Student readiness, engagement and success in entry level undergraduate mathematics courses

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by

Lorraine F. Dame

B.Sc., University of Victoria, 1988 M.Sc., University of Victoria, 2005

A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of

DOCTOR OF PHILOSOPHY

in the Department of Mathematics and Statistics

c

Lorraine F. Dame, 2012 University of Victoria

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Student Readiness, Engagement and Success in Entry Level Undergraduate Mathematics Courses by Lorraine F. Dame B.Sc., University of Victoria, 1988 M.Sc., University of Victoria, 2005 Supervisory Committee

Gary MacGillivray, Supervisor

(Department of Mathematics and Statistics)

Peter Dukes, Departmental Member

Wendy Myrvold, Outside Member (Department of Computer Science)

Joe Parsons, Additional Member (Department of Psychology)

Yvonne Coady, Outside Member (Department of Computer Science)

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Supervisory Committee

Gary MacGillivray, Supervisor

Peter Dukes, Departmental Member

Wendy Myrvold, Outside Member (Department of Computer Science)

Joe Parsons, Additional Member (Department of Psychology)

Yvonne Coady, Outside Member (Department of Computer Science)

ABSTRACT

The results of this thesis can be used to help identify students at risk of an unsuc-cessful ELUM outcome early in their course by using a student’s score on a diagnostic test, using a student’s high school performance or tracking graded homework submis-sions. The quantitative results suggest that optional additional remediation is not addressing students’ need for remediation and that struggling students do not fre-quently engage with departmental supports offered. They also suggest how a suite of pedagogical changes to an ELUM course can be associated with increased student success rates if managed carefully. It is commonly known that prior mathematical knowledge and success in current math courses are strongly linked. Through observa-tional studies at UVic it is found that a significant proportion of students beginning one of several ELUM courses do not demonstrate the high levels of preparation re-quired to succeed and that optional additional remediation is not addressing this issue. In Calculus I, we can infer that more than half of the population of graduating high school students in British Columbia meet or exceed the minimum prerequisite of a B in Principles of Math 12 at UVic. However, entering Calculus I students scored

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a mean of only 51% on a diagnostic test of important topics from Grade 7 to Grade 12 level mathematics. Almost 90% of the Calculus I students that were identified as at risk by results of the diagnostic test had an unsuccessful outcome. In addition, al-though nearly half of Calculus I students who entered with a B in Principles of Math 12 failed the Calculus I final exam, passing the remedial course Precalculus is only required for students with a C+ or less in Principles of Math 12. Thus an insignificant proportion of students are adequately prepared for Calculus I by passing the remedial course while many struggling students do not get the remediation required for them to succeed. The quantitative results indicate that the strength of the link between levels of prior mathematical knowledge and ELUM success varies by course.

According to Astin (1984) increasing student engagement positively correlates with higher satisfaction levels. Thesis results show that a student who misses more than one graded homework (used as a measure of engagement) will very likely fail the final exam. They show that a student who does not express satisfaction with his/her individual performance is also very unlikely to engage frequently with departmental supports offered such as office hours or the Math Assistance Centre. The results of these observational studies influenced pedagogical changes to the course Logic and Foundations that were designed to increase student engagement. These changes included a more accessible textbook, giving in-class quizzes on assigned readings, fostering a positive course experience, intervening with students at-risk and assigning less weight to the final exam. Analysis of course outcomes shows that the failure rate significantly decreased during this term. Student outcomes were not initially improved after similar modifications to Calculus II, but many of these changes to Calculus II have been maintained through subsequent terms in which improved student outcomes have been observed.

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List of Tables

1.1 Provincial Level Distribution of Letter Grades for B.C. High School Math Courses Including School Years 2007/08, 2008/09 and 2009/10 4 1.2 Percentage of Variation in ELUM achievement explained by

assess-ment test. . . 10 1.3 Percentage of Variation in ELUM achievement explained by high

school achievement . . . 11 2.1 Interpretation of P-values . . . 34 3.1 Distribution of High School Grades for Students Analysed (n = 403) 38 3.2 Chi-squared tests for dependence of passing Calculus I on entering

with an .... . . 39 3.3 Kruskal-Wallis Tests on Median Calculus I Letter Grade . . . 40 3.4 Kruskal-Wallis Tests on Median Calculus I Final Exam Percent . . 40 3.5 Distribution of High School Grades for Students Analysed (n = 223) 42 3.6 Chi-squared tests for dependence of passing Calculus for the Social

and Biological Sciences on entering with an... . . 43 3.7 Kruskal-Wallis Tests on Median Calculus for the Social and

Biolog-ical Sciences Letter Grade . . . 43 3.8 Distribution of High School Grades for Students Analysed (n = 136) 45 3.9 Chi-squared tests for dependence of passing Matrix Algebra for

En-gineers on entering with an... . . 46 3.10 Kruskal-Wallis Tests on Median Matrix Algebra for Engineers Letter

Grade . . . 46 3.11 Distribution of High School Grades for Students Analysed (n = 590) 48 3.12 Chi-squared tests for dependence of passing Precalculus on entering

with an... . . 48 3.13 Kruskal-Wallis Tests on Median Precalculus Letter Grade . . . 49 3.14 Distribution of High School Grades for Students Analysed (n = 5000) 50

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3.15 Chi-squared tests for dependence of passing Finite Math on entering with an... . . 51 3.16 Kruskal-Wallis Tests on Median Finite Math Letter Grade . . . 51 3.17 Distribution of High School Grades for Students Analysed (n = 1333) 53 3.18 Chi-Squared tests for dependence of passing Math for Elementary

School Teachers I on entering with an... . . 53 3.19 Kruskal-Wallis Tests on Median Math for Elementary School

Teach-ers I Letter Grade . . . 54 3.20 Percentage of Students who Completed ELUM courses with

Mini-mum Prerequisite that Succeeded . . . 58 5.1 Diagnostic Test Results – Percentage of Correct Responses by Question 68 5.2 Diagnostic Test Results – Mean Scores by Topic and Grade Level

for calculus courses . . . 69 5.3 Diagnostic Test Results – Mean Scores by Topic and Grade Level

for other courses . . . 69 5.4 Mean Diagnostic Test Scores by Course . . . 87 5.5 Success Rates Achieved by At-Risk Students and Others by Course 88 5.6 Median Final Exam Percentage Achieved by At-Risk Students and

Others by Course . . . 88 6.1 Summary of Mean Percentages by Term for Calculus I . . . 93 6.2 Means for Assigned versus Submitted Homework by Term for

Cal-culus I . . . 93 6.3 Percentage of Students Who... by Term for Calculus I . . . 93 6.4 Relative Risk of Earning Lower than 50% on the Final Exam by

Term for Calculus I . . . 93 6.5 Final Exam Scores out of 100 by Term for Calculus I . . . 93 6.6 Percentage of Variation (R2_{) in Final Exam Score Due To Variation}

in Factor for Calculus I . . . 94 6.7 Summary of Mean Percentages by Term for Calculus II . . . 96 6.8 Means for Assigned versus Submitted Homework by Term for

