Abstract
In Chapters 4 and 5, we specifically addressed contexts in which visuoalphanumeric symbols in school algebra and number sense could be interpreted as symbolic entities that have roots in structured visual experiences. We also discussed the significance of progressive symbolization relative to intra- and inter-semiotic transitions that occur from iconic and/to indexical and to symbolic representations. In this chapter, we focus our attention on diagrams that are purposefully constructed to convey visual relationships and mediate in students’ understanding of alphanumeric forms. Examples of such diagrams include tables in pattern generalization, graphs of functions, squares, sticks, and dots in adding and subtracting whole numbers, binary chips and number lines used in understanding integer operations, etc.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
It is difficult to decide between the two definitions of mathematics: the one by its method, that of drawing necessary conclusions; the other by its aim and subject matter, as the study of hypothetical state of things.
(Peirce, 1956, p. 1779)
Mathematics requires an inter-subjectively given object. This is supplied in modern mathematics by conceptual systems and in Greek mathematics by the diagram.
(Netz, 1998, p. 38)
Plainly the movement to accord diagrams a substantial role in mathematics is crucial to a philosophy of real mathematics.
(Sherry, 2009, p. 60)
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
The process of figural parsing and biasing shares many of the characteristics that Gal and Linchevski (2010) identified in relation to the operative apprehension of geometric figures (Duval, 1998), which involves engaging in an appropriate and purposeful “dimensional deconstruction of figures” that leads to “infer[ring relevant] mathematical properties in axiomatic geometry” (Gal & Linchevski, 2010, p. 180).
- 2.
Swafford and Langrall (2000) found differing representational effects between presented and self-generated tables of values in patterning activity among 10 sixth-grade students prior to formal instruction. They note:
Tables seemed to be more useful when students constructed them to make sense of the problem. However, when the interviewer provided a table for the student to complete or to examine, the table seemed to be more of a distraction than an aid, diverting students’ focus from the context of the problem to a string of numbers” (pp. 106–107).
In my 3-year study with my middle school students, however, those student-constructed tables that merely show common difference (e.g., Fig. 5.3b) were problematic because they funneled my students to a particular, narrow form of numerical strategy that encouraged mostly constructive standard generalizations with no room for more creative and other complex forms of generalization (e.g., constructive nonstandard; deconstructive). The proposed inductive-structuring tables (e.g., Fig. 5.3a) as an alternative diagrammatic representation help overcome issues that Swafford and Langrall (2000) identified as unproductive and ineffective actions relevant to table use [“hinder(ing) their abilities to recognize and describe the relationship between dependent and independent variables implicit in the situation;” “cloud(ing) rather than clarify(ing) the students’ recognition of a relationship between the independent and dependent variables;” “draw(ing) students’ attention to the recursive relation between consecutive values of the dependent variable instead of the relation between the independent and dependent variables”; “forc(ing) an artificial relation between the numbers in the table with no regard for the context of the situation” (pp. 105–106)].
References
Ainsworth, S., & Loizou, A. T. (2003). The effects of self-explaining when learning with text or diagrams. Cognitive Science, 27, 669–681.
Bakker, A. (2007). Diagrammatic reasoning and hypostasized abstraction in statistics education. Semiotica, 164(1/4), 9–29.
Barmby, P., Harries, T., Higgins, S., & Suggate. J. (2009). The array representation and primary children’s understanding and reasoning in multiplication. Educational Studies in Mathematics, 70, 217–241.
Batt, N. (2007). Diagrammatic thinking in literature and mathematics. European Journal of English Studies, 11(3), 241–249.
Booth, R., & Thomas, M. (2000). Visualization in mathematics learning: Arithmetic problem-solving and student difficulties. Journal of Mathematical Behavior, 18(2), 169–190.
Brown, E., & Jones, E. (2006). Understanding conic sections using alternate graph paper. Mathematics Teacher, 99(5), 322–327.
Campos, D. (2007). Peirce on the role of poietic creation in mathematical reasoning. Transactions of the Charles S. Peirce Society, 43(3), 470–489.
Carter, J. (2010). Diagrams and proofs in analysis. International Studies in the Philosophy of Science, 24(1), 1–14.
Cellucci, C. (2008). The nature of mathematical explanation. Studies in the History and Philosophy of Science, 39, 202–210.
Cox, R., & Brna, P. (1995). Supporting the use of external representations in problem solving: The need for flexible learning environments. Journal of Artificial Intelligence in Education, 6(2/3), 239–302.
Curriculum Planning and Development Division. (2008). Primary mathematics textbooks. New Industrial Road, Singapore: Marshall Cavendish Education.
Dea, S. (2006). “Merely a veil over the living thought:” Mathematics and logic in Peirce’s forgotten spinoza review. Transactions of the Charles S. Peirce Society, 42(4), 501–517.
Dörfler, W. (2001a). Instances of diagrammatic reasoning. Paper presented to the discussion group on semiotic and mathematics education at the 25th PME conference, University of Utrecht, Netherlands. Retrieved from http://www.math.uncc.edu/~sae/dg3/dorfler1.pdf
Dörfler, W. (2001b). How diagrammatic is mathematical reasoning? Paper presented to the discussion group on semiotic and mathematics education at the 25th PME conference, University of Utrecht, Netherlands. Retrieved from http://www.math.uncc.edu/~sae/dg3/dorfler2.pdf
Dörfler, W. (2007). Matrices as Peircean diagrams: A hypothetical learning trajectory. CERME, 5, 852–861.
Gooding, D. (2006). From phenomenology to field theory: Faraday’s visual reasoning. Perspectives on Science, 14(1), 40–65.
