Kinds of Mathematical Reasoning Addressed in Empirical Research in Mathematics Education: A Systematic Review
Abstract
:1. Introduction
2. Methods
2.1. Identification: Search Procedure
2.2. Screening Basis: Criteria for Inclusion of Articles
- The article was an empirical article. We included empirical articles, in which students’ reasoning was investigated, since we wanted to investigate the kinds of reasoning addressed in empirical research (see research question). With this focus, we wanted to account for the kinds of mathematical reasoning addressed in empirical studies in mathematics education.
- The article focused on school students. Articles with a focus on reasoning at university level [14] were excluded. Our main reason for narrowing our focus to primary and secondary school years was to get a manageable selection of articles to be included in the review. Besides, since the topic of reasoning is mostly related to the curriculum in primary and secondary school years, the restriction to these school years appeared appropriate. This restriction also was in line with our intention to draw conclusions for school practice (see Section 4).
2.3. Screening Process: Focus on Titles and Abstracts
2.4. Eligibility and Inclusion: Scanning and Categorization
3. Results
3.1. Mathematical Reasoning in a Domain-General View
3.1.1. Definitions and Explanatory Statements
3.1.2. Creative and Imitative Reasoning
3.1.3. The Structure of the Learned Outcomes (SOLO) Taxonomy
3.1.4. Hierarchic Interactionism
3.2. Mathematical Reasoning in Domain-Specific Views
3.2.1. Spatial Reasoning
3.2.2. Informal Inferential Reasoning
3.2.3. Additive, Multiplicative, and Distributive Reasoning
3.2.4. Algebraic Reasoning
3.2.5. Proportional and Covariational Reasoning
3.2.6. Quantitative Reasoning
3.2.7. Transformational Reasoning
3.2.8. Combinations of Kinds of Reasoning
4. Discussion
4.1. Discussion of the Results
4.2. Implications for School Practice
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
- Cambridge University Press. Reasoning: Cambridge English Dictionary. Available online: https://dictionary.cambridge.org/dictionary/english/reasoning (accessed on 7 November 2018).
- Yackel, E.; Hanna, G. Reasoning and proof. In A Research Companion to Principles and Standards for School Mathematics; Kilpatrick, J., Martin, W.G., Schifter, D., Eds.; National Council of Teachers of Mathematics: Reston, VA, USA, 2003; pp. 227–236. ISBN 0873535375/9780873535373. [Google Scholar]
- Lithner, J.; Palm, T. Learning Difficulties and Mathematical Reasoning. In The First Sourcebook on Nordic Research in Mathematics Education: Norway, Sweden, Iceland, Denmark, and Contributions from Finland; Sriraman, B., Bergsten, C., Goodchild, S., Pálsdóttir, G., Dahl, B., Haapasalo, L., Eds.; Information Age Pub: Charlotte, NC, USA, 2010; pp. 283–298. ISBN 1617351008/9781617351006. [Google Scholar]
- Lithner, J. A research framework for creative and imitative reasoning. Educ. Stud. Math. 2008, 67, 255–276. [Google Scholar] [CrossRef]
- National Council of Teachers of Mathematics. Principles and Standards for School Mathematics; The Council: Reston, VA, USA, 2000. [Google Scholar]
- Jeannotte, D.; Kieran, C. A conceptual model of mathematical reasoning for school mathematics. Educ. Stud. Math. 2017, 96, 1–16. [Google Scholar] [CrossRef] [Green Version]
- Niss, M.; Højgaard, T. Mathematical competencies revisited. Educ. Stud. Math. 2019, 102, 9–28. [Google Scholar] [CrossRef]
- Skolverket. Läroplan för Grundskolan, Förskoleklassen och Fritidshemme; Lgr 11 (Rev. ed.); Skolverket: Stockholm, Sweden, 2019. [Google Scholar]
- Baker, S.; Gersten, R.; Lee, D.S. A synthesis of empirical research on teaching mathematics to low-achieving students. Elem. Sch. J. 2002, 103, 51–73. [Google Scholar] [CrossRef]
- Eriksson, K.B.; Forsberg, C.; Wengström, Y. Systematiska Litteraturstudier i Utbildningsvetenskap: Vägledning vid Examensarbeten och Vetenskapliga Artiklar; Natur & Kultur: Stockholm, Sweden, 2013; ISBN 9789127134119. [Google Scholar]