Cal-culus II . . . 96 6.9 Percentage of Students Who... by Term for Calculus II . . . 96 6.10 Relative Risk of Earning Lower than 50% on the Final Exam by

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6.11 Final Exam Scores out of 100 by Term for for Calculus II . . . 96 6.12 Percentage of Variation (R2_{) in Final Exam Score Due To Variation}

in Factor for Calculus II . . . 97 6.13 Summary of Mean Percentages by Term for Calculus for the Social

and Biological Sciences . . . 99 6.14 Means for Numbers of Assigned vs. Submitted Homeworks for

Cal-culus for the Social and Biological Sciences . . . 99 6.15 Percentage of Students Who... by Term for Calculus for the Social

and Biological Sciences . . . 99 6.16 Relative Risk of Earning Lower than 50% on the Final Exam by

Term for Calculus for the Social and Biological Sciences . . . 99 6.17 Final Exam Scores out of 100 by Term for Calculus for the Social

and Biological Sciences . . . 100 6.18 Percentage of Variation (R2_{) in Final Exam Score Due To Variation}

in Factor for Calculus for the Social and Biological Sciences . . . . 100 6.19 Summary of Mean Percentages by Term for Logic and Foundations 101 6.20 Means for Assigned versus Submitted Homework by Term for Logic

and Foundations . . . 102 6.21 Percentage of Students Who... by Term for Logic and Foundations 102 6.22 Relative Risk of Earning Lower than 50% on the Final Exam by

Term for Logic and Foundations . . . 102 6.23 Final Exam Scores out of 100 by Term for Logic and Foundations . 102 6.24 Percentage of Variation (R2_{) in Final Exam Score Due To Variation}

in Factor for Logic and Foundations . . . 102 6.25 Summary of Mean Percentages by Term for Finite Math . . . 104 6.26 Means for Assigned versus Submitted Homework by Term for Finite

Math . . . 104 6.27 Percentage of Students Who... by Term for Finite Math . . . 104 6.28 Relative Risk of Earning Lower than 50% on the Final Exam by

Term for Finite Math . . . 105 6.29 Final Exam Scores out of 100 by Term for Finite Math . . . 105 6.30 Percentage of Variation (R2_{) in Final Exam Score Due To Variation}

in Factor for Finite Math . . . 105 6.31 Mean Component Achievement by Course . . . 106 6.32 Mean Final Exam Scores vs. Missed Homeworks by Course . . . . 107

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6.33 Percentage of Students Passing the Final vs. Number of Missed Homeworks by Course . . . 107 7.1 Common Responses for Long Answer Portion by 511 Students in

First Year Math Courses Surveyed in the Spring 2010 Term . . . . 113 7.2 Average Responses for Multiple Choice Portion – ELUM Courses . 114 7.3 Percentage of Responses for Multiple Choice Portion – Calculus I

Spring 2008 (n = 44) . . . 115 7.4 Percentage of Responses for Multiple Choice Portion – Calculus I

Spring 2010 (n = 117) . . . 115 7.5 Average Responses for Multiple Choice Portion – Calculus I . . . . 116 7.6 Strongly Correlated Variables on Multiple Choice Portion for

Cal-culus I . . . 117 7.7 Percentage of Responses for Multiple Choice Portion – Calculus II

Spring 2008 (n = 244) . . . 118 7.8 Percentage of Responses for Multiple Choice Portion – Calculus II

Spring 2010 (n = 271) . . . 118 7.9 Average Responses for Multiple Choice Portion for Calculus II (n =

517) . . . 119 7.10 Strongly Correlated Variables on Multiple Choice Portion for

Cal-culus II . . . 119 7.11 Percentage of Responses for Multiple Choice Portion – Calculus for

the Social and Biological Sciences Spring 2008 (n = 123) . . . 121 7.12 Percentage of Responses for Multiple Choice Portion – Calculus for

the Social and Biological Sciences Spring 2010 (n = 77) . . . 121 7.13 Average Responses for Multiple Choice Portion – Calculus for the

Social and Biological Sciences . . . 122 7.14 Strongly Correlated Variables on Multiple Choice Portion – Calculus

for the Social and Biological Sciences . . . 122 7.15 Percentage of Responses for Multiple Choice Portion – Precalculus

Spring 2008 (n = 21) . . . 124 7.16 Percentage of Responses for Multiple Choice Portion – Precalculus

Spring 2010 (n = 28) . . . 124 7.17 Average Responses for Multiple Choice Portion – Precalculus . . . 125

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7.18 Strongly Correlated Variables on Multiple Choice Portion for

Pre-calculus . . . 126

7.19 Percentage of Responses for Multiple Choice Portion – Calculus I Spring 2008 (n = 77) . . . 128

7.20 Percentage of Responses for Multiple Choice Portion – Logic and Foundations Spring 2010 (n = 75) . . . 128

7.21 Average Responses for Multiple Choice Portion – Logic and Foun-dations . . . 129

7.22 Strongly Correlated Variables on Multiple Choice Portion – Logic and Foundations . . . 129

7.23 Percentage of Responses for Multiple Choice Portion – Finite Math Spring 2008 (n = 234) . . . 131

7.24 Percentage of Responses for Multiple Choice Portion – Calculus I Spring 2010 (n = 94) . . . 131

7.25 Average Responses for Multiple Choice Portion – Finite Math . . . 132

7.26 Strongly Correlated Variables on Multiple Choice Portion – Finite Math . . . 132

7.27 Percentage of Responses for Multiple Choice Portion – Math for Elementary School Teachers II Spring 2008 (n = 78) . . . 133

7.28 Percentage of Responses for Multiple Choice Portion – Math for Elementary School Teachers II Spring 2010 (n = 44) . . . 134

7.29 Average Responses for Multiple Choice Portion – Math for Elemen-tary School Teachers II . . . 134

7.30 Strongly Correlated Variables on Multiple Choice Portion – Math for Elementary School Teachers II . . . 135

8.1 Historical Data for Logic and Foundations (Percent of Enrolment) . 140 8.2 Fall 2009 Logic and Foundations – Students at Risk . . . 143

8.3 Logic and Foundations Comparison with Historical Percentages . . 143

9.1 Preparation Levels for Students of Calculus II . . . 154

9.2 Attrition . . . 155

9.3 Summary of Performance . . . 155

9.4 Percentage of Students Who... . . 155

9.5 Summary of Performance for Instructor A . . . 156

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9.7 Final Exam Performance Spring 2010 vs. Fall 2010 . . . 161

9.8 Final Exam Performance Spring 2010 vs. Fall 2010 for Instructor A 162 9.9 Final Exam Percentages Corresponding to Lower end of Letter Grade Boundaries . . . 163

9.10 Assigned versus Submitted Homework . . . 168

9.11 Percentage of Students Who... . . 168

9.12 Relative Risk of Earning Lower than 50% on Final Exam . . . 168

9.13 Analysis of Final Exam Percentages . . . 169

9.14 Percentage of Variation (R2_{) in Final Exam Score Due To Variation} in Factor for Calculus II . . . 169

9.15 Percentage of Common Responses for Long Answer Portion of Sur-vey by Students in ELUM Courses for Spring 2010 and by Calculus II students in Fall 2010 . . . 172

