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A Boundary of the Second Multiplicative Concept: The Case of Milo

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Title: A Boundary of the Second Multiplicative Concept: The Case of Milo
Language: English
Authors: Hackenberg, Amy J. (ORCID 0000-0002-3406-4761), Sevinc, Serife
Source: Educational Studies in Mathematics. Jan 2022 109(1):177-193.
Availability: Springer. Available from: Springer Nature. One New York Plaza, Suite 4600, New York, NY 10004. Tel: 800-777-4643; Tel: 212-460-1500; Fax: 212-460-1700; e-mail: customerservice@springernature.com; Web site: https://link.springer.com/
Peer Reviewed: Y
Page Count: 17
Publication Date: 2022
Sponsoring Agency: National Science Foundation (NSF), Division of Research on Learning in Formal and Informal Settings (DRL)
Contract Number: 1252575
Document Type: Journal Articles
Reports - Research
Education Level: Elementary Education
Grade 7
Junior High Schools
Middle Schools
Secondary Education
Descriptors: Multiplication, Fractions, Grade 7, Middle School Students, Logical Thinking, Mathematical Concepts, Numbers
DOI: 10.1007/s10649-021-10083-8
ISSN: 0013-1954
Abstract: Students entering sixth grade operate with three different multiplicative concepts that influence their reasoning in many domains important for middle school. For example, students who are operating with the second multiplicative concept (MC2 students) can begin to construct fractions as lengths but do not construct improper fractions as numbers. Students who are operating with the third multiplicative concept (MC3 students) can construct both proper and improper fractions as multiples of unit fractions. This paper is a case study of one seventh grade MC2 student, Milo, who demonstrated the most advanced reasoning of all MC2 students in a large project with 13 MC2 and 9 MC3 students. In working on problems involving fractional relationships between two unknowns, most MC3 students constructed reciprocal reasoning. In contrast, the MC2 students struggled with these problems. Similar to the MC3 students, Milo showed some evidence of reciprocal reasoning, and he used proper fractions as operators on unknowns with a rationale. However, Milo did not construct reciprocal reasoning. We account for his reasoning by showing how he used length meanings for proper fractions and how he coordinated two different two-levels-of-units structures. The study expands the mathematics for MC2 students, showing what learning may be possible for students like Milo, and it suggests a change to the theory of students' multiplicative concepts. Specifically, advanced MC2 students are those who have constructed length meanings for fractions and can coordinate two different two-levels-of-units structures.
Abstractor: As Provided
Entry Date: 2022
Accession Number: EJ1326600
Database: ERIC
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  Value: <anid>AN0154815483;esm01jan.22;2022Jan25.01:51;v2.2.500</anid> <title id="AN0154815483-1">A boundary of the second multiplicative concept: the case of Milo </title> <p>Students entering sixth grade operate with three different multiplicative concepts that influence their reasoning in many domains important for middle school. For example, students who are operating with the second multiplicative concept (MC2 students) can begin to construct fractions as lengths but do not construct improper fractions as numbers. Students who are operating with the third multiplicative concept (MC3 students) can construct both proper and improper fractions as multiples of unit fractions. This paper is a case study of one seventh grade MC2 student, Milo, who demonstrated the most advanced reasoning of all MC2 students in a large project with 13 MC2 and 9 MC3 students. In working on problems involving fractional relationships between two unknowns, most MC3 students constructed reciprocal reasoning. In contrast, the MC2 students struggled with these problems. Similar to the MC3 students, Milo showed some evidence of reciprocal reasoning, and he used proper fractions as operators on unknowns with a rationale. However, Milo did not construct reciprocal reasoning. We account for his reasoning by showing how he used length meanings for proper fractions and how he coordinated two different two-levels-of-units structures. The study expands the mathematics for MC2 students, showing what learning may be possible for students like Milo, and it suggests a change to the theory of students' multiplicative concepts. Specifically, advanced MC2 students are those who have constructed length meanings for fractions and can coordinate two different two-levels-of-units structures.</p> <p>Keywords: Multiplicative concept; Units coordination; Fractions as lengths; Fractions as operators; Equation; Algebra</p> <hd id="AN0154815483-2">Introduction</hd> <p>Students enter sixth grade operating with three different multiplicative concepts that influence their reasoning in many domains important for middle school (Steffe, [<reflink idref="bib24" id="ref1">24</reflink>]). For example, students operating with the first multiplicative concept (MC1 students) can construct fractions as parts within wholes, but for these students fractions do not have a length meaning (Steffe, [<reflink idref="bib21" id="ref2">21</reflink>]). Students operating with the second multiplicative concept (MC2 students) can construct fractions as iterations of parts out of wholes and are beginning to construct a length meaning (Steffe, [<reflink idref="bib22" id="ref3">22</reflink>]), but improper fractions are not yet numbers for these students (Hackenberg, [<reflink idref="bib7" id="ref4">7</reflink>]). Students operating with the third multiplicative concept (MC3 students) can construct both proper and improper fractions as multiples of unit fractions (Hackenberg, [<reflink idref="bib7" id="ref5">7</reflink>]), and fractions do have a length meaning (Steffe, [<reflink idref="bib22" id="ref6">22</reflink>]).</p> <p>These ways of thinking about fractions have a profound influence on students' rational number knowledge. For example, Olive ([<reflink idref="bib19" id="ref7">19</reflink>]) found that only two out of eight third through fifth grade students in a 3-year teaching experiment constructed the "rational numbers of arithmetic" (p. 280) (RNA). The two students were MC3 students, and they made the construction in fifth grade. The RNA is an operational way of thinking about rational numbers that precedes more formal definitions of rational numbers as numbers that can be expressed as a ratio of integers. Specifically, the RNA consists of the all the ways a person has to multiplicatively transform one fraction to another. For example, a student can transform <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mn>3</mn></mfrac></math> </ephtml> to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>4</mn><mn>5</mn></mfrac></math> </ephtml> by thinking as follows: <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mn>3</mn></mfrac></math> </ephtml> times <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>3</mn><mn>2</mn></mfrac></math> </ephtml> is 1, so I need <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>4</mn><mn>5</mn></mfrac></math> </ephtml> of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>3</mn><mn>2</mn></mfrac></math> </ephtml> to scale <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mn>3</mn></mfrac></math> </ephtml> to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>4</mn><mn>5</mn></mfrac></math> </ephtml> .</p> <p>Constructing the RNA requires improper fractions to be numbers for students, and so it is not accessible to MC1 or MC2 students. Steffe ([<reflink idref="bib24" id="ref8">24</reflink>]) has estimated that of the students entering sixth grade, 30% are MC1 students, 30% are MC2 students, and 40% are MC3 students. Thus, constructing the RNA is a possibility for about 40% of incoming middle school students. Yet that number may be smaller based on recent studies that found 29% of 100 sixth grade students in the US were MC2 and MC3 combined (Zwanch & Wilkins, [<reflink idref="bib31" id="ref9">31</reflink>]), and 4.8<emph>%</emph> of 139 fifth through eighth grade students in Turkey were MC3 (Acar & Sevinc, [<reflink idref="bib1" id="ref10">1</reflink>]). So, it is not surprising that in Olive's ([<reflink idref="bib19" id="ref11">19</reflink>]) teaching experiment, only two fifth grade students made the construction.