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Leaping from Discrete to Continuous Independent Variables: Sixth Graders' Science Line Graph Interpretations

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Title: Leaping from Discrete to Continuous Independent Variables: Sixth Graders' Science Line Graph Interpretations
Language: English
Authors: Boote, Stacy K., Boote, David N.
Source: Elementary School Journal. Mar 2017 117(3):455-484.
Availability: University of Chicago Press. Journals Division, P.O. Box 37005, Chicago, IL 60637. Tel: 877-705-1878; Tel: 773-753-3347; Fax: 877-705-1879; Fax: 773-753-0811; e-mail: subscriptions@press.uchicago.edu; Web site: http://www.press.uchicago.edu
Peer Reviewed: Y
Page Count: 30
Publication Date: 2017
Document Type: Journal Articles
Reports - Research
Education Level: Elementary Education
Grade 6
Intermediate Grades
Middle Schools
Secondary Education
Junior High Schools
Descriptors: Elementary School Mathematics, Elementary School Students, Middle School Students, Grade 6, Graphs, Data Interpretation, Protocol Analysis, Interviews, Data Analysis, Data Collection
DOI: 10.1086/690204
ISSN: 0013-5984
Abstract: Students often struggle to interpret graphs correctly, despite emphasis on graphic literacy in U.S. education standards documents. The purpose of this study was to describe challenges sixth graders with varying levels of science and mathematics achievement encounter when transitioning from interpreting graphs having discrete independent variables to graphs having continuous independent variables. Data included think-aloud interviews and written line graph interactions. Data analysis focused on three constituent processes of graph interpretation: (1) encoding salient structures, (2) relating salient structures to each other, and (3) understanding referents in relation to salient structures. Difficulties encoding individual data points influenced interpretations of referents and relationships among data points. Cognitive resources learned for interpreting graphs with discrete independent variables both supported and hindered interpretations of graphs with continuous independent variables. Struggles relating graphs to referents reflected inexperience with data collection and analysis. Recommendations are provided to support students during this transition and to improve their ability to answer different types of graph questions.
Abstractor: As Provided
Entry Date: 2017
Accession Number: EJ1138136
Database: ERIC
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  Value: <anid>AN0121710528;esj01mar.17;2018Oct10.09:36;v2.2.500</anid> <title id="AN0121710528-1">LEAPING FROM DISCRETE TO CONTINUOUS INDEPENDENT VARIABLES. </title> <p>Students often struggle to interpret graphs correctly, despite emphasis on graphic literacy in U.S. education standards documents. The purpose of this study was to describe challenges sixth graders with varying levels of science and mathematics achievement encounter when transitioning from interpreting graphs having discrete independent variables to graphs having continuous independent variables. Data included think-aloud interviews and written line graph interactions. Data analysis focused on three constituent processes of graph interpretation: (<reflink idref="bib1" id="ref1">1</reflink>) encoding salient structures, (<reflink idref="bib2" id="ref2">2</reflink>) relating salient structures to each other, and (<reflink idref="bib3" id="ref3">3</reflink>) understanding referents in relation to salient structures. Difficulties encoding individual data points influenced interpretations of referents and relationships among data points. Cognitive resources learned for interpreting graphs with discrete independent variables both supported and hindered interpretations of graphs with continuous independent variables. Struggles relating graphs to referents reflected inexperience with data collection and analysis. Recommendations are provided to support students during this transition and to improve their ability to answer different types of graph questions.</p> <p>Ensuring that students become proficient graph readers is a significant goal within multiple literacy domains—mathematical literacy, scientific literacy, social scientific literacy, and informational text literacy (Common Core State Standards Initiative, [<reflink idref="bib22" id="ref4">22</reflink>]; Connected Learning Coalition, [<reflink idref="bib23" id="ref5">23</reflink>]; National Council for Social Studies, [<reflink idref="bib57" id="ref6">57</reflink>]; National Governors Association Center for Best Practice & Council of Chief State School Officers, [<reflink idref="bib58" id="ref7">58</reflink>]; NGSS Lead States, 2013). These principles, content standards, and practice standards signal to stakeholders that K–12 students must know how to interpret graphs effectively and use them successfully across disciplines (Friel, Curcio, & Bright, [<reflink idref="bib30" id="ref8">30</reflink>]).</p> <p>In practice, this means that knowing how to interpret a graph, usually taught through the Common Core State Standards for Mathematics (CCSSM) (Common Core State Standards Initiative, [<reflink idref="bib22" id="ref9">22</reflink>]), is prerequisite to using graphs to learn disciplinary content knowledge in domains like science and social studies (Glazer, [<reflink idref="bib35" id="ref10">35</reflink>]; Leinhardt, Zaslavsky, & Stein, [<reflink idref="bib48" id="ref11">48</reflink>]; National Council for Social Studies, [<reflink idref="bib57" id="ref12">57</reflink>]; NGSS Lead States, [<reflink idref="bib61" id="ref13">61</reflink>]). However, students often struggle when applying graph interpretation skills learned in mathematics during science, social studies, or language arts (Glazer, [<reflink idref="bib35" id="ref14">35</reflink>]; Leinhardt et al., [<reflink idref="bib48" id="ref15">48</reflink>]). More specifically, the Next Generation Science Standards (NGSS) presume students will come to science class knowing that different levels of measurement and different analytic purposes require different types of graphs (App. L, NGSS Lead States, [<reflink idref="bib61" id="ref16">61</reflink>]).</p> <p>By the end of fifth grade, most students have experience plotting and reading line plots and bar graphs and have been introduced to the Cartesian coordinate system. In sixth grade, these same students begin working with histograms and scatter plots and start analyzing continuous independent variables in addition to analyzing discrete independent variables. This is an important transition in students' learning about graphs and data analysis. During this learning "leap," students must understand appropriate connections and contrasts between the two forms of independent variables. Whereas bar graphs do not <emph>evolve</emph> into line graphs (both will continue to be used in and after middle school), line graphs do require a different kind of context. Contexts become graph referents, a critical component of interpreting a graph having a continuous independent variable. Both learning new graph forms and how context affects the selection of graph type make the leap from discrete to continuous independent variables precarious.</p> <p>To better understand this transition, the purpose of this study was to describe challenges sixth graders encounter when transitioning from interpreting graphs having discrete independent variables to graphs having continuous independent variables. Protocol analysis (Ericsson & Simon, [<reflink idref="bib28" id="ref17">28</reflink>]) guided the design of our study. Think-aloud data and written interactions with line graphs were recorded while sixth-grade participants individually answered questions excerpted from the Test of Graphing Skills in Science (TOGS) (McKenzie & Padilla, [<reflink idref="bib55" id="ref18">55</reflink>]). Two research questions guided the analysis: (<reflink idref="bib1" id="ref19">1</reflink>) How did sixth graders interact with graphs having continuous independent variables embedded in a science context? (<reflink idref="bib2" id="ref20">2</reflink>) How did sixth graders' interactions inform our understanding of their transition from interpreting discrete to continuous independent variables? Analysis of these data suggests that participants were affected by their previous knowledge of bar graphs and line plots when they interacted with line graphs, and the challenges they encountered varied by graph question level.</p> <hd id="AN0121710528-2">Literature Review and Theoretical Framework</hd> <p>The Measurement and Data domain in K–5 outlines a developmental progression supported by research (Friel et al., [<reflink idref="bib30" id="ref21">30</reflink>]) that leads students from understanding picture graphs to scaled bar graphs to line plots (Common Core State Standards Initiative, [<reflink idref="bib22" id="ref22">22</reflink>]). Such graphs are only appropriate for representing categorical and discrete independent variables, hereafter called <emph>discrete graphs</emph>. Students are introduced to Cartesian graphs as part of the Geometry domain in fifth grade but without explicit connections to data analysis. Beginning in the 6–8 Statistics and Probability domain, students are expected to plot and interpret continuous independent variables, hereafter called <emph>continuous graphs</emph>, and to understand the relationships between independent and dependent variables (2010). Appropriate graphs for representing continuous independent variables include histograms, scatter plots, and line graphs.</p> <p>In the NGSS Science and Engineering Practices, the graph types that students are expected to use in the 3–5 and 6–8 grade bands mirror the types of graphs recognized in the CCSSM Common Core State Standards Initiative (CCSSM, [<reflink idref="bib22" id="ref23">22</reflink>]; NGSS Lead States, [<reflink idref="bib61" id="ref24">61</reflink>]). In the 3–5 grade band, the NGSS indicate that students should use data tables, bar graphs, pictographs, or pie charts, consistent with the CCSSM in 3–5. Then, parallel to the CCSSM in the 6–8 grade band, students following NGSS begin analyzing continuous data, interpreting linear and nonlinear relationships, and distinguishing causal from correlational relationships when reading graphs.</p> <p>A critical difference between discrete and continuous graphs is the increased number of conceptual relationships available in the latter. Research on graph comprehension also emphasizes that different graphs are more appropriate and effective for representing different kinds of relationships (Carswell, [<reflink idref="bib16" id="ref25">16</reflink>]; Carswell & Wickens, [<reflink idref="bib17" id="ref26">17</reflink>]; Shah, Meyer, & Hegarty, [<reflink idref="bib83" id="ref27">83</reflink>]). Discrete graphs are more appropriate for representing univariate measures or absolute differences between measures. Indeed, the K–5 CCSSM and NGSS emphasize students' ability to read dependent variables and identify relative (qualitative) and absolute (quantitative) comparative conceptual relationships (Curcio, [<reflink idref="bib24" id="ref28">24</reflink>]; Kosslyn, [<reflink idref="bib42" id="ref29">42</reflink>]). By contrast, the introduction of continuous graphs affords new conceptual relationships: interpolation between data points, extrapolation beyond the data set, and discerning trends and relationships within a data set (Leinhardt et al., [<reflink idref="bib48" id="ref30">48</reflink>]; Trickett & Trafton, [<reflink idref="bib86" id="ref31">86</reflink>]). Thus, learning to read a scatterplot or line graph is not merely a matter of learning to read the <emph>x</emph>- and <emph>y</emph>-coordinates of a point or a point on a line, or learning the procedures for interpolation and extrapolation. Comprehension with continuous graphs also entails understanding the meaning of inferred points and trends between or beyond measured data points, the meaning of positive and negative slope, and the meaning of curvilinear relationships. Comprehending a graph's meaning is critical in NGSS, since the primary purpose of graph creation and interpretation resides under larger goals of understanding disciplinary core ideas and crosscutting concepts (NGSS Lead States, [<reflink idref="bib61" id="ref32">61</reflink>]).</p> <hd id="AN0121710528-3">Graph Comprehension and Component Cognitive Processes</hd> <p>Within mathematics education, graph <emph>comprehension</emph> emerged to link what we knew about successfully reading texts to successfully reading graphs (Adams, [<reflink idref="bib1" id="ref33">1</reflink>]; Curcio, [<reflink idref="bib24" id="ref34">24</reflink>]; Pinker, [<reflink idref="bib63" id="ref35">63</reflink>]). Similar to texts, reading graphs requires people to know graph types, constituent interpretive processes, communicative purposes, and content knowledge specific to graphs. Beyond mathematics education, researchers in other disciplines have examined how people interpret graphs, including statistics and data analysis (Gal, [<reflink idref="bib31" id="ref36">31</reflink>], [<reflink idref="bib32" id="ref37">32</reflink>]; Shaughnessy, [<reflink idref="bib84" id="ref38">84</reflink>]), science education (Berg & Phillips, [<reflink idref="bib4" id="ref39">4</reflink>]; Glazer, [<reflink idref="bib35" id="ref40">35</reflink>]; Preece & Janvier, [<reflink idref="bib66" id="ref41">66</reflink>]; Roth & Bowen, [<reflink idref="bib76" id="ref42">76</reflink>]), cognitive psychology (Shah & Freedman, [<reflink idref="bib80" id="ref43">80</reflink>]; Shah & Hoeffner, [<reflink idref="bib82" id="ref44">82</reflink>]; Trickett & Trafton, [<reflink idref="bib86" id="ref45">86</reflink>]), and information visualization (Card, [<reflink idref="bib13" id="ref46">13</reflink>]; Card, Mackinlay, & Shneiderman, [<reflink idref="bib14" id="ref47">14</reflink>]).