The Mathematics Educator 2007, Vol. 17, No. 1, 15–23 A Quilting Lesson for Early Childhood Preservice and Regular Classroom Teachers: What Constitutes Mathematical Activity? Shelly Sheats Harkness Lisa Portwood In this narrative of teacher educator action research, the idea for and the context of the lesson emerged as a result of conversations between Shelly, a mathematics teacher educator, and Lisa, a quilter, about real-life mathematical problems related to Lisa’s work as she created the templates for a reproduction quilt. The lesson was used with early childhood preservice teachers in a mathematics methods course and with K-2 teachers who participated in a professional development workshop that focused on geometry and measurement content. The goal of the lesson was threefold: (a) to help the participants consider a nonstandard real-world contextual problem as mathematical activity, (b) to create an opportunity for participants to mathematize (Freudenthal, 1968), and (c) to unpack mathematical big ideas related to measurement and similarity. Participants’ strategies were analyzed, prompting conversations about these big ideas, as well as an unanticipated one. What would happen if the activities we, as physical materials and oral and written languages mathematics teacher educators, use to model best that are used to think about mathematics (Heibert et al., practices and standards-based teaching in mathematics 1997). In the process of doing mathematics, one thinks methods courses and professional development or reasons in logical, creative, and practical ways. workshops honored mathematical activity that is According to Sternberg (1999), American schools have nonstandard in the sense that it is sometimes not even a closed system that consistently rewards students who considered mathematics? Mary Harris (1997) describes are skilled in memory and analytical reasoning, how she uses “nonstandard problems that are easily whether in mathematics or other domains. This system, solved by any woman brought up to make her family’s however, fails to reward, in the sense of grades, clothes” (p. 215) in mathematics courses for both students’ creativity, practical skills, and thinking. In preservice and classroom teachers. To make a shirt, problem-solving situations, students should be “all you need (apart from the technology and tools) is encouraged to use both physical and mental activity to an understanding of right angles, parallel lines, the idea do mathematics in order to (a) identify the nature of the of area, some symmetry, some optimization and the problem; (b) formulate a strategy; (c) mentally ability to work from two-dimensional plans to three- represent the problem; (d) allocate resources such as dimensional forms” (p. 215). Although none of these time, energy, outside help, and tools; and (e) monitor considerations are trivial, making a shirt is not and evaluate the solution (Sternberg, 1999). typically considered mathematical activity. Harris Researchers who studied the consumer and vendor raises questions about why this is true. Is it because the sides of mathematical reasoning found skills revealed seamstress is a woman or because only school by a practical test were not revealed on an abstract- mathematics is valued by our society? And, more analytical, or school-type, test (Lave, 1998; Nunes, broadly, what constitutes mathematical activity? 1994). We contend that mathematical activity is both Too often, mathematics is viewed as the mastery of physical and mental. It requires the use of tools, such bits and pieces of knowledge rather than as sense making or as sensible answers to sensible questions (Schoenfeld, 1994). Sensible questions arise from Shelly Sheats Harkness, Assistant Professor at the University of Cincinnati in the Secondary Education Program, is a mathematics many nonstandard contexts. If we design problems that educator. Her research interests include Ethnomathematics, are based on those questions, model best practices, and mathematics and social justice issues, and the impact of listening elicit mathematical big ideas, our students might begin and believing in mathematics classrooms. to see mathematics as a human endeavor. They may use logical, creative, and practical thinking to solve Lisa Portwood is a self-taught quiltmaker and quilt historian. She has been quilting for the past 21 years and is an active member of those problems. the American Quilt Study Group. She is also a financial secretary at Miami University in the Teacher Education Department. Shelly Sheats Harkness & Lisa Portwood 15 As sometimes happens, an idea for a nonstandard reproducing, one her neighbor owned, had a side real-world lesson took root in an unexpected place. length of about 88 inches; however, the display quilt During lunchroom conversations, Lisa, a member of could not be more than 200 inches in perimeter. She the American Quilt Study Group (AQSG), described wanted to trace the templates for the design from her her work on a reproduction quilt with Shelly, a neighbor’s quilt (see Figure 1) and then use a copy mathematics teacher educator. AQSG members machine to reduce these traced