Second-Graders' Mathematical Practices for Solving

I nv e stigations in Mathematics Learning Fall Edition 2014

Volume 7, Number 1

Editor Vicki J. Schell Pensacola State College

Sheryl A. Maxwell Univ. of Memphis

Editorial Board Anne Reynolds Kent State University

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Alan Zollman Northern Illinois Univ.

Table of Contents Redefining the whole: Common Errors in Elementary Preservice Teachers' Self-authored Word Problems for Fraction Subtraction.......1   Juli K. Dixon, University of Central Florida   Janet B. Andreasen, University of Central Florida   Cheryl L. Avila, University of Central Florida   Zyad Bawatneh, University of Central Florida   Deana L. Deichert, Cedar Crest College   Tashana D. Howse, Daytona State College   Mercedes Sotillo Turner, University of Central Florida Responsive Decision Making from the Inside Out: Teaching Base Ten to Young Children.....................................................................23   Susan B. Empson Second-Graders' Mathematical Practices for Solving Fraction Tasks...........................................................................................57   Patricia S. Moyer-Packenham, Utah State University   Johnna J. Bolyard, West Virginia University   Stephen I. Tucker, Utah State University

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Investigations in Mathematics Learning ©The Research Council on Mathematics Learning Fall Edition 2014, Volume 7, Number 1

Redefining the whole: Common errors in elementary preservice teachers' self-authored word problems for fraction subtraction Juli K. Dixon University of Central Florida [email protected]

Janet B. Andreasen University of Central Florida

Cheryl L. Avila

University of Central Florida

Zyad Bawatneh

University of Central Florida

Deana L. Deichert Cedar Crest College

Tashana D. Howse Daytona State College

Mercedes Sotillo Turner University of Central Florida

Abstract

    A goal of this study was to examine elementary preservice teachers' (PSTs) ability to contextualize and decontextualize fraction subtraction by asking them to write word problems to represent fraction subtraction expressions and to choose prewritten word problems to support given fraction subtraction expressions. Three themes emerged from the data: (a) subtraction problems were represented by an incorrect redefinition of the whole; (b) the type of unit chosen for the whole (e.g. cups, gallons, pounds vs. pizzas, pies) influenced the success of PSTs in representing Separate (Result Unknown) context problems for subtraction; and (c) the structure of the problem influenced PSTs' performance in writing subtraction word problems. -1-

Introduction   "Students learn mathematics through the experiences that teachers provide. Thus, students understanding of mathematics, their ability to use it to solve problems, and their confidence in, an disposition toward, mathematics are all shaped by the teaching they encounter in school. The improvement of mathematics education for all students requires effective mathematics teaching in all classrooms" (NCTM, 2000, 16-17). It is fairly well accepted that teachers need pedagogical content knowledge to teach effectively (Shulman, 1986). In more recent years, mathematical knowledge for teaching (MKT) has been explicated by Ball and her colleagues (Ball & Bass, 2000). While they have sought to define the mathematical knowledge for teaching in general, the task still remains to define what that knowledge should include for specific mathematical concepts, in this case fraction operations. Most would agree that it is not enough to just be able to compute with fractions. The question remains, what must teachers be able to do to convey the concept of fraction operations? Is it enough for teachers to be able to select appropriate word problems for representing operations for fractions or must they also be able to create problems to support such expressions? As expectations of students are increased in light of the Common Core State Standards (CCSS) (National Governors Association (NGA) & Council of Chief State School Officers (CCSSO), 2010) do the expectations of teachers need to be adjusted as well? The purpose of this research was to explicate preservice elementary teachers' knowledge with respect to these aspects. What follows are results of a qualitative study describing elementary preservice teachers' (PSTs) struggles related to selecting and writing appropriate word problems to support fraction subtraction along with a description of scenarios that inhibited or supported success with this task and the research implications for the classroom.

Background   Shulman (1986) brought to the forefront of mathematics education literature the need to describe and examine the pedagogical content knowledge of teachers. Shulman examined the cognitive research on learning and shifted the focus from learner to teacher, describing the knowledge the teacher needs to possess to be an effective teacher of mathematics. Pedagogical content knowledge research examines this specialized type of knowledge which is a need specific to the teaching of content, not necessarily needed directly for either the practitioner or the learner of the subject matter content. Likewise, Ball and Bass (2000) have worked throughout the last two decades to further define pedagogical content knowledge as it relates spe-2-

cifically to mathematics with what they term Mathematical Knowledge for Teaching (MKT). Through their work with elementary school preservice and inservice teachers, they have come to define specific types of knowledge for mathematics in general, namely Knowledge of Content, Knowledge of Students; however, the research is limited when it comes to defining these types of knowledge within specific mathematics domains. This research aims to expand on these efforts to define what that knowledge should include for teaching fraction operations, specifically subtraction of fractions in context.   In the same manner as Shulman, we shift the focus of a known area of mathematics education research from the learner's perspective to that of the teacher. Research on the assessment of the depth of mathematical conceptual knowledge possessed by students has been examined through the evidence of students' abilities to author their own word problems that utilize the operations being studied (Alexander & Ambrose, 2010; Alibali et al., 2009). Extending this idea to teacher education, it is reasonable to use the same strategies to assess PSTs' conceptual understanding of mathematical concepts (Drake & Barlow, 2007; Whittin & Whittin, 2008).  When students are able to create their own word problems related to a new mathematical concept it has a positive influence, not only on their understanding, but also on their problem solving skills and disposition towards mathematics (Barlow & Cates 2007). In addition, original student-authored word problems can reveal a variety of misunderstandings that students may hold (Alexander & Ambrose, 2010). Researchers from the Wisconsin Center for Education Research (Alibali, Brown, Stephens, Kao, & Nathan, 2009) conducted a survey using student-authored story problems focused on middle school students' understanding of equations. The results suggested that middle school students have substantial difficulty generating stories to correspond with algebraic equations. Not surprisingly, students who were more successful generating stories were also more successful solving such equations. Our research sought to apply this research to the arena of preservice teacher preparation by focusing on "student"-authored word problems based on fraction subtraction with the "students" being PSTs.   Friske (2011) used student-authored word problems to assess her sixthgrade students' understanding of fraction operations. Through the process of examining her data related to her students' abilities to write fraction word problems, Friske realized that she did not take into account the variety of problem structures for fraction word problems. She actually was not even aware that there were different types of problems to be represented. Her own misconception, which was connected to the structure of the problem, had inadvertently been conveyed to her students. We wondered, as researchers, if PSTs had similar misconceptions related to fraction operation word problems that might then be inadvertently conveyed to their future students. -3-

Ball (1990) found that PSTs "demonstrated that they wanted to give the pupils what they considered to be meaningful answers, but often they could not do so because their subject matter knowledge ... was insufficient to act on that commitment" (p. 142). In order to improve a teacher's ability to provide multiple explanations, varied instructional strategies, and in the case of this research, varied problem structures, the PST's own knowledge must be improved (Shulman, 1987).   Ma (1999) reported findings related to the pedagogical content knowledge of 23 U.S. teachers and 72 Chinese teachers who were asked "to compute 1 ¾ ÷ ½, and to represent meaning for the resulting mathematical sentence" (p. 55). Only 43% of participating American teachers could calculate this division of fractions problem accurately; additionally, most of them could not accurately represent division of fractions in a word problem. Ma's research supports the notion that many teachers in the United States are unable to provide contextual support for dividing fractions. Is this also the case for other fraction operations, particularly subtraction of fractions?   In examining contextual support for subtraction, it is also relevant to examine the problem structures elucidated by the research of Carpenter et al. (1999) through their work with Cognitively Guided Instruction (CGI). When the PSTs create context problems to model subtraction of fractions, will the problem type make a difference in their ability to create accurate story problems? Are some problem types more problematic?

Purpose   Assessing student learning and deciding on a means of instruction is one of the most critical decisions a teacher has to make (Hiebert et al., 1997). Students' mathematical proficiency is shaped by the learning experiences that result from the tasks assigned by their teachers. When designing these tasks, teachers must create activities that are relevant to their students' lives and believable to students (Gravemeijer, 2004). But more importantly, teachers need to be able to make sense of the mathematics they teach on a level that is deep enough for them to understand their students' thinking and the mathematical activities that support that thinking. "Quality of instruction is a function of teachers' knowledge and use of mathematical content, teachers' attention to and handling of students, and student's engagement in and use of mathematical tasks" (NRC, 2001, p. 424). Ball and Bass (2000) argue that teachers must be able to "work backwards from mature and compressed understanding of the content to unpack its constituent elements" (p. 98).   One goal of this study was to investigate elementary PSTs' conceptual understanding of subtraction of fractions and how those understandings influenced the PSTs' ability to write word problems for subtraction of frac-4-

tions (for a discussion of all of the fraction operations see Dixon and Tobias, 2013). If the teacher is not confident in his/her own understanding of this topic, s/he may have the tendency to address fraction operation concepts devoid of context. Context is vitally important for students to make sense of the mathematics they are learning in the classroom (Gravemeijer, 2004). If the teacher is not able to create, or interpret, relevant and accurate contexts, the students are less likely to make sense of the mathematics in meaningful ways. A second goal of this study was to explore types of problem structures aligned with the Cognitively Guided Instruction model to determine if particular types were more supportive of making sense of fraction subtraction than others (Carpenter et al., 1999; Carpenter, Fennema, & Franke, 1996). This research sought to answer the following questions: 1. What conceptually-based errors occur when preservice elementary teachers write word problems to support subtraction of fractions? 2. What contexts and problem structures are helpful in writing word problems for subtraction of fractions?   For the purposes of this study, prior research related to the use of studentauthored word problems coupled with identified difficulties elementary school teachers tend to encounter with fraction operations guided the research design. Elementary PSTs' knowledge and understanding of fraction subtraction was identified and explicated through the use of PST-authored word problems. What follows is a detailed description of the methodology, data, and analysis.

Procedures   This study consisted of 19 PSTs who were enrolled in a graduate elementary mathematics methods course in a large, urban university in the Southeastern United States. This course was designed for graduate students with degrees in fields other than education and who chose to pursue teaching elementary school. The instructor for the course was an experienced professor of mathematics education and a member of the research team. The course was designed to focus on both methods and content for teaching elementary school mathematics with the methods in the foreground and content knowledge for teaching in the background. Students were frequently situated in small groups, each working on the same tasks. Once students had explored the tasks in small groups, the class would meet as a whole group to discuss their solutions and solution strategies. The instructor facilitated all smalland whole-group discussions. The class met once per week for three hours each week. The major topics of the class were guided by learning trajectories within the content areas of whole number concepts and operations, fraction concepts and operations, geometry, and linear and area measurement. -5-

Instructional Design   The use of learning trajectories in mathematics is not new, but has gained momentum with the introduction of the CCSS and the work of the Center on Continuous Instructional Improvement (CCII). In the document, Learning Trajectories in Mathematics: A Foundation for Standards, Curriculum, Assessment, and Instruction (Daro, Mosher, & Corcoran, 2011), learning trajectories are described as empirically supported hypotheses about the levels . . . of thinking, knowledge, and skill in using knowledge, that students are likely to go through as they learn mathematics and . . . reach or exceed the common goals et for their learning. Trajectories involve hypotheses both about the order and nature of the steps in the growth of students' mathematical understanding, and about the nature of the instructional experiences that might support them in moving step by step toward the goals of school mathematics (p. 12).   Trajectories help the teachers clearly identify interim goals for students in learning a particular mathematical concept and help to define formative assessments relative to the trajectory rather than solely in comparison to their peers. Trajectories provide models of student thinking, which can assist teachers in making sense of student work and providing experiences that develop conceptual understandings of mathematics. Students' experiences with fraction concepts and whole number operations need to be solidified before students start to operate on fractions (Common Core Standards Writing Team, 2011). Using these types of trajectories give teachers informal and formal assessment data to drive instruction; however, at the same time, it requires a higher level of teacher knowledge of curriculum, content, and pedagogy. "It is one thing to talk theoretically about learning trajectories and a whole other thing to understand how to transfer the knowledge from learning trajectory research to practice in a way that teachers can embrace it" (Daro, Mosher, & Corcoran, 2011, p. 35).   When developing fraction operation concepts, a teacher's instruction should build off of previously learned whole number operation concepts. For example, in the case of division of fractions, Sharp and Adams (2002) found that students naturally applied concepts of division of whole numbers to division of fractions. By using similar contexts to whole number division, students were able to invent their own common-denominator algorithm for the division of fractions. It was one goal of the course to focus on the use of trajectories by modeling through instruction, which focused on correcting predictable errors and misconceptions and sense making on the part of the PST.   As with operations with whole numbers, students need to be aware of the units that accompany the given quantities with fraction operations. As -6-

indicated in the Progressions for the Common Core State Standards in Mathematics (draft), 3-5 Number and Operations - Fractions (Common Core Standards Writing Team, 2011), students should attend to precision in examining addition and subtraction of fractions by attending to the underlying unit quantities. "In order to formulate an equation of the form A + B = C or A ˗ B = C for a word problem, the numbers A, B, and C must all refer to the same (or equivalent) wholes or unit amounts" (p. 7). For example, ¾ of a pizza + ½ of a pizza = 1 ¼ of two pizzas. In the latter example, the addends each refer to one pizza as the whole and the sum refers to two pizzas as the whole or unit. If students are expected to attend to this unit quantity, teachers must also be able to attend to the same precision and to identify what units are appropriate for fraction addition and subtraction contexts. The knowledge that teachers should have must not only be focused on fraction operations procedures but on listening to students' mathematical thinking while they construct or solve word problems as it is paramount in understanding and gauging their conceptual knowledge; however, this is not easily accomplished (Fennema et al., 1996; Tirosh, 2000). Through the instructional design of this course, it was hoped that PSTs would experience this type of learning for themselves and then eventually be able to translate that learning back into their own classrooms. Throughout the course, whenever word problems were examined, the work associated with Cognitively Guided Instruction was examined, particularly the problem structures described. Cognitively Guided Instruction   The work of research associated with Cognitively Guided Instruction (CGI) included a clear description of problem structures for word problems along with examples of how students solved word problems with these various structures. The aspect of CGI that was used in this study was the problem structures for addition and subtraction word problems. Word problems dealing with addition and subtraction can be sorted into four classes: Join, Separate, Part-Part-Whole, and Compare (Carpenter et al., 1999). Within the Join and Separate classes there are three types: Result Unknown, Change Unknown, and Start Unknown. Within the Part-Part-Whole class there are two types: Whole Unknown and Part Unknown and within the Compare class there are three types: Difference Unknown, Compare Quantity Unknown, and Referent Unknown. When examining word problems dealing with addition and subtraction it is helpful to be familiar with the different types of problems and how their structure is related to how children solve the problems (Carpenter et al., 1999). Because of this, problem types played an important role in discussion around word problems for whole number and fraction operations. In our study when the PSTs were asked to write word problems they were often directed to write problems with specific structures. -7-

Participants   Of the PSTs who agreed to participate in the study, 17 completed a fraction survey, which required them to select a context to support a mathematical expression involving fractions. The survey served as a pretest for knowledge of contexts to support operations with fractions prior to the unit on fractions and was developed by the researchers with attention to including various problem structures, each of the four operations, and several types of contexts. Results from this survey assisted the researchers in developing an interview protocol, which was adaptive to the participants' responses to the survey items and focused primarily on fraction subtraction due to PST's overwhelming difficulty with this area.   A sample of the PSTs was selected to participate in an interview based upon the participants' willingness and availability to schedule an interview. Nine PSTs were interviewed prior to the unit on fractions. The survey results for the interviewed participants were representative of the overall group. The survey results assisted the researchers in explicating the participants' thinking during the interview.   Each participant interviewed was asked to write a word problem for 4/5 ˗1/2 using a situation involving pizza. They were then asked to write a word problem for 5/6 ˗ 1/3 using a situation involving gallons of iced tea. We required the specific contexts of pizza and gallons of iced tea to examine if using a standard unit of measurement (a gallon instead of a pizza) might affect the results. Following writing the word problems, interviewees were asked to revisit their responses to the two survey items that dealt with subtraction (see Figure 1). For each item, the interviewee was asked to provide a justification for his/her answer choice.   Observation field notes were collected during class instruction on portions of the fraction unit focused on operations with fractions. Prior to this portion of the unit, instruction had focused on using context, manipulatives, and drawings to make sense of fractions in sharing situations; to model equal parts of the whole using area, linear, and set models; and in examining the relationship between the defined whole and the corresponding parts.   Fraction operations were explored by having groups of PSTs write word problems to represent given expressions. The word problems were discussed with the whole class so that misconceptions could be identified and resolved. Resolution of misconceptions often involved using drawings to illustrate solutions to the word problems and contrasting those drawings with those representing solutions to the fraction expressions. The fraction unit spanned two three-hour class sessions, with one class session devoted to fraction concepts and the other to fraction operations. The researchers audio recorded the class session on fraction operations in order to gather data related to the results of the survey and interviews. At the end of the course, the PSTs were administered a final examination. Students were required to -8-

Figure 1: Survey Subtraction Items (correct response in bold type)

respond to 5 of 7 tasks. One task they could select was related to fractions and asked them to "Write a Separate (Result Unknown) word problem for 6/8 ˗ 2/3."   The research team consisted of five mathematics education doctoral students and two mathematics education faculty members. Data from surveys, interview transcripts, class transcripts, class observation notes, and the final examination were analyzed by the researchers using the Constant Comparative Method (Glaser & Strauss, 1967). At least two researchers separately analyzed each type of evidence and differences in analysis were discussed and resolved. The researchers convened as a team to review the analyses. Three themes emerged from the data: a) subtraction problems were represented by an incorrect redefinition of the whole; b) the type of unit chosen for the whole (e.g. cups, gallons, pounds vs. pizzas, pies) influenced the success of PSTs in representing Separate (Result Unknown) context prob-9-

lems for subtraction; and c) the structure of the problem influenced PSTs' performance in writing subtraction word problems. Each theme is further discussed in the results and findings.

