# Supporting Mathematical Discussions: the Roles of Comparison and Cognitive Load

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## Abstract

Mathematical discussions in which students compare alternative solutions to a problem can be powerful modes for students to engage and refine their misconceptions into conceptual understanding, as well as to develop understanding of the mathematics underlying common algorithms. At the same time, these discussions are challenging to lead effectively, in part because they involve complex cognitive acts of identifying structural relationships within multiple solutions and comparing between these sets of relationships. While many of the considerations in leading such discussions have been described elsewhere, we highlight the cognitive challenges for students and the core role of relational reasoning that underpins student learning from these interactions. We review the literature on children’s development of relational reasoning and learning from comparisons to highlight particular challenges for students. We also review literature that suggests pedagogical practices for maximizing the likelihood that children will notice the intended relationships among solutions while minimizing overload to their cognitive resources. These practices include providing explicit comparison cues and labels, sequencing comparison before explicit instruction, using spatially aligned visual representations, and capitalizing on teacher gestures.

## Keywords

Relational reasoning Cognitive load Executive function Mathematical discussions## Notes

### Acknowledgments

This study was funded by the National Science Foundation, Grant Nos. DRL-1313531, SMA-1548292, and SBE-0541957. Funding was also provided by the IES Postdoctoral Research Training Program in the Education Sciences Grant #R305B150014.

### Compliance with Ethical Standards

### Conflict of Interest

The authors declare that they have no conflict of interest.

