The Pedagogical Manifestations: A Driver of Teachers’ Practices in Teaching Algebraic Equations
Mathematics teachers’ instructional strategies lack in-depth knowledge of algebraic systems and hold misconceptions about solving two algebraic .
- Pub. date: January 15, 2023
- Pages: 15-28
- 309 Downloads
- 393 Views
- 0 Citations
Mathematics teachers’ instructional strategies lack in-depth knowledge of algebraic systems and hold misconceptions about solving two algebraic equations simultaneously. This study aimed to gain an in-depth analysis of teachers’ knowledge and perceptions about the promotion of conceptual learning and effective teaching of algebraic equations. The main question was, ‘How do junior secondary school mathematics teachers manifest their pedagogical practices when teaching algebraic equations? This article reports on a qualitative, underpinned by the knowledge quartet model study, that sought to explore how junior secondary school teachers’ pedagogical practices manifested in the teaching of algebraic equations. Data were collected from observations, semi-structured interviews, and document analysis of two mathematics teachers purposely selected from two schools. The collected data were analysed using a statistical analysis software called Atlas-ti. (Version 8) and triangulated through thematic analysis. The study revealed that teachers’ choices of representations, examples, and tasks used did not expose learners to hands-on activities that promote understanding and making connections from the underlying algebraic equation concepts. The study proposed Penta-Knowledge Collaborative Planning and Reflective Teaching and Learning Models to enable teachers to collaborate with their peers from the planning stage to lesson delivery reflecting on good practices and strategies for teaching algebraic equations.
Keywords: Classroom practices, pedagogical practices, penta-knowledge collaborative planning, teacher-centered methods.
References
Ahmat, N., Azmee, N. A., Mohamed, N. H., Zamzamir, Z., Zahari, N. S., Shafie, S., Mohamed, N. A., & Raja Ma'amor Shah, R. N. F. A. (2022). Knowledge, skills and attitude of pre-service mathematics teachers towards higher-order thinking skills. International Journal of Educational Methodology, 8(4), 795-804. https://doi.org/10.12973/ijem.8.4.795
Ary, D., Jacobs, L., Razavieh, A., & Sorensen, C. (2010). Introduction for research in education (8th ed.). Wadsworth Cengage Learning. https://cutt.ly/sNJFIry
Asquith, P., Stephens, A. C., Knuth, E. J., & Alibali, M. W. (2007). Middle school mathematics teachers' knowledge of learners' understanding of core algebraic concepts: Equal sign and variable. Mathematical Thinking and Learning, 9(3), 249-272. https://doi.org/10.1080/10986060701360910
ATLAS.ti Scientific Software Development GmbH. (2022). Atlas.ti [Computer software]. https://atlasti.com
Babbie, E., & Mouton, J. (2011). The practice of social research. University Press.
Ball, D., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389-407. https://doi.org/10.1177/0022487108324554
Baratta, C. (Ed.). (2011). Environmentalism in the realm of science fiction and fantasy literature. Cambridge Scholars Publishing.
Baumert, J., Kunter, M., Blum, W., Brunner, M., Voss, T., Jordan, A., Klusmann, U., Krauss, S., Neubrand, M., & Tsai, Y. M. (2010). Teachers’ mathematical knowledge, cognitive activation in the classroom, and student progress. American Educational Research Journal, 47(1), 133-180. https://doi.org/10.3102/0002831209345157
Bell, E. (2014). Rethinking quality in qualitative research. Australian Journal of Rural Health, 22(3), 90-91. https://doi.org/10.1111/ajr.12119
Black, C. M. (2008). Working for a healthier tomorrow: Dame Carol Black's review of the health of Britain's working age population. The Stationery Office.
Boikhutso, K. (2010). The theory into practice dilemma: Lesson planning challenges facing Botswana student-teachers. Improving Schools, 13(3), 205-220. https://doi.org/10.1177/1365480206061994
Buchbinder, O., Chazan, D., & Fleming, E. (2015). Insights into the school mathematics tradition from solving linear equations. For the Learning of Mathematics, 35(2), 2-8.
Cai, J., Nie, B., & Moyer, J. C. (2010). The teaching of equation solving: Approaches in standards-based and traditional curricula in the United States. Pedagogies: An International Journal, 5(3), 170-186. https://doi.org/ghvb5j
Canobi, K. H. (2009). Concept–procedure interactions in children’s addition and subtraction. Journal of Experimental Child Psychology, 102(2), 131-149. https://doi.org/10.1016/j.jecp.2008.07.008
Carpenter, T. P., Fennema, E., Franke, M. L. L., & Levi, L. (1999). Children's mathematics: cognitively guided Instruction (1st ed.). Heinemann.
Creswell, J. W., & Creswell, J. D. (2017). Research design: Qualitative, quantitative, and mixed methods approaches. Sage Publications.
