The (Activity-)Effect of Manipulatives on Algebraic Generalizations: A Constructivist Teaching Experiment
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| Title: | The (Activity-)Effect of Manipulatives on Algebraic Generalizations: A Constructivist Teaching Experiment |
|---|---|
| Language: | English |
| Authors: | Karen Zwanch (ORCID |
| Source: | Educational Studies in Mathematics. 2025 119(1):41-61. |
| 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: | 21 |
| Publication Date: | 2025 |
| Document Type: | Journal Articles Reports - Research |
| Education Level: | Elementary Education Grade 6 Intermediate Grades Middle Schools |
| Descriptors: | Mathematics Education, Mathematics Instruction, Teaching Methods, Algebra, Manipulative Materials, Educational Experiments, Grade 6, Mathematics Skills, Thinking Skills, Mental Computation, Arithmetic |
| DOI: | 10.1007/s10649-024-10371-z |
| ISSN: | 0013-1954 1573-0816 |
| Abstract: | To understand the ways that manipulatives might support changes in students' reasoning about algebraic generalizations, a constructivist teaching experiment was conducted with two sixth-grade students. The students interpreted numerical situations with units of one and could construct units of units in mental activity. Initially, the students' reasoning about Cuisenaire® rods did not lead to changes in their algebraic generalizations, whereas their reasoning about linking cubes did lead to such changes. The students' learning with linking cubes is explained by their enactment of physical operations with units of one on the linking cubes, which were consistent with their mental operations in numerical situations. Over time, and after learning to generalize with linking cubes, the students also began to attribute meaning to their physical operations with Cuisenaire® rods. Thus, instruction with manipulatives that reflected the student's interpretation of numerical situations supported their construction of algebra as generalized arithmetic. Instructional implications are discussed. |
| Abstractor: | As Provided |
| Entry Date: | 2025 |
| Accession Number: | EJ1469084 |
| Database: | ERIC |
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| FullText | Links: – Type: pdflink Url: https://content.ebscohost.com/cds/retrieve?content=AQICAHj0k_4E0hTGH8RJwT4gCJyBsGNe_WN95AvKlDbXJGqwxwFpt-oTYtKbkSX8GICbhP-AAAAA4zCB4AYJKoZIhvcNAQcGoIHSMIHPAgEAMIHJBgkqhkiG9w0BBwEwHgYJYIZIAWUDBAEuMBEEDIhaZWU3EndSnAfi0gIBEICBm5A_SH7ngXAeo_rNyE9Fmvl_ZemOMGdawJMTx7nHGjI5eQKQaO2_hNLPt2UoWYabE5UIAJg0JKkUfzacpwmC2i4U9AoqRPBwiGa2EIQxJ8MxNF93vY3EGUBEvkAbk6NW9ahRaC3KM1XeQ4Qh0g1FwzrtnFXdeOKUmT422Ohv-V5rt3a6pimn0ZtPycrpnAT3KJ5vnqof86XAj1QE Text: Availability: 1 Value: <anid>AN0184746713;esm01may.25;2025Apr29.02:56;v2.2.500</anid> <title id="AN0184746713-1">The (activity-)effect of manipulatives on algebraic generalizations: a constructivist teaching experiment </title> <p>To understand the ways that manipulatives might support changes in students' reasoning about algebraic generalizations, a constructivist teaching experiment was conducted with two sixth-grade students. The students interpreted numerical situations with units of one and could construct units of units in mental activity. Initially, the students' reasoning about Cuisenaire® rods did not lead to changes in their algebraic generalizations, whereas their reasoning about linking cubes did lead to such changes. The students' learning with linking cubes is explained by their enactment of physical operations with units of one on the linking cubes, which were consistent with their mental operations in numerical situations. Over time, and after learning to generalize with linking cubes, the students also began to attribute meaning to their physical operations with Cuisenaire® rods. Thus, instruction with manipulatives that reflected the student's interpretation of numerical situations supported their construction of algebra as generalized arithmetic. Instructional implications are discussed.</p> <p>Keywords: Algebraic reasoning; Generalizing; Manipulatives; Middle-grades mathematics; Units coordination; Education Specialist Studies In Education</p> <p>Brooke Mullins is no longer affiliated with University of Virginia's College at Wise.</p> <p>Manipulatives are common tools in mathematics classrooms, where manipulatives are physical objects that provide opportunities to enact abstract mathematics concretely (Carbonneau et al., [<reflink idref="bib8" id="ref1">8</reflink>]). There is general agreement that manipulatives positively affect learning (Adom &amp; Adu, [<reflink idref="bib1" id="ref2">1</reflink>]; Carbonneau et al., [<reflink idref="bib8" id="ref3">8</reflink>]) because the kinesthetic activity associated with manipulatives can lead students to internalize activities; students can carry out the activities in imagination (Norton et al., [<reflink idref="bib28" id="ref4">28</reflink>]). Once the focus of mathematics transitions from numerical to algebraic reasoning in the middle grades, however, the role of manipulatives is not well documented.</p> <p>Early algebra can be derived from numerical reasoning, although the relationship between the two is not entirely clear. Furthermore, early algebra is multi-dimensional, and generalizing is a "scarlet thread" woven throughout each dimension (Kieran, [<reflink idref="bib19" id="ref5">19</reflink>]). Thus, the manner by which students apply numerical reasoning when learning to generalize requires further examination.</p> <p>We interpret mathematical learning to result from reflective abstraction (Piaget, 1977/[<reflink idref="bib29" id="ref6">29</reflink>]), which leads to the construction of more sophisticated mental operations (Norton, [<reflink idref="bib27" id="ref7">27</reflink>]). As we cannot observe mental operations directly, we infer changes, called <emph>effects</emph>, to mental operations, called <emph>activities</emph>, based on mathematical reasoning (Simon et al., [<reflink idref="bib33" id="ref8">33</reflink>]). This study focuses on the potential for manipulatives to provide material for reflection and subsequent effects on middle-grades students' mental activities related to algebraic generalizations.</p> <hd id="AN0184746713-2">Manipulatives and learning</hd> <p>Although research generally agrees that manipulatives support learning in elementary-level mathematics, their role in middle-grades algebra is unclear. It has been noted that manipulatives are not magical (Ball, [<reflink idref="bib3" id="ref9">3</reflink>]) nor are they pedagogical in and of themselves (Björklund, [<reflink idref="bib5" id="ref10">5</reflink>]). Rather, "good manipulatives are those that aid students in building, strengthening, and connecting various representations of mathematical ideas" (Sarama &amp; Clements, [<reflink idref="bib32" id="ref11">32</reflink>], p. 146). Manipulatives support these connections with consistent use and instruction that transitions to abstract representations (Laski et al., [<reflink idref="bib21" id="ref12">21</reflink>]), making it critical to consider the environments in which manipulatives may afford the learning of mathematics.</p> <p>Affordances of manipulatives include supporting visual comparisons (Björklund, [<reflink idref="bib5" id="ref13">5</reflink>]), writing and discussion (Kosko &amp; Wilkins, [<reflink idref="bib20" id="ref14">20</reflink>]) and achievement (e.g., Carbonneau et al., [<reflink idref="bib8" id="ref15">8</reflink>]; Liggett, [<reflink idref="bib22" id="ref16">22</reflink>]). For instance, a meta-analysis of manipulative use in K-12 classrooms found small to moderate effects that demonstrate instruction with manipulatives which was more effective than instruction focused on symbols (Carbonneau et al., [<reflink idref="bib8" id="ref17">8</reflink>]). This effect was moderated by the mathematics content, where manipulatives had a greater effect when used in fraction instruction compared to algebra instruction.</p> <p>In contrast, manipulatives did not support greater learning when used as fun activities, rather than to promote conceptual understanding (Moyer, [<reflink idref="bib26" id="ref18">26</reflink>]; Swan &amp; Marshall, [<reflink idref="bib37" id="ref19">37</reflink>]). This was more likely to divide "fun" and "real" math in students' minds than it was to support learning (Moyer, [<reflink idref="bib26" id="ref20">26</reflink>]). Other situations that did not lead to increased learning include situations in which manipulatives were used in a memorized procedure (Dowker, [<reflink idref="bib11" id="ref21">11</reflink>]), and inappropriately complex manipulatives were chosen (Baroody, [<reflink idref="bib4" id="ref22">4</reflink>]).