Results   Results of the initial survey indicated that PSTs experienced difficulty with fraction subtraction, particularly in comparison type problems (see Table 1). Additionally, multiplication and division of fractions was difficult, but this was likely compounded by confusion between multiplication and subtraction contexts as indicated by PSTs during interviews. Table 1: Results of Survey of Selecting Context to Support Operations with Fractions (N = 17)

For the addition expression, 12 of the PSTs chose the correct context. However, for the two subtraction expressions, 10 or fewer of the PSTs were able to choose the correct context. Most notably, for the Compare (Difference Unknown) problem, only five PSTs chose the correct context and nine of the participants chose the same incorrect representation of Separate (Result of Unknown). This is the only problem for which more respondents chose the same incorrect context compared to the correct context. For the multiplication expression, nine PSTs selected the correct context. And lastly, for the division expressions, the results were different based on the type of problem. Eleven participants chose the correct context for division by a whole number while seven of them chose the correct context for division by a fraction. These results guided the choices for interview tasks (see Figure 1) and subsequent focus of data collection. As the data were analyzed, three themes emerged which are explicated below.   Subtraction problems included an incorrect redefinition of the whole. When asked to write subtraction word problems for 4/5 ˗ 1/2 and 5/6 ˗ 1/3, of the nine PSTs interviewed, only one PST was successful. The other eight PSTs all chose to write problems that represented Subtraction (Result Unknown) but were unsuccessful in modeling the given expression. As de- 10 -

scribed in Progressions for the Common Core Standards in Mathematics (draft), 3-5 Number and Operations, Fractions (Common Core Standards Writing Team, 2011), when representing subtractions as A ˗ B, A and B must refer to the same size whole or unit amount. Similarly, when representing such expressions with meaning in context, the quantities must also refer to the same size whole. What we observed in transcripts of student responses was that the whole was actually redefined from the minuend to the subtrahend such that A (the minuend) referred to a whole and B (the subtrahend) represented a scaled version of A with the scale factor of B. In essence, when the eight PSTs attempted to write a word problem to represent A ˗ B, they actually wrote a word problem to represent A ˗ B*A. As such, the expression could have been represented as A ˗ D where both A and D did not refer to the same size whole or unit amount. As a result the whole was redefined. A common incorrect response for 4/5 ˗ 1/2 was: "Sean had 4/5 of a pizza leftover from yesterday. He ate half of his leftover [pizza] for lunch today. How much pizza is left for the dog?" This word problem may seem to represent subtraction but it does not represent the given subtraction expression. The word problem as it was written above incorrectly interprets the operation taking place by beginning with Sean's 4/5 of a pizza and subtracting 1/2 of his leftovers. In order to subtract quantities, the units must be the same. The story problem written above subtracts leftovers from pizza. The units are not the same. The unit or whole of a pizza was redefined as leftovers from a pizza. The word problem, as written, actually represents a multi-step subtraction problem of 4/5 of a pizza minus 1/2 of 4/5 of a pizza, or 4/5 ˗ (1/2 x 4/5).   A representative incorrect response for 5/6 ˗ 1/3 was: "Beth has 5/6 of a gallon of iced tea. If she drinks 1/3 of that, how much will she have left?" As with the earlier example, this word problem depicts 5/6 ˗ (1/3 x 5/6) rather than the given expression.   During class, at the beginning of the fraction subtraction lesson, PSTs were working in small groups to write a word problem for 4/5 ˗ 1/2. This occurred prior to formal instruction on fraction subtraction. The instructor asked students from each group to share their word problems. Remarkably, every single small group made the same error in their word problems. In each case, the word problem actually depicted 4/5 ˗ (1/2 x 4/5). Notice in the transcript that follows how students were able to determine that their problems were similar in structure from one to the next but they did not notice their error.

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PST from First Group: OK. Ours was, I had 4/5 of a cupcake left. I ate a half of my remaining cupcake, how much cupcake will I have left? Instructor: OK. You guys [pointing at another group]. PST from Second Group: Ours was..., hum, Johnny has 4/5 of cake left over from his graduation. He ate half. How much cake is there left over? Instructor: OK, same or different? [Silence] Same, right? [nods and expressions of agreement] Next group. PST from Third Group: I had 4/5 of a pizza. If I eat 1/2 of that pizza, how much is left? Instructor: Same or different? [brief pause] Same. Isn't that interesting? [nods and expressions of agreement] Ok, go ahead. PST from Fourth Group: Mine is the same with more cake. Instructor: And yours, [pointing to the last group] is it the same? PST from Fifth Group: Yes, but with pizza. Instructor: Well, do you notice all the problems are the same? Isn't that interesting? ... Instructor: So, they are all wrong. [there is a paused silence and nervous laughter] Let's look at this one..." PST: So they are all wrong? I can't believe they are all wrong. [there is murmured agreement.]   The students shared all of their responses and analyzed them to determine that they were similar in structure without realizing that the responses were actually incorrect. They had represented the minuend as a part of the whole and the subtrahend as a part of the part. However, with subtraction both the minuend as a part of the whole and the subtrahend as a part of the part. However, with subtraction both the minuend and the subtrahend must be based on the same unit. Consider the first word problem shared, "I had 4/5 of a cupcake left. I ate a half of my remaining cupcake, how much cupcake will I have left?" It would have needed to be worded something like, "I had 4/5 of a cupcake left. If I eat half of a cupcake from what I had remaining, how much cupcake will I have left?" The difference would be to change the "half of my remaining cupcake" to "half of a cupcake from what I had remaining." Grammatically, the sentence seems to have changed very little, but mathematically, the two expressions have completely different meanings.   As students made sense of their error, there was a need to reinvent their understanding within the context of fraction subtraction. It was interesting to observe how they were engaged in constructing viable arguments and critiquing the reasoning of others as described in the Standards for Mathematical Practice of the CCSS (NGA & CCSSO, 2010). Through discussion, PSTs engaged in dialogue with each other related to redefining the whole - 12 -

in subtraction problems. However, they continued to struggle with incorrectly redefining the whole as opposed to keeping the whole consistent as illustrated in the following transcript. Sarah: Well, 4/5... We were all talking about it... like 4/5 of a cookie and then we started talking about taking ½ of a cookie... Instructor: Ok, ½ of what? Sarah: Take ½ of the 4/5 of the cookie, if we do that then we would take 4/5 and divide it into 2 equal parts instead of subtracting ½ of a cookie. Instructor: So, what does it mean? Jessica: I think mistake is that we are dividing 4/5 into half instead of 5/5 Instructor: So, is this a division problem? So dividing 4/5 in half instead of 5/5 in half. What do you mean by that? Jessica: It is supposed to be, there is 4/5, are we taking half of that? Or is it half of a whole? Instructor: That is the question. So, what do you mean? What are you saying?" Zachary: I think I know what Jessica is saying . . . are we . . . like we are dividing that 4/5 into half rather than subtracting half? Instructor: OK, you guys are saying similar things. How are they the same? Zachary: I think that we are subtracting half of that 4/5 instead of 1/2 of the whole. Instructor: And that is what you just said too [pointing to the first students]. So... Sarah: Clearly it's wrong; still, the problems are wrong . . . I am sorry . . . because since the problem is giving you 4/5, then the 4/5 will be the whole. But, I mean, I could be wrong... Instructor: What do you mean? Jessica: I mean, if you are given a number, why would you assume it is another number? That the half refers to another number. Why would we assume it is 5/5, when we are given 4/5? Alex: I know what they mean, wouldn't they say 1 ˗ 1/2 if they wanted us to subtract it from the whole rather than saying from 4/5? Dialogue continues and concludes with: Instructor: Let's go back to those first problems that you wrote: 'I have 4/5 of a pizza. If you eat ½ of my 4/5, how much is left?' Every one of you wrote a word problem that beautifully represented the wrong problem. You represented this problem [pointing to 4/5 ˗ (1/2 x 4/5)'. Isn't that amazing? ... Sarah: So, what you mean, that when we are subtracting 4/5 ˗ ½, the whole is still the pizza, it is not the 4/5? Instructor: This problem is 4/5 of a whole minus ½ of a whole. - 13 -

  PSTs had difficulty identifying their own errors; however, through facilitation by the instructor, PSTs came to the correct conclusion that the whole must remain constant throughout the subtraction problem. The class session related to fraction operations occurred in the 10th week of a 15-week semester. At the end of the semester, PSTs were to answer five of seven given problems on the final examination. One of those tasks was to write a Separate (Result Unknown) word problem for 6/8 ˗ 2/3. Eighteen out of 19 PSTs provided responses to this prompt, demonstrating their confidence with the prompt. Eight of them wrote incorrect word problems, however, and of those eight, 7 incorrectly redefined the whole as in the following example: "I have 6/8 of a pizza in my fridge. I ate 2/3 of the pizza in my fridge. How much pizza do I have left?" This demonstrates that, although more than half of the PSTs demonstrated the ability to write a Separate (Result Unknown) problem, misconceptions related to incorrectly redefining the whole persisted.   When examining the incorrect and correct representations for fraction subtraction in context, it became evident that the choice of unit influenced the ability to make sense of the Separate (Result Unknown) problem structure. This is the second theme identified through analysis of the data.   Units of measure influence success with representing subtraction in context. While students continued to have difficulty using the Separate (Result Unknown) problem structure, the type of unit chosen for the whole (e.g. cups, gallons, pounds vs. pizzas, cupcakes, cookies, tanks of gas) seemed to influence the success PSTs had in writing word problems. Consider the following dialogue, which is a direct continuation from the dialogue in the previous section. Notice how students used the unit of measure, in this case a cup, to assist them in avoiding redefining the whole. Paige: So, if I have 4/5 of a cup of milk and I use 1/2 of a cup of milk. That would be this. Instructor: Would that be this (referring to the expression 4/5 ˗ 1/2)? Say it again. Paige: If I have 4/5 of a cup of milk and I use 1/2 of a cup of milk, so I could say, half of a cup of milk, not just half. Instructor: Half of the milk is what you said earlier. Paige: Half of the milk. Instructor: Right, I do not use half of the milk but 1/2 of a cup of milk, so that keeps them separate, that is how I see it in my head at least. Does that help you? [Noticing a student who seemed confused] Alex: A little. Instructor: Which one makes more sense to you, the pizza or the cup of milk? Alex: Say the pizza like this? - 14 -

Instructor: OK, say the pizza like this. Alex: Well I think I got it! Instructor: So teach this class right now! Alex: OK, I have 4/5 of a pizza. I ate half of [Prolonged pause here] does it have to be food? Paige: I don't think you can do that with food, can you? Instructor: OK, milk is food. So what is the difference between the pizza and the milk? Paige: Mine has a measurement like a cup of milk. Instructor: OK. Paige: And the pizza is pizza, it is not like a cup of pizza. Instructor: OK, so what is your whole? Paige: I can't figure that out.   Even when the students seemed to make sense of their errors in redefining the whole, they had trouble with the language used to define it. Students were more successful representing problems when they used a standard unit of measure such as a cup rather than a unit that was less clearly defined, such as pizza, because they could visualize the whole more clearly. Making sense of the whole is crucial as it is at the center of the concept of fraction subtraction. NCTM posits that, "the concept of 'unit' is fundamental to the interpretation of rational numbers" (2010, p. 19). According to Smith (2002), this is due to a detachment between paper and pencil representations of fractions and mental visualization because, "both are acts of mental construction, but only one is visible" (p. 6).   Students were actually shocked when they finally came to the understanding of the need to refer to the same unit for the subtrahend and minuend. Notice how the students are able to begin to verbalize the need for the same whole and the surprise at this understanding in the transcript that follows. Sarah: I have 4/5 of a gallon of iced tea. I am thirsty so I am going to drink ½ of a gallon of iced tea. How much iced tea will I have left? Instructor: Does that work? Sarah: Yes. Instructor: Why does it work? Sarah: Because I minus, the same Instructor: Because I subtract from the same whole. Any questions about that? Sarah: Oh my gosh! [indicating surprise and understanding]   On the survey and during the interviews prior to class instruction on fraction subtraction, students did not experience success with fraction subtraction regardless of the type of unit used. However, on the final examination, of the ten PSTs who correctly represented 6/8 ˗ 2/3 with a Separate (Result - 15 -

Unknown) structure, seven of them used a standard unit of measure (gallons, pounds, and cups) while three PSTs used units that would not be considered standard (pizza, pie, and cake). The following is representative of a response using standard units: I had 6/8 lbs. of fudge. I ate 2/3 lb. of that fudge. What fraction of a pound of fudge was left? Of the eight PSTs who incorrectly represented the same problem, seven used a nonstandard unit of measure while only one PST used a standard unit of measure. This provides additional support that the type of unit chosen for the whole seemed to influence the success PSTs had with writing Separate (Result Unknown) problems since a majority of the PSTs who wrote the problem correctly used standard units. The structure of the word problem also seemed to be a factor in success with writing problems as described in the next theme.   Problem structure influences success with writing problems.The PSTs in this study associated the operation of subtraction of fractions with the Separate (Result Unknown) structure rather than with the Compare (Difference Unknown) structure. The only question on the survey that included the Compare (Difference Unknown) problem section was 7/8 ˗ 1/4 (see Figure 1). In this task, the correct word problem to support 7/8 ˗ 1/4 was actually the Compare (Difference Unknown) problem. This question on the survey had the least amount of correct responses at 29%. The PSTs overwhelmingly chose the response that contained the Separate (Result Unknown) structure even though it was incorrect. This option received 53% of the responses. During the post survey interviews, we asked the PSTs to explain the reason behind their choice between the correct answer ˗ the Compare (Difference Unknown) structure ˗ and the most popular distractor ˗ the Separate (Result Unknown) structure ˗ without identifying to them whether they were correct or not. Only one PST switched from the incorrect answer to the correct answer and based that change upon a realization of the difference in problem structures between the Compare (Difference Unknown) and the Separate (Result Unknown), seemingly using her earlier experiences making sense of these problem structures with whole numbers to inform her work with fractions. All the other PSTs persisted with their misconception. A representative response from a PST who selected the incorrect answer was "I chose b (indicating incorrect answer). You know why I chose it? Because it was clear to me that it was 7/8 ˗ 1/4 because it says, 'how much cake does Mark have left?' and up here (indicating the correct answer) I remember, how much more pizza did Mark eat than Kenny. I wasn't sure that it was 7/8 ˗ 1/4 but this one (referring to the wrong answer) was very clear that it was 7/8 ˗ 1/4." With these students, the work with problem structures in whole numbers did not seem to transfer to work with fractions. It might be because their earlier school experiences focusing on key words was a stronger influence on their thinking making the Separate (Result Unknown) structure - 16 -

more dominant for them.   This theme of problem structure and keyword association was confirmed when we interviewed PSTs and asked them to create word problems for fraction subtraction expressions. PSTs were asked to write word problems for 4/5 ˗ 1/2 using a context having to do with pizza and 5/6 ˗ 1/3 using a context having to do with gallons of iced tea. Out of the nine PSTs, only one PST used the Compare (Difference Unknown) structure. This PST created word problems for both expressions using this problem structure (see Table 2) and wrote them correctly. All others used the Separate (Result Unknown) structure for both expressions and wrote them incorrectly. Table 2: Correct Responses for fraction subtraction during interview

  Even more interesting, the PST who used the Compare (Difference Unknown) structure as opposed to the Separate (Result Unknown) structure was the only one whose word problems correctly reflected the given expressions. All of the PSTs who used the Separate (Result Unknown) structure wrote incorrect word problems. These word problems that the PSTs created are reflected in the sample provided in Table 3. Table 3: Sample of Incorrect Responses for fraction subtraction during interview

All but one of the PSTs who wrote Separate (Result Unknown) word problems used the key words "left" or leftover" in their responses.   During instruction on whole number addition and subtraction, PSTs explored problem types according to those described in Cognitively Guided - 17 -

Instruction (Carpenter et al., 1999). Problem types continued to play an important role in discussion around word problems for fraction operations. When the instructor drew PSTs' attention to the problem type, they were able to change their problems to both reflect the Compare (Difference Unknown) structure as well as the given expression. Instructor: All of you wrote a separate result unknown problem. Remember those? What if you wrote a compare problem? ... Riley: Ella has 4/5 of a pizza. Ray has 1/2 of a pizza. How much more pizza does Ella have than Ray? Instructor: Does yours sound like that? Alex: Similar   It was interesting that once one student was able to make this jump from incorrectly representing the subtraction expression 4/5 ˗ 1/2 with the Separate (Result Unknown) structure to the Compare (Difference Unknown) structure, the rest of the class seemed to follow.

Discussion   According to the CCSS for Mathematics, "students must be given the opportunity to reason abstractly and quantitatively" (NGA & CCSSO, 2010, p. 6). This includes the need to decontextualize a problem situation as well as to recontextualize computations and solutions. Making connections with context inherently requires that there be a context from which to begin. The context could be generated by commercial resources or be developed by teachers and/or students. Regardless, the teacher's role must include making sense of the context. "Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects" (NGA & CCSSO, 2010, p. 6). Based on the results of this study, this level of reasoning may be difficult for teachers to facilitate without particular attention given to their own knowledge relative to contextualizing and decontextualizing fraction subtraction. In the case of this study, PSTs demonstrated difficult with creating word problems (contextualzing) given an expression as well as with identifying appropriate contexts from a list of options to support a given expression (decontextualizing). When contextualizing, there seemed to be an overreliance on the Separate (Result Unknown) word problem structure. In applying the Separate (Result Unknown) word problem structure, PSTs neglected the need to maintain the unit. Decontextualizing was assessed through the PSTs' need to examine - 18 -

word problems and mathematize them to determine if they correctly represented the expression given. Our conclusions are supported by the findings of Tobias (2009) and Dixon & Tobias (2013) who studied PSTs ability to decontextualize word problems and found that PSTs had difficulty determining if word problems represented fraction multiplication or fraction subtraction. In our study, in addition to difficulties contextualizing and decontextualizing, as PSTs reasoned quantitatively, it became clear that attention to the unit involved and focusing on precision were either problematic or held a key to success with the given tasks.   Similar to research with student-authored word problems (Alexander & Ambrose, 2010), misunderstandings that PSTs held were revealed through their self-authored word problems. These difficulties could inhibit their ability to facilitate student engagement with Standard for Mathematical Practice 2: Reason Abstractly and Quantitatively (NGA & CCSS0, 2010) as it did in Friske's (2011) case. According to Friske, her understanding of Compare (Difference Unknown) was sufficient but as the teacher, she also needed to be able to create contexts supporting other problem types. As such, and consistent with Shulman (1987) and Ball (1990), teachers' pedagogical content knowledge related to teaching mathematics must include an ability to both contextualize and decontextualize fraction subtraction. This supports similar research related to fraction division (Ma, 1999) and also must include attention to problem structures (Carpenter et al., 1999) for fractions specifically.