9.16 Percentage of Responses for Multiple Choice Portion of Survey – Calculus II Fall 2010 (n = 41) . . . 173

9.17 Average Responses for Multiple Choice Portion of Survey . . . 173

9.18 Strongly Correlated Variables on Multiple Choice Portion of Survey 175 10.1 Percentage of Students Missing More than One Homework . . . 198

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List of Figures

3.1 Distribution of Calculus I Final Exam Scores by PoM 12 Grade . . 41 5.1 Distribution of Diagnostic Test Scores for Calculus I Students . . . 74 5.2 Distribution of Diagnostic Test Scores for Students of Calculus for

the Social and Biological Sciences . . . 76 5.3 Distribution of Diagnostic Test Scores for Students of Precalculus . 78 5.4 Distribution of Diagnostic Test Scores for Students of Finite Math . 81 5.5 Distribution of Diagnostic Test Scores for Students of Math for

El-ementary School Teachers . . . 84 6.1 What Percentage of Students who Failed the Final Exam Missed

More Than One Homework? . . . 108 6.2 What Percentage of Students who Missed More Than One

Home-work Failed the Final Exam? . . . 109 9.1 Overall Course Percentages for Fall 2010 A . . . 157 9.2 Overall Course Percentages for Fall 2010 B . . . 158 9.3 Overall Course Percentages for Instructor of Fall 2010 Section A . . 160 9.4 Letter Grades Fall 2010 Section A . . . 165 9.5 Letter Grades Fall 2010 Section B . . . 166 9.6 Letter Grades Fall 2010 Section A – Comparison with Other Sections

by the Same Instructor . . . 167 9.7 Comparing Engagement and Performance . . . 170

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ACKNOWLEDGEMENTS I would like to thank:

Mom, Dad, Karmen, Emily, my dog Sunshine & my cat Summer, for loving, laughing, kissing, hugging, advising, wagging, purring and cuddling.

G. MacGillivray, for friendship, mentoring, support, loyalty, encouragement, pa-tience and funding.

The Learning Community of the University of Victoria, for many enjoyable years of learning and teaching.

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DEDICATION

To all the educators and learners in the Department of Mathematics and Statistics and at the Learning and Teaching Centre at the University of Victoria. I hope we can use this to make the journey to understanding mathematics more enjoyable.

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Introduction

1.1 Why Do This?

When I was in my 20’s and 30’s I was very much afraid of public speaking. The thought of standing up in front of a group of people and saying anything, even “erk?” was pretty terrifying. I started to think of why I was afraid of this and I came to the conclusion that I was actually afraid of doing a bad job of it. The flip side of this coin is that it meant I could get a lot of personal satisfaction out of doing a good job of it. After taking some teacher training and leading a couple of math tutorials, I discovered that I really really really love teaching (just another perk of academic life – learning new things about yourself). I find that being part of a student’s successful learning experience is very exciting and personally satisfying.

A passion for mathematics comes with me as part of the package. I want to share this passion for mathematics by helping to make learning post secondary mathemat-ics more interesting, accessible and enjoyable. For me this journey begins with the Scholarship of Teaching and Learning (SoTL), which is the use of evidence-based re-search to improve teaching and learning. SoTL necessarily involves using statistics to

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study learning. What better world for a mathematics teacher? Using mathematics to study how students learn mathematics? That sounded like an enjoyable challenge. These interests fit very well with the University of Victoria’s goal of increasing stu-dent success and retention. If we can retain more stustu-dents through entry level courses, student satisfaction and enrolment in subsequent courses should naturally increase. One way to increase retention is to change our pedagogy for the benefit of students at risk of failing. This thesis focusses on central themes in the pursuit of increasing success and retention: identifying students at risk of failing entry level undergraduate mathematics (ELUM) courses by using early warning signals, and increasing success rates by changing pedagogy.

1.2 What is the Problem and What Can We Do

About It?

In 2003, Human Resources and Social Development Canada (HRSDC) reported that over half of adult Canadians did not demonstrate levels of mathematical skill and knowledge associated with functioning well in Canadian society (HRSDC, 2003). There are now dwindling numbers of students entering college to study science, tech-nology, engineering, and mathematics disciplines (Turner, 2008). A low retention rate means that a college is always working to replace students that leave, which requires resources that could be used elsewhere. In addition, if students leave before graduating, they are not likely to become donors to their former schools (Jamelske, 2009). From the student perspective, retention (post-secondary) is important for the simple reason that college pays (Jamelske, 2009). In 2003 the median annual salary in the U.S. was $30, 800 for a worker with only a high school diploma. This was significantly lower than the median earnings of $49, 900 for those with a bachelor’s

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degree (Jamelske, 2009).

There is evidence in a 2001 study done by members of the Department of Mathe-matics and Statistics at UVic that high school matheMathe-matics is not always adequately preparing students for success at the university level. In the 2000/2001 academic year, the in-house developed Calculus I Assessment Test was given to all students in Calculus I and their progress through the course was tracked. Students who achieved less than 50% on the Assessment Test were at least three times as likely to fail Cal-culus I as those who scored 50% or higher. Findings from the 2001 study prompted changes to the prerequisites and subsequently a grade of at least B in Principles of Math 12 (PoM 12) or a passing grade on the Assessment Test became the prerequisite for Calculus I. After this change in prerequisites for Calculus I, the failure rate fell by approximately 8-10% (G. MacGillivray, 2010).

Academic year 2009 was the first year that students admitted to UVic from B.C. secondary schools were not required to write a standardized Provincial Exam in PoM 12. In addition, the minimum high school GPA required for admission to UVic was reduced. Failure rates in ELUM courses increased (G. MacGillivray, 2010). It is natural to suspect that perhaps there was a connection. Since students were not required to take the provincial exam, high school math letter grades were the primary measure of a student’s readiness.

B.C. Principles of Math 11 (PoM 11) and PoM 12 could be used as prerequisites for ELUM courses at UVic. A recent report by the B.C. Ministry of Education (2011) provides the distribution of letter grades achieved by students in these courses. Table 1.1 summarizes the distribution of letter grades for these courses over several academic years (the enrolment quantities quoted in Table 1.1 included students who attended Yukon public schools). High school letter grades recorded for students include the values A, B, C+, C, C- and F.

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Table 1.1: Provincial Level Distribution of Letter Grades for B.C. High School Math Courses Including School Years 2007/08, 2008/09 and 2009/10

Course n A(%) B(%) C+/C/C-(%) F(%)

P oM 11 122072 25.0% 25.0% 40.6% 8.7% P oM 12 80430 27.8% 31.0% 35.2% 6.0%

PoM 11 and PoM 12 are offered to students who are considering post-secondary education in engineering, mathematics or the sciences. Similar courses have been offered in B.C. high schools by different names. Any reference to these courses in this document is intended to mean B.C. PoM 11/12 or their equivalents.

