Angling for Students’ Mathematical Agency Author

Angling for Students’ Mathematical Agency Author(s): José Manuel Martínez and Laura Ramírez Source: Teaching Children Mathematics, Vol. 24, No. 7 (May 2018), pp. 424-431 Published by: National Council of Teachers of Mathematics Stable URL: Accessed: 15-08-2018 13:28 UTC REFERENCES Linked references are available on JSTOR for this article: You may need to log in to JSTOR to access the linked references. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to Teaching Children Mathematics This content downloaded from on Wed, 15 Aug 2018 13:28:46 UTC All use subject to FOR STUDENTS’ MAT 424 May 2018 • teaching children mathematics | Vol. 24, No. 7 This contentofdownloaded from on Wed, 15 Aug 2018 13:28:46 UTC Copyright © 2018 The National Council of Teachers Mathematics, Inc. All use subject toor All rights reserved. This material may not be copied or distributed electronically in any other format without written permission from NCTM. José Manuel Mar tínez and Laura Ramírez oward the end of a geometry unit, students in a fourth-grade classroom in Colombia, South America, were working in groups to measure angles. They were recording these measurements to develop a generalization about the sum of the interior angles of triangles. A student, Emily, placed her 180-degree protractor over an angle, as the other three students in her group leaned over to see. “One hundred thirty degrees,” she announced, and all four students wrote that answer without hesitation. Coauthor José Manuel Martínez asked how they knew it was a 130-degree angle. Miguel confidently pointed at the 130, where one of the angle’s rays intersected the numbers in the protractor. When the first author asked, “How would you classify this angle?” Emily started to say, “It’s acu . . .” She smiled and looked down at the paper. A group of fourth graders went from overreliance on protractors to relying on their own reasoning and understanding of how to measure angles. HEMATICAL AGENCY Vol. 24, No. 7 | teaching children mathematics • May 2018 This content downloaded from on Wed, 15 Aug 2018 13:28:46 UTC All use subject to 425 PREVIOUS PAGES: KORAYSA; CAROLINE SCHIFF; JBRYSON/THINKSTOCK THESE PAGES: LABORER (PROTRACTOR); SEB_RA (STUDENT) Students’ mathematical agency 426 Looking puzzled, Juan asked, “How can it be 130 if it’s an acute angle?” “The protractor is telling us lies!” declared Emily. This interaction, during which students were measuring what was a 50-degree angle, raised our awareness about issues of mathematical knowledge authority. That is, the teacher (Laura Ramírez) and the researcher (Martínez) considered students’ confidence to author and assess mathematical ideas, instead of relying only on the teacher or the textbook as possessors of mathematical knowledge. Others have problematized overreliance on mathematical tools as sources of mathematical knowledge authority (Kamii 2006). When the focus of instruction is on learning to use mathematical tools—such as a ruler or a calculator—as a procedure, students tend not to incorporate their own mathematical reasoning. This unreflective use of tools hinders students’ questioning of the reasonableness of their answers. We describe how children expressed their mathematical agency by challenging the knowledge authority of a mathematical tool: the protractor. The first author—a researcher in the classroom—and the second author—the classroom teacher—encouraged students’ critique of the traditional protractor. Students expressed mathematical agency by (1) challenging the protractor’s effectiveness in helping them measure angles and (2) by imagining an alternative tool. The students did not build their imagined tool. Instead they used their imagined alternative tool to help them better understand how to use a conventional protractor. First we define student mathematical agency as an important component of student learning experience. Then we describe how students’ initial use of a traditional tool presented an opportunity to foster the expression of their mathematical agency. We illustrate how, in turn, the expression of their mathematical agency helped students critically consider the traditional tool to make sense of it. We draw on Boaler’s (2002) and Lawler’s (2012) notions of student mathematical agency. Boaler describes high school students’ preference for mathematics learning environments that allow the expression of their agency by “creating initial thoughts and ideas, or by taking established ideas and extending them” (p. 45). For Lawler (2012, p. 167), student agency is learners’ confidence in their own knowledge, which “embraces the learner as an active agent working upon the world, rather than a passive recipient.” Student mathematical agency shifts the locus of knowledge authority from the teacher or the textbook to the students. Students’ awareness of their knowledge authority supports the development of a sense of being authors (instead of only users) of mathematical ideas, capable of “judging the viability of another’s way of knowing” (Lawler 2010, p. 179). Students who can express their mathematical agency can rely on their own thinking and their cooperation with others to expand their understandings. In turn, the ability to rely on their own thinking helps students proactively engage in unknown tasks without depending on others for direction or confirmation. In our example, students did not rely on textbooks or the teacher. They relied on the protractor to determine an answer. The students’ language use (“The protractor is telling us lies”), as well as their own questioning of their initial answer, brought to our attention the knowledge authority that may be bestowed on mathematical tools. Instead, we wanted to support students to rely on their own thinking, which was fostered by their mathematical agency on the basis of their sense making and understandings. In what follows, we describe how the lesson unfolded to position students as authors of mathematical ideas, capable of critiquing, designing, and using measuring tools. Students, the teacher, and the researcher went beyond using the protractor. We promoted student agency by deconstructing the protractor and imagining