</p> <p>Yet, constructing rational number knowledge is a major curricular goal in middle school (Ministry of National Education (MNE), [<reflink idref="bib16" id="ref12">16</reflink>]; National Council of Teachers of Mathematics (NCTM), [<reflink idref="bib17" id="ref13">17</reflink>]). We study middle school students' rational number knowledge and algebraic reasoning together because working on both domains can be mutually supportive. In our project, we engaged groups of 6–9 middle school students across three iterative, 18-episode design experiments to study relationships between their rational number knowledge and algebraic reasoning. Because of the importance of reciprocal reasoning in the construction of the RNA, in each experiment we worked on Two Unknowns Problems. In a Two Unknowns Problem, two unknown heights are related by a whole number or fraction, such as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>3</mn><mn>5</mn></mfrac></math> </ephtml> (Fig. 1a). In the Fern Sunflower Heights Problem, students can draw pictures (Fig. 1b) and reason that each fifth of the fern height is also one-third of the sunflower height, so the fern height is five one-thirds, or <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>5</mn><mn>3</mn></mfrac></math> </ephtml> , of the sunflower height, thereby constructing a reciprocal relationship.</p> <p>Graph: Fig. 1 a and b Fern Sunflower Heights Problem (left) and student picture (right)</p> <p>Across the three experiments in our project, 9 MC3 students and 13 MC2 students participated. We found that seven of the nine MC3 students constructed reciprocal reasoning (Hackenberg & Sevinc, [<reflink idref="bib10" id="ref14">10</reflink>]), while none of the MC2 students did (Hackenberg et al., [<reflink idref="bib11" id="ref15">11</reflink>]). The purpose of this paper is to propose a change in the theory of students' multiplicative concepts based on the case of Milo, an MC2 student who demonstrated the most advanced reasoning on Two Unknowns Problems of all 13 MC2 students. Our research questions are: What reasoning did Milo demonstrate that was more advanced than other MC2 students? Why did he demonstrate more advanced reasoning?</p> <hd id="AN0154815483-3">Theoretical framework</hd> <p></p> <hd id="AN0154815483-4">Students' multiplicative concepts</hd> <p>Students' multiplicative concepts refer to how students coordinate units as they construct number and quantities (Ulrich, [<reflink idref="bib28" id="ref16">28</reflink>]). A <emph>unit</emph> is a discrete item, length, or standard measurement unit, and a <emph>composite unit</emph> is a unit of units. Here we discuss length and fraction units for students operating at each of the three multiplicative concepts typical in middle school.</p> <p>MC1 students can view a length as unit of units, but they have to actually make parts rather than imagine them. Once they break a length into parts, they do not think of the parts as parts of the whole length—they do not maintain that part-to-whole relationship (Steffe, [<reflink idref="bib21" id="ref17">21</reflink>]). As a result, they construct parts-within-wholes fraction schemes (Hackenberg, [<reflink idref="bib8" id="ref18">8</reflink>]).</p> <p>In contrast, MC2 students can imagine a length partitioned into equal parts without making it. A main reason is that they can use a unit of units, or two levels of units, to structure a situation prior to acting. In addition, MC2 students have constructed 1 as an <emph>iterable unit</emph>, which means that a number word like 180 can refer to a unit that is iterated 180 times, rather than to 180 distinct, sequential units (Ulrich, [<reflink idref="bib28" id="ref19">28</reflink>]). MC2 students can also create three levels of units as they solve problems. For example, they can determine that nine 20s is 180. However, they do not maintain this structure as they operate further: 180 becomes 180 1s.</p> <p>MC2 students can use their iterable unit of 1 when partitioning continuous quantities to create fractions (Steffe, [<reflink idref="bib22" id="ref20">22</reflink>]). For example, MC2 students can create <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>3</mn><mn>5</mn></mfrac></math> </ephtml> of a bar by partitioning the bar into five equal parts, disembeddding one part, and iterating the part three times. Yet, MC2 students think of the result as three parts out of five, a part out of a whole, so they have not yet constructed a multiplicative relationship between the unit fraction and the whole (Steffe, [<reflink idref="bib22" id="ref21">22</reflink>]). Their construction, a <emph>partitive fraction scheme</emph>, marks the beginning of a length meaning for fractions (Steffe, [<reflink idref="bib22" id="ref22">22</reflink>]).</p> <p>In contrast, MC3 students can imagine a length partitioned into equal parts, each of which are partitioned into equal parts. They can use a unit of units of units, or three levels of units, to structure a situation prior to acting. MC3 students can use these abilities to construct <emph>iterative fraction schemes</emph> (Hackenberg, [<reflink idref="bib7" id="ref23">7</reflink>]), where there is an explicit multiplicative relationship between a unit fraction, the unit, and any fraction made from it. For example, students who have constructed iterative fraction schemes see <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>9</mn><mn>7</mn></mfrac></math> </ephtml> as 9 times <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>7</mn></mfrac></math> </ephtml> , and also as 1 whole consisting of 7 times <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>7</mn></mfrac></math> </ephtml> and two more sevenths.</p> <hd id="AN0154815483-5">A quantitative approach</hd> <p>We seek to support students to construct fractions knowledge, one piece of rational number knowledge, and algebraic reasoning based on reasoning with quantitative relationships (Thompson, [<reflink idref="bib27" id="ref24">27</reflink>]). A <emph>quantity</emph> is a property of one's concept of an object or phenomenon, along with a measurement unit and process (Thompson, [<reflink idref="bib27" id="ref25">27</reflink>]). For example, to conceive of a person's height as a quantity according to Thompson's definition requires conceiving of a unit and of the distance from the foot to the top of the head subdivided into these units. Counting is one process to enumerate these units. In our approach to algebraic reasoning, we support students to abstract and generalize quantitative relationships, represent them systematically, and later reason with algebraic notation in lieu of quantities (Kaput, [<reflink idref="bib12" id="ref26">12</reflink>]).</p> <hd id="AN0154815483-6">Equation writing</hd> <p>One example of algebraic reasoning is writing equations to relate unknowns, which is challenging for secondary students (e.g., Hackenberg & Lee, [<reflink idref="bib9" id="ref27">9</reflink>]; Kloosterman, [<reflink idref="bib13" id="ref28">13</reflink>]; MacGregor & Stacey, [<reflink idref="bib14" id="ref29">14</reflink>]). Research shows that although many algebra students indicate verbally that they understand relationships between quantities in story problems, far fewer successfully express those relationships with algebraic notation (MacGregor & Stacey, [<reflink idref="bib14" id="ref30">14</reflink>]). Indeed, Olive and Çağlayan ([<reflink idref="bib20" id="ref31">20</reflink>]) studied how two pairs of eighth grade students, one MC2 and one MC3, worked on a problem involving relationships between the number of coins and their values. Only the MC3 students constructed a quantitative structure that supported complete equation writing. Even college students have been found to represent the situation "there are six times as many students as professors" with the equation 6<emph>S</emph> = <emph>P</emph>, where <emph>S</emph> was the number of students and <emph>P</emph> the number of professors (Clement, [<reflink idref="bib3" id="ref32">3</reflink>]). This famous reversal error was resilient and based in part on using 6 to indicate that the number of students was greater than the number of professors. MacGregor and Stacey ([<reflink idref="bib14" id="ref33">14</reflink>]) found similar reversals among secondary students.