</p> <p>In turn, these researchers have employed a variety of learning theories to interpret the successes and struggles of graph readers—behaviorism, cognitivism, sociocultural, and resource-based theories (Delgado & Lucero, [<reflink idref="bib25" id="ref48">25</reflink>]). To understand sixth graders' interactions with continuous graphs, we relied primarily on cognitive theories of graph comprehension (Carpenter & Shah, [<reflink idref="bib15" id="ref49">15</reflink>]; Hegarty, [<reflink idref="bib38" id="ref50">38</reflink>]; Shah & Freedman, [<reflink idref="bib80" id="ref51">80</reflink>]; Shah & Hoeffner, [<reflink idref="bib82" id="ref52">82</reflink>]). This dual processing theory recognizes the importance of visual processes like chunking and pattern matching (i.e., "bottom-up" processes) and the importance of prior knowledge about graph forms and content (i.e., "top-down" processes) (Shah & Freedman, [<reflink idref="bib80" id="ref53">80</reflink>]). Thus, while it is primarily a cognitive theory, it is compatible with constructs borrowed from other theoretical traditions within the graph interpretation literature. In particular, to better understand participants' interactions with graphs, we used <emph>inscriptions</emph> from sociocultural research on graphing (Bowen & Roth, [<reflink idref="bib11" id="ref54">11</reflink>]; Bowen, Roth, & McGinn, [<reflink idref="bib12" id="ref55">12</reflink>]; Latour, [<reflink idref="bib44" id="ref56">44</reflink>]; Roth & McGinn, [<reflink idref="bib78" id="ref57">78</reflink>]) and <emph>interactions</emph> from information visualization research on graphing (Card, [<reflink idref="bib13" id="ref58">13</reflink>]; Card et al., [<reflink idref="bib14" id="ref59">14</reflink>]). We also analyzed how participants used prior experiences with bar graphs as <emph>resources</emph> to scaffold their understanding of line graphs (Delgado & Lucero, [<reflink idref="bib25" id="ref60">25</reflink>]; Elby & Hammer, [<reflink idref="bib27" id="ref61">27</reflink>]; Hammer, Elby, Scherr, & Redish, [<reflink idref="bib37" id="ref62">37</reflink>]).</p> <hd id="AN0121710528-4">Challenges Encountered during Graph Comprehension</hd> <p>Reading a graph is not a single skill; instead, it is a complex set of interrelated skills. Several taxonomies describing component processes have been proffered (see Friel et al., [<reflink idref="bib30" id="ref63">30</reflink>], for a review), most of which stem from Bertin's ([<reflink idref="bib7" id="ref64">7</reflink>]/[<reflink idref="bib7" id="ref65">7</reflink>]) semiotic theory of graph interpretation. For this study's purposes, Shah and Hoeffner's ([<reflink idref="bib82" id="ref66">82</reflink>]) taxonomy was the most comprehensive and synthetic (see also Carpenter & Shah, [<reflink idref="bib15" id="ref67">15</reflink>]). They describe the cognitive processes of graph reading using three sequential <emph>graph component processes</emph> (see Table 1). Especially important from the cognitive perspective is to understand how these component graph interpretation processes are influenced by inherent limitations and biases of human visual processing, encoding, working memory, and information retrieval. Working memory is especially important in graph interpretation, because it helps the graph reader monitor and integrate information explicitly available and inferred from the graph (Carpenter & Shah, [<reflink idref="bib15" id="ref68">15</reflink>]; Halford, Wilsoon, & Phillips, [<reflink idref="bib36" id="ref69">36</reflink>]; Lohse, [<reflink idref="bib50" id="ref70">50</reflink>]). This progression of graph comprehension suggests that graph readers may have challenges within each component process and that challenges during an earlier process may compound difficulties in later processes.</p> <p>Graph</p> <p>Table 1. Graph Component Process Codes for Written Graph Interactions</p> <p> <ephtml> <table><tr valign="top"><td valign="top">Graph Component Processes</td><td valign="top">Written Interactive Behaviors</td></tr><tr valign="top"><td valign="top"><p>1. </p><p>Encoding salient structures</p></td><td valign="top"><p>a</p><p>Highlight individual structures of the graph by circling or underlining axes labels or scales for the purpose of making them stand out</p><p>b</p><p>Draw single or multiple line segments from axis labels to data point(s)</p><p>c</p><p>Draw tick marks or data points on axis/axes or add interval amounts on axis/axes</p></td></tr><tr valign="top"><td valign="top"><p>2. </p><p>Relating salient structures to each other</p></td><td valign="top"><p>a</p><p>Find differences between data points or perform other calculations using the interval data or answer choice data</p><p>b</p><p>Add data point to graph between existing data points (that are serving as "benchmark points")</p><p>c</p><p>Draw line segments to connect existing points or extend points (interpolation or extrapolation)</p><p>d</p><p>Draw line segments to connect all existing data points (to see trends)</p></td></tr><tr valign="top"><td valign="top"><p>3. </p><p>Understanding the referents in relation to salient structures</p></td><td valign="top"><p>a</p><p>Draw hypothetical scales</p><p>b</p><p>Draw hypothetical data (points/bars)</p><p>c</p><p>Draw both hypothetical scales and data</p></td></tr></table> </ephtml> </p> <p>11 The classification of graph component processes was amended from Shah and Hoeffner's taxonomy ([<reflink idref="bib82" id="ref71">82</reflink>]).</p> <p>Discrete graphs learned in K–5 can support students as they learn continuous graphs in 6–8. However, discrete and continuous graphs are different, and these differences may challenge students. The three graph component processes (Shah & Hoeffner, [<reflink idref="bib82" id="ref72">82</reflink>]) highlight continuities and challenges that students will encounter.</p> <hd id="AN0121710528-5">Process 1: identifying and encoding important visual features</hd> <p>Graph readers must first identify and encode important visual features. Encoding important information is influenced by natural biases and limitations of human visual processing and affects recognition of salient information (Trickett & Trafton, [<reflink idref="bib86" id="ref73">86</reflink>]). When students learn to read particular kinds of graphs, they parse information needed to interpret the graph type into visual chunks that are goal-directed, salient pieces of information, and encoded based on gestalt principles. Visual chunking reduces the demand on working memory by reducing the number of cognitive objects being handled, freeing working memory for additional interpretive tasks. Visual chunking also eases transition to the second and even third cognitive processes, when the referent of the graph is familiar and the format of the graph is well practiced. However, if the graph reader is not well practiced with a format, the process of encoding salient features will require deliberate processing (Bowen et al., [<reflink idref="bib12" id="ref74">12</reflink>]; Shah et al., [<reflink idref="bib83" id="ref75">83</reflink>]). This deliberate processing is facilitated by interacting with the graph, including the use of inscriptions to identify or eliminate salient features from consideration (Latour, [<reflink idref="bib44" id="ref76">44</reflink>]; Roth & McGinn, [<reflink idref="bib78" id="ref77">78</reflink>]), reducing the load on working memory (Card, [<reflink idref="bib13" id="ref78">13</reflink>]).</p> <p>Each graph type requires the reader to use different <emph>graph schemas</emph> (Pinker, [<reflink idref="bib63" id="ref79">63</reflink>]) or <emph>resources</emph> (Delgado & Lucero, [<reflink idref="bib25" id="ref80">25</reflink>]) that support encoding. Considerable attention has been devoted to scaffolding students' understanding of graphs during K–5 (Common Core State Standards Initiative, [<reflink idref="bib22" id="ref81">22</reflink>]). During grades K–1, students focus on categorical data by grouping, counting, and comparing groups. In grades 2–3, students represent categorical data using picture and bar graphs and compare categories within the graphs. Grades 4–5 focus on line plots, including operations on line plots. The goal of this progression is for students to read <emph>y</emph>-axis values on bar graphs and <emph>x</emph>-axis values on line plots. While levels of measurement (i.e., nominal/categorical, ordinal, interval, and ratio) are not explicitly introduced in K–5, it is presumed to be background knowledge for the Statistics and Probability domain in sixth grade. However, while there is a progression of skills and concepts in the K–5 standards, the transition to interpreting scatter plots and line graphs in sixth grade seems abrupt.</p> <p>Reading data points on scatter plots and line graphs requires skills similar to and different from reading bar graphs and line plots. Two differences can increase demands on working memory (Kosslyn, [<reflink idref="bib42" id="ref82">42</reflink>]). First, when reading a bar graph or line plot, the student has a visual reference to read one variable (the bar or stacked "<emph>x</emph>"s). Effort is only needed to read the other value by tracing an eye or finger. Second, while scaled bar graphs and line plots provide both independent and dependent variables, the K–5 standards only require finding one value and the difference between two values. By contrast, when reading a scatter plot or line graph, the student must simultaneously coordinate two variable values for each point (Leinhardt et al., [<reflink idref="bib48" id="ref83">48</reflink>]).</p> <hd id="AN0121710528-6">Process 2: relating salient structures to each other</hd> <p>Once salient features of the graph are encoded, the graph reader must interpret conceptual relationships among these salient features. A graph reader's ability to interpret these relationships is influenced by visual biases and limitations of the first process. When a graph type is well practiced and in an appropriate form, graph readers will more readily see the intended conceptual relationships (Hegarty, [<reflink idref="bib39" id="ref84">39</reflink>]; Shah & Hoeffner, [<reflink idref="bib82" id="ref85">82</reflink>]). As mentioned above, visual chunking may replace the encoding process (Trickett & Trafton, [<reflink idref="bib86" id="ref86">86</reflink>]), enabling the graph reader "the ability to automatically associate visual patterns with interpretations" (Shah & Freedman, [<reflink idref="bib80" id="ref87">80</reflink>], p. 563). However, graph readers will often need to interact with a graph to discern conceptual relationships using written or kinesthetic inscriptions (Latour, [<reflink idref="bib44" id="ref88">44</reflink>]; Roth & McGinn, [<reflink idref="bib78" id="ref89">78</reflink>]). These inscriptions externalize memory and cognition, reducing the load on working memory and cuing retrieval from long-term memory (Card, [<reflink idref="bib13" id="ref90">13</reflink>]; Card et al., [<reflink idref="bib14" id="ref91">14</reflink>]).</p> <p>Each conceptual relationship afforded by continuous graphs requires new skills and concepts (Bertin, [<reflink idref="bib7" id="ref92">7</reflink>]/[<reflink idref="bib7" id="ref93">7</reflink>]; Shah & Hoeffner, [<reflink idref="bib82" id="ref94">82</reflink>]). These interpretive processes all entail seeing the independent and dependent variables as continuous and understanding that data could exist abstractly between and beyond visible data points (Leinhardt et al., [<reflink idref="bib48" id="ref95">48</reflink>]). Friel and colleagues call this "reading between the data" ([<reflink idref="bib30" id="ref96">30</reflink>], pp. 130–131). In addition, interpolation and extrapolation require additional interactions or computations with data extracted from a graph, necessitating free cognitive resources and procedural routines.</p> <hd id="AN0121710528-7">Process 3: understanding the referents of the conceptual relationships</hd> <p>Once conceptual relationships among salient features are understood, the graph reader may interpret these relationships using knowledge of graph referents. That is, the graph reader may connect the trends among or differences between variables to meanings of those trends or differences for the phenomena these data represent. Understandings will be influenced by informal experiences, formal knowledge, and the ability to interpret graph relationships. Understandings will also be affected by the graph construction (Card, [<reflink idref="bib13" id="ref97">13</reflink>]) and metacognitive processes for cuing appropriate knowledge from long-term memory (Shah & Hoeffner, [<reflink idref="bib82" id="ref98">82</reflink>]).</p> <p>Once again, new conceptual relationships afforded by continuous graphs support new interpretations between graphs and referents. Especially important is the <emph>meaning</emph> of interpolated or extrapolated points and the <emph>meaning</emph> of trends and relationships. Students' experiences engaging in science inquiry using a variety of data, scientific texts, and field experiences may affect how they read and interpret graphs (Roth, Pozzer-Ardenghi, & Han, [<reflink idref="bib79" id="ref99">79</reflink>]). However, experiences with inquiry in general may not prepare them for specific graph interpretation activities, as it is often difficult to understand how raw data from a study were collected and measured, transformed in data tables, and converted into a graph (McClain & Cobb, [<reflink idref="bib54" id="ref100">54</reflink>]). The graph reader must infer and unpack these multiple transformations.