ABC Problem in Elementary Mathematics Education: Arithmetic 'before' Comprehension
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| Title: | ABC Problem in Elementary Mathematics Education: Arithmetic 'before' Comprehension |
|---|---|
| Language: | English |
| Authors: | Boote, Stacy K., Boote, David N. |
| Source: | Journal of Mathematics Teacher Education. Apr 2018 21(2):99-122. |
| Availability: | Springer. 233 Spring Street, New York, NY 10013. Tel: 800-777-4643; Tel: 212-460-1500; Fax: 212-348-4505; e-mail: service-ny@springer.com; Web site: http://www.springerlink.com |
| Peer Reviewed: | Y |
| Page Count: | 24 |
| Publication Date: | 2018 |
| Document Type: | Journal Articles Reports - Research |
| Education Level: | Higher Education |
| Descriptors: | Mathematics Teachers, Preservice Teachers, Mathematics Instruction, Beliefs, Elementary School Teachers, Problem Solving, Mixed Methods Research, Models, Visual Aids, Error Correction, Equations (Mathematics), Feedback (Response), Comprehension, Mathematics Skills, Content Analysis |
| DOI: | 10.1007/s10857-016-9350-2 |
| ISSN: | 1386-4416 |
| Abstract: | Mathematical habits of prospective teachers affect problem comprehension and success and expose their beliefs about mathematics. Prospective elementary teachers (PSTs) (n = 121) engaged in a problem solving activity each week in class. Data were collected from PSTs enrolled in an undergraduate elementary mathematics methods course at a Southeastern State University over multiple semesters (six semesters, seven classes). PSTs' solution methods for one intentionally misleading mathematics problem were analyzed using a convergent parallel mixed methods content analysis. Two-thirds of PSTs misunderstood the problem scenario and directly translated numbers from the problem text. PSTs who answered correctly used a problem model strategy to comprehend the scenario and were more likely to use multiple models, draw a diagram, and draw a diagram before using another model. However, a large number of PSTs who answered incorrectly also used multiple models and drew diagrams. Self-correction was not common (8 of 121), because their equations did not provide feedback or support comprehension. Three kinds of imprecision also affected problem comprehension and were evident in both correct and incorrect solutions. Intentionally misleading problems helped PSTs see consequences of their mathematical habits and highlighted the importance of sense making and precision when creating problem models. |
| Abstractor: | As Provided |
| Number of References: | 96 |
| Entry Date: | 2018 |
| Accession Number: | EJ1172248 |
| Database: | ERIC |
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| FullText | Links: – Type: pdflink Url: https://content.ebscohost.com/cds/retrieve?content=AQICAHj0k_4E0hTGH8RJwT4gCJyBsGNe_WN95AvKlDbXJGqwxwGDXQcjlOV0weNBEOtdXez-AAAA4zCB4AYJKoZIhvcNAQcGoIHSMIHPAgEAMIHJBgkqhkiG9w0BBwEwHgYJYIZIAWUDBAEuMBEEDA-FTw9Z4RCYc449XgIBEICBm5EMgwusmNiRpvhrDP5cov526FJkjOeedkWrWSm_4AKHK2NDALxVP8QcAXQq-J2n2uyiBrb4g9cUlnK2CuQnry2r3Z4Bx2i05bZ1ZWnsPtratf_FlUf90QLBJ5n5s11FE63mnOmj1X-fkAjyPpL3mssp8cYLIrHhuTwHczApeJ_b1uf021qcmxGA5aDf7H-8QFLkhZqUpxu4oHmS Text: Availability: 1 Value: <anid>AN0128421072;oih01apr.18;2018Mar13.12:27;v2.2.500</anid> <title id="AN0128421072-1">ABC problem in elementary mathematics education: Arithmetic &lt;italic&gt;before&lt;/italic&gt; comprehension </title> <p>Mathematical habits of prospective teachers affect problem comprehension and success and expose their beliefs about mathematics. Prospective elementary teachers (PSTs) (&lt;italic&gt;n&lt;/italic&gt; = 121) engaged in a problem solving activity each week in class. Data were collected from PSTs enrolled in an undergraduate elementary mathematics methods course at a Southeastern State University over multiple semesters (six semesters, seven classes). PSTs’ solution methods for one intentionally misleading mathematics problem were analyzed using a convergent parallel mixed methods content analysis. Two-thirds of PSTs misunderstood the problem scenario and directly translated numbers from the problem text. PSTs who answered correctly used a problem model strategy to comprehend the scenario and were more likely to use multiple models, draw a diagram, and draw a diagram before using another model. However, a large number of PSTs who answered incorrectly also used multiple models and drew diagrams. Self-correction was not common (8 of 121), because their equations did not provide feedback or support comprehension. Three kinds of imprecision also affected problem comprehension and were evident in both correct and incorrect solutions. Intentionally misleading problems helped PSTs see consequences of their mathematical habits and highlighted the importance of sense making and precision when creating problem models.</p> <p>Content analysis; Elementary education; Mathematical proficiencies; Mathematics education; Mathematics teaching practices (MTPs); Mixed methods research; Prospective teachers; Problem solving; Standards for mathematical practice (SMPs)</p> <p>Prospective teachers come to mathematics methods courses with many well-practiced bad habits (Stigler and Hiebert [<reflink idref="bib87" id="ref1">87</reflink>] ; Cooney and Shealy [<reflink idref="bib13" id="ref2">13</reflink>] ; Sleep [<reflink idref="bib83" id="ref3">83</reflink>] ). Of particular concern in elementary mathematics education is the tendency to jump immediately to arithmetic before understanding the problem. If you began to arithmetize the problem in the epigraph before visualizing the log-cutting scenario, you may understand our dilemma. Using this direct translation strategy (Mayer and Hegarty [<reflink idref="bib52" id="ref4">52</reflink>] ) perpetuates one of the biggest challenges in mathematics education—the continued overemphasis on mathematical procedures without any connections to meaning (Stein and Smith [<reflink idref="bib85" id="ref5">85</reflink>] ; National Council of Teachers of Mathematics [<reflink idref="bib61" id="ref6">61</reflink>] ). Problem comprehension is such a concern among educators that this issue has taken center stage in current standards documents (National Council of Teachers of Mathematics [<reflink idref="bib61" id="ref7">61</reflink>] ; Common Core State Standards Initiative (CCSSI) [<reflink idref="bib12" id="ref8">12</reflink>] ; National Research Council [<reflink idref="bib63" id="ref9">63</reflink>] ; National Council for Social Studies [<reflink idref="bib58" id="ref10">58</reflink>] ; NGSS Lead States [<reflink idref="bib66" id="ref11">66</reflink>] ; National Research Council [<reflink idref="bib64" id="ref12">64</reflink>] ).</p> <p>The Standards for Mathematical Practice (SMPs) articulate the aims of the intended US mathematics curriculum (Common Core State Standards Initiative (CCSSI) [<reflink idref="bib12" id="ref13">12</reflink>] ; Porter et al. [<reflink idref="bib71" id="ref14">71</reflink>] ). These aims transcend numerous specific content standards of the K-12 curriculum and provide a more general vision of the mathematically proficient graduates of US school systems ([<reflink idref="bib12" id="ref15">12</reflink>] ). According to the National Research Council (NRC) in their seminal document, Adding It Up ([<reflink idref="bib62" id="ref16">62</reflink>] ), mathematical proficiency includes five specific constructs: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition. As such, the proficiency strands and SMPs motivate mathematics educators with a collective purpose and provide instructional guiderails to address documented, systemic weaknesses in our mathematics education system.</p> <p>One challenge for mathematics teacher educators is to provide engaging and purposeful experiences for prospective teachers (PSTs) to notice their own problem solving behaviors (Mellone [<reflink idref="bib55" id="ref17">55</reflink>] ; Mason [<reflink idref="bib51" id="ref18">51</reflink>] ). In particular, when solving a non-routine problem, PSTs must see the consequences of impulsive arithmetic. After learned habits are surfaced, they can see changes they must make in order to implement content and practice standards in their future classrooms. In essence, before we can change the problem solving habits of elementary school students, we need to change the habits of their future teachers.</p> <p>The purpose of this study was to understand factors affecting PSTs’ abilities to comprehend and accurately answer an intentionally misleading mathematics problem. In particular, our study was guided by the following research question: What patterns emerged from PSTs’ use of mathematical models to solve an intentionally misleading problem? The findings offer valuable suggestions for using intentionally misleading problems in mathematics methods courses as well as describing common mathematical struggles that PSTs in this sample experienced.