Examining pre-service mathematics teachers’ reasoning errors, deficiencies and gaps in the proof process

Enes Demir 1, Tugba Ozturk 2 * , Bulent Guven 2
More Detail
1 Department of Mathematics Education, Siirt University, Siirt, Turkey
2 Department of Mathematics Education, Karadeniz Technical University, Trabzon, Turkey
* Corresponding Author
EUR J SCI MATH ED, Volume 6, Issue 2, pp. 44-61. https://doi.org/10.30935/scimath/9522
OPEN ACCESS   1605 Views   946 Downloads
Download Full Text (PDF)

ABSTRACT

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 constructions 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.

CITATION

Demir, E., Ozturk, T., & Guven, B. (2018). Examining pre-service mathematics teachers’ reasoning errors, deficiencies and gaps in the proof process. European Journal of Science and Mathematics Education, 6(2), 44-61. https://doi.org/10.30935/scimath/9522

REFERENCES

  • Almeida, D. (2003). Engenderingproofattitudes: Can thegenesis of mathematicalknowledgeteach us anything?International Journal of Mathematical Education in ScienceandTechnology, 34(4), 479-488.
  • Altıparmak, K., &Öziş, T. (2005). Matematiksel ispat ve matematiksel muhakemenin gelişimi üzerine bir inceleme [An investigationuponmathematicalproofanddevelopment of mathematicalreasoning]. Ege Eğitim Dergisi, 6(21), 25-7.
  • Anderson, J. (1996). The place of proof in school mathematics. Mathematics Teaching, 155, 33−39.
  • Andrew, L. (2009). Creating a prooferrorevaluationtoolforuse in thegrading of student-generated “Proofs”. PRIMUS: Problems, Resources, andIssues in MathematicsUndergraduateStudies, 19(5), 447-462. doi: 10.1080/10511970701765070
  • Aydoğdu-İskenderoğlu, T., & Baki, A. (2011). Quantitativeanalysis of pre-service elementarymathematicsteachers’ opinionsaboutdoingmathematicalproof. EducationalSciences: TheoryandPractice, 11(4), 2285-2290.
  • Baki, A. (2008). Kuramdan uygulamaya matematik eğitimi[From theory to practice on mathematics education].Ankara, Turkey: Harf Eğitim Publishing.
  • Baki, A. (1999). Öğretmen eğitimi üzerine düşünceler [Thoughts on teacher education]. Türk Yurdu, 19(138), 4-9.
  • Ball, D. L., Hoyles, C., Jahnke, H. N., &Movshovitz-Hadar, N. (2002). Theteaching of proof. In L.I. Tatsien (Ed.), Proceedings of the International Congress of Mathematicians(Vol. III, pp. 907–920). Beijing, China: HigherEducationPress.
  • Birinci, K. S. (2010). Matematik öğretmen adaylarının ispatlama performanslarının süreç-nesne ilişkisi açısından incelenmesi [Investigation of mathematics teacher candidates' proof processes in terms of process-object relationship](Master’sthesis). Marmara University, Istanbul, Turkey.
  • Flores, A. (2002). How do childrenknowthatwhattheylearn in mathematics is true?TeachingChildrenMathematics, 8(5), 269–274.
  • Hanna, G. (2000). Proof, explanationandexploration: An overview. EducationalStudies in Mathematics, 44, 5–23.
  • Haynes, F. (1997). Teachingtothink. AustralianJournal of TeacherEducation, 22(1), 1 – 23.
  • Herbst, P., & Brach, C. (2006). Proving and doing proofs in high school geometry classes: What is it that is going on for students?Cognition and Instruction, 24(1), 73-122.
  • Hoyles, C. (1997). The curricular shaping of students' approaches to proof. For the Learning of Mathematics, 17(1), 7- 16.
  • Jones, K. (2000). Thestudentexperience of mathematicalproof at universitylevel. International Journal of Mathematical Education in ScienceandTechnology, 31(1), 53-60.
  • Knuth, E. J. (2002). Proof as a tool for learning mathematics. Mathematics Teacher, 95(7), 486–490.
  • Lin, F. L., Yang, K. L., &Chen, C. Y. (2004). Thefeaturesandrelationships of reasoning, proving, andunderstandingproof in numberpatterns. International Journal of ScienceandMathematicsEducation, 2, 227–256.