## References

- Alfieri, L., Nokes-Malach, T. J., & Schunn, C. D. (2013). Learning through case comparisons: a meta-analytic review.
*Educational Psychologist, 48*(2), 87–113.CrossRefGoogle Scholar - Alibali, M. W., & Nathan, M. J. (2007). Teachers’ gestures as a means of scaffolding students’ understanding: Evidence from an early algebra lesson. In R. Goldman, R. Pea, B. Barron, & S. J. Derry (Eds.),
*Video research in the learning sciences*(pp. 349–365). Mahwah, NJ: Erlbaum.Google Scholar - Alibali, M. W., Sylvan, E. A., Fujimori, Y., & Kawanaka, T. (1997).
*The functions of teachers’ gestures: What’s the point? Paper presented at the 69th Annual Meeting of the Midwestern Psychological Association*. Chicago: Illinois.Google Scholar - Alibali, M. W., Nathan, M. J., Wolfgram, M. S., Church, R. B., Jacobs, S. A., Martinez, C. J., & Knuth, E. J. (2014). How teachers link ideas in mathematics instruction using speech and gesture: a corpus analysis.
*Cognition and instruction, 32*, 65–100. doi: 10.1080/07370008.2013.858161. - Atkinson, R. K., Catrambone, R., & Merrill, M. M. (2003). Aiding transfer in statistics: examining the use of conceptually oriented equations and elaborations during subgoal learning.
*Journal of Educational Psychology, 95*(4), 762–773.CrossRefGoogle Scholar - Baddeley, A. D., & Hitch, G. (1974). Working memory. In G. H. Bower (Ed.),
*The psychology of learning and motivation: advances in research and theory*(Vol. 8, pp. 47–89). New York: Academic Press.Google Scholar - Ball, D. L. (1993). Halves, pieces, and twoths: Constructing and using representational contexts in teaching fractions. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research (pp. 157–195). Hillsdale, NJ: ErlbaumGoogle Scholar
- Ball, D.L., Lewis, J. & Thames, M.H. (2008). Making mathematics work in school.
*Journal for Research in Mathematics Education, Monograph 14, A Study of Teaching: Multiple Lenses, Multiple Views*. http://www.jstor.org/stable/30037740 - Begolli, K.N., Richland, L.E. (2016) Analog visibility as a double-edged sword.
*Journal of Educational Psychology, 108*(2), 194–213. doi: 10.1037/edu0000056. - Begolli, K. N., & Richland, L. E. (2015). Analog visibility as a double edged sword.
*Journal of Educational Psychology., 107*(3).Google Scholar - Begolli, K. N., Richland, L. E., & Jaeggi, S. (2015), The role of executive functions for structure-mapping in mathematics.
*Proceedings of the Cognitive Science Society Annual Meeting,*Pasadena*.*Google Scholar - Boaler, J., Dweck, C. (2016). Mathematical mindsets: unleashing students’ potential through creative math. Inspiring Messages and Innovative Teaching.Google Scholar
- Booth, J. L., Lange, K. E., Koedinger, K. R., & Newton, K. J. (2013). Example problems that improve student learning in algebra: differentiating between correct and incorrect examples.
*Learning and Instruction, 25*, 24–34.CrossRefGoogle Scholar - Booth, J. L., Cooper, L., Donovan, M. S., Huyghe, A., Koedinger, K. R., & Paré-Blagoev, E. J. (2015a). Design-based research within the constraints of practice: AlgebraByExample.
*Journal of Education for Students Placed at Risk, 20*(1-2), 79–100.Google Scholar - Booth, J. L., Oyer, M. H., Paré-Blagoev, E. J., Elliot, A., Barbieri, C., Augustine, A. A., & Koedinger, K. R. (2015b). Learning algebra by example in real-world classrooms.
*Journal of Research on Educational Effectiveness., 8*(4), 530–551.Google Scholar - Bransford, J. D., Brown, A. L., & Cocking, R. R. (2000).
*How people learn: Brain, mind, experience, and school. National Research Council, Commission on Behavioral & Social Sciences & Education. Committee on Developments in the Science of Learning*. Washington, DC: National Academy Press.Google Scholar - Bunge, S. A., Hauk Helskog, E., & Wendelken, C. (2009). Left, but not right, rostrolateral prefrontal cortex meets a stringent test of the relational integration hypothesis.
*NeuroImage, 46*(1), 338–342.CrossRefGoogle Scholar - Carpenter, T. P., Fennema, E., Franke, M. L., Empson, S. B. (1999). Children’s Mathematics: Cognitively Guided Instruction, Heinemann and the National Council of Teachers of Mathematics, Reston, VA: Heinemann.Google Scholar
- Carpenter, T. P., Fennema, E., Franke, M. L., Empson, S. B., & Levi, L. W. (2014).
*Children’s mathematics: cognitively guided instruction*(2nd ed.). Portsmouth: Heinemann.Google Scholar - Catrambone, R. (1996). Generalizing solution procedures learned from examples.
*Journal of Experimental Psychology: Learning, Memory, and Cognition, 22*(4), 1020–1031.Google Scholar - Catrambone, R., & Holyoak, K. J. (1990). Learning subgoals and methods for solving probability problems.
*Memory and Cognition, 18*(6), 593–603.CrossRefGoogle Scholar - Chi, M. T. H., Feltovich, P. J., & Glaser, R. (1981). Categorization and representation of physics problems by experts and novices.