Conclusion   According to NCTM (2000), "Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn well" (p. 16). Additionally, teachers must provide students with the opportunities to demonstrate each of the Standards for Mathematical Practice specified in the CCSS document. For fifth grade, students must add and subtract fractions with unlike denominators and solve problems involving addition and subtraction of fractions (NGA & CCSSO, 2010). When students are only presented with traditional algorithms for adding or subtracting fractions (i.e. finding a common denominator), it is unlikely the students will have the conceptual understanding of fractions called for in the standards. This kind of deep understanding is only developed when teachers provide rich, meaningful learning activities that are deliberately chosen to meet the goals set by the teacher (Hiebert et al., 1997). The preservice teachers in this study had considerable difficulty selecting and authoring correct word problems to represent given subtraction contexts. Given this difficulty, and without intervention, it is unlikely that - 19 -

preservice teachers with these sorts of misconceptions will become teachers who will adequately support student engagement in reasoning abstractly and quantitatively or attending to precision regarding fraction subtraction. It is unlikely that connections to real word problems will be part of the teachers' planning and instruction. Without a deep understanding of both fraction subtraction procedures and situations that accurately model fraction subtraction, teachers are not able to both provide appropriate learning activities and assess student learning in meaningful ways. A goal, then, is to increase focus on PSTs self-authored word problems to support fraction subtraction as a means of making sense of common misconceptions and preparing PSTs for their future facilitation of students' reasoning abstractly and quantitatively.

References Alexander, C. M. & Ambrose, R. C. (2010). Digesting student-authored story problems. Mathematics Teaching in Middle School, 16(1), 27-33. Alibali, M. W., Brown, A. N., Stephens, A. C., Kao, T. S., & Nathan, M. J. (2009). Middle school students' conceptual understanding of equations: Evidence from writing story problems (WCER Working Paper No. 20093). Retrieved from http://www.wcer.wisc.edu/publications/workingPapers/papers.php. Ball, D. L. (1990). Prospective elementary and secondary teachers' understanding of division. Journal for Research in Mathematics Education, 21(2), 132-44. Ball, D. L., & Bass, H. (2000). Interweaving content and pedagogy in teaching and learning to teach: Knowing and using mathematics. In J. Boaler (Ed.) Multiple perspectives on the teaching and learning of mathematics (p. 83-103). Greenwich, CT: JAI/Ables. Barlow, A., & Cates, J. (2007). The answer is 20 cookies, what is the question? Teaching Children Mathematics, 13(5), 316-18. Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (1999). Children's mathematics: Cognitively guided instruction. Portsmouth, NH: Heinmann. Carpenter, T. P., Fennema, E., & Franke, M. L. (1996). Cognitively guided instruction: A knowledge base for reform in primary mathematics instruction. The Elementary School Journal, 97(1), 3-20. Common Core Standards Writing Team. (2011). Progressions for the common core state standards in mathematics (draft) 3-5 number and operations ˗ fractions. Retrieved from http://commoncoretools.files.storypress. com/2011/08/ccss_progresson_nf_35_2011_08_12.pdf. Council of Chief State School Officers (CCSSO) & National Governors As- 20 -

sociation (NGA). (2010). Common core state standards ˗ mathematics. Retrieved from http://www.corestandards.org/the-standards/mathematics. Daro, P., Mosher, F. A., & Corcoran, T. (2011). Learning trajectories in mathematics: A foundation for standards, curriculum, assessment, and instruction. Philadelphia, PA: Consortium for Policy Research in Education. Dixon, J. K., & Tobias, J. M. (2013). The "whole" story: Understanding fraction computation. Mathematics Teaching in the Middle School, 19(3), 156-163. Drake, J. M., & Barlow, A. T. (2007). Assessing students' levels of understanding multiplication through problem writing. Teaching in the Middle School, 19(3), 156-163. Fennema, E., Carpenter, T. P., Franke, M. L., Levo, L., Jacobs, V. R., & Empson, S. B. (1996). A longitudinal study of learning to use children's thinking in mathematical instruction. Journal for Research in Mathematics Education, 27(4), 403-34. Friske, M. L. (2011). Influence of using context supportive of the area model on sixth grade students' performance when writing word problems for fraction subtraction and multiplication. (Unpublished masters thesis). University of Central Florida, Orlando, Florida. Glaser, B. G., & Strauss, A. L. (1967). The Discovery of Grounded Theory: Strategies for Qualitative Research. New York: Aldine De Gruyter. Gravemeijer, J. (2004). Local instructional theories as a means of support for teachers in reform mathematics education. Mathematical Thinking and Learning, 6(2), 105-28. Hiebert, J., Carpenter, T. P., Fennema, E., Fushon, K. C., Wearne, D., Murray, H., Olivier, A., & Human, P. (1997). Making sense: Teaching and learning mathematics with understanding. Portsmouth, NH: Heinemann. Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates. National Council of Teachers of Mathematics. (2010). Developing essential understanding of rational numbers, grades 3-5. Reston, VA: NCTM. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM. Common Core State Standards Initiative (CCSSI). (2010). Common Core State Standards. Washington, D.C.: The National Governors Association Center for Best Practices and the Council of Chief State School Officers. http://www.corestandards.org/the-standards. National Research Council. (2001). Adding it up: Helping children learn mathematics. J. Kilpatrick, J. Swafford, and B. Findell (Eds.). Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Eduction. Washington, DC: National Academy Press. - 21 -

Sharp, J., & Adams, B. (2002). Children's constructions of knowledge for fraction division after solving realistic problems. The Journal of Educational Research, 95(6), 333-47. Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14. Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57(1), 1-22. Smith, J. P. (2002). The development of students' knowledge of fractions and ratios. In B. H. Litwiller and G. Bright (Eds.), Making sense of fractions, ratios, and proportions, 2002 yearbook. Reston, VA: NCTM. Tirosh, D. (2000). Enhancing prospective teachers' knowledge of students' conceptions: The case of division of fractions. Journal for Research in Mathematics Education, 31(1), 5-25. Tobias, J. M. (2009). Preservice elementary teacher' development of rational number understanding through the social perspective and the relationship among social and individual environments. (Unpublished doctoral dissertation). University of Central Florida, Orlando, Florida. Whittin, P., & Whittin, D. J. (2008). Learning to solve problems in the primary grades. Teaching Children Mathematics, 14(7), 426-32.

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Investigations in Mathematics Learning ©The Research Council on Mathematics Learning Fall Edition 2014, Volume 7, Number 1

Responsive Teaching from the Inside Out: Teaching Base Ten to Young Children1 Susan B. Empson

The University of Texas at Austin [email protected]

Abstract

    Decision making during instruction that is responsive to children's mathematical thinking is examined reflexively by the researcher in the context of teaching second graders. Focus is on exploring how the research base on learning informs teaching decisions that are oriented to building on children's sound conceptions. The development of four children's understanding of base ten over a ten-week period is tracked. "The work of teaching orients teachers to constantly consider their next moves" (Jacobs, Lamb, Philipp, & Schapelle, 2011, p. 98).   A recent agenda-setting document called for more attention to how research on learning can be used by teachers. Its authors argued that there is no set of materials or technology that "can replace careful attention and timely interventions by a well-trained teacher who understands how children learn mathematics" (Daro, Mosher, & Corcoran, 2011, p. 15). Such teachers would .... get students to reveal where they are in terms of what they understand and what their problems might be. They have to have specific ideas about how students might progress ... and how they might be expected to go off track or have problems. And they would need to have, or develop, ideas about what to do to respond helpfully to the particular evidence of progress and problems they observe (Daro, Mosher, & Cocoran, 2011, p. 15). In a nutshell, responsive teaching entails taking into account the evidence provided during instruction about children's thinking and its advancement. Acknowledgement: Sincere thanks to Vicki Jacobs for feedback on this article as I was preparing it, Luz Maldonado for enriching conversations about children's thinking, and the four childrn with whom I worked and their teachers for inviting me into their school and agreeing to work with me.

1

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To teach in ways that are responsive to children's mathematical thinking, teachers need to elicit children's thinking, interpret this thinking, and then "respond helpfully." They need to understand the mathematics children are to learn and know what progress in learning the mathematics look like. And they need to make decisions, often quickly in response to children's thinking during instruction.   In spite of its intuitive appeal, responsive teaching is not widespread or well understood (e.g., Hiebert, Gallimore, Garnier, et al., 2003; Kennedy, 2005). To appreciate the nature of this work, I decided to explore responsive teaching from a first-person perspective by immersing myself in the work of teaching a small group of children over an extended period. My goal was to document what was involved in making teaching decisions that were responsive to children's mathematical thinking during instruction (Carpenter, Fennema, Peterson, Chiang, & Loef, 1989; Fraivillig, Murphy, & Fuson, 1999; Jacobs & Ambrose, 2008; Jacobs, Lamb, Philipp, & Schapelle, 2011; Philipp, Jacobs, Lamb, Bishop, Siegfrid, & Schappelle, 2012). How would I respond "to the particular evidence of progress and problems" I observed? In what sense would my decisions be research-based and in what sense not? What would be involved in using the research on children's mathematical thinking as I was interacting with children?   In this paper, I present a first-person narrative of my work with a group of children over a ten-week period, in which I foreground the challenges I encountered and the kinds of decisions I made to address these challenges. The children with whom I worked were four lively second-graders named Sunny, Daniella, Jack, and Emilio. These children were characterized by their teachers as lacking number sense and, generally, as having difficulties in mathematics. Out of all of the second graders in their school, they had scored the lowest on a benchmark mathematics test. My work with them centered on base-ten concepts and problem solving. We did not start out with this focus; after a short time of working with the children, I decided that their lack of number sense could be more precisely characterized as a lack of usable understanding of base ten. In my decision making I strived to balance sensitivity to the children's current understanding, as I interpreted it, with movement toward the "mathematical horizon" (Ball, 1993), which required balancing multiple, sometimes contradictory, goals. I wanted to document what was involved in drawing upon research into children's mathematical thinking to guide my decision making while also honoring these four children's emerging mathematical understanding.   This report can be read on two levels. On one level, it is an account of four children in the midst of advancing their understanding of base ten, whole-number operations, and problem solving and the teacher's researchbased decision making in service of that advancement. It would be useful for anyone wondering how to support children to learn base ten, and could - 24 -

provide insight into why some children in later grades struggle with more advance operations, such as multi-digit multiplication or subtraction with regrouping. On another level, it is a case study of responsive teaching, from the perspective of a teacher's decision making. As such, it does not make claims about what responsive teaching should look like for other teachers with other children. It provides an account of this approach to instruction in a particular domain ˗ base ten and whole-number operations ˗ involving four second graders and considers, more generally, how teacher decision making might draw on research-based findings about children's thinking in the midst of interactions.   The report is organized chronologically and reflects the threads of understanding to which I was attending in my work with the children. Children's mathematical thinking and its details is foregrounded, to provide an opportunity for teachers to think about how they might respond to similar evidence of children's thinking in their own classrooms. Not all the details of what we did are included. Rather, I include the details that stood out to me as informative during my interactions with the children and which became pertinent to my decision making.

Method   The four children with whom I worked were chosen by teachers on the basis of a mid-year paper-and-pencil mathematics assessment given to all second graders at the school. They had the four lowest scores out of all second graders at the school and their teachers believed they needed additional help in mathematics. Before meeting with the four children, I met with their teachers to learn about the children, although most of what I learned came from my interactions with them during our sessions together.   We met once per week for ten weeks, after school for one hour. During these sessions, I took field notes on the children's thinking and our intentions. After each session, I spent an additional hour alone elaborating the notes and supplementing them with additional information about what happened during our time together. My focus in these research notes was to document my own decision making and the evidence on which it was based (Lampert, 1998). I also often used this time to plan the next session's problems.

Teaching Sessions and Findings Session 1: We Begin Our Work Together   I did not know Sunny, Daniella, Jack, and Emilio before my work with them. To prepare for our first session together, I wrote a set of word prob- 25 -

lems that involved addition and subtraction and represented four distinct problem structures (Carpenter, et al., 1999, in press), including a mix of single-digit and double-digit quantities. I wanted to spend the first session finding out about what the children could do, what they understood, how they expressed it — both orally and in writing — and how confident they were. Not knowing what to expect, I started with fairly small numbers, but made sure to include some double-digit numbers Table 1: Problems given during first instructional session to find out what the children understood.

The first problem that I posed to the children was a Join Result Unknown (Table 1). The children either counted up from 13 to solve it or direct modeled it with cubes or pictures. Nobody used a more advanced strategy such as derived or recalled facts. (Direct modeling strategies involve the physical representation of each item in a story problem, such as with cubes or marks on paper, and the manipulation of these representations in a way that follows the structure of the story. Number-fact strategies represent an advance over direct modeling. A derived fact strategy for this problem would be something like, "I know that 13 plus 7 is 20, plus 1 more is 21." [Carpenter et al., 1999]).

Figure 1: Daniella's written strategy for 28-31.

  The second problem we worked on was a Separate Result Unknown (Table 1). Daniella solved this problem by writing 28 ˗ 13, vertically (Figure 1). - 26 -

She separated the tens and ones into two columns, and subtracted the ones first, then the tens. Her use of this procedure made me curious about what she understood about base-ten concepts, as opposed to the procedure she had used. I began to listen for evidence of base-ten understanding among all the children and noticed that, even though the problem involved doubledigit numbers, none of the children had used base-ten concepts in their strategies. I was listening in particular for children's use of ten as a countable unit ˗ a shorthand way of describing children's understanding that 10 ones can be represented as one 10 (Carpenter, et al, 1999, forthcoming; Cobb & Wheatley, 1988; Fuson, Wearne, Hiebert, et al., 1997). This concept is distinct from the place-value concept in which the value of a digit is determined by its placement in a number ˗ for example, in the number 28, the 2 represents 2 tens. However, understanding the meaning of 2 tens depends on understanding 10 as a unit and thinking flexibly about 2 tens as 20 ones or 1 ten as 10 ones.   I decided to create a third problem, on the spot, to explore children's understanding of base-ten. After ascertaining that they all knew about and liked soccer, I posed a problem that involved groupings of ten: "You've got 3 big bags of soccer balls. Each bag has 10 balls in it. You've also got 2 loose balls. How many balls do you have?" I used hand gestures to indicate the bags were big and repeated the problem to be sure the children heard it. They set to work. Everyone but Emilio was direct modeling the problem by drawing all the balls individually (Figure 2).

Figure 2: Drawing all the individual balls

  There was no use of ten as a unit! Emilio did not appear to be doing anything, so I repeated the problem for him. "Oh," he said. "10 plus 10 is 20." I was so pleased with this insight that I emphasized to him ˗ and for the benefit of the others ˗ that he did not even need to draw any pictures to figure it out. It prompted Jack to remember that he, too, knew that 10 plus 10 was 20. - 27 -

  However neither boy was able to use his knowledge of this fact to solve the problem. Sunny was not sure whether to add or subtract the two loose balls, so I told her she had to decide for herself what made sense; she decided to subtract because, she said, the two "loose" balls could roll away. Emilio got 30 for his answer and when I asked him how he was going to count the 2 loose balls, he changed his answer to 31. And Jack got 28, because one of his bags had the wrong number of balls in it.   There was so much to talk about, but it was time to go, so I made a note to myself to return to problems like this one next time. Session 2: I Discover a "Lack of Number Sense" is Really a Lack of Base-Ten Understanding   I had a hunch that what seemed like a lack of number sense for some of the children was actually little-to-no understanding of base ten. I decided that I wanted to find our more about what the children understood about groupings of 10 and 10 as a unit. I wrote four problems for this session that were intended to both assess and develop base-ten understanding (Table 2).   Children who understand 10 as a unit find a problem in which they have to calculate four groups of 10 easy (e.g., 4 rolls of 10 candies each). They might count by tens to solve the problem or immediately realize that 4 tens is 40. Those who do not understand 10 as a unit would find such a problem just as difficult as any other grouping problem. For example, they would solve 4 groups of 7 in the same way as they would solve 4 groups of 10.   Emilio's thinking about the Valentine's Day problem showed no base-ten understanding. First, he interpreted the context to mean he should add 10 and 4 to get 14 rolls altogether. After questioning him, unsuccessfully, about why he added, I described a context where he was the candy maker and had to put 10 candies into each of 4 boxes. The librarian handed us a roll of Table 2: Problems given during the second session to assess and develop children's understanding of ten as a unit.

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sweet tarts to help Emilio visualize the 10-candies-per-1-roll relationship. I then left him to solve the problem; he solved it by drawing each box and putting 10 single cubes in the interior of each. He counted his answer by twos and got 40. Success, of a limited sort then: he used his knowledge of the context to construct a solution. But he did not use any of the knowledge of tens that was in evidence last week.   The other children performed similarly. I felt reinforced in my hunch that much of their current difficulties in mathematics could be traced to underdeveloped knowledge of ten as a unit and decided to focus on developing this understanding over the next few weeks. At the same time, I would be working on helping the children increase their sense of agency in problem solving and their capacity to "make sense of problems and persevere in solving them" (CCSSM, 2010, p. 6). Session 3: I Accidentally Make the Problems Too Difficult   In my excitement about discovering a possible source of children's difficulties in second-grade mathematics, I wrote a set of problems (Table 3) to address these difficulties that, in hindsight, were too difficult. My aim was to further assess and begin to develop children's understanding of ten as a unit, so I wrote a set of problems that all involved multiples of ten. Table 3: Problems given during the third session to further assess and begin to develop children's understanding of ten as a unit.