</p> <p>A quantitative approach to algebraic reasoning has been found to support students' learning (Ellis, [<reflink idref="bib6" id="ref34">6</reflink>]; Olive & Çağlayan, [<reflink idref="bib20" id="ref35">20</reflink>]). For example, Ellis ([<reflink idref="bib6" id="ref36">6</reflink>]) demonstrated that reasoning with quantities was a pivotal resource in the construction of emergent ratios that formed the basis for students' understanding of slope. Her study is an example of how rational number knowledge is important in algebraic reasoning. Hackenberg and Sevinc ([<reflink idref="bib10" id="ref37">10</reflink>]) have studied a different connection between rational number knowledge and algebraic reasoning: Knowledge of reciprocals, which are needed to construct the RNA, are important in writing equations, an algebraic topic. Hackenberg and Sevinc found that only MC3 students constructed reciprocal reasoning when solving Two Unknowns Problems (Fig. 1). As noted earlier, these students viewed the fern height as five one-thirds of the sunflower height. Then these students wrote equations like <emph>y</emph> = <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>3</mn><mn>5</mn></mfrac></math> </ephtml><emph>x</emph> and <emph>x</emph> = <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>5</mn><mn>3</mn></mfrac></math> </ephtml><emph>y</emph>, where <emph>x</emph> represented the fern height and <emph>y</emph> the sunflower height. To construct reciprocal reasoning, the students made an accommodation, or reorganization, in their iterative fraction schemes.</p> <p>In contrast, we found the MC2 students were challenged to represent multiplicative relationships between unknowns because of the units coordination required to do so (Hackenberg et al., [<reflink idref="bib11" id="ref38">11</reflink>]). MC2 students simplified the units coordination in several ways, including not fixing the multiplicative relationship and using numerical examples in lieu of unknowns. For example, MC2 student Connor worked on a Two Unknowns Problem where one unknown was <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>4</mn></mfrac></math> </ephtml> of another unknown. He assigned 20 inches for one height and 5 inches for the other. His picture showed one height as a 4-part bar and the other as one of those parts. But his equation was <emph>B</emph> ÷ <emph>C</emph> = <emph>D</emph> where <emph>B</emph> was the larger height, <emph>C</emph> the smaller height, and <emph>D</emph> "the answer." That is, neither <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>4</mn></mfrac></math> </ephtml> nor 4 appeared in his equation. MC2 students also did not use fractions as operators on unknowns. So, there appears to be a marked contrast between MC2 and MC3 students' work on Two Unknowns Problems. Yet, when working on Two Unknowns Problems, Milo showed some evidence of reciprocal reasoning, similar to the MC3 students.</p> <hd id="AN0154815483-7">Method</hd> <p>Milo participated in the third of three 18-episode design experiments conducted over three semesters after school with students in a middle school in the USA to study relationships between students' rational number knowledge and algebraic reasoning.</p> <hd id="AN0154815483-8">Participant selection and data collection</hd> <p>To select students for the third experiment, we conducted 30-min interviews with 24 seventh and eighth grade students and asked them to complete a 12-item worksheet. This selection process assessed students' multiplicative concepts and fractions knowledge. We assessed multiplicative concepts by analyzing student work from the interviews, following prior research (e.g., Hackenberg & Lee, [<reflink idref="bib9" id="ref39">9</reflink>]). We used scoring guidelines (Norton et al., [<reflink idref="bib18" id="ref40">18</reflink>]) to analyze student responses on the worksheet, using these responses to triangulate inferences from the interviews. We aimed to have three participants operating with each multiplicative concept in the experiment, but MC1 students declined to participate. So, we selected three MC3 and six MC2 students. Attrition led to three MC3 and three MC2 students who completed the experiment. Across the three experiments, 9 MC3 students and 13 MC2 students participated to completion.</p> <p>The hour-long episodes began 1 week after the selection interviews ended and occurred twice per week for 9 weeks. They were video-recorded with one stationary and two roaming cameras. During episodes, students worked on problems in groups of two or three using a laptop with a program called JavaBars (Biddlecomb & Olive, [<reflink idref="bib2" id="ref41">2</reflink>]). In JavaBars, students could draw and partition bars (rectangles), take parts out of bars, and repeat parts and bars. Students' computer work was recorded with Screenflow (Telestream LLC, [<reflink idref="bib26" id="ref42">26</reflink>]).</p> <p>One researcher (the first author) was the teacher for the episodes. She formulated goals, planned tasks, held class discussions based on goals and students' work, and inquired responsively when students worked in groups. Other research team members operated cameras, took notes, and inquired responsively. After each episode, the team processed video, wrote summaries and conjectures about student reasoning, and met to discuss plans for the next episode. The first author watched all Screenflow (Telestream LLC, [<reflink idref="bib26" id="ref43">26</reflink>]) videos after each episode and kept a research journal. Following each experiment, each student participated in a 45-min interview to assess their current understanding of topics. In Milo's experiment, we spent four episodes on Two Unknowns Problems, so those episodes and Milo's interviews were our main data sources (Table 1, shaded rows).</p> <p>Graph</p> <hd id="AN0154815483-9">Data analysis</hd> <p>To analyze Milo's reasoning, we developed a second-order model (Steffe & Thompson, [<reflink idref="bib25" id="ref44">25</reflink>]) of Milo. A second-order model is generated from researchers' theoretical constructs, models from prior research, and a commitment to use constructs in an orienting but not deterministic way (Clement, [<reflink idref="bib4" id="ref45">4</reflink>]). Because we were comparing Milo to the other 21 students across experiments, we also built second-order models of these students, focusing on the episodes in each experiment devoted to Two Unknowns Problems.</p> <p>To build second-order models, we repeatedly examined video records and student work. We looked for regularities and changes in students' ways of operating. When we observed changes, we traced them back to interactions that may have been involved in the changes (Cobb et al., [<reflink idref="bib5" id="ref46">5</reflink>]). During this process, we wrote analytical memos (Miles et al., [<reflink idref="bib15" id="ref47">15</reflink>]), conjectures, and explanations for our observations. A 6-member research team met regularly to watch video and revise interpretations. Then we wrote a narrative portrait of each student that explains the student's second-order model. Next, we compared the models. We created comparison documents that tracked differences and similarities in the students' schemes and in how the students reasoned on Two Unknown Problems. From these documents, we identified Milo as an important case.</p> <hd id="AN0154815483-10">Initial model of Milo</hd> <p>In this section, we communicate our understanding of Milo at the start of the experiment.</p> <hd id="AN0154815483-11">Milo's multiplicative concept</hd> <p>Our analysis of Milo's work on two problems in his selection interview and nine of 12 worksheet items indicate that Milo was an MC2 student at the start of the study. For example, in the selection interview, we posed the Crate Problem (Fig. 2).