Addressing Calculus Readiness through the Implementation...

Addressing Calculus Readiness

through the Implementation of a

Co-requisite Program

Addressing Calculus Readiness through the

Implementation of a Co-requisite Program

Abstract: This paper explains the structure, implementation and prelimi-

nary results of a program designed to refresh precalculus topics for Calculus I

students. At our university, the Calculus I DFW rates increased at an alarming

rate and the department took action with the implementation of a co-requisite

program that we call Calculus Readiness. We discuss the details of the pro-

gram and conclude with a preliminary statistical analysis of the program’s

outcomes.

Keywords: Curriculum Implementation, Co-Requisite Program, Calcu-

lus Readiness

1 Motivation

In recent semesters at our university the Math and Computer Science

Department has been with faced with students entering into freshman

level Differential Calculus who have insufficient knowledge of prerequi-

site knowledge. This issue is well known; see for example [1]. Students

were not able to recall basic facts like the equation for the circumfer-

ence of a circle, the unit circle and logarithmic properties. They were

having trouble solving quadratic equations and simplifying algebraic ex-

pressions. We speculated that students have forgotten these necessary

skills due to lack of algebra based math courses taken in their last year

of high school.

2 Lori Carmack, Erika Gerhold, and Ryan M. Shifler

The inability to understand and apply precalculus material was caus-

ing Calculus I DFW rates to increase at an alarming rate. We decided

that the current environment was not just unacceptable but it was un-

sustainable. This is a preliminary report on the steps that we have

taken to remedy the serious issues that have arisen in our university’s

Differential Calculus course.

2 Background

Before we dive into the program we will share some information about

our university and its Mathematics and Computer Science Department.

Our university is a regional comprehensive university with approximately

8700 undergraduate students located in Maryland. The department

includes thirty-three mathematics faculty members including adjuncts,

full-time lecturers, and tenure and tenure-track faculty.

The department formed a committee of eleven faculty members to

address the issue at hand. The discussions were extensive. The commit-

tee often found itself back tracking to find a solution that would work

well for our students and our department. We eventually agreed on an

idea that we implemented as our Calculus Readiness Program (CRP)

inspired by the articles [2, 3]. Our department is supportive of its mem-

bers efforts to improve the education in lower level mathematics courses.

We felt that is was important that the department unanimously agreed

on the implementation of the CRP. Moreover, updates on the program

were given through out its implementation at the regularly scheduled

departmental meetings. We describe more details about the CRP in the

following section.

Calculus Readiness 3

3 Calculus Readiness Program

All students enrolled in Calculus I are required to attend weekly student-

led precalculus review sessions and complete assessments on the mate-

rial. In order to have a baseline for what our students’ strengths and

weaknesses are, we administer the Pre-Gateway Assessment during the

first week of classes. Throughout the next seven weeks, students take a

weekly mini-assessment to make sure that they are progressing through

the precalculus material. After seven weeks, students take the Gateway

Assessment with the requirement of needing to pass it in order to earn

a passing grade in Calculus I. We discuss these assessments later in this

article.

4 Calculus Readiness Sessions and Calculus Readiness Lead-

ers

The goal is to strengthen the students’ algebra and trigonometry skills

through active participation in Calculus Readiness Sessions (CRS). The

CRS are led by a tutor that we call a Calculus Readiness Leader (CRL).

During the sessions, the CRL has students work in groups to complete

algebra and trigonometry problems that are relevant to the current cal-

culus material that is being covered in class. Much of the success of the

CRP is to the credit of the CRLs. To facilitate active learning there is

a strong emphasis on recruiting strong and charismatic students to be

CRLs. They work under a faculty member and are given a significant

amount of flexibility to implement active learning in their CRS. For ex-

ample, some students run their CRS with a traditional approach where

they float around to help students as needed. Another CRL has students

in their session break up into teams to compete to see who can complete

problems the fastest at the board. Hiring suitable students to be CRLs


is the most significant challenge to overcome to successfully implement

the CRP. The CRLs are actively recruited outside of the Mathematics

and Computer Science Department since the demographics of Calculus I

at our university also include chemistry, physics, geography majors and

majors from other STEM disciplines. We anticipate increased success

by having more Calculus I students be able to identify with someone

from their own major. Successfully recruiting from across disciplines is

a work in progress.