</p> <p>Uribe-Flórez and Wilkins ([<reflink idref="bib42" id="ref23">42</reflink>]) posit insight into the ways that manipulative use may affect achievement. They conducted cross-sectional and longitudinal analysis of data relating manipulative use to mathematics achievement. Cross-sectional analysis found that in one school year, there was not a significant relationship between manipulative use and achievement. However, longitudinal analysis from kindergarten through fifth grade found that medium and high use of manipulatives was a significant predictor of higher achievement annually. This suggests that consistent use of manipulatives over time may afford greater achievement. In combination, these results contribute to the specification of environments in which manipulatives might afford the learning of mathematics.</p> <hd id="AN0184746713-3">Algebra as generalized arithmetic</hd> <p>Generalizing is interwoven through all dimensions of early algebraic thinking, including analytic, structural, and functional thinking (Kieran, [<reflink idref="bib19" id="ref24">19</reflink>]), and Kaput ([<reflink idref="bib18" id="ref25">18</reflink>]) agrees that generalizing is a key characteristic of early algebra. In generalizing, students recognize a pattern, express it, and manipulate the generality (Driscoll, [<reflink idref="bib12" id="ref26">12</reflink>]); throughout mathematics, students need to generalize by learning to "'see the general through the particular' and also 'to see the particular in the general'" (Mason, [<reflink idref="bib23" id="ref27">23</reflink>], p. 334). Amit and Neria ([<reflink idref="bib2" id="ref28">2</reflink>]) found that generalizing can act as a gateway to algebra, potentially because it promotes algebraic habits of mind (Driscoll, [<reflink idref="bib12" id="ref29">12</reflink>]), and proof and justification (Ellis, [<reflink idref="bib13" id="ref30">13</reflink>]).</p> <p>To reap these benefits, it is paramount that students develop functional rules, rather than recursive (Carraher et al., [<reflink idref="bib9" id="ref31">9</reflink>]), as recursive rules do not necessarily support generalization (Blanton &amp; Kaput, [<reflink idref="bib7" id="ref32">7</reflink>]) or symbolic representation (Amit &amp; Neria, [<reflink idref="bib2" id="ref33">2</reflink>]). Yet, Carraher and colleagues ([<reflink idref="bib9" id="ref34">9</reflink>]) found that teaching students to define functions explicitly is one of the "primary issues" of generalizing in early algebra because recursive rules do not treat independent variables as variables, but as placeholders in a sequence, and skipping ahead in that sequence to think about <emph>any</emph> term can be helpful in moving students toward functional rules.</p> <p>One difficulty that may lead to reliance on recursive rules is that students do not view the relationships to be generalized structurally. For instance, Hackenberg ([<reflink idref="bib14" id="ref35">14</reflink>]) asked students to generalize the number of squares on the border of an <emph>x</emph>-by-<emph>x</emph> grid by examining 10-by-10 and 6-by-6 grids. A student who interpreted numerical situations with units of one explained that there were 36 squares on the border of the 10-by-10 grid because he added 10 + 10 + 8 + 8 and 20 squares on the border of the 6-by-6 grid because he added 6 + 6 + 4 + 4. The student did not express a relationship between 10 and 8, nor between 6 and 4. This student did not structure the numerical relationship between 10 and 8 or 6 and 4 such that he interpreted the smaller quantity as both a part of and separate from the larger; the two quantities were not understood to vary systematically and instead were calculated separately, resembling recursive reasoning.</p> <p>Zwanch ([<reflink idref="bib45" id="ref36">45</reflink>]) also investigated students' generalizations of linear patterns, some of which were represented numerically and others visually as figures containing squares. Similar to Hackenberg ([<reflink idref="bib14" id="ref37">14</reflink>]), Zwanch ([<reflink idref="bib45" id="ref38">45</reflink>]) found that students who interpreted numerical situations with units of one did not explicitly identify functional relationships. They did, however, identify functional relationships implicitly by identifying, for example, outputs corresponding to the one-hundredth input. Additionally, Zwanch ([<reflink idref="bib45" id="ref39">45</reflink>]) found that students who interpreted numerical situations as units of units were apt to explicitly generalize functional relationships because they interpreted each output as a unit containing the input and common difference; recursive reasoning was inefficient for these students.</p> <p>Movement toward functional rules can be supported by examining visual patterns (Rivera, [<reflink idref="bib31" id="ref40">31</reflink>]) or non-consecutive inputs (Moss &amp; London McNab, [<reflink idref="bib25" id="ref41">25</reflink>]), but Kieran ([<reflink idref="bib19" id="ref42">19</reflink>]) suggests that more research is needed into instructional interventions that lead to students' identification of functional generalizations. Additionally, the role of manipulatives is not yet clear in this instruction, because while much of the research represented patterns visually for students, none was identified that indicated students used manipulatives.</p> <p>Much research on generalizing has focused on the progression of generalizations that students make when learning to write functional rules. Inhelder and Piaget (1958/[<reflink idref="bib17" id="ref43">17</reflink>]) described that students begin with non-symbolic and build to symbolic generalizations. Providing additional detail, Radford ([<reflink idref="bib30" id="ref44">30</reflink>]) found that elementary students began by noticing commonalities between consecutive terms in sequences. Next, they predicted near cases, followed by far cases. He designated that identifying far cases (such as the one-hundredth) indicates algebraic thinking because students generalize a functional rule implicitly in these predictions. Then, students may learn to verbalize generalizations for <emph>any</emph> term and finally develop symbolic representations of functional rules. This is an example of research that attends to the progression of generalizations students may advance through in early algebra.</p> <p>Generalizing functional rules is important to early algebra (Kaput, [<reflink idref="bib18" id="ref45">18</reflink>]; Kieran, [<reflink idref="bib19" id="ref46">19</reflink>]), and numerical reasoning can be a foundation to generalizing (Zwanch, [<reflink idref="bib45" id="ref47">45</reflink>]). There is also a need to better understand how students learn to generalize in varied contexts (Blanton et al., [<reflink idref="bib6" id="ref48">6</reflink>]). Inhelder and Piaget (1958/[<reflink idref="bib17" id="ref49">17</reflink>]) found that physical experiences support generalizations, which suggests that manipulatives might also support advanced generalizations. Additionally, research should consider the ways that instruction helps students identify qualities of patterns that lead to functional generalizations (Kieran, [<reflink idref="bib19" id="ref50">19</reflink>]). Therefore, this research will examine the ways that instruction focused on supporting reflections on manipulatives might engender learning in the context of generalizing functional relationships.</p> <hd id="AN0184746713-4">Theoretical framework</hd> <p>Sarama and Clements ([<reflink idref="bib32" id="ref51">32</reflink>]) found that "students may require physically concrete materials to build meaning initially, but they must <emph>reflect</emph> on their actions with manipulatives to do so" (p. 146). Additionally, Norton et al. ([<reflink idref="bib28" id="ref52">28</reflink>]) argue that "the psychological construction of mathematical objects would depend heavily on coordinated activity beginning with the hands" (p. 49), meaning that abstract mathematical objects are constructed through the coordination of kinesthetic activity, potentially involving manipulatives, with mental activity. Thus, attributing mathematical meaning to the actions carried out with manipulatives requires reflections on those actions and coordinations with mental actions.