</p> <p>Graph: Fig. 2 The Crate Problem</p> <p>Milo computed the number of cans by multiplying 4 × 8 and then 32 × 6. Then he drew three separate pictures: a small rectangle with 4 circles in it, a medium-sized rectangle partitioned into 8 parts (a box), and a large rectangle partitioned into 6 parts (a crate). When asked how these pictures were related, he said, "they're all carrying the same thing. They all have cans of juice in them. They're all packaged." When pressed about what was in the parts in each larger rectangle, he answered appropriately: 4 cans in each part in the box, 32 cans in each part in the crate. So, although Milo did not draw a picture with all parts embedded, he successfully solved the problem and indicated embedment with his comments.</p> <p>Yet, in continued work on the problem, Milo dealt primarily with cans, packages, and the crate, omitting boxes, as we will show. The interviewer asked Milo to find the number of packages in the crate, and he divided 192 by 4, getting 48. When asked how he would see that in his picture, he pointed to the 6-part rectangle. When asked what he would see, he said, "it would be a very complicated drawing because you'd have 4 cans would be drawn inside the 8 packages and you'd have to draw 8 packages inside of 6 boxes." When asked to show that, he began to partition one part of the 6-part rectangle, mentioning "48 divided by 6 would be 8 in each one." He partitioned that part into 8 mini-parts. Here, he appeared to produce the 8 packages in each of the 6 parts (boxes) by calculating, even though he had repeatedly mentioned 8 packages in a box during his work on the problem. Then he partitioned each of the other parts of his crate into 6 parts, not 8. Although he corrected this when it was brought to his attention later, he never used 6 × 8 as a way to find the number of packages, even when questioned for a second way. Thus, he appeared to base his drawing of packages on segmenting the 48 packages in a crate into 6 equal groups in his activity, not on viewing the crate a priori as 6 eights.</p> <p>Milo's work is not strong enough evidence to conclude that he was an MC3 student. First, Milo did not draw embedded units beyond the 4 cans in the package without a request. In our experience, MC3 students almost always draw embedded units, usually inside one picture of a crate. Second, he indicated that the crate consisted of 32 cans in each box, but it seemed difficult for him to see that there were also 8 packages in each box and that 6 × 8 could determine the number of packages. Instead, to determine the number of packages, he treated the entire 192 cans as a unit of 4s, somewhat ignoring boxes. Although doing so is not incorrect, it shows a lack of incorporation of more than two levels of units: Milo appeared to be most comfortable thinking about the crate as 48 packages of 4 cans (two levels of units) or as 48 packages sorted into 6 boxes (two levels of units). He did not use 8 packages in a box as what a box contained but instead made that coordination in solving the problem. In our experience, not coordinating packages and boxes is typical of MC2 student work on this problem. Milo's responses on his worksheet corroborated our assessment that he was an MC2 student.</p> <hd id="AN0154815483-12">Milo's fractions knowledge</hd> <p>We drew conclusions about Milo's fractions knowledge from three problems in his selection interview; he did not respond to three fractions items on the worksheet. The Seven-Fifths Problem (Fig. 3a) assessed the construction of an iterative fraction scheme.</p> <p>Graph: Fig. 3 a and b Seven-Fifths Problem (left) and Milo's drawing (right)</p> <p>On this problem, Milo interpreted the bar as 7 inches long and asked for a ruler. When asked if he could draw the other bar without a ruler, he drew "seven and a half" hops along a bar that extended beyond the given bar (Fig. 3b). He identified the solution as <emph>only</emph> this extended bar. So, he appeared to interpret seven-fifths as seven and a half units, where those units did not have a clear relationship to the given bar. Thus, he did not demonstrate an iterative fraction scheme at the start of the study.</p> <p>Milo also shared two equal bars equally among five people. To do so, he partitioned each bar into five equal parts. He said each person got two parts because it was "1 person for 2 pieces." So, he appeared to view the 10 pieces in groups of two for each of the five people. He drew out the 2-part share for one person and called it "two-fifths." Here Milo showed some evidence of a partitive fraction scheme.</p> <hd id="AN0154815483-13">Milo's equation work</hd> <p>During the first episode, students worked on a 3-item pre-assessment that addressed main topics in the experiment, such as relating two unknowns with pictures and equations. Milo worked on the Two Heights Problem (Fig. 4). In part (a), Milo drew a picture of a tree and a school, but no segments to show heights. For both equations, he wrote <emph>q</emph> = <emph>w</emph>, writing that the heights were the same. Here Milo did not show evidence of solving a Two Unknowns Problem.</p> <p>Graph: Fig. 4 Two Heights Problem in pre-assessment</p> <hd id="AN0154815483-14">Milo's reasoning on Two Unknowns Problems with fractional relationships</hd> <p>Yet as noted, Milo made the most progress on Two Unknowns Problems out of all 13 MC2 students. Like many of them, when the relationship was a whole number, he came to regard these problems as involving two equations, one involving whole number multiplication and one involving whole number division (Hackenberg et al., [<reflink idref="bib11" id="ref48">11</reflink>]). Unlike other MC2 students, he appeared to develop this view even with fractional relationships, and he was the only MC2 student across the experiments to use a fraction as a multiplier of an unknown with a rationale. Milo's work shows that he had constructed a length meaning for fractions that distinguished him from the other MC2 students. It also shows that he coordinated two two-levels-of-units structures but not two three-levels-of-units structures, like the MC3 students. We present evidence for these claims from episodes 13 through 15 and Milo's follow-up interview.</p> <hd id="AN0154815483-15">A ¼ relationship</hd> <p>In episode 13, Milo worked on the Dog Heights Problem (Fig. 5a), drawing a picture that showed a <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>4</mn></mfrac></math> </ephtml> relationship from our perspective (Fig. 5b). Initially, he labeled Sadie as the larger height, but when explaining to his two MC2 groupmates, he changed the picture so that Sadie was the smaller height. In contrast, his MC2 groupmates seemed stumped about how to draw a picture or which dog was taller. The teacher spread her hands to pantomime a height for Riley and asked them to show how tall Sadie was. One groupmate put her hand in the middle of the teacher's hands; Milo put his hand about one-fourth of the way above the teacher's lower hand; and the third groupmate eventually agreed with Milo. Shortly after, the teacher asked whether there was a 4 times relationship in the problem. Milo said yes immediately: "Riley's height is four times the height of Sadie's." In contrast, his groupmates did not respond. Thus, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>4</mn></mfrac></math> </ephtml> appeared to imply a four times relationship to Milo. Furthermore, his picture and hand gestures indicate that he had constructed a length meaning for fourths. Our interpretation is that for Milo, one-fourth of a length meant a part that could be taken four times to make the length.</p> <p>Graph: Fig. 5 a and b Dog Heights Problem (left) and Milo's drawing (right)</p> <p>When Milo wrote equations, initially he used 5 as a multiplier, similar to other MC2 students who changed the relationship on these problems (Hackenberg et al., [<reflink idref="bib11" id="ref49">11</reflink>]). However, when asked about how he saw the five times, Milo said, "Times 4—it's 4." Then he revised his equations to "<emph>S</emph> * 4 = <emph>R</emph>"[<reflink idref="bib1" id="ref50">1</reflink>] and "<emph>R</emph> ÷ 4 = <emph>S</emph>," where <emph>S</emph> was "Sadie's height" and <emph>R</emph> "Riley's height."