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Using Problem-Based Pre-ClassActivities to Prepare Studentsfor In-Class LearningFeryal AlayontPublished online: 17 Jan 2014.

To cite this article: Feryal Alayont (2014) Using Problem-Based Pre-ClassActivities to Prepare Students for In-Class Learning, PRIMUS: Problems,Resources, and Issues in Mathematics Undergraduate Studies, 24:2, 138-148, DOI:10.1080/10511970.2013.844510

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PRIMUS, 24(2): 138–148, 2014Copyright © Taylor & Francis Group, LLCISSN: 1051-1970 print / 1935-4053 onlineDOI: 10.1080/10511970.2013.844510

Using Problem-Based Pre-Class Activitiesto Prepare Students for In-Class Learning

Feryal Alayont

Abstract: This article presents a problem-based approach that prepares students forfuture learning in the classroom. In this approach, students complete problem-basedactivities before coming to class to familiarize themselves with the topics to be covered.After the discussion on how the use of these activities relate to the learning and transfertheories, the structure of the activities is described in detail. Specific examples in a vari-ety of topics are provided. Student perceptions of these activities based on anonymouscourse evaluations are also included.

Keywords: Pre-class activities, problem-based learning, in-class activities, collabora-tive work, transfer theory, constructivism.

1. INTRODUCTION

The constructivist theory of learning indicates that people construct new knowl-edge based on their existing knowledge and beliefs [3]. This theory hassignificant implications for teaching. While planning for class, teachers needto take into account students’ previous knowledge and experiences, includingany possible misconceptions, and students’ beliefs. Moreover, teachers needto pay attention to how students interpret the knowledge presented in classbased on their existing knowledge structure. Effective teaching “elicits fromstudents their pre-existing understanding of the subject matter to be taught andprovides opportunities to build on -or challenge- the initial understanding” [3].However, with class time being so precious, how can we, college instructors,find the time to do so while striving to “cover” the material listed in the coursecatalog description?

In my classes, students spend most of the time working collaboratively onin-class activities. Students’ previous knowledge, which they obtain through

Address correspondence to Feryal Alayont, Department of Mathematics, GrandValley State University, Allendale, MI 49401, USA. E-mail: alayontf@gvsu.edu

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formal schooling and real-life experience, is important when they work onthe in-class activities and during class discussions. In this paper, I describea problem-based approach that I use to help to make students’ existingknowledge explicit and to clarify possible misconceptions. Students completeproblem-based assignments before coming to class and discuss their work insmall groups at the beginning of the class. These assignments help to buildcamaraderie among students and to develop students’ metacognitive abilities.I call these assignments “pre-class activities” as they usually tie to the in-classactivities. They help students be better prepared and more motivated. Theseactivities also serve as formative assessment informing me of students’ currentunderstanding and helping me make pedagogical decisions. Science colleagueshave been using similar problem-based activities to prepare students for labsand lectures for a long time [1, 5]. Physics colleagues also use a similar ideacalled “Just-in-Time Teaching” [8]. In this method students submit answersto web-based activities before class so that the instructor can tailor their in-class time according to student responses. In [14], the author describes howusing course preparation assignments helps prepare students for discussions ina sociology class. My own use of pre-class activities was particularly inspiredby the “preview activities” used in the introduction to proofs textbook [11]and an inquiry-based learning workshop focusing on the Moore Method that Iattended in 2006.

In this paper, I describe various aspects of pre-class activities. I start byrelating the use of these activities to learning and transfer theories. I continuewith an overview of the structure of the activities along with three examples ofthese activities from calculus, linear algebra, and discrete mathematics courses.In the last two sections, I provide a reflection on the use of these activities bothfrom the instructor and student perspectives.

2. SUPPORTING INFORMATION FROM LEARNING ANDTRANSFER THEORIES

As a theory of learning, constructivism implies that learners construct theirunderstanding in an iterative cognitive process through reflecting on their pastand current experiences. Therefore, “what a student will understand of teach-ing will be contingent upon their existing ideas and ways of thinking about atopic” [12]. In other words, students’ background knowledge and their cogni-tive skills (such as self-regulation and metacognition skills) will affect the newunderstanding they construct. In fact, educational research supports that:

learning is enhanced when teachers pay attention to the knowledge and beliefsthat learners bring to a learning task, use this knowledge as a starting pointfor new instruction, and monitor students’ changing conceptions as instructionproceeds” [3].

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Similarly, Taber [12] proposes the following

“constructivist principles for teachers:● teaching involves activating relevant ideas already available to learners to

help construct new knowledge;● students will build their new knowledge upon partial, incorrect, or apparently

irrelevant existing knowledge unless carefully guided.”

Pre-class activities that are carefully planned to tie to the lesson of theday can help instructors in achieving these two principles. These activities helpstudents activate prerequisite knowledge relevant to the lesson topic so thatstudents can work more efficiently during in-class activities. Such an examplefor a calculus course is provided in Section 3.1. Having students discuss theirpre-class work at the beginning of class in small groups provides opportuni-ties for mathematical discourse between students, which can enhance studentunderstanding of the material. This discussion time also allows the teacher toassess students’ background knowledge, and to guide the discussion to addressany misconceptions that surface during the pre-class work and to select thefoundational material to on which to build. For example, before introducingconvergence/divergence tests, students can be asked to conjecture whether afew sample series, including the harmonic series, converge or not. Then whenthe nth-term test is introduced, the whole class will actively be involved in thediscussion of how their own conjecture of harmonic series diverging can bereconciled with the nth-term test.

