Abstract
In this chapter, we discuss issues of depth that are relevant to the concept and process of generalization. We clarify the following useful terms that are now commonly used in patterns research: abduction; induction; near generalization and far generalization; and deduction. We also explore nuances in the meaning of generalization that have been used in different contexts in the school mathematics curriculum. In the closing section, we begin to discuss implications of the findings in the chapter on our proposal of a theory of graded pattern generalization that we explore in some detail in Chap. 4.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
As an aside, kindergarten students (ages 5–6 years) in the absence of formal learning experiences appear to consider deductive inferences as being more certain than inductive ones and other guesses (Pillow, Pearson, Hecht, & Bremer, 2010).
- 2.
Certainly, there can be an abduction without induction (i.e. abductive generalizations). Some geometry theorems, for example, do not need inductive verification. We do not explore these situations in this book in light of our interest on patterns. However, it is useful to note the insights of Pedemonte (2007) and Prusak, Hershkowitz, and Schwarz (2012) about the necessity of a structural continuity between an abduction argument process and its corresponding justification in the form of a logical proof. That is, a productive abductive process in whatever modal form (visual, verbal) should simultaneously convey the steps in a deductive proof.
- 3.
Polya (1973, pp. 17–22) recounts the story of Euler’s numeric-driven generalization of the infinite series \({\displaystyle \sum _{n=1}^{\infty }\frac{1}{{n}^{2}}}\)by initially establishing an analogical relationship between two different types of equations (i.e. a polynomial P of degree n having n distinct nonzero roots and a trigonometric equation that can be transformed algebraically into something like P but with an infinite number of terms). Euler’s abductive claim had him hypothesizing an anticipated solution drawn from similarities between the forms of the two equations. Upon inductively verifying that the initial four terms of the two equations were indeed the same, Euler concluded
that \({\displaystyle \sum _{n=1}^{\infty }\frac{1}{{n}^{2}}=\frac{{\pi }^{2}}{6}}\).
- 4.
See Pedemonte and Reid (2011) for a more refined analysis of different types of abductive action that support and hinder the construction of empirical arguments and deductive proofs.
- 5.
Clements and Sarama (2009) note that children from ages 2–7 years developmentally progress in their understanding of, and expertise in, patterns in the following manner: being pre-explicit patterners of everyday things, actions, etc. at age 2; being recognizers of simple repeating sequences of objects and consecutive counting numbers at age 3; being fillers (or fixers) and duplicators of repeating patterns at age 4; being extenders of repeating patterns at age 5; being core unit recognizers of repeating patterns at age 6; and being numeric patterners of growing patterns at age 7. My overall interpretation of this progression deals with transitions in elementary students’ ability to express an interpreted, stable, and unique core unit from the implicit (nonverbal, gestural, aided by concrete objects) to the explicit (verbal with and without aid of concrete objects) stage. I deal with this issue in some detail in Chaps. 5 and 6.
- 6.
We share Davydov’s (2008) view that “(g)eneralization is regarded as inseparably linked with the operation of abstraction” (p. 75). Hence, in this book, abstraction is treated as an operation that constructs conceptual generalizations.
- 7.
Certainly this view should be seen in Davydov’s (2008) larger perspective in which school curricula should aim for higher level theoretical consciousness and cognition beyond, and not merely, the formation of roots of empirical consciousness and cognition. While the empirical grounding is “important,” nonetheless, “at present [is] not the most effective way of developing [students’] minds” (p. 73).
- 8.
See Luria (1976, pp. 53–98) for details of his experimental studies in which he obtained the same patterns of generalized thinking schemes among different groups of adult subjects. In his closing remark relative to the studies, he notes how the “evidence assembled indicates that the processes used to render abstractions and generalizations does not assume an invariable form at all stage of mental growth” but “are themselves a product of socioeconomic and cultural development” (p. 98). Further, he stresses the possibility of transforming from situational, concrete, and practical to theoretical and abstract ways of generalizing via an evolving language whose meanings are enriched via, say, more education, new experiences, and new ideas.
References
Adler, J. (2008). Introduction: Philosophical foundations. In J. Adler & L. Rips (Eds.), Reasoning: Studies of human inference and its foundations (pp. 1–34). New York, NY: Cambridge University Press.