sketches (see Figure 2). participate in efforts to preserve quilt heritage through She needed to decide which reduction factor to use and various publications, an extensive research library, and how much of each color fabric–white and blue–to a yearly seminar. At this seminar, the AQSG invites purchase. She wanted to buy the least possible amount members to make smaller versions, or reproductions, of fabric. These reproduction quilt problems became of antique quilts from a specified time period so that the context for the lesson that Shelly created and used many of these can be displayed in one area. with both early childhood education preservice Lisa and Shelly discussed the mathematical teachers enrolled in a mathematics methods course and problems Lisa encountered as she designed the with K-2 classroom teachers in a professional reproduction quilt. The square quilt she was development workshop. Figure 1: Reproduction (left) and original quilt (right) Figure 2: Templates 16 A Quilting Lesson (Hopkinson, 1993), and mathematics–the properties of Literature Review of Lessons Related to squares, rectangles, and right triangles–in her lesson Mathematics and Quilting for upper elementary school children. Because of a desire to know more about the In their book Mathematical Quilts: No Sewing connections between mathematics and quilting, we Required!, Venters and Ellison (1999) included 51 began by searching for literature related to lessons for activities for giving “pre- and post-geometry students teachers. We found some rich resources that included a practice in spatial reasoning” (p. x). These activities wide range of mathematical topics embedded in these are situated within four chapters: The Golden Ratio lessons. Quilts, The Spiral Quilts, The Right Triangle Quilts, Transformational geometry was the foundation for and The Tiling Quilts. The authors noted that the quilts several lessons. Whitman (1991) provided activities for that inspired their book were created when they were high school students related to Hawaiian quilting teaching mathematics and taking quilting classes in the patterns with a focus on line symmetry and rotational mid-1980s: symmetry. Ernie (1995) showed examples of how Because we had no patterns for our [mathematical] middle school students used modular arithmetic and quilts, we had to draft the design and solve the transformational geometry to create quilt designs. Most many problems that arise in this process involving recently, Anthony and Hackenberg (2005) described an measurement, color, and the sewing skills needed activity for high school students that made “Southern” for construction ... Taking on a project and working quilts by integrating an understanding of planar it through to completion provide invaluable symmetries with wallpaper patterns. experiences in problem solving. (p. ix) The patterns and sequences found in quilt designs These were the same challenges that Lisa faced when provided a basis for mathematical topics in other she created her reproduction quilt. lessons. Rubenstein (2001) wrote about several What was missing from this extensive list of methods that high school students used to solve a resources was any reference to using mathematics and mathematical problem related to quilting: finite quilting in lessons for preservice teachers or differences, the formula for the sum of consecutive professional development workshops for teachers. We natural numbers, and a statistical-modeling approach felt that Lisa’s real-world task would provide the using a graphing calculator. Westegaard (1998) opportunity for the preservice and classroom teachers described several quilting activities for students in to mathematize,1 recognize big mathematical ideas, and grades 7-12 that reinforced coordinate geometry skills consider what constitutes mathematical activity. We and concepts such as identifying coordinates, thought the big ideas that would emerge from this task determining slope as positive or negative, finding included: intercepts, and writing equations for horizontal and vertical lines. Mann and Hartweg (2004) showcased • Measurement is a way to estimate and compare third graders’ responses to an activity in which they attributes. covered two different quilt templates with pattern • A scale factor can be used to describe how two blocks and then determined which template had the figures are similar. greatest area. Within this article, we briefly describe the Reynolds, Cassell, and Lillard (2006) shared preservice and classroom teachers who we worked activities based on a book by Betsy Franco, Grandpa’s with, the quilting lesson that we created–based upon Quilt, which they incorporated into lessons for their the actual mathematical questions that Lisa faced as second-graders. In these activities, students made she created her reproduction quilt–and the mathematics connections to patterns, measurement, geometry, and a that the preservice and classroom teachers used as they “lead-in” to multiplication. In a lesson for third mathematized. We then summarize our follow-up graders, Smith (1995) described how she linked the conversations about participants’ general reactions to mathematics of quilting–problem solving, finding the reading, An Example of Traditional Women’s Work patterns, and making conjectures–with social studies as a Mathematics Resource (Harris, 1997), and our through the use of a children’s book, Jumping the question: What constitutes mathematical activity? Broom. Also with a connection to integration with Finally, we describe the emergence an unanticipated social studies, Neumann (2005) focused on the mathematical big idea, based on the ways that the significance of freedom quilts, the Underground preservice and classroom teachers approached the Railroad, the book Sweet Clara and the Freedom Quilt problem. Shelly Sheats Harkness & Lisa Portwood 17 We piloted this lesson during the first semester of stipend to spend on math books, manipulatives, or the 2004-2005 school year with a class of preservice other items. All but one of the 20 teachers were female. teachers at a large public university in the Midwestern The majority of the classroom teachers described their United States. We then obtained IRB approval to own mathematical experiences as less than pleasant collect data in the form of work on chart paper that and their fear of mathematics was evident from the groups in a second class did the following semester. beginning. Similar to the preservice teachers, they were We used the lesson again during the summer of 2005 very bold about their dislike of mathematics. They with a group of 20 classroom teachers, grades K-2, in a openly discussed their views of mathematics as a set professional development workshop. rules and procedures to be memorized. Although the focus of the workshop was on improving the teachers’ The K-2 Preservice Teachers and Classroom content knowledge in geometry and measurement, we Teachers felt that we also needed to address their beliefs about Teaching Math: Early Childhood (TE300) was a 2- teaching and learning mathematics within that context. credit methods course that preservice teachers enrolled As a springboard for the semester and the in prior to student teaching. The course included a two- workshop, we discussed the meaning of mathematics week field experience in which the preservice teachers and mathematizing. To initiate discussion, we posed wrote one standards-based, “best practices” lesson the following question: “Is mathematics a noun or a plan, and then taught the lesson. During each class verb?” Some thought it was a noun because session throughout the semester, we focused on a mathematics is a discipline or subject you study in chapter from Young Mathematicians at Work (Fosnot school. Others argued that it was a verb because you & Dolk, 2001) and one of the content or process “do” it. About half argued that it could be considered standards from Principles and Standards for School both. Mathematics (PSSM) (National Council of Teachers of When both the preservice and classroom teachers Mathematics [NCTM], 2000). The reading from PSSM read Young Mathematicians at Work (Fosnot & Dolk, for the week of the quilting activity focused on 2001), we again negotiated the meaning of measurement. mathematics and mathematizing (Freudenthal, 1968), All but one of the sixty TE300 preservice teachers reaching a consensus consistent with Fosnot and were female. In mathematical autobiographies written Dolk’s interpretation: “When mathematics is during the first week of the course, about two-thirds of understood as mathematizing one’s world— these preservice teachers said they either disliked or interpreting, organizing, inquiring about, and had mixed feelings about their previous school constructing meaning with a mathematical lens, it mathematical experiences, K-12 and post-secondary. becomes creative and alive” (p. 12-13). These are all Some who reported dislike for mathematics described processes that “beg a verb form” (p. 13) because feeling physically sick before math class, helplessness, mathematizing centers around an investigation of a and lack of self-confidence. Those with mixed feelings contextual problem. wrote about grades of A’s and B’s as “good times” and The Lesson grades of D’s and F’s as “bad times.” Many described board races and timed math tests over basic facts as According to Fosnot and Dolk (2001), situations dreaded experiences. Generally speaking, they hoped that are likely to be mathematized by learners have at to help their own students experience the success that least three components: they did not enjoy in math classes. These mathematical • The potential to model the situation must be built experiences posed a special challenge for us because in. we felt that their beliefs about teaching and learning mathematics had to be addressed. Due to time • The situation needs to allow learners to realize what they are doing. The Dutch used the term constraints, we attempted to address them at the same zich realizern, meaning to picture or imagine time that we talked about best practices and standards- something concretely (van den Heuvel-Panhizen, based methods. 