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Preparation of Students Completing a Core-Plus or Commercially Developed High School Mathematics Curriculum for Intense College Mathematics Coursework

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Title: Preparation of Students Completing a Core-Plus or Commercially Developed High School Mathematics Curriculum for Intense College Mathematics Coursework
Language: English
Authors: Harwell, Michael R., Medhanie, Amanuel, Post, Thomas R., Norman, Ke, Dupuis, Danielle N.
Source: Journal of Experimental Education. 2012 80(1):96-112.
Availability: Routledge. Available from: Taylor & Francis, Ltd. 325 Chestnut Street Suite 800, Philadelphia, PA 19106. Tel: 800-354-1420; Fax: 215-625-2940; Web site: http://www.tandf.co.uk/journals
Peer Reviewed: Y
Page Count: 17
Publication Date: 2012
Document Type: Journal Articles
Reports - Research
Education Level: High Schools
Higher Education
Postsecondary Education
Secondary Education
Descriptors: Secondary School Mathematics, Difficulty Level, College Preparation, College Mathematics, Mathematics Achievement, College Students, Research Universities, Course Selection (Students), STEM Education, Grades (Scholastic)
DOI: 10.1080/00220973.2011.567311
ISSN: 0022-0973
Abstract: The purpose of this study was to examine the college mathematics achievement and course-taking of students at a large public research university who completed a commercially developed or standards-based (Core-Plus) high school mathematics curriculum, and who subsequently completed at least 2 college mathematics courses of difficulty level at or beyond precalculus mathematics. Mathematics course-taking and achievement data across 8 college semesters were analyzed for a sample of 1,588 students. Findings indicated that students (including science, technology, engineering, and mathematics majors) were equally prepared for intense college mathematics coursework regardless of which high school mathematics curriculum they completed. These findings inform high school mathematics curriculum adoption decisions for college-bound students, and college policies and practices for advising students enrolling in mathematics courses. (Contains 2 tables.)
Abstractor: As Provided
Number of References: 65
EIS Cited: ED565641
Entry Date: 2012
Accession Number: EJ949929
Database: ERIC
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  Value: <anid>AN0069659830;jxe01jan.12;2019Mar11.13:15;v2.2.500</anid> <title id="AN0069659830-1">Preparation of Students Completing a Core-Plus or Commercially Developed High School Mathematics Curriculum for Intense College Mathematics Coursework. </title> <p>The purpose of this study was to examine the college mathematics achievement and course-taking of students at a large public research university who completed a commercially developed or standards-based (Core-Plus) high school mathematics curriculum, and who subsequently completed at least 2 college mathematics courses of difficulty level at or beyond precalculus mathematics. Mathematics course-taking and achievement data across 8 college semesters were analyzed for a sample of 1,588 students. Findings indicated that students (including science, technology, engineering, and mathematics majors) were equally prepared for intense college mathematics coursework regardless of which high school mathematics curriculum they completed. These findings inform high school mathematics curriculum adoption decisions for college-bound students, and college policies and practices for advising students enrolling in mathematics courses.</p> <p>Keywords: college mathematics; curriculum; high school; longitudinal</p> <p>THERE IS WIDESPREAD AGREEMENT of the importance of documenting the ability of high school mathematics curricula to prepare college-bound students in the United States for college mathematics (Augustine, [<reflink idref="bib7" id="ref1">7</reflink>]; Mathematical Sciences Education Board, [<reflink idref="bib36" id="ref2">36</reflink>]; National Center for Education Statistics [NCES], [<reflink idref="bib39" id="ref3">39</reflink>]; National Research Council, [<reflink idref="bib43" id="ref4">43</reflink>]) for at least two reasons. First, there is substantial evidence that the mathematics preparation of many college-bound high school students is inadequate. Perhaps the most compelling evidence of inadequate preparation is that approximately one quarter of all college freshmen enroll in at least one developmental mathematics class, typically a non–credit-bearing class that should have been completed in high school (NCES, [<reflink idref="bib38" id="ref5">38</reflink>]). This percentage would be even higher except that many institutions do not allow students to take a developmental class or discourage them from doing so (Conley, [<reflink idref="bib16" id="ref6">16</reflink>]).</p> <p>Second, questions about the preparation of students for college mathematics coincide with growing concern over the inability of the United States to produce sufficient numbers of students majoring in science, technology, engineering, and mathematics (STEM) to meet the country's technological, academic, and intellectual needs. The United States currently ranks 17th among developed nations in the proportion of college students receiving degrees in science or engineering, a fall from third place three decades ago, and ranks 26th in the proportion receiving undergraduate degrees in mathematics. Recent estimates are that the United States will soon graduate 200,000 fewer STEM majors per year than are needed (Committee on Prospering in the Global Economy of the 21st Century, [<reflink idref="bib14" id="ref7">14</reflink>]; Congressional Research Service, [<reflink idref="bib15" id="ref8">15</reflink>]). Because many STEM and STEM-related disciplines require substantial mathematics coursework, college-bound students interested in these majors need to be adequately prepared for college mathematics when they graduate from high school.</p> <p>Empirical studies involving high school mathematics curricula typically include traditional curricula as well as curricula developed with support from the National Science Foundation (NSF) in the late 1990s. However, the introduction of NSF-funded curricula was quickly followed by harsh criticism of these curricula (Schoenfeld, [<reflink idref="bib54" id="ref9">54</reflink>]). A precise measure of the effect of the criticism is unavailable, but there is anecdotal evidence that the criticism has reduced the number of school districts using these curricula in many states (Post, Monson, Dupuis, Medhanie, & LeBeau, [<reflink idref="bib48" id="ref10">48</reflink>]).