Hegarty, M., & Kozhevnikov, M. (1999). Types of visual-spatial representations and mathematical problem solving. Journal of Educational Psychology, 91, 684–689.
Larkin, J., & Simon, H. (1987). Why a diagram is (sometimes) worth ten thousand words. Cognitive Science, 11, 65–99.
Leslie, A., Xu, F., Tremoulet, P., & Scholl, B. (1998). Indexing and the object concept: Developing “what” and “where” systems. Trends in Cognitive Science, 2(1), 10–18.
Lewis, K. (2009). Patterns with checkers. Mathematics Teaching in the Middle School, 14(7), 418–422.
Lomas, D. (2002). What perception is doing, and what it is not doing, in mathematical reasoning. British Journal for Philosophy of Science, 53, 205–223.
Netz, R. (1998). Greek mathematical diagrams: Their use and their meaning. For the Learning of Mathematics, 18(3), 33–39.
Netz, R. (1999). Shaping of deduction in Greek mathematics: A study in cognitive history. Cambridge, MA: Cambridge University Press.
Ng, S. F., & Lee, K. (2009). The model method: Singapore children’s tool for representing and solving algebraic word problems. Journal for Research in Mathematics Education, 40(3), 282–313.
Norman, J. (2006). After Euclid: Visual reasoning and the epistemology of diagrams. Stanford, CA: CSLI Publications.
Pantziara, M., Gagatsis, A., & Elia, I. (2009). Using diagrams as tools for the solution of nonroutine mathematical problems. Educational Studies in Mathematics, online first.
Peirce, C. (1956). The essence of mathematics. In J. Newman (Ed.), The world of mathematics: Volume 3 (pp. 1773–1786). New York: Simon & Schuster.
Peirce, C. (1958b). Lessons of the history of science. In P. Wiener (Ed.), Charles S. Peirce: Selected writings (Values in a universe of chance) (pp. 227–232). New York: Dover.
Peirce, C. (1976). In C. Eisele (Ed.), The new elements of mathematics: Volume IV mathematical philosophy. Atlantic Highlands, NJ: Humanities Press.
Ponce, G. (2008). Using, seeing, feeling, and doing absolute value for deeper understanding. Mathematics Teaching in the Middle School, 14(4), 234–240.
Pylyshyn, Z. (2006). Seeing and visualizing: It’s not what you think. Cambridge, MA: MIT Press.
Rivera, F. (2009). Visuoalphanumeric mechanisms that support pattern generalization. In I. Vale & A. Barbosa (Eds.), Patterns: Multiple perspectives and contexts in mathematics education (pp. 123–136). Portugal: Escola Superior de Educação do Instituto Politécnico de Viana de Coastelo.
Rotman, B. (1995). Thinking diagrams: Mathematics, writing, and virtual reality. South Atlantic Quarterly, 94(2), 389–415.
Sherry, D. (2006). Mathematical reasoning: Induction, deduction, and beyond. Studies in the History and Philosophy of Science, 37, 489–504.
Sherry, D. (2009). The role of diagrams in mathematical arguments. Foundations of Science, 14, 59–74.
Stjernfelt, F. (2000). Diagrams as centerpiece of a Peircean epistemology. Transactions of the Charles S. Peirce Society, XXXVI(3), 357–384.
Stjernfelt, F. (2007). Diagrammatology: An investigation on the borderlines of phenomenology, ontology, and semiotics. Dordrecht, Netherlands: Springer.
Suh, J., Johnson, C., Jamieson, S., & Mills, M. (2008). Promoting decimal number sense and representational fluency. Mathematics Teaching in the Middle School, 14(1), 44–50.
Swafford, J., & Langrall, B. (2000). Grade 6 students’ preinstructional use of equations to describe and represent problem situations. Journal for Research in Mathematics Education, 31(1), 89–112.
Tarlow, L. (2008). Sense-able combinatorics: Students’ use of personal representations. Mathematics Teaching in the Middle School, 13(8), 484–489.
Van Garderen, D. (2003). Visual–spatial representation, mathematical problem solving, and students of varying abilities. Learning Disabilities Research & Practice, 18, 246–254.
Van Garderen, D. (2006). Spatial visualization, visual imagery, and mathematical problem solving of students with varying abilities. Journal of Learning Disabilities, 39(6), 496–506.
Van Garderen, D. (2007). Teaching students with LD to use diagrams to solve mathematical word problems. Journal of Learning Disabilities, 40(6), 540–553.
Zahner, D., & Corter, J. (2010). The process of probability problem solving: Use of external visual representations. Mathematical Thinking and Learning, 12, 177–204.
Zbiek, R. M., & Heid, M. K. (2009). Using computer algebra systems to develop big ideas in mathematics. Mathematics Teacher, 102(7), 540–544.
Zhang, J. (1997). The nature of external representations in problem solving. Cognitive Science, 21(2), 179–217.
Zhang, J., & Norman, D. (1994). Representations in distributed cognitive tasks. Cognitive Science, 8, 87–122.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2011 Springer Science+Business Media B.V.
About this chapter
Cite this chapter
Rivera, F.D. (2011). Visual Thinking and Diagrammatic Reasoning. In: Toward a Visually-Oriented School Mathematics Curriculum. Mathematics Education Library, vol 49. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0014-7_6
Download citation
DOI: https://doi.org/10.1007/978-94-007-0014-7_6
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-0013-0
Online ISBN: 978-94-007-0014-7
eBook Packages: Humanities, Social Sciences and LawEducation (R0)