Research Article
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Eighth Grade Students’ Understanding of Slope Concept

Year 2021, Volume: 10 Issue: 4, 1535 - 1561, 30.12.2021

Emine Aytekin Kazanç , Ece Acar Mine Işıksal

https://doi.org/10.30703/cije.874553

Abstract

In this study, it was aimed to investigate the eight grade students’ performance on drawing graphs of real world situations, to what extent they can interpret unit rate and rate of change concepts with its different representations (geometric and algebraic ratio, functional property), and how they relate different slope representations with each other. In addition, it was examined to what extent they can relate the given situations with the slope concept. Participants consisted of 158 eight-grade students who were selected from two public schools via convenience sampling. First, four-open ended and its sub-problems were used as data collection instrument. One of those problems included static situation and three of them included dynamic real world situations. While formulating sub-questions, different slope representations including physical property, functional property, algebraic ratio, geometric ratio and linear constant were considered. Next, task-based interviews were held with four students selected based on their responses to the four problems. The findings of this study showed that most of the students could calculate the rate of change or unit rate correctly however, they sketched graph by coordinating pairs and they used graphics as figurative while explaining representations of slope. It was also seen that students who had an understanding of geometric ratio in the static situation met a challenge of relating representation of geometric ratio with functional property and linear constant representation in the interviews. In addition, it was found that, students associated the slope concept more with the static context than dynamic contexts. Therefore, to make students to establish relations between different representations of slope, it is important to use the graphical displays of both dynamics and static situations as operative.