</p> <p>Criticism of NSF-funded curricula has several facets but a common thread is that the goal of these curricula, to teach mathematics to all students, fails to adequately support the learning of college-bound students. Specific criticisms of these curricula are that they lack mathematical rigor and encourage misconceptions by the careless manner in which logical arguments are treated, rely too much on realistic problems, and use imprecise language (Wu, [<reflink idref="bib64" id="ref11">64</reflink>]; see also the Mathematically Correct website). Critics have also claimed that the use of group work encouraged by these curricula compromises the learning of more proficient students (Wu, [<reflink idref="bib64" id="ref12">64</reflink>]). These characteristics are argued to lead to poor mathematics performance with potentially dire consequences, such as poor performance on the mathematics portions of standardized tests taken by college-bound students and an increase in the likelihood that college-bound students will not be adequately prepared for rigorous college mathematics coursework such as Calculus I and II (Hill & Parker, [<reflink idref="bib26" id="ref13">26</reflink>]; Klein, [<reflink idref="bib31" id="ref14">31</reflink>], [<reflink idref="bib32" id="ref15">32</reflink>]). Schoenfeld ([<reflink idref="bib54" id="ref16">54</reflink>]) provided additional discussion of these criticisms.</p> <p>The present study examined the preparation of students who completed a traditional (commercially developed) or NSF-funded high school mathematics curriculum for intense college mathematics coursework, including those who completed a STEM major. We build on the definition of academic/curricular intensity used by Adelman ([<reflink idref="bib3" id="ref17">3</reflink>], [<reflink idref="bib4" id="ref18">4</reflink>]), in which intense high school mathematics coursework was characterized as completing an average of almost four Carnegie units (one unit is typically defined to be the content equivalent of an academic year course taught 1 hr per day; Boyer, [<reflink idref="bib9" id="ref19">9</reflink>]), including trigonometry, precalculus mathematics (college algebra, finite mathematics, and coursework at a similar level), or calculus. Adelman's ([<reflink idref="bib3" id="ref20">3</reflink>], [<reflink idref="bib4" id="ref21">4</reflink>]) index combined indicators of high school mathematics with information from other domains such as science and English. Students who completed more (and more advanced) coursework in high school showed greater academic intensity in mathematics and were more likely to earn a college degree.</p> <p>Academic/curricular intensity has been defined in several research contexts as the completion of advanced coursework. For example, Attewell and Domina ([<reflink idref="bib5" id="ref22">5</reflink>]) studied inequality in access to an advanced curriculum in high school and its effect on test scores and college entry, and defined an intensive curriculum as one in which a high school student takes many advanced courses in different content areas. Goldrick-Rab ([<reflink idref="bib22" id="ref23">22</reflink>]) cited Adelman's ([<reflink idref="bib3" id="ref24">3</reflink>]) work in defining academic/curricular intensity as the rigor of a high school student's curriculum across several areas of study as reflected in the difficulty and quality of courses. Other researchers have defined academic/curricular intensity in a similar way (e.g., Attewell, Lavin, Domina, & Levey, [<reflink idref="bib6" id="ref25">6</reflink>]; Doyle, [<reflink idref="bib18" id="ref26">18</reflink>]; Hargrove, Grodin, & Dodd, 2008). Similarly, we defined academically intense college mathematics coursework as having completed a minimum of two college mathematics courses of difficulty level at or beyond precalculus mathematics.</p> <hd id="AN0069659830-2">High School Mathematics Curricula</hd> <p></p> <hd id="AN0069659830-3">Commercially developed curricula</hd> <p>There are three general categories of high school mathematics curricula in the United States. The most widely used are commercially developed curricula, with approximately 75% of U.S. students completing one of these curricula in high school (Education Market Research, [<reflink idref="bib19" id="ref27">19</reflink>]). General characteristics of these curricula include an emphasis on algebra as the fundamental underpinning of high school mathematics, attention to algorithms and repetition, and an expectation that learning takes place in relatively small increments and is heavily dependent on the actions of the teacher (Roitman, [<reflink idref="bib51" id="ref28">51</reflink>]; Schoenfeld, [<reflink idref="bib54" id="ref29">54</reflink>]). Examples of publishers of commercially developed curricula used for the sample of students in our study include Houghton-Mifflin, Glencoe-McGraw Hill, and Saxon. We did not distinguish among the various publishers of commercially developed curricula in our analyses.</p> <hd id="AN0069659830-4">NSF-funded curricula</hd> <p>NSF-funded curricula were aligned with the National Council of Teachers of Mathematics' (NCTM, [<reflink idref="bib41" id="ref30">41</reflink>]) Curriculum and Evaluation Standards for School Mathematics. The NCTM Standards and the follow-up Principles and Standards for School Mathematics published in 2000 (NCTM, [<reflink idref="bib42" id="ref31">42</reflink>]) reflect a framework for mathematics learning that includes students, schools, curricula, assessment, technology, parents, and communities.</p> <p>NSF-funded curricula reflect a significant expansion of the scope of the mathematical content considered each year, along with dramatic changes in the conceptions of school mathematical content from a fixed set of facts and procedures to a dynamic body of knowledge that is continually enriched through conjecture, exploration, analysis, and proof (Smith, [<reflink idref="bib58" id="ref32">58</reflink>]). In addition, these curricula are associated with substantial changes in models of teaching and learning along with a reconceptualized role for the student.</p> <p>Advocates of NSF-funded curricula claim that the potential benefit is deeper and broader student understanding compared with commercially developed curricula (Riordan & Noyce, [<reflink idref="bib50" id="ref33">50</reflink>]). These benefits result from orienting these curricula around (a) problems that actually occur in, or are derived from, real-world experiences, often involving multiple strategies and content domains in their solution, which can enhance student engagement; (b) problems whose solution requires more than a few minutes and more than simply applying previously learned procedures and thus encourages student explanation and investigation; (c) providing opportunities for group discussion and problem solving, thus emphasizing the role of students as active participants in the learning process; (d) using various forms of technology where appropriate. These curricula are also consistent with the cognitive perspective on the nature of human learning (Post et al., [<reflink idref="bib47" id="ref34">47</reflink>]).</p> <p>The present study focuses on one of the NSF-funded curricula: contemporary mathematics in context (or Core-Plus; Coxford et al., [<reflink idref="bib17" id="ref35">17</reflink>]). The Core-Plus curriculum focuses on algebra, geometry, probability, statistics and topics in discrete mathematics every school year in a way that emphasizes problem solving and deemphasizes algorithmic manipulation. As a result, this curriculum represents a significant departure from the traditional Algebra 1, geometry, Algebra 2, precalculus mathematics sequence of high school mathematics.