</p> <hd id="AN0128421072-2">Literature review and theoretical framework</hd> <p>The Mathematics Teaching Practice (MTP), Use and connect mathematical representations, supports a problem solving culture within mathematics education (National Council of Teachers of Mathematics [<reflink idref="bib61" id="ref19">61</reflink>] ). One means of accomplishing this goal is the emphasis on “modeling” within both content and practice standards of CCSSI ([<reflink idref="bib12" id="ref20">12</reflink>] ). However, “[t]he distinction between models of mathematics and mathematical modeling is not always clear in US standards documents and in the mathematics education literature” (Cirillo et al. [<reflink idref="bib11" id="ref21">11</reflink>] ). Several researchers have emphasized the importance of differentiating mathematical modeling and modeling mathematics (Hirsch and McDuffie [<reflink idref="bib32" id="ref22">32</reflink>] ). The former “refers to using representations of mathematics to communicate mathematical concepts or ideas,” whereas the latter “links mathematics and authentic real-world questions” (Cirillo et al. [<reflink idref="bib11" id="ref23">11</reflink>] ). While modeling mathematics is an important focus in the K-8 curriculum, it remains important that elementary students learn to use appropriate models to represent real-world situations. In this paper, our emphasis is on the models PSTs construct to make sense of an intentionally misleading problem that demands comprehension of a real-world scenario.</p> <p>Pervasive instrumentalist beliefs about mathematics are one reason for this emphasis on modeling (Ernest [<reflink idref="bib19" id="ref24">19</reflink>] ; Cooney et al. [<reflink idref="bib14" id="ref25">14</reflink>] ). That is, most people believe that mathematics is a body of procedures and rules that, if followed with fidelity, will consistently lead to correct mathematical answers. They have learned these beliefs about mathematics through years of seeing them modeled and reinforced in K-12 and college mathematics classes (Lampert [<reflink idref="bib42" id="ref26">42</reflink>] ). In turn, their mathematical beliefs become embodied in their own mathematical practices (McGinn and Boote [<reflink idref="bib54" id="ref27">54</reflink>] ). Indeed, examining how someone solves mathematics problems may be the most reliable and valid means of understanding their beliefs about mathematics (Philippou and Christou [<reflink idref="bib68" id="ref28">68</reflink>] ).</p> <p>Not surprisingly, teachers who hold instrumentalist beliefs about mathematics are much more likely to perpetuate instrumentalist views of mathematics in their instruction, including the emphasis on procedures at the expense of conceptual understanding (National Council of Teachers of Mathematics [<reflink idref="bib61" id="ref29">61</reflink>] ; Francis et al. [<reflink idref="bib21" id="ref30">21</reflink>] ). Beliefs about learning outcomes, beliefs about effective instruction, and beliefs about the role of mathematics in school all depend on beliefs about the nature of mathematics (Ernest [<reflink idref="bib19" id="ref31">19</reflink>] ). Changing teachers’ beliefs in isolation will not address the broader systemic and institutional realities that also perpetuate instrumentalist mathematics (McCloskey [<reflink idref="bib53" id="ref32">53</reflink>] ; Hiebert [<reflink idref="bib30" id="ref33">30</reflink>] ; Gill and Boote [<reflink idref="bib25" id="ref34">25</reflink>] ). Nonetheless, one critical step in reforming elementary school mathematics education requires that PSTs become aware of unproductive mathematical habits before changing their broader belief system.</p> <p>Recent research on elementary PSTs has focused on several areas: content knowledge (Ball et al. [<reflink idref="bib4" id="ref35">4</reflink>] ; Kahan et al. [<reflink idref="bib39" id="ref36">39</reflink>] ), pedagogical content knowledge (Hill et al. [<reflink idref="bib31" id="ref37">31</reflink>] ; Isiksal and Cakiroglu [<reflink idref="bib34" id="ref38">34</reflink>] ), beliefs about mathematics and instruction (Ernest [<reflink idref="bib19" id="ref39">19</reflink>] ; Francis et al. [<reflink idref="bib21" id="ref40">21</reflink>] ; Swars et al. [<reflink idref="bib88" id="ref41">88</reflink>] ), noticing (Sherin et al. [<reflink idref="bib82" id="ref42">82</reflink>] ; Mitchell and Marin [<reflink idref="bib57" id="ref43">57</reflink>] ), and motivation and anxiety (Gellert [<reflink idref="bib23" id="ref44">23</reflink>] ; Swars et al. [<reflink idref="bib88" id="ref45">88</reflink>] ; Newton [<reflink idref="bib65" id="ref46">65</reflink>] ). In addition, research on elementary prospective teacher education has examined the effects and effectiveness of various interventions on these constructs. PSTs’ mathematical practices and the beliefs that support those practices receive relatively little attention in current research (National Council of Teachers of Mathematics [<reflink idref="bib61" id="ref47">61</reflink>] ).</p> <p>The importance of fostering mathematical understanding is not new but dates to at least the publication of NCTM’s Curriculum and Evaluation Standards that called for “a shift in emphasis from curriculum dominated by memorization of isolated facts and procedures to one that emphasizes conceptual understanding, multiple representation and connection, mathematical modeling, and mathematical problem solving” ([<reflink idref="bib59" id="ref48">59</reflink>] ). For this article, we focused on three of the five mathematical proficiencies identified in Adding It Up: conceptual understanding, procedural fluency, and strategic competence (National Research Council [<reflink idref="bib62" id="ref49">62</reflink>] ). Reform efforts often emphasize the importance of teaching content standards in ways that promote deeper conceptual understanding. However, within the Common Core State Standards (CCSS), the SMPs synthesize the NRC’s mathematical proficiencies ([<reflink idref="bib62" id="ref50">62</reflink>] ) and NCTM’s Process Standards ([<reflink idref="bib59" id="ref51">59</reflink>] , [<reflink idref="bib60" id="ref52">60</reflink>] ). Within K-12 education, the SMPs are integral to the reform agenda and embody the conceptual foundation of reform mathematics education in the USA.</p> <hd id="AN0128421072-3">Theory of mathematical understanding</hd> <p>We used Mayer and Hegarty’s theory of mathematical understanding ([<reflink idref="bib52" id="ref53">52</reflink>] ) to analyze PSTs’ problem comprehension processes. Mayer and Hegarty contrasted two approaches to solving mathematics word problems: direct translation and problem model strategies. Building on the work of Duncker ([<reflink idref="bib16" id="ref54">16</reflink>] ), Mayer and Hegarty defined a problem as the transition between a “given state” and a “goal state” when one does not immediately know how to achieve the latter. As a cognitive theory, their analytic focus was on strategies used when searching and identifying salient information from the problem text and, subsequently, how that information was represented and modeled. As such, their theory does not discount the importance of mathematical skills or procedures. Instead, it rests on evidence that difficulties with problem comprehension and representation cause greater difficulties (Mayer and Hegarty [<reflink idref="bib52" id="ref55">52</reflink>] ). By focusing on strategies PSTs used to comprehend, represent, and answer an intentionally misleading problem, we were able to examine mathematical proficiencies across the sample.</p> <hd id="AN0128421072-4">Mathematical problem types</hd> <p>Within problem solving research, there is an important distinction between routine and non-routine problems (Pólya [<reflink idref="bib70" id="ref56">70</reflink>] ; English and Sriraman [<reflink idref="bib17" id="ref57">17</reflink>] ; Goldin [<reflink idref="bib26" id="ref58">26</reflink>] ; Schoenfeld [<reflink idref="bib77" id="ref59">77</reflink>] ). In the former, the problem solver understands the given and goal states, and has a well-practiced procedure to move from one to the other. In the latter, there is neither a familiarity with the problem nor a readily accessible procedure available. Thus, increased cognitive effort is required to understand the given and goal states as well as how to move successfully between the two. Within mathematics education, non-routine problems are often seen as more important, because they demand the application of a variety of strategies (Avcu and Avcu [<reflink idref="bib3" id="ref60">3</reflink>] ; Jurdak [<reflink idref="bib38" id="ref61">38</reflink>] ; Lester [<reflink idref="bib49" id="ref62">49</reflink>] ).</p> <hd id="AN0128421072-5">Non-routine problems that appear routine</hd> <p>Mayer and Hegarty ([<reflink idref="bib52" id="ref63">52</reflink>] ) created a class of non-routine problems they called “inconsistent” to investigate problem comprehension strategies. On the surface, these problems look like routine, multi-step problems requiring straightforward calculations. However, inconsistent problems are actually non-routine due to misleading language about the arithmetic operation required:</p> <p>At Lucky, butter costs 65 cents per stick. This is 2 cents less per stick than butter at Vons. If you need to buy 4 sticks of butter, how much will you pay at Vons? (Mayer and Hegarty [<reflink idref="bib52" id="ref64">52</reflink>] )</p> <p>A superficial reading of the word “less” will lead many to subtract 2 cents from the given amount when adding 2 cents is correct. Taking advantage of a problem solver’s tendency to rely on “key words” instead of comprehending the scenario is the hallmark of inconsistent problems ([<reflink idref="bib52" id="ref65">52</reflink>] ). As a result, problem solvers are often unsuccessful when modeling and answering inconsistent problems.