Both Sunny and Daniella had some trouble with the pennies problem. Each used unifix cubes in sticks of 10 to build 22 (Figure 3), which suggested to

Figure 3: Daniella's model of 22 - 29 -

me that they have some understanding of using 10 as a unit to model 22. But after this auspicious start, they were stumped about how to build on to 22 and to get to 50. When I changed the 50 pennies in the problem to 30 pennies, Daniella quickly solved it by counting up from 22 to 30 by ones.   Jack solved the pennies problem handily, although his strategy made no sense of tens. He counted up by ones from 22 using tallies to keep track.   I noticed several of the children using vertical notation for double-digit problems but not making use of tens in their solutions. For instance, Jack wrote a vertical number sentence for the chocolate chips problem (40-15), but actually solved it by drawing the 40 individual "chips" and crossing out 15 of them (Figure 4).

Figure 4: Jack's vertical number sentence for 40-15.

  Daniella wrote something similar for the cards problem (60-26) and got 4 for her answer - a classic "bug" which suggests she does not understand, or is at least not making use of, base-ten concepts.   I decided for next time that I needed to choose numbers that were more conducive to using 10 as a unit. For example, having a start number of 20 for the pennies problem, instead of 22, would make it easier for the children to build up from the start number using unifix cubes in stacks of 10. I also decided to continue to provide the children with materials that are structured in tens (e.g., unifix cubes in sticks of tens or base-ten blocks) and to urge them to use these materials to solve problems. We would reflect on their strategies and record them using numbers to help the children connect the base-ten structure of the materials with their number symbols. I expected that developing an understanding of base-ten concept in a way that these concepts are usable in their strategies would take some time (Hiebert & Wearne, 1996). Session 4: I Find Evidence of Emergent Understanding of Base Ten   For the next session, I wrote a set of problems without a story context to find out if children could connect numbers and context (Table 4). It seemed - 30 -

from previous sessions that the children had a basic understanding of addition and subtraction. Because they have been using strategies based on ones but not tens I used multiples of ten in these problems. Before the children started working on these problems, I made sure each child had 70 unifix cubes in groups of ten in front of him or her, in stacks of ten. Table 4: Number sentences for Session 4 Problems

  I began by holding up 3 sticks of 10 unifix cubes (Figure 5), and asking the children how many I had. All but Emilio, who was sharpening his pencil, said 30. Then I held up 52, in 5 tens and 2 ones. It was a little harder for them to see, but basically they understood the tens and ones combination. (Sunny saw 42, Daniella saw 51, and Jack saw 52.) Yet their understanding of the base-ten structure of double-digit numbers seemed fragile, because they used very little of that understanding to solve the multidigit addition and subtraction I posed to them that day.

Figure 5: Three groups of 10 cubes each.

  I asked the children to begin by providing a story for the first number sentence, 30 -12 = ___. The story they came up with went like this: "Jack and Daniella went to the candy store and bought 30 pieces of Valentine's gum. Jack ate 1 piece and Daniella ate 11 pieces. How many pieces did they have left?" - 31 -

  The children used a variety of incorrect and correct strategies to solve the problem. Emilio solved it by counting back by ones, with no miscount this time. Jack solved it, as usual, by direct modeling by ones — he made 30 tally marks, and crossed out 12 of them. Both boys got 18.   Daniella solved it using the same common buggy algorithm that she had used the week before; her original answer, erased, was 22 (Figure 6).

Figure 6: Daniella's buggy algorithm, with original answer (22) erased and replaced with the correct answer (18)

  When she heard that Emilio and Jack had gotten 18, she erased her "22" and wrote "18."   Despite the story frame that she helped create, Sunny first added 30 and 12. But when I reminded her of the story she easily figured she would subtract.   Reflecting on each other's strategies. I decided to have the children listen to each other's strategies as a way to move their thinking forward. Although they can count by tens and can identify group of tens, they do not readily use this knowledge to solve problems; it is not very flexible knowledge for them. So my goal was to use the group discussion to help them begin to make connections and develop this base-ten understanding.   I had a big piece of newsprint that we could all easily see. I asked Jack to share his strategy first, because it was basic direct modeling. I represented his strategy using tallies. Sunny had the idea of grouping the tallies into tens to make them easier to count. I grouped the tallies and everyone said it was 30 (Figure 7). This was consistent with the very first quick activity we had done with the unifix cubes and it seemed to be a good way for the children to develop an understanding of the ten-ones-is-one-ten relationship.

Figure 7: My representation of Jack's direct modeling strategy

  But when it came to subtracting 12 from this group of 3 tens by subtracting 1 group of ten and then 2 ones, the difficulty of applying knowledge of - 32 -

tens became apparent. None of the children knew spontaneously what 30 take away 10 was. Daniella said 29 but it seemed to be a guess. I told them to use their unifix cubes to figure it out. They did, easily.   The case with which the children solved this problem using manipulatives suggests a clear cognitive distinction between modeling with tens, as they had done, and working with ten as a unit mentally, which they did not do. This distinction is consistent with the research literature (Fuson, et al., 1997; Sophian, 2007; Steffe & Olive, 2010). I decided to continue working with the children to make connections like the ones they made this day between ones grouped into tens and ten as a unit. I believed that repeated opportunities to create and reflect on ten as a unit in their strategies would pay off. Session 5: I Use Money as a Context and Some of the Children Have Difficulty   We continue to work on building a flexible understanding of ten as a unit in this session. I started by reading aloud a book called Only One (Harshman & Garrison, 1993), which emphasizes the mathematical big idea of thinking of several things as one thing. We discussed the big idea that one thing can be the same amount as many things, such as one dozen is 12 eggs and one dime is 10 cents. They seemed to understanding this idea in the abstract, and it gave us a point of reference when talking about tens and ones: "Oh, you mean a dime is the same as ten cents!" We then moved on to some addition and subtraction story problems to continue to work on developing ten as a unit (Table 5). As it turned out, the money context posed some special problems of its own. Table 5: Problems for Session 5

Emilio and Jack both solved problems in ways that showed me they are building an understanding of ten as a unit. I was especially happy to see Jack represent five dimes with 20 cents subtracted out by drawing circles for dimes (Figure 8). In my sessions with him before today, he has been - 33 -

representing tens with ten tally marks or something similar, so to use one circle to represent 10 things was a real advance! I was hoping that a dime would have for the children a "one-ness" and also a "ten-ness."

Figure 8: Jacks representation of 5 dimes take away 20 cents

  However, both Daniella and Sunny struggled with these problems. In fact, Sunny did not solve a single one. She represented the 5 dimes in first problem with 5 cubes. We talked about how much 20 cents was; she knew it was 2 dimes. But when I left her alone to work on the problem, she took all of the tens she had (unifix cubes) and broke them into ones to represent the stars in the sky that Emilio, in the story problem, had decided to buy. Daniella seemed confused about dimes and cents as different units, and how they related. She wrote "Emilio has 0 money now," because 5 of something take away 20 of something leaves you with, at most, 0 (Figure 9). Interestingly, she represented the dimes as units with the numeral "10" on it, suggesting she may know that one dime is 10 cents; but she did not seem able to use that knowledge to solve the problem.

Figure 9: Daniella's written strategy for 5 dimes take away 20 cents

  I think these problems were just right for Jack and Emilio, but too hard for Sunny and Daniella. For children whose knowledge of dimes and other coin denominations is not easily accessed as they reason, these problems must seem like multi-step problems with one of the steps left out ─ something like this: Henry has 4 packages. He eats 6 cookies. How much food does he have left? It does not make sense without the crucial information of how many cookies per package.   I wondered what we should do next. I knew that simply telling or showing these children that one ten is the same as ten ones would not be enough to help them learn to use this knowledge in problem solving. It is a difficulty - 34 -

concept for young children, although the fact that it is a sophisticated mathematical idea is not obvious. I decided to continue to engage the children in problem solving with explicit discussion of how they were using tens and to write problems that involved units of 10 and units of one, to help the children learn to coordinate the two different units. I also planned to engage them in more comparisons of each other's strategies and the differences and similarities in how tens are used. Table 6: Problems for Session 6

Session 6: Progress!   I wrote more problems involving multiples of tens, using bigger numbers in hopes that the children would find the use of individual tallies tedious. It worked, for some.   The third problem, which involved 11 packages of 10 cookies each and 5 extra cookies, proved interesting and productive. It was an appropriate challenge for most of the children. Emilio and Jack both started out by adding up the numbers to get 26 for their answer. Jack then decided on his own that 26 was not the correct answer, and easily direct modeled the problem by drawing groups of 10 ones (Figure 10).   As he was drawing all of this out, I seized an opportunity to extend his thinking. I asked him if, instead of drawing each cookie in every package, he could represent the cookies in the package by writing the numeral "10" in each one. He said that no, he could not; it would be too hard. So I let

Figure 10: Jack's drawn strategy for 11 groups of 10 cookies plus 5 extra cookies - 35 -

him continue to model by ones. But I noticed when he counted the total he counted by tens instead of by ones, so again I asked him if he could represent the cookies by writing "10" in each package instead of drawing each individual cookie. I pointed out to him that the had just counted each group by tens. It seemed to make sense to him this time so I gave him a new but related problem to solve, encouraging him to use this new strategy. The problem was 14 packages of cookies, 10 in each package, and 10 extra cookies. He began, as usual, by representing each individual cookie (Figure 11).

Figure 11: Jack's drawn strategy for 14 groups of 10

He said he forgot to use the strategy we had talked about (and only remembered when his hand started getting tired). Because he had so easily solved this problem, however, I felt sure he could use the more abstract counting approach. So again I posed a new but related problem: How many cookies would be in 12 packages? When I came back, I saw he had successfully represented each package of cookies with a single mark (the numeral 10), rather than drawing each individual cookie (Figure 12).

Figure 12: Jack's drawn strategy for 12 groups of 10

He agreed that this strategy was faster, as well as easier on the hand.   Emilio misinterprets the problem and I try to get him to listen to Daniella's strategy to change his mind. Emilio had trouble getting started on this problem. It is not clear to me why. His initial answer was 26, which he got by adding 11, 10, and 5. I asked him why he decided to add them altogether and whether they were all cookies or packages, but he gave no clear answer. - 36 -

  Daniella, like Jack, direct modeled the entire situation by representing each cookie, but she confounded packages of cookies with single cookies (Figure 13) − just as she had done with dimes and pennies the week before. At first she counted her answer on by ones from the first package of 10, then she went back and counted the total of tens. (She got 160, which she later erased.)

Figure 13: Daniella's strategy for 11 groups of 10 cookies plus 5 extra (with the 5 extra cookies drawn as packages and then erased)

  Because she had accurately represented the packages of cookies and Emilio had not, I decided to ignore her confusion about the 5 extra cookies for the time being and called Emilio over to compare what he was doing with what Daniella was doing. The first difference he saw was in how each of them had represented the package. His was more "realistic" (Figure 14). Daniella saw that he had six cookies in his packages and she had put 10 in each of hers. With some prompting from me to speak directly to Emilio and not me, she was further able to tell Emilio why she had drawn her packages this way. Emilio decided to start over, and at my suggestion, gathered a

Figure 14: Emilio's invalid strategy for 11 groups of 10 cookies plus 5 extra - 37 -

bunch of sticks of unifix cubes in tens. He ended up with 34 sticks of 10 arrayed in front of him but did not solve the problem before it was time to go.   Sunny thinks really hard. Like Jack and Daniella, Sunny started out drawing the packages of cookies with each individual cookie represented. I encouraged her to use the cubes in sticks of ten instead, anticipating that the ten-to-one structure might support a more sophisticated strategy. She decided she wanted a bunch of sticks of four. I was not sure where she was going with it, but decided to let her create them, and even helped her. When we were done, she gathered sticks of 10 and used the sticks of four to stand for the packages! (The number of cubes in each stick of four was probably irrelevant; perhaps the long rectangular shape reminded her of a package.) Then she put 10 cookies ─ a stick of 10 ─ in each package (Figure 15) and counted the total beautifully by tens. During all of this she mentioned that she was thinking so hard she could not even think of anything else. I thought it was a keen observation because at the end I had to remind her of the 5 extra cookies. She included them but counted them as tens, as Daniella had at first.

Figure 15: Sunny's strategy for 11 groups of 10 cookies

  Looking forward. With encouragement, then, Jack, Daniella, and Sunny were able to move from counting by ones to counting by tens. Jack was able to represent groups of 10 by something other than a collection of 10 things ─ a real advance, if he sustains it. I was not sure that Emilio could do or how much he understood of problems like this one. He had solved them in the past. His focus on that day seemed divided so I think these problems did not get his full attention. In fact, he started out wanting to do his spelling homework.   To take advantage of the progress the children made, I decided that in our next session, we would solve more equal groups problems involving groups of 10, like the first and third problems and I would continue to push the children to represent sets of 10 with the numeral 10. Having counted 10 as a unit, I thought that the children would be more likely to use it to solved problems if the quantities were smaller. - 38 -

Session 7: Frustration!   For this session, I used a story context, the Candy Factory, that involved units that were powers of ten. At the Candy Factory (McClain, Cobb, & Bowers, 1998) candies are made, which are packed into rolls of 10 candies each; rolls are packed into cartons of 10 rolls or 100 candies each (Figure 16).

Figure 16: At the Candy Factory, the candy packing machines puts 10 candies in each roll and 10 rolls in each carton. Table 7: Problems for Session 7, in which a roll contains 10 candies

multi-step

  I began by posing some quick problems just to check for understanding of the context, I asked how many candies were in 2 rolls, then in 5 rolls. Jack and Sunny both counted by tens to figure these problems out. Emilio too seemed to understand, although looking back, and knowing what he did later in the session, I am not sure now. (Daniella was absent.)   I limit there use of tallies. I began by reminding them how sometimes they solved problems by making single tallies (Figure 17) and told them that today, I did not want them to use tallies like these. They could use the unifix cubes in sticks of ten or use numbers written on their paper or solve it mentally. My purpose was to push them to work with ten as a unit. Al-

Figure 17: My illustration of using tallies to solve a problem - 39 -

lowing unifix cubes meant that if they needed to count by ones they could; but the structure of groupings of ten would at least be something they had to choose to ignore.   Jack sustains the progress he made last week. As it turned out, Jack began the problem by drawing the rolls without candies (Figure 18a). He then decided to represent the individual candies in (Figure 18b).

(a) Jack draws rolls without candies at first

(b) Jack presents individual candies in each roll but the last Figure 18: Jack's drawn strategy for 6 groups of 10 candies plus 10 individual candies

  Just as he was up to the last roll, I asked him if he needed to show those candies in order to count them. This conversation was just like the one we had the week before. He quickly said "no" and wrote "10" in the last roll. When I asked him later to write a number sentence or write numbers to show how he solved it, he wrote: 10, 20, 30, 40, 50, 60, 70. Progress!   It's not clear what Sunny understands about ten as a unit. Sunny was slow getting started. She seemed to be confusing the idea of 6 rolls with rolls of 6. She easily modeled the 10 loose candies with 10 single cubes. But for the rolls she had a stick of 6 unifix cubes and described it as "a roll of 6." I clarified: "6 rolls of 10, not a roll of 6," and she was off, modeling the rolls with 6 sticks of 10. There was some confusion about how to count the total of 6 sticks of 10 and 10 loose ones; she got 16 at first, but with a discussion in which I asked her to connect it back to rolls and candies, she counted appropriately. I emphasized in my revoicing of what she had done that she could count the rolls ─ 1, 2, 3, 4, 5, 6 ─ or count the candies ─ 10, 20, 30, 40, 50, 60. (Plus the loose ones, which no one has any trouble counting.) Success!   I am frustrated with Emilio. Emilio solved the first problem by adding 6 rolls and 10 candies and got 16. I asked him to solve it a second way, and - 40 -

he drew a stick of 6 and a stick of 10, and counted all to get 16. I asked him to talk with Jack about his strategy, and listen to how Jack solved his, but he declined to do those things. I asked him if the problem was too hard for him, but he did not answer, working instead on figuring out what time it was and when he could go home. I asked Sunny to explain her terrific direct modeling strategy, hoping he would see the difference between 6 rolls (sticks of ten) and 10 loose candies (individual cubes). It seemed like he looked everywhere but at Sunny or her strategy. At each step of Sunny's explanation, I stopped her to ask Emilio a question, trying to get him to make a connection between the cubes arranged in sticks of tens and rolls of candy, trying to get him to make sense of the problem. I judged he was deliberately not engaging.   Maybe, I decided I did not want to assume that he was deliberately avoiding work. Perhaps it was his way of expressing boredom or confusion; maybe he was preoccupied with a personal problem more important to him than counting candies in rolls. So finally, as it was nearing time for our session to be over, I asked him if he wanted me to make him an easier problem. He said he did. So I turned his paper over and wrote "2 rolls, 10 candies, how many candies?" "12," he quickly replied. So I asked him to use cubes to show the rolls and the loose candies. "How many candies in one roll?" I asked him. He put his head down and said he was ready to go home. Feeling resolute, however, I told him he could not leave until he solved this problem. I was remembering the fact that he had solved problems like this in the past. "How many candies in 2 rolls then?" I asked. "20?" he squeaked out, with his head buried under his arms. "So," I continued, "if you put 10 more candies with them─why don't you represent those 10 candies with these cubes, any way you want." He picked up a few sticks of 10 and began to put them with the 2 "rolls." "Now," I pressed him, "show me the 10 loose candies." It took a while but, finally, he pulled one stick of 10 out of what he had grabbed, and put it with the 2 rolls. "How many?" I asked. "30," he said, without even counting.   In hindsight, I do not know if I was engaged in a power struggle with Emilio or helping him make a cognitive leap. I wondered what would be the residue (Hiebert, et al., 1996) of this interaction for Emilio? What did he take away from it? A new understanding of ten as a unit? A feeling of confidence that he can solve problems? A feeling of being forced to do something he did not want to do? I think the answer to that question − which I simply do not know − is much more important than the fact that he answered "30" in the end. Session 8: I work on Extending the Children's Thinking   We continued our work with the Candy Factory again and the use of number sentences to represent the situation. - 41 -

