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Rethinking Precalculus and Calculus: A Learner-Centered Approach

Rachel J. Weir

To cite this article: Rachel J. Weir (2020) Rethinking Precalculus and Calculus: A Learner-Centered Approach, PRIMUS, 30:8-10, 995-1016, DOI: 10.1080/10511970.2019.1686669

To link to this article: https://doi.org/10.1080/10511970.2019.1686669

Published online: 04 Dec 2019.

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PRIMUS, 30(8–10): 995–1016, 2020Copyright © Taylor & Francis Group, LLCISSN: 1051-1970 print / 1935-4053 onlineDOI: 10.1080/10511970.2019.1686669

Rethinking Precalculus and Calculus:A Learner-Centered Approach

Rachel J. Weir

Abstract: Like many math educators, I have spent much of my career bound to tradi-tional methods of instruction and assessment. In recent years, motivated by a growingunderstanding that such approaches may not result in equitable or inclusive classroomenvironments, my teaching philosophy has shifted radically. In this article, I describehow I transformed my Precalculus and Calculus I courses, incorporating mastery grad-ing, inquiry-based learning, and metacognitive and community building activities. Idescribe the motivation and philosophy behind the key components of these courses,the observed effect on student learning and attitudes, and my own responses to this newapproach.

Keywords: Calculus, growth mindset, inquiry-based learning, master grading,mastery-based testing, metacognition, precalculus, self-efficacy, specifications grading,standards-based grading

1. INTRODUCTION

Over the past few years, my views about mathematics teaching have shiftedradically, and this shift has been reflected in major transformations to thecourses I teach and to my approach to teaching in general. This was initiatedby a deeper exposure to the history of racial inequities in the United States [1,33], which led me to explore possible ramifications for my classes [9, 26, 28,29, 34], together with the literature on effective approaches to teaching andlearning [8, 15, 16, 35].

National data shows that “fewer than 40% of the students who entercollege with the intention of majoring in a STEM1 field complete a STEMdegree.” [31, p. 5]. Common reasons cited for leaving STEM fields are “lackor loss of interest in science” and “poor quality teaching by [STEM] faculty,”

Address correspondence to Rachel J. Weir, Department of Mathematics, AlleghenyCollege, 520 N. Main Street, Meadville, PA 16335, USA. E-mail: rweir@allegheny.edu1Science, Technology, Engineering, and Mathematics

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and these concerns were shared both by students who left STEM fields andthose who persisted [32, p. 32]. The loss of students in these fields is particu-larly notable for women [32, p. 3], and African American, Latinx, and NativeAmerican students [18].

For these reasons, rethinking how I teach introductory mathematicscourses became the focus of my recent sabbatical. At the private liberal artscollege where I teach, we have historically offered two Calculus sequences: atraditional sequence that mainly serves students with a natural science majoror minor and a Calculus for Life and Social Sciences sequence. The mathplacement test that we use measures only student preparedness for Calculus I;students who do not earn a high enough score begin either in Precalculus orin Calculus I for Life and Social Sciences, depending on their intended majorand minor. Our precalculus course is designed solely to prepare students totake our traditional Calculus I sequence and does not satisfy any graduationrequirements. The class sizes for these courses are typically capped at 18, withfaculty teaching two of these courses per semester as part of a 3-3 teachingload. Historically, our precalculus course has been taught by non-tenure-trackinstructors but, in order to better understand the barriers to success for studentsin this course, I taught this course for the first time in spring 2018; I taught itagain in fall 2018 and spring 2019.

The time and freedom provided by my sabbatical allowed me to delvemore deeply into educational literature and what I learned motivated andempowered me to move away from more traditional teaching methods. Thechanges that I have made have reenergized my teaching, led me to deeperconnections with my students, and, I believe, created a more positive learningenvironment. By highlighting the ideas and research that have most greatlyinfluenced me, I hope that I can provide encouragement to others to considernew ways to think about their teaching and, in particular, to see the poten-tial benefits, to instructors and students alike, of a mastery grading system.Although I was fortunate to have the time to transform entire courses, I hopethat the different components will spark readers to think about smaller changesthat they can make in their courses and, more generally, to think about theircourses differently.

The primary focus of this article will be a description and discussion ofthe mastery grading scheme that I used in Precalculus. The structure of myCalculus I course was very similar; details can be found in my syllabus [24].In Section 2, I will summarize how I developed the course objectives (seeFigure 1). Each objective and its assessment will be described in Section 3,with a particular focus on Objectives 4 and 5 as they are not standard contentobjectives. The information in this section will lead the reader to an under-standing of the information conveyed in Table 1, which is extracted from myspring 2018 syllabus. In Section 4, I will describe how I managed the workloadin the course. My reflections on this new approach to grading will appear inSection 5.

