Listening Responsively

296 December2011/January2012•teaching children mathematics www.nctm.org

Listening

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www.nctm.org teaching children mathematics • December 2011/January 2012 297

Standards documents, such as the Common Core State Standards for Mathematics (CCSSI 2010) and Principles and Standards for School Mathematics(NCTM 2000), expect teachers to foster mathemat-ics learning by engaging students in meaningful

mathematical discourse to expose students to different ways of thinking about and solving problems (White 2003) and positively infl uence their problem-solving abilities, reasoning skills, and thinking processes (Davis and Maher 1997; Doerr and Tripp 1999; Hiebert et al. 1997; Pirie and Kieren 1994; Martin et al. 2005; NCTM 2000). However, merely getting stu-dents to talk in math class is not enough to assure that learn-ing will occur (Lampert and Cobb 2003; McCrone 2005); once teachers get students to express their ideas, they must decide what to do with those ideas.

Teachers’ follow-up moves, such as evaluating, rebroadcast-ing, acknowledging, or making a related statement or question

(Pierson 2008), will shape “the nature and fl ow of classroom discussions and the cognitive opportunities afforded to stu-dents” (Boaler and Brodie 2004, p. 780). Thus, engaging stu-dents in meaningful mathematical discourse can be rather daunting because it requires teachers to make careful peda-gogical choices and spontaneous decisions in the midst of the instructional process (Atkins 1999; Martin et al. 2005; Pierson 2008; Sherin 2002; Wells 1996); and when students’ ideas are considered, it can be diffi cult to predict and manage the direc-tion of instruction (Heaton 2000; Fennema et al. 1996; Sherin 2002; Silver and Smith 1996). Nevertheless, a teacher’s ability to listen effectively and respond appropriately to students’ think-ing has been shown to play a critical role in effective math-ematics instruction (Carpenter and Fennema 1992; Davis 1997; Empson and Jacobs 2008; Pierson 2008; White 2003).

Empson and Jacobs (2008) assert that teachers can learn to be responsive to their students’ thinking and, by doing so, can

By Kadian M. Callahan

ListeningResponsively

Opportunities to explore prospective elementary teachers’ ideas about the perimeter and area

of polygons require more than simply engaging the teachers in mathematical discourse.

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garner several benefits: (1) improve students’ understanding of mathematics, (2) obtain a means of formative assessment, (3) increase their own mathematical knowledge, and (4) sup-port their own engagement in generative learn-ing. Empson and Jacobs identify three phases of listening that a teacher goes through in the effort to become more responsive to student thinking: (1) directive, (2) observational, and (3) respon-sive. In directive listening (referred to by Davis [1997] as evaluative listening), a teacher listens to a student’s thinking to see if it matches a pre-determined response. When a teacher engages in observational listening, she is genuinely inter-ested in students’ thinking but is unsure of what to do with their ideas, which may result in limited attempts to expound on students’ ideas. In the final phase, responsive listening, a teacher listens carefully to students’ thinking and actively works to support and extend their thinking.

Many teachers are unfamiliar with, and unprepared to provide, sufficient opportuni-

ties for students to learn math in ways that are consistent with standards-

based teaching (NMAP 2008). This is especially true when it comes to engaging students in meaningful discourse that is responsive to their

ideas (Heaton 2000; Sherin 2002; Silver and Smith 1996), and may be

due, in part, to prospective teachers having limited opportunities to learn math in similar ways during their teacher preparation programs (CBMS 2001; MAA 1998; NMAP 2008; NRC 2001). If teacher educators expect prospective teach-

ers to engage their students in mean-ingful mathematical discourse that is responsive to students’ ideas, they

too must find ways to press prospec-tive teachers to articulate their ideas

and use them to support the learning of mathematics.

