Investigating Seventh-Grade Students’ Slope Preconceptions

Crisólogo Dolores-Flores; Martha Iris Rivera-López; Adán Tlalmanalco-Ramírez

doi:10.29333/ejmste/11417

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Crisólogo Dolores-Flores ¹ ^* , Martha Iris Rivera-López ¹ , Adán Tlalmanalco-Ramírez ¹

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¹ Research Center in Mathematics Education, Faculty of Mathematics, Autonomous University of Guerrero, MEXICO^* Corresponding Author

Abstract

This paper reports the results of research in which the objective was to explore the preconceptions of slope in seventh-grade students. Preconceptions are understood as students´ knowledge prior to the formal teaching of a certain concept. For data collection, task-based interviews composed of ten tasks applied to 21 Mexican students were used. The data analysis was carried out using the Thematic Analysis method. Results indicate that the students have several preconceptions in which they consider the slope as any of the following: an intersection with the X or Y-axis, an arithmetic operation, a length, an object, a height, and something to do. These findings pose the challenge of achieving conceptual changes from these preconceptions. In this sense, science education has been the field most exploited in mathematics education; a collaboration between teachers and researchers from both fields could contribute to finding strategies to face this challenge.

Keywords

seventh-grade
slope preconceptions
task-based interviews
thematic analysis

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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article Type: Research Article

EURASIA J Math Sci Tech Ed, Volume 17, Issue 12, December 2021, Article No: em2053

https://doi.org/10.29333/ejmste/11417

Publication date: 12 Dec 2021

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Article Downloads: 1474

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Declaration of Conflict of Interest: No conflict of interest is declared by author(s).

Data sharing statement: Data supporting the findings and conclusions are available upon request from the corresponding author(s).