</p> <p>Milo skipped part (d) of the Dog Heights Problem and worked on part (e), checking his equations with a numerical example. He computed 76 ÷ 4 to get 19, substituted both 76 and 19 in his two equations, and verified them. Then a teacher asked whether he could write an equation with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>4</mn></mfrac></math> </ephtml> . Milo said that he could do <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>4</mn></mfrac></math> </ephtml> × 76 instead of 76 ÷ 4. She asked what an equation with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>4</mn></mfrac></math> </ephtml> would look like, and Milo wrote " <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>4</mn></mfrac><mspace width="0.25em" /></math> </ephtml> * <emph>R</emph> = <emph>S</emph>". When asked why he could multiply, he said when you "multiply the fraction in the equation, and that's the division." The teacher asked why multiplying by a fraction was division, and he said he did not know. The teacher asked if multiplying by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>4</mn></mfrac></math> </ephtml> made the quantity smaller or larger. Milo said smaller. When asked why, he paused. Then he said it was because "it's trying to pull out one-fourth." The teacher asked whether that worked for multiplying by 2, and he said no, "it has to be a fraction."</p> <p>Here, Milo used a fraction as a multiplier of an unknown with a rationale: Multiplying by a fraction "pulled out" that fractional part of the unknown height. Our interpretation is that he knew empirically that multiplying a known quantity by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>4</mn></mfrac></math> </ephtml> produced the same result as dividing by 4. With respect to unknowns, we suggest he made an analogy: If the 4-part bar was Riley's unknown height, then taking <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>4</mn></mfrac></math> </ephtml> of it produced the 1-part bar, Sadie's unknown height. Based on this analogy, he rewrote <emph>R</emph> ÷ 4 = <emph>S</emph> as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>4</mn></mfrac></math> </ephtml> * <emph>R</emph> = <emph>S</emph>. When pressed by the teacher about why that worked, he communicated the actions he was taking to find <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>4</mn></mfrac></math> </ephtml> of a height—pulling it out. However, work on a problem with a <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mn>5</mn></mfrac></math> </ephtml> relationship revealed that Milo did not have this meaning for multiplying by fractions larger than 1.</p> <hd id="AN0154815483-16">A 2/5 relationship</hd> <p>At the end of episode 13, Milo started work on the Tree Heights Problem (Fig. 6a). He drew a tall bar, partitioned it into five equal parts, pulled out two parts, and joined them (Fig. 6b). Similar to the Dog Heights Problem, initially he said that the crabapple tree height was the 5-part bar, a reversal. However, as he explained his drawing to his MC2 groupmates, he corrected his picture. We note that four of the nine MC2 students who worked on this problem across the experiments did not make a drawing without considerable questioning support from a teacher.[<reflink idref="bib2" id="ref51">2</reflink>] For example, just as with the Dog Heights Problem, Milo's groupmates had difficulty determining which tree was taller. Our interpretation is that Milo thought of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mn>5</mn></mfrac></math> </ephtml> of a quantity—even an unknown quantity—as two parts that were each <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>5</mn></mfrac></math> </ephtml> of that quantity. So, his work in episode 13 demonstrated a length meaning for proper fractions, not just unit fractions.</p> <p>Graph: Fig. 6 a and b Tree Heights Problem (left) and Milo's drawing (right)</p> <hd id="AN0154815483-17">Initial equations</hd> <p>During the next episode (<reflink idref="bib14" id="ref52">14</reflink>), Milo wrote his first equation for the Tree Heights Problem: "<emph>m</emph> * <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mn>5</mn></mfrac></math> </ephtml> = <emph>c</emph>," where <emph>m</emph> was "maple tree height" and <emph>c</emph> was "crabapple tree height." He asked a teacher what <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mn>5</mn></mfrac></math> </ephtml> was as a decimal, divided 2 by 5 on a calculator, and wrote "<emph>c</emph> * 0.4 = <emph>m</emph>." When asked to explain his first equation, he said, "Because if you multiply <emph>m</emph> by two-fifths, it pulls out two-fifths." So, he drew directly on his idea from episode 13. When asked to explain "<emph>c</emph> * 0.4 = <emph>m</emph>," he stated the process of finding the decimal for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mn>5</mn></mfrac></math> </ephtml> . The teacher asked if that would be his pulling out idea, and he said, "No, that would be adding it, I think." Later he said, "No, you'd be multiplying. That could be division. That could be <emph>m</emph> divided by oh point 4."</p> <p>Our interpretation is that for Milo, pulling <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mn>5</mn></mfrac></math> </ephtml> of a height out of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>5</mn><mn>5</mn></mfrac></math> </ephtml> of that height was expressed by the equation <emph>m</emph> * <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mn>5</mn></mfrac></math> </ephtml> = <emph>c</emph>. Then he converted the fraction to a decimal and switched the order of the heights to write <emph>c</emph> * 0.4 = <emph>m</emph>. However, he did not see <emph>c</emph> * 0.4 as pulling <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>4</mn><mn>10</mn></mfrac></math> </ephtml> out of <emph>c</emph>, perhaps because he knew he had to increase the crabapple tree height to create the maple tree height. Thus, he suggested "adding it." Finally, he suggested dividing by 0.4, likely using his idea of developing two equations, one with multiplication and one with division.</p> <hd id="AN0154815483-18">Switching the referent unit</hd> <p>The teacher asked Milo how many times the smaller height fit into the larger height in the Tree Heights Problem, Milo said "twice" and then, "it goes in 2.5 times." Then he wrote "2.5 * <emph>c</emph> = <emph>m</emph>." To show how he saw the 2.5 relationship, he dragged the 2-part bar along the 5-part bar two times, stopped before the top part, and said, "then you cut that in half and put that [a 1-part bar] on." Here Milo interpreted the maple tree height as 2 crabapple tree heights and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math> </ephtml> of crabapple tree height. This activity is evidence that he switched the referent unit to the smaller height, measuring the larger height with that referent unit.</p> <p>Milo then used his own numerical example to test his equations: <emph>m</emph> * <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mn>5</mn></mfrac></math> </ephtml> = <emph>c</emph>, <emph>c</emph> * 0.4 = <emph>m</emph>, and 2.5 * <emph>c</emph> = <emph>m</emph>. When his test of <emph>c</emph> * 0.4 = <emph>m</emph> did not work out, he crossed out all equations and wrote "<emph>c</emph> * 2.5 = <emph>m</emph>" and "<emph>m</emph> ÷ 2.5 = <emph>c</emph>". We propose that he believed in the equation <emph>c</emph> * 2.5 = <emph>m</emph> because he could see how 2 small heights and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math> </ephtml> of a small height fit into the large height. Since he believed in that relationship, he rejected the initial equations and used 2.5 to write equations similar to ones he had written before, a multiplication equation and a division equation.</p> <hd id="AN0154815483-19">Writing an equation with a fraction</hd> <p>Then the teacher asked Milo to determine what fraction the maple height was of the crabapple height. Milo said one-half, then two-fifths, and then, after asking the teacher to repeat the question, "five two." When asked what that meant, he said he would "flip the fraction."