Making Thoughts Visible through Formative Feedback in a Mathematical Problem-Solving Process

Year 2021, Volume: 8 Issue: 3, 133 - 151, 03.09.2021

Zerrin Toker

https://doi.org/10.33200/ijcer.845288

Abstract

This study aims to elicit the role of formative feedback in the development of students in a mathematical problem-solving process. For this purpose, the study's primary process is to investigate the development of elementary school students (aged 10 to 11) through feedback given during a problem-solving process. While visually engaged in the sub-processes expressing a problem situation and describing their thinking structures in writing, three different dimensions are addressed: communicating visually what they understood from the problem; expressing their thoughts about solution; and creating explanations regarding their solution process. The six-week embedded mixed method study reveals that students' explanations of their thinking processes developed towards the expectations. They were able to depict the problem and the relationships involved in the problem more clearly in their drawings to understand the problem. They made fewer mistakes in mathematical operations.

Keywords

Explanation, Formative feedback, Mathematics, Problem Solving

References

• Alfieri, L., Brooks, P. J., Aldrich, N. J., & Tenenbaum, H. R. (2011). Does discovery-based instruction enhance learning? Journal of Educational Psychology, 103(1), 1–18.
• Altun, M. (2001). Matematik öğretimi. [Mathematics teaching]. İstanbul, Turkey: Alfa Yayınları Anderson, L. W. and Krathwohl, D. R., et al (Eds.) (2001) A Taxonomy for Learning, Teaching, and Assessing: A Revision of Bloom’s Taxonomy of Educational Objectives. Allyn & Bacon.
• Andrade, H., Du, Y., & Wang, X. (2008). Putting rubrics to the test: The effect of a model, criteria generation, and rubric-referenced self-assessment on elementary school students’ writing. Educational Measurement: Issues and Practices, 27(2), 3–13.
• Attali, Y., & Van der Kleij, F. (2017). Effects of feedback elaboration and feedback timing during computer-based practice in mathematics problem solving. Computers & Education, 110, 154-169.Basu, S., Biswas, G., & Kinnebrew, J. S. (2017). Learner modeling for adaptive scaffolding in a computational thinking based science learning environment. User Modeling and User-Adapted Interaction, 27(1), 5–53.
• Bayazit, İ & Aksoy, Y. (2009). Matematiksel problemlerin öğrenimi ve öğretimi. ilköğretimde karşılaşılan matematiksel zorluklar ve çözüm önerileri (Ed. Bingölbali, E & Özmantar, M,F). [Learning and Teaching of mathematical Problems: Mathematical difficulties in elementary mathematics and suggestions for solutions]. Ankara, Turkey: PegemA Yayıncılık.
• Black, P., & Wiliams, D. (1998). Assessment and classroom learning. Assessment in Education, 5, 7–75.
• Boonen, A. J., Van Wesel, F., Jolles, J., & Van der Schoot, M. (2014). The role of visual representation type, spatial ability, and reading comprehension in word problem solving: An item-level analysis in elementary school children. International Journal of Educational Research, 68, 15-26.
• Brookhart, S. (2008). How to give effective feedback to your students. Alexandria, USA: Association for Supervision and Curriculum Development.
• Butler, A. C., Godbole, N., & Marsh, E. J. (2013). Explanation feedback is better than correct answer feedback for promoting transfer of learning. Journal of Educational Psychology, 105, 290–298.
• Cáceres, M., Nussbaum, M., González, F., & Gardulski, V.(2019). Is more detailed feedback better for problem solving?, Interactive Learning Environments, doi: 10.1080/10494820.2019.1619595.
• Chase, C. C., & Klahr, D. (2017). Invention versus direct instruction: For some content, it's a tie. Journal of Science Education and Technology, doi:10.1007/s10956-017-9700-6.
• Chin, C. (2006). Classroom interaction in science: Teacher questioning and feedback to students’ responses. International Journal of Science Education, 28, 1315–1346.
• Cobb, P. (2000). Conducting teaching experiments in collaboration with teachers. In A. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 307–333). Mahwah: Lawrence Erlbaum Associates, Inc.