Demme, I. (2018, July 17). Six Reasons why we learn algebra. Demme Learning Blog. https://cutt.ly/JNJFX6R
Ellerton, N. F., Kanbir, S., & Clements, M. A. (2017). Historical perspectives on the purposes of school algebra, 40 years on: We are still learning! In A. Downton, S. Livy, & J. Hall (Eds.), Proceedings of the 40th Annual Conference of the Mathematics Education Research Group of Australasia (pp.221–228). MERGA.
Ghanaguru, S., Nair, P., & Yong, C. (2013). Teacher trainers’ beliefs in microteaching and lesson planning in a teacher training institution. The English Teacher, 42(2), 104-116. https://cutt.ly/qNJFNNz
Groth, R. E. (2017). Classroom data analysis with the five strands of mathematica proficiency. The Clearing House: A Journal of Educational Strategies, Issues and Ideas, 90(3), 103-109. https://doi.org/jj7r
Hageraats, E. (2016). Using relation algebra to generate feedback to learners in an intelligent tutoring system for business rules [Unpublished Master's thesis]. Open Universiteit Nederland.
Haggstrom, J. (2008). Teaching systems of algebraic equations in Sweden and China: What is made possible to learn? Acta Universitatis Gothoburgensis.
Hall, R. D. (2002). An analysis of errors made in the solution of simple linear equations. Philosophy of Mathematics Education Journal, 15(1), 1-67.
Huntley, M. A., & Terrell, M. S. (2014). One-step and multi-step linear equations: A content analysis of five textbook series. ZDM, 46, 751–766. https://doi.org/10.1007/s11858-014-0627-6
Kieran, C. (1992). The learning and teaching of school algebra. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning: A project of the national council of teachers of mathematics (pp. 390-419). Macmillan Publishing Company.
Ko, Y.-Y., & Karadag, Z. (2013). Fostering middle school learners’ relational thinking of the equal sign using GeoGebra. Mevlana International Journal of Education, 3(3), 45-49.
Koency, G., & Swanson, H. L. (2000, April 24-28). The special case of mathematics: Insufficient content knowledge a major obstacle to reform. [Paper presentation]. American Educational Research Association Conference 2000, New Orleans, LA.
Kuhs, T., & Ball, D. L. (1986). Approaches to teaching mathematics: Mapping the domains of knowledge, skills, and dispositions. National Center for Research on Teacher Education, Michigan State University.
Kullberg, A., Kempe, U. R., & Marton, F. (2017). What is made possible to learn when using the variation theory of learning in teaching mathematics? ZDM Mathematics Education, 49(4), 559–569. https://doi.org/f94dsc
Mason, L. (2003). High school students' beliefs about maths, mathematical problem solving, and their achievement in maths: A cross-sectional study. Educational Psychology, 23(1), 73-85. https://doi.org/10.1080/01443410303216
Ministry of Education. (2010). Three-year junior certificate syllabus. Curriculum Division.
Moalosi, S. S. (2008). An investigation of teachers’ subject matter knowledge and pedagogical content knowledge in Algebra: Study of Botswana junior secondary schools. [Unpublished Master’s Thesis]. University of Botswana.
Naseer, M. S. (2016). Algebraic content and pedagogical knowledge of sixth grade mathematics teachers [Doctoral dissertation]. Walden University. https://cutt.ly/aNJGrOY
Ng, L. K., & Dindyal, J. (2016). Examples in the teaching of mathematics: Teachers' perceptions. In P. C. Toh., & B. Kaur (Eds.), Developing 21st century competencies in the mathematics classroom: Yearbook 2016, Association of Mathematics Educators (pp. 189-211). World Scientific. https://doi.org/10.1142/9789813143623_0011
Ní Shúilleabháin, A. (2015). Developing mathematics teachers' pedagogical content knowledge through iterative cycles of lesson study. In K. Krainer & N. Vondrová (Eds.), Proceedings of the Ninth Conference of the European Society for Research in Mathematics Education (CERME9, 4-8 February 2015) (pp. 2734-2740). Charles University in Prague, Faculty of Education and ERME.
Novotna, J., & Hospesova, A. (2008). Types of linking in teaching algebraic equations. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano, & A. Sepulveda (Eds.), Proceedings of the 22nd Conference of the International Group for the Psychology of Mathematics Education and PME NA XXX (Vol. 4, pp. 49-56). https://cutt.ly/oNJGuGL
Noris, M., Saputro, S., & Muzazzinah. (2022). The virtual laboratory based on problem based learning to improve students' critical thinking skills. European Journal of Mathematics and Science Education, 3(1), 35-47. https://doi.org/10.12973/ejmse.3.1.35
Osei, C. M. (2006). Student teachers’ knowledge and understanding of algebraic concepts: The case of colleges of Education in the Eastern Cape and Southern KwaZulu Natal, South Africa [Unpublished Doctoral thesis]. University of the Witwatersrand.