</p> <p>Knowledge of the phenomena can also influence a graph reader's ability and willingness to interpret a graph's meaning (Hunter, Crismore, & Pearson, [<reflink idref="bib40" id="ref101">40</reflink>]; Lehrer & Romberg, [<reflink idref="bib47" id="ref102">47</reflink>]; Shah & Hoeffner, [<reflink idref="bib82" id="ref103">82</reflink>]). Inadequate knowledge of a graph's context can lead to superficial attention to graph features, staying in the second interpretive process, or using irrelevant information in the interpretation (Clement, [<reflink idref="bib19" id="ref104">19</reflink>]; Preece & Janvier, [<reflink idref="bib67" id="ref105">67</reflink>]; Shah & Freedman, [<reflink idref="bib80" id="ref106">80</reflink>]).</p> <p>Students vary in their willingness to move from the second to third interpretive process. In their study of middle-school biology students, Preece and Janvier ([<reflink idref="bib67" id="ref107">67</reflink>]) identified two broad approaches to the task—contextual and decontextual. The contextual approach involved understanding the line graph in relation to its referent, the covariation between shrimp and wastewater effluent around a drainage pipe. The decontextual approach involved reading the graph without reference to the science phenomena and remaining in the second process. This finding has been reported in subsequent studies (Roth, [<reflink idref="bib72" id="ref108">72</reflink>]; Roth & Bowen, [<reflink idref="bib76" id="ref109">76</reflink>]; Wu & Krajcik, [<reflink idref="bib89" id="ref110">89</reflink>]). These decontextual behaviors are not aberrant but are, instead, comprehension and interpretation strategies that were learned in the dominant culture of school mathematics and science (Gerofsky, [<reflink idref="bib34" id="ref111">34</reflink>]; Lave, [<reflink idref="bib45" id="ref112">45</reflink>]).</p> <p>Content standards across English language arts, mathematics, science, and social studies offer students many graphing opportunities, such as interpreting continuous graphs to understand embedded conceptual relationships (Common Core State Standards Initiative, [<reflink idref="bib22" id="ref113">22</reflink>]; National Council for Social Studies, [<reflink idref="bib57" id="ref114">57</reflink>]; National Research Council, [<reflink idref="bib59" id="ref115">59</reflink>]). Component processes of graph interpretation suggest specific challenges that sixth graders may face when interpreting continuous graphs. According to published research, we might expect sixth graders to require new visual processes, face greater demands on working memory, need more interactions to externalize memory and cognitive processes, and struggle to understand new conceptual relationships.</p> <p>The purpose of this study was to describe challenges sixth graders encounter when they transition from interpreting discrete to continuous graphs. In particular, we focused on two types of data: think-aloud verbalizations and written interactions with science line graphs. Two research questions guided the analysis: How did sixth graders interact with graphs having continuous independent variables embedded in a science context? How did sixth graders' interactions inform our understanding of their transition from interpreting discrete to continuous independent variables? The next section explains the methods used to answer these questions.</p> <hd id="AN0121710528-8">Methodology</hd> <p>Protocol analysis (Ericsson & Simon, [<reflink idref="bib28" id="ref116">28</reflink>]) was used to study the cognitive challenges sixth graders encounter while interpreting continuous graphs.</p> <hd id="AN0121710528-9">Sample</hd> <p>All sixth graders at one elementary school were asked to participate near the end of the school year (<emph>N</emph> = 125). A purposive sample of 13 was selected using maximum variation sampling (Patton, [<reflink idref="bib62" id="ref117">62</reflink>]). Participant selection was based on results from scores on their state standardized mathematics and science achievement tests (see Table 2). Participants' fifth-grade standardized state mathematics and science achievement levels were ranked on a 5-point scale. Quartiles were only available on the mathematics test. Pseudonyms were assigned based on high (levels 5 and 4), medium (level 3), and low (levels 2 and 1). Mathematics levels (H, M, or L) were used for participants' first names, and science levels were used for their last names. For example, Lucy Mag scored <emph>low</emph> on the mathematics test and <emph>medium</emph> on the science test.</p> <p>Graph</p> <p>Table 2. Participants' Previous Standardized State Mathematics and Science Achievement Levels</p> <p> <ephtml> <table><tr><td valign="bottom">Pseudonym</td><td valign="bottom">Mathematics Score(Level, Quartile)</td><td valign="bottom">Science Score(Level)</td></tr><tr><td valign="bottom">Hans Hazel</td><td valign="bottom">5, 3rd</td><td valign="bottom">5</td></tr><tr><td valign="bottom">Hugh Hickson</td><td valign="bottom">5, 3rd</td><td valign="bottom">5</td></tr><tr><td valign="bottom">Huck Handy</td><td valign="bottom">5, 1st</td><td valign="bottom">4</td></tr><tr><td valign="bottom">Hyde Hegel</td><td valign="bottom">4, 4th</td><td valign="bottom">4</td></tr><tr><td valign="bottom">Heather Miller</td><td valign="bottom">4, 4th</td><td valign="bottom">3</td></tr><tr><td valign="bottom">Hester Luck</td><td valign="bottom">4, 1st</td><td valign="bottom">2</td></tr><tr><td valign="bottom">Hodge Leader</td><td valign="bottom">4, 1st</td><td valign="bottom">2</td></tr><tr><td valign="bottom">Matt Maples</td><td valign="bottom">3, 4th</td><td valign="bottom">3</td></tr><tr><td valign="bottom">Misty Murphy</td><td valign="bottom">3, 4th</td><td valign="bottom">3</td></tr><tr><td valign="bottom">Lucy Mag</td><td valign="bottom">2, 2nd</td><td valign="bottom">3</td></tr><tr><td valign="bottom">Linda Mills</td><td valign="bottom">2, 1st</td><td valign="bottom">3</td></tr><tr><td valign="bottom">Lydia Lynn</td><td valign="bottom">2, 2nd</td><td valign="bottom">1</td></tr></table> </ephtml> </p> <p>21 Participants' fifth-grade standardized state mathematics and science achievement levels on a 5-point scale. Quartiles were only available on the mathematics test. Pseudonyms were assigned based on high (levels 5 and 4), medium (level 3), and low (2 and 1). Mathematics levels were used for participants' first names and science levels were used for their last names.</p> <p>Prior academic achievement in mathematics and science was expected to predict participants' interactions with science line graphs. A sampling matrix ensured the greatest range of development in mathematics and science for sixth graders at this school—from significantly below grade level to significantly above grade level. Participants with lower levels of prior achievement were slightly underrepresented in the final sample, because they were underrepresented in the pool of volunteers. The achievement tests used to choose the sample included short- and long-response items, free-response graphing items, and multiple-choice items.</p> <p>The school district in which these data were collected required that all grade K–2 students participate in class-created science projects and all grade 3–6 students create individual or team science fair projects each year. Participants were sampled from five classes taught by veteran teachers who all used the same district-mandated mathematics and science textbook series.</p> <hd id="AN0121710528-10">Data Collection</hd> <p>Primary data were individual think-aloud interviews and written interactions on the completed tests. The TOGS (McKenzie & Padilla, [<reflink idref="bib55" id="ref118">55</reflink>]) was a validated, multiple-choice test that assessed a large number of specific graphing abilities. Thirteen questions were excerpted from the full TOGS that met several criteria. Questions had to assess skills and concepts related to interpreting scatter plots and had to be developmentally appropriate. While some researchers have questioned the validity of assessing graph creation ability using multiple-choice questions (Berg & Boote, [<reflink idref="bib3" id="ref119">3</reflink>]; Berg & Phillips, [<reflink idref="bib4" id="ref120">4</reflink>]; Berg & Smith, [<reflink idref="bib5" id="ref121">5</reflink>]), we excerpted TOGS questions that focused on graph interpretation, not creation. Additional concerns with validity of multiple-choice questions were addressed by collecting think-aloud data, adding two open-ended questions (i.e., Overall Level Questions), and using retrospective debriefing questions.</p> <p>The standard format was one main question per page with an explanation of the science context at the top, a graph related to the context in the middle, a question below the graph, and four multiple-choice answers below the question. Two sequences of questions shared a common context and were presented in order, with each page re-presenting the context; therefore, participants had all the information needed on one page. Table 3 provides the excerpted TOGS questions with corresponding graph features, participants' success rates, and graph question levels.</p> <p>Graph</p> <p>Table 3. Features of Excerpted Test of Graphing in Science (TOGS) a Questions</p> <p> <ephtml> <table><tr><td valign="bottom">Question Level</td><td valign="bottom">Subtype</td><td valign="bottom">TOGS #</td><td valign="bottom">ParticipantsAnsweringCorrectly(%)</td><td valign="bottom">TOGSQuestion</td><td valign="bottom">DataPoints onGraph</td><td valign="bottom">ScaledAxes(x and y)</td><td valign="bottom">LabeledAxes(x and y)</td></tr><tr><td valign="top">External Identification Stage</td><td valign="top">Scales</td><td char="." valign="top">1</td><td char="." valign="top">43</td><td valign="top">Which set of axes below is the best to use for graphing his results?</td><td valign="top" /><td valign="top">✔</td><td valign="top">✔</td></tr><tr><td valign="top" /><td valign="top">Choose correct axes</td><td char="." valign="top">8</td><td char="." valign="top">57</td><td valign="top">Danny measured the time needed to heat various amounts of water to boiling. Which of the following is correctly labeled for showing the results?</td><td valign="top" /><td valign="top" /><td valign="top">✔</td></tr><tr><td valign="top" /><td valign="top" /><td char="." valign="top">9</td><td char="." valign="top">79</td><td valign="top">Mike wanted to know if the weight of chickens affected the number of eggs they laid each day. Which of the following is correctly labeled for showing his results?</td><td valign="top" /><td valign="top" /><td valign="top">✔</td></tr><tr><td valign="top">Elementary</td><td valign="top">No context</td><td char="." valign="top">6</td><td char="." valign="top">71</td><td valign="top">What are the proper coordinates for point a?</td><td valign="top">✔</td><td valign="top">✔</td><td valign="top" /></tr><tr><td valign="top" /><td valign="top" /><td char="." valign="top">7</td><td char="." valign="top">71</td><td valign="top">Which point is identified by the coordinates (15, 8)?</td><td valign="top">✔</td><td valign="top">✔</td><td valign="top" /></tr><tr><td valign="top" /><td valign="top">Science context</td><td char="." valign="top">3</td><td char="." valign="top">86</td><td valign="top">How much water was given each day to the plant that grew 10 cm tall?</td><td valign="top">✔</td><td valign="top">✔</td><td valign="top">✔</td></tr><tr><td valign="top" /><td valign="top" /><td char="." valign="top">10</td><td char="." valign="top">93</td><td valign="top">How much gas (liters) was used to drive 1 km at 60 km per hour?</td><td valign="top">✔</td><td valign="top">✔</td><td valign="top">✔</td></tr><tr><td valign="top">Intermediate</td><td valign="top">Interpolation</td><td char="." valign="top">2</td><td char="." valign="top">64</td><td valign="top">One plant was given 140 mL of water daily for 3 weeks. What would be the expected height of this plant at that time?</td><td valign="top">✔</td><td valign="top">✔</td><td valign="top">✔</td></tr><tr><td valign="top" /><td valign="top" /><td char="." valign="top">12</td><td char="." valign="top">86</td><td valign="top">At 55 km per hour, how much gas (liters) would the car use?</td><td valign="top">✔</td><td valign="top">✔</td><td valign="top">✔</td></tr><tr><td valign="top" /><td valign="top">Describe relationship between variables</td><td char="." valign="top">4</td><td char="." valign="top">79</td><td valign="top">The following statements describe the relationship between the amount of water given and the height of the plant. Which is the best description?</td><td valign="top">✔</td><td valign="top">✔</td><td valign="top">✔</td></tr><tr><td valign="top" /><td valign="top" /><td char="." valign="top">11</td><td char="." valign="top">86</td><td valign="top">Which of the following is the best description of the relationship shown on the graph?</td><td valign="top">✔</td><td valign="top">✔</td><td valign="top">✔</td></tr><tr><td valign="top" /><td valign="top">Extrapolation</td><td char="." valign="top">5</td><td char="." valign="top">93</td><td valign="top">How tall would you expect plants to grow if given 205 mL of water each day?</td><td valign="top">✔</td><td valign="top">✔</td><td valign="top">✔</td></tr><tr><td valign="top" /><td valign="top" /><td char="." valign="top">13</td><td char="." valign="top">93</td><td valign="top">At 80 km per hour, how much gas (liters) would the car use?