High School Curricula and Collegiate Calculus: Comparative Essay

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Introduction

The transition from high school to college can be a turbulent time for many reasons: students moving out on their own, the state of being in a different environment for potentially the first time in many years, and a greater role in their lives coming with added responsibility all happen simultaneously at the start of a collegiate career. One common course for STEM majors to take in this stressful first semester is College Calculus, but there is a disagreement in education on how to prepare students for this course, to ease their burdens in this stressful time of their lives. Secondary instructors see high school calculus as an essential tool for preparing successful College Calculus students, while collegiate instructors want their students to have a firm grasp of the pre-requisite courses (Sadler & Sonnert, 2018).

Regardless of what side of this argument we align with currently, it is clear that change needs to be made. Major evidence of problems in STEM fields occur from students not being prepared for concepts, and one of the biggest hurdles is calculus. In fact, as students are not prepared for calculus, about 25% of students complete developmental courses when they arrive at college (Harwell, Post, Medhanie, Dupuis, & LeBeau, 2013; Post et al., 2010; Sonnert & Sadler, 2013), which is the equivalent of relearning material that was covered in high school, so this implies that some of the blame lies with high school curricula (Harwell 2013).

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Despite this understanding, surprisingly little is known about the impact that high school curricula have on collegiate coursework, let alone specifically on calculus (Harwell et al., 2013; LeBeau et al., 2012). It is hard to answer whether or not high school calculus has a positive effect without knowing how different calculus curricula impact education, so examining this leads us closer to addressing the necessity of high school calculus. To this end, research is needed on the different curricular types that influence high school mathematics and how they influence calculus success in multiple ways: grades, retention, and the ability to progress to higher mathematics. In order to determine the direction of future research, this paper seeks to characterize themes that emerge from a sample of the literature as well as to see what is missing from the literature.

Literature Review of Research and Practices

In order to gain some insight into the types of research that has been done in regard to curricular effects on collegiate calculus, the author began a literature search of materials that involved how college calculus was affected by different secondary curricula that prepared for the transition to calculus. An additional constraint was that these articles should be modern, which was interpreted as being published after the year 2000. An initial search done by the author for a previous literature review found too few papers that focused on the subject matter. To expand upon the number of studies, the author searched this time in databases like JSTOR and ERIC to find any other literature that would give more credence to the conclusions listed here. Three themes emerged in the literature found in the review, and these shall be separated into sections in which the author will go into detail about them: Effects of Curriculum, Effects of College, and Effects of Achievement.

Effects of Curriculum

Curricula play a vital role in the educational system. More than just a collection of textbooks, curricula represent both those materials, supplemental materials and supports, and the way that they are implemented in the classroom. A choice in curricula is both a choice in the content taught, how it is planned to be taught, and how it is actually taught. With such a pivotal role in education, one cannot immediately dismiss its effect on student educational outcomes. As such, it has become a key focus of much research, even going as far as the research of how studies on curricula are carried out (NRC 2004). As the primary focus of this literature review, the Effects of Curriculum were indeed a major reoccurring theme across these articles. However, the exact nature of these effects is not completely agreed upon in the literature.

On the one hand, there are many studies that conclude that there are no effects of curriculum on Calculus success (Harwell et al., 2009; Klopfenstein & Thomas, 2009; LeBeau et al., 2012; Harwell et al., 2013; Post et al., 2013). An overwhelming majority of these were studies conducted using archived data from colleges in the Midwest, as well as contacting high schools listed in the file in order to determine the curriculum implemented, and utilized final grades in collegiate calculus as the relevant outcome (Harwell et al., 2009; LeBeau et al., 2012; Harwell et al., 2013; Post et al., 2013). The remaining article examined one particular case of curriculum, AP, and used archived data from the Texas Schools Microdata Panel to determine that the AP curriculum had no effect on both grades and second-year retention in their college (Klopfenstein & Thomas, 2009).