Ozgur, I. O., & Aslan, M. (2016). Prescriptions guiding prospective teachers in teaching mathematics educational sciences: Theory & Practice, 16(3), 735-769. https://doi.org/10.12738/estp.2016.3.0371
Pai, G. (2014, December 20). How do we use the transpose method to solve a linear equation? [Video file]. YouTube. https://cutt.ly/3NJGadc
Pappano, L. (2012). The Algebra Problem. How to elicit algebraic thinking in learners before eighth grade. Harvard Education Letter, 28(3), 1-2. https://cutt.ly/rNJGgXW
Patton, M. Q. (2002). Two decades of developments in qualitative inquiry: A personal, experiential perspective. Qualitative Social Work, 1(3), 261-283.
Rabionet, S. E. (2011). How I learned to design and conduct semi-structured interviews: An ongoing and continuous journey. Qualitative Report, 16(2), 63-566. https://doi.org/10.46743/2160-3715/2011.1070
Rittle-Johnson, B., Schneider, M., & Star, J. R. (2015). Not a one-way street: Bidirectional relations between procedural and conceptual knowledge of mathematics. Educational Psychology Review, 27(4), 587-597. https://doi.org/10.1007/s10648-015-9302-x
Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93(2), 346-362. https://doi.org/10.1037/0022-0663.93.2.346
Rittle-Johnson, B., & Star, J. R. (2007). Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology, 99(3), 561. https://doi.org/10.1037/0022-0663.99.3.561
Roller, M. R., & Lavrakas, P. J. (2015). Applied qualitative research design: A total quality framework approach. Guilford Publications.
Ronan, M. A. (2012). Modern algebra. Encyclopedia Britannica. https://www.britannica.com/science/modern algebra
Rosenshine, B. (2012). Principles of instruction: Research-based strategies that all teachers should know. American Educator, 36(1), 12-39.
Saleh, M., & Battisha, M. (2020). A Proposed paradigm for the requirements for designing and using digital games-based learning by educable intellectual disabled children. Technium Social Sciences Journal, 2(1), 37–66. https://doi.org/10.47577/tssj.v2i1.5
Shah, S. (2012). Transition of students from arithmetic to algebra [Unpublished master's dissertation]. Aga Khan University.
Silver, E. A., Ghousseini, H., Gosen, D., Charalambous, C., & Strawhun, B. T. F. (2005). Moving from rhetoric to praxis: Issues faced by teachers in having students consider multiple solutions for problems in the mathematics classroom. The Journal of Mathematical Behavior, 24(3-4), 287-301. https://doi.org/10.1016/j.jmathb.2005.09.009
Skemp, R. R. (1964). Understanding mathematics. University of London Press.
Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77(1), 20-26.
Star, J. R., Caronongan, P., Foegen, A., Furgeson, J., Keating, B., Larson, M. R., Lyskawa, J., McCallum, W. G., Porath, J., & Zbiek, R. M. (2015). Teaching strategies for improving algebra knowledge in middle and high school learners (NCEE 2014-4333). National Center for Education Evaluation and Regional Assistance (NCEE), Institute of Education Sciences, U.S. Department of Education.
Stipek, D. J., Givvin, K. B., Salmon, J. M., & MacGyvers, V. L. (2001). Teachers’ beliefs and practices related to mathematics instruction. Teaching and Teacher Education, 17(2),213-226. https://doi.org/10.1016/S0742-051X(00)00052-4
Swanson, H. L. (2017). Searching for the Best Model for Instructing Students with Learning Disabilities. Focus on Exceptional Children, 34(2), 1-16. https://doi.org/10.17161/foec.v34i2.6785
Tajudin, N. A. M., & Kadir, N. Z. A. (2014). Technological pedagogical content knowledge and teaching practice of mathematics trainee teachers. In AIP Conference Proceedings (Vol. 1605, No. 1, pp. 734-739). American Institute of Physics. https://doi.org/10.1063/1.4887681
Thompson, A. G. (1992). Teachers' beliefs and conceptions: A synthesis of the research. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics (pp. 127–146). Macmillan Publishing Co, Inc.
Turner, F. (2012). Using the Knowledge Quartet to develop mathematics content knowledge: The role of reflection on professional development. Research in Mathematics Education, 14(3), 253-271. https://doi.org/jj7s
Turner, F., & Rowland, T. (2008, March 18). The knowledge quartet: A means of developing and deepening mathematical knowledge in teaching? Seminar presented at the Nuffield Seminar Series on Mathematical Knowledge in Teaching University of Cambridge, UK. https://bit.ly/NuffieldSeminarSeries
Walters, K. (2014). Professional development strategies to support student success in Algebra I. American Institutes for Research.
Zegarell, M. (2017). Basic math & pre-algebra workbook for dummies (3rd ed.). John Wiley & Sons.