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Figure 1. Course learning objectives for Precalculus.

Table 1. Determination of final grades

To earn this grade Do all of the following

A Objective 1: Master 23 Learning ObjectivesObjective 2: Complete 2 application problemsObjective 3: Complete 2 proof problemsObjective 4: Earn 22 out of 24 Habits and Skills pointsObjective 5: Earn 90% of available Classroom Community points

B Objective 1: Master 20 Learning ObjectivesObjective 2: Complete 2 application problemsObjective 3: Complete 1 proof problemObjective 4: Earn 19 out of 24 Habits and Skills pointsObjective 5: Earn 80% of available Classroom Community points

C Objective 1: Master 18 Learning ObjectivesObjective 2: Complete 1 application problemObjective 3: Complete 1 proof problemObjective 4: Earn 17 out of 24 Habits and Skills pointsObjective 5: Earn 70% of available Classroom Community points

D Objective 1: Master 15 Learning ObjectivesObjective 2: Complete 0 application problemsObjective 3: Complete 0 proof problemsObjective 4: Earn 15 out of 24 Habits and Skills pointsObjective 5: Earn 60% of available Classroom Community points

F Do not meet the requirements for a D

2. DEVELOPING THE COURSE OBJECTIVES

When I first began to read literature about teaching and learning, I was drawnto the concepts of fixed and growth mindsets [13] and ramifications for teach-ing and learning mathematics [5], particularly as what I was reading resonatedwith my own mathematical experiences. Here, when someone has a fixed

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mindset, they believe that their abilities are predetermined and fixed, whereassomeone with a growth mindset believes that they can develop their abilitiesthrough appropriate effort. It is not surprising to learn that a fixed mindset pre-dominates in mathematics [22]. Indeed, every mathematician, upon revealingtheir profession, has heard the comment “I’m not a math person” on multi-ple occasions. Conversely, any signs of mathematical ability quickly lead to alabel of “smart” or even “gifted,” which also can have negative consequences[25, 36].

It has also been shown that a fixed mindset about one’s abilities can leadto “performance goals (in which individuals are concerned with gaining favor-able judgments of their competence),” which in turn can result in “avoidanceof challenge and a deterioration of performance in the face of obstacles” [11,p. 256]. On the other hand, a growth mindset can lead to “learning goals (inwhich individuals are concerned with increasing their competence),” whichcan result in “the seeking of challenging tasks and the maintenance of effec-tive striving under failure” [11]. Recent research also suggests that it may bethe combination of negative stereotypes with predominantly fixed mindsets ina field that results in a lack of gender equity within a field [4, 17, 22]. Foster-ing a growth mindset can also help to combat the effects of stereotype threat,a phenomenon in which a student’s anxiety associated with the possibility ofconfirming negative stereotypes can result in lowered performance [2, 33].

All of this confirmed for me that focusing on promoting a growth mindsetin my classes would be crucial for success, especially in introductory mathe-matics courses. The work of Jo Boaler was particularly helpful in this respectas she provides specific evidence-based strategies for promoting growth mind-sets in mathematics classrooms [5, 6, 37]. Several of these strategies guidedthe design of my courses, including: encourage and believe in all of your stu-dents; value struggle and failure; give growth praise and help; value depthover speed; and encourage students to pose questions, reason, justify, and beskeptical.

I was also influenced by several books that focused more specifically onhelping students adjust to the demands of college [15, 27]. I was initially drawnto the concept of self-regulated learning, which “encompasses the monitoringand managing of one’s cognitive processes as well as the awareness of andcontrol over one’s emotions, motivations, behavior, and environment as relatedto learning” [27, p. 5]. As Nilson [27] describes, we, as professors, are “profes-sional learners” and were likely able to pick up the ability to self-regulate ourlearning on our own. On the other hand, many students have not been exposedto these skills and this led me to begin to include activities and assignmentsinvolving goal setting, self-reflection, identification of prior knowledge, andstudent-created review sheets. By doing this, I hoped to create a more equitablestarting point for all students in my classes.

I also embraced the idea of learner-centered teaching in which “[t]eac-hers create conditions that foster growth and learning, but it is the students

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Figure 2. Facilitative teaching: Principles that guide its implementation [35, p. 72].

who master the material and develop the learning skills” [35, p. 60]. In herbook, Weimer [35] provides an overview of how learner-centered teachingconnects with various educational theories and presents evidence that points tothe benefits of its use. The influence of her work permeates my courses, fromthe grading system to the mode of instruction to the day-to-day interactionswithin my classroom. A good summary is provided in the list of principles inFigure 2, which is taken from her book.