• Colton, C. (2010). Justifying answers and providing explanations for mathematical thinking: the impact on student learning in a middle-school classroom: Math in the Middle Institute Partnership Action Research Project Report.
• Countryman, J. (1992). Writing to learn mathematics. Portsmouth, NH: Heinemann.
• Douville, P., & Algozzine, B. (2004). Use mental imagery across the curriculum. Preventing School Failure, 49(1), 36-39.
• Fuson, K. C., & Willis, G. B. (1989). Second graders' use of schematic drawings in solving addition and subtraction word problems. Journal of Educational Psychology, 81, 514–520.
• Fyfe, E. R., & Brown, S. A. (2018). Feedback influences children's reasoning about math equivalence: A meta-analytic review. Thinking & Reasoning, 24(2), 157–178.
• Fyfe, E. R. & Brown, S.A. (2020). This is easy, you can do it! Feedback during mathematics problem solving is more beneficial when students expect to succeed. Instructional Science, 48, 23–44.
• Fyfe, E. R., & Rittle-Johnson, B. (2016). Feedback both helps and hinders learning: The causal role of prior knowledge. Journal of Educational Psychology, 108(1), 82–97.
• Fyfe, E. R., & Rittle-Johnson, B. (2017). Mathematics practice without feedback: A desirable difficulty in a classroom setting. Instructional Science, 45(2), 177–194.
• Frank, B., Simper, N., & Kaupp, J. (2018). Formative feedback and scaffolding for developing complex problem solving and modelling outcomes. European Journal of Engineering Education, 43(4), 552-568.
• Hattie, J., & Gan, M. (2011). Instruction based on feedback. In R. Mayer & P. Alexander (Eds.), Handbook of research on learning and instruction (pp. 249–271). New York: Routledge.
• Hattie, J., & Timperley, H. (2007). The power of feedback. Review of Educational Research, 77, 81-112.
• Hattie, J., & Yates, G. (2014). Using feedback to promote learning. In V. A. Benassi, C. E. Overson, & C. M. Hakala (Eds.), Applying science of learning in education: Infusing psychological science into the curriculum. APA.
• Havnes, A., Smith, K., Dysthe, O., & Ludvigsen, K. (2012). Formative assessment and feedback: Making learning visible. Studies in Educational Evaluation, 38(1), 21-27
• Ho, S. Y., & Lowrie, T. (2014). The model method: Students’ performance and its effectiveness. The Journal of Mathematical Behavior, 35, 87-100.
• Hu, B. Y., Li, Y., Zhang, X., Roberts, S. K., & Vitiello, G. (2021). The quality of teacher feedback matters: Examining Chinese teachers’ use of feedback strategies in preschool math lessons. Teaching and Teacher Education, 98, 103253.
• Huxham, M. (2007). Fast and effective feedback: are model answers the answer? Assessment & Evaluation in Higher Education, 32 (6), 601-611.
• Krawec, J. L. (2014). Problem representation and mathematical problem solving of students of varying Math ability. Journal of Learning Disabilities, 47(2), 103–115.
• Lee, C. (2006). Language for learning mathematics. Berkshire, England: Open University Press.
• Lester, F. K. (1994). Musings about mathematical problem solving research: 1970-1994. Journal for Research in Mathematics Education, 25, 660-675.
• Li, N., Cao, Y., & Mok, I. A. C. (2016). A framework for teacher verbal feedback: Lessons from Chinese mathematics classrooms. Eurasia Journal of Mathematics, Science and Technology Education, 12(9), 2465-2480.
• Luwel, K., Foustana, A., Papadatos, Y., & Verschaffel, L. (2011). The role of intelligence and feedback in children's strategy competence. Journal of Experimental Child Psychology, 108, 61–76. Marshall, S. P. (1995). Schemas in problem solving. New York: Cambridge University Press.
• Mayer, R. E. (1992). Learners as information processors: Legacies and limitations of educational psychology’s second metaphor. Educational Psychologist, 31 (3/4), 151-161.
• Millî Eğitim Bakanlığı (MoNE) (2017). İlköğretim matematik dersi 5-8. Sınıflar öğretim program ve kılavuzu. [Middle School Mathematics Curriculum] Ankara, Türkiye: MEB.
• Minton, L (2007). What if your ABCs were your 123s? Building connections between literacy and numeracy. Thousand Oaks, CA: Corwin Press.