Table 8: Problems for Session 8

  Emilio does something different. As the children worked on the new Candy Factory problems, they fell into their usual patterns, with the exception of Emilio. I sat with him first to get him started. He read the problem to himself then I asked him rephrase it in his own words out loud. No problem; he remembered the quantities in the Candy Factory. I asked him how many candies in a roll; he said 10. So he understood the context and the problem parameters. "So," I asked him, "how many candies does Dr. E. have?" His first response was 40 because, he spontaneously gave the reason, there's candy in the four rolls. When I asked him about the 11 loose ones, he got 52, at first, because he added 10 on to the 40 (nice work!), and counted up somehow to get 52. When I asked him why he added the 10 on, he did not say, and ended up solving the problem by counting up from 40 by ones. What a terrific solution! In contrast to his thinking last week, he did not seem to have any problem distinguishing groups of 10 from singletons; and once he understood the context, he had no problem applying his knowledge of multiples of 10.   I was puzzled by how easily this strategy came to Emilio, considering the struggle the last time we met. I wondered how much of his success, or lack of it, was based on whether he is preoccupied with something more pressing or more interesting than the problem at hand; after all, when our attention is divided, our capacity to reason is compromised (Glenn, 2008).   Jack uses numerals to represent his thinking. Jack direct modeled by representing the groups of 10, showing each candy. But just as he has been doing, he counted the solution by 10s. He has shown that he does not really need to do this and so I asked him to write a number sentence that showed how he solved the problem. He wrote: 10, 20, 30, 40, 10, 1—showing the quantities separately, and not how he combined them by counting tens. I asked him to write another sentence using plus and equals like we had been doing, and he wrote 51 = 52 - 1; 51 = 53 - 2; 51 = 54 - 3; 51 = 55 - 4 — not, as I was hoping, 10 + 10 + 10 + 10 + 11 = 51. I wondered if it would make a difference if I asked him to show the rolls and candies with a number sentence? My goal was for him to articulate (verbally or symbolically) how 51 was related to groups of 10. I kept this goal in mind for our group discussion.   Dramatization helps Sunny. Sunny, as before, seemed to have trouble getting started. She confused rolls and candies, and at one point said there - 42 -

were 10 rolls, instead of 4. She also did not combine the rolls and the loose candies at first. Although her strategy was not clear to me, I think she separated out the 11 loose candies from the 4 rolls. I thought that animating the situation for her, and in particular, putting her in the problem with me, might help her visualize the context. So I dramatized the problem with her as a character asking Dr. E. about the candies she had, just as Jacobs and Ambrose (2008) reported teachers do. It worked. She decided that she needed 4 rolls of 10 and the extra 11, and counted them all by ones to get 51.   Making connections. At this point I decided to gather the children together for a group discussion. They had three different strategies but all of them had in common the use of tens in some way. A number sentence might help tie together the ideas that were in each of these strategies, and extend the children's understanding (Jacobs & Ambrose, 2008). I asked Jack to report his strategy. As he was talking about the sticks of ten, I wrote "10, 10, 10, 10" to represent what he had drawn and to make a connection. Then on big paper, we followed through with: "10 + 10 + 10 + 10 + 11 = 51" to represent the entire situation. I was anticipating that writing the number sentence in reference to the concrete strategy would help children build a connection between the quantities in the problem and the mental use of 10 as a unit. Session 9: I Focus on Number Sentences to Reinforce Children's Understanding   I decided to introduce true/false number sentences (Table 9), to help the children begin to use mental strategies involving tens, and to help them become comfortable working with symbols (Carpenter, Franke, & Levi, 2003). As Jacobs, Franke, Carpenter, and Battey (2007) described in their work with elementary students, I wrote these number sentences on index cards so that I could easily shuffle through them as I as working with the children. Table 9: Problems for Session 9

  I started by asking the children to make up a problem they could solve in their head. I was hoping they would choose numbers that they could mentally manipulate; but as I found out, to do so requires some metacognitive awareness about what is easy and what is hard relative to their own under- 43 -

standing. My example for them was 10 plus 10 plus 10. I was not subtle at all; I wanted the children to use tens in their problems. As usual, however, they had other things in mind.   Sunny wanted a story to go with her number problem. So she made up her own: "Sunny and Daniella go to the store and buy 90 candies. They eat 8 of them. How many are left?" I suspected the problem was too hard for her to solve in her head ─ and it was. She got 80. It seemed she was trying to do something in her head with the standard algorithm, because she mentioned "crossing out." But when I suggested that she solve it another way using the cubes, she easily modeled the problem using tens, and counted what was left, by tens, to get 82.   We moved on. I had my index cards ready. I wanted to start with a number relationship I knew that they knew: 10 + 10 = 5. I asked the children to write the sentence down then put "true" or "false" after it. This one was easy. They all knew it was false.   Then we did: 5 + 5 = 10 + 10. I expected there would not be consensus on this since many children interpret the "=" as "and the answer is . . ." (Knuth, Alibali, Hattikudur, McNeil, & Stephens, 2007; Knuth, Stephens, McNeil, & Alibali, 2006). Under this erroneous interpretation, the number sentence could be construed as true. Sure enough, Jack and Sunny said that it was true and Emilio put a question mark. I introduced the language of "is the same amount as" for the equal sign and we talked a little about whether 10 was the same amount as 20. No problem there. But I did not necessarily expect the children to have fully assimilated the meaning of "=."   Next we did: 10 + 10 + 10 + 3 = 33. This one was really designed to see how they figured the sum. Jack counted the 10s then the 3 and said it was true. Daniella used 10 plus 10 is 20, but then counted up by ones. Sunny modeled it with tens to get 33. They all said it was true.   Next: 22 + 10 = 30. I was so pleased with Sunny's response on this one! She used relational thinking (Carpenter, Franke, & Levi, 2003) and I speculated her prior work with the unfix cubes may have helped. She said that 20 plus 10 is 30, but since the 2 is with the 20 it should "be taken out of the 10" and 22 plus 8 is 30. It was a flash of insight.   Next: 10 + 10 + 10 + 10 + 10 = 100. Jack counted up by tens. It was easy for him to figure out it was false. Sunny wrote that it was true, but I did not find out her reason. Daniella got out 5 tens, said it was 50, but then to prove it, she counted on by ones from 20. It is just not clear to me how much she understands. She is right on the cusp.   Then, a controversial one: 12 + 10 = 10 + 12. At first, they all said it was false, although Jack wavered a little and wanted to put a question mark. So I told them I was going to make up a story problem to help them think about it. It went like this: "Sunny had 12 cents. Then she got 10 cents for her birthday. Daniella had 10 cents. Then she got 12 cents for her birthday. Do they - 44 -

have the same amount of money or not?" The answers were interesting! Daniella added both sides up (counted on by ones from the first numbers for each), got 22 for each. Jack said you could add the numbers in either order, it did not matter. As we talked, I represented each quantity (Figure 19).

Figure 19: My representation of each girl's quantity

  They all agreed it was true that the amounts were the same. Then we went back to the number sentence, 12 + 10 = 10 + 12, and I related the two sums on either side to the money that Sunny and Daniella had (Figure 20).

Figure 20: My equation relating each girl's quantity

  The children seemed more included to believe it was a true number sentence, but wavered ─ not surprising as children's conceptions are often strong!   But the next number sentence brought a nice surprise from Daniella: 10 + 2 = 5 + 5 + 2. Sunny and Jack thought it was false but Daniella argued it was true and her reason was just beautiful. She grabbed the card from me and wrote "12" and "12" underneath each expression (Figure 21).

Figure 21: Daniella's notation to explain why the equation was true

  I was curious about whether Daniella and Sunny could work mentally to combine multiples of 10, such as 20 + 30, and whether Jack could use invented strategies for adding double-digit numbers. We had one more session together. - 45 -

Session 10: I Informally Assess What the Children Have Learned   For our last session together, I wrote a mix of problems that would provide insight into what the children had learned about base-ten concepts and their use in problem solving. This session, as all of them really, intertwined assessment and instruction. Table 10: Problems for Session 10

 Jack. Jack seemed to have made a great deal of progress. He solved the first problem (8 rolls of candy, 10 in each roll) by drawing a rectangle-like representation of each roll. At my suggestion, he wrote "10" above each one. He finished off by drawing the extra 12 candies individually. As he counted them however, he pointed out the extra 10, for a total of 90 and "3, oops, 2 more" (Figure 22).

Figure 22: Jack's drawn strategy for 8 groups of 10 candies each, plus 12 extra candies - 46 -

  I asked him to write a number sentence and he wrote "10, 20, 30, 40, 50, 60, 70, 80, 90, 92" (as before). This strategy is significant because he no longer depends on representing the individual units (each candy) to construct 10. He can use 10 as a unit in his strategies.   I asked Jack if he could solve the second problem (30 pencils, 29 more pencils) in his head; he though for a moment, said no, and proceeded to draw this (Figure 23):

Figure 23: Jack's drawn strategy for 30 pencils plus 29 more

  This time I asked him to write a number sentence using plus and equals to show how he solved it. He wrote "10+10+10+10+10+9=59." As he was writing the tens, I asked him how many tens in 50. "Five," he said. So he understands the place-value relationship between 50 and five 10s.   Jack's solution for the third problem (45 beads, 10 beads per necklace) suggests that his new knowledge of ten as a unit may be somewhat fragile. When I checked in with him, he had written some tens and ones on his paper to represent the total quantity (Figure 24).

Figure 24: Jack's written strategy for 45 beads divided into groups of 10

  The problem seemed to be solved. He seemed to think the answer was 4. Excellent! But as I questioned him about what he had done and why he had done it, his answer changed, first to 5 (pointing to the remainder), then to 40 (the number of beads in 4 necklaces). I continued to ask him questions to help clarify his thinking and to emphasize the context of putting beads on necklaces, and the relationship between beads and necklaces. He finally returned to his original answer of 4 total necklaces. He was in the process of learning to mentally coordinate related units, such as ones and tens, beads and necklaces. - 47 -

  Turning to the open number sentences, I again asked Jack if he could solve the problems in his head. "Yes," he said, for 30 + 40 = ___. He counted on by tens from 30 to get 70. I skipped 25 + 20 in order to see what he would do with another problem that involved multiples of 10 only. "How about 60 - 20?" I asked, "80," he replied. I drew his attention to the minus sign. "So if it's plus," I said, wanting to reinforce his mental strategy, "the answer is 80. What if it's minus?" Jack easily counted back by tens to get 40 and likened the problem to 6 take away 2. If I were to continue to work with Jack, I would give him more problems like all of these, and support him to use more abstract counting strategies consistently. I would expect him to move toward a place-value understanding of multiple groups of ten, as in just knowing that 5 groups of 10 is 50, for example.  Sunny. Sunny's strategies were more concrete than Jack's, but I noticed that the language she was using to describe these strategies suggested an emerging understanding of base-ten concepts.   For instance, to solve the problem involving 8 rolls of candies, with 10 candies per roll, she direct modeled using unifix cubes in sticks of 10. But when she described her solution she said, "It's 80, because 8 tens is 80, when you county 10, eight times, it's the number 80." In her explanation Sunny was making a connection between "counting by tens" a certain number of times and multiples of ten.   The emergent nature of her understanding of the base-ten structure of numbers was also apparent in her strategy for adding 30 pencils and 29 pencils. Again she direct modeled the quantities, using unifix curves in sticks of 10. But beyond this, Sunny made little use of base-ten concepts: to count the total, she counted up by ones from 30. I think she is just arriving at understanding 30 is 3 tens and that applying this knowledge in constructing a solution such as counting on by tens is somewhat beyond her right now. If I were to keep on working with Sunny, I would continue to give her addition and subtraction story problems with double-digit quantities as well as equal groups problems involving four or more groups of 10 to help her develop strategies that made more efficient use of base-ten concepts and processes.  Daniella. Daniella was also making progress in her use of ten as a unit. In particular, in her strategies for problems that involved three or more groups of ten, she moved from adding the first two tens (10 + 10 = 20) and counting the other groups of ten by ones to counting all groups of ten by tens. She was also able keep the difference between tens and ones in mind, in contrast to earlier strategies in which she conflated the two units. For example, to determine the value of 3 dimes, 1 nickel, and 2 pennies, Daniella counted by tens to 30, then counted the other coin denominations by ones. To figure 8 rolls of candy, she counted by tens up to 80, and then counted the 12 extra candies on by ones. However, she found it harder to use this knowledge of ten as a unit to solve the division problem involving groups of ten and to - 48 -

count on by tens from a non-decade number, such 25. If I were to continue working with Daniella, I would give her equal groups problems to solve involving four or more groups of tens, including both multiplication and measurement division, to help her consolidate her new use of ten as a unit in her counting strategies. I would also give her double-digit addition story problems involving the addition of a non-decade number and a small multiple of 10, such as 10 or 20, to provide the opportunity for her to begin to count by tens from a non-decade number.   Emilio. Like Daniella, Emilio solved problems that involved multiple groups of ten by counting by tens and had some difficulty using ten as a unit in other problems, such as the measurement division problems (45 beads, 10 beads per necklace) even when the total number of beads was changed to 30. For the coin problem, he counted the dimes by tens to 30, and then counted the nickel and two pennies by ones. For the rolls of candy problems (8 rolls of candy, 10 candies in each roll, plus 12 extra candies), he successfully direct modeled the 8 rolls using cubes in sticks of ten and counted the total using a combination of skip counting by tens and counting by ones. I would encourage Emilio to continue to solve problems like these and I would look for evidence that he was beginning to transition from direct modeling by tens to skip counting by tens.   Looking back, I was struck by the ebb and flow of the children's advancing understanding. Constructing ten as a countable unit ─ the foundational concept of base-ten and place value understanding ─ was a protracted process for these children. Their understanding did not advance at the same rate, and when advances were made, they were not uniformly sustained. Further, I stopped my work with them with many open questions. Our time together was up, and there was no neat resolution to the most basic learning goal I had for these children, which was the development of more sophisticated understanding of base ten and the ability to use this knowledge flexibly in problem solving. Children do not necessarily learn what we may have planned in the time frames we set. Nonetheless, each child made what I recognized as progress. I knew their learning would continue.   David Cohen (2011) described one of the predicaments of teaching as a paradox: no matter how expert teachers may be, they "frequently have no conclusive expert solutions, even to many basic problems" (p. 5). Because of my work as a professor of mathematics education, my knowledge base of research on children's thinking is fairly extensive. Yet this knowledge base did not contain ready-made answers to the problems of teaching mathematics to a group of four wiggly young children. In my work with Daniella, Emilio, Jack, and Sunny, I encountered many of what Cohen might call basic problems. I worked hard to decide what to do and often felt uncertain of the outcome.   My basic teaching problems centered on decisions about what problem - 49 -

to pose next, when to push a child to use a more sophisticated strategy and when to hold back, how to respond to children's incorrect strategies in a way that supported their thinking rather than took over their thinking (Jacobs & Philipp, 2010), how to help children move from more concrete to more abstract strategies, how to deal with a diversity of understanding among the children, how to foster children's curiosity about their mathematical thinking, and how to manage the group so that they listened to and learned from each other. Research findings on the development of children's understanding of base ten were an important resource for me, as were teaching principles centered on the development of children's mathematical agency and ownership (Barton & Tan, 2010; Turner, Dominguez, Maldonado, & Empson, 2013). Although these children had all been characterized as struggling with mathematics, I wanted to find their strengths ─ what they understood, rather than what they lacked. Research-based knowledge gave me a lens through which to see these things.   How did I decide what to do next in ways that were responsive to children's mathematical thinking? My moment-to-moment decision making was guided by the goal of supporting these children to work from what made sense them. To find out what made sense to them, I elicited their thinking about strategically chosen problems involving both equal groupings of ten and adding and subtracting decade numbers, without dictating that they think about these problems in a certain way. I probed their thinking, I asked questions to support and extend their thinking (Fraivillig, et al., 2010; Jacobs & Ambrose, 2008). Thus, the information on which my decisions were based emerged almost exclusively during instruction as I talked with the children about their thinking. This talk and its teaching moves were intended to provide opportunities for children's thinking to advance. I taught these children by listening to them and, based on what I heard, providing opportunities for them to extend their thinking.   I wrote problems after each session, taking into account what I had learned about one or more of the children. The problems were informed by framework provided in Carpenter and colleagues (1999; in press) and Carpenter, Franke, and Levi (2003). However, I decided the problem contexts (e.g., soccer balls, packages of cookies, beads on necklaces), specific numbers, the sequences in which to present the problems, and what parts of children's strategies to focus on in my conversations with the children. These choices, in turn, were based on my emerging understanding of what these children understood and where their understanding might lead them next (Simon, 1995). For example, in Session 2, I wrote a problem involving multiple groups of ten, because I was curious about how the children would think about these groupings. In Session 3, when I saw that Daniella was stumped about how to use unifix cubes in stacks of ten to solve a problem that involving building on a non-decade number (22) up to a decade num- 50 -

ber (50), I realized I needed to adjust the numbers in the problem to make it easier for her to use a more sophisticated strategy ─ modeling with and counting by tens ─ to solve the problem.   With each teaching move, I aimed to create an opportunity for one or more of the children to make critical connections (Hiebert & Grouws, 2007). Because I saw it as my role to create such opportunities but not to force a particular answer, it meant that sometimes an opportunity was not taken up by a child in a way that was obvious to me. I responded, for the most part, by not insisting that the children provide an answer that accorded with my idea of the connection I wanted them to make. Instead, I listened for the connections that were readily made by the children, so that I could capitalize on them, and I allowed myself to be comfortable with uncertainty about what exactly each child might be taking away from the interaction.   To an observer some of my decisions may have seemed counter intuitive or misguided. At one point or another, for example, I decided to ignore a wrong answer or to not show a more efficient strategy. In Session 6, when Daniella solved a problem involving 11 groups of 10 plus 5 extras by counting the 5 extra singles as 5 extra tens, I decided to momentarily ignore her incorrect answer (160) to focus on how she had modeled and counted the 11 groups (in 11 groups of 10 ones each) so that Emilio might have the chance to reflect on his own, incorrect strategy. Decisions such as these were contextualized in my larger goals of supporting each child to work from what he or she understood, to make the connections that were critical to their mathematical growth at that moment, and in the long run, to develop mathematical power. Keeping my eye on these larger, longer-team goals sometimes required making trade offs in my moment-to-moment work with the children between emphasizing procedural accuracy and building on children's sound conceptions.   What was the role of research-based knowledge in this decision making? My knowledge of research on children's thinking in the domain of number and operations oriented me to attend to the children's strategies, what I thought these strategies indicated about their understanding, and how I could use that understanding in deciding my next steps (Jacobs, Lamb, & Philipp, 2010; Philipp, et al., 2012).2 Certain details, such as how a child used ten as a unit or counted on from 22, drew my attention because they fit into my generalized understanding of what it meant for children to understand base ten. For example, when Jack began to count his groups of individually drawn cookies by tens instead of ones in Session 6, I recognized that he was on the cusp of a significant shift from direct modeling by ones to skip counting by tens. It made sense to urge him to use a more abstract strategy at that point. Without knowledge of the development of children's Vicki Jacobs, Randy Philipp, and colleagues call this phenomenon teacher noticing. In a sense, I am studying my own noticing. For more information about this growing body of work, see Sherin, Jacobs, & Philip (2011).