5 Calculus Readiness Session Materials

Each CRS consists of a mini-assessment on the previous week’s topics,

a short lecture to introduce the current week’s topics, and time for the

students to work together on practice problems with the CRL to assist

if necessary. In order to create these problems, we looked at a standard

Calculus I weekly schedule and aimed to cover precalculus topics that

would be necessary for the next week’s calculus lectures. We have also

included the topics we find most important (e.g. evaluating trig and

solving equations) throughout multiple weeks. Each weekly problem set

contains between ten and twenty problems, many of which have multiple

parts. We don’t expect the students to finish all problems during the

CRS and hope they will use these problem sets as additional practice to

help them prepare for the Gateway Assessments. The following questions

are examples of practice problems from the problem sets.

During the first week of Calculus Readiness we ask the following

question:

Evaluate:

1. cos(0)

2. sin(0)

3. tan(π/4)

4. sec(2π/3)


5. cot(π)

6. sin(3π/2)

7. csc(−π/6)

8. sec(−7π/4)

During the third week we reinforce the topic of evaluating trigono-

metric functions with the following questions:

Evaluate:

1. sin(π/6)

2. cos(−45◦)

3. tan(π)

4. cot(π/3)

5. sec(9π/4)

6. csc(5π/6)

7. sin(0)

8. tan(240◦)

9. csc(5π/6)

10. cos(270◦)

11. sin(−4π/3)

12. sec(8π/3)

During Week 5, students practice the following examples to prepare

to learn the Chain Rule in Calculus I:

The following functions are of the form h(x) = f(g(x)). Determine the

functions f(x) and g(x).

1. h(x) = tan(πx)

2. h(x) = cos(ex)

3. h(x) = ecos(x)

4. h(x) = ex2−x

5. h(x) = sin(cot(x))

6. h(x) = sin(x3)

7. h(x) = (1 + 3x2 + 5x3)9

8. h(x) = ln(2− x2)

9. h(x) =√x− 3

10. h(x) =(x2+1x2−1

)4

In addition to the weekly problem sets, we have also created a re-

source page that every Calculus I student is able to access through their

Calculus I Canvas account. This page contains an overview of the weekly

goals of Calculus Readiness and a breakdown of the topics that they need

to review. Within each weekly breakdown, we include a summary of the

topics covered, their importance, the tasks the student should focus on


that week, the weekly problem set, and additional resources they can

utilize. These additional resources include links to educational websites

and videos.

Figure 1. Sample Canvas Page

6 The Assessments

6.1 Gateway Assessment

The Gateway Assessment is administered on WebAssign two times dur-

ing the semester—the first (Pre-GA) and seventh week (GA) of classes.

In order to receive an A, B, or C in the course the students must per-

form satisfactorily on the Gateway Assessment. If a student is not able

to receive an A, B, or C, they will have the opportunity to enroll in a 7-

week precalculus course for the second half of the semester to give them

an opportunity to satisfy a mathematics general education requirement.

We discuss more about this course later in the article.

The Gateway Assessment consists of twenty-four multiple choice ques-

tions covering precalculus topics that are the most crucial for students to

master to succeed in calculus. Our purpose for making these assessments

multiple choice is due to students’ varying comfort levels with inputting


their answers into WebAssign. These two exams are administered with

a professor in the room to avoid help from outside resources. Students

are not allowed to use a calculator and they are given blank paper for

working through their solutions.

Having students take the Pre-GA provides a baseline to help us gauge

student improvement with the emphasized material over the course of

the semester. Pre-GA scores also give us some indication of the strengths

and weaknesses of our incoming Calculus I students. Figure 2 shows

the percentage of students who correctly answered each question on the

Pre-GA given at the beginning of Spring 2019. Students were weak-

est on questions related to trigonometry, logarithms, and exponentials.

There were only two questions that more than 60% of students answered

correctly—factoring a simple quadratic and finding the slope between

two given points.

Q1 Q4 Q7 Q10 Q13 Q16 Q19 Q22

Pre−GA (Week 1) By Question

Pe

rce

nta

ge

of

Stu

de

nts

with

Co

rre

ct

Re

sp

on

se

02

04

06

08

01

00

Figure 2. Pre-GA results by question


6.2 Mini-Assessments

In order to better prepare students for the Gateway Assessment, stu-

dents take mini-assessments each week running up to the GA. The mini-

assessments cover topics from the previous week’s CRS which aligns with

the calculus material that the students will be learning in the upcom-

ing weeks in class. From those topics, we have written approximately

ten multiple choice (or matching) problems that we think sufficiently

test those topics. From this test bank of ten questions, WebAssign

randomly generates five questions for each students’ mini-assessment.