Fostering a sense of community in my classes also fits well with thelearner-centered approach that so appealed to me. For example, Barr and Taggwrote that “[i]n the Learning Paradigm . . . a college’s purpose is ... to makestudents members of communities of learners that make discoveries and solveproblems” [3, p. 16]. In addition, Weimer states that “[l]earner-centered teach-ing is about creating classrooms in which students begin to mature and actmore responsibly about their own learning and toward the learning of others”(the emphasis is mine) [35, p. 147] and Boaler reports that “[r]esearch showsthat when students work on mathematics collaboratively, which also givesthem opportunities to see and understand mathematics connections, equitableoutcomes result” [7] (as cited on p. 105 in [5]).

These considerations led me to identify the five course objectives seenin Figure 1. The grading system that I subsequently developed was comple-mented by the use of inquiry-based learning (IBL) techniques. In particular,I used an IBL workbook approach, in which students initially work throughan assigned number of problems before each class meeting; classes are thendevoted primarily to student presentations of solutions and subsequent classdiscussions. Research has shown that IBL methods can be beneficial to womenand to students who have previously struggled in mathematics courses [21].The methods of IBL also closely aligned with the principles of learner-centeredteaching such as those in Figure 2.

3. ASSESSING THE COURSE OBJECTIVES

Mastery grading aligns with my course goals in multiple ways and my assess-ment methods combine all three forms, mastery-based testing, specifications

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grading, and standards-based grading. Particularly important is the fact thatmastery grading gives rise to a cycle of submission, feedback, and revisionthat naturally cultivates a growth mindset and provides motivation for studentsto revisit their work and to further their understanding of course material. Thisallows instructors to maintain high standards, as students will have opportuni-ties to revise and resubmit coursework. Interleaving of new and old material isanother natural consequence of this cycle, potentially resulting in deeper, moresustained learning [10, 12, 20].

In addition, whereas the first three objectives listed in Figure 1 couldconceivably have been tested using a traditional points system with quizzes,problem sets, and exams, the last two were less amenable to this. In com-parison, a mastery system is flexible enough to adjust to a variety oflearning goals, while still allowing for points-based components to meet,for example, departmental and institutional grading requirements. Masterygrading also allows instructors to move away from the idea that every-thing must have a point value in order to motivate students to complete thework. Indeed, constructing a mastery grading system requires instructors tomore clearly articulate the objectives of the course, so the students shouldbe able to more easily see how each activity fits into the overall coursestructure.

Finally, identifying specific objectives and assessing student progresstowards these imbues the final grade with much more meaning than in a tra-ditional points system. By the end of the semester, both the instructor and thestudent should have a clear picture of what the student understands and whereany remaining gaps in their knowledge may be. In my case, I was able to set upmy system so that a C grade in Precalculus, which is necessary for progressto Calculus I at my institution, is achieved only when a student meets cer-tain, specific benchmarks, namely, mastering a minimum number of contentobjectives, writing at least one correct mathematical proof, and solving andexplaining at least one application problem.

With these ideas in mind, I will now describe how I assess each of the fivecourse objectives listed in Figure 1.

Objective 1: Mathematical Content

Students’ understanding of the core mathematical content of the course isassessed using mastery-based testing on timed assessments that occur in classevery 2 weeks, with the final assessment taking place during the 3-hour finalexam period. These timed assessments contain one or two problems for eachof the content objectives that we have covered in the course so far (see [24]for lists of core content objectives for my Precalculus and Calculus I courses).Solutions are graded on a Pass/No Pass basis, requiring that a solution mustbe completely correct to earn a Pass, and students have to earn two passes in

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order to Master an objective. One of these passes can be earned by correctlypresenting a related workbook problem in class.2

Near the end of my spring 2018 course, I started allowing students toretake objectives during office hours. Students could attempt each objec-tive once per day, but there was no limit on how many objectives could beattempted each day. As a result, my office hours during the last weeks of thesemester were extremely busy and I had a large number of problems to gradeeach day. To make this more manageable, I have since implemented a “level”system that I learned about at a recent conference [23]. Students begin at Level1, allowing them to retake one objective per week. After five retakes, studentsmove up to Level 2 and are allowed to retake two objectives per week; theymove up to Level 3 after an additional 10 retakes. At each level, students arelimited to two attempts per objective per week.

In my most recent course, I gave students the opportunity to earn bonusretakes by submitting corrections to timed assessment problems. If their cor-rection for a particular problem was correct, then they were entitled to onebonus retake for that objective and this retake would not count against theirusual retake allowance. Students were able to earn up to three bonus retakesper timed assessment, so this did not add significantly to my grading load but,for some students, it appeared to be an effective way to encourage them torevisit their incorrect timed assessment answers.