2

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mathematical thinking, details such as this one would have escaped my attention or seemed irrelevant. I would not have seen a possible advance in the making or I might have rushed children who were not ready to advance.   At best, however, research-based knowledge of children's thinking served as a touchstone for me. It did not prescribe what to do in each lesson and certainly did not provide an answer for what to do next when I was in the midst of interacting with the children. I had to do the work to interpret each child's thinking, by attending to what the child did or said ─ how the child used cubes, what they wrote on paper, or how they counted, for example ─ and connecting these actions with the components of this research-based knowledge, as I understood it; my interpretation of each child's thinking then provided the basis for deciding how to respond (Jacobs, et al., 2010). Researchers who study teachers' thinking have pointed out the seamlessness with which these interconnected parts of decision making occur (Erickson, 2007; Jacobs, et al., 2010; Philip, et al., 2012).   Reading the research on learning, one might be tempted to think that the development of children's thinking follows a clear sequence that can be easily applied by teachers in practice. It's an appealing idea, because we think of learning as cumulative, which seems obvious when we view learning retrospectively. Children's thinking about base ten, for example, tends to progress from modeling quantities by ones, to modeling quantities by tens, to skip counting and adding tens, to immediate place value knowledge (Carpenter, et al., 1998; Hiebert & Wearne, 1996). Teachers who look for these general patterns in their classrooms, however, quickly realize the complexity and apparent non-linearity of children's learning. Children's thinking is not always clear, children advance at different rates, breakthroughs that were made one week seem lost the next, and what worked for one child may not work for the next.   At any given decision point, there are so many possible choices about what to do next, and so many variables related to these choices, that the idea that there might be a single best next step or path is profoundly misleading. Research on children's learning may provide frameworks and mathematics educators may design resources, but the real work of using research on learning in instruction requires an interpretive agent ─ the teacher ─ to do the work of connecting what one child is doing with the more generalized knowledge of how children learn in order to decide how to respond. In this account of my work with four children, my goal as to document what was involved in doing this work in the domain of base-ten number and operations.

Conclusion   Responsive teaching involves new skills and requires that teachers be constantly attentive to children's mathematical understanding as they teach. - 52 -

Teachers are to look for evidence of and attend to this understanding for several purposes ─ to monitor correctness, to diagnose errors and misconceptions, and to build on children's sound conceptions (Daro, et al., 2011). I focused on the decision making that occurs during instruction and is aimed toward building on children's sound conceptions, as children engaged in solving problems and expressing their reasoning. This decision making was responsive to what emerged in children's activity and informed by research on children's learning in a specific content domain.   If instruction provides opportunities for students to engage in conceptually challenging mathematics drawing on what they know, then the possible directions when students reveal their thinking are many. Research on learning can offer resources by not prescriptions. Teachers need to be able to work out for themselves, on the spot, how to respond to the evidence at hand, and they need to be able to do this continuously as they engage children solving problems and reflecting on their solutions. A critical goal for teacher education, then, is to develop teachers' capacity for making and enacting informed, responsive decisions as a continuous feature of instruction. When it comes to teaching children, nothing replaces a teacher as the ultimate decision maker, attending to each child.

References Ball, D. (1993). With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics. The Elementary School Journal, 93(4), 373-397. Barton, A. Calabrese & Tan, E. (2010). We be burnin! Agency, identity, and science learning. Journal of the Learning Sciences, 19(2), 187-229. Carpenter, T. P., Fennema, E., Peterson, P., Chiang, C. & Loef, M. (1989). Using knowledge of children's mathematics thinking in classroom teaching: An experimental study. American Educational Research Journal, 26(4), 499-531. Carpenter, T. P., Fennema, E., Franke, M., Levi, L. & Empson, S. B. (1999). Children's Mathematics: Cognitively Guided Instruction (1st edition). Portsmouth, NH: Heinemann Carpenter, T. P., Fennema, E., Franke, M., Levi, L. & Empson, S. B. (in press). Children's Mathematics: Cognitively Guided Instruction (2nd edition). Portsmouth, NH: Heinemann. Carpenter, T. P., Franke, M. L., Jacobs, V., Fennema, E. & Empson, S. B. (1998). A longitudinal study of invention and understanding in children's use of multidigit addition and subtraction procedures. Journal for Research in Mathematics Education, 29(1), 3-20. - 53 -

Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic & algebra in elementary school. Portsmouth, NH: Heinemann. Cohen, D. (2011). Teaching and its predicaments. Cambridge, MA: Harvard University. CCSSO/NGA. (2010). Common core state standards for mathematics. Washington, DC: Council of Chief State School Officers and the National Governors Association Center for Best Practices. Retrieved November 10, 2012, from http://corestandards.org/ Daro, P., Mosher, F. A., & Corcoran, T. (2011). Learning trajectories in mathematics: A foundation for standards, curriculum, assessment, and instruction (Consortium for Policy Research in Education, Research Report #RR68). Philadelphia, PA: Consortium for Policy Research in Education. Erickson, F. (2007). Some thoughts on 'proximal' formative assessment. In P. Moss (Ed.), Evidence and decision making. Yearbook of the National Society for the Study of Education, 106(1), 186-216. 10.1111/j.17447984.2007.00102. Fraivillig, J. L., & Fuson, K. C. (1999). Advancing children's mathematical thinking in Everyday Mathematics classrooms. Journal for Research in Mathematics Education, 30, 148-170. Fuson, K., Wearne, D., Hiebert, J., Murray, H., Human, P., Olivier, A., Carpenter, T. & Fennema, E. (1997). Children's conceptual structures for multidigit numbers and methods of multidigit addition and subtraction. Journal for Research in Mathematics Education, 28(2), 130-162l Glenn, D. (2010, February 28). Divided attention. The Chronicle of Higher Education, 56. Retrieved http://chronicle.com/article/Scholars-TurnTheir-Attention/63746/ Harshman, M. & Garrison, B. (1993). Only one. New York, NY: Cobblehill Books. Hiebert, J. & Wearne, D. (1996). Instruction, understanding, and skill in multidigit addition and subtraction. Journal for Research in Mathematics Education, 14(3), 251-283. Hiebert, J., Gallimore, R., Garnier, H., Givvin, K. B., Hollingsworth, H., Jacobs, J. et al., (2003). Teaching mathematics in seven countries: Results from the TIMSS 1999 video study (NCES 2003-013). Washington, DC: U.S. Department of Education, National Center for Education Statistics. Hiebert, J. S., & Grouws, D. A. (2007). The effects of classroom mathematics teaching on students' learning. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 371-404). Reston, VA: National Council of Teachers of Mathematics. Jacobs, V. R. & Ambrose, R. C. (2008). Making the most of story problems. Teaching Children Mathematics, 15, 260-266. Jacobs, V. R., Ambrose, R. C., Philipp, R. A., & Martin, H. (2011, April). - 54 -

Exploring One-On-One Teacher-Student Conversations During Mathematical Problem Solving. Paper presented at the 2011 annual meeting of the American Educational Research Association, New Orleans, LA. Jacobs, V., Franke, M., Carpenter, T., Levi, L. & Battey, D. (2007). Exploring the impact of large scale professional development focused on children's algebraic reasoning. Journal for Research in Mathematics Education 38(3), 258-288. Jacobs, V. R., Lamb, L. L. C., & Philipp, R. A. (2010). Professional noticing of children's mathematical thinking. Journal for Research in Mathematics Education, 41(2), 169-202. Jacobs, V. R., Lamb, L. L. C., Philipp, R. A. (2010). Professional noticing of children's mathematical thinking. Journal for Research in Mathematics Education, 41(2), 169-202. Jacobs, V. R., Lamb, L. L. C., Philipp, R. A., & Schappelle, B. P. (2011). Deciding how to respond on the basis of children's understandings. In M. G. Sherin, V. R., Jacobs, & R. A. Philipp (Eds.), Mathematics teacher noticing: Seeing through teachers' eyes (pp. 97-116). New York: Routledge. Kennedy, M. (2005). Inside teaching: How classroom life undermines reform. Cambridge, MA: Harvard University Press. Knuth, E., Alibali, M., Hattikudur, S., McNeil, N., & Stephens, A. (2007). The importance of equal sign understanding in the middle grades. Mathematics Teaching in the Middle School, 13(9), 514-519. Knuth, E., Stephens, A., McNeil, N., & Alibali, M. (2006). Does understanding the equal sign matter: Evidence from solving equations. Journal for Research in Mathematics Education, 37(4), 297-312. McClain, K., Cobb, P., & Bowers, J. (1998). A contextual investigation of three-digit addition and subtraction. In L. J. Morrow & M. Kenny (Eds.), The Teaching and Learning of Algorithms in School Mathematics, NCTM Yearbook. Reston, VA: National Council of Teachers of Mathematics. Philipp, R. A., Jacobs, V. R., Lamb, L. C., Bishop, J. P., Siegfried, J., & Schappelle, B. (2012, April). A study of teachers engaged in sustained professional development. Paper presented at the 2012 annual research presession of the National Council of Teachers of Mathematics, Philadelphia, PA. Sherin, M., Jacobs, V., & Philipp, R. (2011). Mathematics teacher noticing: Seeing through teachers eyes. New York: Routledge. Sophian, C. (2007). The origins of mathematical knowledge in childhood. New York: Lawrence Erlbaum Associates. Steffe, L. P., & Olive, J. (2010). Children's fractional knowledge. New York: Springer. Simon, M. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114-145. - 55 -

Turner, E. E., Dominguez, H., Maldonado, L., & Empson, S. B. (2013). English learners' participation in mathematical discussion: Shifting positionings, dynamic identities. Journal for Research in Mathematics Education, 44(1), 199-234.

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Investigations in Mathematics Learning ©The Research Council on Mathematics Learning Fall Edition 2014, Volume 7, Number 1

Second-Graders' Mathematical Practices for Solving Fraction Tasks Patricia S. Moyer-Packenham Utah State University [email protected]

Johnna J. Bolyard West Virginia University

Stephen I. Tucker Utah State University

Abstract

    Recently, over 40 states in the United States adopted the Common Core State Standards for Mathematics (CCSSM) which include standards for content and eight standards for mathematical practices. The purpose of this study was to better understand the nature of young children's mathematical practices through an exploratory examination of the practices of a group of second-grade students engaged in several mathematical tasks focused on rational number concepts. Twenty-five second-grade students completed three fraction tasks in structured clinical interviews. The interviews and student work were analyzed using an interpretational analysis to examine the data for constructs, themes, and patterns that were useful in explaining children's mathematical practices. The results reveal that children used a variety of mathematical practices during the interviews to respond to the mathematical problems presented. Children's mathematical practices were both a product that they used to solve the mathematical situations, and a process that was developing during the interactions of the interview. The findings lead to new insights about how mathematical practices develop and what promotes their development.   A decade ago, the RAND Mathematics Study Panel (2003) identified mathematical practices as highly important and suggested that a better understanding of the nature of mathematical practices had the potential to have significant impacts on the improvement of student learning. One particular concern of the panel was that mathematical practices are often left - 57 -

implicit in curriculum and standards documents, which can lead to them being overlooked during instruction. Another concern was that there were many unanswered questions about mathematical practices, which has led to them being misunderstood by teachers and curriculum developers.   In 2010, the Common Core State Standards for Mathematics (CCSSM) brought mathematical practices to the fore by laying out eight Standards for Mathematical Practice (National Governor's Association Center for Best Practices, 2010). These practices are: a) make sense of problems and persevere in solving them; b) reason abstractly and quantitatively; c) construct viable arguments and critique the reasoning of others; d) model with mathematics; e) use appropriate tools strategically; f) attend to precision; g) look for and make use of structure; and, h) look for and express regularity in repeated reasoning. The CCSSM note that "The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students" (National Governor's Association Center for Best Practices, 2010, p. 6). In the CCSSM documents, descriptions are provided for each of the eight mathematical practices. But what do mathematical practices "look like" when students are employing them during mathematical tasks? Are there some mathematical practices that are more common or well-developed in children from an early age and other mathematical practices that take a great deal of time and experience to develop? While there are large bodies of research focused on specific mathematical content domains (e.g., number operations, fractions, geometry), there are virtually no studies that place emphasis on examining students' mathematical practices as defined in the CCSSM. The present study sought to begin the conversation on the CCSSM's Mathematical Practices that are employed by children during mathematical tasks. The purpose of this study was to better understand the nature of young children's mathematical practices through an exploratory examination of the practices of a group of second-grade students engaged in several mathematical tasks focused on rational number concepts.

Mathematical Practices as Product and Process   Our theoretical framework for this study was based on the idea that mathematical practices may be thought of as both a product and a process (Li, 2013). By being in the form of a product, we mean that children's mathematical practices may be well-developed and usable by the child; their mathematical practices may be in a well-formed state that can be accessed and employed like a tool for solving different types of problems. However, it is important to note that this "product" that may be employed during one mathematical task is constantly being reformed and reshaped by the indi- 58 -

vidual in preparation for the "product" that is used during the next mathematical task. By mathematical practices being in the form of a process, we mean that children's mathematical practices are in a constant state of formation. This means that their mathematical practices are developing and changing while students are engaged in mathematical tasks and a new or revised mathematical practice may emerge as a result of engaging in each subsequent mathematical task.   Mathematical practices, either in their product or process modality, can be developed and can be developing through Piaget's (1972) idea of reflective abstraction. Reflective abstraction is identified as the mechanism of the development of intellectual thought and a key mechanism involved in learning (Dubinsky & Lewin, 1986). Similarly, we believe that mathematical practices develop and are shaped by reflective abstraction. Reflective abstraction was first introduced by Piaget to describe "the construction of logico-mathematical structures by an individual during the course of cognitive development" (Dubinsky, 1991, p. 95). While Piaget's use of reflective abstraction often focused on examples of children learning mathematics, we propose that reflective abstraction is also an important part of children's development of mathematical practices.   Piaget (1972, 1980, 1985) distinguished reflective abstraction from empirical abstraction. For example, empirical abstract is derived from external experiences with objects which can lead to some knowledge about the common properties of those objects. In contrast, reflective abstraction results when a child forms a generalization that is constructive. While empirical abstraction deals with objects, reflective abstraction deals with action by drawing properties from mental or physical actions. These ideas can also be related to the development of students' mathematical practices. Take for example the CCSSM mathematical practice of "using appropriate tools strategically." A student using a particular mathematical tool (e.g., a protractor) or a particular mathematical action (e.g., measuring an angle) will gain some knowledge about the properties and processes involved in that activity (empirical abstraction). However, to develop the mathematical practice of "using appropriate tools strategically" requires mathematical practices abstracted from mathematical activities and reflection on those activities, with mathematical practices becoming formed and developed through reflection on the use of those practices in mathematical activity (reflective abstraction). This means that to form a generalization that is constructive about the mathematical practice of "using appropriate tools strategically" students will undergo many experiences where they are using many different types of tools, they will find that some tools work better than others for particular types of mathematical situations, and they will begin to recognize the benefits of strategic tool selection. All of these processes (and many more) that come together for the individual will be involved in the process of develop- 59 -

ing this mathematical practice is most closely aligned with constructivist and embodied approaches to mathematical thinking and learning (Davis, Maher, & Noddings, 1990). For example, Nemirovsky, Kelton, and Rhodehamel (2013) propose that "mathematical learning consists of transformations in learners' lived bodily engagement in mathematical practices" (p. 376). Similarly, Clements and Sarama (2007) suggest that children actively and recursively construct knowledge. Aligned with these ideas, we suggest that children actively and recursively construct mathematical practices. Clements and Sarama's theoretical framework of hierarchic interactionalism is a synthesis of previous theoretical frameworks that represents the interaction of innate competencies, internal resources, and experience. In this theoretical framework, Clements and Sarama propose that "Mathematical ideas are represented intuitively, then with language, then metacognitively..." (2007, p. 464). We suggest that mathematical practices progress in a similar fashion ─ first intuitively, then with language, then metacognitively. Clements and Sarama describe hypothetical learning trajectories for children's construction of mathematical concepts. Just as there are hypothetical learning trajectories for learning within particular mathematical domains, there may be hypothetical learning trajectories for learning mathematical practices within or across mathematical domains. Just as learning particular mathematical content follows a hypothetical learning trajectory, the development of mathematical practices may also follow a similar type of learning trajectory. And just as reflective abstraction is important in the development of mathematical learning trajectory, reflective abstraction may also be key to children's developmental trajectories for mathematical practices (von Glasersfeld, 1995).   Until recently, there has been very little focus on the examination of students' mathematical practices. However, in 2013 Li proposed a conceptual framework for observing the status of mathematical practices and evaluating the growth of mathematical practices. In Li's framework, there are three overarching themes that underlie the mathematical practices that one would expect of learners. These include: (a) the level of learner engagement and commitment; (b) the learner's employment and development of knowledge, skills and strategies; and (c) the internalization and habituation of the mathematical practices. As Li (2013) notes, "...learner's mathematical practices exhibit a process-product duality: they make up both a means to and an end of mathematics learning and understanding" (p. 62). Therefore, in Li's framework, mathematical practices can be developing (a process) or developed (a product).   In the present study, we focus on Li's second theme underlying the mathematical practices by examining the learner's employment of knowledge, skills and strategies. Li (2013) includes in this category such actions as logical reasoning, justification techniques, applying knowledge, problem solving, and - 60 -

reasoning. We believe that this theme allows us to see children's mathematical practices in process (as they are forming their mathematical practices) and children's mathematical practices in product form (as they are employing a mathematical practice with which they are confident and knowledgeable).