- Nilsson, P.; Schindler, M.; Bakker, A. The nature and use of theories in statistics education. In International Handbook of Research in Statistics Education; Ben-Zvi, D., Makar, K., Garfield, J., Eds.; Springer: Cham, Switzerland, 2018; pp. 359–386. ISBN 9783319661933. [Google Scholar]
- Dyer, E.B.; Sherin, M.G. Instructional reasoning about interpretations of student thinking that supports responsive teaching in secondary mathematics. ZDM 2016, 48, 69–82. [Google Scholar] [CrossRef]
- Otten, S.; Gilbertson, N.J.; Males, L.M.; Clark, D.L. The mathematical nature of reasoning-and-proving opportunities in geometry textbooks. Math. Think. Learn. 2014, 16, 51–79. [Google Scholar] [CrossRef] [Green Version]
- Dawkins, P.C.; Cook, J.P. Guiding reinvention of conventional tools of mathematical logic: Students’ reasoning about mathematical disjunctions. Educ. Stud. Math. 2017, 94, 241–256. [Google Scholar] [CrossRef]
- Mata-Pereira, J.; da Ponte, J.-P. Enhancing students’ mathematical reasoning in the classroom: Teacher actions facilitating generalization and justification. Educ. Stud. Math. 2017, 96, 169–186. [Google Scholar] [CrossRef]
- Brousseau, G.; Gibel, P. Didactical handling of students’ reasoning processes in problem solving situations. Educ. Stud. Math. 2005, 59, 13–58. [Google Scholar] [CrossRef]
- Pólya, G. Mathematics and Plausible Reasoning: Induction and Analogy in Mathematics; University Press: Princeton, NJ, USA, 1954; Volume 1, ISBN 4871878333/978-4871878333. [Google Scholar]
- Rivera, F.; Becker, J. Algebraic reasoning through patterns. Math. Teach. Middle Sch. 2009, 15, 213–221. [Google Scholar] [CrossRef]
- Hackenberg, A.J.; Lee, M.Y. Students’ distributive reasoning with fractions and unknowns. Educ. Stud. Math. 2016, 93, 245–263. [Google Scholar] [CrossRef]
- Thompson, P.W. Quantitative reasoning, complexity, and additive structures. Educ. Stud. Math. 1993, 25, 165–208. [Google Scholar] [CrossRef]
- Thompson, P.W. Quantitative reasoning and mathematical modeling. In New Perspectives and Directions for Collaborative Research in Mathematics Education; Chamberlin, S., Hatfield, L.L., Eds.; University of Wyoming: Laramie, WY, USA, 2010; pp. 33–57. [Google Scholar]
- Barmby, P.; Harries, T.; Higgins, S.; Suggate, J. The array representation and primary children’s understanding and reasoning in multiplication. Educ. Stud. Math. 2009, 70, 217–241. [Google Scholar] [CrossRef] [Green Version]
- Meletiou-Mavrotheris, M.; Paparistodemou, E. Developing students’ reasoning about samples and sampling in the context of informal inferences. Educ. Stud. Math. 2015, 88, 385–404. [Google Scholar] [CrossRef]
- Bjuland, R.; Luiza Cestari, M.; Borgersen, H.E. The interplay between gesture and discourse as mediating devices in collaborative mathematical reasoning: A multimodal approach. Math. Think. Learn. 2008, 10, 271–292. [Google Scholar] [CrossRef]
- Hohensee, C. Backward transfer: An investigation of the influence of quadratic functions instruction on students’ prior ways of reasoning about linear functions. Math. Think. Learn. 2014, 16, 135–174. [Google Scholar] [CrossRef]
- Leighton, J.P. Defining and describing reason. In The Nature of Reasoning; Leighton, J.P., Sternberg, R.J., Eds.; Cambridge University Press: Cambridge, UK, 2004; pp. 3–11. ISBN 0521009286/978-0521009287. [Google Scholar]
- Christou, C.; Papageorgiou, E. A framework of mathematics inductive reasoning. Learn. Instr. 2007, 17, 55–66. [Google Scholar] [CrossRef]
- Pólya, G. How to Solve it; University Press: Princeton, NJ, USA, 1945. [Google Scholar]
- Polya, G. How to Solve it; University Press: Princeton, NJ, USA, 1957; Original work published 1945. [Google Scholar]
- Presmeg, N. Semiotics and the “connections” standard: Significance of semiotics for teachers of mathematics. Educ. Stud. Math. 2006, 61, 163–182. [Google Scholar] [CrossRef]
- Gravemeijer, K. Local instruction theories as means of support for teachers in reform mathematics education. Math. Think. Learn. 2004, 6, 105–128. [Google Scholar] [CrossRef]
- McClain, K.; Cobb, P.; Gravemeijer, K. Supporting students’ ways of reasoning about data. In Learning Mathematics for a New Century, 2000 Yearbook; Burke, M.J., Curcio, R.F., Eds.; National Council of Teachers of Mathematics: Reston, VA, USA, 2000; pp. 174–187. ISBN 978-0873534796/0873534794. [Google Scholar]