</td><td valign="top">✔</td><td valign="top">✔</td><td valign="top">✔</td></tr></table> </ephtml> </p> <ulist> <item>31 Levels and stages are classified according to Bertin ([<reflink idref="bib7" id="ref122">7</reflink>]).</item> <item>3 TOGS questions were excerpted with permission for use in the original study (McKenzie & Padilla, [<reflink idref="bib55" id="ref123">55</reflink>]).</item> </ulist> <p>Excerpted TOGS questions were categorized into three question levels (Bertin, [<reflink idref="bib7" id="ref124">7</reflink>]/[<reflink idref="bib7" id="ref125">7</reflink>]) that closely align with our theoretical framework of three graph component processes (Shah & Hoeffner, [<reflink idref="bib82" id="ref126">82</reflink>]). Bertin distinguished levels of graph questions—Elementary, Intermediate, and Overall—where each level of question builds on information and skills from lower levels. Elementary Level Questions require extraction of a single piece of information found at one location on the graph. Two of our Elementary Level Questions were "contextual," asking participants to find the <emph>y</emph>-coordinate value of a given point when given the <emph>x</emph>-coordinate but asked using the identification of axis labels of coordinates. The other two Elementary Level Questions were "decontextual," asking participants to identify proper coordinates of a point or to identify the point when given a proper coordinate. Intermediate Level Questions, by contrast, require noticing patterns among groups of elements on a graph, gleaning information from several places, and consolidating information into a more general statement (Bertin, [<reflink idref="bib8" id="ref127">8</reflink>]). Such questions might seek either qualitative (by eye) or quantitative (extracting and manipulating numerical data) answers. Overall Level Questions require using content knowledge, experiences, or inferences to explain the graph's meaning. These questions may also ask for an explanation of data represented within the graph. A fourth question type, External Identification Stage Questions, requires the selection of correct axes for independent and dependent variables represented in a study (Bertin, [<reflink idref="bib7" id="ref128">7</reflink>]/[<reflink idref="bib7" id="ref129">7</reflink>]; Boote, [<reflink idref="bib9" id="ref130">9</reflink>]; Keller, [<reflink idref="bib41" id="ref131">41</reflink>]). External Identification Stage Questions are more challenging than the other three question types, since there are no cues provided within given graphs (Boote, [<reflink idref="bib9" id="ref132">9</reflink>]; Keller, [<reflink idref="bib41" id="ref133">41</reflink>]). For this type of question, graph readers must imagine hypothetical data values emerging from the scenario in order to select corresponding independent and dependent axis labels.</p> <p>The original TOGS (McKenzie & Padilla, [<reflink idref="bib55" id="ref134">55</reflink>]) only included Elementary and Intermediate Level Questions. To assess participants' knowledge and experiences of the science scenarios within the questions, three strategies were used. First, an Overall Level Question, "Why do you think this happened?" was added after two questions. These questions were written on the test page and presented to participants as questions 4b and 11b. From participants' perspectives, these questions were simply a continuation of the test. Second, the researcher asked standardized interview questions at specific points during think-aloud interviews. Some assessed when participants were guessing: "Why did you chose answer _____ instead of answer _____?" These were asked even when participants answered correctly. Standardized questions also elicited additional data about participants' experiences with phenomena represented in graphs. For example, "Do you have any personal experience boiling water?" and "Do you have any experience with chickens?" Third, the researcher asked retrospective debriefing questions when participants' behavior or verbalizations were not clear or when their think-aloud utterances stopped (Ericsson & Simon, [<reflink idref="bib28" id="ref135">28</reflink>]). The second and third kinds of interview questions were only asked after participants had already finished the question. As a result, these additional questions had no effect on think-aloud responses.</p> <p>Even though the cognitive processes (Carpenter & Shah, [<reflink idref="bib15" id="ref136">15</reflink>]; Shah & Hoeffner, [<reflink idref="bib82" id="ref137">82</reflink>]) and question levels (Bertin, [<reflink idref="bib7" id="ref138">7</reflink>]/[<reflink idref="bib7" id="ref139">7</reflink>]) are closely related, for the sake of clarity, we used these two constructs to distinguish question levels from cognitive processes. While others have proposed alternative categorizations for graph question levels (see Friel et al., [<reflink idref="bib30" id="ref140">30</reflink>], for a review), all were based on Bertin's original taxonomy and did not add new insight.</p> <p>In addition to the primary think-aloud and written interaction data, the Graph Interpretation Scoring Rubric (Boote, [<reflink idref="bib9" id="ref141">9</reflink>]) was used to record participants' behaviors during the interview. This instrument indicated the primary categorization of each TOGS question as well as secondary graph interactions that participants exhibited during the interview. It also synthesized errors, previously identified in prior research, that are likely to occur during graph interpretation.</p> <hd id="AN0121710528-11">Analysis</hd> <p>Protocol analysis was used for the data analysis (Ericsson & Simon, [<reflink idref="bib28" id="ref142">28</reflink>]). The first author created verbatim transcriptions from all audiorecorded interviews. All participants' utterances were included except when unintelligible. Pauses longer than 2 seconds were notated using ellipses. Participants' interactions, like drawing on the graph, inserting data points, and using an answer elimination strategy, were recorded in parentheses within transcriptions.</p> <p>Graph interpretation often requires interactions with the graph (Card, [<reflink idref="bib13" id="ref143">13</reflink>]; Latour, [<reflink idref="bib44" id="ref144">44</reflink>]; Roth & McGinn, [<reflink idref="bib78" id="ref145">78</reflink>]), so participants' written interactions provided additional data about their cognitive processes. To answer the first research question, participants' written interactions with each graph were categorized using the three component processes of graph interpretation (Carpenter & Shah, [<reflink idref="bib15" id="ref146">15</reflink>]; Shah & Hoeffner, [<reflink idref="bib82" id="ref147">82</reflink>]). Additional subcategories of graph interactions were added for greater precision (see Table 1). To answer the second research question, patterns emerging from the first analysis were used to guide line-by-line analysis of think-aloud transcripts using the three component processes (Shah & Hoeffner, [<reflink idref="bib82" id="ref148">82</reflink>]). Although many participants answered questions without using written interactions, all provided think-aloud data. In all cases when participants interacted with graphs, their think-aloud verbalizations were consistent with their written interactions, though in many instances one form of data provided clearer insights into their cognitive processes. These insights are reflected in the reporting of the results.</p> <p>Standardized interview and retrospective debriefing questions were used to triangulate, clarify, and elaborate think-aloud data. Trustworthiness of our analysis was checked several ways. First, data were coded by two independent experts who agreed on all but one indicator. Second, after data were collected and transcribed, the same two experts verified four transcriptions against audio data and coded the transcriptions using the scoring instrument (Boote, [<reflink idref="bib9" id="ref149">9</reflink>]). No discrepancies were identified. Third, after data were analyzed, an external auditor verified the coding and interpretation of participants' graph interpretation behaviors. Finally, negative case analysis was used to ensure that all inferences and interpretations were tested against the entire sample of data to determine whether generalizations were correct.</p> <hd id="AN0121710528-12">Results</hd> <p>Analyses of think-aloud data and written tests provided insights into specific challenges this sample of sixth graders experienced while answering four levels of graph questions (Bertin, [<reflink idref="bib7" id="ref150">7</reflink>]/[<reflink idref="bib7" id="ref151">7</reflink>]) excerpted from the TOGS (McKenzie & Padilla, [<reflink idref="bib55" id="ref152">55</reflink>]). Across the sample of participants having varying levels of mathematics and science abilities, performance varied by question level: 80% of Elementary Level Questions were answered correctly, 83.5% of Intermediate Level Questions were answered correctly, and 60% of External Identification Stage Questions were answered correctly. (Overall Level Questions were analyzed qualitatively.) We used Spearman's rho to correlate participants' performance on the previous year's state standardized achievement tests and their performance on the TOGS. TOGS scores and mathematics achievement test scores had a strong positive correlation that was statistically significant, <emph>r</emph>(<reflink idref="bib11" id="ref153">11</reflink>) =.71, <emph>p</emph> =.007. TOGS scores and science achievement test scores had a moderate positive correlation that approached statistical significance, <emph>r</emph>(<reflink idref="bib11" id="ref154">11</reflink>) = 0.51, <emph>p</emph> =.07. Data analysis for the first and second research questions provided insights into participants' interactions with questions and their transition to interpreting continuous graphs.</p> <hd id="AN0121710528-13">How Did Sixth Graders Interact with Graphs Having Continuous Independent Variables Embedded i...</hd> <p>Though not required, all but three participants interacted with graphs in writing while interpreting them. Table 4 shows participants' written interactions with graphs organized by TOGS score. Written interactions were classified according to three component graph interpretation processes (Shah & Hoeffner, [<reflink idref="bib82" id="ref155">82</reflink>]) (see Table 1).</p> <p>Graph</p> <p>Table 4. Coded Written Interactions</p> <p> <ephtml> <table><tr><td valign="bottom" /><td valign="bottom">TOGSScore(%)</td><td valign="bottom">1EISa43%</td><td valign="bottom">2ILQ64%</td><td valign="bottom">3ELQ86%</td><td valign="bottom">4aILQ79%</td><td valign="bottom">5ILQ93%</td><td valign="bottom">6ELQ71%</td><td valign="bottom">7ELQ71%</td><td valign="bottom">8EIS57%</td><td valign="bottom">9EIS79%</td><td valign="bottom">10ELQ93%</td><td valign="bottom">11aILQ86%</td><td valign="bottom">12ILQ86%</td><td valign="bottom">13ILQ93%</td><td valign="bottom">Totalb</td><td valign="bottom">%c</td></tr><tr><td valign="bottom">Hans Hazel</td><td char="." valign="bottom">100</td><td valign="bottom" /><td valign="bottom">1ad 2b</td><td valign="bottom">1b</td><td valign="bottom" /><td valign="bottom">1c 2a</td><td valign="bottom">1a</td><td valign="bottom">1c</td><td valign="bottom">3c</td><td valign="bottom">3b</td><td valign="bottom">1b</td><td valign="bottom" /><td valign="bottom">1b, 1c, 2b</td><td valign="bottom">2c</td><td char="." valign="bottom">10</td><td char="." valign="bottom">77</td></tr><tr><td valign="bottom">Hyde Hegel</td><td char="." valign="bottom">100</td><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom">3c</td><td valign="bottom">3a</td><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td char="." valign="bottom">2</td><td char="." valign="bottom">15</td></tr><tr><td valign="bottom">Heather Miller</td><td char="." valign="bottom">92</td><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td char="." valign="bottom">0</td><td char="." valign="bottom">0</td></tr><tr><td valign="bottom">Hugh Hickson</td><td char="." valign="bottom">85</td><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td char="." valign="bottom">0</td><td char="." valign="bottom">0</td></tr><tr><td valign="bottom">Huck Handy</td><td char="." valign="bottom">85</td><td valign="bottom" /><td valign="bottom" /><td valign="bottom">1b</td><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom">1c</td><td valign="bottom" /><td valign="bottom" /><td valign="bottom">1b</td><td valign="bottom" /><td valign="bottom">1b</td><td valign="bottom">2c</td><td char="." valign="bottom">5</td><td char="." valign="bottom">38</td></tr><tr><td valign="bottom">Hester Luck</td><td char="." valign="bottom">85</td><td valign="bottom" /><td valign="bottom">1b, 2d</td><td valign="bottom">1b, 2d</td><td valign="bottom">1b, 2d</td><td valign="bottom">2d</td><td valign="bottom">1b</td><td valign="bottom">1b</td><td valign="bottom" /><td valign="bottom" /><td valign="bottom">1b, 2d</td><td valign="bottom" /><td valign="bottom">1b, 2d</td><td valign="bottom">1b, 2d</td><td char="." valign="bottom">9</td><td char="." valign="bottom">69</td></tr><tr><td valign="bottom">Matt Maples</td><td char="." valign="bottom">85</td><td valign="bottom" /><td valign="bottom">1b, 1c, 2b</td><td valign="bottom">1b,1c</td><td valign="bottom">1b</td><td valign="bottom">1c</td><td valign="bottom">1b</td><td valign="bottom">1b</td><td