Anderson, K., Casey, B., Thompson, W., Burrage, M., Pezaris, E., & Kosslyn, S. (2008). Performance on middle school geometry problems with geometry clues matched to three different cognitive styles. Mind, Brain, and Education, 2(4), 188–197.
Arzarello, F. (2008). The proof in the 20th century. In P. Boero (Ed.), Theorems in schools: From history, epistemology, and cognition in classroom practices (pp. 43–64). Rotterdam: Sense Publishers.
Becker, J., & Rivera, F. (2005). Generalization strategies of beginning high school students. In H. Chick & J. Vincent (Eds.), Proceedings of the 29th conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 121–128). Melbourne: PME.
Clements, D., & Sarama, J. (2009). Learning and teaching early math: The learning trajectories approach. New York, NY: Erlbaum.
Davydov, V. V. (1990). Type of generalization in instruction: Logical and psychological problems in the structuring of school curricula. In J. Kilpatrick (Ed.), Soviet studies in mathematics education (Vol. 2). Reston, VA: National Council of Teachers of Mathematics.
Davydov, V. (2008). Problems of developmental instruction: A theoretical and experimental psychological study (P. Moxhay., Trans.). New York, NY: Nova Science Publishers.
Dehaene, S. (1997). The number sense. New York, NY: Oxford University Press.
Dehaene, S., & Cohen, L. (1995). Two mental calculational systems: A case study of severe acalculia with preserved approximation. Neuropsychologia, 29, 1045–1074.
Dörfler, W. (1991). Forms and means of generalization. In A. Bishop & S. Mellin-Olsen (Eds.), Mathematical knowledge: Its growth through teaching (pp. 63–85). Netherlands: Kluwer.
Eco, H. (1983). Horns, hooves, insteps: Some hypotheses on three types of abduction. In U. Eco & T. Sebeok (Eds.), The sign of three: Dupin, Holmes, Peirce (pp. 198–220). Bloomington, IN: Indiana University Press.
Ellis, A. (2007). Connections between generalizing and justifying: Students’ reasoning with linear relationships. Journal for Research in Mathematics Education, 38(3), 194–229.
Fischbein, E. (1993). The theory of figural concepts. Educational Studies in Mathematics, 24, 139–162.
Gentner, D. (2010). Bootstrapping the mind: Analogical processes and symbol systems. Cognitive Science, 34, 752–775.
Goswami, U. (2011). Inductive and deductive reasoning. In U. Goswami (Ed.), The Wiley-Blackwell handbook of childhood cognitive development (pp. 399–419). Malden, MA: Wiley-Blackwell.
Hibben, J. (1905). Logic: Deductive and inductive. New York: Charles Scribner’s Sons.
Israel, R. (2006). Projectibility and explainability or how to draw a new picture of inductive practices. Journal for General Philosophy of Science, 37, 269–286.
Josephson, J. (2000). Smart inductive generalizations are abductions. In P. Flach & A. Kakas (Eds.), Abduction and induction: Essays on their relation and integration (pp. 31–44). Netherlands: Kluwer.
Josephson, J., & Josephson, S. (1994). Abductive inference: Computation, philosophy, technology. New York, NY: Cambridge University Press.
Knuth, E. (2002). Proof as a tool for learning mathematics. Mathematics Teacher, 95(7), 486–490.
Luria, A. (1976). Cognitive development: Its cultural and social foundations. Cambridge, MA: Harvard University Press.
Maslow, A. (1970). Motivation and personality. New York: Harper & Row.
Mason, J. (2002). Generalization and algebra: Exploiting children’s powers. In L. Hegarty (Ed.), Aspects of teaching secondary mathematics: Perspectives on practice. London: Routledge Falmer.
Mason, J., & Johnston-Wilder, S. (2004). Fundamental constructs in mathematics education. London: Routledge Falmer.
Mason, J., Stephens, M., & Watson, A. (2009). Appreciating mathematical structures for all. Mathematics Education Research Journal, 21(2), 10–32.
McClelland, J., Botvinick, M., Noelle, D., Plaut, D., Rogers, T., Seindenberg, M., et al. (2010). Letting structures emerge: Connectionist and dynamical systems approaches to cognition. Trends in Cognitive Science, 14, 348–356.