Keywords

Rate of change slope slope conceptualization middle school students steepness

References

  • Birgin, O. (2012). Investigation of eight-grade students’ understanding of slope of the linear function. Bolema, 26(42), 139–162. doi: 10.1590/S0103636X2012000100008
  • Carlson, M., Larsen, S. and Lesh, R. (2003). Integrating models and modeling perspective with existing research and practice. In R. Lesh and H. M. Doerr (Eds.), Beyond constructivism: Models and modelling perspective on mathematics problem solving, learning, and teaching (pp. 465-478). Mahwah, NJ: Lawrence Erlbaum.
  • Carlson, M., Oehrtman, M. and Engelke, N. (2010). The precalculus concept Assessment: A tool for assessing students’ reasoning abilities and understandings, Cognition and Instruction, 28(2), 113-145. doi: 10.1080/07370001003676587
  • Cho, P. and Nagle, C. (2017). Procedural and conceptual difficulties with slope: An analysis of students’mistakes on routine tasks. International Journal of Research in Education and Science, 3(1), 135–150
  • Clement, J. (1985). Misconceptions in graphing. Proceedings of the Ninth International Conference for the Psychology of Mathematics Education. The Netherlands.
  • Clement, J. (2000). Analysis of clinical interviews: Foundation and model viability. In A. E. Kelly and R. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 547-589). New Jersey: Lawrence Erlbaum.
  • Cohen, L., Manion, L. and Morrison, K. (2007). Research methods in education (6th ed.). London: Routledge.
  • Common Core State Standards Initiative (CCSSI). (2010). Common Core State Standards for Mathematics. Washington, DC.
  • Deniz, Ö. and Uygur-Kabael, T. (2017). Students’ mathematization process of the concept of slope within the realistic mathematics education. H.U. Journal of Education, 32(1), 123-142. doi: 10.16986/HUJE.2016018796
  • Dolores-Flores, C., Rivera-Lopez, M. I. and García-García, J. (2019). Exploring mathematical connections of pre-university students through tasks involving rates of change. International Journal of Mathematical Education in Science and Technology, 50(3), 369–389. doi:10.1080/0020739X.2018.1507050
  • Ellis, A., Ely, R., Singleton, B. and Tasova, H. (2020). Scaling-continuous variation: Supporting students’ algebraic reasoning. Educational Studies in Mathematics, 104(1), 87–103. doi: 10.1007/s10649-020-09951-6
  • Goldin, G. (2000). A scientific perspective on structures, task-based interviews in mathematics education research. In A. E. Kelly and R. Lesh (Eds.), Handbook of 21 research design in mathematics and science education (pp. 517–545). New Jersey: Lawrence Erlbaum.
  • Hattikudur, S., Prather, R., Asquith, P., Knuth, E., Nathan, M. and Alibali, M. (2011). Constructing graphical representations: Middle schoolers’ developing knowledge about slope and intercept. School Science and Mathematics, 112(4), 230–240.
  • Herbert, S. and Pierce, R. (2008). An “Emergent Model” for rate of change. International Journal of Computers for Mathematical Learning, 13, 231–249. doi: 10.1007/s10758-008-9140-8
  • Hunting, R.P., (1997). Clinical interview methods in mathematics education research and practice. Journal Of Mathematıcal Behavior, 16 (2), 145-165.
  • Koichu B. and Harel G. (2007). Triadic interaction in clinical task-based interviews with mathematics teachers Educational Studies in Mathematics, 65(3), 349-365. doi: 10.1007/S10649-006-9054-0
  • Lobato, J. and Siebert, D. (2002). Quantitative reasoning in a reconceived view of transfer. Journal of Mathematical Behavior, 21, 87-116.
  • Lobato, J. and Thanheiser, E. (2002). Developing understanding of ratio-asmeasure as a foundation of slope. In B. Litwiller and G. Bright (Eds.), Making sense of fractions, ratios, and proportions, (pp. 162-175). Reston, VA: The National Council of Teachers of Mathematics.
  • Lobato, J., Ellis, A.B. and Muñoz, R. (2003). How “focusing phenomena” in the instructional environment afford students’ generalizations. Mathematical Thinking and Learning, 5, 1-36. doi: 10.1207/S15327833MTL0501_01
  • Maher, C. A. and Sigley, R. (2020). Task-based interviews in mathematics education. In S. Lernman (Ed.), Encyclopedia of Mathematics Education. Dordrecth, Netherlands: Springer.
  • Maher, C. A. and Sigley, R. (2014). Task-Based Interviews in Mathematics Education. In S. Lerman (Ed.), Encyclopedia of Mathematics Education (pp. 579–582). Dordrecht: Springer Netherlands. doi: 10.1007/978-94-007-4978-8_147
  • Miles, M. B. and Huberman, A. M. (1994). Qualitative data analysis (2nd edition). Thousand Oaks, CA: Sage Publications.
  • Milli Eğitim Bakanlığı [MEB] (2018). Ortaokul matematik dersi (5, 6, 7 ve 8. sınıflar) öğretim programı. Ankara, Turkey.
  • Moore, K. C., Stevens, I. E., Paoletti, T., Hobson, N. L.F. and Liang, B. (2019). Pre-service teachers’ figurative and operative graphing actions. The Journal of Mathematical Behavior. 56. doi: 10.1016/j.jmathb.2019.01.008
  • Moore-Russo, D., Conner, A. and Rugg, K. (2011). Can slope be negative in 3-space? Studying concept image of slope through collective definition construction. Educational Studies in Mathematics, 76(1), 3-21.
  • Nagle, C. and Moore-Russo, D. (2014). Slope Across the Curriculum: Principles and Standards for School Mathematics and Common Core State Standards. The Mathematics Educator, 23(2), 40-59.
  • Nagle, C., Martínez-Planell, R. and Moore-Russo, D. (2019). Using APOS theory as a framework for considering slope understanding. Journal of Mathematical Behavior, 54, [100684]. doi: 10.1016/j.jmathb.2018.12.003
  • National Council of Teachers of Mathematics (NCTM). (2000). Principles and Standards for School Mathematics. Reston, VA: NCTM.
  • Planinic, M., Milin-Sipus, Z., Katic, H., Susac, A. and Ivanjek, L. (2012). Comparison of student understanding of line graph slope in physics and mathematics. International Journal of Science and Mathematics Education, 10, 1393–1414. doi: 10.1007/s10763-012-9344-1
  • Reiken, J.J. (2008). Coming to understand slope and the Cartesian connection: An investigation of student thinking (Doktora Tezi). ProQuest Dissertations and Theses veri tabanından erişildi. (UMI No. 9943436)
  • Schreier, M. (2012). Qualitative content analysis in practice. Thousand Oaks, CA: Sage. Smith, T.M., Seshaiyer, P., Peixoto, N., Suh, J.M., Bagshaw, G. and Collins, L.K. (2013). Exploring slope with stairs and steps. Mathematics Teaching in Middle School, 18(6), 370-377. doi:10.5951/mathteacmiddscho.18.6.0370
  • Stanton, M. and Moore‐Russo, D. (2012). Conceptualizations of slope: A review of state standards. School Science and Mathematics, 112(5), 270-277.
  • Strauss, A. and Corbin, J. (1998). Basics of qualitative research: Techniques and procedures for developing grounded theory (2nd ed.). Thousand Oaks, CA: Sage.
  • Stump, S. (2001a). Developing pre service teachers' pedagogical content knowledge of slope. Journal of Mathematical Behaviour, 20, 207-227.
  • Stump, S. L. (1997). Secondary mathematics teachers' knowledge of the concept of slope. .Chicago, IL.
  • Stump, S. L. (2001b). High school precalculus students’ understanding of slope as measure. School Science and Mathematics, 101(2).
  • Tanışlı, D. and Bike-Kalkan, D. (2018). Linear functions and slope: How do students understand these concepts and how does reasoning support their understanding? Croatian Journal of Education, 20 (4), 1193-1260.
  • Teuscher, D. and Reys, R. (2010). Slope, rate of change, and steepness: Do students understand these concepts? Mathematics Teacher, 103, 519-524.
  • Thacker, I. (2020). An embodied design for grounding the mathematics of slope in mid-dle school students’ perceptions of steepness. Research in Mathematics Education, 22(3), 304–28. doi: 10.1080/14794802.2019.1692061
  • Thompson, P. W. and Carlson, M. P. (2017). Variation, covariation, and functions: Foundational ways of thinking mathematically. In J. Cai (Ed.), First compendium for research in mathematics education (pp. 421–456). Reston, VA: National Council of Teachers of Mathematics. U.S. Department of Education (2013). National Assessment of Educational Progress (NAEP): NAEP questions tool. Erişim adresi (20 Mart 2017): http:// nces.ed.gov/nationsreportcard/itmrlsx/search.aspx?subjectmathematics
  • Walter, J. G. and Gerson, H. (2007). Teachers' personal agency: Making sense of slope through additive structures. Educational Studies in Mathematics, 65, 205-233.

