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«PROCEEDINGS OF THE 15TH ANNUAL CONFERENCE ON RESEARCH IN UNDERGRADUATE MATHEMATICS EDUCATION EDITORS STACY BROWN SEAN LARSEN KAREN MARRONGELLE ...»

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A longstanding concern in mathematics education is the balance and relationship between knowing how to do something and knowing why something is the case. The research community has addressed this issue by developing a number of explanatory frames, including procedural versus conceptual understanding (Hiebert, 1986), process versus object conceptions (Breidenbach et al., 1992), concrete versus abstract modes of reasoning (Wilensky, 1991), instrumental versus relational understanding (Skemp, 1976), synthetic versus analytic thinking (Sierpinska, 2000), and operational versus structural reasoning (Sfard, 1991). Although there are differences in these constructs (both nominal and theoretical), there is a general consensus that both modes of reasoning are necessary to develop mathematical proficiency. Indeed, each of these types of understanding is represented in NCTM’s five strands of mathematical proficiency (Kilpatrick, Swafford, & Findell, 2001), which highlights the need for students to develop both forms of reasoning.

In the domain of linear algebra, researchers have expanded on these dual modes of reasoning. For example, Sierpinska (2000) describes different modes of student reasoning as synthetic-geometric, analytic-arithmetic, and analytic-structural. Related to these modes of reasoning, Hillel (2000) describes three modes of representations: geometric (using the language of R2 and R3, such as line segments and planes), algebraic (using language specific to Rn, such as matrices and rank), and abstract (using the language of the general formalized theory such as vector spaces and dimension). A number of student difficulties in linear algebra have also been documented (see Carlson, 1993; Hillel, 2000; Dorier, Robert, Robinet, & Rogalski, 2000;

Sierpinska, 2000; Stewart & Thomas, 2009), with many of these difficulties attributed to the disconnect between various representations and students’ modes of reasoning. For example, some researchers have been interested in how a geometric introduction to linear algebra may (or may not) help students make connections to algebraic and abstract modes of reasoning.

In this study we examine students’ conceptions of linear transformations by analyzing their solutions to a series of tasks involving geometric representations of linear transformations.

These tasks differed in their level of complexity. In increasing order of complexity, the first task was a matching problem, the second was a prediction problem, and the third was a creation problem. The research questions related to these tasks are: (1) What are students’ strategies on these three types of problems? (2) What patterns exist in students’ strategies across the three types of problems? In answering these questions we also sought to account for any patterns that we identified in student reasoning.

Methods Data for this analysis were collected from one extensive, semi-structured problem-solving interview (Bernard, 1988) with 13 undergraduate students. The interview questions were used to gather information related to participants’ understanding of linear transformations, with an emphasis on geometric representations on linear transformations. For this study, the last three questions of the interview were analyzed: a matching question consisting of five parts, a prediction task, and a creation task. These tasks will be discussed in detail below. The students were primarily engineering majors at a large southwestern university. Four of these students received a final grade of a ‘C’ in the linear algebra course, six students received a ‘B’, and three received an ‘A’, and pseudonyms were developed that reflect these grades. The interview was the second of a series of three interviews that was part of a semester-long classroom teaching experiment (Cobb, 2000). The interview was conducted after students had discussed geometric 414 15TH Annual Conference on Research in Undergraduate Mathematics Education Preliminary Report 3 and algebraic interpretations of linear transformations, but before they had begun a unit on eigentheory. Each interview was videotaped, transcribed, and thick descriptions were developed for students’ solutions to each of the tasks that included students’ written work (Geertz, 1994). The videos, transcriptions, and thick descriptions were analyzed through grounded analysis (Corbin & Strauss, 2008).

We analyzed student responses to three tasks from the interview: a matching task, a prediction task, and a creation task. The matching task consisted of five problems of increasing difficulty, beginning with a positive, diagonal matrix and ending with a matrix with no zero entries. The prediction task was created to be slightly more difficult than the matching tasks, and the creation task was thought to be the hardest. This task design was modeled after Artigue’s (1992) interview task design involving student understanding of differential equations.

Interview Tasks. The prompt for each matching task was follows: “In each of the following questions, you are given a matrix transformation and a corresponding set of images.