</p> <p>Accordingly, we take the perspective that reflective abstraction is the mechanism by which mental operations are projected to objects of reflection and once reflected upon can be reorganized into more sophisticated operations (Piaget, 1977/[<reflink idref="bib29" id="ref53">29</reflink>]), which constitutes learning. In the case of manipulatives, students' reflections on their actions and their coordination with manipulatives could support the projection of physical actions to objects of mental reflection and more sophisticated operations. Simon et al. ([<reflink idref="bib33" id="ref54">33</reflink>]) elaborate reflective abstraction to reflection on activity-effect relationships which adds attention to the interplay between learning and instruction. As this study will examine the ways instruction incorporating manipulatives is related to students' learning of algebraic generalizations, we utilize reflection on activity-effect relationships to interpret learning.</p> <hd id="AN0184746713-5">Reflection on activity-effect relationships</hd> <p>Reflective abstraction is elaborated for instruction as reflection on activity-effect relationships. Figure 1 represents the opportunity for instruction to prompt students' goal-directed activity. Simon et al. ([<reflink idref="bib33" id="ref55">33</reflink>]) define four components that may result from instruction: (<reflink idref="bib1" id="ref56">1</reflink>) the learner's goal, (<reflink idref="bib2" id="ref57">2</reflink>) an activity sequence, (<reflink idref="bib3" id="ref58">3</reflink>) the result of each attempt to meet the goal, and (<reflink idref="bib4" id="ref59">4</reflink>) an effect. From this perspective, students learn by engaging in goal-directed activity. Once a goal is identified, the student initiates an activity sequence, which designates the mental operations they carry out in service of the goal. Inevitably, the result of the activity sequence may or may not be expected; this creates an opportunity for learning. If the student identifies the disconnect between their activity sequence and their intended goal through reflection, then they may make adjustments to their activity sequence, called <emph>effects</emph>, to more closely approach their goal.</p> <p>Graph: Fig. 1 Reflection on activity-effect relationships</p> <p>Consider a student's use of base-10 blocks to add two single-digit numbers, and a teacher's lesson designed to extend addition from sums less than 10 to sums greater than 10. We delineate the four potential components of a hypothetical student's reflection on activity-effect relationships within this example. If the teacher poses the problem 8 + 7 and stipulates that the students represent their sum with the fewest number of blocks possible to encourage regrouping, the student's goal (<reflink idref="bib1" id="ref60">1</reflink>) might be to find the sum by joining eight 1s with seven 1s blocks, as they have done previously. Next, the student activates (<reflink idref="bib2" id="ref61">2</reflink>) an activity sequence, which could involve counting out eight 1s and seven 1s blocks, then counting on, saying, "8. 9, 10, 11, 12, 13, 14, 15." This counting activity suggests that the student unitized eight blocks into one unit of eight and created another seven units in mental activity as they counted. The result of this activity (<reflink idref="bib3" id="ref62">3</reflink>) might be dissatisfaction that they used so many blocks despite the teacher's directions, leading them to consider how to incorporate a 10 s block. One possible "conception-based adjustment" (Simon et al., [<reflink idref="bib33" id="ref63">33</reflink>], p. 319) to their activity sequence could be that they unitize the initial eight 1s with two 1s from the seven, making a unit of 10. This constitutes an effect (<reflink idref="bib4" id="ref64">4</reflink>), because the student adjusted their mental activity to generate the desired result.</p> <hd id="AN0184746713-6">Units coordination</hd> <p>Units coordination is a unique predictor of algebra knowledge (Viegut et al., [<reflink idref="bib43" id="ref65">43</reflink>]), and algebraic generalizations can be understood as applications of units coordinations (Hackenberg, [<reflink idref="bib14" id="ref66">14</reflink>]). Therefore, we frame students' numerical reasoning stages in terms of the numerical units that they can interpret, construct, and coordinate (Hackenberg &amp; Sevinc, [<reflink idref="bib15" id="ref67">15</reflink>]; Steffe, [<reflink idref="bib34" id="ref68">34</reflink>]; Ulrich, [<reflink idref="bib38" id="ref69">38</reflink>], [<reflink idref="bib39" id="ref70">39</reflink>]). <emph>Unitizing</emph> is a mental operation in which many are interpreted as one (von Glasersfeld, [<reflink idref="bib44" id="ref71">44</reflink>]). For instance, "seven" can be interpreted as seven individual units of one or as one unit containing seven; the latter is a <emph>composite unit</emph> of seven. <emph>Units coordination</emph> involves distributing one composite unit across the elements of another (Steffe, [<reflink idref="bib34" id="ref72">34</reflink>]). If asked to find how many in 4 groups of 7, students could distribute a composite unit of 7 across each of 4 units within the composite unit of 4, forming 28. The fluidity with which students reason about the result of a units coordination is in part dependent upon the level of units with which they interpret, or <emph>assimilate</emph>, the situation.</p> <p>Students who assimilate numerical situations with units of one can enact a units coordination, as described, but do not anticipate or operate on the result because the units coordination is implicit (Steffe, [<reflink idref="bib34" id="ref73">34</reflink>], [<reflink idref="bib35" id="ref74">35</reflink>]). This student might find 4 groups of 7 by counting to 7 by ones, four times, and may use a known finger pattern for 7 to keep track of their counting. In this way, the units coordination is enacted because they assimilate the situation with units of one and form composite units of 7 in their counting activity. The result of this units coordination, from the student's perspective, is a counting sequence from 1 to 28 that does not retain the 4 groups of 7 structure.</p> <p>In comparison, students who assimilate numerical situations with composite units can anticipate a composite unit of 7, making it possible to find the result of 4 groups of 7 without counting by ones. The perceived result for this student is a composite unit of 28 that can be operated on (Hackenberg &amp; Sevinc, [<reflink idref="bib15" id="ref75">15</reflink>]). For instance, if the student were now asked to find how many total when an additional group of 7 is added, the student might take 3 from 28 (leaving 25) and add it to the 7 to make 10, forming 25 and 10, or 35.</p> <p>Research has identified that students who assimilate numerical situations with units of one have difficulty generalizing functional relationships (Hackeberg, [<reflink idref="bib14" id="ref76">14</reflink>]; Zwanch, [<reflink idref="bib45" id="ref77">45</reflink>]). However, research has not examined instructional situations in which these students might learn to generalize functional relationships. Therefore, this study asks: In what ways can manipulatives in mathematics instruction engender reflections on the activity-effect relationship for sixth-grade students who assimilate numerical situations with units of one, in the context of algebraic generalizations?</p> <hd id="AN0184746713-7">Methodology</hd> <p>This research study utilized a constructivist teaching experiment to examine effects on a sixth-grade student's activity sequence for algebraically generalizing linear patterns using manipulatives. A constructivist teaching experiment is a longitudinal, qualitative methodology in which a teacher-researcher plans mathematical play for a pair of students to engender learning and observe changes in the mental operations that instigate that learning (Steffe &amp; Thompson, [<reflink idref="bib36" id="ref78">36</reflink>]).