Abstract
This chapter deals with the recognition of visual and numeric patterns and their description through the language of functions which, in turn, form algebraic patterns to be recognized and generalized at a higher conceptual level. In the age of computers, this generalization is critical for developing a computational environment within which new numeric patterns can be revealed through a computational experiment. Once a new numeric pattern is discovered, it can be expressed in visual terms to be used as a springboard into new mathematical activities. Through this process, one comes full circle from visual at the basic level to visual at a higher cognitive level. The recognition of a visual pattern often deals with the constructive aspect of perception in which one can discern in an image more information than the image, in the absence of one’s insight, transmits. Such aspect of perception in the context of Gestalt psychology is called reification (Wertheimer, 1938). In addition to the reification of the Gestaltists, this aspect of perception is of importance for developing metacognition (Flavell, 1976). When planning how to apply findings from metacognition to include mathematics instruction, such a cyclic reinterpretation of earlier experienced concepts at a higher abstracted level was recognized as being critical. The process of mentally “following” each subsequent representation, how it followed from the preceding representation and how it led to the next was viewed as a crucial step in developing metacognitive awareness of the concept and its formation (Campione et al., 1989).
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Notes
- 1.
With the advent of computers in education, a computational experiment secured an important niche among big ideas of mathematics and its pedagogy, including the study of patterns and functions in the middle grades.
- 2.
The OEIS® provides other descriptions used to continue the five-number sequence 1, 3, 5, 7, 9. A simple rule is to continue with odd numbers having odd digits only, thus not including 21, 23, 25, 27, 29 and like (odd) numbers. A more complicated rule stems from the fact that base-ten numbers 1, 3, 5, 7, and 9, being trivial (one-digit) palindromes (integers that read forward the same as backward; see Chap. 9, Sect. 9.10, and Chap. 12, Sect. 12.11), are also palindromes in base two. Indeed, \(1_{10} = 1_{2} ,3_{10} = 11_{2} ,5_{10} = 101_{2} ,7_{10} = 111_{2} ,9_{10} = 1001_{2}\). One can use Wolfram Alpha (entering the command “base ten number N in base two” for specific values of N into the input box of its free on-line version) to see that the sequence 1, 3, 5, 7, 9 extends to include 33, 99, and 313, as \(33_{10} = 100001_{2} ,99_{10} = 1100011_{2} ,313_{10} = 100111001_{2}\). In that way, a function that defines the sequence 1, 3, 5, 9, 33, 99, 313, …, is presented in the form of the verbal description of the rule that guides the development of the sequence. Also, see Sect. 4.7.1 of Chap. 4.
- 3.
A free on-line version of Wolfram Alpha provides solutions for various recurrences, both in symbolic and numeric forms; e.g., see Sect. 4.10 of Chap. 4. Also, the OEIS provides closed formulas for a great numbers of integer sequences, given their first few terms.
- 4.
Diophantus of Alexandria—a Greek mathematician of the third century A.D., called “the father of algebra”.
- 5.
This pattern was proposed by a student of one of the authors (SA) in a master’s childhood education program when the topic of patterns in the context of early algebra was discussed. Although the pattern cannot be described as an AB pattern taught already in preschool, teacher candidates taking part in this discussion were advised, in the spirit of Montessori (1917), to see the pattern through the lens of “the recognition of new phenomena, their reproduction and utilization” (p. 73) as important objectives of education.
- 6.
Note that the above comment regarding the need to enter more than three terms of the sequence is due to relations 1 = 21 − 1, 3 = 22 − 1, and 7 = 23 − 1 which appear being more plausible than the values of the quadratic function \(f(n) = n^{2} - n + 1\). Only the presence of the number \(13\;( \ne 2^{4} - 1)\) as the fourth term prompts both OEIS and Wolfram Alpha to recognize the four terms as the values of a quadratic function (although three values do define a quadratic function). This suggests three things: (i) when generalizing from numerical evidence, one should have some idea about what to expect from generalization and in order to guide students to these understandings, the teacher must be aware of and have experience with them; (ii) inductive generalization should be followed by mathematical induction (or any other) proof; and (iii) a wealth of information with an easy access, often requiring more than basic skills to deal with, attests to the duality of positive and negative affordances of technology.
- 7.
Using OEIS®, one gets the sequence \(3n^{2} + 2n + 1\) which looks different from the sequence \(3n^{2} - 4n + 2\). However, the former and the latter sequences generate the same numbers by starting to change n from 0 and from 1, respectively.
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Abramovich, S., Connell, M.L. (2021). Patterns and Functions. In: Developing Deep Knowledge in Middle School Mathematics. Springer Texts in Education. Springer, Cham. https://doi.org/10.1007/978-3-030-68564-5_10
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