It should be noted that the UVic Academic Calendar (The University of Victoria (2011)) currently indicates that Precalculus 11/12 can be used as prerequisites for some math courses. However at the time that the analysis in this thesis was done, Precalculus 11/12 was not available at British Columbia high schools.

The University of British Columbia (UBC) and Simon Fraser University (SFU) responded to the termination of the mandatory Provincial Exam by raising the min-imum PoM 12 grade necessary to enter first year calculus (UBC is using 80%, SFU is using the letter grade A, which corresponds to 86%). A passing grade on an in-house-developed assessment test also suffices. Students without the prerequisite grade in PoM 12 or a pass on the assessment test can satisfy the prerequisite by taking a special calculus class that includes extra hours for review. At UBC this is a two-term course worth twice the credit as regular differential calculus (Mathematics Department at University of British Columbia, 2010). At SFU this is a one term course worth 4/3 as much credit as regular differential calculus (Department of Mathematics at Simon Fraser University, 2012). The UVic Department of Mathematics and Statistics has not changed its prerequisite for entry level calculus in response to the termination of the mandatory Provincial Exam.

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II are an essential and required part of their programs. Improving success in these courses could improve retention across many programs at UVic. During the Spring 2008 and Spring 2009 terms, approximately 50% of the students in Calculus II (see Appendix for course description) achieved 50% or less on their common final exam. How can we improve this outcome while maintaining the topics of Calculus II? Will pedagogical changes positively influence student engagement and student outcomes? Will the success rate and the retention rate be improved in this course?

This research will identify factors strongly correlated with success in entry level undergraduate mathematics (ELUM) courses including those in the categories of stu-dent readiness and stustu-dent engagement. It will also describe the outcomes associated with pedagogical changes designed to influence these factors. The results of this research can be used to improve secondary school mathematics, post-secondary ad-mission policies and post-secondary mathematics education practices so that more students are successful and retained through ELUM courses.

This research supports the University of Victoria (UVic) strategic goal of establish-ing UVic as a recognized cornerstone of our community, committed to the sustainable social, cultural and economic development of our region and our nation (UVic, 2007). Studying measures of student readiness will support UVic’s strategic goal of recruit-ing and retainrecruit-ing a diverse group of exceptionally talented students (UVic, 2007) by providing information on useful and accurate ways to identify students who are less prepared to succeed in mathematics. Studying measures of student engagement will support UVic’s strategic goal of supporting students in ways that allow them to achieve their highest potential (UVic, 2007) since increased engagement is associated with increased performance (Astin, 1984).

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1.3 Thesis Questions

This research will help us learn about the level of preparation of our students, the relationship between preparation and success, the level and importance of student engagement, and what can be accomplished with specific pedagogical changes. We can use what we learn to improve student success by more accurately measuring when they are ready for an ELUM course, recognizing when they are struggling in an ELUM course, intervening with struggling students, designing ELUM courses for success and offering appropriate supports and interventions. Specifically, the following questions were studied using quantitative and qualitative analysis:

Thesis Question 1 What are student levels of achievement in their education before they enter ELUM courses? (Chapter 3)

Thesis Question 2 How well do students’ achievement in previous education pre-dict achievement in ELUM courses? (Chapter 3)

Thesis Question 3 How accurately do current minimum prerequisites predict suc-cess in ELUM courses? (Chapter 3)

Thesis Question 4 How effective are remedial courses at preparing students who need remediation for success in ELUM courses? How well do students’ achieve-ments in remedial courses predict achievement in ELUM courses? (Chapter 4)

Thesis Question 5 What levels of essential mathematical skills do students bring to ELUM courses when they first enter? (Chapter 5)

Thesis Question 6 Can we identify students at risk of failing ELUM courses by measuring their levels of essential skills at course entry? (Chapter 5)

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Thesis Question 7 Can we predict student achievement in ELUM courses using assessed levels of achievement in essential skills at course entry? (Chapter 5) Thesis Question 8 Which quantitative measures of students’ levels of preparation

should we choose to predict ELUM success? (Chapter 10)

Thesis Question 9 What are students’ levels of engagement in ELUM courses? (Chapter 10)

Thesis Question 10 What are student achievement levels in ELUM course compo-nents? (Chapter 6)

Thesis Question 11 How well do students’ levels of engagement predict their achieve-ment in ELUM courses? (Chapter 6)

Thesis Question 12 Can we predict student failure in ELUM courses using mea-sures of engagement? (Chapter 6)

Thesis Question 13 How frequently do students report they engage with academic supports? (Chapter 7)

Thesis Question 14 How well does student engagement with supports predict achieve-ment in ELUM courses? (Chapter 10)

Thesis Question 15 What other resources do students access frequently besides course material provided? (Chapter 7)

Thesis Question 16 What were the outcomes of the project that made pedagogi-cal changes to Logic and Foundations and the project that made pedagogipedagogi-cal changes to Calculus II? (Chapter 10)

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1.4 Literature Survey

The factors correlated with entry level undergraduate mathematics (ELUM) achieve-ment that are included in this review can be divided into three categories: student readiness; student engagement; and pedagogy.

Many of the studies cited here have calculated the coefficient of determination R2

based on sample data. R2 _{is the square of the Pearson-R coefficient R. R}2 _{is often}

expressed as a proportion, but for this thesis it will be expressed as its equivalent percentage. R2 _{is an estimate of the percentage of variation in the dependent variable}

(e.g. ELUM achievement) that is due to variation in the independent variables in a model. Thus values can vary between R2 _{= 0% and R}2 _{= 100% with R}2 _{= 100%}

indicating the two variables are perhaps different ways of measuring the same quantity. For example, if one variable measured a person’s exact height in inches and the other measured exact height in metres then these two variables would have R2 _{= 100%.}

Values of R2 _{are interpreted as follows: R}2 _{< 4% means little or no correlation; 4%}

≤ R2 _{< 9% means weak correlation; 9%} _{≤ R}2 _{< 16% means moderate correlation}

and R2 _{≥ 16% means strong correlation. Thus a value of R}2 _{≤ 16% may indicate a}

less than strong ability of a regression model to predict the value of the dependent variable.

1.4.1 Student Readiness

Student readiness can be measured through the use of assessment or placement tests, or by achievement in other educational contexts. Overall achievement in high school, rigour of high school math/science program, overall achievement in university, achievement in prior math courses (including prerequisite and remedial courses) and other factors are also reported herein. Several studies have related student scores

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on standardized tests to achievement in ELUM courses. These standardized tests in-clude: the Mathematics Association of America Algebra placement Test (MAA, 1996), the American College Test (ACT, 2009) and the Scholastic Aptitude Test (SAT) (Col-lege Board, 2009). The relationship between performance on custom math placement tests and student achievement in ELUM courses has also been studied.

Rodgers and Wilding (1998) studied 266 incoming freshmen in 1996. They con-cluded that approximately 16.3 % of the variation in final grade in freshman algebra could be accounted for by the student’s score on the MAA placement test.