valign="bottom">3a</td><td valign="bottom" /><td valign="bottom">1b</td><td valign="bottom" /><td valign="bottom">1b,1c, 2c</td><td valign="bottom">1b, 1c</td><td char="." valign="bottom">10</td><td char="." valign="bottom">77</td></tr><tr><td valign="bottom">Hodge Leader</td><td char="." valign="bottom">69</td><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td char="." valign="bottom">0</td><td char="." valign="bottom">0</td></tr><tr><td valign="bottom">Misty Murphy</td><td char="." valign="bottom">69</td><td valign="bottom">1c</td><td valign="bottom">1c</td><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom">1b, 1c</td><td valign="bottom" /><td char="." valign="bottom">3</td><td char="." valign="bottom">23</td></tr><tr><td valign="bottom">Linda Mills</td><td char="." valign="bottom">69</td><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom">3b</td><td valign="bottom" /><td valign="bottom" /><td valign="bottom">1b</td><td valign="bottom" /><td char="." valign="bottom">2</td><td char="." valign="bottom">15</td></tr><tr><td valign="bottom">Lucy Mag</td><td char="." valign="bottom">69</td><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom">3c</td><td valign="bottom">1a,1b</td><td valign="bottom" /><td valign="bottom">1c</td><td valign="bottom" /><td char="." valign="bottom">3</td><td char="." valign="bottom">23</td></tr><tr><td valign="bottom">Henrietta Harmon</td><td char="." valign="bottom">62</td><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom">1c</td><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td char="." valign="bottom">1</td><td char="." valign="bottom">8</td></tr><tr><td valign="bottom">Lydia Lynn</td><td char="." valign="bottom">54</td><td valign="bottom" /><td valign="bottom">2a</td><td valign="bottom">1b</td><td valign="bottom" /><td valign="bottom">2a</td><td valign="bottom">1b</td><td valign="bottom">1b</td><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom">1c</td><td char="." valign="bottom">6</td><td char="." valign="bottom">46</td></tr><tr><td valign="bottom">Question totals</td><td char="." valign="bottom" /><td valign="bottom">1e</td><td valign="bottom">5</td><td valign="bottom">5</td><td valign="bottom">2</td><td valign="bottom">4</td><td valign="bottom">4</td><td valign="bottom">6</td><td valign="bottom">3</td><td valign="bottom">4</td><td valign="bottom">5</td><td valign="bottom">0</td><td valign="bottom">7</td><td valign="bottom">5</td><td char="." valign="bottom" /><td char="." valign="bottom" /></tr></table> </ephtml> </p> <ulist> <item>41 Participants sorted by TOGS score.</item> <item>4 Question level designation; EIS = External Identification Stage, ILQ = Intermediate Level, ELQ = Elementary Level.</item> <item>4 Total number of written interactions for each participant.</item> <item>4 Percentage of written interactions for each participant.</item> <item>4 See Table 1 for codes.</item> <item>4 Total number of participants who used written interacts for each question.</item> </ulist> <p>These data suggest that participants were much more likely to interact with graphs to support their ability to encode information (<reflink idref="bib39" id="ref156">39</reflink>). Interactions to support identification of salient structures were less common (<reflink idref="bib16" id="ref157">16</reflink>), and interactions to support understanding referents of those salient structures were less common still (<reflink idref="bib8" id="ref158">8</reflink>). There were 20 written interactions on the four Elementary Level Questions, 23 on the six Intermediate Level Questions, 0 on the two Overall Level Questions, and 8 on the three External Identification Stage Questions. Three participants—Hans Hazel, Hester Luck, and Matt Maples—wrote while interpreting most graphs. Beyond that, the rate of written interactions varied greatly among participants and according to graph question levels.</p> <p>Analysis of interactions also revealed a number of lingering habits learned from interpreting discrete graphs, some of which supported participants' interactions with continuous graphs and others that interfered. We also identified new habits of sense making needed for interpreting continuous graphs that participants struggled to enact. These findings are explored further in the second research question.</p> <hd id="AN0121710528-14">How Did Sixth Graders' Interactions Inform Our Understanding of Their Transition from Interpr...</hd> <p></p> <hd id="AN0121710528-15">Struggling with Cartesian coordinates</hd> <p>Most participants needed to remind themselves, in one way or another, how to encode coordinates of a data point or encode a position when given coordinates. Participants' encoding of data points was aided by axis labels, but they struggled more when they could not rely on axis labels. In addition, some participants emphasized <emph>y</emph>-coordinate values at the expense of <emph>x</emph>-coordinate values.</p> <p>Participants supported their encoding of data points in several ways. Many participants (<emph>n</emph> = 7) across ability levels drew on graphs and explicitly verbalized the procedure to encode the coordinates of points. A few higher-ability participants (<emph>n</emph> = 3) needed to verbalize mnemonics such as "healthy vitamins" on the decontextual questions to remember that the "horizontal" axis is read before the "vertical" axis. These methodical procedures suggest that participants lacked fluency with encoding data from a Cartesian graph. These procedures supported encoding salient structures not only for Elementary Level Questions but also for Intermediate Level and External Identification Stage Questions.</p> <p>Participants were helped by added information in the contextual Elementary Level Questions. This was evident in the think-aloud of Hester Luck as she answered decontextual Question 6 by reversing the typical order of reading data points. She started at the data point, moving first to the <emph>y</emph>-axis and then to the <emph>x</emph>-axis: "Well, I'm going to draw a line all the way ... it goes to the side it's 12 and at the bottom it's 20. So, I'm probably going to say D, 12, 20." By contrast, when she answered contextual Question 3 earlier, she was helped by the axis labels and relieved of the need to remember the convention: "Well, if I looked at centimeters I should look to the side (<emph>y</emph>-axis) where it says Height of Plant centimeters. So, if I draw a line to the centimeters I'm going to connect the graph dots again then you draw a big line it will go to probably about 160. Probably around 160." The performance difference between contextual and decontextual Elementary Level Questions suggests a lack of fluency with encoding data points on Cartesian graphs.</p> <p>In addition to challenges participants had with encoding data points on Cartesian graphs, some (<emph>n</emph> = 5) overgeneralized the importance of <emph>y</emph>-coordinates at the expense of <emph>x</emph>-coordinates when answering Elementary and/or Intermediate Level Questions. For example, while answering Intermediate Level Question 2, Matt Maples started by encoding all of the data points (see Fig. 1, Question 2). He initially read only the <emph>y</emph>-coordinate values of each point: "The dots are at 7, 15, 20, 10, and 5." Only then did he relate the <emph>y</emph>-coordinate of each data point to its associated <emph>x</emph>-coordinate, in the form (<emph>y</emph>, <emph>x</emph>), "5 is at 200, 10 is at 160, 20 is at 120, 15 is 40, and 7 is at 0." By contrast, most participants were more likely to emphasize the standard convention when reading data points and place equal emphasis on <emph>x</emph>- and <emph>y</emph>-coordinates during the initial encoding process. We saw this in the way Hugh Hickson read data points in Question 2: "On 0 grams, I mean 0 mL of water, it went up to 8 cm, just because the plant grew for 3 weeks and on 40 mL it looked like it was 15 cm." Hugh also chose to preserve the units from the graph as he read each data point. Challenges participants had while encoding data in a Cartesian graph and their tendency to privilege the <emph>y</emph>-coordinate when encoding data points affected their ability later to understand relationships among encoded features.</p> <p>Graph: Figure 1. Examples of imprecision while answering Intermediate Level Questions. Question 2, Matt Maples. Question 12, Hester Luck. Adapted from the Test of Graphing in Science (TOGS), copyright 1986, Michael J. Padilla.</p> <hd id="AN0121710528-16">Cumulative effects on relating salient structures</hd> <p>Difficulties in the first interpretive process affected the second interpretive process. Participants' difficulties encoding coordinates led to a lack of precision during interpolation and extrapolation. Unfamiliar interactions were required to understand the conceptual relationships implied by procedures. In addition, participants who privileged <emph>y</emph>-coordinate values tended to calculate differences rather than describe graph trends.</p> <p>Participants' lack of comfort with Cartesian coordinates was also evident in their lack of precision (<emph>n</emph> = 5) as they interpolated or extrapolated. While interpolating in Question 2, Matt Maples did not draw a straight line between the two nearest points on the graph, and the line he drew up from the given <emph>x</emph>-coordinate value stopped rather short of the imagined line between the two nearest points (see Fig. 1, Question 2). The result was that he significantly underestimated the interpolated value. Hans Hazel made a similar error while answering Question 12, though he was able to self-correct. Hester Luck habitually used line segments to connect data points ("dots") on most graphs, though her line segments were noticeably "saggy." However, while she answered Question 12, she corrected her saggy line segment with one that was much straighter before trying to interpolate (see Fig. 1, Question 12). These data cannot help us determine whether lack of precision is the result of lack of practice, a failure to appreciate the conceptual importance of straight line segments when working in Cartesian coordinates, or simply a lack of fine motor control. Nonetheless, lack of precision during graph interactions made it harder for participants to relate correctly the salient structures of these graphs, though only two led to incorrect answers.</p> <p>When participants described the relationship between <emph>x</emph>- and <emph>y</emph>-variables in a graph, their excessive focus on <emph>y</emph>-coordinates made it hard to understand relationships described by data points. Henrietta Harmon was one who privileged <emph>y</emph>-coordinates while reading data points. While she was eventually able to answer Question 4 correctly, it took her a while to relate <emph>x</emph>-coordinate values for each data point (see Fig. 1; Questions 2 and 4 used the same graph). She started by reading <emph>y</emph>-coordinate values for each point and seemed quite confused about how to answer the question: "The graph is showing 7 and then it goes to 15 then to 20 then to 10 then back down to 5 and so um, okay, um, I [rereads answer choice A] I don't get this at all." Then, for no obvious reason, she calculated <emph>y</emph>-coordinate differences between the second and third data points and then the third and fourth data points: "Okay they go up, the graph number go up by 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, they go up by about 10 so they decrease some cuz it goes up to 20 and then it goes back down to 10 so it decreases by 10." Only by rereading the answer choice B did she finally attend to the significance of the <emph>x</emph>-coordinate values: "So oh, okay it goes up to 120 and then it drops about another 120 but the height increased and the amount of water decreased but the more water it decreased. Actually, it could be B. But then it decreased again so it would probably be B although I thought it wouldn't be. Um, wait [reads answer choice D aloud]. I am going to say D." Even though her prior achievement on state mathematics and science achievement tests suggested she should have been working above grade level, her behavior was more like students with lower ability in the sample. This was especially evident in Henrietta's difficulty attending simultaneously to both variables associated with each data point.</p> <p>Also evident in her answer to Question 4 was her tendency to calculate differences in <emph>y</emph>-coordinate values for data points, especially when otherwise unsure how to answer a question. Lydia Lynn also calculated differences between all sequential data points on Question 3, but then proceeded, for no apparent reason, to calculate the sum of those differences. While infrequent, interacting with graphs by calculating differences suggests a behavior carried over from interpreting discrete graphs, where differences are the only meaningful way to understand relationships among salient structures. Only Heather Miller calculated a difference purposefully, when answering Question 2, in order to calculate arithmetically the interpolated data point.</p> <hd id="AN0121710528-17">Cumulative effect on understanding referents</hd> <p>Unfortunately, responses from participants of all science achievement levels provided little additional insight beyond phrases and words provided in the text of the scenario and question. Moreover, participants focused on individual data points rather than discussing trends or relationships evident in graphs. None explored the meaning of possible anomalies evident in graphs. Finally, when interpreting the graph, they treated the <emph>x</emph>-axis as if it were a time-series graph.