Mini-assessments are password protected and must be completed during

a CRS.

The following problems are examples of the multiple choice problems

students see on a mini-assessment. These questions test students on the

composite functions they practiced during Week 1 and the evaluation

of trigonometric functions they completed during a CRS about the unit

circle during Week 3.

For f(x) = ln(x) and g(x) = x2 +x, find the composition, h(x) =

(f ◦ g)(x).

a) h(x) = ex2+x;

b)h(x) = ln(x)(x2 + x);

c) h(x) = (ln(x))2 + ln(x);

d) h(x) = ln(x2) + x;

e) h(x) = ln(x2 + x).

Find the exact value of the trigonometric function at the given

real number:

sin

(7π

3

)a)√32 ; b) −

√32 ; c) 1

2 ; d) π3 ; e) − 1

2 .


7 The 7-week prerequisite back up plan

We have created an option for students who are struggling with the

precalculus and calculus material within the first seven weeks of the

semester. We realize that for some students who are missing larger parts

of the prerequisite material, a one hour per week CRS is not enough

time for adequate learning. These students are able to withdraw from

Calculus I and register for a seven week prerequisite College Algebra and

Trigonometry class. The cost of this course is already included in the

tuition and fees for full time students that are currently enrolled in 15

credits or less. If a student is enrolled in 16 credits or more, they need

special permission and have an additional cost in tuition for one or two

extra credtis over nine-teen.

The course is our standard four credit College Algebra and Trigonom-

etry class. It meets four days a week, for one hour and 40 minute sessions

to cover topics involving but not limited to solving equations, graph-

ing functions, working with exponential and logarithmic functions, and

trigonometry. To avoid conflicts with other classes, we have scheduled

this class in the evening. This course gives the students who choose

to enroll in the seven week College Algebra and Trigonometry class the

opportunity to earn credits for a math course in the same semester, as

well as prepares them to retake Calculus I the following semester.

For the Spring 2019 semester, we had seven out of the eighty-eight

students in Calculus I withdraw and register for the seven-week course.

8 Other university resources

Utilizing university resources to their full benefit of the CRP is another

important aspect of the project. There are two university offices that

we worked with: Instructional Design and Delivery (ID&D), and Center


for Student Achievement (CSA).

The staff of ID&D were fantastic resources and colleagues when it

came to creating materials on Canvas and WebAssign. The staff at

ID&D met with the Calculus Readiness team multiple times to dis-

cuss technical difficulties that arose from trying to use Canvas and We-

bAssign. Having experts on campus that specialize in these topics is

invaluable and significantly contributed to a successful development of

online materials.

The CSA generously offered the CRLs the opportunity to attend

tutor training that is offered to other CSA tutors. It’s our goal to require

the training for all CRLs in the future.

9 Preliminary Results

Thus far we have implemented our CRP for two semesters, Fall 2018 and

Spring 2019. We received IRB approval to assess the program during

Spring 2019 and are currently in the process of collecting and analyzing

data. Although the program is in its early stages, we are seeing some

preliminary results worth mentioning. First, students’ scores on the

Gateway Assessment noticeably improved over the first seven weeks of

the semester (Figures 3 and 4).

The mean scores on the Pre-GA and GA were x̄ = 11.87 and x̄ = 16.25,

respectively, a significant increase of 18.25% (t(54) = 8.27, p < .0001).

It may be argued that the improved performance is a result of students

naturally gaining in proficiency due to the semester being in session, but

this may be countered by comparing the histograms of the GA scores

at Weeks 1 and 7 (Figure 4)—such differences in the shape and skew of

the histograms would not be expected if improved facility were solely a

consequence of students being enrolled in Calculus I.

When breaking down students’ performance by question (Figure 5), on


Pre−GA GA

05

10

15

20

Pre−GA (Week 1) vs. GA (Week 7)

Nu

mb

er

of

Co

rre

ct

Qu

estio

ns o

ut

of

24

Figure 3. Comparison of students’ Gateway Assessment scores

the Pre-GA, there were only 14 of 24 questions that at least half of the

students answered correctly. On the GA, at least half of the students

correctly answered 21 of the 24 questions.


