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“Rising to the challenge”: using generalization in pattern problems to unearth the algebraic skills of talented pre-algebra students

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Abstract

This study focuses on the generalization methods used by talented pre-algebra students in solving linear and non-linear pattern problems. A qualitative analysis of the solutions of three problems revealed two approaches to generalization: recursive–local and functional–global. The students showed mental flexibility, shifting smoothly between pictorial, verbal and numerical representations and abandoning additive solution approaches in favor of more effective multiplicative strategies. Three forms of reflection aided generalization: reflection on commonalities in the pattern sequence’s structure, reflection on the generalization method, and reflection on the “tentative generalization” through verification of the pattern sequence. The latter indicates an intuitive grasp of the mathematical power of generalization. The students’ generalizations evinced algebraic thinking in the discovery of variables, constants and their mutual relations, and in the communication of these discoveries using invented algebraic notation. This study confirms the centrality of generalizations in mathematics and their potential as gateways to the world of algebra.

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Notes

  1. The club, which includes ∼400 students aged 10–16, was founded nine years ago by one of the authors of this paper. The project aimed at addressing the special needs of students (most of whom come from underdeveloped or struggling areas), who possess mathematical ability but are not necessarily considered gifted, and who are interested in and desirous of learning more about mathematics. The club and its activities are designed to help these children by developing their mathematical and creative thinking skills.

  2. The students participate in weekly mathematics workshops in specific topics (such as logic, problem-solving, number theory, etc.), and attend a full day devoted to science and social activity every five weeks. Classes are taught by teachers rich in mathematical knowledge and very experienced in working with talented students—most from the former Soviet Union. The students come from 50 schools in 14 different cities and villages. Some leave the club after one year, but most remain for several years. In its 9 years of existence, Kidumatica has won a string of awards in national and international Olympiads, and its graduates have moved on to prestigious university faculties.

  3. In a pilot study, the students were asked to provide “justifications” for their solution paths, a term that proved misleading and problematic. Students were not familiar with the request to justify in mathematics and associated the instruction only to verbal tasks such as in history or social studies. Some students were confused by the origin of the word “justification,” were unsure what it had to do with mathematics, and did not proceed. As a result, the wording was altered to "What did I do; why did I do it; if I changed my mind—why; if I did not answer—why.” These instructions guided the students in providing justifications without using explicit instructions.

  4. Hanukah is a Jewish holiday on which we light candles on each day of the 8-day holiday. On the first day, two candles are lit (the leading candle and one additional candle); on the second day, three candles are lit (the leading candle and two additional candles); on the third day, four candles are lit (the leading candle and three additional candles); and so on, until the eighth and last day of celebration.

  5. None of the students were familiar with the formula for the sum of arithmetic progression.

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  1. Department of Science and Technology Education, Ben-Gurion University, P.O.B. 653, Beer-Sheva, 84105, Israel

    Miriam Amit & Dorit Neria

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  1. Miriam Amit
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Correspondence to Miriam Amit.

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Amit, M., Neria, D. “Rising to the challenge”: using generalization in pattern problems to unearth the algebraic skills of talented pre-algebra students. ZDM Mathematics Education 40, 111–129 (2008). https://doi.org/10.1007/s11858-007-0069-5

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