  • Martinez, M. V., &Superfine, A. C. (2012).Integratingalgebraandproof in highschool: Students' workwithmultiplevariablesand a singleparameter in a proofcontext.Mathematical Thinkingand Learning, 14(2), 120-148. doi: 10.1080/10986065.2012.657956
  • Milli Eğitim Bakanlığı [MEB]. (2013). Ortaöğretim matematik dersi (9, 10, 11 ve 12. sınıflar) öğretim programı [Curriculum for secondary school mathematics (9th ,10th , 11th and 12th grades]. Ankara, Turkey: Milli Eğitim Press.
  • NationalCouncil of Teachers of Mathematics [NCTM]. (2000). Principlesandstandardsforschoolmathematics. Reston, VA: Author.
  • Pedemonte, B. (2007). How can therelationshipbetweenargumentationandproof be analysed? EducationalStudies in Mathematics, 66, 23-41.
  • Rav, Y. (1999). Why do we prove theorems?Philosophia Mathematica, 7(1), 5-41.
  • Sarı-Uzun, M. (2013). Matematik öğretmen adaylarının kanıtlama becerilerini geliştirmeye yönelik bir öğretme deneyi [A teachingexperiment on development of pre-service mathematicsteachers’ provingskills].Eğitim ve Bilim [Education and Science], 38(169), 372-392.
  • Schoenfeld, A. H. (2009). Serieseditor'sforeword. InD. A. Stylianou, M. L. Blanton& E. J. Knuth (Eds.), Teachingandlearningproofacrossthegrades (pp. xii-xvi). New York, NY: Routledge.
  • Selden, A., & Selden J. (2007). Overcomingstudents’difficulties in learningtounderstandandconstructproofs(Report No. 2007-1). Cookeville, TN: TennesseTechnologicalUniversity.
  • Selden, A., & Selden, J. (2003). Validations of proofsconsidered as texts: Can undergraduatestellwhether an argumentproves a theorem?JournalforResearch in MathematicsEducation, 34(1), 4-36.
  • Selden, A., McKee, K., & Selden, J. (2010). Affect, behaviouralschemasandtheprovingprocess. International Journal of Mathematical Education in ScienceandTechnology, 41(2), 199-215.
  • Selden, J., & Selden, A. (2009). Teachingprovingbycoordinatingaspects of proofswithstudents’abilities. In D. A. Stylianou, M. L. Blanton& E. J. Knuth (Eds.), Teachingandlearningproofacrossgrades: A K-16 perspective (pp. 339-354). New York, NY: Routledge.
  • Sowder, L., &Harel, G. (2003). Case studies of mathematicsmajors' proofunderstanding, production, andappreciation. CanadianJournal of Science, Mathematics, andTechnologyEducation, 3(2), 251-267.
  • Stylianides, A. J., &Stylianides, G. J. (2009). Proofconstructionsandevaluations. EducationalStudies in Mathematics, 72(2), 237-253.
  • Tall, D., &Mejia-Ramos, J. P. (2006, October). Thelong-termcognitivedevelopment of differenttypes of reasoningandproof. Paperpresented at Conference on ExplanationandProof in Mathematics: PhilosophicalandEducationalPerspectives, UniversitatDuisburg-Essen.
  • Weber, K. (2001). Studentdifficulty in constructingproofs: Theneedforstrategicknowledge. EducationalStudies in Mathematics, 48, 101–119.
  • Weber, K. (2004). Traditionalinstruction in advancedmathematicscourses: A casestudy of oneprofessor‘slecturesandproofs in an introductoryrealanalysiscourse. Journal of Mathematical Behavior, 23, 115–133.
  • Weber, K. (2005a). A proceduralroutetowardunderstandingaspects of proof: Case studiesfromrealanalysis, CanadianJournal of Science, MathematicsandTechnologyEducation, 5(4), 469-483.
  • Weber, K. (2005b). Problem-solving, proving, andlearning: Therelationshipbetween problem-solvingprocessesandlearningopportunities in theactivity of proofconstruction. Journal of Mathematical Behaviour, 24, 351-360.
  • Yackel, E., & Hanna, G. (2003). Reasoningandproof. In J. Kilpatrick, W. G. Martin & D. Schifter (Eds.), A researchcompaniontoPrinciples and Standards for School Mathematics (pp. 227-236).Reston, VA: National Council of Teachers of Mathematics.