</p> <p>Teaching experiments are meant to generate and test hypotheses about students' mathematics. Therefore, the teacher-researcher constructs models of students' mathematics related to these hypotheses. Initial models act as a baseline so that changes in students' reasoning can be detected. While working with students, the teacher-researcher may interact intuitively or analytically. Intuitive interactions are such that the teacher-researcher is attempting to understand the students' reasoning and does not have predetermined expectations as to where the interaction will lead. Analytical interactions are those in which the teacher-researcher seeks to compare students' reasoning across contexts (Steffe &amp; Thompson, [<reflink idref="bib36" id="ref79">36</reflink>]). In this study, the teacher-researcher constructed initial models of the students' numerical reasoning. These allowed the observation of effects (i.e., learning) over time.</p> <hd id="AN0184746713-8">Initial models of students' numerical reasoning</hd> <p>In this section, we describe the initial models of numerical reasoning for two sixth-grade students, Lucía and Penny (pseudonyms). They were selected from a general math 6 class. The students' numerical reasoning, described in terms of their operations with units of one and composite units, were utilized as initial models because numerical reasoning is a foundation for generalizing (Zwanch, [<reflink idref="bib45" id="ref80">45</reflink>]). To provide comprehensive, rich descriptions, we focus analysis on Penny, because she was more apt to explain her reasoning to both the teacher-researcher and to Lucía.</p> <p>To illustrate Penny's numerical reasoning, we share her solution to the cupcake task (Fig. 2; Ulrich &amp; Wilkins, [<reflink idref="bib41" id="ref81">41</reflink>]), which she solved independently in a clinical interview prior to the teaching experiment. This was one of 30 tasks that contributed to our initial model of Penny's numerical reasoning.</p> <p>Graph: Fig. 2 Penny's solution to the cupcake task. Note. The cupcake task is taken from the assessment in Ulrich and Wilkins ([<reflink idref="bib41" id="ref82">41</reflink>])</p> <p>Penny counted the three cupcakes printed on the page, saying, "one, two, three," and then drew five more groups of three cupcakes and counted, "one, two, three..." five times (Fig. 2, circled cupcakes). This suggests that she created figurative composite units of three (a box of cupcakes containing three units of one). Then, she stopped and re-counted the cupcakes from 1 through 18 by ones, which suggests that she did not anticipate that 6 groups of 3 contained 18 cupcakes. Next, she drew the remaining cupcakes (Fig. 2, un-circled cupcakes) and counted all cupcakes by ones to 39, which suggests that she was attending to units of 1. The researcher repeated, "How many boxes of cupcakes will you fill?" Penny added a rectangle around each group of three cupcakes and then counted the rectangles from 1 through 13. This suggests that Penny interpreted and solved the task by operating on units of one and using figurative material (drawings) to construct composite units. Her reasoning on the cupcake task and others from the clinical interview indicated Penny assimilated numerical situations with units of one, iterated units of one in mental activity by counting, and could unitize a composite unit. She did not yet anticipate, reflect, or operate on composite units.</p> <hd id="AN0184746713-9">Data collection and analysis</hd> <p>Teaching episodes with Lucía and Penny began in March and occurred weekly until the end of the school year. Each teaching episode lasted 55 min. All teaching episodes were video recorded and transcribed, and written work was digitized. The analysis of each teaching episode occurred prior to the next episode and informed the design of the next episode. Analysis was done collaboratively. The researchers independently viewed the data and created analytical memos (Charmaz, [<reflink idref="bib10" id="ref83">10</reflink>]) detailing Lucía's and Penny's generalizing, evidence of their mental operations with units of one or composite units, and use of manipulatives (Table 1). Next, the researchers discussed the data and memos and created detailed descriptions of Lucía's and Penny's ways of operating and formed hypotheses about how Lucía and Penny might respond to different tasks and manipulatives. These hypotheses were tested in upcoming teaching episodes.</p> <p>Table 1 Units coordination coding scheme</p> <p> <ephtml> &lt;table frame="hsides" rules="groups"&gt;&lt;thead&gt;&lt;tr&gt;&lt;th align="left"&gt;&lt;p&gt;Units coordination codes and operationalized definitions&lt;/p&gt;&lt;/th&gt;&lt;th align="left"&gt;&lt;p&gt;Characteristic behaviors of numerical reasoning&lt;/p&gt;&lt;/th&gt;&lt;th align="left"&gt;&lt;p&gt;Extension of characteristic behaviors to manipulatives&lt;/p&gt;&lt;/th&gt;&lt;/tr&gt;&lt;/thead&gt;&lt;tbody&gt;&lt;tr&gt;&lt;td align="left" rowspan="3"&gt;&lt;p&gt;&lt;bold&gt;Operating on units of 1&lt;/bold&gt;: Student assimilates the situation with units of one and can unitize ones to form a composite unit in mental activity&lt;/p&gt;&lt;/td&gt;&lt;td align="left"&gt;&lt;p&gt;&amp;#8226; Counting by 1s&lt;/p&gt;&lt;p&gt;&amp;#8226; Chunking 1s into groups after the fact&lt;/p&gt;&lt;p&gt;&amp;#8226; Limited awareness of how many 1s and how many groups simultaneously&lt;/p&gt;&lt;/td&gt;&lt;td align="left"&gt;&lt;p&gt;&amp;#8226; Counting individual linking cubes (representative of units of 1)&lt;/p&gt;&lt;p&gt;&amp;#8226; Forming trains of linking cubes after the fact (representative of composite units)&lt;/p&gt;&lt;p&gt;&amp;#8226; Counting either by individual cubes or by trains separately&lt;/p&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align="left" colspan="2"&gt;&lt;p&gt;&lt;bold&gt;Illustration with the cupcake task&lt;/bold&gt;&lt;/p&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align="left"&gt;&lt;p&gt;Penny drew cupcakes and counted by 1s as she drew them: "1, 2, 3. 1, 2, 3..." six times. Then, she re-counted the 18 cupcakes by 1s to check her total against the target (39). After drawing thirty-nine 1s and grouping them into threes, she counted the groups&lt;/p&gt;&lt;/td&gt;&lt;td align="left"&gt;&lt;p&gt;&lt;italic&gt;Linking cubes are provided&lt;/italic&gt;&lt;/p&gt;&lt;p&gt;Student joins 3 linking cubes repeatedly, forming 6 trains of 3, and then re-counts the individual cubes by 1s to 18. Realizing they have not "boxed" all 39 cupcakes, the student counts out 39 linking cubes by 1s and finishes grouping them. After grouping all 39 cubes, the student counts the groups&lt;/p&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align="left" rowspan="3"&gt;&lt;p&gt;&lt;bold&gt;Operating on composite units&lt;/bold&gt;: Student assimilates the situation with composite units and can unitize composite units in mental activity&lt;/p&gt;&lt;/td&gt;&lt;td align="left"&gt;&lt;p&gt;&amp;#8226; Skip counting or multiplying&lt;/p&gt;&lt;p&gt;&amp;#8226; Using a single object to represent many&lt;/p&gt;&lt;p&gt;&amp;#8226; May keep track of groups and 1s simultaneously&lt;/p&gt;&lt;p&gt;&amp;#8226; May decompose numbers strategically&lt;/p&gt;&lt;/td&gt;&lt;td align="left"&gt;&lt;p&gt;&amp;#8226; Skip counting or multiplying without manipulatives&lt;/p&gt;&lt;p&gt;&amp;#8226; Using manipulatives to explain their response, rather than find their answer&lt;/p&gt;&lt;p&gt;&amp;#8226; Using one linking cube to represent many&lt;/p&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align="left" colspan="2"&gt;&lt;p&gt;&lt;bold&gt;Illustration with the cupcake task&lt;/bold&gt;&lt;/p&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align="left"&gt;&lt;p&gt;Student says, "10 boxes would be 30 cupcakes, so add 1 box is 33, 2 boxes is 36, 3 boxes is 39. 