</p> <p>We called the non-routine problem used in this study intentionally misleading. Like inconsistent problems, intentionally misleading problems provoke comprehension evasiveness. Inconsistent problems (Mayer and Hegarty [<reflink idref="bib52" id="ref66">52</reflink>] ) lead the solver to use the wrong operation (e.g., subtraction when addition would be contextually correct). By contrast, intentionally misleading problems incite the solver to focus only on the numbers evident in the problem text instead of the hidden actions within the scenario. Unfortunately, using available numbers will always lead to an incorrect solution when solving intentionally misleading problems. Only considering the embedded actions within the scenario will illuminate the right numbers needed to solve the problem. Unlike inconsistent problems, both sets of numbers (i.e., the overt and inferred) will use the same multi-step operations to solve. The superficial features of intentionally misleading problems appear to be routine problems that are easily modeled, but the most salient aspects of the problem context are camouflaged by numbers that appear to be easily manipulated. This type of question exposes the tendency to focus on number rather than quantity (Thompson [<reflink idref="bib89" id="ref67">89</reflink>] ).</p> <hd id="AN0128421072-6">Direct translation and problem model strategies</hd> <p>When solving inconsistent or intentionally misleading problems, there is a high probability that a superficial reading of the problem text will lead to using a direct translation strategy (Mayer and Hegarty [<reflink idref="bib52" id="ref68">52</reflink>] ). Behaviors associated with this strategy include extracting numerical values and operations from the problem text without making sense of the problem scenario. Values and operations are then translated into an arithmetic or algebraic equation. Such problem solving behaviors are often associated with solving routine problems which only require the application of a previously known algorithm to a recognized mathematical situation (Pólya [<reflink idref="bib70" id="ref69">70</reflink>] ; Gerofsky [<reflink idref="bib24" id="ref70">24</reflink>] ). Students do not spontaneously arrive at these strategies on their own; rather, they are often explicitly or implicitly taught such “key word” strategies in school (Lave [<reflink idref="bib43" id="ref71">43</reflink>] ).</p> <p>By contrast, to solve inconsistent or intentionally misleading problems successfully, the problem solver must use a problem model strategy (Mayer and Hegarty [<reflink idref="bib52" id="ref72">52</reflink>] ). That is, they must first focus on comprehending what the problem is asking before transforming the problem scenario into a representative model using arithmetic, algebra, or a diagram. With intentionally misleading problems, the numbers available in the problem text communicate information that provide important parameters to the problem scenario. However, when creating arithmetic or algebraic equations when solving intentionally misleading problems, these numbers should not be used directly. When creating diagrams, these numbers provide feedback to “see” hidden and easily missed information within the problem text. In both cases, the provided numbers are relevant but require a thoughtful interpretation and inference prior to modeling the problem. If actions to align the problem text with the problem model(s) are overlooked, the likelihood of correctly answering the problem diminishes.</p> <p>Both inconsistent and intentionally misleading problems expose the learned tendency to read superficially and solve the problem before comprehending the problem. These superficial problem solving strategies are learned, explicitly or implicitly, in mathematics classes (Mayer and Hegarty [<reflink idref="bib52" id="ref73">52</reflink>] ) and are reinforced by cultural assumptions about mathematics and mathematics education (Lampert [<reflink idref="bib41" id="ref74">41</reflink>] ; Gill and Boote [<reflink idref="bib25" id="ref75">25</reflink>] ). Instrumentalist beliefs about mathematics encourage people to answer problems quickly and with fidelity because in many classrooms, “doing mathematics means following the rules laid down by the teacher; knowing mathematics means remembering and applying the correct rule when the teacher asks a question; and mathematical truth is determined when the answer is ratified by the teacher” (Lampert [<reflink idref="bib41" id="ref76">41</reflink>] ). These beliefs about mathematics influence students’ problem solving behaviors and can be explained by examining their resources, orientations, and goals (Schoenfeld [<reflink idref="bib76" id="ref77">76</reflink>] , [<reflink idref="bib80" id="ref78">80</reflink>] ).</p> <hd id="AN0128421072-7">Intentionally misleading problems expose the tendency to ignore a problem’s context</hd> <p>The characteristics of intentionally misleading problems make them especially amenable to explicit efforts to encourage the use of modeling (English and Sriraman [<reflink idref="bib17" id="ref79">17</reflink>] ; Lesh and Lehrer [<reflink idref="bib46" id="ref80">46</reflink>] ; Lesh and English [<reflink idref="bib45" id="ref81">45</reflink>] ; Lehrer and Schauble [<reflink idref="bib44" id="ref82">44</reflink>] ). For most people, the main challenge of mathematical problem solving is correctly representing the problem and not correctly executing the procedure (Mayer and Hegarty [<reflink idref="bib52" id="ref83">52</reflink>] ; Jonassen [<reflink idref="bib37" id="ref84">37</reflink>] ; Lester and Kehle [<reflink idref="bib50" id="ref85">50</reflink>] ). The notion and language choice of problem modeling strategies rests on information processing theories of cognition and the idea that comprehension means building mental models of the world or situations (Lehrer and Schauble [<reflink idref="bib44" id="ref86">44</reflink>] ). Subsequent mental processes act upon those mental models. However, more recent approaches to problem solving and modeling acknowledge the important role of externalizing our mental models in diagrams, tables, and equations to make them more amenable to inspection, challenge, and revision (English and Sriraman [<reflink idref="bib17" id="ref87">17</reflink>] ; Lester and Kehle [<reflink idref="bib50" id="ref88">50</reflink>] ; Roth and McGinn [<reflink idref="bib73" id="ref89">73</reflink>] ).</p> <p>Yet, comprehension does not just happen in the initial stage of problem solving just as it does not happen only in the beginning of reading a story. Like reading, problem solving comprehension is an ongoing process, and monitoring one’s comprehension of a problem while solving it can affect the ability to both find and have security in an answer. Students often emerge from K-12 education believing that correctly executing mathematical procedures will lead to their final answers being correct (Lampert [<reflink idref="bib41" id="ref90">41</reflink>] ; Mayer and Hegarty [<reflink idref="bib52" id="ref91">52</reflink>] ). Regrettably, satisfaction with procedural fluency can mask a flawed model and incorrect solution and, unsuspectingly, end the problem solving process.</p> <p>A considerable body of research has examined the role of heuristics and strategies that support problem comprehension (Cai [<reflink idref="bib8" id="ref92">8</reflink>] ; Schoenfeld [<reflink idref="bib75" id="ref93">75</reflink>] ; Cai and Lester [<reflink idref="bib9" id="ref94">9</reflink>] ; Fan and Zhu [<reflink idref="bib20" id="ref95">20</reflink>] ; Goldin [<reflink idref="bib26" id="ref96">26</reflink>] ; Henningsen and Stein [<reflink idref="bib29" id="ref97">29</reflink>] ; Jonassen [<reflink idref="bib37" id="ref98">37</reflink>] ; Schoenfeld [<reflink idref="bib78" id="ref99">78</reflink>] ; Garofalo and Lester [<reflink idref="bib22" id="ref100">22</reflink>] ), but the topic has received less attention recently (Cai [<reflink idref="bib8" id="ref101">8</reflink>] ; Schneider and Artelt [<reflink idref="bib74" id="ref102">74</reflink>] ). The decline in attention to heuristics and strategies is in part attributed to the shift in mathematics education to teaching “through” problem solving instead of teaching “about” or “for” problem solving (Lester [<reflink idref="bib49" id="ref103">49</reflink>] ). In Polya’s model of problem solving, comprehension is the first step, “Understand the problem,” and the only heuristic mentioned is “Draw a figure” ([<reflink idref="bib70" id="ref104">70</reflink>] ). Not surprisingly, many mathematics educators emphasize that drawing a picture or diagram of a scenario is a critical step in problem comprehension. Polya placed more emphasis on the role of heuristics during the second step, “Devise a plan.” However, subsequent research on problem solving instruction placed emphasis on the role of heuristics for a problem solution rather than problem comprehension. Unfortunately, explicit instruction of heuristics does not appreciably improve students’ problem solving performance (Schoenfeld [<reflink idref="bib78" id="ref105">78</reflink>] ).</p> <p>Finally, the characteristics of well-designed mathematics problems are well documented in the literature (Schoenfeld [<reflink idref="bib79" id="ref106">79</reflink>] ; Cai [<reflink idref="bib8" id="ref107">8</reflink>] ; Lesh and Zawojewski [<reflink idref="bib47" id="ref108">47</reflink>] ; Lester [<reflink idref="bib48" id="ref109">48</reflink>] ; Lampert [<reflink idref="bib41" id="ref110">41</reflink>] ; Keller [<reflink idref="bib40" id="ref111">40</reflink>] ). Well-designed problems should be open-ended, allow multiple solution methods, and be engaging (Cai [<reflink idref="bib8" id="ref112">8</reflink>] ). Less often discussed, however, is how students receive feedback during the problem solving process. Schoenfeld praised système didactique for intentionally designing instructional activities that allow students to understand whether they are making progress without needing guidance from the instructor ([<reflink idref="bib79" id="ref113">79</reflink>] ). The related notion that problem solvers should “check their work” (i.e., “look back,” Pólya [<reflink idref="bib70" id="ref114">70</reflink>] ) has been common advice for decades. However, less attention has been given to how the choice of problem model provides feedback and whether this feedback helps improve problem comprehension throughout the process.