Pre−GA Scores (Week 1)

Scores

Count

0 5 10 15 20

05

10

15

20

GA Scores (Week 7)

Scores

Count

0 5 10 15 20

05

10

15

20

Figure 4. Comparison of students’ Gateway Assessment scores

Q1 Q3 Q5 Q7 Q9 Q11 Q14 Q17 Q20 Q23

Pre−GA (Week 1) By Question

Perc

enta

ge o

f S

tudents

with

Corr

ect R

esponse

020

40

60

80

100

Q1 Q3 Q5 Q7 Q9 Q11 Q14 Q17 Q20 Q23

GA (Week 7) By Question

020

40

60

80

100

Figure 5. Pre-GA versus GA results by question.

We also administered the GA to our Calculus II students during the

sixth week of Spring 2019 (Figures 6 and 7); mean scores were 67.7%

for Calculus I students and 65.8% for Calculus II students. While this

difference is not statistically significant, it is encouraging that our Cal-

culus I students appear to have roughly the same precalculus facility

as students who are at least one semester ahead in their mathematical

study and have earned a grade of C or better in Calculus I. We note that


the majority of our Calculus II students experienced the CRP during Fall

2018.

Calc I Pre−GA Calc I GA Calc II GA

05

10

15

20

Pre−GA (Week 1) vs. GA (Week 7) vs. Calc II GA (Week 6)

Num

ber

of C

orr

ect Q

uestions o

ut of 24

Figure 6. Comparison of Calculus I and Calculus II scores


Calc I Pre−GA (Week 1)

Scores

Count

0 5 10 15 20 25

05

10

15

20

Calc I GA (Week 7)

Scores

Count

0 5 10 15 20 25

05

10

15

20

Calc II GA (Week 6)

Scores

Count

0 5 10 15 20 25

05

10

15

20

Figure 7. Comparison of Calculus I and Calculus II scores

10 Conclusion

With administrative and departmental support, we are excited to con-

tinue to implement and improve the CRP after receiving positive student

feedback and promising preliminary statistics. Indeed, in a survey given

to our Calculus I students at the end of Spring 2019, 63.4% of respon-

dents indicated they would recommend incoming students to enroll in a

Calculus I course with the CRP as opposed to one without, even with the

CRP’s stringent requirements. Additionally, 73.2% of students felt the

CRP helped them refresh their precalculus knowledge and skills. Given

that this is the main emphasis of the CRP, this student perception of

the program is another hopeful indication of success. Anecdotally, an

additional benefit of the CRP is that instructors feel they can cover the

Calculus I material more effectively.


ACKNOWLEDGMENTS

The authors would like to thank Dr. Michael S. Scott, Dean of the

Henson School of Science and Technology at Salisbury University for his

generous support of the Calculus Readiness Program.

BIOGRAPHICAL SKETCHES

L. Carmack received a Ph.D. in applied mathematics from the University

of California, Santa Barbara in 1996 and has been teaching at Salisbury

University since 2012. Some of her favorite pastimes include analyzing

the financial markets, watching/playing tennis, and spending time with

her two kids; she welcomes any communication that stems from her

paper.

E. Gerhold received an M.S. in Mathematics from Louisiana State Uni-

versity in 2015. She enjoys being an educator at Salisbury University

and the nearby community college, doing yoga, and hanging out with

her dog, Joey.

R. Shifler received a Ph.D. in Mathematics from Virginia Tech in 2017.

When he isn’t working on the implementation of the Calculus Readiness

Program and the scholarship of teaching, you can find him working with

undergraduates on research questions arising from Schubert Calculus.

His cat’s name is McClane.

REFERENCES

[1] Agustin, Z. and M. Agustin. Algebra and Precalculus Skills and Per-

formance in the First-Semester Calculus. International Journal of

Case Method Research and Applications, 21(3), 2009.


[2] Bressoud, D, V. Mesa, C. Rasmussen. Insights and Recommenda-

tions from the MAA National Study of College Calculus. MAA Press,

2015.

[3] Vestal, S, T. Brandenburger, A. Furth. Improving Student Success

in Calculus I Using a Co-Requisite Calculus I Lab. PRIMUS, 25(4):

381–387, 2015.

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