1996). The K-2 classroom teachers participated in a professional development workshop offered the • The situation prompts learners to ask questions, notice patterns, and wonder why or what if. summer after we implemented the lesson with the preservice teachers. Most opted for free tuition to earn Guided by these components, we planned the lesson graduate credit; each of them also received a $300 within Lisa’s quilting context. 18 A Quilting Lesson Throughout the semester and the workshop, Shelly original quilt to create the pattern pieces for the read part of a picture book, Sweet Clara and the reproduction.) Freedom Quilt by Deborah Hopkinson (1993), in order 2. How much white fabric did she need to buy for to provide a context for the quilting problem. In this the front and back of the quilt? (Please note: book, Sweet Clara is a slave on a large plantation. Her Fabric from bolts measures 44-45 inches wide.) Aunt Rachel teaches her how to sew so that Clara can 3. How much blue fabric did she need to buy for work in the Big House. There, she overhears other the appliqués, borders, and binding around the slaves’ talk of swamps, the Ohio River, the edges? (Use the templates from the original quilt Underground Railroad, and Canada. Listening intently to determine your answer.) to these conversations, Clara visualizes the path to Before they began to work, we also showed them an freedom and creates a quilt that is a secret map from actual bolt of fabric and explained how fabric is sold the plantation to Canada. from the bolt because we were not sure they would To launch the problem, Lisa told the groups about know what this meant (and many did not!). We gave her work in the AQSG and explained why she wanted each group original-sized copies of the templates used to produce a replica of a two-color quilt from the for the appliqué blue pieces on the original quilt (see period 1800–1940. As it happened, her neighbor found figure 2) to use to answer the third question. a quilt in her basement and showed it to Lisa, who We asked them to keep a record on chart paper of both could hardly believe her luck! Not only did Lisa like the mathematics and mathematical thinking or the design, but she liked the two colors, blue and white, processes they used to answer the questions so that as well. She decided to use her neighbor’s quilt as the they could share the results in a whole group original for her reproduction. discussion. Calculators, rulers, meter-yard sticks, tape We then shared the parameters for the reproduction measures, string, scissors, and tape were also available. quilt, as given by the AQSG: We walked around, listened, and watched the • Display: Each participant is limited to one quilt. groups work. Some had questions we had not expected: Each quilt must be accompanied by a color Is there white underneath the blue? Does the back have image of the original and the story of why it was to be all one piece of fabric? Can we round our chosen. numbers? Should we allow for extra fabric? The • Size: The maximum perimeter of the replica is students’ questions made us realize that, even though 200 inches. This may require reducing the size of the three questions we posed might seem trivial for the original quilt. Size is limited to facilitate the some quilters and mathematicians, they served as a display of many quilts. springboard for the rich mathematical discussion that • Color: “Two-color” indicates a quilt with an followed the small group work. overall strong impression of only two colors. A The Mathematics single color can include prints that contain other colors but read as a single color. We assessed the groups’ strategies while they worked to answer the three questions by listening to We also explained how the square-shaped original quilt their discussions and analyzing the chart paper record had side lengths of 88 inches and showed them a photo of their strategies. We noticed that most groups, both in of both the original and reproduction quilts (see Figure the class for preservice teachers and the workshop for 1). classroom teachers, took the directions quite literally In order to help both the preservice and classroom (i.e. that the reproduction perimeter must be exactly teachers immerse themselves in mathematizing and 200 inches) and used similar strategies. consider what constitutes mathematical activity related After the groups posted their chart paper on the to measurement and similarity, we posed the following walls, we began a whole-group discussion by posing three questions that were actual questions that Lisa the question: Are the two quilts, the reproduction and faced as she prepared to create the study quilt: the original, mathematically similar? All agreed that 1. By what percent did she need to reduce the they were but when asked why, their responses focused original quilt to fit the 200 inches measured on the notion that they just looked similar. They knew around all four sides (the perimeter)? The the quilts were not congruent because they were original was 88 inches on one side. (Lisa wanted different sizes. We told them that we would return to to use the copy machine and a scale factor to this question later so that we could negotiate a reduce the pattern pieces she traced from the