</p> <p>To understand factors affecting PSTs’ abilities to comprehend and accurately solve an intentionally misleading mathematics problem, the following research question guided our analysis: What patterns emerged from PSTs’ use of mathematical models to solve an intentionally misleading problem? The next section explains our research methods used to answer this question.</p> <hd id="AN0128421072-8">Methodology</hd> <hd id="AN0128421072-9">Participants</hd> <p>The demographics of the participants were broadly representative of the PSTs in the University’s Bachelor of Arts in Elementary Education degree program. Over the period of the study, the gender demographics for the program were 4.6 % male and 95.4 % female; ethnically, the demographics were 1 % Asian, 5.4 % African American or black, 5 % Hispanic or Latino, 2 % multiple races, and 86 % white (the remainder did not report their ethnicity).</p> <p>Participants (n = 121) were undergraduate Elementary Education majors enrolled in a mathematics methods course in a State University across six semesters in seven classes. Most participants were in the first semester of their senior year. Institutional Review Board (IRB) approval was obtained prior to data collection. Then, at the end of each semester, the purpose of the study was explained to students via a proxy and journals of willing participants were collected. Pseudonyms were used in the analysis.</p> <p>This course was the only required course that focused exclusively on mathematics methods in the program. A prerequisite methods course had a combined focus on teaching science and mathematics conceptually. The program does not require any specific prerequisite mathematics course beyond the general education requirements of the University or a complete Associates of Arts degree, a situation common to many PST education programs (Hart et al. [<reflink idref="bib27" id="ref115">27</reflink>] ). As a result, PSTs come into the program and course having completed a variety of college-level mathematics courses.</p> <hd id="AN0128421072-10">Data sources and procedures</hd> <p>The problem solving journal was a course assignment aligned with the state’s Elementary Education K-6 Mathematics Subject Area Standards. Each week in class, PSTs were given a problem to solve in three parts. After the problem was displayed on the document camera and read aloud, PSTs first worked alone, writing their solutions and strategy explanations in a composition notebook (Phase 1). Second, PSTs shared their solutions in small groups. During these discussions, they recorded peers’ responses in their journals and tried to persuade those who arrived at different answers to see their points of view. When groups finished sharing, the class engaged in a large group discussion to negotiate competing viewpoints. The goal was to arrive at consensus through mathematical reasoning (Phase 2). Finally, PSTs responded in writing to reflective prompts about experiences during the problem solving activity including affective responses, strategy justification and effectiveness, and awareness of solution accuracy (Phase 3).</p> <p>Weekly problems provided PSTs with opportunities to use a variety of strategies and practice productive sociomathematical norms (Yackel and Cobb [<reflink idref="bib94" id="ref116">94</reflink>] ). In addition, problems were constructed to expose PSTs’ well-practiced habits and assumptions about mathematics (Philippou and Christou [<reflink idref="bib68" id="ref117">68</reflink>] ). While problem solving was the main pedagogy used to teach many course outcomes, for the journal activity, intentionally misleading problems were used to reinforce the importance of comprehending the problem scenario before reacting to alluring surface features like numbers, units, and key words. Unless the problem context was thoughtfully considered while answering intentionally misleading problems, an incorrect answer would result.</p> <p>The primary data were 121 written responses to an intentionally misleading problem, the Log Problem:</p> <p>A uniform log can be cut into three pieces in 12 seconds. Assuming the same rate of cutting, how long would it take a similar log to be cut into four pieces?</p> <p>Given during the second class meeting, the language of this problem suggested a typical, multi-step story problem that could be solved using arithmetic calculations and/or an algebraic equation. Participants would have certainly answered such questions in previous mathematics courses when they studied proportional relationships, equivalent fractions, or rates. However, a correct solution required the recognition that the number of cuts (<reflink idref="bib2" id="ref118">2</reflink>) is the salient feature instead of the number of pieces (<reflink idref="bib3" id="ref119">3</reflink>). In this way, the problem was intentionally misleading.</p> <p>Students were asked not to erase in their journals, only strike-through when they felt they had made a mistake. This convention aided data analysis, because we could assume that students generally wrote from the top of the page to the bottom and from left to right. An important exception was when students returned to earlier equations or pictures to amend or append; such amendments or appendages could be distinguished, because they were written in smaller script and/or fit into available spaces.</p> <hd id="AN0128421072-11">Data analysis</hd> <p>Journals were analyzed using a convergent parallel mixed methods analysis, with mixing occurring throughout the analysis (Creswell and Plano Clark [<reflink idref="bib15" id="ref120">15</reflink>] ). The term “parallel” implies that the quantitative and qualitative data collection and analyses happen simultaneously, and then, these findings are compared and contrasted. In doing so, the two sources of data and findings are given equal weight (Johnson et al. [<reflink idref="bib36" id="ref121">36</reflink>] ). When those findings simply corroborate or triangulate each other, no additional analysis is needed (Jick [<reflink idref="bib35" id="ref122">35</reflink>] ). However, it is more common that the analysis of one data source will reveal the need for additional exploration in the other data source. This need to reconcile (i.e., “converge”) the qualitative and quantitative findings often requires additional analyses and sometimes even additional data collection.</p> <p>Our data in this study were problem solving journals, analyzed both quantitatively and qualitatively. We gave equal priority to both quantitative and qualitative analyses, allowing us to understand both the frequency of solution methods and the relationships among those solution methods. Our parallel analyses were iterative: Quantitative analyses suggested questions that needed qualitative analysis; qualitative analyses suggested the need for additional quantitative coding and frequency counts. This iterative approach was consistent with summative qualitative document analysis (Altheide et al. [<reflink idref="bib2" id="ref123">2</reflink>] ; Hsieh and Shannon [<reflink idref="bib33" id="ref124">33</reflink>] ) and continued until the research question was satisfactorily answered.</p> <p>Following the methods of Mayer and Hegarty ([<reflink idref="bib52" id="ref125">52</reflink>] ), we first coded solution methods as correct or incorrect and whether solutions used information evident in the text of the question (i.e., “three pieces in 12 seconds”) or information inferred from the scenario of the problem (i.e., 2 cuts in 12 seconds). We used an Excel spreadsheet to record these codes, keeping track of course semester and student numeric code. Descriptive statistics were calculated for data in the spreadsheet to identify patterns among answers and solution methods that were explored further using qualitative analysis. Differences in modeling choices were later assessed using Chi-squared goodness-of-fit tests, with the null hypotheses of no differences between participants who answered correctly and incorrectly.</p> <p>Our preliminary qualitative analysis sought to understand the reasons behind these preliminary quantitative findings. This qualitative analysis suggested that we should code the strategies used by each participant (i.e., arithmetic and/or algebraic equations, units, diagrams) and the order in which each strategy was used. We coded solutions as an algebraic equation if they used an unknown (“x”) or an arithmetic equation (without an unknown). The few participants who first answered incorrectly but then answered correctly within Phase 1 were coded as self-correcting. During preliminary analyses, we explored the entire corpus of solutions. Later, after the quantitative results highlighted salient differences within the sample, we focused on a subsample of journals to explore differences and similarities among specific strategies. These similarities and differences are presented in the results.