an alternative angle-measuring tool. The setting Several aspects of the setting might resonate with educators in other contexts, such as in the United States. The language of standards about the use of the protractor is similar in Colombia May 2018 • teaching children mathematics | Vol. 24, No. 7 This content downloaded from on Wed, 15 Aug 2018 13:28:46 UTC All use subject to The lesson’s four stages and the United States. In the latter, the fifth of the Common Core’s eight Standards for Mathematical Practice prompts students to “use appropriate tools strategically” (CCSSI 2010, p. 7). Similarly, in Colombia—where the classroom was located—Standards for Mathematical Competencies ask that students “select and use instruments to measure length, area, and angles accurately” (Ministerio de Educación Nacional 2006, p. 134). The lesson took place at an International Baccalaureate English immersion school. In addition to the Colombian Standards for Mathematical Competencies, resources that inform mathematics teaching include a textbook in English and NCTM’s Principles to Actions: Ensuring Mathematical Success for All (2014). Both authors are native Spanish speakers with teaching experience in bilingual elementary schools. Ramírez characterizes her efforts as striving for an inquiry-based class. Students interact to generate and assess mathematical ideas in the context of particular tasks. All twenty-four students in this fourth-grade classroom were native Spanish speakers. The students participating in the small group on Wilson and Adams’s 1992 article informed the unit that the authors taught to native Spanish speakers in a fourth-grade classroom. The unit they describe focused on the four points below. 1. Explore the concept of angles: What are they, and how can we represent them? 2. Compare and classify angles on the basis of attributes (e.g., size). Recognize a right angle. 3. Learn to measure angles by degrees. 4. Observe measurements and deduce rules about the angles of a triangle. which we focus here were Emily, Miguel, Steve, and Juan (pseudonyms). The unit followed four stages informed by Wilson and Adams’s (1992) work (see the sidebar above). First, we were helping students explore the concept of angles. The focus of these lessons was on the questions “What is an angle?” and “How can we represent an angle?” Second, we were comparing angles. The focus here was on relevant attributes of an angle, recognizing a landmark angle—the right angle— and classifying angles by size. Third, we developed a unit that can be used to measure angles: the degree. Finally, we observed measurements and inferred rules about the angles of a triangle. During the lesson that we describe, the teacher focused on the fourth point. Therefore, STUDENT MATHEMATICAL AGENCY SHIFTS THE LOCUS OF KNOWLEDGE AUTHORITY FROM THE TEACHER OR THE TEXTBOOK TO THE STUDENTS. This content downloaded from on Wed, 15 Aug 2018 13:28:46 UTC All use subject to by the time this lesson took place, students had had several opportunities to develop certain understandings about angles and angle measurement. Eliciting students’ mathematical agency F IGURE 1 As narrated before, Emily stated that the protractor was telling lies. Instead of interpreting this interaction as indicating students’ mis­understandings and the need to reteach the procedure to measure an angle with the protractor, the authors encouraged students to explain how the protractor was “lying.” Students proceeded to critique the size of the protractor. They said that the rays of the angle they were trying to measure were too short, forcing students to somehow extend the rays until they intersected the numbers on the protractor. The students went on to question what they described as “too many numbers” on the protractor, claiming it was confusing remembering when to use which. They explained that protractors have a set of numbers at the outer circle, going clockwise from 0 degrees to 180 degrees, and another set of numbers in the inner circle, going from 180 degrees to 0 degrees. The students’ critique of the protractor does not seem unwarranted if we consider that many mathematical tools and representations were not designed for pedagogical purposes. Instead, the design of many mathematical Showing angle as sweeping movement, Miguel placed his hand vertically over one ray and moved until his hand was over the second ray. tools values efficiency in terms of production and usage costs, and in terms of the amount of information condensed (O’Halloran 2005). For these students, in the case of the protractor, tool efficiency came at a price in terms of sense making. Allowing students to express their critique elicited their mathematical agency in the form of daring to challenge a mathematical tool. Listening to their critique allowed us to notice which features of the protractor were salient to these students. The focus was then on what they knew about the protractor, including aspects of it they could justify as questionable. Thus far, eliciting students’ mathematical agency had allowed us to move from simply using a protractor to being critical about it. Going beyond the critical aspect, we encouraged students to author mathematical ideas and prompted them to imagine a tool that would better support their angle-measuring efforts. That is, instead of portraying the protractor as the only and unquestionable tool, we supported students in using their mathematical reasoning and their knowledge about the protractor to imagine an alternative angle-measuring instrument. First, we asked students to brainstorm what they needed to do to measure an angle. Emily: [Pointing at one of the rays in the angle] You start with the first line. Miguel: And then you see how much it turns. It stops at the other line [pointing at the second ray]. Researcher: Can you show us how much this one [pointing at the angle on the paper] turns? Miguel: Like this [placing his hand vertically over one ray and making a sweeping movement until his hand is over the second ray (see fig. 1)]. JOSÉ MANUEL MARTÍNEZ Steve: You start with the first line and the corner [pointing at the vertex]. 