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Graphing as a Means to Improve Middle School Science Learning and Mathematics-Related Affective Domains

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Title: Graphing as a Means to Improve Middle School Science Learning and Mathematics-Related Affective Domains
Language: English
Authors: McHugh, Luisa, Kelly, Angela M. (ORCID 0000-0003-1393-1296), Fisher, Jacqueline Horan, Burghardt, M. David
Source: Research in Science Education. Apr 2021 51(2):301-323.
Availability: Springer. Available from: Springer Nature. One New York Plaza, Suite 4600, New York, NY 10004. Tel: 800-777-4643; Tel: 212-460-1500; Fax: 212-460-1700; e-mail: customerservice@springernature.com; Web site: https://link.springer.com/
Peer Reviewed: Y
Page Count: 23
Publication Date: 2021
Sponsoring Agency: National Science Foundation (NSF)
Contract Number: 0314910
Document Type: Journal Articles
Reports - Research
Tests/Questionnaires
Education Level: Junior High Schools
Middle Schools
Secondary Education
Elementary Education
Grade 8
Descriptors: Graphs, Mathematics Skills, Middle School Students, Science Process Skills, Science Projects, Science Curriculum, Integrated Curriculum, Problem Solving, Psychological Patterns, Grade 8, Scientific Concepts, Thinking Skills, Faculty Development
DOI: 10.1007/s11165-018-9796-6
ISSN: 0157-244X
Abstract: This study evaluated the effectiveness of the Mathematics Infusion into Science Project (MiSP), which integrated science and mathematics in an engaging middle school science curriculum that fostered improvements in science content knowledge, higher-level problem solving, and affective domains related to mathematics. The project design was based upon a framework that suggests cross-curricular designs promote scientific thinking and positive attitudes towards the role of mathematics in learning and communicating science. The curriculum incorporated graphing skills that complemented eighth grade science concepts such as thermal energy transfer, density, and photosynthesis. Using a quasi-experimental wait-list control design, this research explored the impacts of MiSP in terms of students' content knowledge, application ability, reasoning skills, and affective domains over the course of one academic year. Over two academic years, 28 teachers participated in 87 h of professional development and 1135 students experienced mathematics-infused lessons and completed pre- and post-science assessments and attitude surveys. Data analyses utilizing analysis of covariance indicated significant improvements in science disciplinary knowledge, higher-order science process skills, and select affective domains, with small to medium effects. Further analyses indicated that treatment-related improvement in science process skills was not mediated by the significant affective domains including linear equation confidence, graph construction confidence, and mathematics applicability recognition. Quantitative findings support the use of graphing-infused curricula in middle schools to improve student science learning and mathematics-related attitudes. Implications for implementing science-mathematics integrated curricula and assessing reform-based science initiatives are discussed.
Abstractor: As Provided
Entry Date: 2021
Accession Number: EJ1298707
Database: ERIC
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  Value: <anid>AN0150318815;g7201apr.21;2021May18.01:43;v2.2.500</anid> <title id="AN0150318815-1">Graphing as a Means to Improve Middle School Science Learning and Mathematics-Related Affective Domains </title> <p>This study evaluated the effectiveness of the Mathematics Infusion into Science Project (MiSP), which integrated science and mathematics in an engaging middle school science curriculum that fostered improvements in science content knowledge, higher-level problem solving, and affective domains related to mathematics. The project design was based upon a framework that suggests cross-curricular designs promote scientific thinking and positive attitudes towards the role of mathematics in learning and communicating science. The curriculum incorporated graphing skills that complemented eighth grade science concepts such as thermal energy transfer, density, and photosynthesis. Using a quasi-experimental wait-list control design, this research explored the impacts of MiSP in terms of students' content knowledge, application ability, reasoning skills, and affective domains over the course of one academic year. Over two academic years, 28 teachers participated in 87 h of professional development and 1135 students experienced mathematics-infused lessons and completed pre- and post-science assessments and attitude surveys. Data analyses utilizing analysis of covariance indicated significant improvements in science disciplinary knowledge, higher-order science process skills, and select affective domains, with small to medium effects. Further analyses indicated that treatment-related improvement in science process skills was not mediated by the significant affective domains including linear equation confidence, graph construction confidence, and mathematics applicability recognition. Quantitative findings support the use of graphing-infused curricula in middle schools to improve student science learning and mathematics-related attitudes. Implications for implementing science-mathematics integrated curricula and assessing reform-based science initiatives are discussed.</p> <p>Keywords: Affective domains; Middle school; Mediation analysis; Professional development; Science learning; STEM integration</p> <hd id="AN0150318815-2">Introduction</hd> <p>Society increasingly relies upon advances in science, technology, engineering, and mathematics (STEM) to solve global problems, and employment demands in these areas are significant and increasing (National Academies of Sciences [NAS] [<reflink idref="bib36" id="ref1">36</reflink>]). However, the pursuit of study within these fields in the USA remains low in comparison to other disciplines, such as the humanities and social sciences, and in comparison to STEM study in other countries (Kuenzi [<reflink idref="bib26" id="ref2">26</reflink>]). The result of this shortage is that employment demands within STEM fields in the USA remain unmet (National Center for Education Statistics [<reflink idref="bib39" id="ref3">39</reflink>]). The STEM talent pool has had a disproportionately low share of women and individuals from traditionally underrepresented minority backgrounds in STEM (National Academy of Sciences, National Academy of Engineering, and Institute of Medicine [<reflink idref="bib38" id="ref4">38</reflink>]; National Science Board [NSB] [<reflink idref="bib41" id="ref5">41</reflink>]). Promoting engagement in STEM in early academic years has been offered as a strategy for broadening and diversifying the STEM workforce, and integrating STEM concepts and practices in K-12 science education has shown promise for improving student learning, interest, and academic identity (National Academies of Sciences and National Research Council [NAS and NRC] [<reflink idref="bib37" id="ref6">37</reflink>]).</p> <p>Research has indicated that many students lack interest in STEM fields due to weak teaching practices, poor mathematical preparation, and lack of transformative change in STEM education (President's Council of Advisors on Science and Technology [PCAST] [<reflink idref="bib50" id="ref7">50</reflink>]). In order to prepare a STEM-literate citizenry and increase preparation for STEM employment, it is essential that stakeholders invest in educational opportunities that provide high-quality science and mathematics instruction early in the pipeline (National Science Foundation [<reflink idref="bib42" id="ref8">42</reflink>]; PCAST [<reflink idref="bib49" id="ref9">49</reflink>]). To this end, the precollege STEM teaching workforce requires in-service training to enhance their content knowledge and pedagogical skills, which will promote science and mathematical literacy (NSB [<reflink idref="bib35" id="ref10">35</reflink>]; Pringle et al. [<reflink idref="bib51" id="ref11">51</reflink>]).</p> <p>Many issues faced by our global society are multidisciplinary in nature, and much of the work for STEM professionals involves fluid boundaries between disciplines. Consequently, educators, policy makers, and researchers have called for a focus on integration in STEM education (Nathan et al. [<reflink idref="bib34" id="ref12">34</reflink>]; NAS and NRC [<reflink idref="bib37" id="ref13">37</reflink>]; Wang et al. [<reflink idref="bib64" id="ref14">64</reflink>]). Although a lack of consensus currently exists among educators and researchers regarding how to conceptualize curricular integration (Beane [<reflink idref="bib5" id="ref15">5</reflink>]; Miller et al. [<reflink idref="bib33" id="ref16">33</reflink>]), there has been general agreement that it is based upon the idea that when faced with real-world problems, solutions will involve knowledge and skills across a range of disciplines instead of isolated subjects, as is characteristic of many elementary and secondary curricula (Czerniak et al. [<reflink idref="bib18" id="ref17">18</reflink>]; Wang et al. [<reflink idref="bib64" id="ref18">64</reflink>]).</p> <p>The present study aims to contribute to the expanding literature on curriculum integration by examining middle school science achievement and affective domains in the Math Infusion into Science Project (MiSP)—an integrated curriculum that infuses graphing skills and knowledge into science classes—involving <emph>N</emph> = 1135 eighth grade students in schools in both high- and low-performing districts in the USA. The curriculum was implemented by middle school science teachers trained in an 87-h professional development program. Using a quasi-experimental wait-list control design, this study focused on the following research questions: (<reflink idref="bib1" id="ref19">1</reflink>) How has participation in a mathematics-infused science curriculum affected students' middle school science knowledge, process skills, and attitudes towards mathematics as it relates to science over the course of an academic year? and (<reflink idref="bib2" id="ref20">2</reflink>) do affective domains mediate cognitive outcomes for middle school students? It was anticipated that the infusion project would result in improved science content knowledge, higher-level analytical thinking, and appreciation of mathematics in middle school science, which incorporates physical, life, and Earth sciences.