THE IMPORTANCE OF ALGEBRA TEACHING; DAILY LIFE VARIABLES AND NUMBER SYSTEMS CORRESPONDING TO THESE VARIABLES

Abstract

The aim of the study is to determine which number systems do the mathematics teachers use to associate the daily life variables in daily life problems, and if they have associated, to find whether they used the algebraic features of the number systems they associated. In the research, the participants' ability to transform between number sets and daily life variables and their using of the features of the algebraic structure of these number sets were tried to be investigated in detail. With this purpose, the case study method, one of the qualitative research patterns, was employed in the research. The research was carried out with 15 secondary school mathematics teachers, studying at the Post-graduate Department of the Secondary School Mathematics Teaching in the 2014-2015, 2015-2016 and 2016-2017 academic years. For this reason, the study was designed on the use and awareness of knowledge. As the data collection tools, the mid-term and final exams papers were applied. The results suggest that the penny and algebraic features of number systems and the concept of "equation" are not correctly formed, therefore which number systems are the daily life variables valued in is not exactly known.

Keywords: student awareness, algebra teaching, variable, mathematics in daily life, teacher education.

REFERENCES 

Akay, H., Soybaş, D., & Argün, Z. (2006). Problem posing experiences and using open-ended questions in mathematics teaching. Kastamonu Education Journal, 14(1), 129-146.

Akgün, L., & Özdemir, M. E. (2006). Students' understanding of the variable as general number and unknown: A case study. The teaching of mathematics, (16), 45-51.

Akkan, Y., Çakıroğlu, Ü., & Güven, B. (2009). Equation forming and problem posing abilities of 6th and 7th grade primary school students. Mehmet Akif Ersoy University Journal of Education Faculty, 17,  41-55.

Akkaya, R. (2006). The Effectiveness of activity based approach in overcoming misconception in the field of learning algebra appreared among the sixth students (Unpublished master’s thesis).  Abant İzzet Baysal University, Bolu.

Akkaya, R., & Durmuş, S. (2006). Misconceptions of elementary school students in grades 6-8 on learning algebra. Hacettepe University Journal of Education, 31(31), 1-12.

Akkuş, O, & Çakıroğlu, E. (2006). Seventh grade students' use of multiple representations in pattern related algebra tasks. Hacettepe University Journal of Education, 31(31).

Aksu, M. (1997). Student performance in dealing with fraction. The Journal of Educational Research, 90(6), 375- 380.

Apsari, R. A., Putri, R. İ. İ., Sariyasa, S., Abels, M., & Prayitno, S. (2020). Geometry representation to develop algebraic thinking: A recommendation for a pattern investigation in pre-algebra class. Journal on Mathematics Education, 11(1), 45-58.

Arzarello, F., Bazzini, L., & Chiappini, C. (1993). Cognitive processes in algebraic thinking: Towards a theoretical framework. İn Proceedings of the 17th İnternational Conference for the Psychology of Mathematics Education (Vol. 1, pp. 138-145).

Bağdat, O. (2013). Investigation of the 8th grade students' algebraic thinking skills with solo taxonomy (Unpublished master’s thesis). Eskişehir Osman Gazi University, Eskişehir.

Baki, A., & Kartal, T. (2002). Lise öğrencilerinin cebir bilgilerinin kavramsal ve işlemsel bilgi bağlamında değerlendirilmesi. In UFBMEK Proceedings Book, 211.

Baki, A., & Kartal, T. (2004). Characterizing high school students’ algebra knowledge in terms of procedural and conceptual knowledge. The Journal of Turkish Educational Sciences, 2(1), 27-46.

Ball, D. L., & Cohen, D. K. (1996). Reform by the book: What is—or might be—the role of curriculum materials in teacher learning and instructional reform? Educational researcher, 25(9), 6-14.

Barbieri, C. A., Miller-Cotto, D., & Booth, J. L. (2019). Lessening the load of misconceptions: Design-based principles for algebra learning. Journal of the Learning Sciences, 28(3), 381-417.

