Investigating the Variety and Usualness of Correct Solution Procedures of Mathematical Word Problems

Keywords: Formative assessment, inquiry based mathematics education, open approach to mathematics, primary school teachers, solving strategies, word problems

Abstract

The contribution focuses on issues related to the implementation of formative assessment methods into inquiry based teaching, by means of issues related to solving twelve multiple-step arithmetic word problems based on operations with natural and rational numbers. These word problems have multiple correct solution procedures and the presented qualitative exploratory empirical study investigates how varied and how usual might be correct solution procedures provided by diverse groups of solvers – future primary school teachers attending diverse university mathematics courses of diverse forms and/or time extent. According to written data collected from 149 solvers, six notions are introduced in the paper: majority, minority and even solution procedures, and majority, minority and mixed solvers. Issues regarding minority solvers are recognized as an important element for implementing formative assessment methods. All the six notions are illustrated in the paper by samples of solution procedures and diagrams of relative frequency. Implications are given for formative assessment within any kind of education involving multiple-step word problems, regardless of the extent of implemented inquiry. 

References

Artigue, M. and Blomhøj, M. (2013) ‘Conceptualizing inquiry-based education in mathematics’, ZDM Mathematics Education, vol. 45, no. 6, pp. 797-810. http://dx.doi.org/ 10.1007/s11858-013-0506-6

Baptist, P. and Raab, D. (eds.) (2012) Implementing Inquiry in Mathematics Education, Bayreuth: University of Bayreuth, [Online], Available: https://www.fondation-lamap.org/sites/default/files/upload/media/minisites/action_internationale/2%20-%20implementing_inquiry_mathematics_education_web.pdf [26 June 2019].

Black, P. and Wiliam, D. (2009) ‘Developing the theory of formative assessment’, Educational Assessment, Evaluation and Accountability, vol. 21, pp. 5-31. http://doi.org/10.1007/s11092-008-9068-5

Bruder, R. and Prescott, A. (2013) ‘Research evidence on the benefits of IBL’, ZDM Mathematics Education, vol. 45, no. 6, pp. 811-822. http://dx.doi.org/10.1007/s11858-013-0542-2

Bulková, K. and Čeretková, S. (2017) ‘Creativity as assessed attribute in mathematical open ended problem solving’, in Proceedings of EDULEARN17 Conference, Barcelona, pp. 583-590. http://doi.org/10.21125/edulearn.2017.1126

Clarke, D., Cheeseman, J., Roche, A. and van der Schans, S. (2014) ‘Teaching strategies for building student persistence on challenging tasks: insights emerging from two approaches to teacher professional learning’, Mathematics Teacher Educations and Development, vol. 16, no. 2, pp. 46-70.

Clipart Library (2016) Marble Bag, [Online], Available: http://clipart-library.com/image_gallery/n735161.png [19 June 2019].

Dabell, J., Keogh, B. and Naylor, S. (2008) Concept Cartoons in Mathematics Education, [CD-ROM], Sandbach: Millgate House Education.

Dolin, J. and Evans, R. (eds.) (2018) Transforming assessment, Cham: Springer. http://doi.org/10.1007/978-3-319-63248-3

Dorier, J.-L. and Maass, K. (2014) ‘Inquiry based mathematics education’, in Encyclopedia of Mathematics Education, Dordrecht: Springer, pp. 300-304. http://doi.org/10.1007/978-94-007-4978-8_176

Evans, S. and Ayalon, M. (2016) ‘Can designed student responses support teachers to interact with students in a productive way?’, Educational Designer, vol. 3, no. 9, [Online], Available: http://www.educationaldesigner.org/ed/volume3/issue9/article33/ [26 June 2019].

Evans, S. and Swan, M. (2014). ‘Developing students' strategies for problem solving in mathematics: the role of pre-designed “Sample Student Work” ’, Educational Designer, vol. 2, no. 7, [Online], Available: http://www.educationaldesigner.org/ed/volume2/issue7/article25/ [26 June 2019].

Grossman, P., Compton, C., Igra, D., Ronfeldt, M., Shanan, E. and Williamson, P. (2009) ‘Teaching practice: a cross-professional perspective’, Teachers College Record, vol. 111, no. 9, pp. 2055-2100.