Paavola, S. (2011). Diagrams, iconicity, and abductive discovery. Semiotica, 186(1/4), 297–314.
Parker, T., & Baldridge, S. (2004). Elementary mathematics for teachers. Okemos, MI: Sefton-Ash Publishing.
Pedemonte, B. (2007). How can the relationship between argumentation and proof be analyzed? Educational Studies in Mathematics, 66, 23–41.
Pedemonte, B., & Reid, D. (2011). The role of abduction in proving processes. Educational Studies in Mathematics, 76, 281–303.
Peirce, C. (1869). Grounds of validity of the laws of logic: Further consequences of four incapacities. The Journal of Speculative Philosophy, 2, 193–208.
Peirce, C. (1934). Collected papers of Charles Saunders Peirce: Volume 5. Cambridge, MA: Harvard University Press.
Peirce, C. (1960). Collected papers of Charles Saunders Peirce: Volumes I and II. Cambridge, MA: The Belnap Press of Harvard University Press.
Pillow, B., Pearson, R., Hecht, M., & Bremer, A. (2010). Children’s and adults’ judgments of the certainty of deductive inference, inductive inferences, and guesses. Journal of Genetic Epistemology, 171(3), 203–217.
Polya, G. (1957). How to solve it. Princeton, NJ: Princeton University Press.
Polya, G. (1973). Induction and analogy in mathematics: Volume I of mathematics and plausible reasoning. Princeton, NJ: Princeton University Press.
Prusak, N., Hershkowitz, R., & Schwarz, B. (2012). From visual reasoning to logical necessity through argumentative design. Educational Studies in Mathematics, 79, 19–40.
Radford, L., Bardini, C., & Sabena, C. (2007). Perceiving the general: The multisemiotic dimension of students’ algebraic activity. Journal for Research in Mathematics Education, 38(5), 507–530.
Rivera, F. (2011). Toward a visually-oriented school mathematics curriculum: Research, theory, practice, and issues (Mathematics Education Library Series 49). New York, NY: Springer.
Shtoff, V. (1966). Modeling and philosophy. Moscow: Leningrad.
Smith, L. (2002). Reasoning by mathematical induction in children’s arithmetic. Oxford, UK: Elsevier Science Ltd.
Stacey, K. (1989). Finding and using patterns in linear generalizing problems. Educational Studies in Mathematics, 20, 147–164.
Strevens, M. (2008). Depth: An account of scientific explanation. Cambridge, MA: Harvard University Press.
Thagard, P. (1978). Semiosis and hypothetic inference in C. S. Peirce. Versus Quaderni Di Studi Semiotici, 19(20), 163–172.
Varzi, A. (2008). Patterns, rules, and inferences. In J. Adler & L. Rips (Eds.), Reasoning: Studies of human inference and its foundations (pp. 282–290). New York, NY: Cambridge University Press.
Vinner, S. (2011). The role of examples in the learning of mathematics and in everyday thought processes. ZDM Mathematics Education, 43, 247–256.
Vygotsky, L. (1962). Thought and language. Cambridge, MA: MIT.
Watson, A. (2009). Thinking mathematically, disciplined noticing, and structures of attention. In S. Lerman & B. Davis (Eds.), Mathematical action & structures of noticing (pp. 211–222). Rotterdam, Netherlands: Sense Publishers.
Williams, J., & Lombrozo, T. (2010). The role of explanation in discovery and generalization: Evidence from category learning. Cognitive Science, 34, 776–806.
Williams, J., Lombrozo, T., & Rehder, B. (2011). Explaining drives the discovery of real and illusory patterns. In L. Carlson, C. Hölscher, & T. Shipley (Eds.), Proceedings of the 33rd annual conference of the Cognitive Science Society (pp. 1352–1357). Austin, TX: Cognitive Science Society.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Rivera, F. (2013). Contexts of Generalization in School Mathematics. In: Teaching and Learning Patterns in School Mathematics. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2712-0_2
Download citation
DOI: https://doi.org/10.1007/978-94-007-2712-0_2
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-2711-3
Online ISBN: 978-94-007-2712-0
eBook Packages: Humanities, Social Sciences and LawEducation (R0)