</p> <hd id="AN0150318815-3">Curricular Integration</hd> <p>Research evaluating STEM curricular integration has identified positive outcomes in terms of improving students' development of problem-solving skills, knowledge acquisition, and knowledge retention (Burghardt et al. [<reflink idref="bib12" id="ref21">12</reflink>], [<reflink idref="bib13" id="ref22">13</reflink>]; McHugh et al. [<reflink idref="bib32" id="ref23">32</reflink>]; James et al. [<reflink idref="bib25" id="ref24">25</reflink>]). Some studies indicated integration had greater consequences than just positive associations with the content and increases in disciplinary content knowledge, that is, there was added value in scientific and quantitative literacy (Berlin and White [<reflink idref="bib6" id="ref25">6</reflink>]; Bragow et al. [<reflink idref="bib9" id="ref26">9</reflink>]; Sherrod et al. [<reflink idref="bib55" id="ref27">55</reflink>]). Research has suggested that an integrated approach to learning is brain compatible, and the more connections made by the brain, the greater the opportunity for making higher-level inferences (Bransford et al. [<reflink idref="bib10" id="ref28">10</reflink>]; Frykholm and Glasson [<reflink idref="bib21" id="ref29">21</reflink>]; Osborne and Wittrock [<reflink idref="bib48" id="ref30">48</reflink>]; Venville et al. [<reflink idref="bib62" id="ref31">62</reflink>]). Learning is best achieved within authentic experiences utilizing multidisciplinary problem-solving strategies (NRC [<reflink idref="bib40" id="ref32">40</reflink>]). There is sound theoretical rationale underlying this concept yet it has received scant empirical testing to date (NAS and NRC [<reflink idref="bib37" id="ref33">37</reflink>]; Venville et al. [<reflink idref="bib63" id="ref34">63</reflink>]).</p> <p>One possible explanation for the lack of empirical research on mathematics and science integration is a conceptual issue related to the definition of integration, since no universal or commonly understood term has been adopted to describe it (Berlin and White [<reflink idref="bib7" id="ref35">7</reflink>]; Stinson et al. [<reflink idref="bib59" id="ref36">59</reflink>]). The <emph>Framework for K-12 Science Education</emph> incorporated the notion of integration in its definition of crosscutting concepts, which they defined as "concepts that bridge disciplinary boundaries, having explanatory value... these concepts help provide students with an organizational framework for connecting knowledge from various disciplines into a coherent and scientifically based view of the world" (NRC [<reflink idref="bib40" id="ref37">40</reflink>], p. 83). Integration implies unity and a holistic approach rather than separation and fragmentation, and students need to develop integrated skills from personally meaningful questions that build upon their own experiences (Beane [<reflink idref="bib4" id="ref38">4</reflink>]; Smith and Karr-Kidwell [<reflink idref="bib56" id="ref39">56</reflink>]). Miller et al. ([<reflink idref="bib33" id="ref40">33</reflink>]) identified content-specific integration as choosing existing learning objectives from mathematics and science and combining the two objectives, such as using mathematical graphing skills during an ecology unit in science. Lonning and DeFranco ([<reflink idref="bib30" id="ref41">30</reflink>]) described true mathematics-science integration as an equal balance between the subjects in a consistently unified curriculum. However, true curriculum integration is quite challenging in the typical middle school context where subjects are often taught discretely by separate teachers. Consequently, the most frequently used method of integration involves infusion—two or more subject areas brought together in a meaningful curriculum by applying one subject area to a second in order to cultivate a deeper understanding of the second subject area (Fogarty [<reflink idref="bib19" id="ref42">19</reflink>]; Lonning and DeFranco [<reflink idref="bib30" id="ref43">30</reflink>]; Steen [<reflink idref="bib58" id="ref44">58</reflink>]). In the context of the present study, infusion was utilized rather than true integration, since a mathematical focus was incorporated in six specific science units that were part of the curriculum, rather than every science unit during the entire academic year.</p> <hd id="AN0150318815-4">Theoretical Framework</hd> <p>The theoretical framework for this study is based upon the integration of science and mathematics knowledge as a means to develop the cognitive dimensions of science learning in K-12 settings. The current project developed curricula that infused mathematics into middle school science instruction, specifically focusing on graphing, since the representation and interpretation of data are foundational skills in scientific literacy (Glazer [<reflink idref="bib22" id="ref45">22</reflink>]). New York State, the location of this study, recently adopted a slightly modified version of the <emph>Next Generation Science Standards</emph>, which emphasizes quantitative and abstract reasoning through mathematical models that describe phenomena and reveal patterns in the natural world (NGSS Lead States [<reflink idref="bib47" id="ref46">47</reflink>]). In grades 6–8, children are expected to display numerical data in plots, interpret linear functions, analyze relationships between dependent and independent variables, and identify patterns in data through graphs, charts, and images (New York State Education Department [NYSED] [<reflink idref="bib46" id="ref47">46</reflink>]).</p> <p>However, many learners have difficulty constructing and analyzing graphs due to misconceptions regarding unit and scale, inability to recognize the relationships between graphs and algebraic functions, and problems with abstract extrapolation (Leinhardt et al. [<reflink idref="bib28" id="ref48">28</reflink>]). Research has indicated that students often confuse global features of graphs, which represent overall trends, with local details such an independent variable measured at a particular moment in time (Testa et al. [<reflink idref="bib61" id="ref49">61</reflink>]). Teacher training and contextual instruction have shown promise in developing the perceptual demands required for graphical construction and interpretation; a focus on data analysis and visual decoding requires sophisticated pedagogical skills that may not be acquired in pre-service preparation (Friel et al. [<reflink idref="bib20" id="ref50">20</reflink>]; Shah and Hoeffner [<reflink idref="bib54" id="ref51">54</reflink>]). The instruction of graphing and diagrammatic reasoning has been shown to improve content knowledge as well as skill transfer to new domains (Cromley et al. [<reflink idref="bib17" id="ref52">17</reflink>]). Graphical literacy should be taught within the science context given its advantages in reducing cognitive demands, emphasizing links between visual features and real-world phenomena, and promoting scientific reasoning (Shah and Hoeffner [<reflink idref="bib54" id="ref53">54</reflink>]).</p> <p>The integration of traditionally discrete disciplinary concepts is fundamentally important since integration facilitates meaning making, retrieval, and transfer to novel contexts (NAS and NRC [<reflink idref="bib37" id="ref54">37</reflink>]). A dynamic mindset towards learning science will improve skill acquisition and the ability to apply scientific constructs in making inferences about everyday situations (Songer and Linn [<reflink idref="bib57" id="ref55">57</reflink>]); such contextual orientation in science learning has been shown to be an effective teaching strategy (Ruthven [<reflink idref="bib52" id="ref56">52</reflink>]). However, strategic approaches are necessary to maintain baseline competencies while strengthening critical thinking and attitudes. Instruction, assessment, and professional development should be aligned to provide relevant outcome measures when knowledge is integrated (Lee et al. [<reflink idref="bib27" id="ref57">27</reflink>]). In doing so, the overall impact of the infusion model may be evaluated to inform curricular reform and policy change (Weiss et al. [<reflink idref="bib65" id="ref58">65</reflink>]).</p> <p>Improvements in integrated science learning are best measured by assessments that differentiate between cognitive and affective constructs with varying complexity (Littledyke [<reflink idref="bib29" id="ref59">29</reflink>]). Many science assessments measure recall over nuanced scientific reasoning; consequently, science curricula often focus on factual knowledge rather than the application of a repertoire of ideas when evaluating scientific phenomena (Lee et al. [<reflink idref="bib27" id="ref60">27</reflink>]). The TIMMS science assessment was developed to measure students' disciplinary content understanding and the associated cognitive domains necessary to solve problems in science—these cognitive domains include knowledge, application, and reasoning, with a progressively increased emphasis on reasoning as students advanced from fourth to eighth grades. These thinking processes constitute a hierarchy of complexity, with more students scoring higher on knowledge rather than higher-order tasks (Martin et al. [<reflink idref="bib31" id="ref61">31</reflink>]). Also, the link between science learning and attitudes is an important construct in understanding student achievement and informed engagement (Alsop and Watts [<reflink idref="bib1" id="ref62">1</reflink>]). Affective domains in science such as interest, self-efficacy, and perceived relevance have been shown to influence student performance; however, more research is needed to provide empirical evidence for this link (Littledyke [<reflink idref="bib29" id="ref63">29</reflink>]; NAS and NRC [<reflink idref="bib37" id="ref64">37</reflink>]).