</p> <p>We ensured the trustworthiness of the coding and analysis in several ways. Journals were initially coded by each author separately using two journals from each course section; disagreements and inconsistencies were identified and resolved through discussion. For example, we initially disagreed about how to code participants who first answered 16 and then 18 during Phase 1, agreeing to code them as correct but distinguishing them as self-correcting. Once agreement was reached, coding was completed by the first author with the second author auditing one-third of the journals. This audit led to the final decision to group arithmetic, algebraic, and rates as equations in the quantitative analysis to ensure consistency across the sample (Miles et al. [<reflink idref="bib56" id="ref126">56</reflink>] ; Altheide et al. [<reflink idref="bib2" id="ref127">2</reflink>] ). As patterns emerged that extended beyond Mayer and Hegarty’s theoretical framework (e.g., the use of direct translation in diagrams), these patterns were checked against the larger sample through additional quantitative coding of the entire data set. Finally, the iterative nature of convergent parallel mixed methods ensured triangulation between quantitative and qualitative results (Creswell and Plano Clark [<reflink idref="bib15" id="ref128">15</reflink>] ).</p> <hd id="AN0128421072-12">Results</hd> <p>The purpose of this study was to understand factors affecting PSTs’ abilities to comprehend and accurately answer an intentionally misleading mathematics problem. Our convergent parallel mixed methods analyses focused on answering the following research question: What patterns emerged from PSTs’ solutions to an intentionally misleading mathematics problem? Mayer and Hegarty’s theory of mathematical understanding ([<reflink idref="bib52" id="ref129">52</reflink>] ) was used to analyze 121 solutions to the Log Problem in PSTs’ problem solving journals. Quantitative findings are first presented in a list to compare the frequencies of incorrect and correct solution methods. Then, qualitative findings are presented to elaborate and describe relationships among solution methods.</p> <hd id="AN0128421072-13">Quantitative findings</hd> <p>After iterative cycles of coding and analyses, six quantitative findings were evident. Findings 1-4 relate to incorrect solutions, finding 5 describes correct solutions, and finding 6 presents the frequencies of impression across the sample:Table 1 presents descriptive statistics for findings 1-5. Table 2 shows results of Chi-squared goodness-of-fit tests for selected comparisons between correct and incorrect solutions.</p> <p>Two-thirds of participants (n = 80) incorrectly answered the multi-step Log Problem when they used the numbers and units directly evident in the story problem.</p> <p>Among these 80 participants, 47.5 % (38/80) used numbers and units from the story problem when they created diagram models.</p> <p>Of participants who answered incorrectly, 40 % (32/80) only created equations to model their solutions, which significantly limited feedback from their solutions.</p> <p>Thirty (<reflink idref="bib30" id="ref130">30</reflink>) participants who answered incorrectly modeled their solutions using a single model, and fifty (<reflink idref="bib50" id="ref131">50</reflink>) who answered incorrectly modeled their solution using multiple models. In all 50, the second model included the same numbers and units from the problem text.</p> <p>One-third (34 %) of all participants answered correctly (n = 41), inferring the correct numbers and units from the problem text. Of these, 83 % (34/41) created multiple models of the scenario. Eight (<reflink idref="bib8" id="ref132">8</reflink>) of those participants used multiple models but self-corrected between their first and second models, and most were aided by their use of diagrams.</p> <p>Imprecision with calculations, the equals sign, and units were evident in both correct and incorrect solutions. Out of the 41 participants who answered correctly, 4.8 % (n = 2) were imprecise with calculations, 39 % (n = 16) with the equals sign, and 48.7 % (n = 20) with units. Out of the 80 who answered incorrectly, 6 % (n = 5) were imprecise with calculations, 28.7 % (n = 23) with the equals sign, and 68.7 % (n = 55) with units.</p> <hd id="AN0128421072-14">Qualitative findings</hd> <p>Quantitative and qualitative analyses worked iteratively to corroborate and clarify our findings. To exemplify how PSTs used numbers and units directly evident in the story problem (Quantitative findings 1 and 2), we present our analysis of Ann’s direct translation strategy using multiple models (i.e., equations and diagrams). Next, Christy exemplifies PSTs who only created equations. Incorrect solutions from Ann, Christy, and Kameron illustrate how PSTs imitated their first model when creating their second model instead of returning to the original problem scenario to check their comprehension (Quantitative finding 4). We present Ellen’s correct solution using multiple models. Her problem model strategy exemplifies PSTs who did not extract numbers and units evident in the problem text and, instead, made an inference (Quantitative finding 5). Then, we describe solutions of Amber and Pauline who self-corrected between their first and second models after creating or re-examining their diagrams (Quantitative finding 5). Finally, to exemplify levels of imprecision across both correct and incorrect solutions, we refer to previous examples (Quantitative finding 6).</p> <hd id="AN0128421072-15">Directly translating with multiple models did not help</hd> <p>Ann was among two-thirds of PSTs who incorrectly answered the Log Problem (see Fig. 1). She used a direct translation strategy (Mayer and Hegarty [<reflink idref="bib52" id="ref133">52</reflink>] ) when creating both of her models, a diagram and equation, respectively. Ann first drew two logs, mirroring the two sentences in the problem scenario. Each drawing directly translated numbers and units into her diagram. She partitioned the first “log” into three pieces, enumerating each piece with a “4s” underneath to indicate that it would take 4 s to make each piece and adding plus signs between each “4s.” An equals sign followed the last “4s.” Immediately below, she wrote “12 seconds” and added an arrow to show how they were related. Ann drew a second log to the right of the first log and connected them with the conjunction, “so.” She enumerated the second log similar to the first—placing a “4” below each piece (without the accompanying unit “s”) and connected them with plus signs. Below she wrote “16 seconds,” without the accompanying equals sign or arrow.Ann’s solution for the Log Problem</p> <p>Ann modeled the problem a second time using two arithmetic equations in a box to the right of her diagrams. Like her diagram models, she used a direct translation strategy with her equations. She wrote two equations, one above the other, without a clear indication of the values for which she was solving.</p> <p>Similar to Ann, Christie used a direct translation strategy while creating two models, but unlike Ann, both of Christie’s models were equations. First, she used an arithmetic equation (see Fig. 2), getting the answer 16 s. in two short steps. She then remodeled the problem using an equivalent rate format, again directly translating the numbers into the second equation. Both models yielded the same incorrect answer.Christie’s solution for the Log Problem</p> <hd id="AN0128421072-16">Basing the second model on the first (incorrect) model did not help</hd> <p>Fifty participants used multiple models but answered incorrectly (see Table 1). Three patterns emerged from the qualitative analysis. First, many PSTs used the same information evident in the problem text for both the first and second models. This was evident in both of Christie’s solutions (see Fig. 2).</p> <p>Second, several PSTs seemed to model the problem twice, but used information from their first solution in their second model. This was evident in Ann’s second effort to model the problem scenario, using information from the diagrams in her arithmetic equations (see Fig. 1).</p> <p>Third, PSTs modeled the problem using information based on the problem text but not evident in their visible calculations, coded as “mental math.” For example, Kameron (see Fig. 3) first used diagrams to calculate that the second log would take 16 s to cut. He then tried to model the problem with the algebraic equation: 3x = 48. However, the number 48 was not evident in the question text or what he had previously written. The most plausible explanation for this case was that by looking at his second log, he presumed the correct answer was 16. Working backwards, so to speak, he constructed an equation to fit the desired result.Kameron’s solution for the Log Problem</p> <p>While many of the PSTs who answered incorrectly used multiple models, another large proportion tried to answer the question using only one model (30/80, 38 %). Pauline’s solution (see Fig. 4) was typical of PSTs who only used one way of modeling the problem. She modeled the problem using equivalent fractions in a form somewhat like an algebraic equation. Then, in two additional steps, she answered the question. The brevity of her solution was typical of most who used a single model.Pauline’s solution for the Log Problem</p> <hd id="AN0128421072-17">Problem modeling with diagrams and equations supported comprehension</hd> <p>Ellen was among one-third of PSTs who correctly answered the Log Problem (see Fig. 5). She used a problem model strategy (Mayer and Hegarty [<reflink idref="bib52" id="ref134">52</reflink>] ) with both her diagram and equations with units.Ellen’s solution for the Log Problem</p> <p>Ellen started the problem by drawing a “log” trisected by two dashed lines, with the dashed lines depicting that cuts were salient. In contrast to many others who answered the question incorrectly, Ellen’s “log” was rectangular rather than cylindrical. Immediately under her “log,” she wrote “3 pieces = 2 cuts.” Her representation used information inferred from the problem scenario and not in the language of the question, though the “equation” was mathematically imprecise. To the right side of her “log,” Ellen wrote an arithmetic equation. To create the left side of this equation, she had to combine information taken directly from the question text, “12 sec.,” with the inference made from the diagram, “2 cuts.” She then calculated the rate in the form “sec/cut.” Immediately below her arithmetic equation, Ellen converted her rate into a simplified form, “1 cut = 6 sec.” She did not use this simplified form in the next step of her solution; she instead used the rate form of 6 sec/cut.