</p> <hd id="AN0069659830-5">High School Mathematics Curricula and Student Preparation for College Mathematics</hd> <p>It is surprising that literature documenting the ability of high school mathematics curricula to prepare students for college mathematics is sparse and somewhat contradictory. For example, Schoen and Hirsch ([<reflink idref="bib52" id="ref36">52</reflink>]) reported that students who completed a Core-Plus curriculum had higher overall grade point averages in a Calculus I course in college and in subsequent mathematics courses compared with students who completed a commercially developed curriculum. However, Harwell et al. ([<reflink idref="bib25" id="ref37">25</reflink>]) and Post et al. ([<reflink idref="bib47" id="ref38">47</reflink>]) found no relation between high school mathematics curricula (including Core-Plus) and college mathematics course grades.</p> <p>Findings of the effect of high school mathematics curricula on the difficulty level of the college mathematics courses a student completes have also varied. Hill and Parker ([<reflink idref="bib26" id="ref39">26</reflink>]) reported that students completing Core-Plus in high school initially enrolled in less difficult college mathematics courses, and subsequently completed even less difficult college mathematics courses, than students who did not complete Core-Plus. A key deficiency of this study was that the high school mathematics curriculum completed by many students was unknown. Harwell et al. ([<reflink idref="bib25" id="ref40">25</reflink>]) found a relation between a student's high school mathematics curriculum and the difficulty of a students' first college mathematics course in which students completing an NSF-funded curriculum were more likely to take a less difficult course initially compared with those who completed a commercially developed curriculum. However, Post et al. ([<reflink idref="bib47" id="ref41">47</reflink>]) reported that high school curriculum was unrelated to the pattern of course difficulties over multiple college semesters.</p> <p>Other benefits of completing an NSF-funded curriculum have also been documented albeit for middle school and high school students. Webb ([<reflink idref="bib63" id="ref42">63</reflink>]) found that students studying from an NSF-funded curriculum completed more mathematics courses than students studying from a commercially developed curriculum. The results of Cichon and Ellis ([<reflink idref="bib11" id="ref43">11</reflink>]) indicated that about two thirds of the students completing an NSF-funded curriculum in an inner-city school took additional mathematics courses as high school seniors compared with students not completing one of these curricula. There is also evidence that NSF-funded curricula promote more positive student attitudes towards mathematics (Cichon & Ellis, [<reflink idref="bib11" id="ref44">11</reflink>]; Schoen & Pritchett, [<reflink idref="bib53" id="ref45">53</reflink>]).</p> <p>In sum, there is mixed evidence of the preparation of students completing an NSF-funded curriculum in high school for college mathematics coursework. Moreover, no studies have examined the effect of these curricula on student preparation for intense college mathematics coursework.</p> <hd id="AN0069659830-6">Research Questions</hd> <p>Our research questions focused on the relation between high school mathematics curricula (commercially developed, Core-Plus) and course grades and difficulty levels for students completing two or more college mathematics courses whose difficulty level equaled or exceeded that of precalculus mathematics courses:</p> <p>What is the nature and magnitude of the relation between the high school mathematics curriculum a student completed (commercially developed, Core-Plus) and the (a) difficulty level and grade of a student's first college mathematics course, (b) pattern of mathematics grades earned over multiple college semesters, and (c) pattern of difficulty levels of mathematics courses completed over multiple college semesters, taking into account a student's STEM status and student background factors?</p> <p>We defined academically intense college mathematics coursework as having completed a minimum of two college mathematics courses of difficulty level at or beyond precalculus mathematics.</p> <hd id="AN0069659830-7">METHOD</hd> <p></p> <hd id="AN0069659830-8">Research Design</hd> <p>A retrospective cohort (quasi-experimental) design was used in which college mathematics course grades and difficulty levels were examined longitudinally. Based on their high school mathematics curriculum students were categorized into one of two curriculum cohorts: commercially developed or Core-Plus. Because of small numbers students who completed other NSF-funded curricula such as the Interactive Mathematics Program (Fendel, Resek, Alper, & Fraser, [<reflink idref="bib21" id="ref46">21</reflink>]) were omitted from our sample. Mathematics course grades and difficulty level data were collected for five or seven semesters depending on whether a student began college in Fall 2003 or Fall 2002, respectively.</p> <hd id="AN0069659830-9">Population and Sample</hd> <p>The sampled population consisted of college students in a Midwestern state in the United States who graduated from a public high school, completed at least three years of high school mathematics using a commercially developed or Core-Plus curriculum, and completed a minimum of two college mathematics courses of difficulty level at or beyond precalculus mathematics. The requirement that students had to have completed at least three years of high school mathematics is consistent with the material tested on the American College Testing (ACT)/Scholastic Assessment Test (SAT) (ACT, [<reflink idref="bib2" id="ref47">2</reflink>]; College Board, [<reflink idref="bib13" id="ref48">13</reflink>]).</p> <p>The resulting sample of 1,588 students represented approximately 29% of the total number of students enrolled at a large public university in the upper Midwest. The sampled students completed 2–14 college mathematics courses of difficulty level at or beyond that associated with precalculus mathematics, with 45 (21.5%) of the 209 high schools represented in the sample offering both curricula (commercially developed, Core-Plus).</p> <p>Our sample was not random in the classic sense (Cochran, [<reflink idref="bib12" id="ref49">12</reflink>]). However, following the purposive sampling arguments of Shadish, Cook, and Campbell ([<reflink idref="bib56" id="ref50">56</reflink>], pp. 354–372), the sample allowed us to generalize our results at least approximately to students attending other large public research institutions.</p> <hd id="AN0069659830-10">Data Collection</hd> <p>Our data were obtained from college records and reflected a number of variables at the high school and college levels.