The Current Project   The purpose of this project was to better understand the nature of young children's mathematical practices. We chose to examine children's mathematical practices using tasks in the domain of rational numbers. We selected rational numbers for our tasks because the Rational Number Project has a long history of research in this domain with well-developed tasks and instructional materials and sequences (Behr, Lesh, Post, & Silver, 1983; Behr & Post, 1992; Cramer, Wyberg, & Leavitt, 2008). Researchers in this domain have posited five subconstructs of rational number knowledge: part-whole relations, ratios, quotients, measures, and operations (Kieren, 1980), and three partitioning schemes have been identified (Lamon, 1996): (a) halving ─ an early developed partitioning action (Pothier & Sawada, 1983), (b) dealing ─ a primitive form of partitioning which generates equal shares by distributing in a cycle fashion until all shares are given out (Davis & Pitkethly, 1990), and (c) folding or splitting ─ where the number of pieces grow with the number of folds (Confrey, 1998; Kieren, Mason & Pirie, 1992).   We also know a great deal about young children's conceptions of the fraction ½ (Hunting, Davis, & Pearn,1996), and 3- and 4-year-olds' understanding of continuous and discrete quantities involving the fractions ½, ⅓, and ¼ (Hunting & Sharpley, 1988). For example, 4-year-olds can think about the whole or about the parts, but they cannot think about the two concepts simultaneously. This mental action requires the ability to perform two mental actions simultaneously as in the conservation of a whole partitioned into parts (Kamii & Warrington, 1999). This base of knowledge in the domain of rational numbers makes it an excellent site for the study of the CCSSM's mathematical practices within the domain. The following overarching research question guided our inquiry: What is nature of second grader's mathematical practices (as defined in the CCSSM) during mathematical tasks that include a simple quantity, a discrete quantity, and three conflicting empirical models of one-half?

Methods Participants   The participants in this study were 25 second-graders (one age 7, 22 age eight, two age nine) from two intact classes in a small rural elementary - 61 -

school in the southern United States. There were 15 females and 10 males; 17 African American an eight Euro American children. The children represented a range of abilities, as identified by their teachers. The children volunteered for the study, and parent permission was obtained. The children participated in the interviews during the spring of the academic year. The children had little formal instruction in rational number concepts, which allowed the focus of their responses to be based on the contextualized questions rather than on procedures learned during school instruction. Procedures & Instruments   One way to examine children's mathematical practices is through interaction with others through verbal and nonverbal communication, as suggested by Vygotsky (1978) when he proposed that higher mental processes first occur on the social plane (i.e., between people). Therefore, the primary data collection instrument was a structured clinical interview protocol that was used in one-on-one interviews with each participating child. The structured clinical interview allowed students opportunities to explain their own mathematical ideas (Goldin, 2000). Providing children an opportunity to represent a mathematical problem in a way that is meaningful to them helps them to organize their thinking, making the problem more accessible. Each interview was audio taped, photographed, and documented through field notes and children's work samples. Children's responses to the tasks were transcribed along with the interviewer's notes on children's problem solving behaviours. The interviewer photographed children's work and collected work samples, including drawings and writings. Data for analysis included the transcripts, children's written and photographed work, and the interviewer's notes.   The interviews took place in the children's classrooms while other instructional activities were being completed by the classroom teachers, with the exception of two interviews that were conducted in the hallway right outside the classroom. The interviews included a series of structured questions followed by open-ended questions to probe for more information on the children's thinking. One researcher conducted all of the interviews. Prior to the interviews, the researcher had several formal and informal interactions with the children and their teachers during classroom teaching activities; therefore, the children knew and had interacted with the researcher prior to the project. By having the same person administer the interviews, each child essentially participated in the "same interview." This procedure leads to increased replicability of the interview itself, and a basis for drawing inferences from observations during the interview (Goldin, 1997). Problem Solving Tasks and Materials   The content of the interviews focused on the part-whole subconstruct, - 62 -

which is based on the ability to partition a continuous quantity or a set of discrete objects into equal parts. Children were presented with three types of problems: (a) a simple continuous quantity (region models including circles and squares), (b) a discrete quantity (set models using counters), and (c) a task using three conflicting empirical models of one-half. Tasks were posed verbally using contextualized questions. The children were presented with models for partitioning the continuous (region models) and discrete (set models) quantities. The design of the interview tasks followed the recommendations for instruction in the Rational Numbers Project (Cramer, Behr, Post, & Lesh, 1997; Cramer, Behr, Post, & Lesh, 2009). Tasks were developed with the belief that children can solve a wide range of unfamiliar problems when they are able to represent the problems with objects, actions, and familiar situations (Carpenter, Ansell, Franke, Fennema, & Weisbeck, 1993). We classified our tasks as easy, medium, and hard based on the expectations for Grade 2 in the CCSSM. These classifications are described below.   Task #1: A simple continuous quantity (region models including circles and squares). Children were asked to divide a circle region and a square region into ½, ⅓, and ¼. Based on the Grade 2 CCSSM, second-grade students should successfully complete all aspects of Task #1, therefore, we rated questions from this task as "easy."   Task #2: A discrete quantity (set models using counters). An important milestone for children's numerical development is understanding how to decompose numbers (Van de Walle, 1990). Children were given set models with 12 counters and with seven counters and asked to divide the counters in half. Based on the Grade 2 CCSSM, second-grade students would not have learned the fraction content of Task #2. However, students would have experience "pairing objects or counting them by two's" (National Governor's Association Center for Best Practices, CCSSM 2.OA.3, p. 19). These skills would help second graders solve the set model task with 12 counters, so we rated this task as "medium." Second grade students have experience identifying and pairing wholes and identifying fractions, but the CCSSM does not explicitly include mixed numbers until Grade 4. To divide the seven counters, identifying three pieces plus half of a piece is "medium" difficulty, while correctly naming the outcome "three and a half" was rated as "hard.   Task #3: A task using three conflicting empirical models of one-half. Children were asked to generalize the concept of ½ across different models. They were shown a paper circle cut in half, a paper square cut in half, and 12 counters divided in half from the previous tasks. The interviewer asked the children to name each of the pieces. After the children responded that each piece was "one-half," the interviewer asked, "These models do not look the same . . . Why are all of the models called one-half?" The aim of this task was to promote reflective abstraction through children generalizing the concept of one-half in the face of conflicting empirical models. - 63 -

Based on the Grade 2 CCSSM, second-grade students should identify the half as part of a whole for each model ("medium" difficulty), but may not be expected to generalize across the three models ("hard").   We placed materials, including pictures of circles and squares, one set of fraction circles, crayons, scissors, cut-outs of circles and squares, 12 twocolor counters, and extra paper and pencil, on the interview table. Children received materials for some tasks (e.g., shading or drawing a fraction portion on paper). For other tasks, children could select any of the materials on the interview table to provide an explanation. The interviewer proceeded through the tasks on the protocol and asked follow-up questions to encourage children to explain their thinking. Interview ranged from 25-45 minutes depending on the length of each child's responses. From these interactions we identified children's mathematical practices. Data Analysis   Interview transcripts were read and coded separately by three independent readers using an interpretational analysis to examine the data for constructs, themes, and patterns that were useful in explaining children's mathematical practices (Gall, Borg, & Gall, 1996). Researchers coded mathematical practices from the transcripts using a modified constant comparative method to identify behaviours and verbalizations that indicated the practices (Strauss, 1987). Themes were used to develop emic (those that were derived directly from the children's own words) and etic (those that were inferred from the children's responses) categories (Maxwell, 1996).   Microsoft Excel PivotTables were used to summarize, organize and sort categories of data, allowing the researchers to cross-reference and group large amounts of data in a summary format for ease of comparison and interpretation. There were nine main category codes for the interview tasks and these were shorthanded with the following abbreviated titles: Task 1a) Region:Circle:Half, 1b) Region:Circle:Third, 1c) Region:Circle:Fourth, 1d) Region:Square:Half, 1e) Region:Square:Third, 1f) Region:Square:Fourth; Task 2a) Set:12:Half, 2b) Set:7:Half; and, Task 3) Conflicting:Half. Overall, five of the CCSSM Mathematical Practices were employed in these children's interviews (National Governor's Association Center for Best Practices, 2010, pp. 9-10).

Results   The results that follow present a brief summary of the overall accuracy of the group on each of the interview tasks. Following this brief summary, we present examples from the transcripts that reveal the mathematical practices that students employed during the interviews. - 64 -

Table 1: Students' Overall Percent Accuracy by Question Difficulty on Three Mathematical Tasks

Note: na indicates "not applicable," meaning no portion of the task was in this category.

Overall Accuracy   Table 1 summarizes the overall percentage of the 25 students who were successful on each portion of the interview tasks. Some tasks contained only an "easy" portion (e.g., Task 1a and Task 1f), while other tasks contained portions with more than one level of difficulty (e.g., Task 2b and Task 3). Students seemed to have the most difficulty dividing a circle region and a square region into thirds. In contrast 100% of the children were successful dividing a circle region, a square region, and a set of 12 counters in half. Over half of the children (56%) said that a square region could be divided in half in more than one way. Children's high levels of accuracy on most of the tasks allowed us to delve more deeply into the mathematical practices that the children used in their explanations. Mathematical Practices: Task #1 with a Simple Continuous Quantity (Region Models)   During Task #1, children divided circle and square region models into halves, thirds, and fourths. Pictorial representations of children's common solutions appear in Figure 1, showing children's facility at modelling the fraction amounts. Children who had difficulty with the "thirds" division often drew two parallel lines or divided the circle into fourths and said to ignore one of the pieces (see e.g., 1.b.2). Students said it was difficult to draw because thirds "was an odd number" or because they "had never seen one like that before." The comments showed that children were reasoning to - 65 -

Successful Region:Circle:Half Region:Circle:Third Region:Circle:Fourth Region:Circle:Half Region:Circle:Third Region:Circle:Fourth

Unsuccessful

1.a 1.b.1

1.b.2

1.c 1.d 1.e.1 1.f

1.e.2

Figure 1: Most Common Region Model Responses

find a solution. Four students, who drew two parallel lines in an unsuccessful attempt to divide the circle region into thirds (see e.g., 1.b.2.), were able to use the same two parallel lines to successfully divide the square region into thirds (see e.g., 1.e.1).   During the square region tasks, children divided squares into ½, ⅓, and ¼ by drawing lines and coloring the amounts. Students said the square could be divided in half in more than one way (see e.g., 1.d). Once again, these comments showed children's reasoning abilities. Some children had difficulty and divided the square into three unequal parts ─ one half and two fourths (see e.g., 1.e.2) ─ or they told the interviewer that the square could not be divided into thirds. They said it was difficult to draw because they could not "get the pieces to be the same size." When children are unable to divide a region into the specified quantity, it is known as nonexhaustive distribution, and is very common for children of this age. Comments about the importance of the pieces being the same size reveal children's understanding of equivalence and their attention to precision.   The most prominent mathematical practices in children's responses during Task #1 was modelling with mathematics (referring to models) and attending to precision. Children's reflections on familiar models allowed them to respond to the tasks with confidence; while children who could not access a familiar model had difficulty with the task. The following interview excerpts with four different children reveal students modelling with mathematics.

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Interview Task #1 Example #1 Interviewer: Can you shade one-third in the next circle? Student: (draws third of a circle accurately and shades one-third of the circle) I: How did you know to draw it like that? S: I know how to draw this one because my dad works for Mercedes. Interview Task #1 Example #2 I: Let's say that we had this big cookie and we were going to share it between the four of us. How would you take your pencil and divide it to show fourths? S: (draws fourths of a circle accurately) I: And shade in your piece. S: (shades in one-fourth of the circle); I though about an X. I: You thought about an X? S: And a circle around the X to make four. Interview Task #1 Example #3 I: Do you know other ways that you might cut the square in half? S: Like a sandwich (child uses a diagonal hand motion). Interview Task #1 Example #4 I: What do you know about fractions? S: Well, some come in halves. I: Can you show me what you're talking about when you say some come in halves? (hands the child paper and pencil) S: (child draws a circle with a line down the middle) I: So you drew a circle and you put a line in it. What does that mean? S: A half. I: What part is the half? S: This part here (pointing to half of the circle) and the other part too (pointing to the other half of the circle). I: What if someone said to cut it into thirds and three people would share it? S: It would look kind of like a peace sign. It wouldn't be exactly equal though.   As these examples from the interviews show, familiar models (eg., a sandwich, a peace sign, or a Mercedes emblem (played an important role in these young students' ability to model with mathematics. When the children were asked for a generalization that explained how to divide a circular or square region in half, they gave responses that fell into four categories: (a) use of text-based definitions, (b) the idea of two pieces, (c) use of the word "half" to describe the concept half, and (d) a focus on equality (see Table 2). - 67 -

Table 2: Children's Responses When Asked Howto Partition a Circle or Square Region

“...the bottom number shows how many parts are in the circle...and the top number tells us how to shade.”

The word “half” to describe half

Six children used one category in their explanation, and 19 children used more than one category in their explanation. As the summary in Table 2 shows, a majority of the children (18 of 25) were able to reason abstractly and construct a viable argument that focused on the concept of equality. Mathematical Practices: Task #2 with a Discrete Quantity (Set Model)   During Task #2, children divided a set of 12 counters and a set of seven counters in half. The interviewer contextualized the tasks by asking children to think of the counters as pieces of candy to be shared between two people. Pictorial representations of children's common solutions appear in Figure 2. Overall, children's responses included spontaneous and prompted distributions (or systematic cyclic procedures) and nonsystematic procedures.   When second-graders divided 12 counters in half (Figure 2.a), responses focused on: (a) properties of the number (such as its "evenness," or that six plus six equals twelve), and (b) distribution or systematic cyclic procedures (such as dealing cards, "one for you, one for you, and one for you"). Examples of responses that focused on properties for the number included comments such as "This is even," "Because six plus six equals twelve," and "Because I know what's twelve take away six". Examples of responses that focused on prompted distributions included, "You get one and I get another one and I get another..." and "One, one, two, two, three, three, four, four, five, five, six, six" as the children distributed the counters. These responses show a variety of mathematical practices including making sense of the task, attending to precision, and reasoning quantitatively.   When second-graders divided seven counters in half, five children immediately told the interviewer to divide the seventh counter in half (Figure 2.b.3). The interviewer's follow-up questions encouraged the other children to persist and divide all seven counters in half. One child suggested remov- 68 -

Set: 12

2.a

Set: 7

2.b.1 2.b.2 2.b.3

Figure2: Pictorial Representations of Most Common Set Model Responses

ing a counter (Figure 2.b.1) and giving the "piece of candy to someone else," and another suggested adding one counter (Figure 2.b.2.), both options that would produce an even number of counters. These solutions were all based on children's notions of "equal parts." Seven of 20 students used the terminology "three and a half" to correctly name the half portion.   The most prominent mathematical practices in children's responses during Task #2 were attending to precision, reasoning quantitatively, and making sense of problems and persevering in solving them. Their responses used words, drawings and hand gestures that communicated the importance that each half of the counters needed to be exactly equal. They reasoned aloud about what "half" would mean quantitatively for the given amounts of 12 and seven counters, showing a solid understanding of the concept of equivalence. Their mathematical practices were most evident when the children attempted to divide seven counters in half. The following interview excerpts with five different children reveal these practices. Interview Task #2 Example #1 I: Let's pretend that you were sharing between two friends and they should get the same because they want to share. S: Oh... (makes chopping motion and cutting sound) I: What does that mean when you went like that? S: Split it (pointing to the seventh counter). I: Split it? S: Right in the middle I: Then how many would this friend get? S: Four (the child calls the half-counter piece the fourth piece) I: And how many would this friend get? S: Four (the child calls the half-counter piece the fourth piece) Interview Task #2 Example #2 I: What would happen if we just had seven pieces of candy and we had - 69 -

to share? How could we make half? S: Okay, we could half the piece of candy in half. I: What do you mean? Do you have three and I have three? S: Mm-hmm (nodding head yes). And then we will have four if we half it. You will get a piece and I will get a piece. I: So you will have four... (moving four counters toward the child) and then how will I have four? S: Now we will half this piece... (moving one counter away from the four in front of the child). One half. Three wholes and one half. Interview Task #2 Example #3 I: What's a way that we could share it (referring to the 7 counters/pieces of candy) S: Break it. I: What do you mean 'break it'? S: Oh...okay (child grabs paper and begins drawing a circle with a line to divide it in half) I: What are you drawing there? S: A circle. I: A circle. What does a circle mean? S: That's that cherry candy. I: And what does the line mean? S: That's where you break it. Like, this is my half and that's your half. (child points to the two halves of the circle drawn on the paper) I: So if we each get half of this (pointing to the child's drawing) and we also get these (referring to the six counters), then how many candy do we each get? S: Three and a half. Interview Task #2 Example #4 I: What happens if we have seven counters and we were going to share those in half? How would you share those? S: (moves three counters to self and three counters to interviewer; picks up the seventh counter) That's a leftover one. You could break it in half. I: What do you mean? S: If it was like a cookie or something, you could break it in half. I: Oh, okay. S: And then we could both have three and a half. Interview Task #3 Example #5 I: What would you do if we have seven pieces and we were going to share it so you get half and I get half? - 70 -