</p> <p>With the first step completed, Ellen returned to the left side of the page and drew another “log,” this time with three dashed lines to divide it into four pieces. Under the log, she again inferred “4 pieces = 3 cuts.” Using the information calculated from the first step, she wrote her equation with units and answered the question: “It takes 18 seconds to cut a similar log into 4 pieces.”</p> <p>Ellen preserved the units (i.e., “cut” and “sec”) throughout the solution, preserving the meaning of the situation. She also crossed out the units “cut” from the multiplier (written as a whole number) and from the denominator of the multiplicand (written as a fraction). Doing so provided her with immediate feedback, independent of correct procedures, that the product of her equation resulted in a meaningful unit, “sec”—the time necessary to cut 4 pieces. Ellen’s explicit use of dimensional analysis was atypical for this sample.</p> <hd id="AN0128421072-18">Self-correcting with a second model</hd> <p>Among the few who self-corrected before discussing their solutions with peers, most (6/8, 75 %) modeled the problem first using arithmetic or algebraic equations and then by drawing a diagram. For these six, the diagram enabled them to notice their mistake. Amber’s initial arithmetic solution (see Fig. 6) was quite simple, leading her to the incorrect answer of 16 s. However, she modeled the scenario again using a diagram, enabling her to see importance of cuts rather than pieces. Her second, diagrammatic solution was equally simple, but led to the correct solution.Amber’s solution for the Log Problem</p> <p>In contrast to Ellen, Amber’s use of units is less precise. For example, in her second equation she wrote “4 sec. × 4 pieces = 16 seconds.” More precise would have been 4 s/piece x 4 pieces = 16 s.</p> <p>Another participant who self-corrected drew a diagram prior to creating an equation. However, she reported that she recognized her error when she looked back at her diagram: “At first I made an equation and used a solution. Then looking at the picture I realized there are only 2 cuts so my first answer must be wrong.” These data suggest the importance of diagrams to model the problem scenario correctly and to self-correct when necessary.</p> <hd id="AN0128421072-19">Pervasive imprecision</hd> <p>Across the sample and woven within multiple parts of the problem solving process was an overt lack of precision, evident and equally troubling with both PSTs who answered correctly and those who answered incorrectly. Lack of precision was evident in:</p> <p>Imprecise calculations</p> <p>Imprecise use of units</p> <p>Imprecise use of the equals sign</p> <p>Imprecision in calculations was evident when participants were careless or sloppy with their presentation of their calculations. After first presenting a diagrammatic model of cutting a log into 3 pieces in 12 s, Alexa attempted to model it with an arithmetic equation (see Fig. 7). From the diagram, she had already determined that each piece would take 4 s, but she wrote the rate in the form of 3 pieces per 12 s. This form of the rate calculation would not yield the whole number 4 that she was expecting. Rather than rewriting the rate fraction in the form of 12 s per 3 pieces, she simply inserted what she already believed to be the correct answer. Such imprecision in calculations was relatively rare in this sample; all but two instances occurred in similar situations when PSTs inserted expected values into their calculations.Alexa’s solution for the Log Problem</p> <p>Alexa’s calculations also provided evidence of her misuse of units and rates. She used units to calculate her initial rate fraction, the “pieces/seconds” which was then equated with “sec.” When she continued the calculation, she introduced this rate as “4 sec. per log” and then a variation in that form, “seconds it takes to cut a piece,” in her final arithmetic calculation. Her inconsistency with the rate fractions and units meant that she could not perform the kind of systematic unit analysis seen in Ellen’s solution (see Fig. 5). This lack of precision in handling units was very common in this sample and evident in much of the data already presented, especially Ann’s inconsistent use of seconds and absent rate notation (see Fig. 1) as well as Amber’s mishandling of units throughout her calculations (see Fig. 6). By contrast, Ellen’s very careful unit analysis (see Fig. 5) enabled her to determine systematically that her final answer was measured in the correct units.</p> <p>Imprecision in the use of the equals sign ranged from relatively innocuous to egregious. For example, in her otherwise thorough and systematic solution, Ellen noted “3 pieces = 2 cuts” (see Fig. 5). This imprecise use of an equals sign denoted that “2 cuts” yielded “3 pieces.” Pauline (see Fig. 4) modeled the problem using equivalent fractions, “3 pcs/4 pcs = 12 sec/? sec.” However, like several others in the sample solving for an unknown, she continued her calculation by moving horizontally rather than vertically, and equated the initial pair of equivalent fractions with “48/3.”</p> <hd id="AN0128421072-20">Discussion</hd> <p>The purpose of this study was to understand factors affecting PSTs’ abilities to comprehend and accurately answer an intentionally misleading mathematics problem. One of our methodological assumptions was PSTs' journals reflect, to some extent, their thinking while solving the Log Problem. The data analyses revealed a number of patterns evident in the modeling of an intentionally misleading problem. These patterns helped us better understand how a lack of mathematical proficiencies (National Research Council [<reflink idref="bib62" id="ref135">62</reflink>] ) within this sample of PSTs undermined their mathematical comprehension (Mayer and Hegarty [<reflink idref="bib52" id="ref136">52</reflink>] ).</p> <p>Key word strategies focus on surface features and ignore the deep structure of relationships within the question text (Sowder [<reflink idref="bib84" id="ref137">84</reflink>] ; Walkington [<reflink idref="bib90" id="ref138">90</reflink>] ; Parmar [<reflink idref="bib67" id="ref139">67</reflink>] ) and are antagonistic to building the mathematical proficiency of conceptual understanding (National Research Council [<reflink idref="bib62" id="ref140">62</reflink>] ). Even when told the problem was intentionally misleading, two-thirds of PSTs’ solutions exhibited a direct translation strategy as described by Mayer and Hegarty ([<reflink idref="bib52" id="ref141">52</reflink>] ). When creating models, PSTs who used numbers and units directly evident in the question text did not make inferences or connections to the meaning about the problem scenario (Stein and Smith [<reflink idref="bib85" id="ref142">85</reflink>] ). The quantitative data analyses were corroborated by qualitative analyses, such as Ann who was satisfied only to extract numbers and operations from the question text (Gerofsky [<reflink idref="bib24" id="ref143">24</reflink>] ; Lave [<reflink idref="bib43" id="ref144">43</reflink>] ). This pervasive use of direct translation is troubling, because key word instructional strategies are not an effective means of solving story problems (Xin et al. [<reflink idref="bib93" id="ref145">93</reflink>] ; Walkington et al. [<reflink idref="bib91" id="ref146">91</reflink>] ).</p> <p>The remaining one-third of PSTs who answered the log problem correctly were much more likely to use diagrams and multiple representations. Consistent with a problem model strategy (Mayer and Hegarty [<reflink idref="bib52" id="ref147">52</reflink>] ) and the strategic competence proficiency (National Research Council [<reflink idref="bib62" id="ref148">62</reflink>] ), our data showed that accurate comprehension of the intentionally misleading problem scenario had to occur to yield correct diagrams, arithmetic or algebraic equations. The analysis of Ellen’s solution suggested that she correctly understood the meaning of the problem scenario before she modeled it with diagrams and equations. Her use of diagrams and clear reasoning demonstrated that she understood the relationship between cuts and resulting pieces.</p> <p>Our quantitative analyses corroborated that using diagrams and multiple representations helped PSTs overcome limitations of ineffective key word strategies. The data from the few PSTs who self-corrected show that diagrams provided visual feedback to correctly model the problem (Schoenfeld [<reflink idref="bib80" id="ref149">80</reflink>] ; Fan and Zhu [<reflink idref="bib20" id="ref150">20</reflink>] ; Poch et al. [<reflink idref="bib69" id="ref151">69</reflink>] ); visual feedback supported qualitative comprehension that PSTs could then model quantitatively (Thompson [<reflink idref="bib89" id="ref152">89</reflink>] ; Roth and McGinn [<reflink idref="bib73" id="ref153">73</reflink>] ). In all likelihood, the precise moment probably differed when PSTs correctly comprehended that cuts mattered instead of pieces. Some may have understood by reading the problem text multiple times, and others may have “seen” the hidden action after drawing their first log diagram. Either way, comprehension had to occur at some point during the problem solving process before multiple models, including drawing a diagram, could matter.