Barlow, A. T., & Cates, J. M. (2006). The impact of problem posing on elementary teachers' beliefs about mathematics and mathematics teaching. School Science and Mathematics, 106(2), 64-73.

Barnard, J. J. (1989). Poor concept formation in mathematics: A diagnostic perspective. (Report No. RGN/HSRC-P-105). Pretoria, South Africa: Human Sciences Research Council. (ERİC Document Reproduction Service. No. ED 310 926)

Başgün, M., & Ersoy, Y. (2000). Sayılar ve aritmetik-I: kesir ve ondalık sayıların öğretilmesinde bazı güçlükler ve yanılgılar [Numbers and arithmetic-i: some difficulties and mistakes in teaching fractions and decimals]. In IV. Fen Bilimleri Eğitimi Kongresi (ss. 604-608). Ankara: MEB Yay.

Bedel, A., & Arı, R. (2012). The effect of interpersonal problem solving skills training on the adolescents\'constructive problem-solving level and the level of trait anger. Elementary Education Online, 11(2), 440-451.

Birenbaum, M., Kelly, A. E., & Tatsuoka, K. K. (1993). Diagnosing knowledge states in algebra using the rule-space model. Journal for Research in Mathematics Education, 24(5), 442–459. doi: 10.2307/749153

Birgin, O., & Demirören, K. (2020). Investigation of the preservice science teachers’ views and suggestions on the application of the cooperative learning model in distance education environments. International Journal of Social and Educational Sciences, 7(14), 233-247. doi: 10.20860/ijoses.797472

Birgin, O., & Gürbüz, R. (2009). İlköğretim İİ. kademe öğrencilerinin rasyonel sayilar konusundaki işlemsel ve kavramsal bilgi düzeylerinin incelenmesi [Primary education II. investigation of level students' operational and conceptual knowledge levels on rational numbers]. Journal of Uludag University Faculty of Education, 22(2), 529-550.

Blanton, M. L., & Kaput, J. J. (2005). Characterizing a classroom practice that promotes algebraic reasoning. Journal for Research in Mathematics Education, 36(5), 412-446.

Blume, G. W., & Heckman, D. S. (2000). Algebra and functions. In E. Silver & P. Kenney (Ed.), Results from the seventh mathematics assessment (pp. 269-306). Reston, VA: National Council of Teachers of Mathematics.

Brown, S. I., & Walter, M. I. (1993). Problem posing in mathematics education. In S. I. Brown & M. I. Walter (Eds.), Problem posing: Reflection and applications (pp. 16-27). Hillsdale, NJ: Lawrence Erlbaum Associates.

Cai, J. (2003). Singaporean students' mathematical thinking in problem solving and problem posing: an exploratory study. İnternational Journal of Mathematical Education in Science And Technology, 34(5), 719-737.

Cankoy, O., & Darbaz, S. (2010). Effect of a problem posing based problem solving instruction on understanding problem. Hacettepe University Journal of Education, 38, 11-24.

Cenkseven, F., & Akar Vural, R. (2006). Comparing adolescents percieved problem solving skills according to needfor cognition and gender. Eurasian Journal of Educational Research (EJER), (25).

Çağdaşer, B. T. (2008). The effects of constructivist algebra education on the algebraic thinking levels of the 6th grade students (Unpublished master’s thesis). Uludağ University, Bursa.

Çelik, D. (2007). Analytical examination of the preservice teachers' algebraic thinking skills (Unpublished master’s thesis). Karadeniz Technical University, Trabzon.

Dede, Y., & Argün, Z. (2003). Why do students have difficulty with algebra? Hacettepe University Journal of Education, 24, 180-185.

Dede, Y., & Peker, M. (2007). Students’ errors and misunderstanding towards algebra: Pre-service mathematics teachers’ prediction skills of error and misunderstanding and solution suggestions. Elementary Education Online, 6(1), 35-49.