S: Three (child pulls three counters toward herself) I: Any my half would be? S: Three (child pushes three counters to the interviewer) I: What about this one? (pointing to the seventh counter) S: Cut it in half. I: What do you man? S: Cut it down the middle and it would be two halves. I get a half and you get a half. I: Then how many pieces would you have? S: Three half.   As these interview examples reveal, although each child's language for three-and-a-half is still developing, they are all reasoning quite well quantitatively. Additionally, their drawings, verbalizations and hand gestures show that they are attending to precision when they explain "half" as two equal parts. In Interview Example #3, #4, and #5 we can see the children using tools strategically in their explanations when one child (#3) spontaneously grabs the paper and begins drawing an explanation and two other children (#4 and #5) begin moving the counters during the explanation. Mathematical Practices: Task #3 with Conflicting Empirical Models of One-Half   During Task #3, children were presented with three conflicting empirical models of one-half and asked to generalize the concept across the different models. The interviewer showed the children the items from the previous tasks (a circle cut in half, a square cut in half, and the 12 counters divided in half), and posed the following dilemma: All of these pieces are called one-half...These models do not look the same...Why are all of the models called one-half? This question prompted the children to reason abstractly construct viable arguments, model with mathematics, and use appropriate tools strategically.   Overall, children's explanations for Task #3 were categorized in terms of the justification schemes identified by Sowder and Harel (1998) and adopted for elementary children by Flores (2002). (See Table 3.) These justifications included (a) externally-based proof schemes, (b) empirical proof schemes, and (c) analytic proof schemes. Children were placed in one category based on their most salient responses. Most children could be sorted into one category, with the exception of two students unable to articulate a proof.   There was a fairly even distribution of the children among the three justification categories. In the category of external justification schemes, children relied on information they had seen before, like an outside source of knowledge such as teacher, a parent, or a textbook. Children who provided - 71 -

Table 3: Children's Justifications for Three Empirical Models of One-half

Circle and square are half “because I cut it up into two halves” and candy is half because “I gave you six and I gave myself six.” All of the models are called half because “each piece is the same size as the other.”

empirical justification schemes constructed arguments based on specific examples. They focused on a procedure that applied to a specific problem or situation, and often provided a separate explanation for each task that did not generalize to all models. Children who provided analytical justification schemes offered a more generalized explanation of the concept of one-half that applied to all three models (circle, square, counters).   There were a variety of mathematical practices in children's responses during Task #3, including reasoning abstractly, constructing viable arguments, modelling with mathematics, and using appropriate tools strategically. For example, the following interview excerpts with four different children demonstrate their abstractions and justifications for why different shapes and objects can all be called "one-half." Interview Task #3 Example #1 I: We took the cookie (a paper circle) and we cut it in half; and we took a piece of cake (a paper square) and we cut it in half. So we said this was half (pointing to the half circle) and this was half (pointing to the half square). But they don't look the same. How can they both be called half? S: Well, they're both, like, they're, there was two pieces, like we cut this pieces, um, two pieces that would make one half. I: So this is called half... S: (child interrupts interviewer) We...because you because it's out of one piece into two pieces. I: Why... S: (child interrupts interviewer) Are you saying because why did I do it? I: No, no, I'm saying why do we call it the word 'one-half'? S: Because, like...Let me use these things for an example (child grabs eight crayons) - 72 -

I: Okay. S: Like we got...Let me see if I can think something out (counts the eight crayons). Like, I got this...all of these things are just one big chicken nugget put together. I: One big chicken nugget put together? S: Yeah (laughs)...and then we just split it. Like, this is the chicken nugget cutter and we just split it into two pieces (separates four crayons and four crayons), and you know two pieces is half. Interview Task #3 Example #2 I: You told me that this is half of a circle and this is half of a square, what is half of the candy? (referring to the 12 counters) S: It's half of a rectangle I: Half of a rectangle? What do you mean? S: This is a rectangle (child pushes the counters together to form a three by four rectangular array)... but when you split it in half it's two rectangles. (Child spontaneously draws a rectangle on a piece of paper.) I messed up (Child draws a second rectangle on the piece of paper.) (See Figure 3.) I: So why did you draw this rectangle? S: Because that's a rectangle (child points to the 12 counters in the three by four rectangular array). I: Oh...okay... S: When you divide it in half (child draws a line down the middle of the third rectangle that she drew on the piece of paper) it's not a rectangle, it's half a rectangle. I: So when we take this (pointing to the 12 counters in the three by fourrectangular array) in two pieces like this (pointing to the child's picture of a rectangle with a line down the middle) it's half a rectangle? S: It has like six pieces here and six pieces here (child moves the two groups of six counters onto the rectangle that she drew on the piece of paper). As Examples #1 and #2 show, the children spontaneously used tools and models to help the interviewer understand what children were visualizing about the 12 counters. The children wanted the interviewer to understand that they saw the counters as a whole (e.g., chicken nugget or rectangle) and that they were relating that whole to the two region models as wholes (circle region and square region). Interview Task #3 Example #3 I: My question is, let's take a look at all of the things we have here (referring to a paper circle and paper square cut in half and a group of 12 - 73 -

Figure 3: Child's drawing to explain 12 counters as a "rectangle."

counters in half). This is a half of the circle, and this is a half of the square, and this is a half of the candy, right? Why are they all called half when they don't look the same? S: Because the circle is the same piece (puts half paper circle on top of other half paper circle), the square is the same piece (puts half paper square on top of other half paper square), and these are the same pieces (puts half of counters on top of other half of counters), because I took all of the pieces of candy and set them on top of each other and they would equal the same. I: Why aren't they called something different? Why are they each called one-half? S: Because we cut them into halves. I: What do you mean? S: That these two are going to be the same (referring to the circle halves). They are going to be the same because this piece came from that piece (referring to the two circle halves) and that piece came from that piece (referring to the two square halves) and these from these (referring to the two groups of six counters each that she stacked on top of each other). Interview Task #3 Example #4 I: This piece is called half (referring to the half paper circle), and this is called half (referring to the half paper square), an this is half of - 74 -

the candy (referring to half of the counters). Even though they are all called one-half, they look different. S: This is red (referring to half of the counters) and this is white (referring to the half paper circle) and this is white (referring to the half paper square). I: But look at how the shapes are ... this is just a piece of a circle and this is just a piece of a square and this is a bunch of circles. How can they all be called one-half? S: Because you have two and they're cut. So that's a half and you have two of these (referring to the paper and paper circle). I: But there are six counters there... how can... S: (child interrupts interviewer) They're not the same... but why they're all the same is because..., like this is two (touching the two circle halves with two hands), two (touching the two square halves with two hands), two (touching the two groups of six counters with two hands). They all have two parts. They all have partners. I: They all have partners? S: This is his partner...his partner...his partner (child matches up two half paper circles, two half paper squares, and two groups of 6 counters by pushing them together). As Examples #3 and #4 show, the children tried to get the interviewer to understand the ideas that from one whole comes two one-half portions and two halves make one whole. As the interview examples from Task #3 show, children were reasoning abstractly and constructing a viable argument to explain why each different model was called one-half. In the last example (Example #4), when the child says "They're not the same..." the child is referring to the objects not being the same empirically; this is the empirical one-half. Next the child says, "but why they're all the same is because... they all have two parts." Here the child is referring to the abstract one-half. This example shows distinctly how the child is reflecting on the question to generalize ideas. This shows children's ability to see structual-similarity relationships, even though the examples of one-half are presented as dissimilar objects (Alexander, White, & Daugherty, 1997).   In task #3, children restructured their thinking to develop one method or concept for all models. An example of this response was when a student explained that all of the models were called one-half because "each piece is the same size as the other." The child in Example #2 drew a rectangle around the 12 counters to explain to the interviewer how to think of the set of counters as a whole. These analytical justifications demonstrated students' ability to generalize explanations that fit all of the models, demonstrating flexibility in understanding how to define a whole, and showing the ability to reason abstractly. Children's verbalizations and gestures - 75 -

demonstrate perceptual reasoning, in which the child is able to abstract the similarity among the representations of one-half and draw inferences about the meaning of one-half. Children's reasoning and justifications show that they have abstracted the concept of one-half.

Discussion   The purpose of this study was to better understand the nature of young children's mathematical practices through an exploratory examination of second-grade students engaged in several mathematical tasks. The results demonstrate that young children have and are developing mathematical practices from an early age. This exploration has brought us some new insights about children's mathematical practices, but it also leaves us with a variety of questions for future inquiry. Below we submit these insights as questions for future research. Insights about How Mathematical Practices Develop   Mathematical practices are both a product and a process. The interviews show that children's mathematical practices are a product that they employ to solve the mathematical tasks. An example of this might be when the child adds or subtracts a counter to make the counters an even number in an effort to make sense of the task. Because the child has a well-establish idea of that would make sense, they use a sense-focused response in their explanation. The interviews also show that children's mathematical practices are a process that is developing and changing as they are solving the mathematical tasks. An example of this might be when the child employs different strategies (e.g., drawing or moving objects) to support their argument. This may show the process of developing mathematical practices such as learning to give evidence to make their argument more viable to the interviewer.   Children come to school with a foundation of mathematical practices that are inchoate. Clements and Sarama (2007) describe children's premathematical and general cognitive competencies and dispositions as "initial bootstraps" (p. 465). They describe these abilities in young children as supporting and constraining the development of mathematical knowledge, when experiences interact with the child's inborn capabilities. Throughout all of the tasks, students used concrete objects or pictures to conceptualize and make sense of tasks posed in the interviews. Students had access to manipulatives and space to illustrate, using them to demonstrate splitting the whole into equal parts in concrete and pictorial representations. Even without prompting from the interviewer on the set model tasks, students often mentioned cutting or dividing the last object to share it evenly. Providing tasks in a contextualized way helped children to make sense of the tasks. - 76 -

Perhaps children's mathematical practices are also in this initial bootstraps form and children's experiences allow their mathematical practices to develop beyond this initial form.   The development of mathematical practices may follow a hypothetical trajectory similar to mathematical learning trajectories. Just as Clements and Sarama (2007) have proposed that children's learning of mathematical concepts follows a hypothetical learning trajectory, the development of each mathematical practice may also follow a hypothesized path of learning. For example, very young children to seem to intuitively know the importance of constructing a viable argument. They have had many non-mathematical experiences in which they have constructed a successful argument for family members long before they come to school. The development of constructing viable arguments begins prior to school and prior to formal school mathematics. Specific skills of constructing an argument are refined and, then during schooling, children learn to apply this to the argument of a mathematical position or proof. Over many years of school mathematics children have multiple opportunities to develop and refine their expertise in the process of constructing a viable argument. Perhaps there are trajectories for learning each of the mathematical practices.   Different mathematical practices may be related to different points along the child's developmental trajectory. Our interviews in this study elicited six of the eight CCSSM Mathematical Practices. One explanation could be that the mathematical practices that children employ are connected with greater emphasis at different points along the child's mathematical development. Additionally, these practices may develop at different rates for different learners. This emphasis may be similar to the emphasis of mathematics content outlined in the NCTM content standards (NCTM, 2000). In this document, NCTM proposed that some of the mathematical content standards that students learn will be taught with increasing emphasis across the grade levels and some standards will be taught with decreasing emphasis across the grade levels. For example, in the NCTM Standards, Number & Operations, Measurement, and Data Analysis & Probability generally decrease in emphasis, and Algebra and Geometry standards generally increase in emphasis between Kindergarten and Grade 8. In our study we did not observe examples of two of the CCSSM Mathematical Practices: look for and make use of structure and look for an express regularity in repeated reasoning. Perhaps there is a greater emphasis on some practices over others during different points of development, with some practices more prominently used by children in earlier grades and other practices used rarely by younger children but developing and increasing over time among older children. Insights about What Promotes the Development of Mathematical Practices   Mathematical tasks promote the development of mathematical practices. The children were not taught the mathematical practices that they used dur- 77 -

ing the interviews. Children's mathematical practices emerged and developed during the interviews. One explanation for this occurrence is that the mathematical tasks, themselves, promoted the development of children's mathematical practices. Perhaps when children engage in a particular type of mathematical task, and this task is repeated, engagement in the task helps the child to develop different mathematical practices that can be used across a variety of mathematical situations.   We know from the literature on design-based research that excellent well-developed tasks promote mathematics learning (Diefes-Dux, Hjalmarson, Miller, & Lesh, 2008). Additionally Rau, Aleven, and Rummel (2009) suggest that multiple graphic representations of fractions, presented consecutively, can aid students' understanding of fractions, especially when combined with self-explanation. These examples and others point to the importance of presenting children with tasks that provide some cognitive dissonance. For example, children often have difficulty developing proportional reasoning skills, particularly with discrete units (Boyer, Levine, & Huttenlocher, 2008). Comparing the conflicting models was more challenging, as students were simultaneously presented with continuous and discrete wholes. Forcing children to examine conflicting models and to talk about this with an interviewer prompts more generalized understandings of concepts (Buschman, 2001). Perhaps excellent well-developed tasks promote mathematical practices.   Questioning promotes the development of mathematical practices. During the interviews, there were often responses provided by the children that could be characterized as surface answers. In these surface answers, the children rarely employed their mathematical practices. However, when the interviewer used follow-up questioning strategies, the children often employed a mathematical practice to respond to the interviewer and to think more deeply about the mathematics that was being discussed. Perhaps specific questioning techniques prompted mathematical practices, such as when the interviewer asked the children to explain something or asked the children to show the interviewer what they meant by a response. For example, Armstrong and Novillis Larson (1995) found that fourth, sixth, and eighth grade students who participated in clinical interviews discovered mathematical ideas as they constructed their responses to the interview questions.   Research has shown that students sometimes react to the context in which the question is posed. In this study, the interviewer often suggested that students think of the models as cookies, cakes, or candy, and students sometimes used this language in their explanations. In other instances, students suggested models such as apples or chicken nuggets to share, making their own real-world connections. Other studies support questioning in relatable contexts as a way to help hold students' interest while demonstrating the usefulness of the mathematics concepts (Barnes & Venter, 2011; Turtiainen, - 78 -

Glignaut, Els, Laine, & Sutinen, 2009). Perhaps the right questioning promotes children to employ and develop their mathematical practices.   Specific mathematical tasks and follow-up questions may align with specific mathematical practices. During the analysis of children's interviews it seemed that there were particular mathematical tasks and follow-up questions that promoted some mathematical practices more than others. For example, when children were asked to partition region models and to divide 12 and seven counters in half, this prompted children to focus on being precise in their partitioning. When children were presented with the three conflicting empirical models in the third task, this prompted them to focus on constructing a viable argument. When the interviewer's question asked the child to explain something more clearly, this promoted greater precision. Or when the interviewer asked the child to show what the child meant, this prompted the child to model with mathematics.   The tasks that we chose for these interviews clearly elicited six of the eight CCSSM Mathematical Practices with the children in our study. For example, when children were asked to divide seven counters in half this specific task seemed to elicit the mathematical practice of making sense of the problem; then the interviewers repeated follow-up questions seemed to elicit children to persevere in solving the problem. When the task required students to make significant attempts to understand the problem, use concrete objects or pictures to conceptualize and solve the problem, and determine if the problem and the solution makes sense, this involved selfexplanation, which is trying to make sense of a concept by forming one's own explanation (Chi, Bassok, Lewis, Reimann, & Glaser, 1989; Rau et al., 2009). Perhaps types of mathematical tasks and questions are more aligned with the type of mathematical practice that gets developed.   Certain mathematical practices may be more difficult to develop that others. Perhaps there are some mathematical practices that children develop easily, while other practices are only developed with a great amount of experience and support. For example, reasoning abstractly and quantitatively requires that students understand quantities and their relationships in a problem. They must be able to decontextualize and contextualize, understanding the meaning of the quantities and flexibly reason to represent their thinking about a problem. This can be difficult for students to develop, as Lee, Brown, and Orrill (2011) found that even teachers often struggle to flexibly develop and apply fraction knowledge and strategies.   Constructing viable arguments and critiquing the reasoning of others involves students' abilities to justify their solutions, as well as construct and evaluate conjectures. Nicolaou and Pitta-Pantazi (2010) state that justifications "provide an insight into students' understanding of fractions" (p. 3). Throughout the interview tasks, some students struggled to mathematically justify their answers, even when successfully modelling the given fraction. - 79 -

The interview tasks did not require manipulating symbolic representations of fractions in a decontextualized manner but some students showed developing contextualization skills. For example, one student explained a response to one fourth of a circle in Task 1 by saying, "the bottom number is four; that tells me how many pieces are in the whole thing." Students also referred to equal or fair portions in all tasks, though explanations in tasks involving set models mostly focused on operational thinking using counting by twos or adding six and six to make twelve (Mack, 1995). Researchers report that whole number knowledge can be used effectively to develop fraction knowledge by connecting key conceptual understandings between the two number systems (Olive, 1999; Steffe, 1992, 2002; Steffe & Olive, 1996, 2010). Perhaps some mathematical practices are more difficult to develop and children need more help developing some of the mathematical practices than others.

Conclusion   Children's problem solving strategies develop in the toddler and preschool years and allow children to solve problems of increasing complexity (Clements & Sarama, 2007). These developing strategies can lead to mathematical practices that become polished and refined for specific purposes in later mathematics development. The children in this study employed six mathematical practices in response to our tasks. For example, students used their understanding of whole numbers for discrete quantity tasks for making sense of problems. Students used both empirical evidence and analytical thinking to construct arguments. When modeling with mathematics, students were able to model different representations of one-half and created a variety of models of their own. Students reasoned by making connections among the three conflicting models. Attending to precision proved difficult for children as they struggled to use their knowledge of dividing a circle or square in half and apply it to dividing a circle or square in thirds; as when the child recognized that the responses of a "peace sign" would not be precise. And children using tools strategically was evident when they spontaneously grabbed paper and pencil to draw an explanation or began moving counters to clarify their thinking to the interviewer. These results demonstrate that the children in this study had a foundation of mathematical practices that they developed from formal and informal experiences, and that they were able to apply these practices to the mathematical tasks in this study. However, our investigation has left us asking many more questions with which we began. Perhaps ending with a number of questions is a good place to begin to understand the complexity of students developing mathematical practices and teachers developing them in their students. - 80 -

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