On the other hand, there are a few studies that show an effect of some curricular programs on calculus success in some manner (Hill & Parker, 2006; Harwell, Medhanie, et al., 2013). These articles both feature archived data from schools in the Midwest as well, though it should be noted that the article by Hill and Parker (2006) was specifically stated to be conducted with data from Michigan State University while Harwell, Medhanie, et al. (2013) was more ambiguous (Hill & Parker, 2006; Harwell, Medhanie, et al., 2013). The striking difference between these articles and the ‘no effect’ articles is the difference in choice of what measures Calculus success. For the article of Hill and Parker (2006), this measure was the level of mathematics taken, which was found to decline for students enrolled in a Core-Plus curriculum. Harwell, Medhanie, et al. (2013) went with an alternative approach to success, instead of evaluating retention in mathematics courses past calculus. While there was no effect on curriculum from retention, the researchers observed that students in the NSF curriculum were more likely to begin college with a developmental math course, a prerequisite to calculus, than their peers in commercially developed or Chicago Math curricula.

One other article addressed the effects of curriculum, but in a broader sense than the previous articles, tying back to the question of whether calculus is good rather than just looking at curriculum. Sadler and Sonnert (2018) addressed the difference in students nationally who took high school calculus versus those who just had mastery of the prerequisite courses, captured by grades in these courses as well as SAT scores. Their results concluded that having good outcomes in the prerequisites had more than double the effect on college calculus performance than those who took calculus (Sadler & Sonnert, 2018). However, this same study found that students with lower achievement in the prerequisites benefited more from being placed into high school calculus (Sadler & Sonnert, 2018).

There was one article that did not fit into the above categories as neatly as the articles already mentioned yet indicated some subtle effects from the curriculum implemented that should be kept in mind. Teuscher and Reys (2012) examined how high school students in single-subject versus integrated calculus curricula did with both multiple-choice and open-ended tasks involving related rates. In this study, there was generally little difference in the low scores received by both groups on problems in standardized tests involving related rates, but there was a significant increase in performance on tasks involving piecewise functions and rates of change for the single-subject curricular cohorts (Teuscher & Reys, 2012).

Effects of College

Although the focus of this search was the effects of high school curriculum on collegiate calculus outcomes, the effects of college were also a reoccurring theme in the literature examined. This should be a more direct effect on the classes as this is connected to the same institution, but the amount of effect that this has when compared to the high school curriculum is unclear without research. The articles included in this section examined many different aspects of college effects, including college curriculum, college placement tests for math courses, and even the characteristics of the college.

Sahin, Cavlazoglu, and Zeytuncu (2015) examined the effects of collegiate curriculum on college calculus success, in the specific case of a flipped classroom case study. It was found that student quiz scores increased significantly, and students preferred the format of the flipped classroom (Sahin et al., 2015). Sonnert and Sadler (2014) examined archived data from the Midwest like other similar studies from before to determine the efficiency of the placement tests into college math courses. This study shows that while mathematical preparation is significant, taking college pre-calculus is not, even for weakly prepared students (Sonnert & Sadler, 2014). Harwell (2013) examined the effects of many college characteristics, such as size and selectivity, on mathematics success and retention. No significance was determined from any of the college characteristics though, leading to the conclusion that high school factors must be a bigger influence on college mathematics success (Harwell, 2013).

Effects of Achievement

As this review covers primarily a comparison of different curricular samples, one important thing to keep in mind is the comparability of samples, and is given as a recommendation of the National Research Council’s 2004 Curricular Evaluation Report (NRC 2004). Perhaps unsurprisingly, the effects of high school achievements for students play a unanimously vital role in their success in college calculus. With this recommendation, it comes as little surprise that quite a few articles that had some measure of achievement as a categorical variable in their study (Harwell et al., 2009; Post et al., 2010; LeBeau et al., 2012; Harwell et al., 2013). However, this is still a fairly small amount compared to the number of studies in the last section, which goes against the recommendations stated earlier (NRC 2004). Perhaps unsurprisingly, the articles that included effects of achievement all indicated a positive correlation between the said measure of achievement and the grade earned in collegiate calculus, regardless of the measure was high school GPA (Harwell et al., 2009; Post et al., 2010; LeBeau et al., 2012) or ACT score (Harwell et al., 2009; Post et al., 2010; LeBeau et al., 2012; Harwell et al., 2013).