Dede, Y., Yalın, H. İ., & Argün, Z. (2002, Eylül). İlköğretim 8. sınıf öğrencilerinin değişken kavramının öğrenimindeki hataları ve kavram yanılgıları. Paper presented in V. Ulusal Fen Bilimleri ve Matematik Eğitimi Kongresi.

Demirtaş, H. ve Dönmez, B. (2008). Secondary school teachers’ perceptions about their problem solving abilities. İnönü University Journal of the Faculty of Education, 9(16), 177-198.

Driscoll, M. (1999). Fostering Algebraic Thinking: A Guide for Teachers, Grades 6-10. Heinemann, 361 Hanover Street, Portsmouth, NH 03801-3912.

Durmuş, S. (2005). Primary school students’ conception about rational number division. Sakarya University Journal of Education Faculty, 9, 97- 109.

English, L. D. (1997). The development of fifth-grade children's problem-posing abilities. Educational studies in Mathematics, 34(3), 183-217.

Erbaş, A. K., Çetinkaya, B., & Ersoy, Y. (2009). Student difficulties and misconceptions in solving simple linear equations. Education and Science, 34(152), 44-59.

Erbaş, A. K., Kertil, M., Çetinkaya, B., Cakiroglu, E., Alacaci, C., & Bas, S. (2014). Mathematical modeling in mathematics education: basic concepts and approaches. Educational Sciences: Theory and Practice, 14(4), 1621-1627.

Erbaş, A. K., Çetinkaya, B., & Ersoy, Y. (2010). Student difficulties and misconceptions in solving simple linear equations. Education and Science, 34(152), 44-59.

Erdem, Z. Ç., & Gürbüz, R. (2017). An analysis on students’ mistakes and misconceptions: the case of equations. YYU Journal of Education Faculty, 14(1), 640-670. doi: 10.23891/efdyyu.2017.25

Ersoy, Y., & Erbaş, A. K. (2005). Kassel projesi cebir testinde bir grup Türk öğrencinin genel başarısı ve öğrenme güçlükleri [The overall achievement and learning of a group of Turkish students in the Kassel project algebra test difficulties]. Elementary Education Online, 4(1), 18-40.

Etcuban, J. O., Campanilla, B. S., & Horteza, A. D. (2019). The use of Mathcad in the achievement of education students in teaching College Algebra in a university. İnternational Electronic Journal of Mathematics Education, 14(2), 341-351. doi: 10.29333/iejme/5718

Ferretti, F. (2020). The manipulation of algebraic expressions: deepening of a widespread difficulties and new characterizations. İnternational Electronic Journal of Mathematics Education, 15(1), Article No: em0548. doi: 10.29333/iejme/5884

Fidan, N. K. (2008). Teachers' Views with regard to the use of tools and materials in the Primary level. Journal of Theoretical Educational Science, 1(1), 48-61.

Genç, S. Z. ve Kalafat, T. (2007). The research on evaluatıon of prospectıve teachers’ democratıc attıtudes and problem solvıng skılls according as different. Pamukkale University Journal of Education, 22 (2), 10-22.

Gravemeijer, K., & Stephan, M. (2002). Emergent models as an instructional design heuristic. İn Symbolizing, modeling and tool use in mathematics education (pp. 145-169). Springer, Dordrecht.

Gülpek, P. (2006). Development of algebraic thinking levels of primary of school 7th and 8th grade students (Unpublished master’s thesis). Uludağ University, Bursa.

Haines, C., & Crouch, R. (2001). Recognizing constructs within mathematical modelling. Teaching Mathematics and İts Applications: İnternational Journal of the İMA, 20(3), 129-138.

Haines, C., & Crouch, R. (2007). Mathematical modelling and applications: Ability and competence frameworks. İn Modelling and applications in mathematics education (pp. 417-424). Springer, Boston, MA.