</p> <p>However, we were surprised so many PSTs used multiple models, including diagrams, but persisted in using direct translation. Although using multiple forms of representation is an effective way of improving problem comprehension (Brenner et al. [<reflink idref="bib7" id="ref154">7</reflink>] ; Zawojewski [<reflink idref="bib95" id="ref155">95</reflink>] ), creating multiple models cannot guarantee a correct response. The qualitative analyses showed PSTs incorrectly modeled their diagrams by using numbers and units directly from the problem text. The qualitative analyses also showed, whether they drew diagrams first or later, they created subsequent models based on an earlier model rather than returning to the question text to check their comprehension. This finding is consistent with earlier studies, suggesting that people have difficulties making substantial use of multiple models or translating mathematical ideas between models (Ainsworth [<reflink idref="bib1" id="ref156">1</reflink>] ; Schoenfeld et al. [<reflink idref="bib81" id="ref157">81</reflink>] ). Problem solvers need to know which kind of diagrams to use and how to reason systematically while using them (English and Sriraman [<reflink idref="bib17" id="ref158">17</reflink>] ). In addition, many PSTs in this study focused on algebraic notation at the expense of algebraic thinking, in contrast to the findings of Zazkis and Liljedahl ([<reflink idref="bib96" id="ref159">96</reflink>] ). This finding is likely the result of viewing equations and even diagrams as merely a procedural means of deriving the solution rather than a means of modeling a situation or generalizing the result (Ernest [<reflink idref="bib19" id="ref160">19</reflink>] ). An important consequence of this view is that our PSTs' use of diagrams and equations greatly limited feedback they received from those models (Schoenfeld [<reflink idref="bib76" id="ref161">76</reflink>] , [<reflink idref="bib79" id="ref162">79</reflink>] ).</p> <p>Feedback not only improves comprehension, but also stimulates perseverance (Common Core State Standards Initiative (CCSSI) [<reflink idref="bib12" id="ref163">12</reflink>] ; Schoenfeld [<reflink idref="bib79" id="ref164">79</reflink>] ). While working individually, each PST decided whether to continue working. Many were satisfied to get a whole number solution after using a direct translation strategy, which was an odd choice when the problem was described as “intentionally misleading.” Their use of diagrams, multiple representations, and imprecision also suggested that most did not view their solution methods as a source of feedback to signal potential errors, an indication of a lack of strategic competence proficiency (National Research Council [<reflink idref="bib62" id="ref165">62</reflink>] ). PSTs who exhibited greater strategic competence, such as Ellen, increased the likelihood they could confirm their assumptions, modeling, and calculations were correct (Schoenfeld [<reflink idref="bib79" id="ref166">79</reflink>] ). PSTs who demonstrated less strategic competence seemed to use diagrams, multiple representations, and units because they were expected. The majority did not seem to expect their models to provide additional insights. These PSTs were accustomed to answering mathematics problems for teachers who were mainly or entirely concerned with the correct answers (Lampert [<reflink idref="bib41" id="ref167">41</reflink>] ; National Council of Teachers of Mathematics [<reflink idref="bib61" id="ref168">61</reflink>] ). Additionally, they were likely unaccustomed to anticipating potential objections to their mathematical reasoning or potential counter-arguments (Yackel and Cobb [<reflink idref="bib94" id="ref169">94</reflink>] ; English and Sriraman [<reflink idref="bib17" id="ref170">17</reflink>] ; Walshaw and Anthony [<reflink idref="bib92" id="ref171">92</reflink>] ), so they were casual and careless in their calculations and presentation of solutions.</p> <p>Imprecision evident in solutions reflected imprecision in thinking as well as a lack of procedural fluency proficiency (National Research Council [<reflink idref="bib62" id="ref172">62</reflink>] ). We coded and quantified three kinds of imprecision and found them across the sample, among both correct and incorrect solutions. The imprecision within PSTs’ calculations often occurred when they substituted what they deemed to be the “correct” value for an intermediate calculation after initially mismodeling the problem scenario. The imprecision evident in PSTs’ casual use of the equals sign suggests a casual disregard (or simply ignorance) for the concept of equality (Stephens [<reflink idref="bib86" id="ref173">86</reflink>] ). The imprecise use of units suggests, at minimum, most neither viewed units as worthy of careful computation nor valued the ability of dimensional analysis to provide meaningful feedback. Taken together, the analysis of imprecision reaffirms that PSTs held instrumentalist beliefs about mathematics (Ernest [<reflink idref="bib19" id="ref174">19</reflink>] ; Cooney et al. [<reflink idref="bib14" id="ref175">14</reflink>] ) and accepted the social norms of school mathematics (Gerofsky [<reflink idref="bib24" id="ref176">24</reflink>] ; Lave [<reflink idref="bib43" id="ref177">43</reflink>] ).</p> <p>While these findings are interesting, we also note limitations of our methods. Foremost, while we are comfortable these journals are a valid source of data about PSTs' problem solving habits and some of the cognitive processes behind those habits, journals provide limited data about cognitive processes. For example, our data could not measure how frequently problem solvers read and re-read the problem text to ensure precise alignment when creating representative models. As a result, we could not examine the possible correlation between the cognitive effort used to align the problem text with the representative model and the accuracy of the response. A research design using a think-aloud protocol and/or eye tracking would be the most direct means of collecting data about PSTs’ cognitive processes (Ericsson and Simon [<reflink idref="bib18" id="ref178">18</reflink>] ; Boote [<reflink idref="bib5" id="ref179">5</reflink>] ; Hegarty et al. [<reflink idref="bib28" id="ref180">28</reflink>] ; Boote and Boote in press). During the in-class activity, PSTs also discussed solutions with peers, another useful source of data about their cognitive processes. These discussions were partially documented in their journals but not audio-recorded, so an analysis of conversations was not possible. In addition, while some evidence about productive dispositions (National Research Council [<reflink idref="bib62" id="ref181">62</reflink>] ) emerged from analyses of strategic competence, these data could not support robust analyses of PSTs’ dispositions. Future research should examine these dispositions more closely.</p> <hd id="AN0128421072-21">Conclusion and implications</hd> <p>The Log Problem had excellent discriminate validity (Campbell and Fiske [<reflink idref="bib10" id="ref182">10</reflink>] ). Every PST who answered incorrectly gave the same answer, 16 s. Across the entire sample, this problem authentically exposed habits of mathematical sense making.</p> <p>Most PSTs were only one semester away from their final internship where they would take full classroom responsibility for mathematics instruction. Within this methods course, solutions to this problem and the ensuing conversations clarified the danger of arithmetic before comprehension. Yet, there was not always enough time during class to satisfy PSTs’ frustrations when their answers or solutions were incorrect. Since intentionally misleading problems are often provocative and “sticky” (Keller [<reflink idref="bib40" id="ref183">40</reflink>] ; Prensky [<reflink idref="bib72" id="ref184">72</reflink>] ), PSTs sometimes shared these problems with friends and family or continued thinking about them by themselves. One instance comes to mind.</p> <p>After getting the wrong answer in class, Robert wanted to continue defending his (incorrect) solution, because he believed his correct arithmetic and algebraic procedures were being overlooked. However, the following week, Robert started class by sharing his epiphany. He admitted that he was frustrated in the previous class and left unconvinced that he really was wrong. He then shared a story that happened between class meetings. At his job, he was cutting a patron’s submarine sandwich when he envisioned the Log Problem. As he was slicing the bread, he got it—he finally understood concretely the causal relationship implied by the problem scenario. The action of cutting takes a certain amount of time versus a piece of bread/log taking the time. Having a chance to act out the problem in an authentic setting provided clarity and details needed to persuade Robert to change his point of view. It is interesting to note that in the six semesters of journals analyzed for this study, no one used an “Act it Out” strategy (Pólya [<reflink idref="bib70" id="ref185">70</reflink>] ) during Phase 1. Acting out the Log Problem could easily have been improvised using a piece of paper.