Generalization and Justification: The Challenge of Introducing Algebraic Reasoning Through Patterning Activities

Lannin, J.K. (2005). Generalization and justification: the challenge of introducing algebraic reasoning through patterning activities. Mathematical Thinking and Learning, 7(3), 231-258.

Presenters: Wei-Chih Hsu Presenters: Wei-Chih Hsu Professor: Ming-Puu ChenProfessor: Ming-Puu ChenDate: 04/17/2008Date: 04/17/2008

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Introduction

Students be introduced to algebraic ideas at earlier grade levels Creates new challenges The possibility for developing student understanding.

Use computer spreadsheets as an instructional tool Allow students to examine sequences using recursive or explicit

reasoning. Aid the transition to formal algebra.

Developing algebraic understanding through patterning activities Creates considerable difficulties as students move from a focus on

particular examples toward creating generalizations. Help students construct and justify generalizations.

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Literature review Dienes (1961) emphasized that, for students to comprehend

algebraic generalizations, they must see that an algebraic rule is a generalization. A mathematical statement that models a situation for any value in t

he defined domain of the variable (s).

Justification becomes a critical component of the generalization process

Mathematical Teaching Cycle (Simon, 1995)

Be used as the theory to inform instructional decision making. Initial instructional goal is formulated. Potential learning activities and a hypothetical learning trajectory f

or how instruction might proceed.

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Literature review Generalization

Statements of generality and discovering generality are at the very core of mathematical activity (Kaput, 1999; Mason 1996).

Generalizing numeric situations is viewed as one means for transitioning students into formal algebra (Lee, 1996). provide a connection to referential contexts

• can aid student understanding of symbolic representations link to students’ prior knowledge of arithmetic.

The typical patterning activity provides A context (e.g., see Figure 1) Asks students to generate a rule or rules that could be used to determ

ine other particular instances of the pattern. Encourages students to construct a variety of generalizations. (Kenney,

Zawojewski, & Silver, 1998; Stacey, 1989; Swafford & Langrall, 2000)

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Literature review

Generalization Strategies The framework was divided into two categories (see Table 1).

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Literature review

Justification In the mathematics classroom, justifications are deemed

acceptable when they meet the criteria that are established in the mathematical community (Hanna,1990).

A framework (see Table 2).

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Literature review

Spreadsheets as an Instructional Tool Can help students make connections between their info

rmal ideas and the formal representations (Sutherland & Rojano, 1993).

Permit students to reason flexibly, allowing them to reason recursively or explicitly.

Connect mathematical ideas relating their symbolic representations to graphical representations (Drier, 2001), iconic representations (Abramovich, 2000; Healy & Hoyles, 1999), and the problem context itself.

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Method

Participants 25 sixth-grade students.

10 instructional sessions. Each of the class sessions was videotaped, and a classroom observer t

ook field notes. Specific attention was given to four target students’ strategies and

understandings. A video camera focused on these four students during small-group di

scussions These students were interviewed at the end of every other class sessio

n by either the observer or me. Data Analysis

Analyzed using a data reduction approach (Miles & Huberman, 1994)

All transcripts from the videotapes and the interviews were coded using an initial list of codes for students’ generalizations and justifications .

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Method

Instructional Design The initial two instructional sessions

Familiarize with how to use the spreadsheet program ExcelTM,

Enter and organize information, enter a formula, and use parentheses to alter the order of operations.

The remaining eight instructional sessions Develop students’ understanding of generalizing problem situ

ations. Instructional Tasks

Cube sticker. Theater seats. Pizza sharing. Phone cost.

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Instructional Tasks

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Discussion

Justification of Generalizations Students tended to use two types of justification in this

study: empirical justification and generic examples.Empirical evidence may provide some reassurance that a particular

generalization is correct but that such evidence lacks the explanatory power (Hanna, 1990) that is desirable in a general argument.

Generic examples, raises some important questions about the awareness that the students had between this argument and that of an empirical justification.

The role of the teacher was instrumental in referring students back to the context of the situation.

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Discussion

Support and Inhibitors That Justifications Provide for Generalizations The most powerful form of justification linked the rule to a gener

al relation that existed in the context through the use of a generic example.

Students should initially engage in tasks with an obvious geometric connection.

As stated by Mason (1996), students demonstrate an awareness of generality when they are capable of “seeing a generality through a particular and seeing the particular in the general”.

These three strategies (rate-adjust, whole-object, and guess-and-check strategies) emphasize the particular rather than the general, encouraging students to examine only the empirical results provided by the computer spreadsheet.

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Discussion

Use of the Spreadsheet as an Instructional Tool Reduce the need to perform tedious calculations and focus

more on the process of generalization (Maxim & Verhay, 1991).

Implement more quickly a guess-and-check strategy without reflecting on the process employed or why a particular generalization was valid.

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Conclusion and Future directions

Instructional tasks Raise questions about the viability of various generalization

strategies The validity and power of various justifications are necessary for

these ideas to become the focal point of classroom discussion. Future research

Examine the types of tasks that encourage students to examine of the variety of justifications and generalization strategies that other students use.

Utilize tasks of similar structure to encourage students to reflect on the mathematical power of the various generalization strategies and justifications that students provide. varying the rate of change or initial number of seats in the theater se

ats problem. Students need to understand the type of justifications that are

mathematically acceptable and powerful.

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Expansion-Student strategies by task

Lannin, J.K., Barker, D.D. & Townsend, B.E. (2006). Recursive and explicit rules: How can we build student algebraic understanding. The Journal of Mathematical Behavior, 25(4), 299-317.

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Expansion-Student strategies by task

Cube Sticker Iconic Recursive Iconic Explicit Numeric Explicit Numeric whole-object

Theater Seats Iconic Recursive Numeric Explicit

Pizza sharing Iconic Explicit Numeric Explicit

Phone Cost Iconic Explicit Numeric Explicit