*Cognitive Science, 5*, 121–152.CrossRefGoogle Scholar - Cho, S., Holyoak, K. J., & Cannon, T. D. (2007). Analogical reasoning in working memory: Resources shared among relational integration, interference resolution, and maintenance.
*Memory & Cognition, 35*(6), 1445–1455.Google Scholar - Christie, S., & Gentner, D. (2010). Where hypotheses come from: Learning new relations by structural alignment.
*Journal of Cognition and Development, 11*(3), 356–373.Google Scholar - Common Core State Standards Initiative. (2010). Common core state standards for mathematics. Retrieved from: http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf
- Diamond, A. (2013). Executive functions.
*Annual Review of Psychology, 64*, 135–168. doi: 10.1146/annurev-psych-113011-143750.CrossRefGoogle Scholar - Durkin, K., & Rittle-Johnson, B. (2012). The effectiveness of using incor- rect examples to support learning about decimal magnitude.
*Learning and Instruction, 22*, 206–214. doi: 10.1016/j.learninstruc.2011.11.001. - Engle, R. W., Tuholski, S. W., Laughin, J. E., & Conway, A. R. A. (1999). Working memory, short-term memory and general fluid intelligence: a latent variable approach.
*Journal of Experimental Psychology: General, 128*, 309–331.CrossRefGoogle Scholar - Gadgil, S., Nokes-Malach, T. J., & Chi, M. T. H. (2012). Effectiveness of holistic mental model confrontation in driving conceptual change.
*Learning and Instruction, 22*(1), 47–61.Google Scholar - Gentner, D. (1983). Structure-mapping: A theoretical framework for analogy.
*Cognitive Science, 7*, 155–170. doi: 10.1207/ s15516709cog0702_3. - Gentner, D., & Rattermann, M. J. (1991). Language and the career of similarity. In S. A. Gelman & J. P. Byrnes (Eds.),
*Perspectives on thought and language: interrelations in development*(pp. 225–277). London: Cambridge University Press.CrossRefGoogle Scholar - Gentner, D., & Loewenstein, J. (2002). Relational language and relational thought. In E. Amsel & J. Byrnes (Eds.),
*Language, literacy, and cognitive development: the development and consequences of symbolic communication*(pp. 87–120). Mahwah: Erlbaum.Google Scholar - Gentner, D., Holyoak, K. J., & Kokinov, B. N. (2001).
*The analogical mind: Perspectives from cognitive science*(pp. 437–470). Cambridge: MIT Press.Google Scholar - Gerjets, P., Scheiter, K., & Catrambone, R. (2004). Designing instructional examples to reduce intrinsic cognitive load: molar versus modular presentation of solution procedures.
*Instructional Science, 32*, 33–58.CrossRefGoogle Scholar - Gick, M. L., & Holyoak, K. L. (1980). Analogical problem solving.
*Cognitive Psychology, 12*, 306–355. doi: 10.1016/0010-0285(80)90013-4. - Gick, M. L., & Holyoak, K. L. (1983). Schema induction and analogical transfer.
*Cognitive Psychology, 15*, 1–38. doi: 10.1016/0010-0285(83)90002-6. - Goldin-Meadow, S. (2003).
*Hearing gesture: How our hands help us think*. Cambridge, MA: Harvard University Press.Google Scholar - Goswami, U., & Brown, A. L. (1989). Melting chocolate and melting snowmen: analogical reasoning and causal relations.
*Cognition, 35*, 69–95.CrossRefGoogle Scholar - Große, C. S., & Renkl, A. (2006). Effects of multiple solution methods in mathematics learning.
*Learning and Instruction, 16*(2), 122–138.CrossRefGoogle Scholar - Holyoak, K. J., Novick, L., & Metz, E. R. (1994). Component processes in analogical transfer: mapping, pattern completion, and adaptation. In K. J. Holyoak & J. A. Barnden (Eds.),
*Advances in connectionist and neural computation theory: Vol 2. Analogical connections*(pp. 113–180). Norwood: Ablex.Google Scholar - Humphreys, C., & Parker, R. (2015). Making Number Talks Matter: Developing Mathematical Practices and Deepening Understanding, Grades 4-10, Portland, ME: Stenhouse Publishers.Google Scholar
- Kazemi, E., & Hintz, A. (2014).
*Intentional talk: how to structure and lead productive mathematical discussions*. Portland: Stenhouse Publishers.Google Scholar - Kurtz, K. J., & Gentner, D. (2013). Detecting anomalous features in complex stimuli: The role of structured comparison.
*Journal of Experimental Psychology: Applied, 19*(3), 219–232.Google Scholar - Loewenstein, J., & Gentner, D. (2005). Relational language and the development of relational mapping.
*Cognitive Psychology, 50*, 315–353.Google Scholar - Ma, L. (1999).
*Knowing and teaching elementary mathematics: teachers' understanding of fundamental mathematics in China and the United States*. Mahwah, N.J.: Lawrence Erlbaum Associates.Google Scholar - Matlen, B. J., Vosniadou, S., Jee, B., & Ptouchkina, M. (2011). Enhancing the comprehension of science text through visual analogies. In L. Carlson, C. Holscher, & T. Shipley (Eds.),