The selection of the core content objectives is an important and time-consuming task based on multiple factors. First, the instructor (or department)must decide on the content they deem to be most important, both in terms ofa comprehensive introduction to a topic and in terms of prerequisite materialfor subsequent courses. My experience has been that 20 to 30 objectives canbe reasonably tested during a semester, with some variability depending onhow many passes are required for each objective and how many opportunitiesa student will have to attempt a problem.

Second, it is important to consider which content can be tested effectivelyvia a timed assessment approach. Problems that involve multiple parts or thatrequire substantial higher-order thinking could best be tested elsewhere, whichis how I use my Applied and Proof Problems (see Objectives 2 and 3). Finally,it will be necessary to construct multiple problems of a similar level for eachcontent objective to allow for reassessments. If a particular objective doesnot lend itself to this approach then it is better tested elsewhere or, perhaps,omitted.

Even with this advance planning, I found myself having to make mid-semester adjustments to the list of objectives when I taught my precalculuscourse for the first time. I began with 30 objectives, but eventually whittled

2In the instructor version of the workbook, problems connected to objectives wereflagged; the student version did not contain these flags.

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Figure 3. Sample timed assessment problems.

these down to 25, adjusting the Objective 1 requirements in Table 1 accord-ingly. Typically, I made these cuts so that we could spend more time focusingon what I felt were more essential topics. For example, I removed the objec-tive “I can graph transformations of the sine and cosine function and determinethe equation of a trigonometric function from its graph or a table of values.”Although I value this skill, I felt that ensuring that, for example, my studentscould use the unit circle to calculate the values of the trigonometric functionsat special angles was a more important first step and would serve them betterin calculus courses. Since I was reducing the number of requirements that mystudents needed to meet, I did not create any extra work for them and, as aresult, I did not observe any increased anxiety as a result of these changes.

My objectives continue to evolve with each iteration of my precalculuscourse. Common changes include: adjusting the wording of an objective tomore accurately reflect what I wanted to test; splitting an objective into mul-tiple objectives so that I could assess each piece separately, giving a moreaccurate picture of students’ understanding and motivating them to develop abroader understanding of the topic (see the example in the next paragraph); andreordering objectives based on previous semester results so that early, “easier”objectives can serve as confidence boosters.

Figure 3 shows two questions taken from the final timed assessment inmy spring 2018 precalculus course; here, each question tests a different partof objective FT.6. (In later versions of this course, I split these skills into sep-arate objectives.) As shown, the statement of the objective is provided withthe question so these problems are not necessarily testing students’ ability toselect the correct approach. Rather, the idea is to check whether students havethe foundational skills that they need.

Objective 2: Applied Problems

Assessment of students’ ability to apply the mathematics of precalculus isachieved via out-of-class assignments, using specifications grading and with

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Figure 4. Sample Precalculus applied problems [19].

Figure 5. Specifications for applied and proof problems.

weekly revisions allowed. I began the spring 2018 semester with five cat-egories of applied problems (Linear Functions, Polynomials, ExponentialFunctions, Trigonometric Functions, and Optimization). In the end, this wasreduced to just two (Linear Functions and Exponential Functions) to allow usto focus more on depth and quality; these are the categories that I continueto use. (Sample problems are shown in Figure 4.) Within each category, eachstudent solves different problems, giving students ownership of their problemswhile also reducing the possibility of unwanted collaboration. To earn creditfor a problem, students need to satisfy the specifications for Applied Problemsand Proof Problems shown in Figure 5.

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Figure 6. Sample precalculus proof problems.

As described in Table 1, in order to earn a C in the course and move onto Calculus I, students only need to complete one applied problem. On theother hand, students who want to earn an A or a B need to complete two prob-lems successfully, one from each category. Thus, students can make decisionsabout these problems based on their own goals for the course. Such autonomypaired with a mastery approach has the potential to lead to increased studentmotivation to learn [30, 35].

As the specifications in Figure 5 indicate, these problems serve two mainpurposes. First, they are an opportunity for students to gain experience andconfidence solving “word problems.” Second, they require students to thinkabout how to explain their work clearly and completely, using the conven-tions of mathematical writing. For most students, this is the first time theyhave been asked to complete a task of this type and, for this reason, the cycleof submission–feedback–revision is an essential part of the process. As addi-tional scaffolding, we also spend time in class talking about the specifications,examining sample solutions, and discussing how students can transform theirscratch work into an appropriate submission.

Objective 3: Proof Problems

The proof problems are assigned and graded in a manner similar to the appliedproblems, again with two categories, Trigonometric Identities and Propertiesof Functions, and using the specifications listed in Figure 5. As can be seenfrom the examples in Figure 6, the proof problems are chosen to be accessi-ble enough that students can develop their proofs without the use of outsidesources. To provide additional support, I post correct proofs on our Classroompage as they come in, so that other students can use these a guide. If a studentdoes not earn credit for their initial submission, they can revise and resubmittheir proof, using my comments as a guide.