8. Sınıf Öğrencilerinin Eğim Kavramına Yönelik Kavrayışları

Year 2021, Volume: 10 Issue: 4, 1535 - 1561, 30.12.2021

Emine Aytekin Kazanç , Ece Acar Mine Işıksal

https://doi.org/10.30703/cije.874553

Abstract

Bu çalışmada, 8. Sınıf öğrencilerinin gerçek yaşam durumlarını grafik üzerinde gösterme performanslarının, eğim kavramının temelindeki birim oran veya değişim oranını farklı temsilleriyle (geometrik ve cebirsel oran, fonksiyonel özellik) ne derece anlamlandırabildiklerinin ve farklı eğim temsillerini birbiri ile nasıl ilişkilendirdiklerinin incelenmesi amaçlanmıştır. Ek olarak, öğrencilerin verilen durumları eğim kavramı ile ilişkilendirme düzeyleri incelenmiştir. Katılımcılar, iki devlet okulundan kolay ulaşılır örneklem yöntemiyle seçilmiş; 158 sekizinci sınıf öğrencisinden oluşmaktadır. İlkin, veri toplama aracı olarak; dört açık uçlu problem ve alt soruları kullanılmıştır. Bu problemlerden biri durağan durumu, diğer üçü dinamik gerçek yaşam durumlarını içermektedir. Alt sorular oluşturulurken; farklı eğim temsilleri (fiziksel özellik, fonksiyonel özellik, cebirsel oran, geometrik oran ve doğrusal sabit) dikkate alınmıştır. Daha sonra, bu problemlere verdikleri cevaplara göre seçilen dört öğrenci ile görev temelli görüşmeler yapılmıştır. Çalışmanın bulguları, öğrencilerin çoğunun, değişim oran veya birim oran’ı doğru hesaplayabildiklerini fakat grafiği, noktaları koordine ederek çizdiklerini ve eğim temsillerini açıklarken grafiği görsel olarak kullandıklarını göstermiştir. Ayrıca görüşmelerde, durağan durumda, geometrik oran temsilini anlamlandırabilen öğrencilerin geometrik oran ile fonksiyonel özellik temsilini ilişkilendirmede ve doğrusal sabit temsilinde zorlandıkları görülmüştür. Ek olarak, öğrencilerin eğim kavramını dinamik durumlara nispeten durağan durum ile daha fazla ilişkilendirdikleri bulunmuştur. Bu nedenle, öğrencilerin farklı eğim temsilleri arası ilişkiyi kurabilmeleri için, hem dinamik hem de durağan durumların grafik gösterimlerinin işlevsel olarak kullanımı önemlidir.

