds.), Mathematical knowledge: Its growth through teaching (pp. 13-36). Dordrecht, The Netherlands: Kluwer Academic Publishers.
  2. Cambridge Conference on Teacher Training. (1967). Goals for mathematical education of elementary school teachers. Boston, MA: Houghton Mifflin Company.
  3. Cooney, T. J. (1994). Teacher education as an exercise in adaptation. In D. B. Aichele, & A. F. Coxford (Eds.), Professional Development for Teachers of Mathematics (1994 Yearbook of the National Council of Teachers of Mathematics). Reston, VA: National Council of Teachers of Mathematics.
  4. Dubinsky, E. (1986). Teaching mathematical induction I. Journal of Mathematical Behavior, 5, 305-317.
  5. Dubinsky, E. (1990). Teaching mathematical induction II. Journal of Mathematical Behavior, 8, 285-304.
  6. Dubinsky, E., & Lewin, P. (1986). Reflective abstraction and mathematics education: The generic decomposition of induction and compactness. Journal of Mathematical Behavior, 5, 55-92.
  7. Fawcett, H. P. (1938). The nature of proof (1938 Yearbook of the National Council of Teachers of Math- ematics). New York: Bureau of Publications, Teachers College, Columbia University.
  8. Fischbein, E. (1982). Intuition and proof. For the Learning of Mathematics, 3(2), 9-18, 24.
  9. Hanna, G. (2000). Proof, explanation and exploration: an overview. Educational Studies in Mathematics, 44, 5-23.
  10. Harel, G. (2002). The development of mathematical induction as a proof scheme: A model for DNR-based instruction. In S. Campbell, & R. Zaskis (Eds.), Learning and teaching number theory: Research in cognition and instruction (pp. 185-212). New Jersey: Ablex Publishing Corporation.
  11. Healy, L., & Hoyles, C. (2000). Proof conceptions in algebra. Journal for Research in Mathematics Edu- cation, 31, 396-428.
  12. Knuth, E. J. (2002). Secondary school mathematics teachers' conceptions of proof. Journal for Research in Mathematics Education, 33, 379-405.
  13. Magnusson, S. (1996). Building a pedagogical content knowledge base for elementary science teacher education. Paper presented at the annual meeting of the National Association for Research in Science Teaching, St. Louis, MO.
  14. Maher, C. A., & Martino, A. M. (1996). The development of the idea of mathematical proof: A 5-year case study. Journal for Research in Mathematics Education, 27, 194-214.
  15. Marrades, R., & Gutie ´rrez, A. (2000). Proofs produced by secondary school students learning geometry in a dynamic computer environment. Educational Studies in Mathematics, 44, 87-125.
  16. Martin, W. G., & Harel, G. (1989). Proof frames of preservice elementary teachers. Journal for Research in Mathematics Education, 20, 41-51.
  17. Movshovitz-Hadar, N. (1993). The false coin problem, mathematical induction and knowledge fragility. Journal of Mathematical Behavior, 12, 253-268.
  18. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
  19. Patton, M. Q. (1990). Qualitative evaluation and research methods. Newbury Park, CA: Sage.
  20. Reid, D. A. (2002). Elements in accepting an explanation. Journal of Mathematical Behavior, 20, 527-547.
  21. Schoenfeld, A. H. (1994). What do we know about mathematics curricula?. Journal of Mathematical Behavior, 13, 55-80.
  22. Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14.
  23. Simon, M. A., & Blume, G. W. (1996). Justification in the mathematics classroom: A study of prospective elementary teachers. Journal of Mathematical Behavior, 15, 3-31.
  24. Smith, D. C. (2000). Content and pedagogical content knowledge for elementary science teacher educators: Knowing our students. Journal of Science Teacher Education, 11(1), 27-46.
  25. Sowder, L., & Harel, G. (1998). Types of students' justifications. The Mathematics Teacher, 91, 670-675.
  26. Steiner, H. (1989). The nature of theoretical concepts in physics and mathematics-implications for the fragility of knowledge in the educational context. In S. Vinner (Ed.), Proceedings of the Second International Jerusalem Convention on Education: Science and Mathematics Education -Interaction between Research and Practice (pp. 387-396). Israel: University of Jerusalem.
  27. Stylianides, A. J., Stylianides, G. J., & Philippou, G. N. (2004). Undergraduate students' understanding of the contraposition equivalence rule in symbolic and verbal contexts. Educational Studies in Mathe- matics, 55, 133-162.
  28. Sullivan, P. (2003). Editorial: Incorporating knowledge of, and beliefs about, mathematics into teacher education. Journal of Mathematics Teacher Education, 6, 293-296.