All of the trigonometric identity problems require students to rewrite othertrigonometric functions in terms of sine and cosine and then to make use of thePythagorean trigonometric identity in some form, so all of the problems are ata similar level. With larger classes, it would be necessary to generate a greatervariety of identities in order to create a large enough collection. To create a

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similar uniformity for the other category, I focused on properties involvingeven and odd functions. Such problems have the added advantage of requiringstudents to make use of mathematical definitions in a formal way, somethingfirst-year students typically struggle with initially.

For the proof and applied problems, as with other assignments, studentscould learn from mistakes without penalty and were motivated to strive forsolutions that were both completely correct and met all of the specificationsfor mathematical writing, as the following quote from a Productive Failureassignment (see Objective 4 for more details) illustrates. Regarding his workon an applied problem, the student writes:

When I first attempted this I didn’t quite have a clear understanding of what itwas supposed to mean and how it was supposed to be phrased. However, throughmultiple attempt [sic] and corrections, I now have a better understanding of howproblems like these should be phrased to make the most sense.

Objective 4: Habits and Skills

The revelation that students may struggle in first-year college-level coursesbecause they have not been exposed to effective strategies for college success(see, for example, [15, 16, 35]) was transformative for me. In particular, ifstudents are not completing assignments as I might hope, I now think abouthow I can better scaffold those assignments, rather than finding fault withmy students. With this in mind, I created Objective 4 in order to foster amore equitable learning environment by: introducing all students to effectivehabits and skills; motivating them to implement them regularly; and rewardingthem for these positive behaviors. In my first precalculus course, I includedthe following Habits and Skills (H&S) assignments, each worth one H&Spoint: Weekly Goals, Learning Reflections, Productive Failure, and ProblemNotebook. These points contributed to their final grade as described in Table 1.

The Weekly Goals assignments require students to reflect on their previ-ous week’s goal and to set a new goal for the coming week. They are a greatway for me to keep track of how students are feeling about the course and theassignments on which they plan to focus on that week. Also, because studentssubmit them electronically through Google Classroom, I am able to quicklyrespond to each submission using private comments.

I typically assign five or six Learning Reflections each semester (seeFigure 7 for sample prompts). The prompts, accompanied by related subques-tions, are designed to focus students’ attention on certain aspects of the course,promoting skills such as self-reflection, time management, goal setting, andmetacognition.

I sometimes adjust the prompts to address questions I have about myclass or topics that have come up in conversations with students. For exam-ple, when I wanted to learn more about the factors that might prevent

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Figure 7. Sample learning reflection prompts.

students from keeping up with homework, I used the prompt “How do youmanage your work in this course?” which included subquestions like thefollowing:

• What do you do if you have trouble with a workbook problem?• Do you tend to work on your own or with a study group or study partner?• Do you find it difficult to motivate yourself to work on Math 159 problems?

If so, what methods have you tried to help to motivate yourself?

The Productive Failure assignments are an opportunity for students toreflect on moments of struggle that ultimately resulted in success. To earn aProductive Failure credit, students must write up short descriptions of suchexperiences and I grade these solely for completion. These prove to be par-ticularly enjoyable to read, providing me with insight into the challenges thatmy students face in my class, while also demonstrating how they are able toovercome these challenges.

Lastly, the Problem Notebook assignments are opportunities for me tocheck students’ notes for organization and completion. (In the parlance of thecourse, the Problem Notebooks are students’ collections of solutions to theworkbook problems.) These checks are conducted during office hours so theyhelp to get everyone into my office and, hopefully, reduce the hesitancy theymay have about coming to see me.

The Learning Reflection and Productive Failure assignments help to pro-vide insight into student thinking and also validate my new approach to thecourse. For example, I learned that, although they may be nervous or hesi-tant about the presentations at the start of the semester, students find that theydevelop an appreciation for them over the course of the semester. One studentnoted that the course had helped her overcome her fear of public speaking.Another student wrote about presenting problems when he was not sure thathe had the correct answer. He concluded that “through [his] potential failure[he] would learn more than getting the response right’ and this would lead toa “more rewarding feeling at the end of the day.”

Many of my students begin the semester feeling as though they areincapable of learning mathematics, often as a result of prior negative class-room experiences, and it is a joy to see their confidence blossom over thecourse of the semester. The final Learning Reflection prompt, shown in

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Figure 8, is designed to give students space to reflect on their achievementsand to highlight for them their own agency in this work. Some studentswrite about realizing how much time and effort is necessary for success. Forexample:

My approach to studying changed a lot because it went from barely doing most ofthe work to going to office hours every day to get help. I think this class not onlyhelped me with math 159 but showing [sic] that if you work hard and get the helpthat you can do anything you put your mind too [sic]. The lesson I will take intofuture semesters is hard work leads to a lot of open doors.