Herbert, K., & Brown, R. H. (1997). Patterns as tools for algebraic reasoning. Teaching Children Mathematics 3(6), 340-344.

Herscovics, N., & Linchevski, L. (1994). A cognitive gap between arithmetic and algebra. Educational studies in mathematics, 27(1), 59-78.

Işık, C., Kar, T., Yalçın, T., & Zehir, K. (2011). Prospective teachers’ skills in problem posing with regard to different problem posing models. Procedia-Social and Behavioral Sciences, 15, 485-489. doi: 10.1016/j.sbspro.2011.03.127

İspir, O. A., & Palabıyık, U. (2011). The effects of pattern-based algebra ınstruction on students’ algebraic thinking and attitude towards mathematics. Pamukkale University Journal of Education, 30(30), 111-123.

Kaf, Y. (2007). The effects of using model in mathematics on sixth grade students? algebra achievement (Unpublished master’s thesis). Hacettepe Universty, Ankara.

Karataş, İ. & Güven, B. (2003). 8. sınıf öğrencilerinin problem çözme sürecinde kullandığı bilgi türlerinin analizi. Matematikçiler Derneği Bilim Köşesi. [www. matder.org.tr.]

Kaş, S. (2010). The effects of teaching with worksheets on the eighth grades students? algebraic thinking and problem solving skills (Unpublished master’s thesis). Marmara Universty, İstanbul.

Kaya, D., & Keşan, C. (2014). The importance of the algebraic thinking and algebraic reasoning skills for primary school students. İnternational Journal of New Trends in Arts, Sports & Science Education (İJTASE), 3(2), 38-48.

Kertil, M. (2008). Investigating problem solving ability of pre-service mathematics teachers in modeling process (Unpublished master’s thesis). Marmara Universty, İstanbul.

Keşan, C., & Akbulut, E. S. (2019). The effect of using smart boards on eliminating 7th grade middle school students misconceptions in algebra. İnternational Journal of New Trends in Arts, Sports & Science Education (İJTASE), 8(1), 17-33.

Kieran, C. (1992). The Learning and Teaching of School Algebra. In D. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 390-419). New York: Macmillan Publishing Company.

Kieran, C., & Chalouh, L. (1993). Prealgebra: The transition from arithmetic to algebra. In D. T. Owens (Ed.), Research ideas for the classroom: Middle grades mathematics (pp. 179-198). New York: Macmillan.

Kim, Y. (2020). Relationship of mathematical knowledge for teaching and mathematical quality in instruction: Focus on high schools. The Mathematical Education, 59(3), 237-254.

Korkmaz, E., & Gür, H., (2006). Determining of prospective teachers’ problem posing skills. Journal of Balıkesir University Institute of Science and Technology, 8(1), 65-74.

Küchemann, D. (1978). Children’s understanding of numerical variables. Mathematics in Schools, 7, 24-28.

Lavy, İ.& Bershadsky, İ. (2003). Problem posing via ‘‘what if not?’’ strategy in solid geometry a case study. Journal of Mathematical Behavior, 22, 369–387.

Lawrence, A., & Hennessy, C. (2002). Lessons for Algebraic Thinking(Grades 6-8). Math Solutions Publications: Sausalito, CA.

Lehrer, R., & Schauble, L. (2003). Origins and evaluation of model-based reasoning in mathematics and science. İn R. Lesh, & H. M. Doerr (Eds.), Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching (pp. 59-70). Mahwah, NJ: Lawrence Erlbaum.

Lesh, R., & Doerr, H. M. (2003). Foundations of a models and modeling perspective on mathematics teaching, learning, and problem solving. İn R. Lesh, & H. M. Doerr (Eds.), Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching (pp. 3-33). Mahwah, NJ: Lawrence Erlbaum.

Linchevski, L. (1995). Algebra with numbers and arithmetic with letters: A definition of pre-algebra. Journal of Mathematical Behavior,14, 113–120.