References

Alonso, V., González, A., & Sáenz, O. (1988). Estrategias operativas en la resolución de problemas matemáticos en el ciclo medio de EGB [Operational strategies in solving mathematical problems in the middle cycle of EGB]. Enseñanza de las Ciencias, 6(3), 251-264. https://doi.org/10.5565/rev/ensciencias.5102
Assad, D. A. (2015). Task-based interviews in mathematics: Understanding student strategies and representations through problem solving. International Journal of Education and Social Science, 2(1), 17-26.
Barr, G. (1981). Some student ideas on the concept of gradient. Mathematics in School, 10(1), 14-17.
Beichner, R. J. (1994). Testing student interpretation of kinematics graphs. American Journal of Physics, 62(8), 750-762. https://doi.org/10.1119/1.17449
Bell, A., & Janvier, C. (1981). The interpretation of graphs representing situations. For the Learning of Mathematics, 2(1), 34-42.
Birgin, O. (2012). Investigation of eighth-grade students’ understanding of the slope of the linear function. Bolema, 26(42), 139-162. https://doi.org/10.1590/S0103-636X2012000100008
Braun, V., & Clarke, V. (2012). Thematic analysis. In H. Cooper (Ed.), Handbook of Research Methods in Psychology (pp. 57-71), American Psychological Association. https://doi.org/10.1037/13620-004
Brown, T. L., & Hirschfeld, G. (2007). Students’ conceptions of assessment and mathematics: Self-regulations raises achievement. Australian Journal of Education & Development Psychology, 7, 63-74.
Byerley, C., & Thompson, P. W. (2017). Secondary mathematics teachers’ meanings for measure, slope, and rate of change. The Journal of Mathematical Behavior, 48, 168-193. https://doi.org/10.1016/j.jmathb.2017.09.003
Campanario, J. M., & Otero, J. (2000). Más allá de las ideas previas como dificultades de aprendizaje: las pautas de pensamiento, las concepciones epistemológicas y las estrategias metacognitivas de los alumnos de Ciencias. Enseñanza de las Ciencias, 18(2), 155-169. https://doi.org/10.5565/rev/ensciencias.4036
Carpenter, T. P., Corbitt, M. K., Kepner, H. S., Lindquist, M. M., & Reys, R. (1980). Results of the second NAEP mathematics assessment: Secondary school. The Mathematics Teacher, 73(5), 329-338. https://doi.org/10.5951/MT.73.5.0329
Carpenter, T. P., Fennema, E., & Franke, M. L. (1996). Cognitively guided instruction: A knowledge base for reform in primary mathematics instruction. Elementary School Journal, 97, 3-20. https://doi.org/10.1086/461846
Cheng, D., & Sabinin, P. (2008). Elementary students’ conceptions of steepness. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano, & A. Sepúlveda (Eds.). Proceedings of the Joint Meeting of PME 32 and PME-NA XXX, International Group for the Psychology of Mathematics Education (Vol. 2, pp. 297-304). Cinvestav IPN, UMSNH.
Chhabra, M., & Baveja, B. (2012). Exploring minds: alternative conceptions in science. Procedia Social and Behavioral Sciences, 55(2012), 1069-1078. https://doi.org/10.1016/j.sbspro.2012.09.599
Chi, M. T. H. (2008). Three types of conceptual change: Belief revision, mental model transformation, and categorical shift. In S. Vosniadou (Ed.), Handbook of research on conceptual change (pp. 61-82). Erlbaum.
Cho, P., & 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., Brown, D. E., & Zietsman, A. (1989). Not all preconceptions are misconceptions: Finding ‘anchoring conceptions’ for grounding instruction on students’ intuitions. International Journal of Science Education, 11(5), 554-565. https://doi.org/10.1080/0950069890110507
Cobb, P., Wood, T., & Yackel, E. (1994). Discourse, mathematical thinking, and classroom practice. In Contexts for learning: Sociocultural dynamics in children’s development. Oxford University Press
Confrey, J. (1990). A review of research on student conceptions in mathematics, science, and programming. Review of Research in Education, 16, 3-55. https://doi.org/10.3102/0091732X016001003
Crawford, A. R., & Scott. W. E. (2000). Making sense of slope. The Mathematics Teacher, 93(2), 114-118. https://doi.org/10.5951/MT.93.2.0114
Deniz, Ö., & Kabael, T. (2017). 8th grade students’ processes of the construction of the concept of slope. Education and Science, 42(192), 139-172. https://doi.org/10.15390/EB.2017.6996
Dolores, C., Alarcón, G., & Albarrán, D. F. (2002). Concepciones alternativas sobre las gráficas cartesianas del movimiento: el caso de la velocidad y la trayectoria [Alternative conceptions on cartesian graphs of motion: The case of velocity and path]. Revista Latinoamericana de Investigación en Matemática Educativa, 5(3), 225-250.
Dolores, C., Chi, A. G., Canul, E. R., Cantú, C. A., & Pastor, C. G. (2009). De las descripciones verbales a las representaciones gráficas. El caso de la rapidez de la variación en la enseñanza de la matemática [From verbal descriptions to graphic representations. The case of rapid variation in mathematics teaching]. UNION. Revista Iberoamericana de Educación Matemática, 18, 41-57.
Dolores, C., García, J., & Gálvez, A. (2017). Estabilidad y cambio conceptual acerca de las razones de cambio en situación escolar [Stability and conceptual change about the reasons for change in school situation]. Educación Matemática, 29(2), 125–158. https://doi.org/10.24844/em2902.05