Related papers

PROOF BY MATHEMATICAL INDUCTION: PROFESSIONAL PRACTICE FOR SECONDARY TEACHERS
Gregory Hine

The Australian Association of Mathematics Teachers, 2017

Mathematical induction is a proof technique that can be applied to establish the veracity of mathematical statements. This professional practice paper offers insight into mathematical induction as it pertains to the Australian Curriculum: Mathematics (ACMSM065, ACMSM066) and implications for how secondary teachers might approach this technique to students. In particular, literature on proof – and specifically, mathematical induction – will be presented, and several worked examples will outline the key steps involved in solving problems. After various teaching and learning caveats have been explored, the paper will conclude with some mathematical induction example problems that can be used in the secondary classroom.

View PDFchevron_right
PRESERVICE SECONDARY MATHEMATICS TEACHERS' PERCEPTIONS OF PROOF IN THE SECONDARY MATHEMATICS CLASSROOM
Gregory Hine

Despite the recognized importance of mathematical proof in secondary education, there is a limited but growing body of literature indicating how preservice secondary mathematics teachers (PSMTs) view proof and the teaching of proof. The purpose of this survey research was to investigate how PSMTs in Australia and the United States perceive of proof in the context of secondary mathematics teaching and learning. PSMTs were able to outline various mathematical and pedagogical aspects of proof, including: purposes, characteristics, reasons for teaching, and imposed constraints. In addition, PSMTs attended to differing, though overlapping, features of proof when asked to determine the extent to which proposed arguments constituted proofs or to decide which arguments they might present to students.

View PDFchevron_right
Pre-service Secondary Mathematics Teachers' Behaviors in the Proving Process
Melike Yigit Koyunkaya

Pre-service secondary mathematics teachers' (PSMTs) understanding and ability of constructing a proof is not only important for their own learning process, but also important for these PSMTs to help their future students learn how to do proofs. Therefore, this study is focused on and explains PSMTs' behaviors that they revealed throughout the proving process of a proposition. In this qualitative case study, the participants were fifteen volunteer PSMTs from a public university in Turkey. The participants were given a proposition which they were asked to think aloud and prove it on the blackboard. The findings of this study show that PSMTs' behaviors and thoughts regarding the given proposition were limited. In detail, PSMTs had difficulties in application of mathematical language and notations, understanding the meaning of the given proposition, knowing where to get started on a proof, using examples efficiently, using appropriate and efficient methods to construct the proof, and defining logical structures of the proposition to construct the proof.

View PDFchevron_right
Baştürk, S. (2011). Perspectives of Turkish pre-service mathematics teachers of elementary level on mathematical proof. In Robert V. Nata (Ed.), Progress Education Vol.27 (pp. 61-82). Nova Science Publishers, Inc: New York.
Savaş Baştürk

The aim of this study is to investigate elementary pre-service mathematics teachers’ perspectives on proof in mathematics and mathematics teaching. A 5-point Likert-type questionnaire was administered in order to gather data. The sample of the study included 40 senior class pre-service mathematics teachers. The data collected were analyzed and interpreted using the methods of qualitative and quantitative analysis. To minimize the known negative consequences of close-ended questions and to go deeper in our survey by allowing respondents to include more information, eight pre-service teachers were selected to participate in follow-up interviews after the analysis of their answers to the questionnaire. The results have revealed that the pre-service teachers believe that mathematical proof plays a crucial role in the permanence of mathematical knowledge and at the same time provides students with understanding the sense of rules and formulas they should learn. However, teaching with proof does not guarantee an education without memorization as in their university courses. On the other hand, in their teaching the pre-service teachers think giving place to proofs with some conditions such as: proofs should be simple, understandable and students have to be willing and to have an adequate mathematics background.

View PDFchevron_right
Examining Pre-Service Mathematics Teachers’ Reasoning Errors, Deficiencies and Gaps in the Proof Process
Tugba OZTURK, Enes Demir, Bülent GÜVEN

Proving is a process that has important roles in terms of learning and teaching in almost all the areas of mathematics. Because the process of proof construction is an extensive process that includes skills as mathematical thinking, reasoning and making connections. Reasoning is one of the most important components of this process. However, most students have difficulty in making a good reasoning and they make various reasoning errors in the process. The purpose of the study is to investigate the reasoning errors that pre-service mathematics teachers exhibit during proof construction. This study was carried out with 80 university students from second, third, fourth and fifth grade levels. An open-ended exam based on abstract mathematics and algebra was used. To deeply examine reasoning errors in the proving process, clinical interviews were conducted with pre-service teachers. A scale was developed by considering the literature review and the expert opinions; this was used to analyse the data about the reasoning errors. The results illustrate that the reasoning errors mostly do not show differences for all grade levels. However, the percentages of reasoning errors according to the grade levels and to the upper classes these errors show resistance to decrease the deficiencies. It is important to design a learning environment enabling students to experience proof construction in order to reduce or eliminate the reasoning errors.