</p> <p>Data Excerpt: Milo and the teacher talk about "five two."</p> <p>T: Why? How does that work? Can you explain that in your picture?M: Maybe.T: Okay. That's what I'm really curious about.M: This goes with this [moving the 2-part bar along the 5-part bar]. Wait, what did I say?T: You said "five two."M: Five two-ths. I hate saying that, when you have something over what the number is. You have like sixteen fifteenths. Or you have sixteen one-hundredths. It's just like...T: It's cumbersome to say it? How would you write it?M: Five over two [" <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>5</mn><mn>2</mn></mfrac></math> </ephtml> "].T: So how do you see that fraction in your picture?M: It's like putting this [5-part bar] inside of this [2-part bar]. And then what you're putting in you always put on the bottom. Or what you're putting in is on the top, and what is there is what you need for the bottom.T [10 s later]: Some people say this number [ <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>5</mn><mn>2</mn></mfrac></math> </ephtml> ] is "five-halves."[Milo frowns and shakes his head no.]</p> <p>Our interpretation is that Milo switched the numerator and denominator in the fraction and seemed to believe that was an appropriate response to the question of what fraction the maple tree height was of the crabapple tree height. However, he did not seem to have a meaning for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>5</mn><mn>2</mn></mfrac></math> </ephtml> beyond this switch in numbers—it was not five one-halves to him, and in explaining how he saw the fraction in the picture he focused on putting parts inside of other parts and mapping that to notation. Therefore, he did not demonstrate a length meaning for "five two," and the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>5</mn><mn>2</mn></mfrac></math> </ephtml> did not appear to come from measuring the larger height with the smaller one, as 2.5 had.</p> <p>Nevertheless, at the end of the episode, Milo wrote " <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>5</mn><mn>2</mn></mfrac></math> </ephtml> * <emph>c</emph>" as an "equation" and showed the teacher. She asked, "Okay, equals what?" He said softly, "It comes out to 2.5." He wrote " <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>5</mn><mn>2</mn></mfrac></math> </ephtml> * <emph>c</emph> = 2" and proclaimed, "I did it!" Here, Milo wrote an expression for the maple tree height, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>5</mn><mn>2</mn></mfrac></math> </ephtml> * <emph>c</emph>, that appeared correct to us. However, since <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>5</mn><mn>2</mn></mfrac></math> </ephtml> did not indicate to him the size of the maple tree height as measured by the crabapple tree height, the expression <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>5</mn><mn>2</mn></mfrac></math> </ephtml> * <emph>c</emph> did not represent the maple tree height to him. Instead, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>5</mn><mn>2</mn></mfrac></math> </ephtml> seemed to be something to compute to a number that was more understandable, like 2.5. Thus, our interpretation is that he completed his equation by recording how he interpreted the number <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>5</mn><mn>2</mn></mfrac></math> </ephtml> , at least partially, since he wrote 2 and not 2.5 in " <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>5</mn><mn>2</mn></mfrac></math> </ephtml> * <emph>c</emph> = 2."</p> <hd id="AN0154815483-20">Follow-up work with Milo</hd> <p>In episode 15, students completed an exit card about Two Unknowns Problems that was tailored to their work during episodes 13 and 14. To Milo, we posed the Fern Sunflower Heights Problem (Fig. 1). He drew a picture similar to those from episodes 13 and 14 but with the fern height <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>3</mn><mn>5</mn></mfrac></math> </ephtml> of the sunflower height, a reversal. He wrote only one equation, " <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>3</mn><mn>5</mn></mfrac></math> </ephtml> * <emph>s</emph> = <emph>f</emph>," which matched his picture but not the problem. He did not respond to the question about what fraction the fern height was of the sunflower height. At this point, from our perspective, Milo drew a picture for each Two Unknowns Problem in which the quantities were reversed. In each case, he labeled the larger height as the height that was stated first in the problem, which was always the fractional part of the larger height. Yet, he corrected the first two reversals when he explained his pictures to others. So, we suggest he was focused on drawing the stated fraction, and he assigned meanings of the heights based on the order in which he read them. We do not think the reversals indicate a significant conceptual issue because he corrected them himself when explaining to others. In episode 15, he did not explain his picture and so did not have a chance to revise it.</p> <p>In Milo's follow-up interview, about 1 month after the Two Unknowns Problems, we were eager to assess whether he had constructed reciprocal reasoning. We posed the Plant Heights Problem (Fig. 7a), and Milo drew a 7-part bar and 3-part bar (Fig. 7b). Then he operated similarly to episode 14: He wrote "<emph>S</emph> * <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>3</mn><mn>7</mn></mfrac></math> </ephtml> = <emph>L</emph>" and "<emph>L</emph> *.42 = <emph>S</emph>," where <emph>S</emph> was Steve's plant height, <emph>L</emph> was Lia's plant height, and the.42 came from converting <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>3</mn><mn>7</mn></mfrac></math> </ephtml> to a decimal.</p> <p>Graph: Fig. 7 a and b Plant Heights Problem (left) and Milo's drawing (right)</p> <p>When asked how many times Lia's height fit into Steve's height, he said two full times and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>7</mn></mfrac></math> </ephtml> , not two and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>3</mn></mfrac></math> </ephtml> . So, he named the remaining part <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>7</mn></mfrac></math> </ephtml> in relation to <emph>S</emph> as the referent unit and did not complete the switch in referent unit as he had in episode 14. His work contrasts with MC3 students who all made complete switches (Hackenberg & Sevinc, [<reflink idref="bib10" id="ref53">10</reflink>]). Our interpretation is that Milo made a complete shift of referent unit in episode 14 because operating with halves of a 2-part bar is more intuitive than operating with other fractional parts. We have seen some MC3 students begin to switch the referent unit and name the fractional part in relation to the original referent unit. However, with questioning, MC3 students complete the switch. In contrast, even under repeated questioning, Milo continued to view <emph>S</emph> as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mfrac><mn>1</mn><mn>7</mn></mfrac></math> </ephtml> times <emph>L</emph>.</p> <p>In addition, during the follow-up interview, we posed the Seven-Fifths Problem to Milo (Fig. 3a). He drew <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>5</mn><mn>7</mn></mfrac></math> </ephtml> of the given bar. He solved two more problems similarly. Thus, Milo had not constructed an iterative fraction scheme. However, later in the interview he showed evidence that he had constructed a <emph>reversible</emph> partitive fraction scheme. He did so by solving a problem in which he was given an unmarked bar that was <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>3</mn><mn>5</mn></mfrac></math> </ephtml> of A, and he was to draw bar A. He partitioned the bar into three equal parts and added two more of those parts onto the bar. Our assessments of his multiplicative concept at the end of the study indicate that he was still an MC2 student.