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Association: Name: North American Chapter of the International Group for the Psychology of Mathematics Education URL: http://www.pmena.org
Citation: URL: http://www.allacademic.com/meta/p117551_index.html Direct Link: HTML Code:

Hackenberg, Amy. and Tillema, Erik. "Quantitative Operations as a Basis for Algebraic Reasoning and Teaching Practices" Paper presented at the annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Delta Chelsea Hotel, Toronto, Ontario, Canada, Oct 21, 2004 <Not Available>. 2009-05-26 <http://www.allacademic.com/meta/p117551_index.html>

Hackenberg, A. J. and Tillema, E. S. , 2004-10-21 "Quantitative Operations as a Basis for Algebraic Reasoning and Teaching Practices" Paper presented at the annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Delta Chelsea Hotel, Toronto, Ontario, Canada Online <.PDF>. 2009-05-26 from http://www.allacademic.com/meta/p117551_index.html

Publication Type: Conference Paper/Unpublished Manuscript Review Method: Peer Reviewed Abstract: Purpose: In this presentation we describe initial analysis of data from a long-term, on-going constructivist teaching experiment whose purpose is to understand how sixth grade students can build algebraic reasoning out of their evolving quantitative operations and quantitative reasoning. We focus on two of the quantitative operations we identified as important to students’ generation and solution of basic linear equations and the ways in which we worked to help students build these operations. Theoretical Framework and Modes of Inquiry: Following von Glasersfeld (1995), we view learning as modifications and reorganizations in ways of operating that occur in response to perturbations (disturbances) brought about by the person’s current ways of operating. The essential medium for these perturbations is interaction among people, between people and objects, and of ideas within a person. Resolution of perturbations opens the way to modifications and reorganizations that we call acts of learning. This perspective grounds our conceptions of research because we view teaching (one form of interaction among people) as a method to investigate learning. In a conjecture-driven constructivist teaching experiment (Confrey & Lachance, 2000; Steffe & Thompson, 2000), researchers focus on understanding and explaining how students operate mathematically and how their ways of operating change in the context of teaching. Since teacher-researchers’ knowledge of mathematics may be insufficient to understand students’ ways of operating, teacher-researchers aim to learn mathematical ways of thinking from students. Simultaneously, teacher-researchers engage in conceptual analysis of how students might operate in the context of mathematical interactions. Out of this interplay between learning from students and conceptual analysis, teacher-researchers construct conjectures and formulate tasks for teaching episodes in order to test their conjectures. Teaching practices include presenting students with problem situations, assessing students’ responses as indications of students’ current operations, and determining new problem situations that might allow students to build other operations. We use Thompson’s (1994) definition of a quantity as a conceptual entity that consists of an object, a quality of the object, an appropriate unit for the quality, and a process by which to assign a numerical value to the unit. From this point of view, an unknown can be thought of as the measurement of a quantity that is as yet unspecified. We are interested in quantitative operations that involve actions performed mentally in building and analyzing relationships between known and unknown quantities. However, physical actions such as drawing often profitably accompany quantitative operations. Quantitative reasoning is a process of engaging in quantitative operations so as to model relationships among quantities and determine unknown quantities. We believe quantitative reasoning becomes more algebraic as unknowns are operated upon explicitly and reasoning is notated with algebraic symbols. The quantitative operations we discuss in this presentation are reversibility of basic fraction schemes and sharing with distribution, because we believe these operations are important in the generation and solution of basic linear equations. (Note that we do not claim that these two operations are exhaustive of such operations.) We indicate how these operations are important for reasoning through a problem such as: Candy Bar Problem. Draw a collection of seven candy bars each 1-inch long. What’s the total length of the collection? That collection is 3/5 of another collection of candy bars. Draw the other collection of bars. What’s the total length of this new collection? Solving this type of problem using quantitative reasoning was an immediate goal for some of our students and a long-term goal for others. A further goal was how students might come to solve the problem algebraically, including the generation and solution of equations like (3/5)x = 7. Methods (continued) and Data sources: Eight sixth grade students at a rural middle school in Georgia have participated