Getting them to talk about mathematicsTeacher educators (TEs) are often

challenged to foster mathematical dis-course among prospective elementary

school teachers who have had limited opportunities to explore mathematical

ideas and articulate their thinking to others. Fullerton (1995) found that the preservice teach-

ers (PSTs) in her study had never learned how to communicate mathematical ideas. She specu-lated that this was due in large part to the fact that none of the women had participated in any form of mathematical discourse. Jansen’s (2008) research considered PSTs’ motivations for participating in mathematical discourse and found that many consider it to be intimidating. However, her results also suggest that partici-pating in classroom discussions is important to PSTs because of their desire to learn math-ematics, demonstrate competence, help others, and become teachers. Callahan, Hillen, and Watanabe (2009) found that after taking a math-ematics course where discourse was expected, PSTs felt more confident in their ability to do math and more comfortable discussing math with others.

Mathematical discourse most often occurs in a face-to-face setting; however, online dis-cussion boards can be used to engage PSTs in discussions about math. Both forums offer opportunities for teacher educators to listen to PSTs’ thinking and respond to their mathemati-cal ideas. Limited research suggests that online learning communities promote active participa-tion as teachers construct, share, and reorganize their knowledge (e.g., Chang, Chen, and Li 2006; Goos and Benninson 2005; Lin, Lin, and Huang 2007; Linares and Valls 2010; Yeh 2010). Given some PSTs’ hesitation to engage in classroom discussions, online discussion boards may encourage them to express their thinking in a more private, less intimidating forum.

This article describes mathematical dis-course that occurred over a two-week period during face-to-face class meetings and on an asynchronous, online discussion board (using Blackboard) and a teacher educator’s efforts to listen and be responsive to PSTs’ ideas about mathematics. The mathematical goal for the unit was to help the PSTs realize that the perime-ter and area of different polygons are not always directly related. The course was structured such that the twenty-six female PSTs could earn participation points for engaging in in-class dis-cussions or contributing to online discussions. Thus, the online discussion boards furnished an alternative for PSTs who may have been uncom-fortable articulating their ideas during in-class discussions to still express the sense they were making of the mathematics.


Polygons with fixed perimeterAfter spending some time together finding perimeters of different shapes drawn on grid paper, the TE asked the PSTs, either individu-ally or in small groups, to use one-inch squares to create as many polygons as they could that have a perimeter of twelve inches. The TE called on PSTs to describe their polygons as she re-created them on the document camera. Several PSTs—including Sara and Mariel (all names are pseudonyms)—quickly identified the 1 × 5, 2 × 4, and 3 × 3 rectangles; other polygons came more slowly. Tiffany and Kendall described two dif-ferent L-shapes. Heather identified an “almost” 2 × 4 rectangle: “It’s almost like the two-by-four rectangle that we already have, but one square is missing from the top right.” Other PSTs went on to describe a few other polygons each with a perimeter of twelve inches (see fig. 1).

Some PSTs commented that they were surprised that so many shapes have the same perimeter but a different number of one-inch squares. Tiffany commented that it was surpris-ing that you could take away one square from a shape—the 2 × 4 rectangle—and still have a shape that has the same perimeter, the “almost” 2 × 4 rectangle.

Mariel then asked: “Does that always work? Can you always take away squares and still have the same perimeter?” The TE responded by sug-gesting that the class consider the 3 × 3 square and check whether they could remove any one-inch squares and still maintain the perimeter.

Mariel: Yes, you can take away the one in the top right corner [removing the one-inch square from the top right corner and continuing to remove squares identified by the PSTs as the discussion continued (see fig. 2)].

Why is it OK to do that?Mariel: Because you’re taking away the same amount that you are getting.

What do you mean?Kendall: There are two sides of the square that you are taking away that are part of the perime-ter, and when you take them away, there are two sides that were not part of the perimeter before that are a part of it now.

OK, so can any other squares be taken away

while maintaining the perimeter?Mariel: You could take away the one in the top left corner.Tiffany: You can take away the next two down, too [referring to the two squares below the top right square and the top left square].

Can you take away any more? [Some heads shake no.] Why not?Mariel: Because if you do, your perimeter will get smaller. If you tried to take the one in the bot-tom right corner, you would lose three parts of the perimeter, but get only one part back.