Others write about the value of learning how to set goals:

Throughout the semester goal setting has become more and more important to me.During the beginning of the semester I would write goals but I never really caredabout them. As the semester went I really started to care about them. Actuallywriting down a goal helped me stay focused on what I had to do amongst theclutter of school. My study attitudes have definitely changed since I started math159, the way that I study hasn’t changed but I definitely have started setting timeaside each day for just studying. Some lessons that I am going to take away fromthis class are definitely going to be about how to balance learning the materialduring class and the out of class work.

The development of effective study skills and increased confidence are alsomentioned:

Looking back at my learning reflections and weekly goals, I have noticed a growth.This semester I did not feel embarrassed of [sic] going to office hours every week,which was helpful because I got to understand the outcomes better. Honestly, Iam not a fan of math, but I actually enjoyed this class. I feel that I learned morethis semester than the last one. I have dedicated more study time to this courseand have been doing my homework everyday. Also, this course has taught me toseek help from other classmates. For example, I had a study group with anothertwo classmates in order to prepare ourselves for the final exam. This study groupwas very helpful because we worked together to solve problems that we werestruggling with. A lesson that I will take for next semester is that math could bedifficult, but it is not impossible. Also, that I should not give up on myself becauseI know I am capable of getting good grades.

Here is another example:

Throughout this semester I set goals and I was able to meet most of them. Otherthan a few exceptions. Following this math class I feel more confident for futureclasses. When learning new material, I need to remember to ask questions, go tooffice hours, and review in my free time. This is the most effective ways I learnedthis semester. Taking math 159 I had constant homework assignments I had to stayaware of. I had to stay on top of my work to not fall behind. For the future, I willdefinitely take the goal setting habit as well as making a weekly planner to keeptrack of my assignments. I wasn’t expecting to take Math 159, but I learned a lotmore than I originally thought I would.

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Figure 8. The final learning reflection assignment.

Although the Habits and Skills assignments have remained reasonablyconsistent over the past few years, I have made some adjustments. For exam-ple, in my most recent course I replaced the Problem Notebook assignmentwith a Prior Knowledge Solutions assignment. This assignment builds on thePrior Knowledge problems that appear at the start of each chapter of myworkbook, activating and informally assessing students’ prior knowledge. Toillustrate, the start of Chapter 1 of my workbook is shown in Figure 9. Inorder to earn credit for a Prior Knowledge Solutions assignment, students mustwrite up solutions to all of the prior knowledge problems from a given chap-ter and they must annotate their solutions with comments that will remindthem of important ideas when they revisit these solutions as part of theirstudy process. I introduced this assignment because I knew that most of mystudents that semester were retaking the course and I felt that this assign-ment would help them to see the value of developing a collection of samplesolutions.

I have also offered opportunities for students to earn bonus H&S points,giving me a great chance to further promote positive behaviors such as settinga goal for Spring Break or attending class when I am away at a conference.In the latter case, I was encouraged by colleagues to not cancel class on thesedays but, rather, to have one of the students run the class, soliciting volunteersfor presentations, taking photos of student solutions on the board, and taking aclass photo that I could use to record attendance. These days turned out to bevery successful, with reasonably high attendance, even when the days when Iwas absent were the day before and the day after Spring Break. Particularly inPrecalculus, I sensed that the class formed a stronger connection as a result ofthese days and I think it increased their confidence in their ability to learn andprogress on their own.

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Figure 9. Beginning of Chapter 1: Linear functions.

Objective 5: Classroom Community

As described in Section 1, my goal was to foster a cooperative classroomenvironment in which the contributions of all students were valued. This wasintegrated into my grading system via my Classroom Community (CC) pointsystem; students could earn CC points from the following activities:

• presenting workbook solutions in class,• creating review sheets for the class collection,• submitting timed assessment solutions to the class collection, and• contributing to the class collection of biographies of mathematicians.

These points contributed to their final grade as described in Table 1.As described earlier, the presentations are a crucial component of the IBL

approach that I selected for this course, with class meetings typically consist-ing of student presentations and related discussions. In concert with the goalof valuing productive failure and destigmatizing mistakes, presenters receivea CC point whether or not their solution is correct. As I often remind them,

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the best discussions often arise from incorrect solutions and so, by presentingsolutions about which they were uncertain, they were helping the entire classbetter understand a problem. Ideally, at some point during the semester stu-dents begin to openly admit that they cannot completely answer a problem,but that they are still willing to volunteer and I always thank them for theircourage.

