QUILT DESIGN AS INCUBATOR FOR GEOMETRIC IDEAS AND MATHEMATICAL HABITS OF MIND

 

 

Christopher E. Hartmann

Richard Lehrer

 

University of Wisconsin-Madison

University of Wisconsin-Madison

 

cehartma@students.wisc.edu

rlehrer@facstaff.wisc.edu

 

 

 

We investigated student design activity as an incubator of important mathematical ideas and as a forum for developing mathematical habits of mind. We report a case study of one second grade classroom where students used transformations in the plane to design quilts (Watt and Shanahan, 1994). The case study was conducted with an eye toward characterizing the relationship between collective and individual forms of activity, because such a relationship is at the heart of design. Our investigation was guided also by examination of how design activity and teacher assistance jointly crafted mathematical learning.

 

In earlier research, we found that second-grade students engaged in quilt design learned important habits of mind, like the limits of case-based generalization (Lehrer et al., 1998), and important ideas, like composition of transformations (Jacobson & Lehrer, 2000). Here we employ a case study approach (Stake, 1994), to track the experiences of two students. Data sources included videotape of the classroom activities, observation notes prepared by both participant and non-participant observers, teacher interviews, artifacts of student work including portfolios and journals, and pencil and paper assessments.

Theoretical Framework

Scientific and mathematical inquiry through design enables exploration of the space of potential solutions, engages students in problem solving through successive approximation, and highlights the functionality and the acceptability of solutions (Lehrer, Schauble, Carpenter, & Penner, 2000). Design, as a process of invention, requires representational capacities that enable feedback loops to promote reflection and revision (Goodman, 1976). The forms of assistance provided to accompany design activity play a major role in the development of mathematical reasoning and clarify that design is a form of mediated activity (Tharp & Gallimore, 1988; Wertsch, 1998). The development of student ideas and the acquisition of habits of mind are continuous during activity, as students are enculturated in a community of designers (Brown, Collins, & Duguid, 1989). The nature of the tasks and the nature of the teacher's interventions jointly shape the form and the sophistication of the mathematical practices that the students undertake during this apprenticeship (Henningsen & Stein, 1997; Stein, Grover, & Henningsen, 1996).

The Quilting Curriculum

            The quilting curriculum employed in this classroom employs transformations using different base designs (e.g., a core square design or a rectangular strip design) to create quilts (Watt & Shanahan, 1994). The teacher promotes the investigation of mathematical concepts by varying the pallet of shapes and colors available to the students, thereby changing the complexity of the design space. Consistent with design as a craft activity, the students continually revise their quilt designs during the course of instruction. The process of design and revision provides a meaningful venue for the class to engage in mathematical discourse about transformations in the plane. The linguistic and representational resources developed provide students with the means to raise conjectures and to consider arguments pertaining to the mathematical properties of figures in the plane, including symmetry and congruence.

Results

            Our interpretative analysis of classroom activity suggested several themes. First, much of the structure of everyday activity resulted from the dialectic between public display and discussion on the one hand and pursuit of particular individual (or small group) goals on the other. Public forums provided much of the space of reflection, where the teacher orchestrated student discussions about the mathematical consequences and implications of the explorations of particular students or groups of students.  For example, during an early conversation the teacher encouraged the students to consider the functional role of flips and turns in the act of quilt design.  After demonstrating a quilt design produced using only rotations of a core square, the teacher asked the students to consider using flips in producing quilt designs.

Teacher: If we proved we can do it [make a design] with only turning, why would we need to flip?  What do you think Susan?

Susan: If you turn it or flip it then you can get different 2x2's... Just by turning you can't get...you can't get all the different 2x2's//

[5 second cut]

Susan: //right, but you can only turn it from the outside corners or something like that.  But, if you flip it, you will get the same exact pattern only... the same exact core square, only it will give you a different place to put it.

Teacher: okay, and just refresh my memory, why would it be important to get that core square in different places (December 1, 1994).

 

During this exchange, Susan's remarks portray her burgeoning understanding of the functional role of flips and turns in creating designs and expanding the space of potential designs.  However, the italicized statements suggest that she regarded flips as a physical relocation rather than a reflection of a core square. The teacher did not attempt to change this conception at this point in time but simply concluded by recruiting students to the original goal of examining the functions of reflection in this design space.

Discussions like these were often accompanied by public display of various quilting products. Both the physical display of student designs and the patterns of discourse in the classroom highlighted the students’ learning histories.  The students appreciated this norm and followed it by recalling their own designs to provide warrants for the conjectures that they offered during class discussions:

Teacher: could there be a core square that looked the same after a sideways flip as it did after an up-down flip?

Students: yeah, yes [several students]

Eugene: yeah [with his hand raised]

Teacher: are you thinking of one Eugene?

Eugene: I remember the time that I made... this one... that had... that it could only... that every way you turned it it would only look the same.  And, I bet if you flipped it up and flipped it sideways I bet it would still look the same.

Teacher: that was a pinwheel design wasn't it, four triangles of yellow and four triangles of blue.

Eugene: yeah  (December 1, 1994).

 

The teacher extended this particular interaction by providing Eugene with Polydron™ manipulatives and asking him to produce copies of his original pinwheel design (figure 1) so that the class could test his conjecture about the invariance of his design under both flips and turns. 