</p> <p>Hugh Hickson's response was typical of students with high achievement, answering Question 4b with very informal language: "I think this happened because 120 mL is probably the correct amount of water which makes it go up to 20 cm and 160 mL and 200 ml was overwatering and flooding the plant which gave it too much water and it pretty much choked."</p> <p>Hester Luck, with much lower science achievement, answered similarly: "It got overwatered. ... I think 120 was just right but once you put 160 and 200 in it in milliliters, it just went and overflowed it so it just was not going to grow at all." Likewise, Matt Maples explained that "the plant was getting too much water and could not take all of it in so it started shrinking or shriveling." Lucy Mag described:</p> <p>I think this happened because as the amount of water started to go up, I think it got too much water so if the amount of water it got 0 amount of water, it still grew about 6 or 7 feet [should be cm]. If it got 40, it grew about 15. 120 would be 20, 160 went back down to 10 and 200 went way back down. I think because since it needed about 120 amounts of water because if you gave it 200 it would go down and 0 to 40 it was not a lot of water but enough to keep it going but it got too much water so I think that's why.</p> <p>Several issues are evident in data from these and other participants.</p> <p>Participants' efforts to explain what happened in the graph were either loose metaphors—"choked" and "shrinking or shriveling"—or mere tautologies. They may not have been previously taught that plants absorb oxygen through their roots, in addition to water and nutrients, but students who were above grade level might reasonably have been expected to offer some conjectures using their knowledge of plant growth, respiration, systems, and structures taught and assessed in fifth grade. Eleven participants reported having experience with growing plants at home or in school, including some who had previously completed science fair projects that studied plant growth. Even though these experiences were mentioned, they did not help participants elaborate on the meaning of data points or trends within graphs.</p> <p>In addition, all participants' think-aloud responses to Overall Level Question 4b emphasized a few points rather than evident trends. Most noted that 120 mL was "the correct amount" and that at 160 mL and 200 mL the plants' height decreased. However, only two participants, Heather Miller and Lucy Mag, discussed plants receiving less than 120 mL of water. Lucy Mag was the only participant who discussed the 0 mL plant, noting that it was "not a lot" but "enough to keep going." No other participants noted the seeming anomaly of a plant growing to 8 cm over 3 weeks despite receiving no water.</p> <p>Finally, even though the question text says the graph represents several plants, each receiving a different amount of water for 3 weeks, all participants used the singular pronoun "it" in their answers. This use of a singular pronoun and other aspects of their explanations suggest that they imagined the graph represented a single plant, rather incongruously, receiving different amounts of water over time. Their overemphasis on <emph>y</emph>-coordinates in previous questions focused their attention primarily on plant height, seeing the graph describing one plant's growth pattern over time instead of correctly seeing different plants receiving different amounts of water. Mistakenly seeing only one plant also led mistakenly to seeing the <emph>x</emph>-axis representing two measures—the measure of the passing of time <emph>and</emph> the volume of water given to the plant.</p> <hd id="AN0121710528-18">Scaffolding independent and dependent relationships</hd> <p>Participants really struggled with External Identification Stage Questions. Most participants relied on decontextual approaches, but almost half used contextual cognitive resources in their answers. The most supportive resources were knowledge of bar graphs, explicit knowledge of science phenomena and science inquiry, and their inscriptional practices. Answering these questions correctly, without guessing, required participants to coordinate all of these resources successfully.</p> <p>Henrietta Harmon was typical of seven participants who were reduced simply to eliminating answer choices for lack of an appropriate method. The wording of Question 8 was difficult for most participants, and the complex sentence prompted her initially to think there were three possible variables: "he'd have to have a time bar [axis] and he'd have to have the amount of water and the temperature of the water." By reading carefully, she eliminated "temperature of the water" as a variable, enabling her to dismiss two multiple-choice options. Both answers A and C contained the same phrases; even though she did correctly choose A over C, her reasoning was not evident. That is, she guessed correctly. Henrietta and six other participants did not understand that answering this question required them to use knowledge not present in the graph or question text.</p> <p>By contrast, Linda Mills used her knowledge of bar graphs to scaffold her understanding of the relationship between independent and dependent variables. Immediately after reading the text of Question 9 and axis labels, she declared her answer choice, "I think B is the correct answer." She defended it by noting that axis labels within the other answer choices "make no sense." She then verified her answer by modeling the problem. "What stood out about B was the weight of chickens in grams on the bottom and then you had the number of eggs at the top and that would make the most sense. If you had the weight of chickens like 1, 2, 3, 4, 5 grams and then the number of eggs that would make the most sense." Concurrently, she drew two rectangles overlapping one another on answer choice A to show that it had the independent and dependent variables in the wrong locations (see Fig. 2, Question 9). One bar extended vertically from the <emph>x</emph>-axis and the other extended horizontally from the <emph>y</emph>-axis. Her choice to create hypothetical data points to understand the situation, "weight of chickens like 1, 2, 3, 4, 5 grams," led to her claim that B would make the most sense. While her reasoning was opaque and incomplete, it was sufficient to satisfy her. Hans Hazel also drew a bar on the graphs for Questions 8 and 9 to determine the independent and dependent variables, but his reasoning was more complete. Linda and Hans were both able to use the shape of the bars to infer that weight of chickens might plausibly influence the number of eggs laid, and not the other way around. This was notable, because they did not plot data points to see this relationship.</p> <p>Graph: Figure 2. Written interactions with External Identification Stage Questions to determine independent and dependent variables. Hyde Hegel, Question 8; Linda Mills, Question 9. Adapted from the Test of Graphing in Science (TOGS), copyright 1986, Michael J. Padilla.</p> <p>Hyde Hegel also used his understanding of referents to model the situation represented in Question 8 (see Fig. 2, Question 8). However, after drawing a set of scales, he drew points on the graphs to imagine likely data and to remember the convention. Hyde Hegel was an anomaly in this sample because he was successfully able to coordinate his knowledge of science phenomena and Cartesian graphs. Three other participants tried to use their knowledge of phenomena in Questions 8 and 9. Matt Maples also drew plausible bars, like Linda Mills and Hans Hazel, but decided that time must be the independent variable. Two others tried to reason through the scenarios, but did not imagine or draw scales, data points, or bars on the graphs, and were unable to answer the questions correctly.</p> <p>The differences in performance between Questions 8 and 9 (57% vs. 79%) provided additional insight into challenges associated with choosing correct axes for a graph. Among six participants who tried to use content knowledge to answer these questions, all had personal experience boiling water, usually at home in the kitchen, and only one had personal experience with chickens. However, only Hyde Hegel, Hans Hazel, and Matt Maples used these experiences to help them understand concepts of thermodynamics. The rest struggled with relationships among abstract variables of time, volume, and temperature. Five incorrectly inferred that "temperature of water" needed to represent one of the axes. By contrast, only one participant had personal experience with chickens. Even though Linda Mills had no personal experience with chickens, her comfort with mass and number as variables were evident in her willingness to create hypothetical bar graphs and infer a causal relationship between mass of chickens and number of eggs laid. Other participants were also better able to answer Question 9, because they better understood the concrete concepts and variables represented in the question scenario. Unfamiliarity with abstract measures and science concepts in Question 8 hindered most who tried to understand its embedded cause-and-effect relationships.</p> <hd id="AN0121710528-19">Discussion</hd> <p>Data revealed challenges that sixth graders experienced while interpreting science graphs with continuous independent variables. Many challenges were rooted in participants' transition from interpreting discrete graphs in grades K–5 to continuous graphs in sixth grade. Earlier studies have emphasized the importance of graph question levels and identified errors students make when answering questions at each level (Bertin, [<reflink idref="bib7" id="ref159">7</reflink>]/[<reflink idref="bib7" id="ref160">7</reflink>]; Carswell, [<reflink idref="bib16" id="ref161">16</reflink>]; Curcio, [<reflink idref="bib24" id="ref162">24</reflink>]; Friel et al., [<reflink idref="bib30" id="ref163">30</reflink>]; McKnight, [<reflink idref="bib56" id="ref164">56</reflink>]; Wainer, [<reflink idref="bib88" id="ref165">88</reflink>]). However, these prior studies did not focus on component graph interpretation processes (Carpenter & Shah, [<reflink idref="bib15" id="ref166">15</reflink>]; Shah & Hoeffner, [<reflink idref="bib82" id="ref167">82</reflink>]) associated with each question level nor did they focus on students in the early stages of learning to interpret line graphs.</p> <p>Our approaches to data collection and analysis enabled us to identify several important findings about sixth-grade students' transition to interpreting continuous graphs. These findings are grouped into three main insights. First, participants' difficulties while encoding individual data points influenced their interpretations of relationships among data points and referents of those data. Second, cognitive resources learned for interpreting discrete graphs both supported and interfered with their interpretations of continuous graphs. Third, participants' struggles to relate graphs to referents reflected inexperience with data collection and analysis.</p> <hd id="AN0121710528-20">Encoding Process Difficulties Influence the Other Two Processes</hd> <p>Nearly 2 years after being introduced to Cartesian coordinates and line graphs, our participants were still learning procedures and conceptual relationships embedded in this new way of representing data. Analysis of data for our first research question showed that participants' written interactions most often supported encoding salient features on graphs. These interactions were used to support Intermediate Level and External Identification Questions much less often than Elementary Level Questions. Every graph interpretation question type should require encoding of salient features (Carpenter & Shah, [<reflink idref="bib15" id="ref168">15</reflink>]; Shah & Hoeffner, [<reflink idref="bib82" id="ref169">82</reflink>]), so it was surprising to see no interactions while interpreting Overall Level Questions.</p> <p>Only one participant, Hyde Hegel, was able consistently and correctly to encode coordinates on Cartesian graphs without ever slowing to use a mnemonic or other procedure. Indeed, for other participants, reading data points often required conspicuous effort. Lacking automated processes for encoding salient features and relationships made graph interpretation difficult (Trickett & Trafton, [<reflink idref="bib86" id="ref170">86</reflink>]), consuming working memory needed to attend to the meaning of data (Shah & Hoeffner, [<reflink idref="bib82" id="ref171">82</reflink>]). That participants were helped by axis labels further suggests this lack of fluency. In addition, five participants overemphasized <emph>y</emph>-coordinates during encoding, perhaps carried over from reading the "height" of bar graphs (Cleveland & McGill, [<reflink idref="bib20" id="ref172">20</reflink>]) and line plots during grades K–5 (Common Core State Standards Initiative, [<reflink idref="bib22" id="ref173">22</reflink>]). By contrast, Planinic, Milin-Sipus, Katic, Susac, and Ivanjek ([<reflink idref="bib64" id="ref174">64</reflink>]) found that tenth graders could more easily interpret slope with decontextual mathematics questions compared to analogous physics questions. What helped our participants was seeing and coordinating units within the questions, graphs, and answers rather than inferred values like velocity on a displacement-time graph (Planinic et al., [<reflink idref="bib64" id="ref175">64</reflink>]). Our participants' lack of fluency with data points affected their ability to relate salient features and graph referents.