To help them to prepare for timed assessments, students sign up to cre-ate review sheets for the different course objectives, to be added to a shared,online repository that is only accessible to students in the course. I providethem with a review sheet template and start off the collection by creating andsharing one of the review sheets myself. Students can also sign up to submitsolutions to timed assessment problems, which are also posted to an onlinecollection. For these solutions, they are required to include more explanationthan they would typically include on the timed assessment so that studentswho are struggling with a particular problem can use the posted solution toincrease their understanding. This also solidifies the writer’s comprehensionof the solution by requiring them to think about how to best explain theirsolution. As with other assignments, students can revise and resubmit theirwork if I feel that there are changes or additions that would enhance theirwork, and review sheets and solutions are not posted until this process iscomplete.

Creating review sheets and timed assessment solution keys could havebeen included in the Habits & Skills section of the grading scheme asthese are important parts of the study cycle, of which, in my experience,students are not always aware. However, I decided that these also pro-vided a great opportunity to reinforce the goal of establishing the class asa learning community. In particular, by tasking the entire class with creat-ing these collections, students can begin to view each other as resources.In addition, making correct submissions available to the entire class viashared, online repositories provides students who may be struggling tocomplete these assignments with an ever-growing collection of exemplarsubmissions.

Lastly, inspired by the approach of one of my colleagues in Biology andan online list of “mathematical rebels” [14], I compiled two lists of math-ematicians, Pre-20th Century and 20th/21st Century, both of which werediverse in terms of national origin, race/ethnicity, gender, and sexual ori-entation. Students are able to select one mathematician from each list andwrite up short biographies to be added to the class collection, discussingtheir mathematicians’ contributions to mathematics, any connections to pre-calculus or calculus, barriers that their mathematician may have overcome,and any other interesting facts. I grade these on a completion basis anddo not correct the writing at all, as the goal is for students to be exposedto the details of the lives of mathematicians, rather than to improve theirwriting.

Rethinking Precalculus and Calculus 1011

4. COURSE MANAGEMENT

All work in the course is submitted, graded, and returned via Google Class-room, which also allows me to communicate with each student via a privateconversation section connected to each assignment. My approach is to com-ment briefly on each submission, offering feedback, advice, or encouragement,as appropriate.

Everything with a set, one-off deadline is termed an Assignment; thisincludes goal setting, learning reflections, and bonus retakes corrections. Toemphasize the distinction, all work that can be revised and resubmitted iscalled an Opportunity: this includes the applied and proof problems, reviewsheets, timed assessment solutions, and prior knowledge solutions.

I post goal setting assignments on Fridays and these are due at the startof class on Monday. I try to respond briefly to students’ goals by the end ofthe day on Monday; this task generally takes me 1–2 minutes per student, ata maximum. All other Assignments and Opportunities are posted on Fridaysand are due the following Friday. Once an Opportunity has been posted forthe first time, it is reposted every Friday until the end of the semester or untileveryone in the class has successfully completed it; students are able to submitrevisions every week.

Although this sounds like it would result in a heavy grading load, in prac-tice I have not found this to be the case. First, with a Credit/No Credit approachto grading each Opportunity, I do not need to spend time thinking about par-tial credit. Instead, I just need to provide feedback to help students revise theirwork. Since they are allowed to resubmit their work every week, I do not needto comment on everything that is wrong with a submission; rather, it is aniterative process that leads them to a final, complete, submission.

Second, students are completing the Opportunities at different times overthe course of the semester, so it is never the case that everyone turns somethingin during a given week. As might be expected, the grading load does increaseas we near the end of the semester, but I have still found it to be manageable.

Finally, because I am using an IBL workbook approach, my class prepa-ration time is less than it would be if I were lecturing or using some otherapproach. Although I may make some revisions to each chapter of my work-book before posting it online, for the most part it is ready to use and I just needto tell students the problems on which to work for the next class meeting. Thisfrees me up to focus more on course management, grading, and working withindividual students.

Admittedly, our small classes allow me to implement the components ofthis grading system more easily. For larger classes, some adjustments to thisapproach may be necessary, perhaps by incorporating a more limited num-ber of the components that I have described. The use of Google Classroom orsome similar online platform certainly helps with workload management. Inaddition, some work could be delegated to teaching assistants at institutions

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that have such positions. It may also be necessary to lessen the personalizedcontact that instructors have online with students or to institute an alterna-tive such as standardized emails to groups of students, perhaps making use offeatures like Canned Responses in Gmail.