</p> <p>The most important independent variable was the high school mathematics curriculum a student completed (commercially developed, Core-Plus). Students were assigned to a curriculum cohort (commercially developed, Core-Plus) after carefully collecting information on the titles of the high school mathematics courses a student had completed and contacting high schools via phone calls, emails, and/or in person to identify the mathematics textbooks used when the sampled students were in attendance there. Details of this process appear in Harwell et al. ([<reflink idref="bib25" id="ref51">25</reflink>]).</p> <p>Another important independent variable was a student's college major. This information was needed because examining student mathematics course-taking required that we take into account the mathematics requirements of each campus admitting unit represented in our sample. An examination of the mathematics requirements of the majors in our student sample led us to generate a trichotomous variable to represent college major: students who completed a STEM major (e.g., biology, engineering; 47%), those whose majors were in humanities (e.g., political science, education; 32%), and other (e.g., trades, military, culinary; 21%).</p> <p>Other independent variables were as follows: ACT mathematics score, high school mathematics grade point average (GPA), number of years of high school mathematics (<reflink idref="bib3" id="ref52">3</reflink>, 4, 5), sex, ethnicity (African American, Hispanic, Asian, White), and the date a student enrolled in college (Fall 2002 or Fall 2003). Fall 2002 was chosen as the first term to study college mathematics performance because it allowed for significant numbers of students to be sampled who had completed at least three years of the Core-Plus curriculum. Sampling students who enrolled in Fall 2003 essentially doubled our sample size, and there was no evidence in our analyses that these students differed systematically from those enrolling in Fall 2002. Students who enrolled in Fall 2002 provided seven semesters of data and those who enrolled in Fall 2003 provided five semesters of data.</p> <p>One dependent variable reflected student achievement captured through mathematics course grades. This dependent variable was examined to explore the possibility that the cohorts tended to achieve at different levels in their college mathematics courses. The grade scale was A (scale value of 4.0), A– (3.70), B+ (3.30), B (3.00), B– (2.70), C+ (2.30), C (2.0), D (1.0), and F (0.0) and was treated as showing an interval scale of measurement.</p> <p>We also created a variable reflecting the difficulty level of the college mathematics courses students completed. In some analyses, difficulty level served as a dependent variable; in others, it served as a covariate. This variable was constructed after examining mathematics course descriptions and produced a Likert-type scale:</p> <p></p> <ulist> <item> Level 1: This level includes courses that should have been completed at the high school level (i.e., developmental).</item> <p></p> <item> Level 2: This level includes courses that would be considered precalculus mathematics and include college algebra, finite mathematics, and coursework at a similar level.</item> <p></p> <item> Level 3: This would be the typical entry level for mathematically well-prepared high school students, who would start their college mathematics coursework with beginning calculus.</item> <p></p> <item> Level 4: Beyond Calculus I. This would be the typical entry level for very well-prepared high school students who have successfully completed a high school AP Calculus course, normally with a score of 4 or 5.</item> </ulist> <p>We examined college mathematics achievement (grades) and course-taking (difficulty level) using longitudinal data. Only students completing college mathematics courses of difficulty level 2 or higher were included in the analyses.</p> <hd id="AN0069659830-11">Data Analyses</hd> <p>We used descriptive analyses to describe patterns in the data and two-level multilevel modeling (Raudenbush & Bryk, [<reflink idref="bib49" id="ref53">49</reflink>]) with longitudinal grade and difficulty level data to address the research questions. We began by reformulating the semester variable (1–5 or 1–7) to mathematics courses completed (first mathematics course completed, second mathematics course completed, and so on) to facilitate interpretation. Following the advice of Biesanz, Deeb-Sossa, Papadakis, Bollen, and Curran ([<reflink idref="bib8" id="ref54">8</reflink>]), we recoded these values so that the first mathematics course completed was coded 0, the second course completed was coded 1, and so on, which allowed the grade and difficulty level of a student's first college mathematics course to be studied in the intercept model in the multilevel analyses.</p> <p>Few students completed more than seven mathematics courses (1.7%), and those beyond the seventh were excluded. This was done because larger values of number of mathematics courses completed (e.g., 12), although sparse in our data, would disproportionately influence the findings (Neter, Kutner, Nachtsheim, & Wasserman, [<reflink idref="bib45" id="ref55">45</reflink>]).</p> <p>For the multilevel analyses of grade data we fitted a within-student model of the form:</p> <p>Graph</p> <p>where is the mathematics grade of the ith (i = 1, 2, ...,N) student in the pth (p = 1,2,...,P) college mathematics course completed, is a covariate indicating the mathematics course completed (e.g., first, second), is a covariate reflecting the difficulty level of a mathematics course that served as a control variable, is the slope for the ith student reflecting the linear trajectory of grades over mathematics courses completed adjusted for difficulty level, is the slope for the ith student reflecting the effect of difficulty level on grades over mathematics courses completed, and is within student error.</p> <p>The , , and varied significantly across students for the grade data and between-student models were constructed to predict variation in these parameters:</p> <p>Graph</p> <p>Graph</p> <p>Graph</p> <p>where is a slope capturing the effect of the covariate on intercepts, is an error term, and is a slope capturing the effect of on growth trajectories (slopes). A generalized hierarchical proportional odds model for ordinal data was fitted to difficulty level in the same way as the grade data except that the within-student model did not contain the difficulty level covariate. Grand mean centering was used for between-student covariates.</p> <p>The covariates in Equations (<reflink idref="bib2" id="ref56">2</reflink>) through (<reflink idref="bib4" id="ref57">4</reflink>) were curriculum cohort (commercially developed, Core-Plus), college major (STEM, humanities, other), ACT mathematics score, high school mathematics GPA, years of high school mathematics (<reflink idref="bib3" id="ref58">3</reflink>, 4, 5), sex, ethnicity (African American, Asian American, Hispanic, White), year enrolled (Fall 2003, Fall 2002), and difficulty level (2–4). Thus the main analyses consisted of fitting the within- and between-student models in Equations (<reflink idref="bib1" id="ref59">1</reflink>) through (<reflink idref="bib4" id="ref60">4</reflink>) separately to the grade- and difficulty-level dependent variables. Multilevel analyses were performed with the R 2.10.1 lme4 package (The R Foundation for Statistical Computing, [<reflink idref="bib61" id="ref61">61</reflink>]).