</p> <p>Robert’s anecdote suggests that comprehending and solving a problem could easily take more time than is available in a typical class. Answering an intentionally misleading problem provided Robert an opportunity to intellectually struggle in class, leave and find relevance in an everyday activity, and then feel the satisfaction of a problem genuinely resolved. The stickiness and necessity of multiple solution methods are two pedagogical attributes of intentionally misleading problems that make them valuable.</p> <p>The characteristics (Cai [<reflink idref="bib8" id="ref186">8</reflink>] ) of intentionally misleading problems are also effective for surfacing PSTs’ well-practiced bad habits, in particular the overwhelming tendency in these data to use a direct translation instead of a problem model strategy (Mayer and Hegarty [<reflink idref="bib52" id="ref187">52</reflink>] ). The mathematics needed to successfully answer the Log Problem was within the K-5 content standards (Common Core State Standards Initiative (CCSSI) [<reflink idref="bib12" id="ref188">12</reflink>] ). However, the apparent ease of the mathematics led two-thirds of PSTs to read the problem superficially. It would be interesting to investigate whether an intentionally misleading problem requiring harder mathematics would catalyze a more thoughtful consideration of the problem scenario.</p> <p>Intentionally misleading problems also have hidden actions embedded within the scenario that must be understood in order to attain a correct solution. Creating a model, like a diagram, can surface these hidden actions, so long as the focus is comprehension. While not as prolonged and intricate as a mathematical modeling activity, intentionally misleading problems demand comprehending the real-world context and can provide a transition from modeling mathematics to mathematical modeling (see Hirsch and McDuffie [<reflink idref="bib32" id="ref189">32</reflink>] ).</p> <p>Mathematics teacher educators must pay close attention to how their PSTs solve problems, since most have spent nearly two decades focusing primarily on numbers and operations. Drawing their attention to the cognitive space between comprehending a problem text and creating a problem model to “understand[ing] the relations among the variables in the problem” (Mayer and Hegarty [<reflink idref="bib52" id="ref190">52</reflink>] ) will help PSTs confront and adapt unproductive mathematical habits. Focusing on this cognitive space also furthers the agenda of Principles to Actions by helping us understand how PSTs “use and connect mathematical representations” (National Council of Teachers of Mathematics [<reflink idref="bib61" id="ref191">61</reflink>] ).</p> <p>Surfacing these habits to the exclusion of institutional and cultural realities of schools will not, by itself, address all challenges of improving mathematics instruction (McCloskey [<reflink idref="bib53" id="ref192">53</reflink>] ; Hiebert [<reflink idref="bib30" id="ref193">30</reflink>] ; Gill and Boote [<reflink idref="bib25" id="ref194">25</reflink>] ). A 16-week methods course will not be enough to break these habits completely, but it is an important step. Since “the quality of teachers is increasingly recognized as critical to student learning,” what PSTs espouse as appropriate problem solving behaviors is “essential to preparing ‘well started beginners’” (National Research Council [<reflink idref="bib63" id="ref195">63</reflink>] ).</p> <p>A uniform log can be cut into three pieces in 12 seconds. 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| Header | DbId: eric DbLabel: ERIC An: EJ1172248 AccessLevel: 3 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
| IllustrationInfo | |
| Items | – Name: Title Label: Title Group: Ti Data: ABC Problem in Elementary Mathematics Education: Arithmetic 'before' Comprehension – Name: Language Label: Language Group: Lang Data: English – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Boote%2C+Stacy+K%2E%22">Boote, Stacy K.</searchLink><br /><searchLink fieldCode="AR" term="%22Boote%2C+David+N%2E%22">Boote, David N.</searchLink> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="SO" term="%22Journal+of+Mathematics+Teacher+Education%22"><i>Journal of Mathematics Teacher Education</i></searchLink>. Apr 2018 21(2):99-122. – Name: Avail Label: Availability Group: Avail Data: Springer. 233 Spring Street, New York, NY 10013. Tel: 800-777-4643; Tel: 212-460-1500; Fax: 212-348-4505; e-mail: service-ny@springer.com; Web site: http://www.springerlink.com – Name: PeerReviewed Label: Peer Reviewed Group: SrcInfo Data: Y – Name: Pages Label: Page Count Group: Src Data: 24 – Name: DatePubCY Label: Publication Date Group: Date Data: 2018 – 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="%22Higher+Education%22">Higher Education</searchLink> – Name: Subject Label: Descriptors Group: Su Data: <searchLink fieldCode="DE" term="%22Mathematics+Teachers%22">Mathematics Teachers</searchLink><br /><searchLink fieldCode="DE" term="%22Preservice+Teachers%22">Preservice Teachers</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Instruction%22">Mathematics Instruction</searchLink><br /><searchLink fieldCode="DE" term="%22Beliefs%22">Beliefs</searchLink><br /><searchLink fieldCode="DE" term="%22Elementary+School+Teachers%22">Elementary School Teachers</searchLink><br /><searchLink fieldCode="DE" term="%22Problem+Solving%22">Problem Solving</searchLink><br /><searchLink fieldCode="DE" term="%22Mixed+Methods+Research%22">Mixed Methods Research</searchLink><br /><searchLink fieldCode="DE" term="%22Models%22">Models</searchLink><br /><searchLink fieldCode="DE" term="%22Visual+Aids%22">Visual Aids</searchLink><br /><searchLink fieldCode="DE" term="%22Error+Correction%22">Error Correction</searchLink><br /><searchLink fieldCode="DE" term="%22Equations+%28Mathematics%29%22">Equations (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Feedback+%28Response%29%22">Feedback (Response)</searchLink><br /><searchLink fieldCode="DE" term="%22Comprehension%22">Comprehension</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Skills%22">Mathematics Skills</searchLink><br /><searchLink fieldCode="DE" term="%22Content+Analysis%22">Content Analysis</searchLink> – Name: DOI Label: DOI Group: ID Data: 10.1007/s10857-016-9350-2 – Name: ISSN Label: ISSN Group: ISSN Data: 1386-4416 – Name: Abstract Label: Abstract Group: Ab Data: Mathematical habits of prospective teachers affect problem comprehension and success and expose their beliefs about mathematics. Prospective elementary teachers (PSTs) (n = 121) engaged in a problem solving activity each week in class. Data were collected from PSTs enrolled in an undergraduate elementary mathematics methods course at a Southeastern State University over multiple semesters (six semesters, seven classes). PSTs' solution methods for one intentionally misleading mathematics problem were analyzed using a convergent parallel mixed methods content analysis. Two-thirds of PSTs misunderstood the problem scenario and directly translated numbers from the problem text. PSTs who answered correctly used a problem model strategy to comprehend the scenario and were more likely to use multiple models, draw a diagram, and draw a diagram before using another model. However, a large number of PSTs who answered incorrectly also used multiple models and drew diagrams. Self-correction was not common (8 of 121), because their equations did not provide feedback or support comprehension. Three kinds of imprecision also affected problem comprehension and were evident in both correct and incorrect solutions. Intentionally misleading problems helped PSTs see consequences of their mathematical habits and highlighted the importance of sense making and precision when creating problem models. – Name: AbstractInfo Label: Abstractor Group: Ab Data: As Provided – Name: Ref Label: Number of References Group: RefInfo Data: 96 – Name: DateEntry Label: Entry Date Group: Date Data: 2018 – Name: AN Label: Accession Number Group: ID Data: EJ1172248 |
| PLink | https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=eric&AN=EJ1172248 |
| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1007/s10857-016-9350-2 Languages: – Text: English PhysicalDescription: Pagination: PageCount: 24 StartPage: 99 Subjects: – SubjectFull: Mathematics Teachers Type: general – SubjectFull: Preservice Teachers Type: general – SubjectFull: Mathematics Instruction Type: general – SubjectFull: Beliefs Type: general – SubjectFull: Elementary School Teachers Type: general – SubjectFull: Problem Solving Type: general – SubjectFull: Mixed Methods Research Type: general – SubjectFull: Models Type: general – SubjectFull: Visual Aids Type: general – SubjectFull: Error Correction Type: general – SubjectFull: Equations (Mathematics) Type: general – SubjectFull: Feedback (Response) Type: general – SubjectFull: Comprehension Type: general – SubjectFull: Mathematics Skills Type: general – SubjectFull: Content Analysis Type: general Titles: – TitleFull: ABC Problem in Elementary Mathematics Education: Arithmetic 'before' Comprehension Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Boote, Stacy K. – PersonEntity: Name: NameFull: Boote, David N. IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 04 Type: published Y: 2018 Identifiers: – Type: issn-print Value: 1386-4416 Numbering: – Type: volume Value: 21 – Type: issue Value: 2 Titles: – TitleFull: Journal of Mathematics Teacher Education Type: main |
| ResultId | 1 |