- Boesen, J.; Lithner, J.; Palm, T. The relation between types of assessment tasks and the mathematical reasoning students use. Educ. Stud. Math. 2010, 75, 89–105. [Google Scholar] [CrossRef]
- Hershkowitz, R.; Tabach, M.; Dreyfus, T. Creative reasoning and shifts of knowledge in the mathematics classroom. ZDM 2017, 49, 25–36. [Google Scholar] [CrossRef]
- Lithner, J. Principles for designing mathematical tasks that enhance imitative and creative reasoning. ZDM 2017, 49, 937–949. [Google Scholar] [CrossRef] [Green Version]
- Jurdak, M.; El Mouhayar, R. Trends in the development of student level of reasoning in pattern generalization tasks across grade level. Educ. Stud. Math. 2014, 85, 75–92. [Google Scholar] [CrossRef]
- Biggs, J.B.; Collis, K.F. Evaluating the Quality of Learning: The Solo Taxonomy; Academic Press: New York, NY, USA, 1982; ISBN 0120975505. [Google Scholar]
- Clements, D.; Sarama, J. Learning and Teaching Early Math: The Learning Trajectories Approach; Routledge: New York, NY, USA, 2009; ISBN 978-0-415-82850-5. [Google Scholar]
- Khan, S.; Francis, K.; Davis, B. Accumulation of experience in a vast number of cases: Enactivism as a fit framework for the study of spatial reasoning in mathematics education. ZDM 2015, 47, 269–279. [Google Scholar] [CrossRef]
- Hallowell, D.A.; Okamoto, Y.; Romo, L.F.; La Joy, J.R. First-graders’ spatial-mathematical reasoning about plane and solid shapes and their representations. ZDM 2015, 47, 363–375. [Google Scholar] [CrossRef]
- Herbst, P. Interaction with diagrams and the making of reasoned conjectures in geometry. Zent. Didakt. Math. 2004, 36, 129–139. [Google Scholar] [CrossRef] [Green Version]
- Chen, C.-L.; Herbst, P. The interplay among gestures, discourse, and diagrams in students’ geometrical reasoning. Educ. Stud. Math. 2013, 83, 285–307. [Google Scholar] [CrossRef] [Green Version]
- Gómez-Chacón, I.M.; Romero Albaladejo, I.M.; del Mar García López, M. Zig-zagging in geometrical reasoning in technological collaborative environments: A Mathematical working space-framed study concerning cognition and affect. ZDM 2016, 48, 909–924. [Google Scholar] [CrossRef]
- Lakatos, I. Proofs and Refutations: The Logic of Mathematical Discovery; Cambridge University Press: New York, NY, USA, 1979; ISBN 0521290384/9780521290388. [Google Scholar]
- Gil, E.; Ben-Zvi, D. Explanations and context in the emergence of students’ informal inferential reasoning. Math. Think. Learn. 2011, 13, 87–108. [Google Scholar] [CrossRef]
- Pfannkuch, M. The role of context in developing informal statistical inferential reasoning: A classroom study. Math. Think. Learn. 2011, 13, 27–46. [Google Scholar] [CrossRef] [Green Version]
- Makar, K. Young children’s explorations of average through informal inferential reasoning. Educ. Stud. Math. 2014, 86, 61–78. [Google Scholar] [CrossRef] [Green Version]
- Pfannkuch, M.; Arnold, P.; Wild, C. What I see is not quite the way it really is: Students’ emergent reasoning about sampling variability. Educ. Stud. Math. 2015, 88, 343–360. [Google Scholar] [CrossRef]
- Makar, K.; Rubin, A. A framework for thinking about informal statistical inference. Stat. Educ. Res. J. 2009, 8, 82–105. [Google Scholar]
- Nunes, T.; Bryant, P.; Evans, D.; Barros, R. Assessing quantitative reasoning in young children. Math. Think. Learn. 2015, 17, 178–196. [Google Scholar] [CrossRef]
- Kaput, J.J. Transforming algebra from an engine of inequity to an engine of mathematical power by “algebrafying” the K–12 curriculum. In the Nature and Role of Algebra in the K–14 Curriculum, Proceedings of a National Symposium Washington, DC, USA, 27–28 May 1997; National Council of Teachers of Mathematics & Mathematical Sciences Education Board, Ed.; National Academies Press: Washington, DC, USA, 1998. [Google Scholar] [CrossRef]
- Bishop, J.P.; Lamb, L.L.; Philipp, R.A.; Withacre, I.; Schappelle, B.P.; Lewis, M.L. Obstacles and affordances for integer reasoning: An analysis of children’s thinking and the history of mathematics. J. Res. Math. Educ. 2014, 45, 19–61. [Google Scholar] [CrossRef]