Through this task and discussion, the PSTs were beginning to consider similarities and differences in the perimeters and areas of poly-gons. Tiffany’s observation about the relation-ship between the perimeter of a 2 × 4 rectangle and the almost 2 × 4 rectangle prompted Mariel’s

After finding perimeters of different shapes drawn on grid paper, the pSts used one-inch squares to create as many polygons as they could with a perimeter of twelve inches.

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Does it always work to take away squares and still have the same perimeter?

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Heather’s response was similar to those given by a majority of the PSTs, including Justine, who believed that perimeter and area increase or decrease together. Other PSTs offered ideas that conflicted with Heather’s. Janet shared her thoughts about rectangles:

No relationships necessarily exist between perimeter and area, because if rectangle ABCD is 5 × 5 and rectangle EFGH is 20 × 1, ABCD has a perimeter of 20 and an area of 25. Rectangle EFGH has a perimeter of 42 and an area of 20. So, even though rectangle ABCD has a shorter perimeter, it still has more area. Just because an object has a larger perimeter doesn’t mean that the area will be greater.

Kendall’s response was similar to Janet’s, but she took her exploration further to include dif-ferent cases of rectangles:

I don’t think there is a direct correla-tion between the two. If you look at basic squares—2 × 2, 3 × 3, 4 × 4—then yes, the greater the area, the greater the perimeter. But comparing dimensions such as 3 × 6 and 1 × 10, the 3 × 6 has the greater area but a smaller perimeter. I do believe it is safe to say that when comparing rectangles that contain a like side (e.g., 3 × 4, 3 × 5, 3 × 6, etc.), the higher you go up, both will increase.

Although the prompt suggested that students consider relationships between the perimeter and area of rectangles, Mariel and Tiffany found the process of removing one-inch squares while maintaining the perimeter of a polygon to be particularly intriguing. Mariel elaborated:

I think the most interesting thing I have learned about area and perimeter in class was when we took the corner squares away from a 9-square group, leaving it with only 5 squares shaped like a t. I had never thought of this before, but after taking 4 whole squares away, the perimeter did not change at all, but the area decreased significantly. I [had] always assumed that if the area of an object changed up or down, then the same would be true of the object’s perimeter and vice versa. I know now that they can be unique and indepen-dent of one another. Fe

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question about whether this idea is generaliz-able. Instead of answering her question (one possible teacher move) or perhaps ignoring it and moving on (another possible teacher move), the TE chose to respond by posing a related task (a third possible teacher move) that opened up an opportunity for the PSTs to explore changes in a polygon’s perimeter that do or do not occur as parts of it are removed. Pressing the PSTs to explain their reasoning may have also helped other PSTs further solidify their own under-standing of how removing squares might influ-ence the perimeter of a polygon.

relationships between perimeter and areaThe PSTs had considered several examples of different polygons that have the same perimeter but a different area, but they had not discussed what happens to the area as parts of a polygon are removed. The TE decided to continue this discussion online to encourage the PSTs to think more deliberately about the relationship between perimeter and area. She hoped that the PSTs would read and respond to one another’s online postings so as to shift from a PST-to-teacher discussion to a PST-to-PST one. She directed the PSTs to respond to either answer the question posed on the online discussion board or comment on a classmate’s response before the next class meeting:

What relationships exist, if any, between perimeter and area? For example, if you start with a rectangle and you adjust its length or width, or both, what happens to perimeter, and what happens to area? Fool around with this a bit, and see if you can come up with some conjectures.

Heather responded,

There is definitely a relationship between perimeter and area. They are both used to measure shapes or objects. Perimeter is the distance around a shape, and area is the space within a shape.… If you increase the length or width of a shape, the perimeter and area increase.