</p> <p>To check the role of high school characteristics, we fitted preliminary two level models to grade and difficulty level data that included, in the between-student model, the percentage of students eligible for a free or reduced-price lunch at a student's high school and the percentage of non-White students at a student's high school. None of these covariates was a significant predictor of college mathematics grades or course-taking and high school level variables were not subsequently considered. To control for compounding of Type I error rates in the multilevel analyses, we used an adjusted Type I error rate attributed to Sidak ([<reflink idref="bib57" id="ref62">57</reflink>]). The adjusted error rate was computed as α′=1−(1−α)<sups>1/k</sups>, where <emph>α</emph> is the unadjusted Type I error rate, which was set equal to.10, k = number of statistical tests, and <emph>α</emph>′ is the adjusted Type I error rate. Unadjusted error rates above the traditional value of.05, such as.10, are frequently recommended as part of a familywise error rate control strategy (Keselman, Cribbie, & Holland, [<reflink idref="bib30" id="ref63">30</reflink>]).</p> <hd id="AN0069659830-12">RESULTS</hd> <p></p> <hd id="AN0069659830-13">Descriptive Analyses</hd> <p>Among students in our sample 87% completed a commercially developed curriculum, and for high school outcomes these students tended to outperform students in the Core-Plus cohort. For example, scores on the ACT mathematics test for the commercially developed cohort were on average significantly higher (<emph>p</emph> <.05) compared with the Core-Plus cohort (<emph>M</emph> = 26.04, <emph>SD</emph> = 5.1 and <emph>M</emph> = 24.70, <emph>SD</emph> = 4.9, respectively). This pattern is consistent with results reported in Harwell et al. ([<reflink idref="bib25" id="ref64">25</reflink>]) in which students completing a commercially developed curriculum in high school scored on average about one third of a standard deviation higher on a state-mandated mathematics test administered in eighth grade than students completing NSF-funded curricula.</p> <p>For college mathematics outcomes, differences between the cohorts tended to be small or statistically equal to zero. For example, the percentages of students completing different numbers of college mathematics courses showed that they were on the whole fairly equally distributed across the commercially developed and Core-Plus cohorts. For two courses completed, the percentages for the commercially developed and Core-Plus cohorts were 41.8% and 44.1%, respectively; for three courses, the percentages were 29.5% and 25.2%, respectively; for four courses, these percentages were 20% and 21%, respectively. Among the commercially developed and Core-Plus cohorts the percentages of students completing precalculus mathematics were 59.1% and 68.6%, respectively, and for Calculus I, these percentages were 43.1% and 57.5%, respectively. The cohorts on average completed similar numbers of college mathematics courses regardless of years of high school mathematics completed (<emph>p</emph> >.05 for all cohort comparisons).</p> <p>The sample was predominately male (63.8%) and White (71.1%), with far fewer African American (6.7%) and Hispanic (3.3%) students. There was relatively little difference in college mathematics GPA between female (<emph>M</emph> = 2.59, <emph>SD</emph> = 0.92) and male (<emph>M</emph> = 2.51, <emph>SD</emph> = 0.94) students but much larger differences between African American (<emph>M</emph> = 2.11, <emph>SD</emph> = 0.98), Asian (<emph>M</emph> = 2.35, <emph>SD</emph> = 0.89), Hispanic (<emph>M</emph> = 2.36, <emph>SD</emph> = 0.87) and White (<emph>M</emph> = 2.64, <emph>SD</emph> =.93) students.</p> <p>Bivariate correlations between curriculum cohort and other variables showed that the largest value was for ACT mathematics scores (<emph>r</emph> = −.16) and difficulty level of a student's first college mathematics course (<emph>r</emph> = −.14). Among covariates the largest correlations were between ACT mathematics scores and (a) difficulty of a student's first college mathematics course (<emph>r</emph> =.70), (b) whether a student was a STEM major (<emph>r</emph> =.58), (c) high school mathematics GPA (<emph>r</emph> =.52), (d) sex (<emph>r</emph> = −.28). All of these correlations were based on a sample exceeding 1,500 and were statistically significant at <emph>p</emph> <.01.</p> <p>We also examined our data for evidence of the effect of missing values on covariates such as ACT mathematics score and high school mathematics GPA. Most covariates had small percentages of missing data (e.g., ACT mathematics scores, 3.6%) with the most missing data observed for the high school mathematics GPA variable (7.7%). Comparing students who did and did not provide information on our variables (e.g., ACT mathematics scores, college major) did not provide any evidence of differences.</p> <hd id="AN0069659830-14">Multilevel Analyses</hd> <p></p> <hd id="AN0069659830-15">Grades</hd> <p>The fixed effects results for grades are reported in Table 1. For the within-student models the linear grade trajectory for mathematics courses completed was −0.07, meaning that, on average, student mathematics grades decreased approximately 0.07 units on a 0–4 scale from one course to the next after statistically adjusting for the effect of difficulty level.</p> <p>TABLE 1 Multilevel Results for Grade Data for the Commercially Developed Curricula and Core-Plus Cohorts</p> <p> <ephtml> <table><thead valign="bottom"><tr><td>Within-student model</td></tr><tr><td>Effect</td><td><italic>B</italic></td><td><italic>SE B</italic></td></tr></thead><tbody><tr><td>Intercept</td><td char=".">2.78*</td><td char=".">.05</td></tr><tr><td>Math difficulty</td><td char=".">−0.32*</td><td char=".">.05</td></tr><tr><td>Linear</td><td char=".">−0.07*</td><td char=".">.03</td></tr><tr><td><italic>Between-student model</italic></td></tr><tr><td><italic>Effect</italic></td><td>B</td><td>SE B</td></tr><tr><td>Intercepts (<inline-graphic href="vjxe_a_567311_o_ilm0015.gif" />)</td><td /><td /></tr><tr><td> Years high school math</td><td char=".">0.14</td><td char=".">.07</td></tr><tr><td> American College Testing math</td><td char=".">0.04*</td><td char=".">.01</td></tr><tr><td> High school math grade point average</td><td char=".">0.60*</td><td char=".">.10</td></tr><tr><td> Gender</td><td char=".">0.08</td><td char=".">.10</td></tr><tr><td> African American</td><td char=".">0.42</td><td char=".">.36</td></tr><tr><td> Asian</td><td char=".">−0.05</td><td char=".">.12</td></tr><tr><td> Hispanic</td><td char=".">−0.02</td><td char=".">.37</td></tr><tr><td> STEM</td><td char=".">−0.06</td><td char=".">.19</td></tr><tr><td> Humanities</td><td char=".">0.00</td><td char=".">.19</td></tr><tr><td> Year a student enrolled in college</td><td char=".">0.03</td><td char=".">.09</td></tr><tr><td> Commercially developed curricula versus Core-Plus</td><td char=".">−0.03</td><td char=".">.12</td></tr><tr><td>Linear growth slopes (<inline-graphic href="vjxe_a_567311_o_ilm0016.gif" />)</td><td /><td /></tr><tr><td> Years high school math</td><td char=".">−0.03</td><td char=".">.03</td></tr><tr><td> American College Testing math</td><td char=".">0.01</td><td char=".">.01</td></tr><tr><td> High school math grade point average</td><td char=".">−0.02</td><td char=".">.05</td></tr><tr><td> Gender</td><td char=".">0.03</td><td char=".">.05</td></tr><tr><td> African American</td><td char=".">−0.16</td><td char=".">.21</td></tr><tr><td> Asian</td><td char=".">0.05</td><td char=".">.06</td></tr><tr><td> Hispanic</td><td char=".">0.15</td><td char=".">.18</td></tr><tr><td> STEM</td><td char=".">0.19</td><td char=".">.10</td></tr><tr><td> Humanities</td><td char=".">0.14</td><td char=".">.10</td></tr><tr><td> Year a student enrolled in college</td><td char=".">−0.03</td><td char=".">.04</td></tr><tr><td> Commercially developed curricula versus Core-Plus</td><td char=".">0.06</td><td char=".">.06</td></tr><tr><td>Difficulty slopes (<inline-graphic href="vjxe_a_567311_o_ilm0017.gif" />)</td><td /><td /></tr><tr><td> Years high school math</td><td char=".">−0.02</td><td char=".">.07</td></tr><tr><td> American College Testing math</td><td char=".">−0.01</td><td char=".">.01</td></tr><tr><td> High School math grade point average</td><td char=".">0.08</td><td char=".">.11</td></tr><tr><td> Gender</td><td char=".">0.11</td><td char=".">.10</td></tr><tr><td> African American</td><td char=".">−0.11</td><td char=".">.41</td></tr><tr><td> Asian</td><td char=".">−0.18</td><td char=".">.13</td></tr><tr><td> Hispanic</td><td char=".">−0.46</td><td char=".">.40</td></tr><tr><td> STEM</td><td char=".">−0.29</td><td char=".">.21</td></tr><tr><td> Humanities</td><td char=".">−0.24</td><td char=".">.20</td></tr><tr><td> Year a student enrolled in college</td><td char=".">−0.07</td><td char=".">.09</td></tr><tr><td> Commercially developed curricula versus Core-Plus</td><td char=".">−0.01</td><td char=".">.13</td></tr><tr><td><italic>Note</italic>. * = significant at α′=1−(1−α)<sup>1/m</sup> =.00142 where <italic>α</italic> =.10 is the unadjusted Type I error rate, m = number of statistical tests = 36, and <italic>α</italic>′ is the adjusted error rate. Years of high school mathematics completed: 3, 4, 5; gender: 1 = male, 0 = female; African American: 1 = yes, 0 = no; Asian: 1 = yes, 0 = no; Hispanic: 1 = yes, 0 = no so White students served as the reference group; STEM: 1 = yes, 0 = no, humanities: 1 = yes, 0 = no so other served as the reference group; year a student enrolled in college: 2002, 2003; commercially developed: 1 = yes, 0 = no.</td></tr><tr><td>*<italic>p</italic> <.009 (<italic>α</italic>′ =.0085).</td></tr></tbody></table> </ephtml> </p> <p>The most important finding in Table 1 is the absence of a relation between curriculum cohort and mathematics grade trajectory or between curriculum cohort and mathematics course difficulty trajectory represented by difficulty level slopes. That is, the linear pattern of college mathematics grades and the difficulty level of these courses were the same for the commercially developed and Core-Plus cohorts. Curriculum cohort was also unrelated to a student's first college mathematics grade as seen in the intercept model. Significant effects for the intercept model showed that higher ACT mathematics scores (.04 on a 4-point scale) and high school mathematics GPAs (.60) were associated with higher grades earned in a student's first college mathematics course.</p> <p>Table 1 also shows that student-level covariates such as years of high school mathematics and whether a student was a STEM major were not significant predictors of students' linear growth slopes. In fact, none of the student-level covariates were related to the pattern of college mathematics course grades students earned.</p> <hd id="AN0069659830-16">Difficulty data</hd> <p>A generalized hierarchical proportional odds model for ordinal data was fitted to difficulty level (2 = precalculus mathematics; 3 = Calculus I; 4 = Calculus II/beyond Calculus II). The models fitted to the difficulty data were the same as those fitted to grade data, except that difficulty level was not a covariate in the within-student model.</p> <p>The results for difficulty level are displayed in Table 2. For the within-student model, the linear slope difficulty trajectory (expressed in logits) for mathematics courses completed was 1.87. Exponentiating this value (Raudenbush & Bryk, [<reflink idref="bib49" id="ref65">49</reflink>]) such that exp(1.87) = 6.5 means that students are, as would be expected, more likely (6.5 times) to enroll in a more difficult course relative to a less difficult course as they progress.</p> <p>TABLE 2 Multilevel Results for Difficulty Levels for Commercially Developed Curricula and Core-Plus Cohorts</p> <p> <ephtml> <table><thead valign="bottom"><tr><td>Within-student model</td></tr><tr><td>Effect</td><td><italic>B</italic></td><td><italic>SE B</italic></td></tr></thead><tbody><tr><td>Intercept 2</td><td char=".">−0.23*</td><td char=".">.08</td></tr><tr><td>Intercept 3</td><td char=".">−2.28*</td><td char=".">.11</td></tr><tr><td>Linear</td><td char=".">1.87*</td><td char=".">.07</td></tr><tr><td><italic>Between-student model</italic></td></tr><tr><td><italic>Effect</italic></td><td>B</td><td>SE B</td></tr><tr><td>Intercepts (<inline-graphic href="vjxe_a_567311_o_ilm0018.gif" />)</td><td /><td /></tr><tr><td> Years high school math</td><td char=".">0.29*</td><td char=".">.12</td></tr><tr><td> American College Testing math</td><td char=".">0.35*</td><td char=".">.03</td></tr><tr><td> High school math grade point average</td><td char=".">0.60*</td><td char=".">.22</td></tr><tr><td> Gender</td><td char=".">−0.38</td><td char=".">.17</td></tr><tr><td> African American</td><td char=".">0.31</td><td char=".">.50</td></tr><tr><td> Asian</td><td char=".">0.46</td><td char=".">.24</td></tr><tr><td> Hispanic</td><td char=".">0.74</td><td char=".">.48</td></tr><tr><td> STEM</td><td char=".">0.62</td><td char=".">.40</td></tr><tr><td> Humanities</td><td char=".">−0.55</td><td char=".">.43</td></tr><tr><td> Year a student enrolled in college</td><td char=".">0.09</td><td char=".">.14</td></tr><tr><td> Commercially developed versus Core-Plus</td><td char=".">−0.28</td><td char=".">.19</td></tr><tr><td>Linear growth slopes (<inline-graphic href="vjxe_a_567311_o_ilm0019.gif" />)</td><td /><td /></tr><tr><td> Years high school math</td><td char=".">0.10</td><td char=".">.09</td></tr><tr><td> American College Testing math</td><td char=".">−0.09*</td><td char=".">.02</td></tr><tr><td> High school math grade point average</td><td char=".">0.18</td><td char=".">.14</td></tr><tr><td> Gender</td><td char=".">0.49*</td><td char=".">.15</td></tr><tr><td> African American</td><td char=".">0.25</td><td char=".">.36</td></tr><tr><td> Asian</td><td char=".">−0.35</td><td char=".">.18</td></tr><tr><td> Hispanic</td><td char=".">−0.64</td><td char=".">.44</td></tr><tr><td> STEM</td><td char=".">0.64</td><td char=".">.28</td></tr><tr><td> Humanities</td><td char=".">0.21</td><td char=".">.26</td></tr><tr><td> Year a student enrolled in college</td><td char=".">−0.20</td><td char=".">.12</td></tr><tr><td> Commercially developed versus Core-Plus</td><td char=".">0.12</td><td char=".">.19</td></tr><tr><td><italic>Note</italic>. * = significant at α′=1−(1−α)<sup>1/m</sup> =.00421 where <italic>α</italic> =.10 is the unadjusted Type I error rate, m = number of statistical tests = 25, and <italic>α</italic>′ is the adjusted error rate. Years high school math: 3, 4, 5; gender: 1 = male, 0 = female; African American: 1 = yes, 0 = no; Asian: 1 = yes, 0 = no; Hispanic: 1 = yes, 0 = no so White students served as the reference group; STEM: 1 = yes, 0 = no, humanities: 1 = yes, 0 = no so other served as the reference group; year a student enrolled in college: 2002, 2003; commercially developed: 1 = yes, 0 = no.