13 boxes"&lt;/p&gt;&lt;/td&gt;&lt;td align="left"&gt;&lt;p&gt;&lt;italic&gt;Linking cubes are provided&lt;/italic&gt;&lt;/p&gt;&lt;p&gt;Student says, "Each block is 1 box. So 10 boxes would be 30 cupcakes," while retrieving 10 linking cubes. "One more is 33," adds 1 more cube to the 10, "Two more is 36," adds 1 more cube to the 11, and "Three more is 39," while adding the thirteenth cube. "13 box&lt;/p&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/tbody&gt;&lt;/table&gt; </ephtml> </p> <p>Operational definitions are based on Hackenberg and Sevinc ([<reflink idref="bib15" id="ref84">15</reflink>]), Steffe ([<reflink idref="bib34" id="ref85">34</reflink>], [<reflink idref="bib35" id="ref86">35</reflink>]), and Ulrich ([<reflink idref="bib38" id="ref87">38</reflink>], [<reflink idref="bib39" id="ref88">39</reflink>], [<reflink idref="bib40" id="ref89">40</reflink>]) The cupcake task is shown in Fig. 2. Penny's solution is from our data. The other solutions are hypothetical and based on Ulrich and Wilkins's ([<reflink idref="bib41" id="ref90">41</reflink>]) data</p> <hd id="AN0184746713-10">Teaching activity</hd> <p>The teaching activity was a variation of the game 'guess my rule'. The game was played with either Cuisenaire® rods or linking cubes. Cuisenaire® rods are color-coded blocks ranging in length from one (white) to 10 cm (orange). They are continuous, meaning that there are no marks to indicate each centimeter. Linking cubes are 1-inch plastic blocks that can be snapped together to form a train. The students began with 20 one-cm Cuisenaire® rods or 20 single linking cubes, which could be used in any combination as inputs. The teacher-researcher returned a Cuisenaire® rod or linking cube train that reflected the corresponding output. Once the students inputted all 20 single rods or cubes, they were prompted to make a prediction related to the secret rule (Fig. 3). For instance, consider a secret rule of <emph>x</emph> + 3. If the students specified an input of five cubes, then the teacher-researcher would return the 8-cm Cuisenaire® rod or train of eight linking cubes. The students would have 15 cubes remaining to gather additional information about the secret rule. The teacher-researcher started by asking students to predict near cases, then far cases, and verbal generalizations, as the students demonstrated success (see Zwanch and Broome ([<reflink idref="bib46" id="ref91">46</reflink>]) for a full description of the game).</p> <p>Graph: Fig. 3 Progression of predictions with example questions</p> <hd id="AN0184746713-11">Initial hypothesis</hd> <p>The initial hypothesis was that Cuisenaire® rods could support these students to algebraically generalize by engendering reflections on their activity-effect relationships. We aimed to support predictions of far cases, if not verbal generalizations. This hypothesis was based on existing literature that finds students who assimilate with units of one may generalize by finding near and far cases, but do not generalize verbally or symbolically (Zwanch, [<reflink idref="bib45" id="ref92">45</reflink>]). Therefore, we posited that Cuisenaire® rods might provide concrete representations of composite units that Lucía and Penny construct in mental activity, thereby providing material for reflection. We present Lucía's and Penny's reasoning on one-step additive rules, although in later teaching episodes, they also solved rules involving subtraction.</p> <hd id="AN0184746713-12">Results and analysis</hd> <p>The initial hypothesis was not initially supported by data. We provide evidence of this result, then present the new hypothesis, adjustments to teaching activities, and evidence that these adjustments engendered effects to Penny's activity sequence for generalizing.</p> <hd id="AN0184746713-13">Excerpt 1</hd> <p>Excerpt 1 documents Penny's initial way of reasoning when predicting near cases. The teacher-researcher's secret rule was <emph>x</emph> + 2, and the students had inputted 10 white blocks, then 3, 2, 1, and 4; they had exhausted their 20 white blocks and were asked to predict. Figure 4 shows the students' inputs and corresponding outputs, arranged by the students in numerical order by input after exhausting their 20 blocks. The digitally added notations (e.g., I3/O5) represent the input of three white blocks and the output of a yellow block (5 cm), although the students were not told the lengths of any blocks.</p> <p>Graph: Fig. 4 Cuisenaire® Rods used during Excerpt 1. Note. Black circle indicates approximate location that Penny touched in Excerpt 1, Line 14</p> <p></p> <ulist> <item> 1 Teacher-researcher (T): Your job is, can you figure out how the [rule determines] which block to give you back?</item> <p></p> <item> 2 Penny (P): However many units down each one.</item> <p></p> <item> 3 T: OK. Tell me about that. What do you mean?</item> <p></p> <item> 4 P: Like, the length of like, however many blocks you did [input], you take that off [the output] and get that block [common difference].</item> <p></p> <item> 5 T: Okay. [To Lucía (L)] Do you understand what she's saying? [L shakes head no.] Not quite. Okay. [To P] Can you show her with the blocks what you're talking about?</item> <p></p> <item> 6 P: I don't know how I would do that. I'm just like, like, you know, however many blocks there are [points to cyan block in I4/O6], like, say we took four off of this [cyan]. When you take however many, four blocks, that would be [uses two fingers to span a length of approximately 4 centimeters along the cyan block (Figure 5)]. And you have that many [shifts index fingers so they span a length of approximately 2 centimeters (i.e., the common difference) along the cyan block].</item> </ulist> <p>Graph: Fig. 5 Penny's tacit reasoning about the common difference in Excerpt 1, Line 6</p> <p></p> <ulist> <item> 7 T: .... Is there any kind of a pattern here? When you put in three blocks [points to I3/O5], you got the yellow one.</item> <p></p> <item> 8 L: It's getting higher and higher.</item> <p></p> <item> 9 P: Yeah. It's getting one higher each time.</item> <p></p> <item> 10 T: Okay. It's getting one higher each time. What do you mean by one higher each time?</item> <p></p> <item> 11 P: Like, see, that was one, two, three [points to inputs from I1/O3, I2/O4, I3/O5]. It's getting higher by each one.</item> <p></p> <item> 12 L: Instead of one, two [points to inputs from I1/O3, I2/O4], it's three, four [points to outputs]...</item> <p></p> <item> 13 T: If you put five white blocks in, what block do you think you would get back?</item> <p></p> <item> 14 P: Five blocks goes [<emph>sic</emph>] there [points to location of black dot in Fig. 4].</item> <p></p> <item> 15 L: [Talking over P] Yeah, this yellow one.</item> <p></p> <item> 16 T: [Responding to L] So if you put five white blocks in, it would give you a yellow block back?</item> <p></p> <item> 17 L: Yes.</item> <p></p> <item> 18 T: [To P] Okay. Do you agree?</item> <p></p> <item> 19 P: Wait, I don't know. I feel like longer than the yellow one because it would have to be longer than this one [cyan], because this is four [input from I4/O6].</item> <p></p> <item> 20 T: So you were saying it has to be longer than this one [cyan] because this one's four?</item> <p></p> <item> 21 P: So, like, it would be one unit longer than that one [cyan]. .... It's either green or this one [picking up a black (<reflink idref="bib7" id="ref93">7</reflink>)]. It's one of those. Let me see this green one real quick [compares lengths of cyan (<reflink idref="bib6" id="ref94">6</reflink>) and black (<reflink idref="bib7" id="ref95">7</reflink>)]. It could be the black one. [To L] Look at the black one. I think it's the black one, cause that's like one unit [longer than cyan].</item> </ulist> <p>Next, the teacher-researcher asked them to predict the output if they inputted 20 blocks, and they erroneously predicted 21 units. Penny explained, "because it kept getting one higher the whole time. So it wouldn't just be 20." Penny conflated the increase of one unit between consecutive outputs with the common difference.</p> <hd id="AN0184746713-14">Analysis of Excerpt 1</hd> <p>Excerpt 1 demonstrates Penny's initial ways of operating with Cuisenaire® rods. This reasoning was consistent over several teaching episodes; she did not make progress toward predicting far cases. We infer that no effects were engendered on her activity sequences through work with Cuisenaire® rods. These claims are elaborated next.</p> <p>In Lines 1–6, Penny alluded to a common difference when she said, "however many blocks you did [input], you take that off [the output] and get that block [common difference]." This suggests that her goal was to find the difference between the input and output and that she understood the task as identifying a common relationship between inputs and outputs. We infer that the mental activity in which she engaged was subdividing the output into the lengths of the input and common difference. This is based on her explanation in which she visually subdivided the cyan block with her fingers into about 4-cm and 2-cm lengths (Fig. 5). Penny did not quantify the common difference, however, which suggests the subdivision was tacit. This is consistent with the initial model of her numerical reasoning.</p> <p>In Lines 7–12, Penny's goal shifted, which suggests that she was dissatisfied that she had not explicitly identified the common difference. When Penny stopped making progress reasoning about the I4/O6 pair, the teacher-researcher re-directed the students' attention to the I3/O5 pair. In response, Lucía's reasoning suggested a different goal—to explain the relationship between consecutive outputs. Penny shifted to Lucía's goal, and she executed an activity sequence of comparing lengths, as indicated by her explanation that "it's getting one higher each time."