Algebraic Thinking 0004

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Vol.2-118 PME-NA 2006 Proceedings

SUPPORTING ALGEBRAIC THINKING AND GENERALIZING

ABOUT FUNCTIONAL RELATIONSHIP THROUGH PATTERNING

IN A SECOND GRADE CLASSROOM

Susan London McNab

OISE/[email protected]

This paper reports on a teaching study in a second grade classroom, in which functional

relationship was explored through an investigation of growing patterns by explicitly integrating

visual/spatial and numeric representations of pattern to promote algebraic thinking. Findings focus on three aspects of generalization: integration of representations, translation and

application across representations, and generalization as also the abstraction of abstractions.

Introduction

The ability to generalizethat is, to distill from a collection of particular instances a

relational abstraction transferable to new applicationshas been ascribed to algebraic thinking,itself a term that Kieran (1996) explicitly broadened beyond algebra to the use of any of avariety of representations that handle quantitative situations in a relational way. Recentresearch has challenged the assumed hierarchy of representations of mathematical ideas that hasconventionally ranked numeric over visual/spatial (e.g.Noss & Healy, 1997; Lee, 1996; Mason,1996; Nemirovsky, 1996). Case (e.g. Moss & Case, 1999; Case, 1998; Griffin & Case, 1997;Case, 1985) further contended that it is the integration of visual/spatial and numeric schemaswithin a given mathematical domain that allows children to establish what he referred to as anew central conceptual structure.

The study reported here sought to explore the notion that childrens full understanding of andability to engage in mathematical generalization may in fact rely on a critical integration of more

than one form of representation of a mathematical idea. This may more specifically be describedas involving childrens ability to move fluidly and fluently back and forth across multiplerepresentations in both interpreting and applying a mathematical generalization. Further,generalization may go beyond the directly experiential quantitative instances described byKieran, to include abstractions as instances themselves where a generalization describes therelationship amongst these abstractions; this relies for illumination on Piagets (2001[1977])differentiation of empirical abstraction from reflecting abstraction. An exploration of thesethree aspects of generalization (integration of representations, translation and application acrossrepresentations, and generalization as the abstraction of abstractions) will be the focus of thispaper, drawn from a teaching study of algebraic thinking about functional relationship in patternwork with second grade children.

Context and Methods

This classroom teaching intervention took place in an intact second grade classroom of 22students at a university laboratory school. This study is part of an larger ongoing internationalresearch project exploring algebraic thinking of students in second through sixth grades.

The twelve research lessons were presented during regularly scheduled math periods as partof the normal school day, three times a week over a period of four weeks. All lessons werevideotaped and written transcriptions made. Digital photographs were taken of childrens

_____________________________Alatorre, S., Cortina, J.L., Siz, M., and Mndez, A.(Eds) (2006). Proceedings of the 28th annual meeting of the

North American Chapter of the International Group for the Psychology of Mathematics Education. Mrida, Mxico:

Universidad Pedaggica Nacional.
mailto:[email protected]:[email protected]


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activities and constructions, and their classroom work was collected as artifacts for datainterpretation. Field notes were made by the researcher, classroom teacher and researchassistants.

Prior to the start of the research lessons, Number Knowledge Task (Case & Okamoto, 1996)was administered individually as an assessment of numeracy level. Pre- and post-assessments of

nine patterning items in multiple representations were administered in individual interviews.Further post-interviews were conducted with pairs of students attempting two standard algebraicreasoning tasks; these interviews were video-taped and written transcriptions made.

The research lessons: Integration of representations

The lessons began with visual/spatial representations by presenting the students with asequence of positions in a geometric growing pattern. These were made of square tiles placed inarrays that grew by a constant coefficient. The children were not taught multiplication prior to orduring this study; however, they invented it as needed over the four weeks of research lessons(Schliemann, Carrahar & Brizuela, 2001). To introduce integration of numeric with geometricrepresentations, an ordinal position number was placed below the geometric array thatrepresented that position of the pattern. This helped to make clear the functional relationship between, for example, the position number 1 and one row of 3 square tiles, and the positionnumber 2 and two rows of 3 square tiles each or 6 tiles altogether.

Numeric representations of functions were then explored using a function machine (Carrahar& Earnest, 2003; Rubenstein, 2002; Willoughby, 1997). Students took turns creating functionalrules, creating non-sequential examples as clues, to challenge their classmates to guess myrule. The children solving the challenge recorded on T-tables the input and output numbers,and their conjectures for what the rule might be. These numeric examples were non-sequentialto allow a focus on the across (on a T-table) or functional rule rather than on the down pattern or what comes next differencing strategy identified as interfering in reaching afunctional generalization in numeric patterns (Schliemann, Goodrow & Lara-Roth, 2001; Orton& Orton, 1998; Orton, Orton & Roper, 1998).

The students then integrated all aspects of the previous activities, by building non-sequentialgeometric pattern positions from a secret rule (composite function) on a pattern sidewalk, alarge counting line with ordinal position numbers on each section of the sidewalk. They wouldbuild, for example, positions 2, 4 and 9; other students would then guess the rule by trying tobuild the pattern correctly on, for instance, position 7.

Finally, the students in pairs made up their own mystery rules, and built several sequencedpositions of their own patterns out of a variety of construction materials, for other students toguess the rule. These were photographed and a booklet made; students reasoned in writing aboutwhat the pattern rules might be, agreeing, disagreeing, or elaborating on one anothers writtenconjectures. In this activity, an unexpected revelation was the spontaneous introduction by someof the students of the zero position of the pattern, which they explained was a big clue to

guessing the rule because it isolated the bump or the constant.