</p> <p>Various cognitive constructs have been considered in this research by measuring the impacts of a mathematics-infused science curriculum that involves disciplinary knowledge and the cognitive processes involved in solving science problems in middle school. This is particularly important since STEM integration in real-world contexts often presents cognitive demands that may interfere with understanding and learning (NAS and NRC [<reflink idref="bib37" id="ref65">37</reflink>]). We hypothesize that knowledge integration applied to mathematics and science instruction will improve students' science achievement and critical thinking in two main cognitive domains—(<reflink idref="bib1" id="ref66">1</reflink>) knowledge and (<reflink idref="bib2" id="ref67">2</reflink>) process skills, consisting of application and reasoning. We also hypothesize student improvement in affective domains, including students' self-efficacy, interest, and their appreciation of mathematical applications in the learning of science, which may mediate science cognitive gains.</p> <hd id="AN0150318815-5">Method</hd> <p></p> <hd id="AN0150318815-6">Research Design</hd> <p>The MiSP was designed to improve middle school student achievement in mathematics via implementation of an instructional model and prototypical materials that infused mathematics into middle school science. MiSP embodied the notion of connecting science and mathematics education to develop an engaging curriculum of problem solving and inquiry that fostered interest and confidence in STEM at a pivotal point in students' academic development. The project involved a long-term professional development component to prepare science teachers for increasing the inclusion of mathematics in their traditional instruction, consistent with research that emphasized this component of curricular reform (Schoen et al. [<reflink idref="bib53" id="ref68">53</reflink>]; Supovitz and Turner [<reflink idref="bib60" id="ref69">60</reflink>]).</p> <p>MiSP employed a wait-list control design where recruited teachers attended professional development, yet some of the group did not implement the treatment until the second year of the project. This design has been used in education research where positive student outcomes are anticipated but controlled trials are required to strengthen empirical validity (for example, Horner et al. [<reflink idref="bib24" id="ref70">24</reflink>]). Teachers who served as control teachers during the first year (2010–2011) became infusion teachers during the second year (2011–2012), at which point a new set of comparison teachers were recruited.</p> <hd id="AN0150318815-7">Participants</hd> <p>Teachers were assigned to either the infusion or comparison groups using a randomized, quasi-experimental block design. Teachers were matched based on variables identified as important to the research and then randomized in order to ensure equal distribution of that variable to experimental and control groups. For MiSP, the variables of importance were the number of teachers per school, the subjects taught by each teacher, and the teacher content knowledge ratings using the <emph>Teacher Mathematical Content Knowledge and PCK Activity</emph> completed at the initial project meeting (see sample tasks in Appendix 1). This activity had two parts: (<reflink idref="bib1" id="ref71">1</reflink>) teachers completed graphing problems as a measure of prior knowledge and (<reflink idref="bib2" id="ref72">2</reflink>) teachers were asked to look at samples of student work in mathematics-science integration and were asked what feedback they would provide to students. The mean score for comparison teachers (<emph>M</emph> =.60) was equivalent to the mean for infusion teachers (<emph>M</emph> =.64), with no statistical difference (<emph>t</emph>(<reflink idref="bib2" id="ref73">2</reflink>,<reflink idref="bib27" id="ref74">27</reflink>) <emph>=</emph> 0.13, <emph>p =</emph>.45).</p> <p>Over the course of two years, 2010–2012, 28 middle school science teachers participated in the study. During 2010–2011, eight schools with two to four teachers at each school participated. When there were two or more teachers per school, half of the teachers were randomly assigned to the infusion group and the other half were in the control group. During 2011–2012, the comparison teachers became infusion teachers and five new schools were recruited. Many of these districts had high poverty levels and low achievement scores. Although such districts were often highly concerned about improving test scores and hesitant to take academic time away from test preparation, interest in mathematics-infused science lessons was strong.</p> <p>In 2010–2011, the control group included 22 general science classes, seven living environment classes, and three Earth science classes. The infusion group included 31 general science classes, three living environment classes, and no Earth science classes. In 2011–2012, 84 math-infused sections of science classes were taught: 58 general science classes, 17 living environment classes, seven Earth science classes, and two science research classes. During 2010–2012, there were 672 infusion students and 463 comparison students, all in grade 8 (generally 13–14 years of age), who completed pre- and post-assessments. The sample sizes varied somewhat in statistical analyses due to some missing responses for individual students; revised sample sizes are noted in the data analysis.</p> <hd id="AN0150318815-8">Teacher Professional Development</hd> <p>This study examined student outcomes in both treatment classes (infusion) and control classes (comparison). During the first year of the study (2010–11), the teachers were trained to teach the MiSP topics during a two-week (10-day) professional development workshop the summer prior to implementation, as well as follow up workshops during the academic year. In total, teachers participated in 87 h of professional development. The workshops were run by project staff in coordination with the mathematics and science curriculum developers who designed the mathematics-infused science topic lessons.</p> <p>The training initially provided teachers with instruction about different mathematical concepts (e.g., graphing, slope of a line, and linear equations). Prior to the summer workshop, teachers selected the topics they would teach to align with their curricula. There were 29 different units that were developed to align with the New York State Intermediate Science Curriculum (NYSED [<reflink idref="bib43" id="ref75">43</reflink>]). The teachers typically selected six units based on the individual courses taught (e.g., general eighth grade science, Earth science, or living environment). The summer workshop also provided teachers with time to practice the science lessons within the unit topics they planned to implement.</p> <hd id="AN0150318815-9">Infusion Curriculum</hd> <p>The mathematics-infused science curriculum was taught in eighth grade science classes with varying levels of complexity. In level 1, students were required to graphically represent data. Level 2 included the mathematical examination of slope and visual understanding of linear vs. non-linear data relationships. At level 3, students contrasted linear and non-linear relationships and developed a linear equation for predictive value. Since the mathematics was typically introduced during the laboratory section of the lesson, the science content remained the same, but the complexity with which students explored and interpreted science data varied. These mathematical concepts were chosen because they were typically challenging concepts yet were considered foundational for understanding more advanced mathematics, particularly algebra. In addition, these concepts fit naturally within the eighth grade science curriculum. Sample topics included lessons in density, simple machines, global warming, and photosynthesis.</p> <hd id="AN0150318815-10">Example of Infusion vs. Comparison Lesson: Thermal Conduction</hd> <p>To exemplify the instructional differences between the comparison and infusion classes, the following middle school thermal conduction unit is described. All students were required to perform an initial experiment where hot water and cold water were poured into separate Styrofoam cups sealed with lids and thermometers. The level 1 tasks for all students included (<reflink idref="bib1" id="ref76">1</reflink>) graphing temperature vs. time for each cup over a period of 10 min, (<reflink idref="bib2" id="ref77">2</reflink>) correctly labeling <emph>x-</emph> and <emph>y-</emph>axes, (<reflink idref="bib3" id="ref78">3</reflink>) drawing lines connecting the data points with different colors and writing a key for the graphs, and (<reflink idref="bib4" id="ref79">4</reflink>) explaining how the graph of the cold water compared to the graph of the hot water. The infusion group was required to meet higher standards with regard to applying mathematical principles to their analysis. Level 2 and 3 tasks included (<reflink idref="bib1" id="ref80">1</reflink>) calculating the slopes of the two lines from lines of best fit, (<reflink idref="bib2" id="ref81">2</reflink>) indicating whether the slopes were positive or negative, (<reflink idref="bib3" id="ref82">3</reflink>) calculating the <emph>y</emph>-intercepts, and (<reflink idref="bib4" id="ref83">4</reflink>) making predictions by extrapolation the graphs while identifying factors affecting real-world outcomes. These tasks are described in more detail in Table 1.</p> <p>Table 1 Comparison of tasks in thermal conduction unit</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th><p>Level</p></th><th><p>Mathematics applied to thermal energy observations and analysis</p></th></tr></thead><tbody><tr><td><p>1</p><p>Comparison and infusion</p></td><td><p>Graph the recorded data to show the relationship between time (minutes) and the temperature (°C) in each cup.</p><p>Label the <italic>x</italic>-axis.</p><p>Label the <italic>y</italic>-axis.</p><p>Connect the dots for each cup's data set (hot water cup, cold water cup). Use two different colors and write a key for the graph.</p><p>How did the graph of the cold water compare with the graph of the hot water?</p></td></tr><tr><td><p>2</p><p>Infusion only</p></td><td><p>Look at the graph you drew. Notice that as time passed, the temperature in each cup changed. You will compare the temperature changes in the second 5 min (from 5 to 10 min) of the experiment in the hot water cup and the cold water cup by calculating the unit rate of change (slope) of each line. Use the information from the graph to calculate the unit rates of change (slopes) for the cold water data and the hot water data on the heat transfer graph. If your data points from 5 to 10 min all lie on a line, determine the unit rates of change (slopes) of the lines. If your data points do not produce lines, determine the unit rates of change (slopes) of best-fit lines from 5 to 10 min.