428 These interactions helped us elicit students’ geometric thinking to inform teaching (Mack 2007). Our discussion about angle measurement elicited a dynamic notion of angles as rotation (Browning, Garza-Kling, and Sundling 2007). Students were able to work collaboratively to express their understandings about how to measure an angle. On the basis of those understandings, we asked students what a tool should do to help them measure an May 2018 • teaching children mathematics | Vol. 24, No. 7 This content downloaded from on Wed, 15 Aug 2018 13:28:46 UTC All use subject to FIGU R E 2 Emily: Like a compass. A compass that does what Miguel just did. With a corner and two hands, and then we put the corner over the angle’s corner. And we leave one hand over one line, and we move the other hand until it gets to the angle’s other line. Miguel drew this adaptation of a compass to measure angles. FIGURE 3 angle. The beginning point was not the tool as an authority guiding how to measure angles, but students’ mathematical assertions about an instrument that would serve them. Juan drew the fan angle-measuring tool. Researcher: How do we know the measurement of the angle? Miguel: The hand that stays can have an arc, and the arc can have all the angles from zero to one hundred eighty [drawing an image (see fig. 2)]. Steve: What about a fan? We only need the part that shows the angle we are measuring. Emily: What do you mean [by] “a fan”? Juan: I get it. Like this. This is the corner [putting his two wrists together]. The two hands start together. And we move one [making a sweeping movement with his left hand, keeping both wrists together]. When we open the hand, a little paper, like one of those fans that women use when it’s hot, shows, and it has the numbers in it [drawing the image (see fig. 3)]. Other authors (Wilson and Adams 1992) have suggested that after students have developed initial ideas about angles, teachers guide students through a series of steps, usually involving wedges, to create a traditional protractor. In the interactions we describe here, we did not narrow options to a protractor. Instead, we fostered students’ expression of their mathematical agency so that they could author mathematical ideas, including an imagined angle-measuring tool that gestured rotation. The role of students’ mathematical agency, however, did not mean ignoring already existing mathematical artifacts that are widely used. Instead of building their imagined tool, at the end of the lesson, students used their tool to better explore the traditional protractor. We asked students to keep in mind their imagined angle-measuring tool when using the traditional protractor on the angle they had initially measured as 130 degrees. Placing the protractor over the angle, students gestured the sweeping movement, starting at one of the angle’s rays. Juan: He [referring to the protractor] wants to play tricks on us and is saying one hundred thirty again, but it also says fifty. Emily: But it’s fifty because if we had our fan with the numbers, it would start with zero, then ten, then twenty, then thirty. Steve: Yeah, and it cannot be one hundred thirty because it’s acute. This example illustrates a component of students’ mathematical agency: relying on their own mathematical thinking. This time, students approached the angle-measuring activity aware that the information they were to extract from the protractor was subject to their interpretation. This interpretation had to rely on the understandings about angles they had developed through the unit plan described above. Instead of uncritically relying on what seemed to be procedural knowledge about how Vol. 24, No. 7 | teaching children mathematics • May 2018 This content downloaded from on Wed, 15 Aug 2018 13:28:46 UTC All use subject to 429 DIEGO_SERVO/THINKSTOCK Rather than rely on a mathematical tool, students learned to rely on their own mathematical agency. to use a protractor, students drew on their conceptual understanding about angles. a protractor. The example we described here focuses on a small group. We, however, encourage teachers to pursue this as a whole-class task in which students can refine a protractor or other mathematical tools, justifying on the basis of their understandings. Our students’ daring to critique and reimagine a frequently used tool positioned them as authors of mathematical ideas. In doing so, students were able to not only use an appropriate tool but also to better understand it and to ground their use of the tool on their reasoning. This approach supported educators and students alike in experiencing the power of mathematical creativity and using divergent thinking to overcome fixations (Mann 2006). We embraced a “productive struggle” (NCTM 2014), helping students grapple with ideas and relationships among angles, measurement, and the protractor. Common Core Connections Eliciting student reasoning We have explored possibilities that open up when educators support the expression of student mathematical agency. We have described a group of students’ initial uncritical reliance on a tool to approach a mathematical task. Instead of focusing on correcting students’ use of the protractor, we focused on what the interaction said about student mathematical agency. Students seem to have placed mathematical knowledge authority on the protractor, to the point of blaming the protractor for telling lies. In our case, the focus of the mathematical task was not on using the protractor but on using their reasoning to measure angles. That shift in focus implied supporting students to critically analyze and assess a traditional mathematical artifact in light of their own understandings. The issue of overreliance on mathematical tools seems to expand across contexts, as we have seen evidence both in the United States and in Colombia. Supporting expression of mathematical agency can help students in different contexts determine which tools make more sense, even if they have not typically been used in classrooms. For example, our students’ imagined angle-measuring tool more closely resembles an angle ruler or a goniometer than 430 SMP 5 REF EREN C ES Boaler, Jo. 2002. “The Development of Disciplinary Relationships: Knowledge, Practic…
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