Figure 1: Pinwheel                    Figure 2: Checkerboard

 

 

After the class tested both the pinwheel design and a checkerboard design as cases that fit Eugene's conjecture about invariance under vertical and horizontal flips (reflections), Susan observed that Eugene's conjecture could be extended to diagonal flips for the pinwheel design, but not for the checkerboard design (figure 2):

Teacher: [after Susan demonstrates a diagonal flip] That one [checkerboard] didn’t fit our rule, did it? I wonder why that is? I mean, this one [pinwheel] all of the ways, diagonal flip, sideways flip, up-down flip ended up looking exactly the same//

Susan: //I knew it was not going to be the same because [points to two shaded squares]//

Teacher: //these two were, you mean you knew the diagonal flip wasn’t going to be the same?//

Susan://…right, because these are going to be right here [points to shaded squares on core flipped diagonally], these are going to be here [points to two unshaded squares on core flipped up], and these are going to be right here [points to unshaded squares on core flipped sideways]. Diagonal flip, this is how we started out [demonstrates original position], but a diagonal flip is going to be exactly like that because the red [points out that red is opposite to red] (December 1, 1994).

 

Susan’s generalization to other cases suggests an increasing orientation toward understanding reflection as a transformation of an image about an axis.  We find further evidence of growth in her knowledge five days later when she describes how a core square that she has designed (figure 3) is invariant for diagonal flips, but changes under sideways or up-down flips:

Susan: after an up-down flip this corner [shaded triangle in top left] would be where this corner is [unshaded triangle in bottom left]…ta dah! But a diagonal flip, it [the core square] would just be exactly the same. Either a sideways flips or an up-down flip it would be…different (December 6, 1994).

Figure 3: Susan’s Core Square              Figure 4: Eugene’s Invariant Design

 

 

Collective activity structures, such as the class investigation of Eugene's pinwheel conjecture, were complemented by individual and small-group investigations, where the teacher modified the nature of activity in accordance with her understanding of how individuals or small groups were thinking. Often, the teacher constrained the design space and asked children to consider the mathematical implications of these constraints. For example, early in the cycle of designs, she asked students to consider the effects on design of limiting their choice of color to only two. Later, she asked students to explore the consequences of using three colors rather than two for quilt designs. Changes in the variables constraining the design space often resulted in the production and investigation of new conjectures.  The teacher’s explicit promotion of students’ attention to their own histories of design (as evidenced by the above interaction between the teacher and Eugene in the pinwheel segment), enabled student exploration to be governed not just by the moment but also in light of past activities and products.

Student histories provided a nice window to the interplay between the social and individual planes of thought (Vygotsky & Cole, 1978) because conversations about particular histories first elicited publicly were later prominent features of the conversations of individual children (recall the development of Susan's ideas about flips). As they talked about their designs with the participant observers, children spontaneously included “lessons learned” as a platform for current activity.  For example, using computer software designed to accompany the curriculum Eugene demonstrated for a researcher that a core square (figure 4) with four lines of symmetry and two colors could only be used to produce a single quilt design, noting that “It’s because… it matters on the outside of it…the blue keeps on going in the blue places. So it never gets in a place…you could have …a yellow diamond [in the middle] and then blue, blue, blue, and then red [triangles bordering the diamond]. That would be good, but, you can’t …just have like all blues on the outside of it (December 13, 1994).”

Eugene’s investigation of the relationship between symmetries and the space of designs led him to conclude that base designs with four lines of symmetry close the design space and resulted in "boring" quilt designs. This privately arrived at conjecture was later discussed in a group design activity as a potential “design standard” (Erickson & Lehrer, 1998) governing “interesting quilts”:

Researcher: Eugene, [can you explain] why you like this design?

Eugene: Because you can turn it [an asymmetrical 2x2 design] and it looks different//

Carol: //The part that Eugene hates about mine [a different 2x2 design that has four lines of symmetry] is that no matter how you turn it or flip it, it will always look the same (December 15, 1994).

 

Although some students conjectured that multiple forms of reflection symmetry might limit the space of potential designs, other students, like Carol, argued that symmetrical designs have attractive qualities. Carol investigated how different transformations or compositions of transformation might result in different degrees of (reflection) symmetry, given base designs of particular patterns. Consideration of competing design standards like these were grounded in mathematical exploration and justification.

            In summary, design in this second grade classroom had a dual aspect. The collective design space was populated by reflection about the mathematical consequences of design activity and by public products that encapsulated the history of design in this particular community. We found that articulation of design standards fostered reflection and prompted further exploration. The semi-private design spaces of individual and small-group activity were forums for generation and test of conjecture, and for pursuit of design guided by personal interests and proclivities. Learning was mediated in this design context by the affordances of the tasks posed (or those generated by children), the material means (paper, glue, and electronic design tools) used to generate the various designs, the inscriptions and notations children invented or appropriated, and by cycles of conjecture and refutation. Teacher orchestration of each of these resources made the dialectic between collective and private design spaces mathematically fruitful and enabled children to participate meaningfully in mathematical activity.

References

 

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Jacobson, C., & Lehrer, R. (2000). Teacher Appropriation and Student Learning of Geometry Through Design. JRME, 31(1), 71-88.

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