</p> <p>In Lines 13–21, the students predicted a near case. Although Lucía initially identified an incorrect output, Penny pointed to the location where an input of five would be if they continued to increase the inputs by one. This suggests that Penny was applying her goal to find the output corresponding to the "next" input. Penny's reasoning was directed toward increasing the length of the cyan block by one, which suggests that her activity sequence included a visual comparison. We infer this from Penny's statement that the black block "would be one unit longer." Penny interpreted the result of her activity as expected, given her explanation to Lucía. As Penny's goal was met, no reflection on her activity sequence was engendered, and no effects generated with the Cuisenaire® rods.</p> <hd id="AN0184746713-15">Excerpt 2</hd> <p>Lucía and Penny consistently used recursive reasoning to identify near cases with Cuisenaire® rods, so we revised our hypothesis to state that linking cubes could support algebraic generalization of linear patterns. Whereas the initial hypothesis utilized Cuisenaire® rods to provide material for reflection on composite units, the revised hypothesis incorporated linking cubes to reflect the students' assimilatory units of one. Excerpt 2 documents effects to Penny's activity sequence when she set the goal of finding a common difference. The teacher-researcher's secret rule is <emph>x</emph> + 3. Lucía and Penny had inputted five then three cubes, which generated I5/O8 and I3/O6, represented by linking cube trains (Fig. 6).</p> <p>Graph: Fig. 6 Linking cubes used during Excerpt 2</p> <p></p> <ulist> <item> 1 P: You're adding three. You're adding three to each one.</item> <p></p> <item> 2 T: Okay. You think you know the [secret rule]?</item> <p></p> <item> 3 P: I don't know.</item> <p></p> <item> 4 T: Okay. Tell Lucía what you're thinking.</item> <p></p> <item> 5 P: Well, because of this one [points to the fifth block in the eight train (black dot)], we put five so we got three [taps the sixth, seventh, and eighth blocks], then in this one, we put three [points to the third block in the six train (black dot)] and we got three more [taps the fourth, fifth, and sixth blocks]...</item> <p></p> <item> 6 T: What if you put in 20 white blocks? How many blocks do you think you would get back?</item> <p></p> <item> 7 L &amp; P (together): Twenty-three.</item> <p></p> <item> 8 T: So how'd you get 23 so quickly?</item> <p></p> <item> 9 P: Since I had noticed that you put three on every single block that we put in. Just 20 plus 3 equals 23...</item> <p></p> <item> 10 T: Let's say we call [a friend named] Taylor[<reflink idref="bib1" id="ref96">1</reflink>] down here and we're like, "Hey Taylor, Lucía and Penny are going to tell you how this rule works so that no matter how many blocks you input, you can predict the output." What would you tell Taylor to do?</item> <p></p> <item> 11 P: I would say, how would I put that? I'll be like...I don't know how I would say that. [Pause.] I would probably just say that each time we did it, we got three back. No matter what answer you put in, you add three to that and you'd get your answer.</item> </ulist> <p>Following Excerpt 2, Lucía and Penny predicted near and far cases for single-operation addition and subtraction rules with the linking cubes across this and future teaching episodes. Penny also verbally generalized rules.</p> <hd id="AN0184746713-16">Analysis of Excerpt 2</hd> <p>In Lines 1–5, without prompting, Penny said, "you're adding three." This indicates that she set a goal of identifying the common difference, and that she met this goal. Penny's reasoning involved touching a block to represent the input embedded within the output and then tapping the three additional blocks. We infer that her activity sequence involved subdividing the output into two embedded composite units (input and common difference), thus unitizing the input, as indicated by her statement of "five" being a single tap. Next, she iterated a unit of one in activity to quantify the common difference, demonstrated as she tapped the three blocks. We perceive that Penny was aware that these activities yielded a positive result because she justified the result to Lucía.</p> <p>Moreover, we take these lines as evidence of an effect to Penny's activity sequence (i.e., learning) because this was the first indication of changes to her activity sequence in service of resolving the goal of identifying a common difference. Although Penny predicted near cases with Cuisenaire® rods, this was accomplished with an activity sequence focused on recursive reasoning. Penny's learning to quantify a common difference was supported by her attribution of mathematical meaning to the linking cubes, which was possible because she could see and count units of one.</p> <p>In Lines 6–11, Penny predicted a far case and verbally generalized, which she had not done with Cuisenaire® rods. We infer that to predict far cases, she set a goal to apply the "adding three" common difference to the 20 white blocks. To meet this goal, she added 3 to 20, which resulted in a correct prediction. Penny verbally generalized when she explained how to tell a friend to predict for <emph>any</emph> input. These results document the first instance of Penny's algebraic reasoning (Radford, [<reflink idref="bib30" id="ref97">30</reflink>]) and suggest the power of the effects to Penny's activity sequence. Identifying a common difference supported Penny's identification of far cases and verbal generalizations, where her previous activity sequence did not. This demonstrates that Penny's abstraction of the common difference allowed her to apply the result in contexts of algebraic reasoning.</p> <hd id="AN0184746713-17">Excerpt 3</hd> <p>We observed stability in Penny's reasoning with the linking cubes over several teaching episodes. Thus, we returned to our initial hypothesis, that Cuisenaire® rods could support Penny's algebraic generalizations. The purpose of returning to the initial hypothesis was to test whether Penny's newly constructed activity sequence was powerful enough to support generalizations without visible units of one. Excerpt 3 documents this result.</p> <p>In this variation of the teaching activity, Lucía and Penny took turns drawing a card with a secret rule written on it in words. The student who drew the card returned output blocks and the other student gave inputs and guessed the rule. This structure allowed the students to reason separately, thereby demonstrating the extent to which effects on their activity sequences were independent of their partner. In Excerpt 3, Lucía selected a card with the rule, "add three." Penny is giving inputs. Penny had inputted three, five, and two white blocks, and Lucía had returned the black (7; this was an error), brown (<reflink idref="bib8" id="ref98">8</reflink>), and yellow (<reflink idref="bib5" id="ref99">5</reflink>) blocks (Fig. 7).</p> <p>Graph: Fig. 7 Cuisenaire® Rods used in Excerpt 3</p> <p></p> <ulist> <item> 1 T: [To P] That was a confused face. All right, well, talk to me. What's going on?</item> <p></p> <item> 2 P: At first, I thought she was adding four...Like, each time I thought she would just add four to it because I gave you three and you gave me back seven.</item> <p></p> <item> 3 T: [To L] Is that your rule?</item> <p></p> <item> 4 L: Sort of.</item> <p></p> <item> 5 T: Sort of? What do you mean? Explain what you mean.</item> <p></p> <item> 6 L: [To P] What did you say again?</item> <p></p> <item> 7 P: You're adding four each time.</item> <p></p> <item> 8 L: [Shakes head no.] I'm giving her three more back...[Lucía and Penny align the inputs next to their outputs (Figure 8). They whisper to each other inaudibly. Penny touches the "empty" space that spans the distance from the last white block (I3) to the end of the black block four times and then the "empty" space that spans the distance from the last white block (I2) to the end of the yellow block three times.]</item> </ulist> <p>Graph: Fig. 8 Re-creation of Penny's block structures in Excerpt 3, Line 8</p> <p></p> <ulist> <item> 9 L: Ohhh. Okay, I messed up. [Replaces black block (<reflink idref="bib7" id="ref100">7</reflink>) with a cyan block (<reflink idref="bib6" id="ref101">6</reflink>).]