View PDFchevron_right

Related topics

  • Mathematics
  • Mathematics Teacher Education
  • Teacher knowledge
  • Preservice Teachers
  • Curriculum and Pedagogy

    • Cited by

    Beliefs and beyond: hows and whys in the teaching of proof
    FULVIA FURINGHETTI

    Zdm, 2011

    In this paper, we report on a study aimed at describing the way secondary school teachers treat proof and at understanding which factors may influence such a treatment. This study is part of a wider project on proof carried out for many years. In our theoretical framework, we combine references from research on proof with those from research on teachers in relation to their beliefs. The study was carried out through interviews with secondary school teachers aimed at learning how they describe their work with proof in the classroom, and to elicit beliefs and other factors that shape this work. Through the interviews we were able to detect reasons behind teachers’ choices in planning their work in the classroom. In the present paper, we concentrate on four cases that, among other factors, offer elements suitable to unravel the problem of inconsistencies using the construct of leading beliefs, i.e., beliefs (whose nature may vary from teacher to teacher) that seem to drive the way each teacher treats proof.

    View PDFchevron_right
    Türkiye’de İspat Üzerine Yapılan Çalışmaların Analizi: Bir Meta-Sentez Çalışması
    Damla Kutlu

    Pamukkale University Journal of Education

    Öz İspat yapma, üst düzey becerilerin aynı anda işe koşulmasına aracı olarak yaşamımızda karşılaşılan problemleri çözme ve olaylar arasında ilişki kurma gibi gerekliliklerin gerçekleşmesinde büyük bir rol oynamaktadır. Öğretim sürecinde matematiksel gerçeklerin sebepleri ile birlikte anlaşılması, matematiksel bilginin yapılandırılması ve bu bilginin kalıcı hale gelmesini sağlaması ispatın matematik eğitiminin merkezinde olduğunu gösterir niteliktedir. Bu önem, matematik eğitimine yön verme noktasında ispat çalışmalarının bütünsel bir bakış açısıyla incelenmesini anlamlı kılmaktadır. Bu araştırmanın amacı, Türkiye'de ispat çalışmalarını sistematik derleme yöntemiyle analiz etmek ve matematik eğitimi alanındaki eğilimi ortaya çıkarmaktır. Araştırma kapsamında amaçlı örnekleme yöntemiyle seçilen 49'u makale ve 55'i tez olmak üzere toplam 104 çalışma analiz edilmiştir. Çalışmaların seçiminde YÖK Ulusal Tez Merkezi, TÜBİTAK ULAKBİM DergiPark, Google Akademik arama motorundan faydalanılmıştır. Çalışmalar; çalışmanın yılı, amacı, yaklaşımı/yöntemi, örneklemi ve veri toplama araçları olmak üzere beş boyutta incelenmiştir. Araştırmanın verileri, frekans ve yüzdeye dayalı olarak yorumlanmış ve tablo ile grafiklerin kullanımıyla mevcut durum gösterilmiştir. Araştırmanın bulguları, son yıllarda ispat çalışmalarının sayısında kayda değer bir artış olduğunu göstermiştir. Ayrıca çalışmaların önemli bir bölümünün ispat sürecini ve ispata yönelik bakış açısını incelemeye yönelik olduğu tespit edilmiştir. En çok nitel yaklaşımın kullanıldığı, öğretmen adaylarıyla yapılan çalışmaların ağırlıkta olduğu, veri toplama aracı olarak çoğunlukla açık uçlu sınav/test ve mülakatın tercih edildiği şeklinde sonuçlara varılmıştır. Bu sonuçlara dayanarak her bir öğretim kademesi için ispat sürecinde yaşanan zorluklar ve bu zorlukların giderilmesine yönelik önlemler ile ispat öğretimine ilişkin sürecin tasarımına yönelik konuların aydınlatılması gerektiği önerilmiştir.

    View PDFchevron_right
    The Development of Students’ Mathematical Argumentation in a Primary Classroom
    Pi-Jen Lin

    Educação & Realidade, 2018

    The Development of Students' Mathematical Argumentation in a Primary Classroom 1. The use of valid argument does not come naturally. It is acquired only through practice. This study was accomplished to provide students intended opportunities of engaging activities for launching argumentation in primary classrooms. The focus of the paper is how argumentation was evolved when students engaged in the conjecturing incorporated into regular mathematics instruction over two consecutive years. Working with a group of 6 teachers was to develop conjecturing tasks and pedagogical strategies to support them in teaching, and then to enhance the quality of students' argumentation when 24 students were in grades 3 and 4. The collected data mainly consisted of conjecturing tasks, audio-and video-taped recordings of classroom observations, and students' worksheets. Results indicate that the evolution of argumentation was identified in two aspects: the characteristics and the quality of the argumentation.

    View PDFchevron_right
    580 California St., Suite 400
    San Francisco, CA, 94104
    © 2025 Academia. All rights reserved