Lingefjard, T. (2002). Mathematical modeling for preservice teachers: A problem from anesthesiology. İnternational Journal of Computers for Mathematical Learning, 7, 117-143. doi: 10.1023/A:1021122431218

Lingefjard, T. (2004). Assessing engineering student’s modeling skills. Retrieved from http://wvvw.cdio.org/files/document/file/assess_model_skls.pdf.

Lingefjard, T. (2006). Faces of mathematical modelling. Zentralblatt Für Didactik Der Mathematic, 38(2), 96-112. doi: 10.1007/BF02655884

Lingefjard, T., & Holmquist, M. (2005). To assess students’ attitudes, skills and competencies in mathematical modeling. Teaching Mathematics and İts Applications, 24(2-3), 123-133.

Litke, E. (2020). The nature and quality of algebra instruction: using a content-focused observation tool as a lens for understanding and improving instructional practice. Cognition and İnstruction, 38(1), 57-86.

Macnair, R. R., & Elliot, T. R. (1992). Self-perceived problem-solving ability, stress appraisal, and coping over time. Journal of Research in Personality, 26, 150-164.

McMillan, J. H. (2000). Educational research: Fundamentals for the consumer (4th ed.). White Plains, NY: Addison Wesley Longman, İnc.

Mayer, R. E. (1982). Memory for algebra story problems. Journal of educational psychology, 74(2), 199.

Moses, B.M., Bjork, E. & Goldenberg, E. P. (1993). Beyond problem solving: problem posing. İn S. İ. Brown & M. İ. Walter, (Eds.), Problem posing: reflections and applications (1st ed.) (pp. 178-188). USA: Lawrence Erlbaum Associates

Moses, B.M., Bjork, E., & Goldenberg, E.P. (1990). Beyond problem solving: Problem po-sing. İnT. J. Cooney (Ed.), Teaching and learning mathematics in the 1990’s (pp. 82-91). Reston, VA: National Counsil of Teachers of Mathematics.

Moss, D. L., & Lamberg, T. (2019). Conceptions of expressions and equations in early algebra: A learning trajectory. İnternational Journal for Mathematics Teaching and Learning, 20(2), 170-192.

Nardone, C. F., & Lee, R. G. (2010). Critical inquiry across the disciplines: Strategies for student-generated problem posing. College Teaching, 59(1), 13-22.

Niss, M., Blum, W., & Galbraith, P. L. (2007). İntroduction. İn W. Blum, P. Galbraith, H. Henn, & M. Niss (Eds.), Modelling and applications in mathematics education: The 14th İCMİ study (pp. 3-32). New York: Springer

Nixon-Ponder, S. (1995). Using problem-posing dialogue: İn adult literacy education. Adult learning, 7(2), 10-12.

Palabıyık, U. (2010). The effects of pattern based algebra teaching on students? algebraic thinking and attitude towards mathematics (Unpublished master’s thesis). Hacettepe Universty, Ankara.

Pawley, D., Ayres, P., Cooper, M. & Sweller, J. (2005). Translatingwords into equation: A cognitive load theory approach. Educational Psychology, 25, 27-97

Polat, R. H. & Tümkaya, S. (2010). An investigation of the students of primary school problem solving abilities depending on need for cognition. Elementary Education Online, 9(1), 346-360.

Powell, S. R., Gilbert, J. K., & Fuchs, L. S. (2019). Variables influencing algebra performance: Understanding rational numbers is essential. Learning and İndividual Differences, 74, 101758.

Sahal, M., & Özdemir, A. S. (2019). Examination of activities and problems oriented toward algebra readiness indicators proposed by pre-service mathematics teachers. Acta Didactica Napocensia, 12(1), 165-176.

Saracaloğlu, A. S., Yenice, N. & Karasakaloğlu, N. (2009). The relationship between communication and problem solving skills and reading interest and habits of candidate teachers’. Van Yuzuncu Yil Universty Journal of Education, 6 (2), 187-206.