in our teaching experiment since October 2003, and we expect to work with them through their seventh grade, academic year 2004-2005. We teach them biweekly in pairs for approximately two weeks, followed by a week off from teaching. Each teaching episode occurs during school hours, lasts approximately 30 minutes, and is videotaped with two cameras. One camera captures the interaction between the teacher-researcher and the pair of students, while the other camera focuses on the students’ computer or written work. In many episodes we use computer software, TIMA: Sticks and Javabars, that is intended to help students build quantitative operations with fractions. We perform ongoing analysis during the weeks away from teaching, and we will engage in the first round of detailed, retrospective analysis in summer 2004. Preliminary Analysis and Discussion: The quantitative operation reversibility of basic fraction schemes is involved in solving a problem like: Sub Sandwich Problem. This drawing shows 3/5 of a whole sub sandwich. Can you use it to make the whole sandwich? With four students, we are attempting to build up to them solving this problem by working on basic fraction operations like partitioning and iterating (i.e., make 1/5 of a whole sandwich; what fraction of the sandwich is three times that amount?) and by first using unit fractions (e.g., use 1/5 of the sandwich to make the whole.) We believe our other four students had already constructed the operations necessary for solving this problem at the start of the teaching experiment: They marked the given sandwich into three equal parts, pulled out one of the parts (a feature of the computer tools), and repeated the part five times. However, two of these four students had not necessarily built operations to solve this variation: Pencil Problem. The 12 pencils on the table are 4/3 of the number in a box. How many pencils are in a box? Originally these two students solved the problem by thinking of 12 as 1/3 of the number in a box. They also split the pencils into four groups of three, but didn’t identify the number of pencils in three groups as a solution. In a continuous context (e.g., this drawing shows a candy bar that’s 7/5 of another candy bar; make the other candy bar) they eventually operated similarly to their reasoning in the Sub Sandwich Problem. Returning to a discrete context (14 pencils are 7/5 of the number in a box), they formed the pencils into 7 groups of 2 and took 5 groups, or 10 pencils. Reversibility of basic fraction schemes is important in solving the Candy Bar Problem because of the need to make a whole (the unknown length of the other candy bar) given a fractional part of it. Students need to build up ways to reverse their operations in order to be able to posit the unknown length as made from a combination of partitioning and iterating the known length (cf. Steffe, 2002). The second quantitative operation, sharing with distribution, is involved in solving a problem like: Special Chocolate Bar Problem. A chocolate bar is 4 inches long, but each inch is a different flavor—for example, one is dark chocolate, one is milk chocolate, one is raspberry chocolate, etc. Make fair shares for three people, if everyone wants to get an equal share of all the flavors. How long is each person’s share? Four of our students solved this problem by partitioning each of the four inches into three equal pieces and then giving each person one third of each of the four inches. (Note that from a perspective of conceptual analysis, distribution is involved in this operation: (1/3)(1) + (1/3)(1) + (1/3)(1) + 1/3(1) = (1/3)(1 + 1 + 1 + 1) = (1/3)(4) = 4/3.) The students said that each person would get a candy bar 1 and 1/3 inch long. Our other four students have not yet worked explicitly on this problem, but they have engaged in numerous sharing tasks in order to build up partitioning operations. Examples include making and comparing individual shares of one candy bar (e.g., 1/9 and 1/10), using fraction language to describe shares, and sharing discrete objects (coins, eggs, pencils) among a group of people. This operation is important in the Candy Bar Problem because seven inches is THREE-fifths of the length of another collection. Students who have built up reversibility of basic fraction schemes aim to know ONE-fifth of the length of the collection, but as one of our students said initially, “you can’t divide 7 by 3!” (Then she admitted you could split up candy bars.) To solve the Candy Bar Problem, students need to build operations that allow them to divide seven inches into three parts, based on their ability to divide 1 inch into three parts (a basic partitioning operation.) Our research shows that building these operations takes concerted, long-term efforts in which students need specific kinds of support that they may not get in traditional middle school mathematics courses. Because our work indicates pathways that students might take to build and use these operations in order to engage in algebraic reasoning in situations like the Candy Bar Problem, our research opens new possibilities for teaching algebraic reasoning in middle schools.

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