To allow the PSTs time and space to respond to their classmates’ ideas, the TE intentionally did not participate in the online discussions. However, as she read the discussion thread, she noticed that many PSTs did not seem to consider one another’s ideas. She also noticed that the PSTs who claimed that perimeter and area have a direct relationship were not backing up their claims with evidence and were not making con-nections to the work they had done in class. Nev-ertheless, many PSTs who had not participated in the in-class discussion (including Justine and Janet) were willing to share their mathematical thinking in the online forum. They were drawing on the work they had done in class and backing up their claims with specific examples.

responding to their ideasThe TE decided that she would not tell the PSTs whether their thinking was correct or incorrect but would instead generate more evidence dur-ing the next class meeting to dispel the idea that perimeter and area are always directly related. She would then rebroadcast the PSTs’ con-jectures in the next online discussion and ask them to comment on those ideas. This teacher move was made to build on the PSTs’ ideas and encourage them to consider different types of rectangles and why perimeter and area might work differently for each type. The TE hoped that this strategy would give the PSTs addi-tional evidence that perimeter and area do not necessarily change together. She expected that considering different cases would broaden PSTs’ experience with reasoning about mathematics.

During the following class, the TE engaged the PSTs in a similar task. Each small group was given one copy of the same rectangle. The TE told the PSTs to trace five copies of the rectangle for their group and cut them out. Then they were to make a single cut from one side to another side and rearrange each pair of pieces into a new polygon. The PSTs were then to make predictions about the polygons’ perimeters and order them from least to greatest perimeter. One PST spoke up and called it a trick question: “They should all have the same perimeter since they all came from the same rectangle.” Several other PSTs agreed. A few said that they thought some of the polygons would be different. The TE did not con-firm or contest their positions; instead, she asked them to record their predictions and a rationale.

Next, the PSTs were to measure the perim-eters of their polygons using string and then compare the lengths of the strings. Many PSTs were amazed to discover that the shapes that are more like the original rectangle have almost the same perimeter as the rectangle, and those that are longer and have more sides showing have a greater perimeter. The PSTs had just enough time to write their discovery before class ended.

reconsidering relationshipsThe TE wanted to find out how the PSTs’ think-ing was changing because of the additional evidence generated from the Rearranging Rectangles task. She also wanted them to think about their classmates’ ideas. So, on the online discussion board, she posed another question that rebroadcast PSTs’ ideas from the previous online discussion. Again the PSTs were asked to either respond to the question posed online or comment on a classmate’s response before the next class meeting:

After reading through your responses, I hear a conjecture: As the perimeter of a rectangle increases, the area also increases, and as the perimeter of a rectangle decreases, the area also decreases. And I hear two refutations: (1) This is not always the case; you can find two rectangles—one with a large perimeter and small area, and one with a large area but small perimeter; and (2) This is not always the case, because you can find two rectangles with the same perimeter but different areas, or two rectangles with the same area but dif-ferent perimeters.

I also hear another conjecture: If you fix one dimension of a rectangle and adjust the other, as the perimeter increases, the area will also increase; and as the perimeter decreases, the area will also decrease. [Here

the preservice teachers suspected a trick question when asked if perimeter would change when area is held constant while a rectangle is rearranged into a new polygon.

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is] a question: Is there something special about perimeters and areas of squares? What do you think? Do you agree or disagree with each of these proposals? What are your thoughts [about] squares? Explain your reasoning.

The in-class work and the online discussions seemed to move Heather’s thinking forward. She no longer believed that perimeter and area are always directly related. In the following posting, she describes explorations that she carried out with squares.

I think that there is a pattern associated with how much area is in a square to how much perimeter is around the square. For example, I took a square that consisted of an area of 4 cm2 and figured the perimeter to be 8 units. I then took another square that consisted of an area that was 9 cm2 and I figured the perimeter to be 12 units. Also, a square with an area of 16 cm2 had perimeter of 16 cm. However, when I removed one of the units from the square where the area totaled 16 cm2, the area became 15 cm2 and the perimeter stayed the same at 16. Then [after removing another one-inch square, it became] an area of 14 cm2 with perimeter of 16. It seems as long as you are decreasing the area by 1 unit, the area is changing; however, the perimeter stays the same.