</td></tr><tr><td>*<italic>p</italic> <.009 (<italic>α</italic>′ =.0085).</td></tr></tbody></table> </ephtml> </p> <p>The results in Table 2 indicate that there was no relation between the high school mathematics curricula a student completed and the pattern of difficulty levels of the college mathematics courses they completed as captured with a linear slope. That is, the linear patterns of difficulty levels of college mathematics courses were the same for the commercially developed and Core-Plus cohorts. Also, curriculum cohort and whether a student was a STEM major were unrelated to the difficulty level of their first college mathematics course as seen in the intercept model. Among the significant effects for the intercept model the largest were for high school mathematics GPA and ACT mathematics scores. For example, Table 2 shows that the odds of enrolling in a more difficult course relative to the odds of enrolling in a less difficult course was exp(0.35) = 1.42 for every one unit increase in ACT mathematics scores.</p> <p>For the linear growth slope model the significant sex effect of exp(.49) = 1.63 means that male students were 1.63 times more likely to enroll in a more difficult course relative to a less difficult course than were female students. An explanation of this pattern is difficult as the literature on sex and college mathematics performance is mixed (e.g., see Bridgman & Lewis, [<reflink idref="bib10" id="ref66">10</reflink>]; Hyde, Fennema, & Lamon, [<reflink idref="bib28" id="ref67">28</reflink>]; Hyde, Lindberg, Linn, Ellis, & Williams, [<reflink idref="bib29" id="ref68">29</reflink>]).</p> <p>In quasi-experimental designs like the one we used, it is important to probe the findings for possible alternative explanations. Descriptive results presented earlier showed that the commercially developed cohort had on average higher ACT mathematics scores compared with the Core-Plus cohort. Although the effect of ACT mathematics scores was controlled for statistically through its inclusion in the multilevel analyses, we further explored its effect on the relation between curriculum cohort and college mathematics outcomes for additional insight. In particular, we polytimized ACT mathematics scores into three strata using cutoffs associated with being minimally college ready in mathematics (≤22), ready for mathematics at the precalculus mathematics level (≥23 and ≤27), or possessing skills consistent with Calculus I or a more difficult course (≥28; ACT, [<reflink idref="bib1" id="ref69">1</reflink>]). The construction of strata is consistent with the "elimination" (pp. 213–214) strategy for data analysis suggested by Pedhazur and Schmelkin (1991) and helps to minimize the effect of preexisting differences in mathematics achievement. We then refitted our multilevel models (except for the ACT mathematics score variable) separately to the longitudinal grade and difficulty data in each stratum. The overall pattern of findings was unchanged, providing additional evidence that our findings are not sensitive to the mathematics achievement of students before their entering college.</p> <hd id="AN0069659830-17">DISCUSSION</hd> <p>Our main conclusion is that students completing a commercially developed or Core-Plus high school mathematics curriculum were equally prepared for intense college mathematics coursework, a finding that was not affected by whether students completed a STEM major in college. Students who completed the commercially developed or Core-Plus curricula, on average, completed a first college mathematics course of similar difficulty and earned similar grades in this course. These students continued to complete mathematics courses of similar difficulty and earn similar grades across multiple college semesters.</p> <p>That two of the student-level covariates (ACT mathematics scores, high school mathematics GPA) were significant predictors of students' first college mathematics grade but not their pattern of college mathematics grades suggests that their effect is relatively short-lived. A slightly different pattern emerged for the difficulty level data. One student-level covariate (ACT mathematics scores) was a significant predictor of the difficulty level of the first college mathematics course as well as the pattern of difficulty levels. However, the strongest effect appeared for male students, who, on average, took more difficult courses than did female students.</p> <p>The absence of a relation between curriculum cohort and course difficulty levels or grades earned helps to refute criticism that the Core-Plus curriculum does not adequately prepare college-bound students (including those with a STEM major) for intense college mathematics coursework. We believe that the smaller degree of congruence between the Core-Plus curriculum, college mathematics placement tests, and traditional college mathematics instructional practices is important and suggests that students completing this curriculum might have performed better had closer alignment existed.</p> <p>Our results also suggest that high schools should continue to make curriculum adoption decisions on the basis of factors believed to enhance mathematics learning and achievement such as available teaching expertise, state-mandated testing requirements, and resources, and not on differential preparation for college mathematics. Moreover, the findings suggest that college policies and practices related to advising students on enrollment in intense mathematics coursework should, other things being equal, not depend on whether a student completed a commercially developed or Core-Plus mathematics curriculum in high school.</p> <p>Despite the pattern of nonsignificant findings for curriculum cohort our view is that the lack of evidence favoring one curriculum does not suggest that it would be wise to stay the course and to use only commercially developed curricula. That most students have completed a commercially developed curriculum over the past four decades is likely related to the disappointing mathematics achievement of U.S. students during this time (NCES, [<reflink idref="bib37" id="ref70">37</reflink>]; National Commission on Excellence in Education, [<reflink idref="bib40" id="ref71">40</reflink>]; Stevenson & Stigler, [<reflink idref="bib59" id="ref72">59</reflink>]).