Dolores, C., Rivera, M. I., & 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. https://doi.org/10.1080/0020739X.2018.1507050
Dolores, C., Rivera, M. I., & Moore-Russo, D. (2020). Conceptualizations of slope in Mexican intended curriculum. School Science and Mathematics, 120(2), 104-115. https://doi.org/10.1111/ssm.12389
Driver, R., Squires, A., Rushworth, P., Valerie, A., & Robinson, W. (1994). Making sense of secondary science. research into children’s ideas. Routledge. https://doi.org/10.4324/9780203823583
Flick, U. (2004). Triangulation in qualitative research. In U. Flick, E. von Kardoff, & I. Steinke (Eds.), A companion to qualitative research (pp. 178-183). Sage.
Fujii, T. (2014). Misconceptions and alternative conceptions in mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 625-627). Springer. https://doi.org/10.1007/978-94-007-4978-8_114
Fuson, K. C., Kalchman, M., & Bransford, J. D. (2005). How students learn: Mathematics in the classroom. In M. S. Donovan & J. D. Bransford (Eds.), Mathematical understanding: An introduction (pp. 217-398). National Academies Press.
Gess-Newsome, J. (1999). Pedagogical content knowledge: An introduction and orientation. In Examining pedagogical content knowledge (pp. 3-17). Springer.
Goldin, G. A. (2000). Scientific perspective on structured, task-based interviews in mathematics education research. In A. E. Kelly & R. A. Lesh, (Eds.), Handbook of research design in mathematics and science education (pp. 517-545). Lawrence Erlbaum Associates.
Greenes, C., Chang, K. Y., & Ben-Chaim, D. (2007). International survey of high school students’ understanding of key concepts of linearity. In J. H. Woo, H. C. Lew, K. S. Park, & D. Y. Seo (Eds.), Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 273-280). PME.
Hattikudur, S., Prather, R. W., Asquith, P., Alibali, M. W., Knuth, E. J., & Nathan, M. (2012). Constructing graphical representations: Middle schoolers’ developing knowledge about slope and intercept. School Science and Mathematics, 112(4), 230-240. https://doi.org/10.1111/j.1949-8594.2012.00138.x
Hoban, R. A. (2021). Introducing the slope concept. International Journal of Mathematical Education in Science and Technology, 52(6), 879-895. https://doi.org/10.1080/0020739X.2020.1723726
Johnston, J. (2005). Early explorations in science (2nd ed.). Open University Press.
Kambouri, M. (2016). Investigating early years teachers’ understanding and response to children’s preconceptions. European Early Childhood Education Research Journal, 24(6), 907-927. https://doi.org/10.1080/1350293X.2014.970857
Kambouri, M., Briggs, M., & Cassidy, M. (2011). Children’s misconceptions and the teaching of early years science: A case study. Journal of Emergent Science, 2(2), 7-16.
Knuth, E. J. (2000). Student understanding of the Cartesian connection: An exploratory study. Journal for Research in Mathematics Education, 31(4), 500-507. https://doi.org/10.2307/749655
Leinhardt, G., Zaslavsky, O., & Stein, M. K. (1990). Functions, graphs, and graphing: Tasks, learning, and teaching. Review of Educational Research, 60(1), 1-64. https://doi.org/10.3102/00346543060001001
Lin, J. W., Yen, M. H., Liang, J. C., Chiu, M. H., & Guo, C. J. (2016). Examining the factors that influence students’ science learning processes and their learning outcomes: 30 years of conceptual change research. Eurasia Journal of Mathematics Science and Technology Education, 12(9), 2617-2646. https://doi.org/10.12973/eurasia.2016.000600a
Mahmud, M. C., & Gutiérrez, O. A. (2010). Estrategia de enseñanza basada en el cambio conceptual para la transformación de ideas previas en el aprendizaje de las Ciencias [Teaching strategy based on conceptual change for the transformation of previous ideas in science learning]. Formación universitaria, 3(1), 11-20. https://doi.org/10.4067/S0718-50062010000100003
McDermott, L. C., Rosenquist, M. L., & van Zee, E. H. (1987). Student difficulties in connecting graphs and physics: Examples from kinematics. American Journal of Physics, 55(6), 503-513. https://doi.org/10.1119/1.15104
Merenluoto, K., & Lehtinen, E. (2002). Conceptual change in mathematics: understanding the real numbers. In M. Limon, & L. Mason (Eds.), Reconsidering conceptual change: Issues in theory and practice (pp. 233-258). Kluwer Academic Publishers.
Moore-Russo, D., Conner, A., & 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. https://doi.org/10.1007/s10649-010-9277-y
Moschkovich, J. (1990). Students’ interpretations of linear equations and their graphs. In R. Speiser, C. Maher, & C. Walter. (Eds.), Proceedings of the 14th Annual Meeting of the International Group for the Psychology of Mathematics Education (pp. 109-116). North American Chapter.
Nadelson, L. S., Heddy B. C, Jones, S., Taasoobshirazi, G., & Johnson, M. (2018). Conceptual change in science teaching and learning: Introducing the dynamic model of conceptual change. International Journal of Educational Psychology, 7(2), 151-195. https://doi.org/10.17583/ijep.2018.3349
Nagle, C., & Moore-Russo, D. (2013). The concept of slope: Comparing teachers’ concept images and instructional content. Investigations in Mathematics Learning, 6(2), 1-18. https://doi.org/10.1080/24727466.2013.11790330