Harlen, W. (2003). Enhancing inquiry through formative assessment, San Francisco: Exploratorium.

Harrison, C., Constantinou, C.P., Correia, C.F., Grangeat, M., Hänkiöniemi, M., Livitzis, M., Nieminen, P., Papadouris, N., Rached, E., Serret, N., Tiberghien, A. and Viiri, J. (2018) ‘Assessment on-the-fly: promoting and collecting evidence of learning through dialogue’, in Dolin, J. and Evans, R. (eds.) Transforming assessment, Cham: Springer, pp. 83-107. http://doi.org/10.1007/978-3-319-63248-3_4

Hattie, J. (2008) Visible learning. A synthesis of over 800 metaanalyses relating to achievement, London: Routledge. http://doi.org/10.4324/9780203887332

Hattie, J. and Timperley, H. (2007) ‘The power of feedback’, Review of Educational Research, vol. 77, no. 1, pp. 81-112. http://doi.org/10.3102/003465430298487   

Herbst, P. and Chazan, D. (2011) ‘On creating and using representation of mathematics teaching in research and teacher development’, ZDM Mathematics Education, vol. 43, no. 1, pp. 1-5. https://doi.org/10.1007/s11858-011-0306-9

Hino, K. (2007) ‘Toward the problem-centered classroom: trends in mathematical problem solving in Japan’, ZDM Mathematics Education, vol. 39, no. 5-6, pp. 503-514. http://doi.org/10.1007/s11858-007-0052-1  

Hohensee, C. (2016) Student noticing in classroom settings: a process underlying influences on prior ways of reasoning,’ Journal of Mathematical Behavior, vol. 42, pp. 69-91. http://doi.org/10.1016/j.jmathb.2016.03.002

Hošpesová, A. and Žlábková, I. (2016) ‘Assessment in inquiry based education in primary mathematics’, in Proceedings of ERIE 2016, Prague, pp. 194-201.

Hoth, J., Döhrmann, M., Kaiser, G., Busse, A., König, J. and Blömeke, S. (2016) ‘Diagnostic competence of primary school mathematics teachers during classroom situations’, ZDM Mathematics Education, vol. 48, no. 1-2, pp. 41-53. http://doi.org/10.1007/s11858-016-0759-y

Jahodová Berková A. (2017) ‘Effect of the use of computer-aided assessment system in the teaching of mathematical analysis with regard to students' approaches to learning’, Journal on Efficiency and Responsibility in Education and Science, vol. 10, no. 3, pp. 71-75. http://doi.org/10.7160/eriesj.2017.100302

Jiang, F. and McComas, W. F. (2015) ‘The effect of inquiry teaching on student science achievement and attitudes: evidence from propensity score analysis of PISA data’, International Journal of Science Education, vol. 37, no. 3, pp. 554-576. http://dx.doi.org/10.1080/09500693.2014.1000426

Kaiser, G., Blum, W., Borromeo Ferri, R. and Stillman, G. (eds.) (2011) Trends in teaching and learning mathematical modelling, Dordrecht: Springer. http://doi.org/10.1007/978-94-007-0910-2

Kwon, O. N., Park, J. S. and Park, J. H. (2006) ‘Cultivating divergent thinking in mathematics through an open-ended approach’, Asia Pacific Education Review, vol. 7, no. 1, pp. 51-61. http://dx.doi.org/10.1007/bf03036784

Le Hebel, F., Constantinou, C.P., Hošpesová, A., Grob, R., Holmeier, M., Montpied, P., Moulin, M., Petr, J., Rokos, L., Stuchlíková, I., Tiberghien, A., Tsivitanidou, O. and Žlábková, I. (2018) ‘Students' perspectives on peer assessment’, in Dolin J., Evans R. (eds.) Transforming assessment, Cham: Springer, pp. 141-173. http://doi.org/10.1007/978-3-319-63248-3_6

Levav-Waynberg, A. and Leikin, R. (2012) ‘The role of multiple solution tasks in developing knowledge and creativity in geometry’, Journal of Mathematical Behavior, vol. 31, no. 1, pp. 73-90.  http://doi.org/10.1016/j.jmathb.2011.11.001

Maaß, K. and Reitz-Koncebovski, K. (eds.) (2013) Inquiry-based learning in maths and science classes. What it is and how it works – examples – experiences. Freiburg: Pädagogische Hochschule Freiburg, [Online], Available: http://primas-project.eu/wp-content/uploads/sites/323/2017/11/primas_final_publication.pdf [26 June 2019].