</item> <p></p> <item> 10 T: That's all right. That's okay, it's going to happen. [To P] Does that make more sense now?</item> <p></p> <item> 11 P: Yeah.</item> <p></p> <item> 12 T: Okay, so can you both predict then? If Penny were to put these last 10 [white blocks] in, what would you give her back? You can write down the number or the color. [Both write 13.] Okay, say it on three. What would you give her back? One, two, three.</item> <p></p> <item> 13 L and P: 13!</item> </ulist> <hd id="AN0184746713-18">Analysis of Excerpt 3</hd> <p>Penny applied the same goal as previously because she said, "I thought she would just add four to it because I gave you three and you gave me back seven." She identified a common difference by comparing an input and corresponding output. Evidence that Penny continued to apply the activity sequence of subdividing the output into the input and the common difference and iterating units of one to quantify the difference was taken from her work with the manipulatives (Fig. 8). Although units of one were not visible, Penny's activity sequence was stable, and she applied the common difference to find near and far cases with Cuisenaire® rods, with addition and later subtraction patterns.</p> <hd id="AN0184746713-19">Summary</hd> <p>This study asked: In what ways can mathematics instruction with manipulatives engender reflection on the activity-effect relationship for sixth-grade students who assimilate numerical situations with units of one, in the context of algebraic generalizations? The excerpts presented took place across approximately 2 months and document conception-based adjustments to Penny's activity sequence for algebraic generalizations. Using Cuisenaire® rods in early teaching episodes, Penny set a goal to identify a common difference, but did not quantify the difference due to the tacit embedding of the input and common difference, and she reasoned recursively instead. The introduction of linking cubes supported Penny's quantification of the common difference by unitizing the input and iterating a unit of one because she could see and count the difference between inputs and outputs. This effect to her activity sequence constitutes conceptual learning (Simon et al., [<reflink idref="bib33" id="ref102">33</reflink>]) and was stable because Penny applied it across teaching episodes and manipulatives. This suggests stability in her evolved form of reasoning and power in the ways that Penny could apply units of one to reason algebraically. Thus, Penny's work with linking cubes engendered reflection on, and subsequent effects to, her activity sequence for generalizing.</p> <hd id="AN0184746713-20">Discussion</hd> <p>This study makes three contributions to the mathematics education literature. First is a deeper understanding of the key mental activities that supported Penny's generalizing and, also, implications for manipulative use and an instantiation of instruction that supported algebraic reasoning among students whose numerical operations were focused on units of one.</p> <hd id="AN0184746713-21">Mental activity supporting generalized arithmetic</hd> <p>The first contribution of this study is a delineation of the key mental activities that supported generalization of explicit patterns. These were critical to supporting Penny's mathematical thinking because they are indicators of algebraic thinking (Radford, [<reflink idref="bib30" id="ref103">30</reflink>]). These mental activities were subdividing a composite unit into embedded composite units—the input and common difference. Then, iterating units of one to quantify the difference. This is in contrast to her activity sequence in excerpt 1, which focused on visual length comparisons that limited her generalizations to near cases.</p> <p>Previous research found that students who interpret numerical situations with units of one did not verbalize generalizations. This was attributed to the decay of composite units following mental activity (Zwanch, [<reflink idref="bib45" id="ref104">45</reflink>]) and informed the driving hypothesis of this teaching experiment that Penny might learn to generalize by reflecting on visual representations of a composite unit. This study confirms the role of composite units in Penny's generalizing and further specifies an instructional sequence that supported effects to Penny's activity sequence for generalizing. This contributes to the field's understanding of the mental activities that may support generalized arithmetic, and early algebraic thinking suggests a rationale that may contribute to the "issue" (Carraher et al., [<reflink idref="bib9" id="ref105">9</reflink>]) of overreliance on recursive patterns and suggests a method by which instruction may engender effects related to generalizing.</p> <hd id="AN0184746713-22">Choice in manipulatives</hd> <p>The manipulative chosen to support Penny's reasoning was critical. Cuisenaire® rods did not support reflection that engendered effects to her activity sequence, but linking cubes did. This adds a theoretical framing to Baroody's ([<reflink idref="bib4" id="ref106">4</reflink>]) finding that some manipulatives are inappropriately complex. Penny's focus on units of one supported attribution of mathematical meaning to the linking cubes. Her work with the Cuisenaire® rods only led to algebraic generalizations after the noted effects to her activity sequence. One of the components of manipulatives that supports connections among mathematical ideas is transitioning to abstract representations (Laski et al., [<reflink idref="bib21" id="ref107">21</reflink>]), and we contend that Penny would not have made that transition without the matching of linking cubes to her numerical units.</p> <p>Additionally, Zwanch ([<reflink idref="bib45" id="ref108">45</reflink>]) found that in clinical interviews, students whose numerical reasoning was consistent with Penny's operations with units of one generalized near and far cases but did not verbalize generalizations because they did not operate on composite units. Therefore, we hypothesized that the Cuisenaire® rods might act as material for Penny's reflections on composite units because they could stand in as figurative composite units. This was not initially true; rather, advancing to verbal generalizations was dependent on effects to Penny's activity sequence that were engendered by an appropriately complex manipulative, indicated by consistency with her assimilatory units of one. Thus, building on previous findings, although Penny did not initially verbalize generalizations, she learned to do so through work with appropriate manipulatives.</p> <p>Finally, Norton and colleagues ([<reflink idref="bib28" id="ref109">28</reflink>]) brought to light the importance of using manipulatives in research and instruction to make apparent the mental activities students rely on in abstract mathematical thinking. They called on researchers to examine "the kinds of manipulatives [that] afford opportunities for sensorimotor activity, and coordinations thereof, that might induce internalized actions..." (p. 51). This study addressed this call in the context of algebraic generalizations. Penny's use of manipulatives made visible her activity sequence for generalizing by unitizing and iterating units of one to quantify the common difference. This provides mathematics educators with insight into the kinds of manipulatives that may support students who operate with units of one to algebraically generalize.</p> <hd id="AN0184746713-23">Instruction supporting algebraic thinking</hd> <p>It may not be uncommon for teachers to incorporate manipulatives or games such as guess my rule in middle-grades algebra; neither are the presumed contribution of this study. The instructional contribution of this study is a manner by which early algebraic thinking can be made accessible to a wider range of learners. Kieran ([<reflink idref="bib19" id="ref110">19</reflink>]) called for research that attends to the instructional interventions that are "pivotal to developing ... students' early algebraic thinking" (p. 1146), and this study contributes one such instructional intervention. This was accomplished by framing Penny's learning in terms of reflections on activity-effect relationships and is relevant for teachers trying to understand the mathematical thinking of students whose primary numerical operations are focused on units of one.