The results for whether or not the ACT Math tests are significantly correlated with achievement in ELUM are mixed. In a study of 1215 first-time university students in 1988, Hudson (1989) concluded that ACT Math scores could account for only 2.9% of the variation in achievement for ELUM courses. In a study of 789 students with repeated sampling, Edge and Friedberg (1984) demonstrated that 13.0% of the variation in calculus achievement can be accounted for by variation in ACT math score. House (1995) concluded that ACT Composite score could account for 3.7% of grade performance in college mathematics (p < 0.01). A possible question for future study is whether ACT Math test results are better predictors for some ELUM courses, e.g. calculus, than others.

Many studies have examined the correlation of SAT quantitative scores with achievement in ELUM courses. Some examples of such studies are: Jull and McK-inney (2002) with R2 _{= 7.3%, Thorndike-Christ (1998) with R}2 _{= 22%, Henderson}

and Landesman (1986) with R2 _{= 9.1%, Goldstein and High (1992) R}2 _{= 4.8% and}

Goolsby (1988) with R2 _{= 2.0%.}

Other studies involve math placement tests that are specific to a local government or school. Researchers have found somewhat higher correlations with achievement in first year math than SAT, ACT or MAA scores.

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Table 1.2: Percentage of Variation in ELUM achievement explained by assessment test.

Study Test N R2 _Course

Kent (1928) Mental Alertness 524 38.0 Algebra Abeyaskeera (1989) Grade 12 assessment 304 33.5− 61.0 ELUM Kaeley and Kaeley (1995) Arith. – distance 85 32.5 ELUM Edge and Friedberg (1984) Algebra Placement 789 30.3 Calculus Kent (1928) Mental Alertness 524 28.0 Trig Kaeley and Kaeley (1995) Arith. – on campus 88 23.0 ELUM Langlie (1928) Prior Math Achvmnt 160 20.2 ELUM Thorndike-Christ (1998) Math Placement 214 16.2 ELUM

Langlie (1928) Math Ability 160 13.0 ELUM

Hailikari et al. (2008) Prior Knowledge 139 11.6_{− 25.0} ELUM

One study of 430 students enrolled in a first year business quantitative methods course by Chansarkar and Michaeloudis (2001) inferred that level of entry qualification and performance on entry qualification are associated with performance in the ELUM. Research at the University of Victoria by professors D. Hewgill, G. MacGillivray and D. Leeming included over 600 first semester calculus students and determined that students who scored less than 50% on an in-house developed assessment test were over three times as likely to have an unsuccessful outcome in Calculus I (received a failing grade or dropped the course) than students who scored at least 50%. These findings strongly suggest that an instrument designed specifically for the locale or the post-secondary institution and includes some arithmetic and assessment of prior knowledge may result in a more accurate placement system than tests that are not specifically designed for the locale or the post-secondary institution.

An important measure of the level of student preparation is overall high school achievement (i.e. GPA or rank). Many studies (see Table 1.3) have demonstrated using statistical analysis of sample data that high school grade point average (GPA) or rank is a significant component in predicting achievement in ELUM courses. In his study, Kent (1928) states that it can be predicted that the higher the high school

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quartile (a measure of rank for a student of first, second, third or fourth quartile) the better the grades will be, and those freshman from the highest quartile are unlikely to achieve an average freshman math grade below D.

Table 1.3: Percentage of Variation in ELUM achievement explained by high school achievement

Study N R2

Goolsby (1988) 118 4.0 Jull and McKinney (2002) > 1500 12.0 Henderson and Landesman (1986) 238 24.0 Edge and Friedberg (1984) 235 10.0 Edge and Friedberg (1984) 397 14.0

A study by Ziomek and Maxey (1995) examined the percentage of students in various categories who had at least a 50% chance of receiving a B or better in col-lege algebra (ready for colcol-lege algebra) and calculus (ready for colcol-lege calculus). Of 103, 322 US public high school students in 1994 who were planning to major in sci-ence, 36% of those students who took three or more years of science and math in high school were ready for college calculus and 58% were ready for college algebra. Only 7% of students planning to major in science with less than three years of math and science in high school were ready for college calculus and 16% were ready for college algebra. A further 617, 787 students reported that they did not plan to major in science. Seventeen percent of these 617, 787 students who completed three or more years of science and math in high school were ready for college calculus and 36% were ready for college algebra. Three percent of students who reported that they did not plan to major in science and who had less than three years of math and science in high school were ready for college calculus and 9% were ready for college algebra. Thus a rigorous high school program remains a significant correlate with success in college algebra and calculus.

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deter-mined that students who had an aggregate PoM 12 grade below B were approximately 2.6 times more likely to have an unsuccessful outcome in first semester calculus than students with an aggregate PoM 12 grade of A. Based on this and their study of the results of an in-house assessment test, a new prerequisite for Calculus I was adopted starting in Fall 2002, i.e. that students with less than a B (including those for whom the grade was unavailable) would either have to pass a remedial math class or pass an assessment test before being admitted to first semester calculus. Prior to 2002, students scoring below a B in their grade 12 aggregate math score were approximately two times more likely to have an unsuccessful outcome in first semester calculus if they scored less than 50% on the assessment test than students scoring below a B in their grade 12 aggregate math score who scored at least 50% on the assessment test. Several other measures of student preparedness were significant contributors to predicting achievement in ELUM courses including overall achievement in university courses, e.g. Hailikari et al. (2008) and Schenker (2007), more introductory college courses, e.g. Gupta et al. (2006), taking a lighter credit load, e.g. Gupta et al. (2006), more college math courses, e.g. Gupta et al. (2006), fewer remedial college math classes, e.g. Gupta et al. (2006), taking high school math by correspondence, e.g. Childs (1956). An interesting result from a study of 91 students by Herman (1993) is that there was no significant relationship between achievement in prerequisite courses and achievement in an applied statistics program. A complication of this study is that students who achieved a low level in the prerequisite courses did not make it into the applied statistics program and thus were not included in the study. A similar result was indicated by Jull and McKinney (2002), i.e. that grades in prior math courses were poor predictors of grades in current math courses. On the other hand, a study by Yushau (2004) demonstrated using multiple regression that proficiency in the language of instruction and math aptitude together accounted for 41% of the variation

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in achievement in precalculus algebra for male students. Students who achieved a high proficiency in English (their language of instruction) performed significantly better than those who had a low proficiency in English according to Yushau’s study. This study is particularly relevant at UVic since students from abroad show a strong interest in attending UVic and in the future our students will continue to come from farther afield nationally and internationally (UVic, 2007).

1.4.2 Student Engagement

One hypothesis of the theory of student involvement by Astin (1984) is that the effectiveness of any educational policy or practice is directly related to the capacity of that policy or practice to increase student involvement, thus increasing the amount of student learning and personal development. In addition, students who become intensely involved in their college studies experience considerably greater satisfaction (Astin, 1984) than students who are less intensely involved.

