%0 Journal Article %1 Zazkis02 %A Zazkis, Rina %A Liljedahl, Peter %D 2002 %J Educational Studies in Mathematics %K algebra ijtme2006 learning mathematics mythesis number patterns postviva sequences sequences-ictmt7 %N 3 %P 379-402 %T Generalization of patterns: The tension between algebraic thinking and algebraic notation %U http://www.sfu.ca/~zazkis/publications/PatternsESM.pdf %V 49 %X This study explores the attempts of a group of preservice elementary school teachers to generalize a repeating visual number pattern. We discuss students emergent algebraic thinking and the variety of ways in which they generalize and symbolize their generalizations. Our results indicate that students ability to express generality verbally was not accompanied by, and did not depend on, algebraic notation. However, participants often perceived their complete and accurate solutions that did not involve algebraic symbolism as inadequate.

@article{Zazkis02, abstract = {This study explores the attempts of a group of preservice elementary school teachers to generalize a repeating visual number pattern. We discuss students emergent algebraic thinking and the variety of ways in which they generalize and symbolize their generalizations. Our results indicate that students ability to express generality verbally was not accompanied by, and did not depend on, algebraic notation. However, participants often perceived their complete and accurate solutions that did not involve algebraic symbolism as inadequate.}, added-at = {2008-05-30T05:41:17.000+0200}, author = {Zazkis, Rina and Liljedahl, Peter}, biburl = {https://www.bibsonomy.org/bibtex/2f7a03b86e20d1f6f4f21f385ab714ef2/yish}, citeulike-article-id = {378534}, comment = {p. 380 Stacey (1989) focused her exploration on linear patterns, presented pictorially as expanding ladders or trees. Participants were asked to determine the number of matches needed to make a ladder with 20 or 1000 rungs, or the number of lights in a Christmas tree of a given size. These patterns are labeled 'linear' because the nth element can be expressed as an+b. Stacey found these problems were challenging for 8-13 year old students. The constant difference property was largely recognized and enabled students to find the nth element of a pattern from the (n-1)th element. However, in an attempt to generalize, a significant number of students used an erroneous direct proportion method, that is, determining the nth element as the nth multiple of the difference. Stacey also reported inconsistencies in the methods chosen by students for 'near generalization' tasks (e.g., find the twentieth term), and 'far generalization' tasks (e.g., find the thousandth term). Similar results were reported by Zazkis and Liljedahl (2001, 2002) in their investigations of arithmetic sequences with preservice elementary school teachers. Orton and Orton (1999) extended investigations of linear patterns (arithmetic sequences) to other sequences of numbers. They reported the tendency of students to use differences between the consecutive elements in a sequence as their preferred method. This method was successfully extended to quadratic patterns by taking the second differences, but led to a dead end in instances such as the Fibonacci sequence. Among the obstacles to successful generalization Orton and Orton mentioned students' arithmetical incompetence and fixation on a recursive approach. Although allowing students to generate the next element in a sequence based on a previous one, this approach prevented them from seeing the general structure of all the elements. A recursive approach was also mentioned by English and Warren (1998) as an approach that students preferred and often reverted to when more challenging patterns were presented to them. p. 382 Attention to patterns is acknowledged in its importance as an introduction to algebra. Mason (1996) describes "expressing generality" as one of the roots of, and routes into, algebra. The use of patterns as a route to expressing generality has become popular over the past decade within school mathematics curricula in the UK (Orton and Orton, 1999). "Understanding patterns, relations and functions" is a continuous theme of the Algebra standard in the Principles and Standards for school mathematics (NCTM, 2000) at all grade levels. p. 398 When the term algebra is used it encompasses two distinct concepts: algebraic thinking and algebraic symbolism. There is a lack of agreement among researchers as to the relationship between the two. Some view the algebraic symbols as a necessary component of algebraic thinking, while others consider them as an outcome or as a communication tool. Further, different perspectives are argued on the relationship between algebraic reasoning and generalization.}, interhash = {0297e13ebf8d2b3705b4205f43f033e6}, intrahash = {f7a03b86e20d1f6f4f21f385ab714ef2}, journal = {Educational Studies in Mathematics}, keywords = {algebra ijtme2006 learning mathematics mythesis number patterns postviva sequences sequences-ictmt7}, number = 3, pages = {379-402}, priority = {2}, timestamp = {2008-08-21T13:43:38.000+0200}, title = {Generalization of patterns: The tension between algebraic thinking and algebraic notation}, url = {http://www.sfu.ca/~zazkis/publications/PatternsESM.pdf}, volume = 49, year = 2002 }