</p> <hd id="AN0154815483-21">Account of Milo's work on Two Unknowns Problems</hd> <p>Now we make an account of Milo's work to demonstrate why Milo is an example of a well-defined advanced MC2 student.</p> <p>Milo's work on the Tree Heights Problem (Fig. 6a) indicates the following: For Milo, <emph>m</emph> was a height, and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>5</mn></mfrac></math> </ephtml> of <emph>m</emph> was a height that could be iterated to make <emph>c</emph>, another height. This view was made possible by Milo's length meaning for fractions and his units coordination: <emph>m</emph> was a unit of five units, and two of those units made <emph>c</emph>. A length meaning for fractions of unknowns and his a priori coordination of two levels of units supported him to write the equation <emph>c</emph> = <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mn>5</mn></mfrac></math> </ephtml> * <emph>m</emph> and speak about this equation as pulling out <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mn>5</mn></mfrac></math> </ephtml> of <emph>m</emph> to make <emph>c</emph>. In contrast, the other MC2 students across experiments found the Tree Heights Problem challenging because they did not have well-developed length meanings for fractions.</p> <p>Then, Milo used <emph>c</emph> as a unit of two length units to structure <emph>m</emph> as two units of <emph>c</emph> with a leftover part that was one-half of <emph>c</emph>. This second view of <emph>m</emph> requires creating three levels of units: <emph>m</emph> is a unit of more than two units, and each of those units consists of two units. We infer that Milo made this structure as he worked on the problem because he measured <emph>m</emph> with <emph>c</emph> and got 2 <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math> </ephtml><emph>c</emph>. However, we conclude that Milo was making this structure in his activity because he did not also see <emph>m</emph> as measured by units of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math> </ephtml> of <emph>c</emph>. In other words, five <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math> </ephtml> units of c was not a length for him—he did not have a length meaning for fractions once he got into the terrain of improper fractions. So, he did not hold out the three-levels-of-units structure that he had created for <emph>m</emph> as a mixed number, a unit of more than two units each containing two units, and rearrange it to view <emph>m</emph> as five units, any of which could be iterated twice to make <emph>c</emph>. If Milo had created <emph>m</emph> as a three-levels-of-units structure prior to operating, as MC3 students did, we argue that he could have viewed <emph>m</emph> in these two ways. That is precisely what interiorizing three levels of units allows—that a student can hold the three levels of units out for reflection and then switch to a different structural view.</p> <p>Instead, we propose that Milo switched from viewing <emph>m</emph> as a unit of five units, one two-levels-of-units structure, to a unit of two units of <emph>c</emph> with an implicit leftover part, another two-levels-of-units structure. In other words, the three levels of units that Milo used to measure <emph>m</emph> with <emph>c</emph> turned into measuring <emph>m</emph> with only whole units of <emph>c</emph>. This may be another reason that Milo wrote "2" as what "5/2*<emph>c</emph>" was equal to. In other words, his equation expressed <emph>m</emph> as 2 units of <emph>c</emph>, where the leftover part was somewhat "negligible" for him. Thus, ultimately Milo coordinated two different two-levels-of-units structures. We claim that his partitive fraction scheme with well-developed length meanings for fractions and his ability to switch between two two-levels-of-units structures mark him as an advanced MC2 student.</p> <hd id="AN0154815483-22">Discussion</hd> <p>A natural question is why Milo was able to demonstrate these abilities when other MC2 students did not. He worked hard, and he was very engaged with us and the problems. As Ulrich ([<reflink idref="bib29" id="ref54">29</reflink>]) has stated about Adam, an advanced MC1 student, students who work hard at their stage of units coordination for an extended period of time may develop advanced ways of using their current operations. We think that was true of Milo. Thus, this study points to the learning that is possible for MC2 students. It suggests that MC2 students can develop fairly general ways of solving Two Unknowns Problems with whole number and some fractional relationships: unit fractions and fractions that consist of two unit fractions (e.g., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mn>5</mn></mfrac></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mn>7</mn></mfrac></math> </ephtml> ) whose reciprocals are a whole number and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math> </ephtml> (e.g., 2 <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mn>3</mn><mfrac><mn>1</mn><mn>2</mn></mfrac></math> </ephtml> ). As shown with Milo, measuring the larger height with whole units of the smaller height is possible for these students, but the leftover part is not integrated into the multiplicative structure—it is implicit. Yet because students have intuition with halves, they can name the multiplicative relationship appropriately as a mixed number.</p> <p>Indeed, our analysis suggests that a better follow-up problem for Milo in episode 15 would have been a Two Unknowns Problem with a <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mn>7</mn></mfrac></math> </ephtml> relationship to test whether he could generalize his ideas about measuring the larger height with iterations of the smaller height and one-half of the smaller height. In addition, we wonder whether Milo could learn to fully switch the referent unit and develop mixed number multipliers for fractional relationships during design episodes. For example, with dedicated time during an episode, could he have developed 2 <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>3</mn></mfrac></math> </ephtml> as a multiplier in the Plant Heights Problem? These questions point to a difference between Milo and most of the MC3 students in the experiments who, by constructing reciprocal reasoning, created a way to solve all Two Unknowns Problems, not particular cases of them.</p> <p>Yet, Milo is not alone: Two other MC2 students with mathematical activity more advanced than expected have been identified in other studies, sixth grade Bridget (Hackenberg, [<reflink idref="bib7" id="ref55">7</reflink>]) and seventh grade Samantha (Hackenberg & Lee, [<reflink idref="bib9" id="ref56">9</reflink>]). Like Milo, both of these students showed evidence of a reversible partitive fraction scheme. Hackenberg's ([<reflink idref="bib7" id="ref57">7</reflink>]) explanation is that when MC2 students construct this scheme it requires coordinating two different two-levels-of-units structures. For example, Bridget, who participated in an 8-month teaching experiment, solved a problem in which $16 was <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>4</mn><mn>5</mn></mfrac></math> </ephtml> of David's money. Bridget divided $16 into four parts and added on $4 to get David's money. Bridget appeared to view the $16 as a unit of four units, and David's money was a unit of five units, which required adding on one of those units. Yet, Bridget did not become an MC3 student or construct an iterative fraction scheme during the 8 months (Hackenberg, [<reflink idref="bib7" id="ref58">7</reflink>]). Similarly, Samantha, who participated in an interview study, did the most advanced work of six MC2 students in solving Two Unknowns Problems (Hackenberg & Lee, [<reflink idref="bib9" id="ref59">9</reflink>]), although her work was not as advanced as Milo's. However, in these studies, neither Bridget nor Samantha were identified as a "class" of students.