Sara was less confident in her thinking, but her exploration of general rectangles and squares led her to conclude that perimeter and area are not always directly related for rect-angles. She also shared her thoughts on what makes squares special:

I started with the idea of finding two rect-angles, one with a large perimeter and small area, and one with a large area but small perimeter, but found that only the area changed; the perimeters did not differ. So, from that I figured that the special thing about squares is that since all sides are equal length, no matter how much adjusting, a square will always remain within the same area or perimeter. If a 4 × 4 square is adjusted to 4 × 6, it is no longer a square; it is a rect-angle. For now, these are my thoughts, and I am still confused!

Although many PSTs’ understanding was changing as a result of the tasks and discussions in which they engaged, others, like Justine, held that perimeter and area are directly related. Nev-ertheless, evidence existed that Justine was no longer certain about this relationship. Unfortu-nately, her post reveals that she did not seem to have considered the examples that were offered by her classmates:

Area and perimeter are special. They relate to each other in a way that when perimeter increases, area will increase. The same is true if perimeter decreases. I am confused about the two rectangles with same perimeter and different area; I would like to see what these look like. I would also like to find the rectangle with large perimeter and small area and the rectangle with small perimeter and large area.

DiscussionGiven the nature of teaching and learning, the opportunities that students have to engage in mathematical discourse and the opportunity for teachers to listen responsively to students’ ideas are tied closely to pedagogical practices (Atkins 1999; Fennema et al. 1996; Hiebert and Wearne 1993; Martin et al. 2005; Pierson 2008; Sherin 2002; White 2003). Providing forums for PSTs and encouraging them to express their mathematical ideas in class and online created different opportunities for them to broaden and deepen their understanding of mathematics by exploring mathematical relationships, an expe-rience that is akin to the Common Core State Standards for Mathematics (CCSSI 2010) and NCTM’s (2000) recommendations for school mathematics. Although it sometimes required that the TE rebroadcast, the specific cases and examples that some PSTs shared helped other PSTs move away from their initial assumptions about relationships between perimeter and area toward considering those relationships for different categories of shapes. Other PSTs even extended the exploration to think about the characteristics of particular shapes that might influence the relationship between perimeter and area. Clearly, some PSTs benefited more than others from the discussions, perhaps partly because some PSTs did not consider their peers’ ideas. To address this concern, TEs should work to encourage PSTs to take more ownership of


their learning by listening to, considering, trying to make sense of, and responding to ideas put forth by anyone in the classroom.

Listening to student thinking creates an opportunity for teachers (and TEs) to learn to make appropriate instructional decisions that are responsive to their students’ learning needs (Atkins 1999; Cady 2006; Sherin 2002). By listen-ing responsively to a PST’s idea during class, the TE was able to engage PSTs in a task that helped them construct their own understanding about the effect that removing square units has on the perimeter of a polygon. By attending to PSTs’ ideas online, the TE determined what the PSTs were gaining from the work they had been doing in class, what they understood, and what they were still unsure of. She was able to use PSTs’ thinking about relationships between the perimeter and area of polygons to help her iden-tify an appropriate task to explore in class, and she used the PSTs’ ideas and examples to push their thinking further in the online discussion.

Mathematics content courses for PSTs pro-vide a powerful opportunity to engage PSTs in a math-talk learning community—“a classroom community in which the teacher and students use discourse to support the mathematical learning of all participants” (Hufferd-Ackles, Fuson, and Sherin 2004, p. 82)—as they do math-ematics, ask questions, investigate ideas, con-sider different problem-solving strategies, and make connections among mathematical ideas. Such experiences will not only strengthen their understanding of mathematics (CBMS 2001; MAA 1998; NCTM 1989, 1991, 2000) and prepare them to supply opportunities for their future stu-dents to learn meaningful mathematics in ways that are consistent with Standards-based teach-ing (NMAP 2008) but will also provide valuable information to TEs that they can use to make instructional decisions that are responsive to and in support of PSTs’ learning of mathematics.

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Kadian M. Callahan, [email protected], teaches mathematics content courses for prospective elementary and secondary school teachers at kennesaw State university in georgia. She is interested in the interdependence of mathematics teaching and learning and how actively engaging students in the learning process influences their understanding of mathematics.

For more information or to place an order, please call (800) 235-7566 or visit www.nctm.org/catalog.

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Listening Responsively