*Proceedings of the 34th annual conference of the Cognitive Science Society*(pp. 2910–2915). Austin, TX: Cognitive Science Society.Google Scholar - Matlen, B. J., Gentner, D., & Franconeri, S. (2014, July).
*Struc- ture mapping in visual comparison: Embodied correspondence lines? Poster presented at the 37th Annual Conference of the Cognitive Science Society*. CA: Pasadena.Google Scholar - Miyake, A., Friedman, N. P., Emerson, M. J., Witzki, A. H., Howerter, A., & Wager, T. D. (2000). The unity and diversity of executive functions and their contributions to complex “frontal lobe” tasks: a latent variable analysis.
*Cognitive Psychology, 41*(1), 49–100.CrossRefGoogle Scholar - Loewenstein, J., & Gentner, D. (2005). Relational language and the development of relational mapping.
*Cognitive Psychology, 50*, 315–353.CrossRefGoogle Scholar - Namy, L. L., & Gentner, D. (2002). Making a silk purse out of two sow's ears: Young children's use of comparison in category learning.
*Journal of Experimental Psychology: General, 131*, 5–15.Google Scholar - National Mathematics Advisory Panel (2008).
*Foundations for success: the final report of the National Mathematics Advisory Panel*. Washington DC: U.S. Department of Education.Google Scholar - Paas, F., Renkl, A., & Sweller, J. (2003). Cognitive load theory and instructional design: recent developments cognitive load theory and instructional design: recent developments.
*Educational Psychologist, 38*(1), 1–4. doi: 10.1207/S15326985EP3801.CrossRefGoogle Scholar - Rattermann, M. J., & Gentner, D. (1998). More evidence for a relational shift in the development of analogy: children’s performance on a causal-mapping task.
*Cognitive Development, 13*, 453–478.CrossRefGoogle Scholar - Richland, L. E. (2015). Cross-Cultural Differences in Linking Gestures during instructional Analogies.
*Cognition and Instruction, 33*(4), 295–321. doi: 10.1080/07370008.2015.1091459. - Richland, L. E., Holyoak, K. J., & Stigler, J. W. (2004). Analogy generation in eighth-grade mathematics classrooms.
*Cognition and Instruction, 22*(1), 37–60.CrossRefGoogle Scholar - Richland, L. E., & McDonough, I. M. (2010). Learning by analogy: Discriminating between potential analogs.
*Contemporary Educational Psychology, 35*, 28–43.Google Scholar - Richland, L. E., & Burchinal, M. (2013). Early executive function predicts reasoning development.
*Psychological Science, 24*, 87–92.CrossRefGoogle Scholar - Richland, L. E., Morrison, R. G., & Holyoak, K. J. (2006). Children’s development of analogical reasoning: insights from scene analogy problems.
*Journal of Experimental Child Psychology, 94*, 249–271.CrossRefGoogle Scholar - 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*, 561–574. doi: 10.1037/0022-0663.99.3.561. - Rittle-Johnson, B., Star, J., & Durkin, K. (2009). The importance of prior knowledge when comparing examples: Influences on conceptual and procedural knowledge of equation solving.
*Journal of Educational Psychology, 101*(4), 836–852.Google Scholar - Sherin, M. G. (2002). A balancing act: Developing a discourse community in a mathematics classroom.
*Journal of Mathematics Teacher Education, 5*, 205–233.Google Scholar - Smith, M. S., Hughes, E. K., Engle, R. A., & Stein, M. K. (2009). Orchestrating discussions.
*Mathematics Teaching in the Middle School, 14*(9), 548–556.Google Scholar - Smith, M., & Stein, M. K. (2011).
*5 practices for orchestrating productive mathematics discussions*. Reston: National Council of Teachers of Mathematics.Google Scholar - Star, J. R., Pollack, C., Durkin, K., Rittle-Johnson, B., Lynch, K., Newton, K., & Gogolen, C. (2014). Learning from comparison in algebra.
*Contemporary Educational Psychology*. doi: 10.1016/j.cedpsych.2014.05.005.Google Scholar - Stein, M. K., Kaufman, J., & Tekkumru-Kisa, M. (2014). Mathematics teacher development in the context of district managed curriculum. In Y. Li & G. Lappan (Eds.),
*Mathematics curriculum in school education*(pp. 351–376). Dordrecht: Springer Science + Business Media. doi: 10.1007/978-94-007-7560-2. - Stigler, J. W., & Hiebert, J. (1999).
*The teaching gap: Best ideas from the world’s teachers for improving education in the classroom*. New York, NY: Free Press.Google Scholar - Tekkumru Kisa, M., & Stein, M. K. (2015). Learning to see teaching in new ways: a foundation for maintaining cognitive demand.
*American Educational Research Journal, 52*(1), 105–136. doi: 10.3102/0002831214549452.CrossRefGoogle Scholar - Vendetti, M., Matlan, B., Richland, L., & Bunge, S. (2015). Analogical Reasoning in the Classroom: Insights from Cognitive Science.
*Mind, Brain, and Education, 9*(2), 100–106.CrossRefGoogle Scholar - Waltz, J. A., Lau, A., Grewal, S. K., & Holyoak, K. J. (2000). The role of working memory in analogical mapping.
*Memory and Cognition, 28*, 1205–1212.CrossRefGoogle Scholar