Nagle, C., & Moore-Russo, D. (2014). Slope across the curriculum: Principles and standards for school mathematics and common core state standards. Mathematics Educator, 23(2), 40-59.
Nagle, C., Martínez-Planell, R., & Moore-Russo, D. (2019). Using APOS theory as a framework for considering slope understanding. Journal of Mathematical Behavior, 54, 100684. https://doi.org/10.1016/j.jmathb.2018.12.003
Narjaikaewa, P. (2013). Alternative conceptions of primary school teachers of science about force and motion. Procedia Social and Behavioral Sciences, 88, 250-257. https://doi.org/10.1016/j.sbspro.2013.08.503
NCTM. (2000). Principles and standards for school mathematics.
Piaget, J. (1985). The equilibration of cognitive structures. University of Chicago Press.
Planinic, M., Milin-Sipus, Z., Katic, H., Susac, A., & 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. https://doi.org/10.1007/s10763-012-9344-1
Prediger, S. (2008). The relevance of didactic categories for analyzing obstacles in conceptual change: Revisiting the case of multiplication of fractions. Learning and Instruction, 18(1), 3-17. https://doi.org/10.1016/j.learninstruc.2006.08.001
Rivera, M. I., & Dolores, C. (2021). Preconcepciones de pendiente de estudiantes de educación secundaria [Preconceptions of slope of secondary education students]. Enseñanza de las ciencias, 39(1), 1-24. https://doi.org/10.5565/rev/ensciencias.3045
Rivera, M. I., Salgado, G., & Dolores, C. (2019). Explorando las conceptualizaciones de la pendiente en estudiantes universitarios [Exploring the conceptualizations of slope in college students]. Bolema, 33(65), 1027-1046. https://doi.org/10.1590/1980-4415v33n65a03
Rizo, C., & Campistrous, L. (1999). Estrategias de resolución de problemas en la escuela. Revista Latinoamericana de Investigación en Matemática Educativa, 3(2), 31-45.
SEP. (2011a). Programas de estudio 2011. Guía para el maestro. Educación básica secundaria [Study programs 2011. Teacher’s guide. Basic secondary education]. Matemáticas. https://www.gob.mx/cms/uploads/attachment/file/18394/Programa._Secundaria_tercer_grado_Matematicas_guia_para_maestros.pdf
SEP. (2011b). Programas de estudio 2011. Guía para el maestro. Educación básica primaria. Sexto grado [Study programs 2011. Teacher’s guide. Primary basic education. Sixth grade]. http://edu.jalisco.gob.mx/cepse/sites/edu.jalisco.gob.mx.cepse/files/sep_2011_programas_de_estudio_2011.guia_para_el_maestrosexto_grado.pdf
Sfard, A. (2008). Thinking as communicating. Cambridge University Press. https://doi.org/10.1017/CBO9780511499944
Simons, P. R. J. (1999). Transfer of learning: Paradoxes for learners. International Journal of Educational Research, 31(7), 577-589. https://doi.org/10.1016/S0883-0355(99)00025-7
Stanton, M., & Moore‐Russo, D. (2012). Conceptualizations of slope: A review of state standards. School Science and Mathematics, 112(5), 270-277. https://doi.org/10.1111/j.1949-8594.2012.00135.x
Stump, S. (1999). Secondary mathematics teachers’ knowledge of slope. Mathematics Education Research Journal, 11(2), 124-144. https://doi.org/10.1007/BF03217065
Stump, S. (2001). High school precalculus students’ understanding of slope as measure. School Science and Mathematics, 101(2), 81-89. https://doi.org/10.1111/j.1949-8594.2001.tb18009.x
Suparno, P. (2005). Miskonsepsi dan perubahan konsep dalam pendidikan fisika [Misconceptions and concept changes in physics education]. PT Grasindo.
Teuscher, D., & Reys, R. (2010). Slope, rate of change, and steepness: do students understand the concepts? Mathematics Teacher, 3(7), 519-524. https://doi.org/10.5951/MT.103.7.0519
Thacker, I. (2019). An embodied design for grounding the mathematics of slope in middle school students’ perceptions of steepness. Research in Mathematics Education, 22(3), 304-328. https://doi.org/10.1080/14794802.2019.1692061
Thompson, F., & Logue, S. (2006). An Exploration of common student misconceptions in science. International Education Journal, 7(4), 553-559.
Thompson, P. W. (1990). Un modelo teórico basado en el razonamiento cuantitativo en aritmética y en álgebra [A theoretical model of quantity-based reasoning in arithmetic and algebra] [Unpublished manuscript]. Center for Research in Mathematics & Science Education, San Diego State University, San Diego, CA.
Vaara, R. L., & Gomes, D. G. G. (2019). Teaching kinematic graphs in an undergraduate course using an active methodology mediated by video analysis. LUMAT: International Journal on Math, Science and Technology Education, 7(1), 1-26. https://doi.org/10.31129/LUMAT.7.1.374
Vosniadou, S., & Verschaffel, L. (2004). Extending the conceptual change approach to mathematics learning and teaching. Special Issue of Learning and Instruction, 14(5), 445-451. https://doi.org/10.1016/j.learninstruc.2004.06.014
Walter, J. G., & Gerson, H. (2007). Teachers’ personal agency: making sense of slope through additive structures. Educational Studies in Mathematics, 65(2), 203-233. https://doi.org/10.1007/s10649-006-9048-y
Yanik, H. B. (2011). Prospective middle school mathematics teachers’ preconceptions of geometric translations. Educational Studies in Mathematics, 78(2), 231-260. https://doi.org/10.1007/s10649-011-9324-3