Keywords

Değişim oranı eğim eğim temsilleri ortaokul öğrencileri

References

  • Birgin, O. (2012). Investigation of eight-grade students’ understanding of slope of the linear function. Bolema, 26(42), 139–162. doi: 10.1590/S0103636X2012000100008
  • Carlson, M., Larsen, S. and Lesh, R. (2003). Integrating models and modeling perspective with existing research and practice. In R. Lesh and H. M. Doerr (Eds.), Beyond constructivism: Models and modelling perspective on mathematics problem solving, learning, and teaching (pp. 465-478). Mahwah, NJ: Lawrence Erlbaum.
  • Carlson, M., Oehrtman, M. and Engelke, N. (2010). The precalculus concept Assessment: A tool for assessing students’ reasoning abilities and understandings, Cognition and Instruction, 28(2), 113-145. doi: 10.1080/07370001003676587
  • Cho, P. and Nagle, C. (2017). Procedural and conceptual difficulties with slope: An analysis of students’mistakes on routine tasks. International Journal of Research in Education and Science, 3(1), 135–150
  • Clement, J. (1985). Misconceptions in graphing. Proceedings of the Ninth International Conference for the Psychology of Mathematics Education. The Netherlands.
  • Clement, J. (2000). Analysis of clinical interviews: Foundation and model viability. In A. E. Kelly and R. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 547-589). New Jersey: Lawrence Erlbaum.
  • Cohen, L., Manion, L. and Morrison, K. (2007). Research methods in education (6th ed.). London: Routledge.
  • Common Core State Standards Initiative (CCSSI). (2010). Common Core State Standards for Mathematics. Washington, DC.
  • Deniz, Ö. and Uygur-Kabael, T. (2017). Students’ mathematization process of the concept of slope within the realistic mathematics education. H.U. Journal of Education, 32(1), 123-142. doi: 10.16986/HUJE.2016018796
  • Dolores-Flores, C., Rivera-Lopez, M. I. and García-García, J. (2019). Exploring mathematical connections of pre-university students through tasks involving rates of change. International Journal of Mathematical Education in Science and Technology, 50(3), 369–389. doi:10.1080/0020739X.2018.1507050
  • Ellis, A., Ely, R., Singleton, B. and Tasova, H. (2020). Scaling-continuous variation: Supporting students’ algebraic reasoning. Educational Studies in Mathematics, 104(1), 87–103. doi: 10.1007/s10649-020-09951-6
  • Goldin, G. (2000). A scientific perspective on structures, task-based interviews in mathematics education research. In A. E. Kelly and R. Lesh (Eds.), Handbook of 21 research design in mathematics and science education (pp. 517–545). New Jersey: Lawrence Erlbaum.
  • Hattikudur, S., Prather, R., Asquith, P., Knuth, E., Nathan, M. and Alibali, M. (2011). Constructing graphical representations: Middle schoolers’ developing knowledge about slope and intercept. School Science and Mathematics, 112(4), 230–240.
  • Herbert, S. and Pierce, R. (2008). An “Emergent Model” for rate of change. International Journal of Computers for Mathematical Learning, 13, 231–249. doi: 10.1007/s10758-008-9140-8
  • Hunting, R.P., (1997). Clinical interview methods in mathematics education research and practice. Journal Of Mathematıcal Behavior, 16 (2), 145-165.
  • Koichu B. and Harel G. (2007). Triadic interaction in clinical task-based interviews with mathematics teachers Educational Studies in Mathematics, 65(3), 349-365. doi: 10.1007/S10649-006-9054-0
  • Lobato, J. and Siebert, D. (2002). Quantitative reasoning in a reconceived view of transfer. Journal of Mathematical Behavior, 21, 87-116.
  • Lobato, J. and Thanheiser, E. (2002). Developing understanding of ratio-asmeasure as a foundation of slope. In B. Litwiller and G. Bright (Eds.), Making sense of fractions, ratios, and proportions, (pp. 162-175). Reston, VA: The National Council of Teachers of Mathematics.
  • Lobato, J., Ellis, A.B. and Muñoz, R. (2003). How “focusing phenomena” in the instructional environment afford students’ generalizations. Mathematical Thinking and Learning, 5, 1-36. doi: 10.1207/S15327833MTL0501_01
  • Maher, C. A. and Sigley, R. (2020). Task-based interviews in mathematics education. In S. Lernman (Ed.), Encyclopedia of Mathematics Education. Dordrecth, Netherlands: Springer.
  • Maher, C. A. and Sigley, R. (2014). Task-Based Interviews in Mathematics Education. In S. Lerman (Ed.), Encyclopedia of Mathematics Education (pp. 579–582). Dordrecht: Springer Netherlands. doi: 10.1007/978-94-007-4978-8_147