5. DISCUSSION

Given that I work at a small college where only one or two sections of Precal-culus are offered per semester, I am hesitant to share specific information aboutgrade distributions or to draw strong conclusions based on the three times thatI have taught the course in this way so far. Indeed, the results so far are mixed,with a higher than average pass rate the first semester, followed by an averagepass rate the following semester.3

I do feel that my final grades accurately reflect student progress in thiscourse and I feel confident that students who earn a C in the course are preparedfor Calculus I, although I have not formally tracked their progress to know forsure if this is true. Again based on limited data, it does appear that my gradedistributions are less uniform, with more As and Fs than in other sections. Isuspect that the greater number of Fs results from the lack of partial credit inmy course; to earn a passing grade, students need to submit a certain amountof work that is completely correct. On the other hand, students have a clearerpath to an A with this system, perhaps resulting in more As being awarded.However, students who earn As have not come by them easily, so I have noqualms about awarding multiple top grades.

This new approach has allowed me to develop closer connections with mystudents, making me more aware of their individual strengths and challenges.This awareness comes partly from their work on assignments such as the learn-ing reflections and partly from the fact that they seem more comfortable talkingto me about the course. In addition, our conversations during office hours aremore focused, because they typically come to ask about a particular objectiveor opportunity.

My impression is that, generally speaking, students enjoy the course. Ourclassroom atmosphere feels more relaxed than before and I have had severalstudents tell me that they chose my section because they had heard good thingsabout my approach through the student grapevine. It is also not uncommon forthem to tell me that they have never enjoyed math before, but that they lookforward to coming to my class. It is comments like this that tell me I am on theright track particularly because I know that the mastery-based set-up makesit a challenging course. Moreover, as illustrated by the quotes in Section 3, itis clear to me that students are making progress in terms of their approach to

3I am comparing these courses with the historical average for this course at myinstitution.

Rethinking Precalculus and Calculus 1013

math courses and college in general, in addition to their mathematical progress,and this is very encouraging.

Given the freedom that students have in this system, one challenge is tomotivate them to begin work on the various assignments earlier in the semesterand to work steadily throughout the semester. The choices inherent in the grad-ing system are designed to give students ownership and independence, but thisalso places responsibility on them to maintain a certain pace of work duringthe semester. Managing time on a long-term project like this is a useful skill forstudents to have as they move into the workforce and I continue to experimentwith ways to guide them with this. For example, I have created worksheets thataccompany my grading system so that students can keep track of their progress(see, for example, my fall 2018 syllabus for Precalculus [24]). Most recently,I created a handout entitled “My Path to a C,” that focused specifically on therequirements for a C grade, the minimum grade necessary to continue on tocalculus [24]. Given how often I have seen my students referring to this sheetin class and in office hours, I think it has been an effective addition.

Not unexpectedly, I have found that the students who are struggling themost are the same students who often fail to regularly send in weekly goals andlearning reflections and I try to carefully highlight this connection to students.More submissions from these students would also provide me with a betterunderstanding of their frame of mind over the course of the semester, so that Ican better support them.

For the proof and applied problems, students usually do not receive creditfor a problem on their first attempt, meaning that they need to allow severalweeks for the submission–feedback–revision cycle for each assignment. I haveincorporated Applied Problem examples into my updated workbook, so thatwe can work through these as a class and discuss how solutions should bestructured in order to meet the specifications. I hope that these class discus-sions motivate students to start working on these problems sooner and providethem with examples of the level of writing that is expected.

6. CONCLUSION

Despite the challenges discussed in Section 5, I feel as though the changesthat I have made and continue to make have fostered a more positive and sup-portive learning environment for my students. Although student acquisitionof content knowledge is central, there are other course components that I feltare equally important. My strategy is to foster a growth mindset, destigma-tize mistakes and failure, promote effective habits, and create a cooperativeclassroom environment, all while upholding high standards. Mastery gradingis an important tool as it provides me with the flexibility to create a gradingsystem that supports my course goals and contributes to a positive classroomatmosphere.

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ACKNOWLEDGEMENTS

The author would like to thank Yousuf George and Matt Jones for sharing theirworkbooks. Thanks also to all of the members of the IBL and Mastery Gradingcommunities for invaluable advice and support. This work stemmed from theauthor’s Teacher-Scholar Professorship at Allegheny College, which providedrelease time over the course of 3 years.

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BIOGRAPHICAL SKETCH

Rachel Weir is Professor of Mathematics at Allegheny College in Meadville,PA. Originally from New Zealand, she earned her B.Sc. (Hons) in mathematicsfrom the University of Auckland, followed by a Ph.D. in mathematics from theUniversity of Michigan. Her mathematical area of interest is function-theoreticoperator theory but, in recent years, her focus has shifted to improving thesuccess of students in college mathematics courses through the use of non-traditional approaches such as flipped learning, mastery grading, and inquiry-based learning.