</p> <p>We think that the Core-Plus curricula have two features that provide an attractive alternative to CD curricula. First, and most important, is that these reflect various cognitively oriented theories of the development of mathematics knowledge and the nature of student learning as we understand it today. We believe that this can promote greater mathematics learning for a larger and more diverse population of high school students. Second, these curricula are based on a variety of methodological and pedagogical approaches that are designed to initiate and sustain student interest including the seamless use of technology, expanded evaluation techniques, sophisticated use of real-world problem solving, and the use of cooperative approaches. However, it is important to emphasize that a compelling body of work specifically documenting the effect of these features is not currently available, although these variables are well established in the research literature embraced by the mathematics education community (English, [<reflink idref="bib20" id="ref73">20</reflink>]; Grouws, [<reflink idref="bib23" id="ref74">23</reflink>]; Lester, [<reflink idref="bib33" id="ref75">33</reflink>]).</p> <hd id="AN0069659830-18">Limitations</hd> <p>It is important to recognize the limitations inherent in research of this nature. An important limitation is that the retrospective cohort design limits the causal inferences that can be drawn. Another limitation is that we did not have the data to examine the possibility of a relation between high school mathematics curricula and high school track (e.g., precollege track). The presence of such a relation might help to explain the relation between high school mathematics curricula and college mathematics course-taking and performance.</p> <p>A third limitation is our inability to assess the role of the content and format of college mathematics placements tests in initial mathematics course enrollment or to explore the effect of academic counseling recommendations on the basis of placement test scores on course-taking patterns. We were also unable to formally assess the alignment of high school mathematics curricula and the content and structure of many college mathematics courses, although anecdotal evidence suggests that they are generally more consistent with commercially developed curricula. For example, a review of several Calculus I texts used in college courses (Himonas & Howard, 2003; Lial, Greenwell, & Ritchey, [<reflink idref="bib34" id="ref76">34</reflink>]; Stewart, 2007) provides evidence that these texts are consistent with the orientation and presentation of commercially developed mathematics curricula in high school. If true, this provides an additional advantage for commercially developed students in college mathematics. However, there is apparently no work studying the alignment of high school mathematics curricula and the content and structure of college mathematics texts and courses.</p> <p>Teacher quality is also frequently cited as an important factor to consider in studying mathematics achievement, but the literature is mixed on key contributors to this effect and their magnitude (Wayne & Youngs, [<reflink idref="bib62" id="ref77">62</reflink>]). The teacher quality effect for our sample was minimized somewhat by the mathematics department's policy of using common student assignments, examinations, and a single course-specific distribution of scores for assigning grades. It is unfortunate that we had no direct data capturing the effect of teacher quality on student achievement. These limitations suggest directions for future research that examines factors related to curricula choices at the high school and college levels.</p> <hd id="AN0069659830-19">AUTHOR NOTES</hd> <p> <bold>Michael R. Harwell</bold> received his Ph.D. in educational psychology (1983) from the University of Wisconsin Madison and is now a professor in the Department of Educational Psychology at the University of Minnesota. His research focuses on methodological issues in educational data analysis. <bold>Amanuel Medhanie</bold> is a graduate student in the Department of Educational Psychology at the University of Minnesota. His research interests include longitudinal data analysis and propensity scores in education. <bold>Thomas R. Post</bold> received his Ph.D. in mathematics education (1967) from Indiana University and is a professor in the Department of Curriculum and Instruction at the University of Minnesota. His research interest focus on middle grades mathematics and the teaching and learning of rational number concepts. <bold>Ke Norman</bold> is an assistant professor in the Department of Mathematical Sciences at the University of Montana. She received her Ph.D. in curriculum and instruction (2008) from the University of Minnesota. Her research focuses on the effect of high school mathematics curricula on college mathematics performance and the use of college mathematics placement tests. <bold>Danielle N. Dupuis</bold> is a graduate student in the Department of Educational Psychology at the University of Minnesota. Her research interests include educational data analysis and statistics education.</p> <hd id="AN0069659830-20">ACKNOWLEDGMENTS</hd> <p>This research was supported by the National Science Foundation under Small Grants Exploratory Research 0533460 and Education and Science Indicators 0627986. 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  Data: Preparation of Students Completing a Core-Plus or Commercially Developed High School Mathematics Curriculum for Intense College Mathematics Coursework
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  Data: <searchLink fieldCode="SO" term="%22Journal+of+Experimental+Education%22"><i>Journal of Experimental Education</i></searchLink>. 2012 80(1):96-112.
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  Data: Routledge. Available from: Taylor & Francis, Ltd. 325 Chestnut Street Suite 800, Philadelphia, PA 19106. Tel: 800-354-1420; Fax: 215-625-2940; Web site: http://www.tandf.co.uk/journals
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  Data: 10.1080/00220973.2011.567311
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  Data: The purpose of this study was to examine the college mathematics achievement and course-taking of students at a large public research university who completed a commercially developed or standards-based (Core-Plus) high school mathematics curriculum, and who subsequently completed at least 2 college mathematics courses of difficulty level at or beyond precalculus mathematics. Mathematics course-taking and achievement data across 8 college semesters were analyzed for a sample of 1,588 students. Findings indicated that students (including science, technology, engineering, and mathematics majors) were equally prepared for intense college mathematics coursework regardless of which high school mathematics curriculum they completed. These findings inform high school mathematics curriculum adoption decisions for college-bound students, and college policies and practices for advising students enrolling in mathematics courses. (Contains 2 tables.)
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        PageCount: 17
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      – SubjectFull: Secondary School Mathematics
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