  • Miles, M. B. and Huberman, A. M. (1994). Qualitative data analysis (2nd edition). Thousand Oaks, CA: Sage Publications.
  • Milli Eğitim Bakanlığı [MEB] (2018). Ortaokul matematik dersi (5, 6, 7 ve 8. sınıflar) öğretim programı. Ankara, Turkey.
  • Moore, K. C., Stevens, I. E., Paoletti, T., Hobson, N. L.F. and Liang, B. (2019). Pre-service teachers’ figurative and operative graphing actions. The Journal of Mathematical Behavior. 56. doi: 10.1016/j.jmathb.2019.01.008
  • Moore-Russo, D., Conner, A. and Rugg, K. (2011). Can slope be negative in 3-space? Studying concept image of slope through collective definition construction. Educational Studies in Mathematics, 76(1), 3-21.
  • Nagle, C. and Moore-Russo, D. (2014). Slope Across the Curriculum: Principles and Standards for School Mathematics and Common Core State Standards. The Mathematics Educator, 23(2), 40-59.
  • Nagle, C., Martínez-Planell, R. and Moore-Russo, D. (2019). Using APOS theory as a framework for considering slope understanding. Journal of Mathematical Behavior, 54, [100684]. doi: 10.1016/j.jmathb.2018.12.003
  • National Council of Teachers of Mathematics (NCTM). (2000). Principles and Standards for School Mathematics. Reston, VA: NCTM.
  • Planinic, M., Milin-Sipus, Z., Katic, H., Susac, A. and Ivanjek, L. (2012). Comparison of student understanding of line graph slope in physics and mathematics. International Journal of Science and Mathematics Education, 10, 1393–1414. doi: 10.1007/s10763-012-9344-1
  • Reiken, J.J. (2008). Coming to understand slope and the Cartesian connection: An investigation of student thinking (Doktora Tezi). ProQuest Dissertations and Theses veri tabanından erişildi. (UMI No. 9943436)
  • Schreier, M. (2012). Qualitative content analysis in practice. Thousand Oaks, CA: Sage. Smith, T.M., Seshaiyer, P., Peixoto, N., Suh, J.M., Bagshaw, G. and Collins, L.K. (2013). Exploring slope with stairs and steps. Mathematics Teaching in Middle School, 18(6), 370-377. doi:10.5951/mathteacmiddscho.18.6.0370
  • Stanton, M. and Moore‐Russo, D. (2012). Conceptualizations of slope: A review of state standards. School Science and Mathematics, 112(5), 270-277.
  • Strauss, A. and Corbin, J. (1998). Basics of qualitative research: Techniques and procedures for developing grounded theory (2nd ed.). Thousand Oaks, CA: Sage.
  • Stump, S. (2001a). Developing pre service teachers' pedagogical content knowledge of slope. Journal of Mathematical Behaviour, 20, 207-227.
  • Stump, S. L. (1997). Secondary mathematics teachers' knowledge of the concept of slope. .Chicago, IL.
  • Stump, S. L. (2001b). High school precalculus students’ understanding of slope as measure. School Science and Mathematics, 101(2).
  • Tanışlı, D. and Bike-Kalkan, D. (2018). Linear functions and slope: How do students understand these concepts and how does reasoning support their understanding? Croatian Journal of Education, 20 (4), 1193-1260.
  • Teuscher, D. and Reys, R. (2010). Slope, rate of change, and steepness: Do students understand these concepts? Mathematics Teacher, 103, 519-524.
  • Thacker, I. (2020). An embodied design for grounding the mathematics of slope in mid-dle school students’ perceptions of steepness. Research in Mathematics Education, 22(3), 304–28. doi: 10.1080/14794802.2019.1692061
  • Thompson, P. W. and Carlson, M. P. (2017). Variation, covariation, and functions: Foundational ways of thinking mathematically. In J. Cai (Ed.), First compendium for research in mathematics education (pp. 421–456). Reston, VA: National Council of Teachers of Mathematics. U.S. Department of Education (2013). National Assessment of Educational Progress (NAEP): NAEP questions tool. Erişim adresi (20 Mart 2017): http:// nces.ed.gov/nationsreportcard/itmrlsx/search.aspx?subjectmathematics
  • Walter, J. G. and Gerson, H. (2007). Teachers' personal agency: Making sense of slope through additive structures. Educational Studies in Mathematics, 65, 205-233.
There are 41 citations in total.

Details

Primary Language Turkish
Journal Section Research Article
Authors

Emine Aytekin Kazanç 0000-0003-1068-4306

Ece Acar 0000-0002-4248-7561

Mine Işıksal 0000-0001-7619-1390

Publication Date December 30, 2021
Published in Issue Year 2021Volume: 10 Issue: 4

Cite

APA Aytekin Kazanç, E., Acar, E., & Işıksal, M. (2021). 8. Sınıf Öğrencilerinin Eğim Kavramına Yönelik Kavrayışları. Cumhuriyet Uluslararası Eğitim Dergisi, 10(4), 1535-1561. https://doi.org/10.30703/cije.874553
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