Implications for Future Studies and Use of Curriculum

For high school curricular comparisons, it seems appropriate that those papers follow the recommendations of the National Research Council (NRC 2004). Although there are a few recommendations not followed, only two will be discussed. The first was the recommendation for more pure experimental studies to strive for better balance with quasi-experimental studies (NRC 2004), yet almost all of the curricular comparisons were quasi-experimental. The other was the recommendation for data to be collected about implementation fidelity (NRC 2004), as the curriculum is more than a book but also includes how faithfully it was adapted. However, almost all of the studies took the institution’s word for the curriculum used. These were studies published after the recommendations and yet did not follow them, which is problematic for the field.

Overall, the answer to the best curricular choice for collegiate calculus at present is uncertain, depending on definitions of success, and weakened by a lack of variety in the research. Without broadening the research in the field, the truth will remain hidden, so this review should be a call for this subfield to come together and coordinate research to tackle these issues.

References

  1. Harwell, M., Post, T. R., Cutler, A., Maeda, Y., Anderson, E., Norman, K. W., & Medhanie, A. (2009). The Preparation of Students From National Science Foundation-Funded and Commercially Developed High School Mathematics Curricula for Their First University Mathematics Course. American Education Research Journal, 46(1), 203-231. doi: 10.3102/0002831208323368
  2. Harwell, M., Post, T. R., Cutler, A., Maeda, Y., Anderson, E., Norman, K. W., & Medhanie, A. (2009). The Preparation of Students From National Science Foundation-Funded and Commercially Developed High School Mathematics Curricula for Their First University Mathematics Course. American Education Research Journal, 46(1), 203-231. doi: 10.3102/0002831208323368
  3. Harwell, M. (2013). The Impact of Institutional Factors on the Relationship Between High School Mathematics Curricula and College Mathematics Course-Taking and Achievement. Educational Research Quarterly, 36(3), 22-46.
  4. Harwell, M., Medhanie, A., Dupuis, D. N., Post, T. R., & LeBeau, B. (2013). A Multisite Study of High School Mathematics Curricula and the Impact of Taking a Developmental Math Course in College. Educational Research Quarterly, 37(3), 3-22.
  5. Harwell, M., Post, T. R., Medhanie, A., Dupuis, D. N., & LeBeau, B. (2013). A Multi-Institutional Study of High School Mathematics Curricula and College Mathematics Achievement and Course Taking. Journal for Research in Mathematics Education, 44(5), 742-774.
  6. Hill, R., & Parker, T. (2006). A Study of Core-Plus Attending Michigan State University. The American Mathematical Monthly, 113(10), 905-921.
  7. Klopfenstein, K., & Thomas, M. K. (2009). The Link Between Advanced Placement Experience and Early College Success. Southern Economic Journal, 75(3), 873-891.
  8. LeBeau, B., Harwell, M., Monson, D., Dupuis, D., Medhanie, A., & Post, T. R. (2012). Student and high-school characteristics related to completing a science, technology, engineering, or mathematics (STEM) major in college. Research in Science & Technological Education, 30(1), 17-28. doi: 10.1080/02635143.2012.659178
  9. National Research Council (2004). On evaluating curricular effectiveness: Judging the quality of
  10. K-12 mathematics evaluations. National Academies Press.
  11. Post, T. R., Medhanie, A., Harwell, M., Norman, K. W., Dupuis, D. N., Muchlinski, T., Anderson, E., & Monson, D. (2010). The Impact of Prior Mathematics Achievement on the Relationship Between High School Mathematics and Postsecondary Mathematics Performance, Course-Taking, and Persistence. Journal for Research in Mathematics Education, 41(3), 274-308.
  12. Sadler, P., & Sonnert, G. (2018). The Path to College Calculus: The Impact of High School Mathematics Schoolwork. Journal for Research in Mathematics Education, 49(3), 292-329.
  13. Sahin A., Cavlazoglu, B., & Zeytuncu, Y. E. (2015). Flipping a College Calculus Course: A Case Study. Educational Technology & Society, 18 (3), 142–152.
  14. Sonnert, G., & Sadler, P. M. (2014). The impact of taking a college pre- calculus course on students’ college calculus performance. International Journal of Mathematical Education in Science and Technology, 45(8), 1188-1207. doi: 10.1080/0020739X.2014.920532
  15. Teuscher, D., & Reys, R. E. (2012). Rates of Change: AP Calculus Students’ Understandings
  16. and Misconceptions After Completing Different Curricular Paths. Social Science and
  17. Mathematics, 112(6), 359-376. doi:10.1111/j.1949-8594.2012.00150.x
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