Preliminary findings: Translation and application across representations

There is some debate in the field regarding whether to use correct mathematical terminologyright from the start, or to rely on invented informal language with young children. In this study,because the concept of a function was first presented through geometric arrays, the informal termbump evolved for the constant because it appeared as an incomplete row above the array, that


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looked like a bump. The strong visual/spatial reference and experiential grounding that gave riseto this term supported the decision to stay with this, and see where it led.

In post assessments and interviews, all children were able to recognize and describe areasonable general functional rule for the pattern that was presented, and had strategies forapplying their rule to find extensions of the pattern, in all representations except one: skip

counting (by 3s). This was despite the fact that throughout the intervention many patterns hadfrequently been described by the students as a counting by [3]s pattern. This invites conjectureregarding the potential for interference of the conventional rote approach to skip counting donefrom Kindergarten. Other representations with which the students were more successfulincluded arrays, drawings, and T-tables with which they were familiar, as well as narrative, two-dimensional standard algebraic reasoning task (square tables problem) and three-dimensionalstandard algebraic reasoning task (cube sticker problem) representations with which they werenot familiar.

Within the narrative format (which describes a child with $10 saved for a scooter, who walksa neighbours dog to earn $5 each day), all but two children showed an understanding of the ruleas being a composite function with some recognition of the constant, so clearly the concept of a

bump had transcended its geometric beginnings. One child explained, The bumps are theextra ones that will always stay there. This understanding cut across all numeracy levels asdetermined by the Number Knowledge Task. However, the childrens strength in applying thisunderstanding varied. Seven children expressed clear correct generalizations, in more and lessformal language, that identified both the constant and the coefficient. One of these children, whowas considered highly distractible and low achieving, responded, Its counting by 5s with a 10bump, even though he lost interest in calculating the far transfer positions. A mid-level childresponded, Oh, I get itits a groups of 5 pattern with a 10 bump. A highly capable highachieving student went on to notice that the constant was larger than the coefficient (not part ofthe original geometric representation): Its always the day [ordinal position number] times 5, plus 10. So theres 10 bumps and 5 normal things, more bumps than normal thingsthatsweird!

A further eight students were able to apply their conceptual understanding of an implicitfunction rule to predict near and far positions, without being able articulate the general rule theywere nonetheless expressing in working through the particular positions asked for. Theremaining five included a constant at first, but lost sight of it as the magnitude of the numbersthey were working with increased (Stacey, 1989), and incorrectly applied a whole objectstrategy in doubling the 5th to get the 10th position (Lannin, 2002; Orton & Orton, 1998).

The two children who did not recognize the constant at any point in this task, explicitly orimplicitly, were still able to identify the correct coefficient and make the generalization that thenarrative presented a counting by 5s pattern. They were able to apply their incomplete rulecorrectly to both near and far positions.

Interviews: Generalization as the abstraction of abstractions

The children were interviewed in pairs, and presented with first the square tables problemwhich asks if square tables are arranged in a line, with one chair at each open side of a table, howmany chairs would there be for increasing numbers of tables. All pairs of students, organized byeither same numeracy level or friendships (with the aim of student comfortability to promotediscussion), were able to articulate a general rule, in informal or more formal language. Several pairs immediately saw the pattern; these students were encouraged to consider what wouldhappen if the tables were trapezoids instead of squares (this was drawn). Responses included a


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surprising sophisticated consideration of multiple ways of expressing the functional rule: Itcould be the number [of tables] plus 1 more, then times by 3, but then you have to take away 1,in recognition of the 2 end seats that were 1 seat shy of being the same as an extra table.

The same pairs were then presented with the cube sticker problem which asks if cubes arelinked together, and a sticker applied to each cube face that was still showing, how many stickers

would there be for increasing numbers of cubes. All pairs were also able to work this out, withsolutions ranging from linking actual cubes and counting sides, to clear generalizations (Its agroups of 4, and then 2 at the ends. In a very interesting leap to an abstraction of abstractions,one child recognized that these problems represented two-dimensional and three-dimensionalversions of the same type of generalization: Its [cubes] like the other one [tables], except times4, because theres 4 sides.

Conclusion

Implications of this study for future work in understanding the role of generalizing andalgebraic thinking in the mathematics learning of young children are many. Among them is theconceptual illumination of skip counting through a three-tiered pattern sidewalk, where thefunctional relationship between the position number and the number of elements in that positionis made clear through the medium of geometric constructions. Further, the link betweenrepeating and growing patterns has yet to be explicitly explored within the integrative frameworkof this research, where it would seem that repeating patterns can be thought of as more complexarticulations of growing patterns. The rich territory of mathematical modeling, largelyunexplored for elementary mathematics students (e.g. London McNab, Moss, Woodruff &Nason, 2004; Van den Heuvel-Panhuizen, 2004; Lesh & Doerr, 2000), is described in many ofthe same ways that help us to understand key aspects of algebraic reasoning; clearly theimportance of multiple representations stands out. It seems a natural further direction to considerhow these two approaches may be merged to the greater benefit of the learner.

Finally, for reasons that bear further thought, this approach supported engagement inactivities that the students found meaningful by including what Weininger (1981) woulddescribe as educational play. As one child explained, I dont really feel like its math. I think itkind of feels like its some fun stuff. Its kind of like youre half man, half horse; its kind oflike half fun, half math. Its like you change the gear into fun!.

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Algebraic Thinking 0004