</p> <p>Additionally, such framing initiates an act of equity in mathematics instruction, in that it demonstrates respect for the student's ways of thinking and "can support a teacher-researcher to disrupt the norm of students being tasked with learning the mathematical knowledge of their teacher" (Hackenberg et al., [<reflink idref="bib16" id="ref111">16</reflink>], p. 480) by beginning to build epistemic subjects in the context of algebraic generalizations.</p> <p>Finally, the framing of Penny's thinking about generalized arithmetic in terms of reflections on activity-effect relationships builds on research that studies the generalizations of students who operate with units of one. Primary findings are that students who operate with units of one did not identify the structural relationships between related quantities (Hackenberg, [<reflink idref="bib14" id="ref112">14</reflink>]) nor did they verbalize generalizations (Zwanch, [<reflink idref="bib45" id="ref113">45</reflink>]), both of which impeded generalizing. This research adds that the identification of structural relationships and verbalizations may be supported by instruction that incorporates increasingly complex manipulatives that engender reflections on units of one and later, composite units.</p> <hd id="AN0184746713-24">Conclusion</hd> <p>Manipulatives are not magical (Ball, [<reflink idref="bib3" id="ref114">3</reflink>]), and "good manipulatives are those that aid students in building, strengthening, and connecting various representations of mathematical ideas" (Sarama &amp; Clements, [<reflink idref="bib32" id="ref115">32</reflink>], p. 146). For Penny, linking cubes provided an entry point to algebraic generalizations that leveraged the operations of her numerical reasoning, thereby building and strengthening connections between mathematical ideas. Furthermore, there is a call for more research to address the conditions under which manipulatives should be utilized in instruction (Carbonneau et al., [<reflink idref="bib8" id="ref116">8</reflink>]; McNeil &amp; Jarvin, [<reflink idref="bib24" id="ref117">24</reflink>]), and this study suggests conditions under which manipulatives supported learning to generalize algebraically for two middle-grades students. We call on researchers and practitioners alike to consider the role that manipulatives should play in middle-grades mathematics instruction, and how the inclusion of manipulatives that are responsive to students' mathematical thinking might support all students to access otherwise abstract algebra concepts.</p> <hd id="AN0184746713-25">Funding</hd> <p>This research was supported by Oklahoma State University, Research Jumpstart/Accelerator Grant program, funded by the Office of the Vice President for Research.</p> <hd id="AN0184746713-26">Declarations</hd> <p></p> <hd id="AN0184746713-27">Conflict of interest</hd> <p>The authors declare no competing interests.</p> <hd id="AN0184746713-28">Publisher's Note</hd> <p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p> <ref id="AN0184746713-29"> <title> References </title> <blist> <bibl id="bib1" idref="ref2" type="bt">1</bibl> <bibtext> Adom G, Adu EO. 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| Items | – Name: Title Label: Title Group: Ti Data: The (Activity-)Effect of Manipulatives on Algebraic Generalizations: A Constructivist Teaching Experiment – Name: Language Label: Language Group: Lang Data: English – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Karen+Zwanch%22">Karen Zwanch</searchLink> (ORCID <externalLink term="http://orcid.org/0000-0001-9500-5186">0000-0001-9500-5186</externalLink>)<br /><searchLink fieldCode="AR" term="%22Brooke+Mullins%22">Brooke Mullins</searchLink> (ORCID <externalLink term="http://orcid.org/0000-0003-3258-9764">0000-0003-3258-9764</externalLink>) – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="SO" term="%22Educational+Studies+in+Mathematics%22"><i>Educational Studies in Mathematics</i></searchLink>. 2025 119(1):41-61. – Name: Avail Label: Availability Group: Avail 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/ – Name: PeerReviewed Label: Peer Reviewed Group: SrcInfo Data: Y – Name: Pages Label: Page Count Group: Src Data: 21 – Name: DatePubCY Label: Publication Date Group: Date Data: 2025 – Name: TypeDocument Label: Document Type Group: TypDoc Data: Journal Articles<br />Reports - Research – Name: Audience Label: Education Level Group: Audnce Data: <searchLink fieldCode="EL" term="%22Elementary+Education%22">Elementary Education</searchLink><br /><searchLink fieldCode="EL" term="%22Grade+6%22">Grade 6</searchLink><br /><searchLink fieldCode="EL" term="%22Intermediate+Grades%22">Intermediate Grades</searchLink><br /><searchLink fieldCode="EL" term="%22Middle+Schools%22">Middle Schools</searchLink> – Name: Subject Label: Descriptors Group: Su Data: <searchLink fieldCode="DE" term="%22Mathematics+Education%22">Mathematics Education</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Instruction%22">Mathematics Instruction</searchLink><br /><searchLink fieldCode="DE" term="%22Teaching+Methods%22">Teaching Methods</searchLink><br /><searchLink fieldCode="DE" term="%22Algebra%22">Algebra</searchLink><br /><searchLink fieldCode="DE" term="%22Manipulative+Materials%22">Manipulative Materials</searchLink><br /><searchLink fieldCode="DE" term="%22Educational+Experiments%22">Educational Experiments</searchLink><br /><searchLink fieldCode="DE" term="%22Grade+6%22">Grade 6</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Skills%22">Mathematics Skills</searchLink><br /><searchLink fieldCode="DE" term="%22Thinking+Skills%22">Thinking Skills</searchLink><br /><searchLink fieldCode="DE" term="%22Mental+Computation%22">Mental Computation</searchLink><br /><searchLink fieldCode="DE" term="%22Arithmetic%22">Arithmetic</searchLink> – Name: DOI Label: DOI Group: ID Data: 10.1007/s10649-024-10371-z – Name: ISSN Label: ISSN Group: ISSN Data: 0013-1954<br />1573-0816 – Name: Abstract Label: Abstract Group: Ab Data: To understand the ways that manipulatives might support changes in students' reasoning about algebraic generalizations, a constructivist teaching experiment was conducted with two sixth-grade students. The students interpreted numerical situations with units of one and could construct units of units in mental activity. Initially, the students' reasoning about Cuisenaire® rods did not lead to changes in their algebraic generalizations, whereas their reasoning about linking cubes did lead to such changes. The students' learning with linking cubes is explained by their enactment of physical operations with units of one on the linking cubes, which were consistent with their mental operations in numerical situations. Over time, and after learning to generalize with linking cubes, the students also began to attribute meaning to their physical operations with Cuisenaire® rods. Thus, instruction with manipulatives that reflected the student's interpretation of numerical situations supported their construction of algebra as generalized arithmetic. Instructional implications are discussed. – Name: AbstractInfo Label: Abstractor Group: Ab Data: As Provided – Name: DateEntry Label: Entry Date Group: Date Data: 2025 – Name: AN Label: Accession Number Group: ID Data: EJ1469084 |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1007/s10649-024-10371-z Languages: – Text: English PhysicalDescription: Pagination: PageCount: 21 StartPage: 41 Subjects: – SubjectFull: Mathematics Education Type: general – SubjectFull: Mathematics Instruction Type: general – SubjectFull: Teaching Methods Type: general – SubjectFull: Algebra Type: general – SubjectFull: Manipulative Materials Type: general – SubjectFull: Educational Experiments Type: general – SubjectFull: Grade 6 Type: general – SubjectFull: Mathematics Skills Type: general – SubjectFull: Thinking Skills Type: general – SubjectFull: Mental Computation Type: general – SubjectFull: Arithmetic Type: general Titles: – TitleFull: The (Activity-)Effect of Manipulatives on Algebraic Generalizations: A Constructivist Teaching Experiment Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Karen Zwanch – PersonEntity: Name: NameFull: Brooke Mullins IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 05 Type: published Y: 2025 Identifiers: – Type: issn-print Value: 0013-1954 – Type: issn-electronic Value: 1573-0816 Numbering: – Type: volume Value: 119 – Type: issue Value: 1 Titles: – TitleFull: Educational Studies in Mathematics Type: main |
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