Identify any images that correspond to the image of the unit square (as shown below on the left) !"##$%&'()*+#,-./&)0*1!!"#!$%&'!()!*'$!)(++(,-#.!/0$1*-(#12!3(0!%4$!.-5$#!%!6%*4-7!

*4%#1)(46%*-(#!%#8!%!&(44$19(#8-#.!1$*!()!-6%.$1:!!"8$#*-)3!%#3!-6%.$1!*'%*!&(44$19(#8!*(!

–  –  –





%=! =!

! ! ! ! !

!"##$%&'()*+#,-./&)0*1!!"#!$%&'!()!*'$!)(++(,-#.!/0$1*-(#12!3(0!%4$!.-5$#!%!6%*4-7!

*4%#1)(46%*-(#!%#8!%!&(44$19(#8-#.!1$*!()!-6%.$1:!!"8$#*-)3!%#3!-6%.$1!*'%*!&(44$19(#8!*(!

*'$!-6%.$!()!*'$!0#-*!1/0%4$!;%1!1'(,#!$+(,!(#!*'$!+$)*=!0#8$4!*'$!.-5$#!*4%#1)(46%*-(#:!!

!

! !

!

&=! 8=!

!

!

!

?!

!

!

!

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$=!(#$!()!*'$1$:!

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–  –  –

!

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&=! 8=!

!

–  –  –

Table 1. Student Strategies Structural A student categorized as using an Se strategy reasoned by treating the two by two entries (Se) matrix as being composed of four pieces, the entries of the matrix.

Structural A student categorized as using an Sv strategy reasoned by treating the two by two vector (Sv) matrix as being composed of two pieces: the two column vectors of the matrix.

Structural A student categorized as using an So strategy attended to the visual and/or orientation geometric properties of the original shape/graph as opposed to properties of the (So) matrix. So often appeared when the students discussed the orientation of the box as well as how the colors of the sides should be oriented.

Operational A student categorized as using an Oi strategy reasoned by performing identify (Oi) multiplication with the identity matrix.

Operational A student categorized as using an Ou strategy reasoned by performing unit-vector multiplication dealing with the unit vectors. In the matching tasks, the unit vector (Ou) (1,0) was colored green, and the unit vector (0,1) was colored yellow, and thus students who performed operations on the ‘green’ and ‘yellow’ vectors were considered to be employing this strategy.

Operational A student categorized as using an Ov strategy reasoned by performing vector (Ov) multiplication dealing with a non-unit vector, such as (1,1).

–  –  –

Frequently students’ overall strategies for solving these tasks involved many substrategies; for example a student may solve a task by using an overall strategy of SeOuOv (first using the entries of the matrix, then performing computations on both unit vectors and non unit vectors). In Table 2, we report students’ overall strategies for each task. Sub-strategies were coded in order of use, and a green sub-strategy indicates that this strategy was used correctly, and a red sub-strategy indicated that it was used incorrectly. For example, on matching task d, Alex used an overall strategy of SeOvSo, indicating that he first used the entries of the matrix to inform his solution (correctly), then performed a computation using a non-unit vector correctly, and last reasoned about the orientation or colors of the matrix incorrectly. Entries that are highlighted in blue indicate that these strategies relied only on structural strategies, and those highlighted green indicate that a purely operational strategy was employed. The times under each entry represent the amount of time the student spent on the task.

This table was the main data source used for the analysis of these tasks. These tasks were grouped as follows: the matching tasks into three groups (the diagonal matrices (a and b), the non-diagonal matrices with at least one zero entry (c and d), and the matrix with no zero entries (e). The analysis of the data was conducted in two ways: first we looked for patterns within each of the individual tasks, and then we looked at the individual student strategies across the tasks.

–  –  –

Discussion One of the clearest patterns that we saw in the data was transition from predominantly structural to a combination of structural and operational reasoning. Sfard (1991) described concept development as a shift from an operational conception to a structural conception. Thus, we may explain this shift in student strategies as indicative of students’ stronger understanding of the geometric implications of linear transformations represented by diagonal matrices versus transformations with matrix representations that contain non-zero entries on the non-diagonals, prediction tasks or creation tasks. This is not surprising, especially considering the geometric results of diagonal matrices versus non-diagonal matrices, and the visual ease of understanding stretching compared to skewing.