MacGregor, M. and Stacey, K. (1998) ‘Cognitive models underlying algebraic and non-algebraic solutions to unequal partition problems’, Mathematics Education Research Journal, vol. 10, no. 2, pp. 46-60. http://doi.org/10.1007/bf03217342

McComas, W.F. (ed.) (2002) The nature of science in science education, Boston: Kluwer. http://doi.org/10.1007/0-306-47215-5

Medová, J., Bulková, K. and Čeretková, S. (2018) ‘Identification of crucial competencies in mathematical inquiry’, in Proceedings of ERIE 2018, Prague, pp. 203-212.

Miles, M.B., Huberman, A.M. and Saldaña, J. (2014) Qualitative data analysis. A methods sourcebook, Thousand Oaks, CA: SAGE.

Milewski, A. and Strickland, S. (2016) ‘(Toward) developing a common language for describing instructional practices of responding’, Mathematics Teacher Educator, vol. 4, no. 1, pp. 126-144. http://doi.org/10.5951/mathteaceduc.4.2.0126

Minner, D., Levy, A. and Century, J. (2010) ‘Inquiry-based science instruction – what is it and does it matter? Results from a research synthesis years 1984 to 2002’, Journal of Research in Science Teaching, vol. 47, no. 4. pp. 474-496. http://doi.org/10.1002/tea.20347

Naylor, S. and Keogh, B. (2007) ‘Active assessment: thinking, learning and assessment in science’, School Science Review, vol. 88, pp. 73-79.

Nohda, N. (2000) ‘Teaching by open-approach method in Japanese mathematics classroom’, in Proceedings of PME 24, Hiroshima, pp. 35-45.

Novotná, J. and Vondrová, N. (2017) ‘Pupils' strategies for missing value proportional problems’, in Proceedings of ERIE 2017, Prague, pp. 279-286.

Pehkonen, E. (ed.) (1997) Use of open-ended problems in mathematics classroom, Helsinki: University of Helsinki, [Online], Available: http://files.eric.ed.gov/fulltext/ED419714.pdf [26 June 2019].

Rokos, L. and Závodská, R. (2015) ‘Formative assessment and other assessment methods in biology education and pre-service biology teacher training in the Czech Republic’, International Journal of Assessment and Evaluation, vol. 23, no. 2, pp. 17-27. http://doi.org/10.18848/2327-7920/CGP/v23i02/48388

Samková, L. (2017) ‘Planning and conducting inquiry based mathematics course for future primary school teachers’, in Proceedings of SEMT ‘17, Prague, pp. 354-364.

Samková, L. (2018a) ‘Assessing future teachers' knowledge on fractions: written tests vs Concept Cartoons’, Journal on Efficiency and Responsibility in Education and Science, vol. 11, no. 3, pp. 45-52. http://doi.org/10.7160/eriesj.2018.110301

Samková, L. (2018b) ‘Concept Cartoons as a representation of practice’, in Buchbinder, O. and Kuntze, S. (eds.) Mathematics Teachers Engaging with Representations of Practice, Cham: Springer, pp. 71-93. http://doi.org/10.1007/978-3-319-70594-1_5

Samková, L. (2019) ‘Majority and minority correct procedures for solving mathematical word problems’, in Proceedings of ERIE 2019, Prague, pp. 243-250.