mathematical definition of similarity. Shelly Sheats Harkness & Lisa Portwood 19 Figure 3: A solution to Question 1 Question 1 perimeters or side lengths. In fact, 32.3% was approximately the square of the ratio for the perimeter All but two groups thought of the perimeter comparison, (50/88)2. parameter as exact and created scale factors to reduce This led us to rethink our questions about the copy the quilt so it would have a perimeter as near to 200 machine. Does the word “reduction” lead to inches as possible. These groups said that the pattern mathematical misconceptions? How does the reduction should be reduced on the copy machine by either 57% scale factor change the perimeter and the area? For or 43%; this led to an interesting conversation about instance, is the original image reduced by the selected how these were related and which one made more percentage or does the machine create an image that is sense. Would we enter 57% or 43% into the copy that percentage of the original? Experimentation with machine? Which number makes the most sense based the copy machine reduction function helped us answer on what we know about copying machines and how this question (we leave it to the reader to explore). they reduce images? Interestingly, even though most groups considered The two groups that did not use the method the perimeter parameter as strict, Lisa knew that the adopted by the majority used the same scale factor as reproduction quilt could be no larger than 200 inches Lisa, approximately 67.7%. We asked these groups to so she decided to use a 50% reduction—this made her share their thinking (see Figure 3) because it seemed study quilt 44 inches per side with a perimeter of 176 like their mathematical calculations and reasoning inches, which was “close enough.” Like Lisa, two were also valid. How could there be different answers? groups decided that 50% was a reasonable and Instead of focusing on perimeter, these groups created “friendly” number to use, making other calculations for a ratio of the total area of the reproduction quilt to the the quilt less cumbersome. This led to a conversation total area of the original quilt, 2500/7744 (assuming about when close enough is sufficient for measurement the quilt would measure 50 inches by 50 inches); the and other uses of mathematics. We felt this was ratio was 32.3%. So, the area needed to be reduced by especially important because many of the preservice 67.7%. At first, we wondered why the percents differed and classroom teachers experienced mathematics as when groups compared areas instead of perimeters for problems with one exact answer. The idea that the same geometric figure. This led to the opportunity measurement can be precise but not exact was to discuss an unexpected big idea, something we had to something they needed to think about. think deeply about ourselves before we realized why the results for the groups were different: When the Questions 2 and 3 perimeter of a rectangle is reduced by a scale factor, For the second and third questions, the chart paper the area is not reduced by the same scale factor. In fact, revealed that the groups had a wide range of answers the ratio of the areas is the square of the ratio of the and some mathematical misconceptions. Most had perimeters. In addition, this is true for any size of answers close to 3 yards of white fabric and 1.5 yards rectangular quilt or similar figures. of blue fabric. Some groups drew sketches of the fabric We had to encourage the preservice and classroom (see Figure 4). Groups that drew sketches or teachers to think about why the two percents were representations had the best estimates for conserving different. When pressed, they realized that, because fabric. Even though one might think that determining area is a square measure, taking the ratio of two areas resulted in a different value than taking the ratio of two 20 A Quilting Lesson Figure 4: A solution for Question 2 the amount of white material would result in trivial you must divide by 144 to convert square inches to mathematical conversations, we noticed that most square feet and by 9 to convert square feet to square groups tried various ways to overcome the fact that the yards. This mistake is one that could have been width of the material (44-45”) posed a real contextual predicted with out-of-context problems, but Lisa had dilemma, as it was shorter than the width of the quilt. shown her reproduction quilt before they began In other words, they had to consider both area and working in their groups. What was most disturbing length in their attempt to minimize the amount of about this answer was the fact that 47 square yards fabric needed. made no sense given the size of one square yard. For the second question, one group decided that Similarity Lisa needed to buy 47 yards of white fabric. This group felt the sides of the quilt should measure 41 inches Returning to the definition of similarity, we again because the fabric was 44-45 inches wide (see Figure posed the question: Why are the two quilts 5). This was similar to Lisa’s thinking and within the mathematically similar? The preservice and classroom 