</p> <p>Intermediate Level Questions required encoding data points but also required participants to see and understand relationships between and among points—interpolating, extrapolating, and finding the graph's shape (Bertin, [<reflink idref="bib7" id="ref176">7</reflink>]/[<reflink idref="bib7" id="ref177">7</reflink>]; Friel et al., [<reflink idref="bib30" id="ref178">30</reflink>]). Answering Intermediate Level Questions always requires inferred information not visible in the graph, a process hampered by participants' lack of fluency encoding data points. Struggles with encoding may have led to imprecise interpolation and extrapolation. Except for a few participants with high mathematics and science achievement, we saw little evidence of "visual chunking" to support pattern-finding (Trickett & Trafton, [<reflink idref="bib86" id="ref179">86</reflink>]), suggesting that participants lacked experience reading the overall shape of continuous graphs.</p> <p>Participants' efforts to relate each graph's shape to its referents were more akin to reading discrete graphs. Answering Overall Level Questions should have required participants to engage in behaviors from all three graph component processes (Bertin, [<reflink idref="bib7" id="ref180">7</reflink>]/[<reflink idref="bib7" id="ref181">7</reflink>]; Friel et al., [<reflink idref="bib30" id="ref182">30</reflink>]). However, participants never used written interactions with graphs when answering Questions 4b and 11b. Almost all participants ignored the left side of the flower experiment graph and focused on data from the right side—data already encoded and related in previous questions. The focus on the right side of the graph may also be evidence of rudimentary understanding of continuous graphs, especially time-series graphs, where it is often most important to understand what happens at the end of the graph. Yet, as we discuss below, this was <emph>not</emph> a time-series graph. Instead, all focused on a few points, especially the highest point of the graph representing Rose's flower experiment, much like finding the tallest bar on a bar graph. Their focus was undoubtedly framed (Hammer et al., [<reflink idref="bib37" id="ref183">37</reflink>]) by Question 4a, but that question asked participants to describe the overall shape, not just a few points. Only Lucy Mag even noticed the anomalous data point of a plant receiving no water, but she did not critically question that point (Chinn & Brewer, [<reflink idref="bib18" id="ref184">18</reflink>]). Interestingly, critically questioning data points and relationships in graphs is not supported by the K–5 content standards related to graphing (Common Core State Standards Initiative, [<reflink idref="bib22" id="ref185">22</reflink>]).</p> <p>Taken together, these findings suggest that difficulties with encoding salient features from a Cartesian graph may affect the ability of sixth graders to identify relationships among salient features of a graph or understand referents of a graph.</p> <hd id="AN0121710528-21">"Mis"understanding How the New Connects with the Old</hd> <p>Our participants did not have well-developed schemas (Pinker, [<reflink idref="bib63" id="ref186">63</reflink>]) for interpreting Cartesian graphs. Like the undergraduate students who were less capable, studied by Shah and Freedman ([<reflink idref="bib80" id="ref187">80</reflink>]), our participants relied more extensively on working memory and interactions with graphs (Shah & Hoeffner, [<reflink idref="bib82" id="ref188">82</reflink>]). In addition, participants used cognitive resources acquired in and outside of school and tried to discern which were relevant for the task (Hammer et al., [<reflink idref="bib37" id="ref189">37</reflink>]).</p> <p>One resource evident in the data was "prior graphing experience" (Delgado & Lucero, [<reflink idref="bib25" id="ref190">25</reflink>]), especially experience with graphs learned in previous grades. These prior graphing experiences sometimes helped, but, unfortunately, more often hindered. Recognizing a unit of height in Question 3 relieved Hester Luck of remembering a convention; without this resource in Question 6, she floundered. Hans Hazel and Linda Mills were helped by activating prior graphing experiences with bar graphs and placing independent and dependent variables on the axes for Questions 8 and 9. Participants were also hindered when questions activated prior graphing experiences that were not helpful. When asked to describe the shape of the graph in Question 4a, Henrietta Harmon instead calculated differences among <emph>y</emph>-values for each data point, suggesting her prior graphing experiences with discrete graphs. Prior graphing experience with time-series graphs certainly undermined Matt Maples's efforts to answer Question 8, even after he used his prior graphing experience to draw helpful bars on the axes. Similarly, prior experiences with time-series graphs also led most participants, incongruously, to interpret time as the independent variable when answering Overall Level Question 4b to explain what had happened to Rose's flowers in Questions 2, 3, and 4a. They interpreted it as a time-series graph even after interpreting correctly that the independent variable was the amount of water given each plant each day for the other questions. Time-series graphs were historically first used to represent trends in data (Tufte, [<reflink idref="bib87" id="ref191">87</reflink>]) and are often the first continuous graphs to which students are introduced (Leinhardt et al., [<reflink idref="bib48" id="ref192">48</reflink>]).</p> <p>Scatter plots and line graphs are very different from bar graphs and support different interpretive goals (Kosslyn, [<reflink idref="bib42" id="ref193">42</reflink>]; Shah & Freedman, [<reflink idref="bib80" id="ref194">80</reflink>]). However, some skills and concepts overlap: axis labels describe the measures, data points in line graphs are similar to the top of a bar in a bar graph, and independent variables are conventionally placed on the <emph>x</emph>-axis. These similarities sometimes helped participants. However, sixth-grade participants in this sample were either (<reflink idref="bib1" id="ref195">1</reflink>) not certain which skills and concepts were and were not relevant, or (<reflink idref="bib2" id="ref196">2</reflink>) not fluent with Cartesian graphs and forced to rely on prior knowledge of discrete graphs.</p> <p>Inscriptions was the second resource used by participants (Latour, [<reflink idref="bib44" id="ref197">44</reflink>]; Roth & McGinn, [<reflink idref="bib78" id="ref198">78</reflink>]). Most participants created inscriptions to externalize cognition and reduce load on working memory, especially while encoding data points. Data from five participants who interacted with graphs while answering External Identification Stage Questions 8 and 9 corroborate the suggestion that inscriptions can act as emerging resources (Nemirovsky, Tierney, & Wright, [<reflink idref="bib60" id="ref199">60</reflink>]). Emerging resources seemed to be rooted in episodic memory (Martin, [<reflink idref="bib51" id="ref200">51</reflink>]) and helped participants reconstruct the convention of placing independent and dependent variables correctly on the axes. Other participants who did not interact in writing did not have access to these emerging resources. Data also suggest that these inscriptions evoked "bodies of life experience" and "experiential domains" instead of "isolated rules" and "memorized examples" (Nemirovsky et al., [<reflink idref="bib60" id="ref201">60</reflink>], p. 120).</p> <p>A third resource evoked by participants while answering TOGS questions (McKenzie & Padilla, [<reflink idref="bib55" id="ref202">55</reflink>]) was their numerous experiences with standardized tests, especially high-stakes standardized tests. These data remind us that answering test questions is an everyday, authentic activity for our elementary students (Lave, [<reflink idref="bib45" id="ref203">45</reflink>]). Advantages of activating prior resources were evident throughout the data, especially for Elementary Level and Intermediate Level Questions. However, for mathematics and science educators who want deeper engagement with concepts, activating test-taking resources greatly inhibited participants when answering Overall Level and External Identification Stage Questions. Our data also corroborate questions about the validity of multiple-choice graphing tests (Berg & Boote, [<reflink idref="bib3" id="ref204">3</reflink>]; Berg & Phillips, [<reflink idref="bib4" id="ref205">4</reflink>]; Berg & Smith, [<reflink idref="bib5" id="ref206">5</reflink>]), at least for questions intended to assess Overall Level and External Identification Stage graph interpretation concepts and skills.</p> <p>Learning to read continuous graphs builds on some prior concepts and skills from discrete graphs, but this is not an example of <emph>misconception constructivism</emph> (Elby, [<reflink idref="bib26" id="ref207">26</reflink>]; Strike & Posner, [<reflink idref="bib85" id="ref208">85</reflink>]), where students need to replace alternative conceptions and theories with more sophisticated ones. Instead, the leap from interpreting discrete graphs to continuous graphs requires a new construct to be introduced alongside an old one. We might mistakenly presume that sixth-grade students have learned discrete graphs well enough to be able to connect and contrast them with continuous graphs. Our findings amplified how this transition challenged participants of all mathematics and science achievement levels.</p> <hd id="AN0121710528-22">Struggles Inferring Relationships between Independent and Dependent Variables</hd> <p>Consistent with earlier studies, most of our participants used a "decontextual" approach for External Identification Stage Questions rather than use their referent knowledge (Mayer, Lewis, & Hegarty, [<reflink idref="bib53" id="ref209">53</reflink>]; Preece & Janvier, [<reflink idref="bib66" id="ref210">66</reflink>]; Roth, [<reflink idref="bib72" id="ref211">72</reflink>]). This behavior is undoubtedly a manifestation of sociomathematical norms in mathematics education (Gerofsky, [<reflink idref="bib34" id="ref212">34</reflink>]; Lave, [<reflink idref="bib46" id="ref213">46</reflink>]; Yackel & Cobb, [<reflink idref="bib90" id="ref214">90</reflink>]), but our data suggest additional factors may be at play.</p> <p>External Identification Stage Questions only contained referents, but they required more than simply remembering a convention for independent and dependent variables. Participants who successfully answered these questions without guessing—Hans Hazel, Hyde Hegel, and Linda Mills—had facility with all three component graph interpretation processes (Shah & Hoeffner, [<reflink idref="bib82" id="ref215">82</reflink>]). Understanding even a single data point on a graph may require knowledge of science phenomena, context of data collection, procedures for measurement, and even how data were transformed (Roth, [<reflink idref="bib74" id="ref216">74</reflink>]; Roth & Bowen, [<reflink idref="bib75" id="ref217">75</reflink>]). Participants imagined salient structures to encode, relationships among those salient structures, and how encoded relationships would represent referents. However, consistent with norms of parsimony in scientific communication (Bazerman, [<reflink idref="bib2" id="ref218">2</reflink>]; Fang, [<reflink idref="bib29" id="ref219">29</reflink>]), these External Identification Stage Questions were terse, with important information veiled and presumed as tacit knowledge (Polanyi, [<reflink idref="bib65" id="ref220">65</reflink>]). Participants needed to infer a whole chain of inscriptions and conventions. Even practicing scientists interpreting their own data will struggle with this problem (Roth, [<reflink idref="bib74" id="ref221">74</reflink>]).</p> <p>Differences in performance between Questions 8 and 9 suggest specific challenges when inferring the meaning behind data points and relationships in line graphs. The wording of Question 9 clearly connected the relationship between the independent variable, "weight of chickens," and the dependent variable, "number of eggs." In addition, it was easy for participants to imagine how these data would be measured. However, the wording of Question 8 was harder for participants to tease apart the relationship between "volume of water" and "time needed to boil." Five incorrectly inferred that "temperature of water" would need to be measured. Participants' nature of science knowledge also played a role. Their informal experiences with boiling water in the kitchen did not help them infer that the rate of heat transfer and pot size would have to be controlled (Roth, [<reflink idref="bib74" id="ref222">74</reflink>]; Roth & Bowen, [<reflink idref="bib76" id="ref223">76</reflink>]). Difficulties understanding the contexts implied by scenarios were similar to students answering "inconsistent" word problems (Mayer & Hegarty, [<reflink idref="bib52" id="ref224">52</reflink>]), "intentionally misleading" problems (Boote & Boote, [<reflink idref="bib10" id="ref225">10</reflink>]), and comprehending terse language of school science (Fang, [<reflink idref="bib29" id="ref226">29</reflink>]; Gee, [<reflink idref="bib33" id="ref227">33</reflink>]).