Samková, L. and Tichá, M. (2016) ‘On the way to develop open approach to mathematics in future primary school teachers’, Journal on Efficiency and Responsibility in Education and Science, vol. 9, no. 2, pp. 37-44. http://doi.org/10.7160/eriesj.2016.090202

Savelsbergh, E.R., Prins, G.T., Rietbergen, C., Fechner, S., Vaessen, B.E., Draijer, J.M. and Bakker, A. (2016) ‘Effects of innovative science and mathematics teaching on student attitudes and achievement: a meta-analytic study’, Educational Research Review, vol. 19, pp. 158-172. http://doi.org/10.1016/j.edurev.2016.07.003

Schack, E.O., Fisher, M.H. and Wilhelm, J.A. (Eds.) (2017) Teacher noticing: bridging and broadening perspectives, contexts, and frameworks, Cham: Springer. http://doi.org/10.1007/978-3-319-46753-5 

Schoenfeld, A. (2016) ‘Learning to think mathematically: problem solving, metacognition, and sense making in mathematics’, Journal of Education, vol. 196, no. 2, pp. 1-38. http://doi.org/10.1177/002205741619600202

Schoenfeld, A.H. and Kilpatrick, J. (2013) ‘A US perspective on the implementation of inquiry-based learning in mathematics’, ZDM Mathematics Education, vol. 45, no. 6, pp. 901-909. http://doi.org/10.1007/s11858-013-0531-5

Shavelson, R.L., Young, D.B., Ayala, C.C., Brandon, P.R., Furtak, E.M., Ruiz-Primo, M.A., Tomita, M.K. and Yin, Y. (2008) ‘On the impact of curriculum-embedded formative assessment on learning: a collaboration between curriculum and assessment developers’, Applied Measurement in Education, vol. 21, no. 4, pp. 295-314. http://doi.org/10.1080/08957340802347647

Simpson, A. and Vondrová, N. (online first, 2019) ‘Developing pre-service teachers’ professional vision with video interventions: a divergent replication’, Journal of Education for Teaching, vol. 45, no. 1, pp. 567-584. http://doi.org/10.1080/02607476.2019.1674563

Spangler, D.A. and Hallman-Thrasher, A. (2014) ‘Using task dialogues to enhance preservice teachers' abilities to orchestrate discourse’, Mathematics Teacher Educator, vol. 3, no. 1, pp. 58-75. http://doi.org/10.5951/mathteaceduc.3.1.0058  

Thevenot, C. and Oakhill, J. (2006) ‘Representations and strategies for solving dynamic and static arithmetic word problems: the role of working memory capacities’, European Journal of Cognitive Psychology, vol. 18, no. 5, pp. 756-775. http://doi.org/10.1080/09541440500412270 

Topping, K. (2013) ‘Peers as a source of formative and summative assessment’, in McMillan, J.H. (de.) SAGE handbook of research on classroom assessment, London: SAGE, pp. 395-412. http://doi.org/10.4135/9781452218649.n22  

Ulm, V. (2011) Inquiry-based mathematics education for gifted children in primary school. Augsburg: University of Augsburg, [Online], Available: http://www.fondation-lamap.org/en/page/27955/fibonacci-resources [26 June 2019].

van Es, E.A. and Sherin, M.G. (2008) ‘Mathematics teachers' “learning to notice” in the context of a video club’, Teaching and Teacher Education, vol. 24, no. 2, pp. 244-276. http://doi.org/10.1016/j.tate.2006.11.005

Verschaffel, L., Greer, B. and de Corte, E. (2000) Making sense of word problems, Lisse: Swets & Zeitlinger.

Visnovska, J. and Cobb, P. (2015) ‘Learning about whole-class scaffolding from a teacher professional development study’, ZDM Mathematics Education, vol. 47, no. 7, pp. 1133-1145. http://doi.org/10.1007/s11858-015-0739-7 

Webel, C., Conner, K. and Zhao, W. (2018) ‘Simulations as a tool for practicing questioning’, in Buchbinder, O. and Kuntze, S. (eds.) Mathematics Teachers Engaging with Representations of Practice, Cham: Springer, pp. 95-112. http://doi.org/10.1007/978-3-319-70594-1_6  

Published
2020-03-31
How to Cite
Samková, L. (2020) ’Investigating the Variety and Usualness of Correct Solution Procedures of Mathematical Word Problems’, Journal on Efficiency and Responsibility in Education and Science, vol. 13, no. 1, pp. 10-26 https://doi.org/10.7160/eriesj.2020.130102.
Section
Research Paper