15TH Annual Conference on Research in Undergraduate Mathematics Education 417 Preliminary Report 6 What is surprising is that C-students overall exhibited a much higher frequency of purely structural strategies. Do C-students have fuller concept development of the geometric implications of linear transformations than A and B-students? Or do C-students have a weaker operational understanding of matrices and thus instead rely on their structural conceptions? In these tasks we were not specifically interested in how strong students’ procedural competency was, and thus have no way to assess if this explains C-students’ preference for structural strategies. However, a weak understanding of matrix multiplication certainly would result in a low grade in any linear algebra course. These differences suggest that further investigation into the differences between A, B, and C-students’ operational and structural conceptions is needed.

Discussion Questions

1. How do you think these results could be best leveraged in a classroom environment?

2. Are the differences between the codes well understood and effectively differentiated?

3. A main result is the difference in student reasoning between grade categories. Other than differences in concept development, what else may explain these differences?

–  –  –

Artigue, M. (1992). Cognitive difficulties and teaching practices. In G. Harel & E. Dubinsky (Eds.), The concept of function: Aspects of epistemology and pedagogy (pp. 109-132).

Washington, DC: The Mathematical Association of America.

Bernard R. H. (1988). Research methods in cultural anthropology. London: Sage.

Breidenbach, D., Dubinsky, E., Hawkes, J., & Nichols, D. (1992). Development of the process conception of function. Educational Studies in Mathematics, 23, 247-285.

Carlson, D. (1993). Teaching linear algebra: Must the fog always roll in? The College Cobb, P. (2000). Conducting teaching experiments in collaboration with teachers. In A.E. Kelly & R.A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 307-334). Mahwah, NJ: Lawrence Erlbaum Associates.

Corbin, J., & Strauss, A. (2008). Basics of qualitative research: Techniques and procedures for doing grounded theory research. Los Angeles: Sage Publications.

Dordrecht: Kluwer. Harel, G. (2000). Three principles of learning and teaching mathematics:

Particular reference to linear algebra—old and new observations. In J.-L. Dorier (Ed.), On the Teaching of Linear Algebra (pp. 177-189). Dordrecht: Kluwer.

Dorier, J-L., & Sierpinska, A., (2001) Research into the teaching and learning of linear algebra.

In Holton, D. (Eds.), The Teaching and Learning of Mathematics at University Level: an ICMI study. Dortrecht, The Netherlands: Kluwer Academic Publishers, pp. 255-274 Dorier, J.L., Robert, A., Robinet, J., & Rogalski, M. (2000). The obstacles of formalism in linear algebra. In J.L. Dorier (Ed.), On the teaching of linear algebra (pp. 85-124).

Geertz, C. (1994). Thick description: Toward an interpretive theory of culture. In M. Martin & L.

C. McIntyre (Eds.), Readings in the philosophy of social science (pp. 213-232).

Cambridge, MA: The MIT Press.

Hiebert, J. (1986). Conceptual and procedural knowledge: The case of mathematics. Hillsdale, NJ: Lawrence Erlbaum.

Hillel, J. (2000). Modes of description and the problem of representation in linear algebra. In J.L. Dorier (Ed.), On the teaching of linear algebra (pp. 191-207). Dordrecht: Kluwer Academic Publisher.

Johnston-Wilder, Susan and Mason, John (2004). Fundamental constructs in mathematics education. Routledge Falmer in association with the Open University.

Kilpatrick, J., Swafford, J., & Findell, B. [Eds.] (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.

Mason, J. (2002). Researching your own practice: the discipline of noticing. UK: Routledge Mathematical Journal, 24(1), 29-40.

Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational studies in mathematics, 22(1), 1– 36.

Sfard, A. (1998). On two metaphors for learning and the dangers of choosing just one.

Educational Researcher, 27(2), pp. 4-13 Sierpinska, A. (2000). On some aspects of students’ thinking in linear algebra. In J.-L. Dorier (Ed.), On the teaching of linear algebra (pp. 209-246). Dordrecht: Kluwer Academic Publisher.



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