</p> <p>The contribution of Milo's case study is (<reflink idref="bib1" id="ref60">1</reflink>) the corroboration of advanced MC2 students identified in prior research into a well-defined group, and (<reflink idref="bib2" id="ref61">2</reflink>) the identification of a partitive fraction scheme, with length meanings for proper fractions, in the profile of this student. Thus, this study points to an expansion of the theory of units coordination in a way similar to Ulrich ([<reflink idref="bib29" id="ref62">29</reflink>]). She proposed that there is a group of advanced MC1 students who are characterized by being able to assimilate with composite units prior to the construction of an iterable unit of 1, which marks becoming an MC2 student. We propose that there is a group of advanced MC2 students who are characterized by partitive fraction schemes with length meanings for proper fractions, as well as the ability to coordinate two different two-levels-of-units structures. These ways of operating contrast with coordinating two three-levels-of-units structures, which marks being an MC3 student and is necessary to construct iterative fraction schemes. The evidence across three studies points to the reasonableness of this conjecture.</p> <p>This study does not allow us to make an estimate of the number of advanced MC2 students among middle school students. Ulrich and Wilkins ([<reflink idref="bib30" id="ref63">30</reflink>]) have created a written assessment to tease out the numbers of advanced MC1 students in larger populations. Similar work needs to be done to determine the prevalence of advanced MC2 students. In addition, longitudinal qualitative studies with advanced MC2 students across their middle school years could help us understand more about how long students may remain in this sub-stage. This research would allow us to develop mathematics for these students (Steffe, [<reflink idref="bib23" id="ref64">23</reflink>]) and design interventions that may promote a transition to the third multiplicative concept.</p> <hd id="AN0154815483-23">Funding</hd> <p>The research reported in this manuscript was supported by the National Science Foundation (grant no. DRL-1252575). The findings and statements in the paper do not necessarily represent the views of the National Science Foundation.</p> <hd id="AN0154815483-24">Publisher's note</hd> <p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p> <ref id="AN0154815483-25"> <title> References </title> <blist> <bibl id="bib1" idref="ref10" type="bt">1</bibl> <bibtext> Acar, F, & Sevinc, S. (2021). Investigation of middle school students' unit coordination levels in mathematics problems involving multiplicative relations. Elementary Education Online, 20(1), 90–114. https://doi.org/10.17051/ilkonline.2021.01.016</bibtext> </blist> <blist> <bibl id="bib2" idref="ref41" type="bt">2</bibl> <bibtext> Biddlecomb, B, & Olive, J. (2000). JavaBars [Computer software]. 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Hackenberg and Serife Sevinc</p> <p>Reported by Author; Author</p> </aug> <nolink nlid="nl1" bibid="bib24" firstref="ref1"></nolink> <nolink nlid="nl2" bibid="bib21" firstref="ref2"></nolink> <nolink nlid="nl3" bibid="bib22" firstref="ref3"></nolink> <nolink nlid="nl4" bibid="bib19" firstref="ref7"></nolink> <nolink nlid="nl5" bibid="bib31" firstref="ref9"></nolink> <nolink nlid="nl6" bibid="bib16" firstref="ref12"></nolink> <nolink nlid="nl7" bibid="bib17" firstref="ref13"></nolink> <nolink nlid="nl8" bibid="bib10" firstref="ref14"></nolink> <nolink nlid="nl9" bibid="bib11" firstref="ref15"></nolink> <nolink nlid="nl10" bibid="bib28" firstref="ref16"></nolink> <nolink nlid="nl11" bibid="bib27" firstref="ref24"></nolink> <nolink nlid="nl12" bibid="bib12" firstref="ref26"></nolink> <nolink nlid="nl13" bibid="bib13" firstref="ref28"></nolink> <nolink nlid="nl14" bibid="bib14" firstref="ref29"></nolink> <nolink nlid="nl15" bibid="bib20" firstref="ref31"></nolink> <nolink nlid="nl16" bibid="bib18" firstref="ref40"></nolink> <nolink nlid="nl17" bibid="bib26" firstref="ref42"></nolink> <nolink nlid="nl18" bibid="bib25" firstref="ref44"></nolink> <nolink nlid="nl19" bibid="bib15" firstref="ref47"></nolink> <nolink nlid="nl20" bibid="bib29" firstref="ref54"></nolink> <nolink nlid="nl21" bibid="bib30" firstref="ref63"></nolink> <nolink nlid="nl22" bibid="bib23" firstref="ref64"></nolink>
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  Data: A Boundary of the Second Multiplicative Concept: The Case of Milo
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  Data: <searchLink fieldCode="AR" term="%22Hackenberg%2C+Amy+J%2E%22">Hackenberg, Amy J.</searchLink> (ORCID <externalLink term="http://orcid.org/0000-0002-3406-4761">0000-0002-3406-4761</externalLink>)<br /><searchLink fieldCode="AR" term="%22Sevinc%2C+Serife%22">Sevinc, Serife</searchLink>
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  Data: <searchLink fieldCode="SO" term="%22Educational+Studies+in+Mathematics%22"><i>Educational Studies in Mathematics</i></searchLink>. Jan 2022 109(1):177-193.
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  Data: Springer. Available from: Springer Nature. One New York Plaza, Suite 4600, New York, NY 10004. Tel: 800-777-4643; Tel: 212-460-1500; Fax: 212-460-1700; e-mail: customerservice@springernature.com; Web site: https://link.springer.com/
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  Data: 17
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  Data: <searchLink fieldCode="EL" term="%22Elementary+Education%22">Elementary Education</searchLink><br /><searchLink fieldCode="EL" term="%22Grade+7%22">Grade 7</searchLink><br /><searchLink fieldCode="EL" term="%22Junior+High+Schools%22">Junior High Schools</searchLink><br /><searchLink fieldCode="EL" term="%22Middle+Schools%22">Middle Schools</searchLink><br /><searchLink fieldCode="EL" term="%22Secondary+Education%22">Secondary Education</searchLink>
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  Data: <searchLink fieldCode="DE" term="%22Multiplication%22">Multiplication</searchLink><br /><searchLink fieldCode="DE" term="%22Fractions%22">Fractions</searchLink><br /><searchLink fieldCode="DE" term="%22Grade+7%22">Grade 7</searchLink><br /><searchLink fieldCode="DE" term="%22Middle+School+Students%22">Middle School Students</searchLink><br /><searchLink fieldCode="DE" term="%22Logical+Thinking%22">Logical Thinking</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+Concepts%22">Mathematical Concepts</searchLink><br /><searchLink fieldCode="DE" term="%22Numbers%22">Numbers</searchLink>
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  Data: 10.1007/s10649-021-10083-8
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  Data: 0013-1954
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  Data: Students entering sixth grade operate with three different multiplicative concepts that influence their reasoning in many domains important for middle school. For example, students who are operating with the second multiplicative concept (MC2 students) can begin to construct fractions as lengths but do not construct improper fractions as numbers. Students who are operating with the third multiplicative concept (MC3 students) can construct both proper and improper fractions as multiples of unit fractions. This paper is a case study of one seventh grade MC2 student, Milo, who demonstrated the most advanced reasoning of all MC2 students in a large project with 13 MC2 and 9 MC3 students. In working on problems involving fractional relationships between two unknowns, most MC3 students constructed reciprocal reasoning. In contrast, the MC2 students struggled with these problems. Similar to the MC3 students, Milo showed some evidence of reciprocal reasoning, and he used proper fractions as operators on unknowns with a rationale. However, Milo did not construct reciprocal reasoning. We account for his reasoning by showing how he used length meanings for proper fractions and how he coordinated two different two-levels-of-units structures. The study expands the mathematics for MC2 students, showing what learning may be possible for students like Milo, and it suggests a change to the theory of students' multiplicative concepts. Specifically, advanced MC2 students are those who have constructed length meanings for fractions and can coordinate two different two-levels-of-units structures.
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