</p><p>ΔTemperature (°C) = ∆<italic>y</italic> = (<italic>y</italic><sub>2</sub> – <italic>y</italic><sub>1</sub>)</p><p>ΔTime (min) ∆<italic>x</italic> = (<italic>x</italic><sub>2</sub> – <italic>x</italic><sub>1</sub>)</p><p>Unit rate of change = ∆<italic>y</italic>/∆<italic>x</italic></p><p>Analysis question: Which set of data had a positive/+ unit rate of change (slope)? What does that tell you about the changes in temperature as time passes?</p></td></tr><tr><td><p>3</p><p>Infusion only</p></td><td><p>Use the equation for a line to calculate the <italic>y</italic>-intercept. The equation for a line is <italic>y</italic> = m<italic>x</italic> + <italic>b</italic>, where <italic>m</italic> is the unit rate of change (slope) and <italic>b</italic> is the <italic>y</italic>-intercept.</p><p>Using each equation above, calculate the predicted temperature of the water at 40 min.</p><p>Reasoning question: The temperatures you calculated for 40 min likely would not be reached if the experiment was allowed to continue for 40 min. Why not? Refer to your graph in your response.</p></td></tr></tbody></table> </ephtml> </p> <hd id="AN0150318815-11">Instruments</hd> <p>Two assessments—one of science content knowledge and process skills and one measuring student affective domains—were administered to students in both the infusion and comparison groups at the beginning and end of the academic year. The measures are described in detail below.</p> <hd id="AN0150318815-12">Science Assessment</hd> <p>Students completed the Science Assessment (see sample items in Appendix 2) at the start and end of the academic year in their science classes to measure their understanding mathematics-infused science principles and skills. To ascertain students' gain in cognitive ability, MiSP structured these assessments by applying the framework utilized in the TIMSS, which is based upon three cognitive domains: knowledge, application, and reasoning (Martin et al. [<reflink idref="bib31" id="ref84">31</reflink>]). <emph>Knowledge-based</emph> questions focused on recall and asked students specifically about facts and procedures. <emph>Application</emph> questions assessed understanding of larger concepts and the ability to make connections among concepts. This type of question required students to apply their knowledge to solve a problem or explain an observation or phenomena. <emph>Reasoning</emph> questions required students to combine content knowledge with reasoning skills to construct explanations, including analyzing and interpreting data, evaluating experimental designs and results, solving complex problems with multiple steps, and applying their knowledge to new situations (see Table 2). These questions required students to think logically beyond established examples and predict outcomes given new scenarios. To do this, students evaluated data and formulated were required to propose sound explanations. For the purpose of this study, application and reasoning questions were combined into one composite to represent higher-level thinking, since it was hypothesized that these <emph>process skills</emph> would be differentiated in the treatment and comparison groups. For teachers, higher-level science tasks are insightful views into a students' thought processes and their ability to make connections to the real world (Chinn and Brewer [<reflink idref="bib15" id="ref85">15</reflink>]; Wilson et al. [<reflink idref="bib66" id="ref86">66</reflink>]).</p> <p>Table 2 Knowledge, application, and reasoning classifications (adapted from Centurino and Jones, [<reflink idref="bib14" id="ref87">14</reflink>])</p> <p> <ephtml> <table frame="hsides" rules="groups"><tbody><tr><td colspan="2"><p>Knowledge</p></td></tr><tr><td colspan="2"><p> Recall/recognize: Identify or state facts, relationships, and concepts; identify the characteristics or properties of specific organisms, materials, and processes; identify the appropriate uses for scientific equipment and procedures; and recognize and use scientific vocabulary, symbols, abbreviations, units, and scales.</p></td></tr><tr><td colspan="2"><p> Describe: Identify descriptions of properties, structures, and functions of organisms and materials, and relationships among them and processes and phenomena.</p></td></tr><tr><td colspan="2"><p> Provide or identify examples: Of organisms, materials, and processes that possess certain specified characteristics; and clarify statements of facts or concepts with appropriate examples.</p></td></tr><tr><td colspan="2"><p>Application</p></td></tr><tr><td colspan="2"><p> Compare/contrast/identify: Identify or describe similarities and differences between groups of organisms, materials, or processes; and distinguish, classify, or sort them based on given properties.</p></td></tr><tr><td colspan="2"><p> Relate: Knowledge of an underlying science concept to an observed or inferred property, behavior, or use of objects, organisms, or materials.</p></td></tr><tr><td colspan="2"><p> Use models: Use a diagram or other model to demonstrate knowledge of science concepts, to illustrate a process cycle relationship, or system, or to find solutions to science problems.</p></td></tr><tr><td colspan="2"><p> Interpret information: Use knowledge of science concepts to interpret relevant textual, tabular, pictorial, and graphical information.</p></td></tr><tr><td colspan="2"><p> Explain: Provide or identify an explanation for an observation or a natural phenomenon using a science concept or principle.</p></td></tr><tr><td colspan="2"><p>Reasoning</p></td></tr><tr><td colspan="2"><p> Analyze: Use relevant information, concepts, relationships, and data patterns to answer questions and solve problems.</p></td></tr><tr><td colspan="2"><p> Synthesize: Answer questions that require consideration of a number of different related concepts.</p></td></tr><tr><td colspan="2"><p> Formulate questions/hypothesize/predict: Use evidence and conceptual understanding to make predictions about the effects of changes in biological or physical conditions.</p></td></tr><tr><td colspan="2"><p> Design investigations: Plan investigations or procedures appropriate for answering scientific questions or testing hypotheses; and describe or recognize the characteristics of well-designed investigations in terms of variables to be measured and controlled and cause-and-effect relationships.</p></td></tr><tr><td colspan="2"><p> Evaluate: Evaluate alternative explanations; weigh advantages and disadvantages to make decisions about alternative processes and materials; and evaluate results of investigations with respect to sufficiency of data to support conclusions.</p></td></tr><tr><td colspan="2"><p> Draw conclusions: Make valid inferences on the basis of observations, evidence, and/or understanding of science concepts; and demonstrate understanding of cause and effect.</p></td></tr></tbody></table> </ephtml> </p> <p>The selection of the MiSP pre-/post-Science Assessment questions was an iterative and highly collaborative process that involved the project and evaluation teams, experts in the areas of mathematics, assessment and psychometrics, and educational psychology, along with master mathematics and science teachers. Questions were developed, piloted, and selected after a review of state mathematics and science assessments (for example, NYSED [<reflink idref="bib44" id="ref88">44</reflink>], [<reflink idref="bib45" id="ref89">45</reflink>]), detailed inventories of mathematics and science textbooks, and consultation with mathematics experts and educators. Although not all assessment questions were contextualized in science specifically, they all were modeled after the state standardized science exam process skill questions (see examples of graphing-related NYSED science assessment items in Appendix 3).</p> <p>To assess the validity of the Science Assessment, educators with at least five years of teaching experience were asked to identify the domain of each question. Consensus was established in differentiating lower-level, knowledge-based questions from higher-level, application and reasoning questions in a mix of multiple choice and open-ended questions (<emph>κ</emph> =.81). Strong internal consistency for student responses was indicated (Cronbach's <emph>α</emph> =.93). On the assessments of science content knowledge and process skills, dichotomous scores were summed for each construct except for five process skills items which were scored with partial credit earned for scores of 0, 1, or 2 points. This resulted in scores ranging from 0 to 9 on the pre- and post-test assessments of content knowledge and 0–23 on the assessments of process skills. The rubric for making these distinctions was adapted from the TIMMS Science Framework (Centurino and Jones [<reflink idref="bib14" id="ref90">14</reflink>]) and is summarized in Table 2.</p> <hd id="AN0150318815-13">Affective Domains Related to Mathematics and Science</hd> <p>The MiSP Attitudinal Survey (Appendix 4) was pilot-tested, revised, and refined during prior work conducted by the research team. The items elicited student responses on the importance of mathematics in their science classes, as well as their confidence they would be able to complete mathematical tasks relevant to science (e.g., measuring and calculating with units, utilizing graphs to show trends, and solving linear problems). Students also indicated their level of agreement with statements about mathematics infusion and science, their interest and ability in mathematics, and perceptions of themselves as mathematics or science students.</p> <p>A factor analysis with the Attitudinal Survey identified five subscales along with respective internal consistencies (Appendix 4): confidence using linear equations (<emph>α</emph> =.89), graph construction confidence (<emph>α</emph> =.78), mathematics self-efficacy (<emph>α</emph> =.81), mathematics interest (<emph>α</emph> =.78), and mathematics applicability recognition (<emph>α</emph> =.72). These subscales were used individually in the general linear model to assess project impacts on attitude subdomains. Confidence questions were rated on a 1–10 scale, with 1 representing <emph>not at all confident</emph> and 10 <emph>highly confident</emph>. Self-efficacy, interest, and mathematics applicability questions were rated on a scale of 1–5, with 1 rated as <emph>strongly disagree</emph>, 3 as <emph>moderately agree</emph>, and 5 as <emph>strongly agree</emph>.