How to cite this article

APA

Dolores-Flores, C., Rivera-López, M. I., & Tlalmanalco-Ramírez, A. (2021). Investigating Seventh-Grade Students’ Slope Preconceptions. Eurasia Journal of Mathematics, Science and Technology Education, 17(12), em2053. https://doi.org/10.29333/ejmste/11417

Vancouver

Dolores-Flores C, Rivera-López MI, Tlalmanalco-Ramírez A. Investigating Seventh-Grade Students’ Slope Preconceptions. EURASIA J Math Sci Tech Ed. 2021;17(12):em2053. https://doi.org/10.29333/ejmste/11417

AMA

Dolores-Flores C, Rivera-López MI, Tlalmanalco-Ramírez A. Investigating Seventh-Grade Students’ Slope Preconceptions. EURASIA J Math Sci Tech Ed. 2021;17(12), em2053. https://doi.org/10.29333/ejmste/11417

Chicago

Dolores-Flores, Crisólogo, Martha Iris Rivera-López, and Adán Tlalmanalco-Ramírez. "Investigating Seventh-Grade Students’ Slope Preconceptions". Eurasia Journal of Mathematics, Science and Technology Education 2021 17 no. 12 (2021): em2053. https://doi.org/10.29333/ejmste/11417

Harvard

Dolores-Flores, C., Rivera-López, M. I., and Tlalmanalco-Ramírez, A. (2021). Investigating Seventh-Grade Students’ Slope Preconceptions. Eurasia Journal of Mathematics, Science and Technology Education, 17(12), em2053. https://doi.org/10.29333/ejmste/11417

MLA

Dolores-Flores, Crisólogo et al. "Investigating Seventh-Grade Students’ Slope Preconceptions". Eurasia Journal of Mathematics, Science and Technology Education, vol. 17, no. 12, 2021, em2053. https://doi.org/10.29333/ejmste/11417