- Lobato, J.; Hohensee, C.; Rhodehamel, B.; Diamond, J. Using student reasoning to inform the development of conceptual learning goals: The case of quadratic functions. Math. Think. Learn. 2012, 14, 85–119. [Google Scholar] [CrossRef]
- Hilton, A.; Hilton, G.; Dole, S.; Goos, M. Promoting middle school students’ proportional reasoning skills through an ongoing professional development programme for teachers. Educ. Stud. Math. 2016, 92, 193–219. [Google Scholar] [CrossRef]
- Johnson, H.L. Secondary students’ quantification of ratio and rate: A framework for reasoning about change in covarying quantities. Math. Think. Learn. 2015, 17, 64–90. [Google Scholar] [CrossRef]
- Johnson, H.L.; McClintock, E.; Hornbein, P. Ferris wheels and filling bottles: A case of a student’s transfer of covariational reasoning across tasks with different backgrounds and features. ZDM 2017, 49, 851–864. [Google Scholar] [CrossRef]
- Thompson, P.W. The development of the concept of speed and its relationship to concepts of rate. In The Development of Multiplicative Reasoning in the Learning of Mathematics; Harel, G., Confrey, J., Eds.; State University of New York Press: Albany, NY, USA, 1994; pp. 181–234. ISBN 978-0791417645/0791417646. [Google Scholar]
- Nunes, T.; Bryant, P.; Evans, D.; Bell, D.; Barros, R. Teaching children how to include the inversion principle in their reasoning about quantitative relations. Educ. Stud. Math. 2012, 79, 371–388. [Google Scholar] [CrossRef]
- de Beer, H.; Gravemeijer, K.; van Eijck, M. Discrete and continuous reasoning about change in primary school classrooms. ZDM 2015, 47, 981–996. [Google Scholar] [CrossRef] [Green Version]
- Simon, M.A. Beyond inductive and deductive reasoning: The search for a sense of knowing. Educ. Stud. Math. 1996, 30, 197–209. [Google Scholar] [CrossRef]
- Noll, J.; Shaughnessy, J.M. Aspects of students’ reasoning about variation in empirical sampling distributions. J. Res. Math. Educ. 2012, 43, 509–556. [Google Scholar] [CrossRef]
- OECD. Pisa 2012 Results: Creative Problem Solving: Students’ Skills in Tackling Real-Life Problems. In PISA 2012 results: Organisation for Economic Co-operation and Development, Programme for International Student Assessment; OECD Publishing: Paris, France, 2014; Volume 5, Available online: http://www.oecd.org/pisa/keyfindings/PISA-2012-results-volume-V.pdf (accessed on 15 October 2020).
Kind of Reasoning | Articles (Reference No.) | Numbers of Articles per Kind of Reasoning |
---|---|---|
Definitions and explanatory statements | 15, 22, 24, 25 | 4 |
Creative and imitative reasoning | 33, 34, 35 | 3 |
SOLO-taxonomy | 36 | 1 |
Hierarchic interactionism | 23 | 1 |
Kind of Reasoning | Articles (Reference No.) | Numbers of Articles per Kind of Reasoning |
---|---|---|
Spatial reasoning | 39, 40, 42, 43 | 4 |
Informal Inferential Reasoning (IIR) | 45, 46, 47, 48 | 4 |
Additive, multiplicative and distributive reasoning | 19, 50, 61 | 3 |
Algebraic reasoning | 19, 52 | 2 |
Proportional and covariational reasoning | 53, 54, 55, 56, 61 | 5 |
Quantitative reasoning | 19, 50, 53, 56, 58, 59 | 6 |
Transformational reasoning | 53, 55 | 2 |
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Hjelte, A.; Schindler, M.; Nilsson, P. Kinds of Mathematical Reasoning Addressed in Empirical Research in Mathematics Education: A Systematic Review. Educ. Sci. 2020, 10, 289. https://doi.org/10.3390/educsci10100289
Hjelte A, Schindler M, Nilsson P. Kinds of Mathematical Reasoning Addressed in Empirical Research in Mathematics Education: A Systematic Review. Education Sciences. 2020; 10(10):289. https://doi.org/10.3390/educsci10100289
Chicago/Turabian StyleHjelte, Alexandra, Maike Schindler, and Per Nilsson. 2020. "Kinds of Mathematical Reasoning Addressed in Empirical Research in Mathematics Education: A Systematic Review" Education Sciences 10, no. 10: 289. https://doi.org/10.3390/educsci10100289
APA StyleHjelte, A., Schindler, M., & Nilsson, P. (2020). Kinds of Mathematical Reasoning Addressed in Empirical Research in Mathematics Education: A Systematic Review. Education Sciences, 10(10), 289. https://doi.org/10.3390/educsci10100289