In a study of results of the National Survey of Student Engagement between 2000 and 2003 using data from 18 baccalaureate-granting colleges and universities, Kuh et al. (2008) inferred that student engagement (measured using the results of the National Survey on Student Engagement) in educationally purposeful activities had a low but statistically significant relationship to first-year grades (an increase in GPA of .06 for every standard deviation increase in their participation in educationally purposeful activities). Several measures of student involvement have been discussed in the literature including attendance, submitting course work and utilizing supports. In a study of 522 pre-algebra and algebra students at the University of Memphis, Glover (1996) found seven measures of student effort that were significantly related to course grade. In a study that extends Glover’s work, Thomas and Higbee (2000) looked at 119 university students enrolled in developmental algebra over two academic

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quarters and concluded that the level of student engagement in educationally pur-posefully activities measured by number of days absent was consistently negatively correlated to the achievement variables of homework average, computer test average, test average, final exam score and course grade.

Attendance appears to be a factor in achievement in ELUM courses. In a study of 23 adult students enrolled in a remedial mathematics course by Immerman (1982), 24% of the variation in final course score was attributed to the number of classes attended. Gupta et al. (2006) concluded that students missing fewer classes were likely to receive a better grade. However, Thomas and Higbee (2000) concluded from a study of 119 students of developmental algebra over two academic quarters that the number of days absent accounted for only 4% of the variation in final exam score. Jull and McKinney (2002) similarly concluded that only approximately 2.6% of the variation in course score could be attributed to number of days missed (as reported by students).

Student involvement in homework also appears to be an important factor in achievement. In a study involving 1163 students in Spring 2008, Dame et al. (2009) concluded that students in first year math courses who missed more than one graded homework were between 1.4 and 6.8 times as likely to achieve less than 50% on the final exam than those who missed at most one graded homework (see Chapter 6). Kaeley (1989) concluded in 1989 that the number of homework assignments turned in had a significant relationship to mean achievement. Students who attempted al-most all homework had a mean score of 58.9% and students who attempted a quarter or less had a mean score of 49.4%. Mathematics homework was also identified as a predictor of mathematics performance by Kaeley (1990) (R2 _{< 6.2%). Kaeley (1989)}

also found that students who found adequate time to work on mathematics after classes/work scored significantly higher (sample mean 59.4%) than those who did not

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(52.2%).

Another measure of a student’s involvement in their ELUM course is the frequency with which he, or she, utilizes supports such as tutoring or math assistance centres. A study by Gupta et al. (2006) concluded that higher grades in ELUM courses are associated with utilizing less tutoring. In contrast, Collins (2008) concluded that a student using their supplemental instruction program regularly could expect to outperform the occasional user by three-fourths of a letter grade. Students in a study by Kaeley (1989) who found their tutors most helpful scored significantly higher than those who did not. Finally, a study by Jull and McKinney (2002) concluded that only about 1.2% of the variation in achievement could be accounted for by frequency of tutor visits. These results suggest that it is not enough to measure the quantity of tutoring accessed, but also what happens in the tutoring sessions and the quality of the tutoring.

The existing literature reviewed suggests that to promote success in ELUM courses, post-secondary institutions can encourage and reward student efforts such as attend-ing class and turnattend-ing in all homework assignments. They can also provide a high quality of support through math assistance centres and tutors that will help students achieve success without repeated visits (i.e. repeated visits may imply lower quality tutoring and are inferred to be associated with lower grades by Gupta et al. (2006)).

1.4.3 Pedagogy

Pedagogy that can influence a student’s performance in an ELUM courses can be divided into two categories. The first is remedial or supplemental instruction outside the ELUM classroom and the second are pedagogical practices inside the ELUM class-room. Studies relating technology usage, assessment strategies, attendance improve-ment measures, the personalized system of instruction and other factors correlated

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with achievement inside the ELUM classroom are also reviewed in this section. For purposes of this discussion two simplistic definitions are adopted. Remedial instruction is defined as a course that an at risk student must take before or instead of an ELUM course, and supplemental instruction is defined as instruction that can be taken concurrently with an ELUM course by any student.

Studies of remedial math education at university demonstrate encouraging results. Some suggest that remedial courses are equalizers for at risk students. For example, Zoller et al. (1987) wrote that there is little or no evidence that at risk students who took remedial education were different than students who had not taken the remedial education, i.e. the students without remediation did not significantly outperform the at risk students with remediation. In a five year study of 40, 000 students at California State University, Guthrie (1992) concluded that retention rates for at risk students taking remedial education were comparable to that of the general population. An additional study by Jull and McKinney (2002) indicated that taking a functions and algebra course at Western Washington University had a positive contribution (R2 _{= 13%) to achievement in subsequent ELUM courses. Guthrie and Guthrie}

(1988) studied the Intensive Learning Experience (ILE) remedial program that offers remediation in mathematics to students scoring in the lowest quartile of the Entry Level Mathematics examination via a full academic year of math in small classes, along with academic advising. Overall, they found that nearly 65% of math grades for other students in ELUM courses were a C or above while only 50% of the grades for ILE students were a C or above. Although in this sense the ILE students did not catch up to other students, they have on average improved their ranking since now only at most half of these students have grades comparable to the lowest 35% of other students (before remediation they were all in the bottom quartile). Thus the literature suggests that remedial instruction can act as an equalizer for students who

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are identified as being at risk of failing a course.

Studies of supplemental instruction (SI) indicate that it can act as an equalizer or better for students enrolled in SI who typically are weaker than their counterparts, e.g. Bayliss (2008). For example, Weinstein (1995) demonstrated that finite math students who elected to take a linked math survival course focussing on math learning strategies, communication of math concepts and affective factors performed signifi-cantly better than those who did not take the linked math survival course. Bayliss (2008) concluded from a study of 1904 students that neither students of precalculus nor calculus demonstrated significantly different levels of achievement if they were also enrolled in SI, although SI students had significantly lower SAT math scores on entry. This implies that SI had an equalizing effect. A third study by Doyle and Hooper (1997) demonstrated that the over 2200 math students who chose to partici-pate in a structured learning assistance (SLA) program had significantly better pass rates than those not in SLA courses (ranging from 24% better in one course to 45% better in another).

An effect size is a measure of the strength of the relationship between two statis-tical variables (Wikipedia Contributers, 2012). Effect size estimation plays an impor-tant role in meta-analysis studies which summarize findings from a specific area of research (Wikipedia Contributers, 2012). Schenker (2007) used 46 studies to compute an average achievement effect size of 0.239 of the use of technology in the classroom on student achievement in Introductory Statistics courses. The results indicated that technology was modestly effective in improving students statistics achievement. Sim-ulations were significantly more effective than other technology types, while online learning was no more or less effective than traditional instruction. Schenker (2007) also found that students appear to benefit most in statistics courses that are enhanced with technology that allow them to interact with and manipulate data, and that do

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not hinder the instructor’s availability and presence to answer questions, clarify con-cepts, and otherwise provide further assistance when necessary.