</p> <p>Prior research on knowledge of graph referents during interpretation suggests that novices are more likely to be influenced by prior knowledge of graph content and ignore relationships depicted in a graph (Shah & Freedman, [<reflink idref="bib80" id="ref228">80</reflink>]). Graph readers with expertise in both content and graph form are better able to critically analyze a graph (Roth, [<reflink idref="bib73" id="ref229">73</reflink>]; Shah, Freedman, & Vekiri, [<reflink idref="bib81" id="ref230">81</reflink>]). The novice status of our participants was especially evident when they answered Overall Level Questions 4b and 11b. Most ignored large portions of the graphs and a possible anomalous data point. It was also troubling that participants with high science achievement and experiences with phenomena provided interpretations no more sophisticated than those from participants with low science achievement. As we discussed above, participants' difficulties with encoding may explain some difficulties with Overall Level Questions. Yet, most participants with high science and mathematics achievement struggled less with encoding but provided simple and terse interpretations of graphs.</p> <p>Taken together, our findings suggest subtle underlying graph interpretation skills necessary to support the transition from discrete to continuous graphs. Better articulation of these skills within mathematics and science standards documents may be necessary (Common Core State Standards Initiative, [<reflink idref="bib22" id="ref231">22</reflink>]; NGSS Lead States, [<reflink idref="bib61" id="ref232">61</reflink>]). As currently written, a student who is successful with the K–5 CCSSM standards may not be prepared for the expectations in grades 6–8. Disjuncture among standards may leave many students floundering. It is especially important for fifth- and sixth-grade teachers to understand this important transition. More explicit and proactive efforts during grades K–5 will help students more easily move from fifth to sixth grade.</p> <hd id="AN0121710528-23">Implications</hd> <p>The purpose of graph interpretation, ultimately, is to understand important natural and social concepts that lead us to see patterns and ask additional questions. It would be very easy when reading the K–5 Common Core Mathematics content standards ([<reflink idref="bib22" id="ref233">22</reflink>]) to miss this larger intent and focus too narrowly on graph types, reading data points, and calculating differences between points. As a result, participants in this sample had little practice or inclination to read science graphs critically to find their meanings (Lai et al., [<reflink idref="bib43" id="ref234">43</reflink>]).</p> <p>At the end of sixth grade, these participants lacked experience explaining the meaning of interpolated, extrapolated, or trend data in connection with their knowledge of those referents. Although the K–5 graph interpretation standards focus mostly on Elementary and Intermediate Level Questions, students in sixth grade would benefit from more practice with Overall Level Questions. Even with younger students, this can be accomplished through discussions around data-collection practices. For example, when surveying a class about pizza topping preferences, a question like, "Why do you think anchovies were not selected?" could lead to a discussion about sample and population characteristics in a developmentally appropriate way. Then, when students read a similar graph where artichoke hearts represent a modal value, they will be better able to analyze the authenticity of the data point and graph. These instructional practices support the expectation in sixth grade that students relate graph referents to their salient features. Experience and fluency with these skills will help students read graphs effectively and critically.</p> <p>While graph <emph>types</emph> get considerable attention in elementary curricula, graph <emph>question levels</emph> should also be explicitly taught in elementary grades (Friel et al., [<reflink idref="bib30" id="ref235">30</reflink>]). To teach question levels, elementary teachers can adapt Question Answer Relationships (QARs), a model widely used in reading and language arts (Raphael, [<reflink idref="bib68" id="ref236">68</reflink>], [<reflink idref="bib69" id="ref237">69</reflink>], [<reflink idref="bib70" id="ref238">70</reflink>]; Raphael & Au, [<reflink idref="bib71" id="ref239">71</reflink>]). By understanding graph question levels, students can understand the intent of a question or task prior to reading the graph. In turn, they will better understand when their knowledge of science and social science is relevant and when they can focus simply on the question text and graph. Elementary students also need to understand the importance of using content knowledge and disciplinary practices when answering Overall Level and External Identification Stage Questions, suggesting one more reason for students to engage in science and mathematical investigations and modeling in elementary classrooms (Connected Learning Coalition, [<reflink idref="bib23" id="ref240">23</reflink>]; National Council for Social Studies, [<reflink idref="bib57" id="ref241">57</reflink>]; National Governors Association Center for Best Practice & Council of Chief State School Officers, [<reflink idref="bib58" id="ref242">58</reflink>]; National Research Council, [<reflink idref="bib59" id="ref243">59</reflink>]).</p> <hd id="AN0121710528-24">Conclusion</hd> <p>Proficiency with reading graphs is a critical component across all domains of literacy in the elementary curriculum: mathematics, science, social science, and English language arts literacy (Common Core State Standards Initiative, [<reflink idref="bib22" id="ref244">22</reflink>]; Connected Learning Coalition, [<reflink idref="bib23" id="ref245">23</reflink>]; National Council for Social Studies, [<reflink idref="bib57" id="ref246">57</reflink>]; National Governors Association Center for Best Practice & Council of Chief State School Officers, [<reflink idref="bib58" id="ref247">58</reflink>]; NGSS Lead States, [<reflink idref="bib61" id="ref248">61</reflink>]). Just as it is unacceptable for a student to decode a text but not comprehend its meaning, so too is it unacceptable to read data points, identify axis labels, and calculate categorical differences while disregarding the graph's meaning. Recent emphasis on sophisticated graphing has stressed the value of graph creation and interpretation in naturalistic settings, especially how students and professionals use graphs to understand specific natural and social phenomena and to support evidentiary reasoning (Berland & Hammer, [<reflink idref="bib6" id="ref249">6</reflink>]; Roth, Bowen, & McGinn, [<reflink idref="bib77" id="ref250">77</reflink>]; Wu & Krajcik, [<reflink idref="bib89" id="ref251">89</reflink>]). Naturalistic approaches reinforce that constructing and using graphs and data tables are social practices where group members use inscriptions as tools that can be shared and amended for effective communication (Cobb, [<reflink idref="bib21" id="ref252">21</reflink>]; Roth & McGinn, [<reflink idref="bib78" id="ref253">78</reflink>]; Wu & Krajcik, [<reflink idref="bib89" id="ref254">89</reflink>]).</p> <p>In addition to comprehending graphs during classroom inquiry, students must also coordinate texts and graphs while reading trade books, textbooks, and electronic sources (Bowen & Roth, [<reflink idref="bib11" id="ref255">11</reflink>]; Bowen et al., [<reflink idref="bib12" id="ref256">12</reflink>]; Lemke, [<reflink idref="bib49" id="ref257">49</reflink>]; Roth et al., [<reflink idref="bib77" id="ref258">77</reflink>]). Taken together, an emerging consensus suggests that "<emph>Graphical literacy skills should be taught in the context of science and social science</emph>" (Shah & Hoeffner, [<reflink idref="bib82" id="ref259">82</reflink>], p. 63, emphasis in the original). To interpret texts and images successfully, students must use a variety of comprehension strategies to understand each individual source and then coordinate meanings across sources. Although signs are ubiquitous, students must learn to make contextual sense of these signs. Scaffolding students during their leap from interpreting discrete to continuous graphs will make this jump more meaningful.</p> <ref id="AN0121710528-25"> <title> Footnotes </title> <blist> <bibl id="bib1" idref="ref1" type="bt">1</bibl> <bibtext> Stacy K. Boote is associate professor of mathematics and science education in the Department of Childhood Education, Literacy, and TESOL at the University of North Florida in Jacksonville. David N. 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Items – Name: Title
  Label: Title
  Group: Ti
  Data: Leaping from Discrete to Continuous Independent Variables: Sixth Graders' Science Line Graph Interpretations
– 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="%22Elementary+School+Journal%22"><i>Elementary School Journal</i></searchLink>. Mar 2017 117(3):455-484.
– Name: Avail
  Label: Availability
  Group: Avail
  Data: University of Chicago Press. Journals Division, P.O. Box 37005, Chicago, IL 60637. Tel: 877-705-1878; Tel: 773-753-3347; Fax: 877-705-1879; Fax: 773-753-0811; e-mail: subscriptions@press.uchicago.edu; Web site: http://www.press.uchicago.edu
– Name: PeerReviewed
  Label: Peer Reviewed
  Group: SrcInfo
  Data: Y
– Name: Pages
  Label: Page Count
  Group: Src
  Data: 30
– Name: DatePubCY
  Label: Publication Date
  Group: Date
  Data: 2017
– 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><br /><searchLink fieldCode="EL" term="%22Secondary+Education%22">Secondary Education</searchLink><br /><searchLink fieldCode="EL" term="%22Junior+High+Schools%22">Junior High Schools</searchLink>
– Name: Subject
  Label: Descriptors
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Elementary+School+Mathematics%22">Elementary School Mathematics</searchLink><br /><searchLink fieldCode="DE" term="%22Elementary+School+Students%22">Elementary School Students</searchLink><br /><searchLink fieldCode="DE" term="%22Middle+School+Students%22">Middle School Students</searchLink><br /><searchLink fieldCode="DE" term="%22Grade+6%22">Grade 6</searchLink><br /><searchLink fieldCode="DE" term="%22Graphs%22">Graphs</searchLink><br /><searchLink fieldCode="DE" term="%22Data+Interpretation%22">Data Interpretation</searchLink><br /><searchLink fieldCode="DE" term="%22Protocol+Analysis%22">Protocol Analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Interviews%22">Interviews</searchLink><br /><searchLink fieldCode="DE" term="%22Data+Analysis%22">Data Analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Data+Collection%22">Data Collection</searchLink>
– Name: DOI
  Label: DOI
  Group: ID
  Data: 10.1086/690204
– Name: ISSN
  Label: ISSN
  Group: ISSN
  Data: 0013-5984
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: Students often struggle to interpret graphs correctly, despite emphasis on graphic literacy in U.S. education standards documents. The purpose of this study was to describe challenges sixth graders with varying levels of science and mathematics achievement encounter when transitioning from interpreting graphs having discrete independent variables to graphs having continuous independent variables. Data included think-aloud interviews and written line graph interactions. Data analysis focused on three constituent processes of graph interpretation: (1) encoding salient structures, (2) relating salient structures to each other, and (3) understanding referents in relation to salient structures. Difficulties encoding individual data points influenced interpretations of referents and relationships among data points. Cognitive resources learned for interpreting graphs with discrete independent variables both supported and hindered interpretations of graphs with continuous independent variables. Struggles relating graphs to referents reflected inexperience with data collection and analysis. Recommendations are provided to support students during this transition and to improve their ability to answer different types of graph questions.
– Name: AbstractInfo
  Label: Abstractor
  Group: Ab
  Data: As Provided
– Name: DateEntry
  Label: Entry Date
  Group: Date
  Data: 2017
– Name: AN
  Label: Accession Number
  Group: ID
  Data: EJ1138136
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=eric&AN=EJ1138136
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  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1086/690204
    Languages:
      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 30
        StartPage: 455
    Subjects:
      – SubjectFull: Elementary School Mathematics
        Type: general
      – SubjectFull: Elementary School Students
        Type: general
      – SubjectFull: Middle School Students
        Type: general
      – SubjectFull: Grade 6
        Type: general
      – SubjectFull: Graphs
        Type: general
      – SubjectFull: Data Interpretation
        Type: general
      – SubjectFull: Protocol Analysis
        Type: general
      – SubjectFull: Interviews
        Type: general
      – SubjectFull: Data Analysis
        Type: general
      – SubjectFull: Data Collection
        Type: general
    Titles:
      – TitleFull: Leaping from Discrete to Continuous Independent Variables: Sixth Graders' Science Line Graph Interpretations
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            NameFull: Boote, Stacy K.
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            NameFull: Boote, David N.
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            – D: 01
              M: 03
              Type: published
              Y: 2017
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            – TitleFull: Elementary School Journal
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