</p> <hd id="AN0150318815-14">Data Analysis</hd> <p>Schools provided demographic information on the students who participated in MiSP, including gender, eligibility for free or reduced-cost lunch, English language proficiency, and enrollment in special education. Preliminary analyses generated descriptive statistics for the sample, though inconsistent school reporting resulted in some missing demographic data (see Table 3). Although split roughly evenly by gender, the proportion of boys was somewhat higher in the infusion group than in the control group. Chi-square analysis indicated distribution differences among representation in various demographic subgroups; also, more students were enrolled in infusion classes as opposed to comparison classes.</p> <p>Table 3 Demographics for infusion and comparison groups</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th rowspan="2"><p>Demographic</p></th><th><p>Infusion (<italic>N</italic> = 672)</p></th><th><p>Comparison (<italic>N</italic> = 463)</p></th><th rowspan="2"><p><italic>χ</italic><sup>2</sup></p></th></tr><tr><th><p><italic>n</italic> (%)</p></th><th><p><italic>n</italic> (%)</p></th></tr></thead><tbody><tr><td colspan="4"><p>Gender</p></td></tr><tr><td><p> Male</p></td><td><p>362 (55.1)</p></td><td><p>224 (48.4)</p></td><td char="." align="char" rowspan="2"><p>23.69***</p></td></tr><tr><td><p> Female</p></td><td><p>295 (44.9)</p></td><td><p>239 (51.6)</p></td></tr><tr><td colspan="4"><p>Socioeconomic status</p></td></tr><tr><td><p> Free/reduced lunch</p></td><td><p>179 (31.2)</p></td><td><p>113 (24.4)</p></td><td char="." align="char" rowspan="2"><p>7.60</p></td></tr><tr><td><p> Non-FRL</p></td><td><p>394 (68.8)</p></td><td><p>350 (75.6)</p></td></tr><tr><td colspan="4"><p>Language proficiency</p></td></tr><tr><td><p> Limited English (LEP)</p></td><td><p>51 (4.1)</p></td><td><p>15 (3.2)</p></td><td char="." align="char" rowspan="2"><p>37.27***</p></td></tr><tr><td><p> Non-LEP</p></td><td><p>626 (95.9)</p></td><td><p>448 (96.4)</p></td></tr><tr><td colspan="4"><p>Special education</p></td></tr><tr><td><p> IEP</p></td><td><p>66 (11.4)</p></td><td><p>37 (8.0)</p></td><td char="." align="char" rowspan="2"><p>5.21</p></td></tr><tr><td><p> Non-IEP classified</p></td><td><p>514 (88.6)</p></td><td><p>426 (92.0)</p></td></tr></tbody></table> </ephtml> </p> <p>***<emph>p</emph> <.001</p> <p>Univariate generalized linear modeling was used to examine differences in the change between pre- and post-infusion scores of science content knowledge and application/reasoning skills between the infusion and comparison groups. Data were combined for both years of the project. The dependent variables were science content knowledge, science process skills, and attitudes towards mathematics in relation to science disaggregated by factor. The independent variable (fixed factor) was group (treatment or control). Covariates included pre-test scores in science content knowledge and attitudes, and state standardized scores in mathematics and science. This process tested the null hypotheses that the mathematics-infused curricular treatment would not have an effect on student cognitive and affective domains in science.</p> <p>Mediation analysis was performed using the three-step process designed by Baron and Kenny ([<reflink idref="bib3" id="ref91">3</reflink>]). First, the direct effect of the treatment on science process skills was measured using a simple regression model, with infusion or comparison designation as the independent categorical variable. Secondly, a regression analysis was performed to measure the effect of the independent variable (group) on affective domains. Third, a multiple regression was conducted with group and affective domains as independent variables and science process skills as the dependent variable. If all three conditions were significant, with the treatment effect weaker in the multivariable model, then results were consistent with the hypothesis that affective domains mediated the relationship between the treatment, mathematics-infusion science learning, and the outcome, gains in higher-level science cognitive comprehension.</p> <hd id="AN0150318815-15">Results</hd> <p></p> <hd id="AN0150318815-16">Changes in Science Cognitive and Affective Domains as Discrete Outcomes</hd> <p>Results indicated that students in the infusion group increased their content knowledge, process skills, and select attitudes towards mathematics in relation to science to a greater extent than students in the comparison group. Descriptive and inferential statistics for the main study constructs are presented in Table 4. Univariate general linear models were used to examine student outcomes. Levene's test showed that there was homogeneity of variance in achievement scores between the two groups (<emph>p</emph> >.05). Effect sizes correspond to partial eta squared values of.0099 (small),.0588 (medium), and.1379 (large), benchmarks suggested by Cohen ([<reflink idref="bib16" id="ref92">16</reflink>]). The first model controlled for scores on the Science Assessment pre-test at the beginning of the academic year, and mathematics and science scores on the state standardized tests. Students who participated in MiSP had higher scores on the assessments of content knowledge (<emph>F</emph>(<reflink idref="bib1" id="ref93">1</reflink>,1043) <emph>=</emph> 18.58, <emph>p</emph> <.001, <emph>η</emph><subs>p</subs><sups>2</sups> =.011) and process skills (<emph>F</emph>(<reflink idref="bib1" id="ref94">1</reflink>,1043) <emph>=</emph> 60.10, <emph>p</emph> <.001, <emph>η</emph><subs>p</subs><sups>2</sups> =.047), with small to medium effects, compared to students' exposed to the traditional science curriculum. The affective domain models controlled for pre-survey attitudes responses in each factor. The treatment group exhibited larger gains in linear equation confidence (<emph>F</emph>(<reflink idref="bib1" id="ref95">1</reflink>,<reflink idref="bib842" id="ref96">842</reflink>) <emph>=</emph> 16.61, <emph>p</emph> <.001, <emph>η</emph><subs>p</subs><sups>2</sups> =.019), and they declined to a lesser extent than the comparison group in math applicability recognition (<emph>F</emph>(<reflink idref="bib1" id="ref97">1</reflink>,<reflink idref="bib942" id="ref98">942</reflink>) <emph>=</emph> 49.86, <emph>p</emph> <.001, <emph>η</emph><subs>p</subs><sups>2</sups> =.050), with small to medium effects. Graph construction confidence (<emph>F</emph>(<reflink idref="bib1" id="ref99">1</reflink>,<reflink idref="bib962" id="ref100">962</reflink>) <emph>=</emph> 4.21, <emph>p</emph> <.05, <emph>η</emph><subs>p</subs><sups>2</sups> =.004) was significantly improved for the infusion students; however, the effect was negligible. Math interest (<emph>F</emph>(<reflink idref="bib1" id="ref101">1</reflink>,<reflink idref="bib972" id="ref102">972</reflink>) <emph>=</emph> 1.47, <emph>p</emph> =.225) and math self-efficacy (<emph>F</emph>(<reflink idref="bib1" id="ref103">1</reflink>,<reflink idref="bib893" id="ref104">893</reflink>) <emph>=</emph> 0.038, <emph>p</emph> =.845) were not differentiated between groups.</p> <p>Table 4 Statistical analyses for measures of content knowledge, process skills, affective domains</p> <p> <ephtml> <table frame="hsides" rules="groups"><tbody><tr><td rowspan="2"><p>Infusion group</p></td><td colspan="2"><p>Pre-MiSP</p></td><td colspan="2"><p>Post-MiSP</p></td><td colspan="2"><p>Between-groups ANCOVA</p></td></tr><tr><td><p>Range</p></td><td><p><italic>M</italic> (SD)</p></td><td><p>Range</p></td><td><p><italic>M</italic> (SD)</p></td><td><p><italic>F</italic></p></td><td><p>Effect size (<italic>η</italic><sup>2</sup>)</p></td></tr><tr><td><p>Content knowledge</p></td><td><p>0–9</p></td><td><p>3.27 (1.68)</p></td><td><p>0–9</p></td><td><p>5.32 (1.55)</p></td><td char="." align="char"><p>18.58***</p></td><td><p>.011</p></td></tr><tr><td><p>Process skills</p></td><td><p>0–23</p></td><td><p>8.77 (3.53)</p></td><td><p>0–23</p></td><td><p>12.39 (4.82)</p></td><td char="." align="char"><p>60.10***</p></td><td><p>.047</p></td></tr><tr><td colspan="7"><p>Affective domains</p></td></tr><tr><td><p> Linear equations confidence</p></td><td><p>0–70</p></td><td><p>49.83 (15.7)</p></td><td><p>0–70</p></td><td><p>59.73 (14.6)</p></td><td char="." align="char"><p>16.61***</p></td><td><p>.019</p></td></tr><tr><td><p> Graph construction confidence</p></td><td><p>0–50</p></td><td><p>38.35 (8.05)</p></td><td><p>0–50</p></td><td><p>41.76 (7.55)</p></td><td char="." align="char"><p>4.21*</p></td><td><p>.004</p></td></tr><tr><td><p> Mathematics self-efficacy</p></td><td><p>1–25</p></td><td><p>14.06 (2.50)</p></td><td><p>1–25</p></td><td><p>14.34 (2.53)</p></td><td char="." align="char"><p>0.04</p></td><td><p>n/a</p></td></tr><tr><td><p> Mathematics interest</p></td><td><p>1–25</p></td><td><p>12.67 (2.47)</p></td><td><p>1–25</p></td><td><p>13.16 (2.68)</p></td><td char="." align="char"><p>1.47</p></td><td><p>n/a</p></td></tr><tr><td><p> Math applicability recognition</p></td><td><p>1–20</p></td><td><p>14.26 (3.29)</p></td><td><p>1–20</p></td><td><p>13.76 (3.49)</p></td><td char="." align="char"><p>49.86***</p></td><td><p>.050</p></td></tr><tr><td rowspan="2"><p>Comparison group</p></td><td colspan="2"><p>Pre-MiSP</p></td><td colspan="2"><p>Post-MiSP</p></td><td /><td /></tr><tr><td><p>Range</p></td><td><p><italic>M</italic> (SD)</p></td><td><p>Range</p></td><td><p><italic>M</italic> (SD)</p></td><td /><td /></tr><tr><td><p>Content knowledge</p></td><td><p>0–9</p></td><td><p>4.02 (1.46)</p></td><td><p>0–9</p></td><td><p>5.01 (1.59)</p></td><td /><td /></tr><tr><td><p>Process skills</p></td><td><p>0–23</p></td><td><p>9.39 (3.79)</p></td><td><p>0–23</p></td><td><p>11.14 (4.27)</p></td><td /><td /></tr><tr><td colspan="7"><p>Affective domains</p></td></tr><tr><td><p> Linear equations confidence</p></td><td><p>0–70</p></td><td><p>48.94 (18.1)</p></td><td><p>0–70</p></td><td><p>55.50 (16.1)</p></td><td /><td /></tr><tr><td><p> Graph construction confidence</p></td><td><p>0–50</p></td><td><p>39.42 (7.31)</p></td><td><p>0–50</p></td><td><p>41.11 (7.44)</p></td><td /><td /></tr><tr><td><p> Mathematics self-efficacy</p></td><td><p>1–25</p></td><td><p>14.21 (2.41)</p></td><td><p>1–25</p></td><td><p>14.51 (2.37)</p></td><td /><td /></tr><tr><td><p> Mathematics interest</p></td><td><p>1–25</p></td><td><p>12.91 (2.55)</p></td><td><p>1–25</p></td><td><p>13.49 (2.69)</p></td><td /><td /></tr><tr><td><p> Math applicability recognition</p></td><td><p>1–20</p></td><td><p>13.71 (3.43)</p></td><td><p>1–20</p></td><td><p>12.38 (3.59)</p></td><td /><td /></tr></tbody></table> </ephtml> </p> <p>*<emph>p</emph> <.05; **<emph>p</emph> <.01; ***<emph>p</emph> <.001</p> <hd id="AN0150318815-17">Affective Domains as a Mediating Factor for Science Understanding</hd> <p>To investigate the hypothesis that affective domains mediated science learning, a three-step regression process was employed (Baron and Kenny [<reflink idref="bib3" id="ref105">3</reflink>]). The affective domains of linear equation confidence, graph construction confidence, and mathematics applicability recognition were selected due to their significance in the general linear models, and they were given equal weights in a post-treatment composite for each group. Science process skills were the outcome variable with group (infusion or comparison) as the predictor. Results indicated that group was a significant predictor of science process skills (<emph>β</emph> = 0.134, SE = 0.255, <emph>p</emph> <.001). Group also predicted affective domains (<emph>β</emph> = 0.135, SE = 1.424, <emph>p</emph> <.001). A multiple regression model revealed independent variables including group and affective domains predicted science process skills (group <emph>β</emph> = 0.134, SE = 0.274, <emph>p</emph> <.001; affective domains <emph>β</emph> = 0.315, SE = 0.006, <emph>p</emph> <.001). Since the indirect effect of group (<emph>c</emph>′ <emph>=</emph> 0.134) was not weaker than the direct effect (<emph>a =</emph> 0.134), the mediational hypothesis was rejected. Results are summarized in Table 5.</p> <p>Table 5 Mediation model statistics</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th><p>Testing path</p></th><th><p><italic>β</italic></p></th><th><p><italic>B</italic></p></th><th><p>SE (<italic>B</italic>)</p></th><th><p>95% CI</p></th><th><p><italic>p</italic></p></th></tr></thead><tbody><tr><td><p>Path c</p><p> Group (IV) ➔ sci process skills (DV)</p><p><italic>R</italic><sup>2</sup> =.018, <italic>F</italic>(1,1101) = 24.00</p></td><td><p>0.134</p></td><td><p>1.25</p></td><td><p>0.255</p></td><td><p>0.75, 1.75</p></td><td><p><.001</p></td></tr><tr><td><p>Path a</p><p> Group (IV) ➔ affective domains (med)</p><p><italic>R</italic><sup>2</sup> =.018, <italic>F</italic>(1,977) = 18.01</p></td><td><p>0.135</p></td><td><p>6.04</p></td><td><p>1.424</p></td><td><p>3.25, 8.84</p></td><td><p><.001</p></td></tr><tr><td><p>Path b</p><p> Affective domains (med) ➔ sci process skills (DV)</p><p> Multivariable model: <italic>R</italic><sup>2</sup> =.128, <italic>F</italic>(2,950) = 69.91</p></td><td><p>0.315</p></td><td><p>0.063</p></td><td><p>0.006</p></td><td><p>0.051, 0.075</p></td><td><p><.001</p></td></tr><tr><td><p>Path c′</p><p> Indirect effect</p></td><td><p>0.134</p></td><td><p>1.20</p></td><td><p>0.274</p></td><td><p>0.66, 1.74</p></td><td><p><.001</p></td></tr></tbody></table> </ephtml> </p> <hd id="AN0150318815-18">Discussion</hd> <p>Although previous studies have indicated overall science achievement and attitudes improved when using integrated curricula (Czerniak et al. [<reflink idref="bib18" id="ref106">18</reflink>]; McHugh et al. [<reflink idref="bib32" id="ref107">32</reflink>]), this study differentiated between content knowledge and process skills to identify positive impacts on science understanding and higher-order thinking. The results from this study suggest that the infusion of mathematics into science is a successful approach for improving science content knowledge, application ability, and reasoning skills. Although students in the treatment group outperformed the comparison group on general content questions, this effect was small, likely due to the fact that both treatment and control groups were expected to improve reasonably on general descriptive and identification tasks. However, emphasis on knowledge-based teaching and recall assessments does not allow students to develop ways of thinking that are consistent with scientific practices. Therefore, a more nuanced approach and analysis were necessary. Improved higher-order scientific process skills for the infusion students were indicated by performance on science reasoning and application questions. The students taught with an infused curriculum demonstrated greater proficiency in applying mathematical concepts when solving science problems. In doing so, they expanded their acquisition of conceptual understanding by making connections between mutually reinforcing disciplinary applications.</p> <p>Student's attitudes towards mathematics showed significant improvement in two areas—linear equation confidence and, to a lesser extent, graph construction confidence. The largest positive change was observed in students' confidence in interpreting linear equations, which had a small to medium effect. Small effect sizes (<emph>η</emph><subs>p</subs><sups>2</sups> ≥.0099) are considered of merit when assessing large-scale innovations and their promise for systemic reform (Hedges and Hedberg [<reflink idref="bib23" id="ref108">23</reflink>]). Related skills included their confidence in identifying variables, determining linear relationships, and evaluating slope and the equation of a line. This technical knowledge was necessary to interpret the relationships among variables, and students felt more confident in their ability to do so. Significant yet weak gains were noted in graph construction confidence for the treatment group. This suggests students should be given more opportunities to construct their own graphs in addition to requiring them to interpret existing graphs. This may correct misconceptions involving selection of appropriate axes, determining scales, and assigning data points.</p> <p>Conflicting results were noted in other affective domain factors. There was a significant group difference observed in students' beliefs about the applicability of mathematics. This measure declined for both groups but significantly less so for the treatment group. These results suggest that students could not always relate to the contexts chosen for instruction, and they did not always see how mathematics was improving their understanding of science. Perhaps more frequent use of formative assessment would improve students' attitudes in these areas, since they would be more likely to make explicit connections between knowledge of mathematics and their improvement in science. Students' interest in mathematics and their mathematics self-efficacy were relatively unchanged in both the treatment and control groups. This suggests that the treatment did not mitigate typical declines in mathematical interest among middle school students. Since these findings did not match their significantly improved performance in science knowledge and process skills, students should be given more opportunities for critical feedback that communicates their improved science understanding. Knowledge of their achievement in infused science and mathematics is necessary to improve self-efficacy in future tasks. Since subject interest is a leading predictor of persistence in post-secondary study and careers (PCAST [<reflink idref="bib50" id="ref109">50</reflink>]), modifying the treatment may motivate students to remain in the STEM pipeline or pursue mathematics to improve their quantitative literacy.</p> <p>In order to explore the relationship between cognitive and affective domains, further data analysis revealed that these factors were distinct and non-related constructs. Affective domains did not mediate improvements in application and reasoning skills. This is an important consideration since measurement of affective domains has often been used as a proxy for determining the effectiveness of curricular reforms (NAS and NRC [<reflink idref="bib37" id="ref110">37</reflink>]). Although self-reported confidence and self-efficacy provide insights into students' thought processes, these measures may not be appropriate substitutes for accurately assessing student learning outcomes in reformed instruction. Other research has suggested affective domains such as self-efficacy mediating the relationship between pedagogical innovations and science comprehension in undergraduate students (Ballen et al. [<reflink idref="bib2" id="ref111">2</reflink>]); however, more research is necessary in precollege settings to explore potential causal relationships between affective factors and science learning (NAS and NRC [<reflink idref="bib37" id="ref112">37</reflink>]). This may provide further empirical support for expansion of the mathematics-science infusion model to full integration in middle school settings. In doing so, students will be more likely to experience science through a challenging curriculum that allows them to improve quantitative literacy.</p> <hd id="AN0150318815-19">Limitations</hd> <p>This study had several limitations. One such limitation was the lack of control for curricular fidelity in the research design. Teachers were provided with mathematics integrated science lessons and professional development training, though they were not directly observed to assess the degree to which they implemented the curriculum in alignment with the program's core objectives. Variations in instructional practices were not taken into account in the quantitative analysis; however, measures were taken during professional development to minimize instructional variation by providing detailed lessons and accompanying student tasks. Future research is necessary to determine the extent of curricular fidelity among teachers in large-scale studies, since teacher quality and training have been shown to be critical variables in student science achievement (Bolyard and Moyer-Packenham [<reflink idref="bib8" id="ref113">8</reflink>]). A second limitation is the missing demographic and test data. With <emph>N</emph> = 1135 students in the study, it was difficult to ensure complete data collection with teacher-reported demographics and skipped item responses, particularly on attitudes assessments. Multiple data imputation was performed and the results did not change significantly from the original analysis; therefore, the results were reported with list-wise deleted cases. A third limitation was the underrepresentation of high needs students in the overall sample. Students in traditionally underserved schools may achieve the largest value added by participation in transformative programs such as MiSP. The study did target low-resource middle schools, with 31% of the treatment group and 24% of the control group qualifying for free or reduced lunch. However, future studies in knowledge integration should target a larger percentage of schools with limited resources to provide more equitable opportunities for STEM learning.