Using ANOVA analysis, Johnson (1989) inferred no significant difference in per-formance for entry level math students on midterm and final exams across exactly one of the four following additional evaluation strategies: (i) tested twice weekly by short quizzes; (ii) given twice weekly graded homework; (iii) no extra testing; and (iv) given four in-class tests. Another study of assessment techniques (Glover (1995)) for entry level math students inferred a significant main effect favouring students tested by traditional work-the-problem methods versus those tested by multiple choice, but no significant effect on future college math grades (Glover (1995)). A study by Haeck et al. (1997) indicated that between 6.0% and 17.4% of ELUM score could be ac-counted for by various developmental test topics. A developmental test assesses a student’s ability to learn from typical math texts and explore their new knowledge in a variety of ways.

In a study of 1300 students by Budig (1991), an attendance notification system was introduced to approximately half of these students (experimental group). Students participating in the experimental group were issued warning tickets for missing classes and dropped from class on the third notification. The result was a decrease of up to 17% in D/F grades and as much as 13.7% increase in success rates for test sections over control sections.

The personalized system of instruction (PSI) is a self-paced instruction model in which mastery is required for advancement to new material, course content is communicated primarily through the written word, and proctors (advanced peers) are employed as course assistants to allow individualized scoring of repeated quizzes. Klopfenstein (1977) concluded from a study that 53 students in the PSI section of a calculus course had slightly higher mean scores on the ACT and SAT mathematical

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aptitude tests on entry however their mean course scores were not significantly higher than those of the non-PSI control group and their withdrawal and failure rates were abnormally high. It should be noted that PSI has been show to work well in a variety of courses by Kulik et al. (1979).

In a further study of 353 students, a computer assisted personalized system of instruction (CAPSI) was used in an online learning environment (OLE) for a discrete math/logic course. Brinkman (2008) noted that in 2004, the students in the CAPSI version of the class scored on average 14 points higher on a scale of 100 points than those in the control group. In 2005 the difference was 20 points. He also noted that OLE usage was a predictor of a low proportion of academic achievement in the course; however, instructional videos and OLE self-tests alone do not have a high impact on student performance. The main conclusion of this study included the finding that CAPSI can be used to enhance achievement in a course that is supported by OLE.

For several pedagogical techniques, there was little or no evidence of improvement in academic achievement in ELUM courses. These included: type of instructor, e.g. Jull and McKinney (2002), a study skills and systematic desensitization training pro-gram, e.g. Rushing (1996), training in making predictions and post-test estimations of test scores, e.g. Stottlemyer (2002), use of graphing calculators versus Maple, e.g. Alkhateeb (2002) or the use of specific collaborative techniques, e.g. Rousseau and Glover (1998). However, Gupta et al. (2006) did demonstrate that students of lower ranked instructors were more likely to receive a better grade. The same study also indicated that having class only once a week was correlated with higher grades.

Cartledge and Sasser (1981) inferred from their controlled pretest post-test study of 30 volunteer students that there is a tendency for properly assigned and evaluated weekly homework to improve gains on an algebra test. However, they admit that their research and past research do not provide a clear-cut endorsement for either

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homework or no-homework groups. In a study of 108 students of intermediate algebra at a four-year university in western Tennessee by Weems (1998), it was determined that an experimental class of students in which homework was collected earned more A’s than the class with no homework collection. It was noted that student comments from the control group favoured homework collection. It was also concluded by Weems (1998) that keeping homework organized in a notebook resulted in more A’s than a control group. The author also speculated that a possible reason for more students withdrawing from the experimental class was that the students in the experimental class had a higher level of involvement than students in the control group. Weems (1998) speculated that students with a higher level of involvement were more likely to be aware of their course standing and chose to withdraw and begin again in another semester.

In making a synthesis of homework studies, Paschal et al. (1984) concluded that much of the voluminous literature on homework was opinionated and polemical. They analysed 85 “methodologically adequate” studies and concluded from these that there is a moderately high average effect of assigned homework that is commented upon or graded, on course outcomes.

The existing literature strongly suggests that using technology to allow the stu-dents to interact with or manipulate data; assessment that places significant emphasis on long answer questions; tracking attendance with consequences and using computer aided instruction (possibly CAPSI) can improve student success rates.

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1.5 Mathematics at the University of Victoria

1.5.1 The UVic Academic Year

At the time that this research was performed, the academic year at UVic was divided into three standard terms. It began with the Fall term from September through December, followed by the Spring term from January through April and then the Summer term from May through August. At approximately the start of the last month of each term, lectures ended and the final exam period began. There were some courses available at UVic in other term formats but they were not studied for this thesis. Data for ELUM courses were analysed for eight standard terms at UVic from 2005 through 2010.

Each of the courses studied had a duration of one standard term. In any term, each course was implemented as a number of sections. For example, suppose it was anticipated that up to 1000 students would register for Calculus I in a Fall term. The 1000 available seats in Calculus I would then be divided into e.g 10 sections of 100 students each. Each section was assigned an instructor and teaching assistants (an instructor may teach more than one section). Each section had a designation that began with a letter (A for the standard term) followed by a two digit number, e.g. A01. Each class could then be uniquely identified using term, year, course name and section number, e.g. Fall 2010 Calculus I A01. Each section had its own lecture schedule and students could choose to register in any section that had available seats. Many programs of study (e.g. Engineering) required taking Calculus I in the first Fall term of the program followed by Calculus II in the Spring term. As a result, the population of Calculus I in the Fall term was typically more than the population for Calculus I in the Spring term and the population of Calculus II in the Spring term was typically more than the population of Calculus II in the Fall term. Populations

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of Calculus I in the Spring term and Calculus II were thought to contain a higher proportion of students who were repeating Calculus. Populations also tended to differ between terms for other courses. For this and other reasons, results are reported by term for each course.

1.5.2 ELUM Courses Studied

Courses analysed included Calculus I (Math 100), Calculus II (Math 101), Calcu-lus for the Social and Biological Sciences (Math 102), Matrix Algebra for Engineers (Math 110 or 133), Precalculus (Math 120), Logic and Foundations (Math 122), Finite Math (Math 151), Math for Elementary School Teachers I (Math 161 or previously Math 160A), Math for Elementary School Teachers II (Math 162 or previously Math 160B) and Mathematical Skills (Math 099). Students without formal prerequisites were sometimes allowed to register with permission of the Department of Mathemat-ics and StatistMathemat-ics. The student records analysed were all those that were readily available, forming a significant proportion of all students enrolled in these courses during these terms. For courses with multiple sections, all sections were taught using a very similar lecture schedule on required topics using the same textbook. Different sections may have different midterms, and they may have common homeworks. A brief description of each course (from the UVic Academic Calendar The University of Victoria (2011)) follows.

Course Coordination

Each of the ELUM courses studied was assigned a course coordinator by the De-partmental Chair (a normal part of the operation of an ELUM course at UVic). The course coordinator was responsible for operationally coordinating all sections of a course towards the goal of a common final exam. Coordination included reviewing

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drafts of course outlines for each section before they were submitted to the Depart-mental Chair for final approval (these are like a blueprint for how each section will operate during the term) and providing information and advice to instructors as needed. However, the responsibilities of instructors to the course coordinator were unclear. For example, were instructors responsible for taking direction from a course coordinator, or was it acceptable for them to operate counter to these instructions?