Saracaloğlu, A., S., Serin, O. & Bozkurt, N. (2001). The relationship between the problem solving skills and the achievement of graduate students of the graduate school of educational sciences. Marmara University Atatürk Education Faculty Journal of Educational Sciences, 14, 121-134.

Serin, O. (2006). The examination of primary school teachers’ problem solving skills ın terms of various variables. Education and Science, 31 (142), 80-88.

Silver, E. A. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing. Zdm, 29(3), 75-80.

Silver, E. A., & Cai, J. (1996). An analysis of arithmetic problem posing by middle school students. Journal for research in mathematics education, 27(5)521-539.

Steele, D. F., & Johanning, D. İ. (2004). A schematic–theoretic view of problem solving and development of algebraic thinking. Educational Studies in Mathematics, 57(1), 65-90.

Stein, M. K., & Henningsen, M. (1997). Mathematical tasks and student cognition: Classroom-based. Journal for research in mathematics education, 28(5), 524-549.

Stewart, S., Troup, J., & Plaxco, D. (2019). Reflection on teaching linear algebra: Examining one instructor’s movements between the three worlds of mathematical thinking. ZDM, 51(7), 1253-1266.

Styers, J. L., Nagle, C. R., & Moore-Russo, D. (2020). Teachers’ noticing of students’ slope statements: attending and interpreting. Canadian Journal of Science, Mathematics and Technology Education, 20(3), 504-520.

Stylianides, A. J., & Stylianides, G. J. (2008). Studying the classroom implementation of tasks: High-level mathematical tasks embedded in ‘real-life’contexts. Teaching and Teacher Education, 24(4), 859-875.

Sun, X. H., Xin, Y. P., & Huang, R. (2019). A complementary survey on the current state of teaching and learning of Whole Number Arithmetic and connections to later mathematical content. ZDM, 51(1), 1-12.

Tatar, E. & Soylu, Y. (2006). A study to determine effect of the achievement of reading-comprehension on the mathematics achievements. Kastamonu Education Journal, 14(2), 503-508

Toluk-Uçar, Z. (2009). Developing pre-service teachers understanding of fractions through problem posing. Teaching and Teacher Education, 25(1), 166-175.

Vance, J. H. (1998). Number operations from an algebraic perspective. Teaching Children Mathematics, 4(5), 282-286.

Verschaffel, L., & De Corte, E. (1997). Teaching realistic mathematical modeling and problem solving in the elementary school. A teaching experiment with fifth graders. Journal for Research in Mathematics Education, 28(5), 577-601.

Verschaffel, L., Greer, B., & De Corte, E. (2002). Everyday knowledge and mathematical modeling of school word problems. In K. P. Gravemeijer, R. Lehrer,H. J. van Oers, & L. Verschaffel (Eds.), Symbolizing, modeling and tool use in mathematics education (pp. 171-195). Dordrecht, The Netherlands: Kluwer Academic Publishers.

Wagner, S. (1983). What are these things called variables? The mathematics teacher, 76(7), 474-479.

Walle, J. A., Karp, K., & Bay-Williams, J. (2013). Teaching mathematics in the 21st century. Elementary and middle school mathematics: Teaching developmentally, 1-12.

Wilkie, K. J. (2019). The challenge of changing teaching: investigating the interplay of external and internal influences during professional learning with secondary mathematics teachers. Journal of Mathematics Teacher Education, 22(1), 95-124.

Yıldırım, A. & Şimşek, H. (2008) Sosyal bilimlerde nitel araştırma yöntemleri, 9. Baskı, Seçkin Yayıncılık, Ankara.

Yıldız, D. G., & Akyüz, D. (2020). Mathematical knowledge of two middle school mathematics teachers in planning and teaching pattern generalization. Elementary Education Online, 19(4), 2098-2117. doi: 10.17051/ilkonline.2020.763457

Yüksel, D., & Argün, Z. (2003). Why do students have difficulty with algebra? Hacettepe University Journal of Education, 24, 180-185.