</p> <hd id="AN0150318815-20">Implications and Future Research</hd> <p>The goal of this research study was to provide empirical evidence for the pedagogical view that mathematics and science should be taught in an integrated fashion at the middle school level. Unlike previous studies, this research delineated student performance in content and process skills, consisting of application and reasoning skills, as well as changes in affective domains. Goals of the program were for students to improve science performance, knowledge of ways in which mathematics might be applied to scientific contexts, and attitudes towards the use of mathematics in science. Students in the infused classrooms demonstrated improved academic performance and attitudes, which suggests treatment outcomes may lead to long-term impacts related to STEM engagement and achievement (Brown et al. [<reflink idref="bib11" id="ref114">11</reflink>]). Curricula that incorporate knowledge integration hold promise for improving middle school students' persistence in mathematics and science in the high school years and beyond. Furthermore, more positive attitudes towards mathematics in science may directly influence science performance if the curriculum were taught in a completely integrated manner. The relationship between attitudinal measures and achievement may be more interrelated in classrooms that seamlessly incorporate mathematics in scientific contexts. This study also provides assessment exemplars that incorporate disciplinary content knowledge and crosscutting concepts in mathematics and science. Such assessments are increasingly important as more states adopt the NGSS and design accountability measures.</p> <p>The research design in this study is also informative for future work in the field. The National Academies of Sciences reported that STEM integration research has focused on learning, interest, and identity, though the few studies that have been done typically utilized small sample sizes (NAS and NRC [<reflink idref="bib37" id="ref115">37</reflink>]). This study demonstrates the relevance of large-scale controlled designs in providing empirically rigorous evidence to support reform efforts. Furthermore, student outcomes should be differentiated between cognitive and affective domains since the results from this study do not support a causal link; however, additional research with more nuanced attitudes assessments may reveal mediating factors in student academic performance. The quasi-experimental wait-list control design ensured that students in the control group had access to the treatment in subsequent years, an important equity consideration when implementing promising educational reforms. By selecting both treatment and control teachers at each school, the model was more likely to be sustained and institutionalized. One aspect of the research design that could be added in future studies is direct observations of the trained teachers. Future research should also include repeated measure observations of classroom settings to triangulate findings with curricular fidelity.</p> <p>STEM integration is a promising strategy for improving the academic achievement and mathematics-related affective domains of middle school students. The results from this study demonstrate that students exposed to this type of curricular design improved science knowledge, application, and reasoning ability. If students are performing at a higher level in STEM, they may be more interested in taking additional mathematics and science coursework as they progress through the academic pipeline. Such STEM reforms hold promise for improving scientific and quantitative literacies for future generations.</p> <hd id="AN0150318815-21">Funding Information</hd> <p>This research was supported in part by the National Science Foundation #0314910.</p> <hd id="AN0150318815-22">Appendix 1. Teacher mathematical content knowledge and PCK activity</hd> <p>Instructions: Please solve the math question. This is the type of math problem which students might be asked to solve. The question comes from a science perspective. As you solve the problem feel free to write notes about the problem on the paper. Notes can be comments or reactions to the questions or suggestions for changing the questions.</p> <p>Your science teacher tells you that the relationship between distance and time for an object moving at constant speed shares many qualities with the relationship between mass and volume for a uniform substance.</p> <p>Below is a table for distance and time for an object moving 8 meters per second.</p> <p></p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th><p>Time (sec)</p></th><th><p>0</p></th><th><p>1</p></th><th><p>2</p></th><th><p>3</p></th><th><p>4</p></th><th><p>5</p></th></tr></thead><tbody><tr><td><p>Distance (m)</p></td><td char="." align="char"><p>0</p></td><td char="." align="char"><p>8</p></td><td char="." align="char"><p>16</p></td><td char="." align="char"><p>24</p></td><td char="." align="char"><p>32</p></td><td char="." align="char"><p>40</p></td></tr></tbody></table> </ephtml> </p> <p>Participant Tasks</p> <p></p> <ulist> <item> Graph the data and label the axis (grid provided).</item> <p></p> <item> What is the slope and <emph>y</emph>-intercept of this line? <emph>Show your work</emph>.</item> <p></p> <item> Explain how the speed of 8 meters per second is represented in this table.</item> <p></p> <item> Write an equation relating the variable distance (<emph>y</emph>) to time (<emph>x</emph>).</item> <p></p> <item> Describe how you know this is a linear relationship based on the table, graph, and equation.</item> </ulist> <p>Thinking about the Questions</p> <p>Where do you think your students might struggle when answering these questions?</p> <p>After Reviewing Student Work</p> <p></p> <ulist> <item> Look at the three student responses to the distance-time graphing question (provided).</item> <p></p> <item> Which student do you feel showed greatest mastery of the material? Least mastery of the material?</item> <p></p> <item> List the conceptual errors made by each student.</item> <p></p> <item> For each error, describe how you would work with the student to help her understand the math.</item> </ulist> <hd id="AN0150318815-23">Appendix 2. Integrated Science Assessment</hd> <p>Graph</p> <hd id="AN0150318815-24">Appendix 3. Sample graphing items from state standardized exams (NYSED, Grade 8 Intermediate...</hd> <p>Sample Question 1. Base your answers to the following questions on the information and table below and on your knowledge of science.</p> <p>A car traveled a distance of 240 kilometers between 8:00 a.m. and 11:00 a.m. The data table below shows the car's distance from the starting location at 0.5-hour intervals during the trip.</p> <p>Distance Traveled vs. Time</p> <p></p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th><p>Time</p></th><th><p>Time (hours)</p></th><th><p>Starting Location (kilometers)</p></th></tr></thead><tbody><tr><td><p>8:00</p></td><td><p>0.0</p></td><td><p>0</p></td></tr><tr><td><p>8:30</p></td><td><p>0.5</p></td><td><p>55</p></td></tr><tr><td><p>9:00</p></td><td><p>1.0</p></td><td><p>90</p></td></tr><tr><td><p>9:30</p></td><td><p>1.5</p></td><td><p>90</p></td></tr><tr><td><p>10:00</p></td><td><p>2.0</p></td><td><p>142</p></td></tr><tr><td><p>10:30</p></td><td><p>2.5</p></td><td><p>200</p></td></tr><tr><td><p>11:00</p></td><td><p>3.0</p></td><td><p>240</p></td></tr></tbody></table> </ephtml> </p> <p>On the grid below, make a graph using the data in the table. Place an <bold>X</bold> to show the distance from the starting location of the car for each 0.5-hour interval. Connect the <bold>X</bold>s with a line.</p> <p>Graph</p> <p>Give <emph>one</emph> possible explanation for the car's distance from the starting location at 9:00 a.m. and at 9:30 a.m.</p> <p>Sample Question 2. The graph below shows the solubility curves for three sold substances.</p> <p>Graph</p> <p>As the water temperature is increased from 30°C to 90°C, how many more grams of potassium bromide will dissolve in 100 grams of water?</p> <p>Sample Question 3. Base your answers to the following questions on the experiment described below.</p> <p>Heat was applied at a constant rate to a solid substance under controlled conditions. The temperature of the substance was recorded every 3minutes. These data are recorded in the table below.</p> <p>Graph</p> <hd id="AN0150318815-25">Appendix 4</hd> <p>Table 6 Mathematics affective domains in relation to science—survey and factor analysis</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th><p>Item</p></th><th><p>Linear equations confidence <italic>α</italic> = .89</p></th><th><p>Graph construction confidence <italic>α</italic> = .78</p></th><th><p>Mathematics self-efficacy <italic>α</italic> = .81</p></th><th><p>Mathematics interest <italic>α</italic> = .78</p></th><th><p>Mathematics applicability recognition <italic>α</italic> = .72</p></th></tr></thead><tbody><tr><td><p>I can use a graph to write the equation of a line.</p></td><td char="." align="char"><p>0.781</p></td><td /><td /><td /><td /></tr><tr><td><p>I can use a data table to write the equation of a line.</p></td><td char="." align="char"><p>0.759</p></td><td /><td /><td /><td /></tr><tr><td><p>I can use the equation of a line to answer questions.</p></td><td char="." align="char"><p>0.697</p></td><td /><td /><td /><td /></tr><tr><td><p>I can use a data table to find the slope of a line.