Mathematical Skills (Math 099)

This is a non-credit course offered by the Department of Continuing Studies. The goals of the course are to develop an understanding of basic math skills up to and including the Pre-calculus grade 11 level, to sharpen memory and reasoning power and to foster good work habits, responsibility and self-reliance. Students who pass this course can apply to register for Finite Math, Precalculus or Math for Elementary School Teachers I. There is no prerequisite. Topics include fractions, ratio and pro-portions; shape, space and geometry; right triangle trigonometry; exponents; lines, polynomials and factoring; functions; co-ordinate systems, graphing functions and equations; and solving equations and inequalities.

Precalculus (Math 120)

The prerequisite for Precalculus was a pass in either PoM 11, Pre-calculus 11 or equivalent, or a pass on a pre-test. We can infer from table 1.1 that approximately 91.3% of B.C. high school students who initially enrolled in PoM 11 became eligi-ble for Precalculus by passing PoM 11. This course included the following topics: elementary functions with emphasis on the general nature of functions; polynomial, rational, exponential, logarithmic, and trigonometric functions; conic sections; and plane analytic geometry. Since passing Precalculus satisfied the prerequisite for

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Cal-culus I or CalCal-culus for the Social and Biological Sciences, it was considered a remedial course for some students. However Precalculus was also a course that students could enter after they successfully finished their secondary education and it was available for credit since 1993. Thus for many students, Precalculus was an ELUM course.

Calculus I (Math 100)

Calculus I included the following topics: a review of analytic geometry; func-tions and graphs; limits; derivatives; techniques and applicafunc-tions of differentiation; antiderivatives; the definite integral and area; logarithmic and exponential functions; trigonometric functions; and Newton’s, Simpson’s and trapezoidal methods. The pre-requisites were: a minimum grade of B in PoM 12 or equivalent; a pass in Precalculus; or a pass on the Calculus I pretest. Thus using table 1.1, we can infer that approxi-mately 58.8% of B.C. high school students who initially enrolled in PoM 12 became eligible for Calculus I by achieving an A or B in PoM 12. The course was primarily intended for students planning to continue in math, statistics, science, engineering, or one of a few other topics.

Calculus II (Math 101)

Calculus II included the following topics: volumes; arc length and surface area; techniques of integration with applications; polar coordinates and area; l’Hospital’s rule; Taylor’s formula; improper integrals; series and tests for convergence; power series and Taylor series; and complex numbers. Students who enrolled in this course had already successfully completed Calculus I or its equivalent. As with Calculus I, the course was intended for students planning to continue in math, science engineer-ing, or one of a few other topics.

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Calculus for the Social and Biological Sciences (Math 102)

Calculus for the Social and Biological Sciences focused on the calculus of one vari-able with applications to the social and biological sciences. The following topics were included: limits; continuity; differentiation; applications of the derivative; exponen-tial and logarithmic growth; and integration. Students who enrolled in this course must have successfully completed PoM 12 or equivalent, or Precalculus, or passed a pre-test. Thus, using table 1.1, we can infer that approximately 94% of B.C. high school students who initially enrolled in PoM 12 become eligible for Calculus for the Social and Biological Sciences by passing PoM 12. This course was intended to be the only calculus required by programs in the social and biological sciences.

Matrix Algebra for Engineers (Math 110)

Admission to this course at UVic required admission to the Bachelor of Engi-neering program or Bachelor of Software EngiEngi-neering program at UVic (admissions requirements for these programs include achieving at least 73% in PoM 12). Thus using table 1.1, approximately 58.8% of students who initially enrolled in PoM 12 became eligible for Matrix Algebra for Engineers by achieving an A or B in PoM 12. This is a required course for several engineering programs. The topics covered in this course were: complex numbers; matrices and basic matrix operations; vectors; linear equations; determinants; eigenvalues and eigenvectors; linear dependence and independence; and orthogonality.

Logic and Foundations (Math 122)

Logic and Foundations included the following topics: basic set theory; counting; solution to recurrence relations; logic and quantifiers; properties of integers; mathe-matical induction; asymptotic notation; introduction to graphs; and trees. Students

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who enrolled in this course had successfully completed Calculus I, Calculus for the Social and Biological Sciences or Finite Math, or scored higher than 90% in PoM 12 and obtained permission from the department. The majority of students who enrolled for this course used Calculus I as their prerequisite. The majority of students who took Logic and Foundations during a Fall term were in the second year of their pro-gram. It was also reported by the course coordinator that the majority of students indicated on their course evaluation forms that they enrolled in this course because it was a requirement of their degree program.

Finite Math (Math 151)

Finite Math included the following topics: geometric approach to linear pro-gramming; linear systems; Gauss-Jordan elimination; matrices; compound interest and annuities; permutations and combinations; basic laws of probability; conditional probability; independence; urn problems; tree diagrams and Bayes formula; random variables and their probability distributions; Bernoulli trials and the binomial distri-bution; hypergeometric distridistri-bution; expectation; applications of discrete probability; and Markov chains. Students who enrolled in this course had successfully completed PoM 11 or equivalent, Foundations of Mathematics 12, or 1.5 units of credit in Math courses numbered 100 or higher, or passed a pre-test. We can infer from table 1.1 that approximately 91.3% of B.C. high school students who initially enrolled in PoM 11 became eligible for Finite Math by passing PoM 11.

Math for Elementary School Teachers I (Math 161 or 160A)

The prerequisites for Math for Elementary School Teachers I were PoM 11 or 12, or Foundations of Mathematics 12, or a pass on an pre-test. We can infer from table 1.1 that approximately 91.3% of B.C. high school students who initially enrolled in

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PoM 11 became eligible for Math for Elementary School Teachers I by passing PoM 11. This course included the following topics: number systems and their properties; the set of real numbers and its subsets; the interpretation of numerical operations with applications including combinations and permutations; standard computation algorithms; basic statistics including simple sampling and design issues; and problem solving. It was intended for and required of students who were enrolled in an educa-tion program.

Math for Elementary School Teachers II (Math 162 or 160B)

The prerequisite for Math for Elementary School Teachers II was successful com-pletion of Mathematics for Elementary School Teachers I or permission of the depart-ment. This course was intended for and required of students who were enrolled in an education program. Topics included: mental computation and estimation; non-standard computation algorithms; basic set theory; probability; basic algebra and functions; two- and three-dimensional objects; symmetry; similarity; compass and straight edge constructions; transformational geometry; and measurement topics in-cluding length, area and volume. Problem solving was emphasized throughout.

Course Letter Grades

Upon completion of a course, a student would receive one of the following letter grades, each of which had an associated grade point value that could be used in the computation of overall grade point average:

Student readiness, engagement and success in entry level undergraduate mathematics courses

Contents

List of Tables

List of Figures

Introduction

1.1

Why Do This?

1.2

What is the Problem and What Can We Do

About It?

1.3

Thesis Questions

1.4

Literature Survey

1.4.1

Student Readiness

1.4.2

Student Engagement

1.4.3

Pedagogy

1.5

Mathematics at the University of Victoria

1.5.1

The UVic Academic Year

1.5.2

ELUM Courses Studied