• Mory, E. H. (2004). Feedback research revisited. In D. Jonassen (Ed.), Handbook of research on educational communications and technology (2nd ed., pp. 745–783). Mahwah, NJ: Erlbaum.
• NCTM (2000). Principles and Standards for School Mathematics. Reston, Va. NCTM.
• Nicol, D. J., & Macfarlane‐Dick, D. (2006). Formative assessment and self‐regulated learning: A model and seven principles of good feedback practice. Studies in higher education, 31(2), 199-218.
• Núñez-Peña, M. I., Bono, R., & Suárez-Pellicioni, M. (2015). Feedback on students’ performance: A possible way of reducing the negative effect of math anxiety in higher education. International Journal of Educational Research, 70, 80-87.
• Öztürk, Z,F., Kişi, E, Öztaş, E & Oruç A. (2011). İlköğretim Matematik 4. Öğretmen Kılavuz Kitabı. [Mathematics Grade 4 Teachers’ Handbook]. Ankara: MEB Yayınları.
• Pape, S. J. (2003). Compare word problems: Consistency hypothesis revisited. Contemporary Educational Psychology, 28, 396–421.
• Pimta S., Tayruakham, S & Nuangchalerm, P. (2009). Factors influencing the mathematic problem solving ability of sixth-grade students. Journal of Social Sciences, 5(4),381-385.
• Reimers, F. M., & Chung, C. K. (2016). Teaching and learning for the twenty-first century: Educational goals, policies, and curricula from six nations. Cambridge, MA: Harvard Education Press. Rittle-Johnson, B. (2006). Promoting transfer: Effects of self-explanation and direct instruction. Child Development, 77(1), 1-15.
• Rittle-Johnson, B. (2016). Feedback both helps and hinders learning: The causal role of prior knowledge. Journal of Educational Psychology, 108(1), 82–97.
• Rubie-Davies, C. M. (2017). Teacher expectations in education. New York: Routledge.
• Schaeffer, L. M., Margulieux, L. E., Chen, D., & Catrambone, R. (2016). Feedback via educational technology. In L. Lin & R. Atkinson (Eds.), Educational technologies: Challenges, applications, and learning outcomes (pp. 59–72). New York, NY: Nova Science Publishers.
• Shute, V. J. (2008). Focus on formative feedback. Review of Educational Research, 78(1), 153–189.
• Soylu, Y., & Soylu, C. (2006). Matematik derslerinde başarıya giden yolda prolem çözmenin rolü. [The role of problem solving for success in mathematics]. Eğitim Fakültesi Dergisi, 7(11), 97-111.
• Steele, D. (2005). Using writing to access students’ schemata knowledge for algebraic thinking. School Science and Mathematics, 105(3), 142.
• Stevenson, C. E. (2017). Role of working memory and strategy-use in feedback effects on children’s progression in analogy solving: An explanatory item response theory account. International Journal of Artificial Intelligence in Education, 27(3), 393–418.
• Stone, N. J. (2000). Exploring the relationship between calibration and self-regulated learning. Educational Psychology Review, 12, 437–475.
• Toker, Z. (2020). Preservice Mathematics Teachers' Views on Using Instructional Rubrics in Materials Development Process. Başkent University, Journal of Education, 7(1), 102-109.
• Van der Kleij, F. M., Feskens, R. C. W., & Eggen, T. J. H. M. (2015). Effects of feedback in a computer-based learning environment on students’ learning outcomes. Review of Educational Research, 85(4), 475–511.
• Van de Walle, J. A., Karp, K. S., & Bay-Williams, J (2013). Elementary and middle school mathematics: Teaching developmentally (8th ed.). New York: Longman.Van Garderen, D., Scheuermann, A., & Jackson, C. (2013). Examining how students with diverse abilities use diagrams to solve mathematics word problems. Learning Disability Quarterly, 36(3), 145-160.
• Van Meeuwen, L. (2013). Self-directed learning in adaptive training systems: A plea for shared control. Cognition and Learning, 9, 193–215.
• Verschaffel, L. Schukajlow, S., Star, J., & Dooren, W.V. (2020). Word problems in mathematics education: a survey. ZDM, 52, 1–16.
• Yazgan, Y., & Bintaş, J. (2005). İlköğretim dördüncü ve beşinci sınıf öğrencilerinin problem çözme stratejilerini kullanabilme düzeyleri: Bir öğretim deneyi.[4th and 5th Grade students’ levels of using problem solving strategies: A teaching experiment]. Hacettepe Üniversitesi Eğitim Fakültesi Dergisi, 28, 210-218.