</p></td><td char="." align="char"><p>0.681</p></td><td /><td /><td /><td /></tr><tr><td><p>I can use an equation to determine if data are linear, non-linear or approximately linear.</p></td><td char="." align="char"><p>0.667</p></td><td /><td /><td /><td /></tr><tr><td><p>I can use a graph to find the slope of a line.</p></td><td char="." align="char"><p>0.655</p></td><td /><td /><td /><td /></tr><tr><td><p>I can use a graph to determine if data are linear, non-linear or approximately linear.</p></td><td char="." align="char"><p>0.607</p></td><td /><td /><td /><td /></tr><tr><td><p>I can draw a line of best fit.</p></td><td char="." align="char"><p>0.596</p></td><td /><td /><td /><td /></tr><tr><td><p>I can correctly set up a graph.</p></td><td /><td char="." align="char"><p>0.771</p></td><td /><td /><td /></tr><tr><td><p>I can correctly label graph axes and titles.</p></td><td /><td char="." align="char"><p>0.670</p></td><td /><td /><td /></tr><tr><td><p>I can set up the scale for a graph.</p></td><td /><td char="." align="char"><p>0.624</p></td><td /><td /><td /></tr><tr><td><p>I can identify the independent and dependent variables.</p></td><td /><td char="." align="char"><p>0.531</p></td><td /><td /><td /></tr><tr><td><p>I can answer questions about a graph.</p></td><td /><td char="." align="char"><p>0.439</p></td><td /><td /><td /></tr><tr><td><p>I think I can handle more difficult math.</p></td><td /><td /><td char="." align="char"><p>0.779</p></td><td /><td /></tr><tr><td><p>I am able to solve complex math problems.</p></td><td /><td /><td char="." align="char"><p>0.696</p></td><td /><td /></tr><tr><td><p>I am not good at math.</p></td><td /><td /><td char="." align="char"><p>0.595</p></td><td /><td /></tr><tr><td><p>I can get good grades in math.</p></td><td /><td /><td char="." align="char"><p>0.582</p></td><td /><td /></tr><tr><td><p>I find math confusing.</p></td><td /><td /><td char="." align="char"><p>0.502</p></td><td /><td /></tr><tr><td><p>Math is boring.</p></td><td /><td /><td /><td char="." align="char"><p>0.681</p></td><td /></tr><tr><td><p>I enjoy learning about math.</p></td><td /><td /><td /><td char="." align="char"><p>0.658</p></td><td /></tr><tr><td><p>Math is interesting.</p></td><td /><td /><td /><td char="." align="char"><p>0.607</p></td><td /></tr><tr><td><p>I do not understand why I need to study math.</p></td><td /><td /><td /><td char="." align="char"><p>0.473</p></td><td /></tr><tr><td><p>The math I learn in school has no relevance to my life.</p></td><td /><td /><td /><td char="." align="char"><p>0.398</p></td><td /></tr><tr><td><p>Math is important for completing tasks in science.</p></td><td /><td /><td /><td /><td char="." align="char"><p>0.676</p></td></tr><tr><td><p>Being able to do math makes learning science easier.</p></td><td /><td /><td /><td /><td char="." align="char"><p>0.553</p></td></tr><tr><td><p>Math courses are very helpful no matter what I decide to study.</p></td><td /><td /><td /><td /><td char="." align="char"><p>0.550</p></td></tr><tr><td><p>Math skills can be used in a variety of ways.</p></td><td /><td /><td /><td /><td char="." align="char"><p>0.544</p></td></tr></tbody></table> </ephtml> </p> <ref id="AN0150318815-26"> <title> References </title> <blist> <bibl id="bib1" idref="ref19" type="bt">1</bibl> <bibtext> Alsop S, Watts M. 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  Label: Title
  Group: Ti
  Data: Graphing as a Means to Improve Middle School Science Learning and Mathematics-Related Affective Domains
– Name: Language
  Label: Language
  Group: Lang
  Data: English
– Name: Author
  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22McHugh%2C+Luisa%22">McHugh, Luisa</searchLink><br /><searchLink fieldCode="AR" term="%22Kelly%2C+Angela+M%2E%22">Kelly, Angela M.</searchLink> (ORCID <externalLink term="http://orcid.org/0000-0003-1393-1296">0000-0003-1393-1296</externalLink>)<br /><searchLink fieldCode="AR" term="%22Fisher%2C+Jacqueline+Horan%22">Fisher, Jacqueline Horan</searchLink><br /><searchLink fieldCode="AR" term="%22Burghardt%2C+M%2E+David%22">Burghardt, M. David</searchLink>
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  Label: Source
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  Data: <searchLink fieldCode="SO" term="%22Research+in+Science+Education%22"><i>Research in Science Education</i></searchLink>. Apr 2021 51(2):301-323.
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  Data: Springer. Available from: Springer Nature. One New York Plaza, Suite 4600, New York, NY 10004. Tel: 800-777-4643; Tel: 212-460-1500; Fax: 212-460-1700; e-mail: customerservice@springernature.com; Web site: https://link.springer.com/
– Name: PeerReviewed
  Label: Peer Reviewed
  Group: SrcInfo
  Data: Y
– Name: Pages
  Label: Page Count
  Group: Src
  Data: 23
– Name: DatePubCY
  Label: Publication Date
  Group: Date
  Data: 2021
– Name: SourceSuprt
  Label: Sponsoring Agency
  Group: SrcSuprt
  Data: National Science Foundation (NSF)
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  Label: Contract Number
  Group: NumCntrct
  Data: 0314910
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  Label: Document Type
  Group: TypDoc
  Data: Journal Articles<br />Reports - Research<br />Tests/Questionnaires
– Name: Audience
  Label: Education Level
  Group: Audnce
  Data: <searchLink fieldCode="EL" term="%22Junior+High+Schools%22">Junior High Schools</searchLink><br /><searchLink fieldCode="EL" term="%22Middle+Schools%22">Middle Schools</searchLink><br /><searchLink fieldCode="EL" term="%22Secondary+Education%22">Secondary Education</searchLink><br /><searchLink fieldCode="EL" term="%22Elementary+Education%22">Elementary Education</searchLink><br /><searchLink fieldCode="EL" term="%22Grade+8%22">Grade 8</searchLink>
– Name: Subject
  Label: Descriptors
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Graphs%22">Graphs</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Skills%22">Mathematics Skills</searchLink><br /><searchLink fieldCode="DE" term="%22Middle+School+Students%22">Middle School Students</searchLink><br /><searchLink fieldCode="DE" term="%22Science+Process+Skills%22">Science Process Skills</searchLink><br /><searchLink fieldCode="DE" term="%22Science+Projects%22">Science Projects</searchLink><br /><searchLink fieldCode="DE" term="%22Science+Curriculum%22">Science Curriculum</searchLink><br /><searchLink fieldCode="DE" term="%22Integrated+Curriculum%22">Integrated Curriculum</searchLink><br /><searchLink fieldCode="DE" term="%22Problem+Solving%22">Problem Solving</searchLink><br /><searchLink fieldCode="DE" term="%22Psychological+Patterns%22">Psychological Patterns</searchLink><br /><searchLink fieldCode="DE" term="%22Grade+8%22">Grade 8</searchLink><br /><searchLink fieldCode="DE" term="%22Scientific+Concepts%22">Scientific Concepts</searchLink><br /><searchLink fieldCode="DE" term="%22Thinking+Skills%22">Thinking Skills</searchLink><br /><searchLink fieldCode="DE" term="%22Faculty+Development%22">Faculty Development</searchLink>
– Name: DOI
  Label: DOI
  Group: ID
  Data: 10.1007/s11165-018-9796-6
– Name: ISSN
  Label: ISSN
  Group: ISSN
  Data: 0157-244X
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: This study evaluated the effectiveness of the Mathematics Infusion into Science Project (MiSP), which integrated science and mathematics in an engaging middle school science curriculum that fostered improvements in science content knowledge, higher-level problem solving, and affective domains related to mathematics. The project design was based upon a framework that suggests cross-curricular designs promote scientific thinking and positive attitudes towards the role of mathematics in learning and communicating science. The curriculum incorporated graphing skills that complemented eighth grade science concepts such as thermal energy transfer, density, and photosynthesis. Using a quasi-experimental wait-list control design, this research explored the impacts of MiSP in terms of students' content knowledge, application ability, reasoning skills, and affective domains over the course of one academic year. Over two academic years, 28 teachers participated in 87 h of professional development and 1135 students experienced mathematics-infused lessons and completed pre- and post-science assessments and attitude surveys. Data analyses utilizing analysis of covariance indicated significant improvements in science disciplinary knowledge, higher-order science process skills, and select affective domains, with small to medium effects. Further analyses indicated that treatment-related improvement in science process skills was not mediated by the significant affective domains including linear equation confidence, graph construction confidence, and mathematics applicability recognition. Quantitative findings support the use of graphing-infused curricula in middle schools to improve student science learning and mathematics-related attitudes. Implications for implementing science-mathematics integrated curricula and assessing reform-based science initiatives are discussed.
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  Label: Abstractor
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  Data: As Provided
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  Label: Entry Date
  Group: Date
  Data: 2021
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  Label: Accession Number
  Group: ID
  Data: EJ1298707
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      – Type: doi
        Value: 10.1007/s11165-018-9796-6
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      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 23
        StartPage: 301
    Subjects:
      – SubjectFull: Graphs
        Type: general
      – SubjectFull: Mathematics Skills
        Type: general
      – SubjectFull: Middle School Students
        Type: general
      – SubjectFull: Science Process Skills
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      – SubjectFull: Thinking Skills
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      – SubjectFull: Faculty Development
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      – TitleFull: Graphing as a Means to Improve Middle School Science Learning and Mathematics-Related Affective Domains
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