200-inch parameter for total perimeter. teachers negotiated a definition that made sense to However, we were shocked by their answer of 47 them. They talked about “not the same size but the yards! They did not take into account the notion that same shape” in terms of scale factors and created a Figure 5: A solution for Question 2 Shelly Sheats Harkness & Lisa Portwood 21 working definition: The scale factors or ratios of the of Harris (1997), however, they began to talk about corresponding sides of the two quilts are equal (or doing mathematics within traditional women’s work proportional) and the ratio of the areas is the square of such as measuring and hanging wallpaper, cooking, the ratio of the side lengths. creating flower garden blueprints, playing musical Generally speaking, we felt as though this task instruments, and determining the number of gallons of provided opportunities to talk about many paint needed to paint a room. We did not discuss the mathematical notions related to measurement and nature of mathematical activity as both physical and similarity including exactness versus precision, mental activity but, after thinking more deeply about it estimation, ratio, proportion, percent, scale factor, ourselves, we now realize that this was a missed perimeter, and area. We briefly discussed the kinds of opportunity. It seems that our conversation should also symmetry—reflection (flip), rotation (turn), and focus on the logical, creative, and practical ways in translation (slide)—in the quilt but this was not a which we think and reason while doing mathematics. focus. The preservice and classroom teachers also Concluding Remarks noted that they used all five NCTM process standards (problem solving, communication, reasoning and Through our collaborative effort to create a lesson proof, connections, and representation) as they worked with real-world applications and a reading related to in their groups and during our class discussion. what constitutes mathematical activity, the preservice and classroom teachers saw mathematics as something What Constitutes Mathematical Activity? you do outside of school. They were mathematizing, As a way to help the preservice and classroom organizing, and interpreting the world through a teachers consider what constitutes mathematical mathematical lens as they made conjectures about the activity, we gave them copies of An Example of same questions that Lisa faced when she created her Traditional Women’s Work as a Mathematics Resource reproduction study quilt. (Harris, 1997) to read before our next class or Analysis of the student strategies revealed professional development session. According to Harris, opportunities to discuss big ideas related to in mathematical activity, women are disadvantaged in measurement and similarity and what constitutes two ways: (a) until very recently, female mathematical activity. It also prompted Lisa to take mathematicians were barely mentioned; and (b), in a another picture of the two quilts, to illustrate the big world where women's intellectual work is not taken idea that emerged from the groups’ sense-making: very seriously, the potential for receiving credit for reducing the perimeter by 50% created a reproduction thought in their practical work is severely limited. In quilt with one-fourth the area of the original quilt (see her book, Harris showed her students a Turkish flat Figure 6). woven rug, called a kilim, and her students explored Within this lesson, we modeled the kind of the mathematics involved in its construction. She also teaching we hope these preservice and classroom displayed a right cylindrical pipe created by an teachers will think about and use in their classrooms: engineer and a sock knitted by a grandmother. She then helping students mathematize, make connections to big posed the following questions: Why is it that the ideas and real-world mathematics, and question what geometry in the kilim is not usually considered serious constitutes mathematical activity. As Harris (1997) mathematics? Is it because the weaver has had no noted, the role of mathematics teachers should not be schooling, is illiterate, and is a girl? How do we know to teach some theory and then look for applications, that the weaver is not thinking mathematically? Why is but to analyze and elucidate the mathematics that designing the pipe considered mathematical activity grows out of the students' experience and activity. but knitting the heel of the sock is not? Using nonstandard contextual problems creates This reading helped create an opportunity for the opportunities to honor school mathematics and K-2 preservice and classroom teachers, groups that are mathematical activity that exists within the real world mostly female, to talk about their own beliefs regarding of everyday activity. By doing so, we also honor and what constitutes mathematical activity in the context of respect our students’ logical, creative, and practical women’s work. Many of them had considered school thinking. We give voice to their mathematics. mathematics as the only kind of mathematics. 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