Additionally, pre-class activities can include application problems orexamples to serve as a starting point for students’ construction of new math-ematical concepts. In [4], the authors suggest that such instructional activitiessatisfy two pedagogical goals one of which is that:

students’ interpretations of and actions in these [activities] should constitutehighly situated, intuitive bases from which they might abstract as they constructincreasingly sophisticated mathematical conceptions.

These experiences provide students with “personally meaningful contextsfrom which they subsequently abstract” [4]. Before introducing combinationsand permutations in the classroom, students can solve real-life counting prob-lems whose solutions can then be used in the in-class activity to lead to thegeneral formulas.

Another important factor in planning for teaching comes from the theoryof transfer. Transfer of learning refers to “when learning in one context or withone set of materials impacts on performance in another context or with otherrelated materials” [9]. The main purpose of education is transfer. We want our

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students to be able to apply and extend their classroom learning to novel situ-ations ranging from the simplest case of solving similar problems to applyingtheir critical thinking and problem-solving skills in an ever-changing world.In other words, we want our students to be “adaptive experts”, as discussedin [3]. Adaptive experts have well-structured rich knowledge bases that theycan retrieve fluently and possess developed metacognition skills. Furthermore,they demonstrate great flexibility in learning in and about new situations.Pre-class activities can be used to emphasize the rich structure of the math-ematical knowledge by highlighting connections between various topics, andthus they help students in building their own well-structured rich knowledgebase. The calculus example provided in Section 3.1 is one such activity. Pre-class activities also help students develop metacognition skills by giving thema chance to grapple with ideas of conjecturing and inventing, to recall and/orrelearn previously acquired knowledge, and to check their understanding ofa topic, all without the threat of a grade. An example of an activity focusingon the conjecturing skill is provided in Section 3.2. The example activity inSection 3.1 expects students to recall the tangent line definition and use thisdefinition.

3. THE STRUCTURE OF THE PRE-CLASS ACTIVITIES

I use pre-class activities in various courses ranging from calculus to linear alge-bra, and from number theory to differential equations. The activities includequestions and problems that build the groundwork for the learning that willtake place in the classroom. Students should be able to solve these problemsusing their knowledge of previous material. Whenever possible, I strive to makethe problems entertaining, concrete, relevant, or intriguing to make them moreenticing to students. This could even be achieved by the way the questions areasked. For example, a question can describe a seemingly contradictory situationand ask the students to resolve the situation.

Students receive the pre-class activity at the end of the class period beforethe due date. I design most of the activities to take about 15–20 minutes to com-plete so as not to overwhelm students with this additional homework. If thereare 2 to 3 days between classes, students are able to find the time to completethe activities before the due date. I found that assigning pre-class activities duein a day does not work well. Also, having a regular schedule for the pre-classactivities works better as it helps students remember those assignments. If theproblems are too difficult, students are easily discouraged since the assign-ment is due soon and the problems are completely new to them. If a problemexpects students to use a different perspective than they are used to, a clarifyingexample will help alleviate any possible student confusion. One such exampleI used was in a pre-class activity aimed to explore the connection between

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matrix multiplication and composition of linear transformations. Initially notmany students were able to complete the activity in which they were asked tofind matrices corresponding to compositions of linear transformations. After Iincluded an example of the composition of two specific transformations, moststudents were able to complete the activity successfully.

On days when there is pre-class work, I ask students to discuss their pre-class work in small groups at the beginning of the class. Students usuallydiscuss their work with another classmate before class begins, so the discus-sion time goes very quickly. Students are expected to reach a consensus intheir small groups, which helps create uniform background knowledge amongstudents. Sometimes I ask groups to turn in a sheet with their answers to selectquestions for participation points. This enables students to keep their own pre-class work and use it along with their in-class work when they review the day’smaterial. By circulating around the classroom during discussion time, I can seewhether there are any students who have not completed their pre-class work.At the beginning of the semester, I might collect individual pre-class work toincrease student motivation for completing the activities. Students receive fullcredit if their work is mostly complete and they provided a good faith effortto complete the activity. This minimizes grading time (about 10 minutes for a30-person class) and helps alleviate student concerns about solving problemsthat “they were not taught.”

After the small group discussion, the consensus knowledge built duringthe discussion is used as a springboard for a whole-class discussion or an inter-active lecture. I make sure that the lecture is clearly built upon the pre-classactivity. After completing a few activities, students see that these activities arerelated to the lecture material and that doing the activities helps them learn thematerial better. To further motivate students, I heavily advertise that my paststudents commented in course evaluations that they found these activities to bevery helpful.

Pre-class activities can contain different types of problems for differentpurposes. An activity can ask students to review a previous mathematical con-cept so that the connection between this concept and the lecture material canbe made successfully. For example, before introducing the substitution method,students can recall and practice the chain rule with the purpose of determiningwhat types of functions result as the derivative functions. Students can also beasked to make conjectures based on a large number of samples. An examplewould be the calculation of the first 20 Fibonacci numbers to conjecture divis-ibility relationships between Fibonacci numbers. Activities can also includemathematical problems whose solutions lead students to the concepts intro-duced in class, such as problems about tiling 1 × n boards with dominoesand squares before introducing recurrence relations. In the next sub-sections,I describe specific examples in detail to give a more complete picture of thepre-class activities.

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3.1. Pre-class Activity: Finding Polynomial Approximations

In calculus, pre-class activities can be used to highlight connections betweenthe current topic and pre-calculus and/or previous calculus topics. These con-nections help students form a well-structured network of calculus knowledge,a necessary step in students’ becoming calculus experts. When students havea chance to think about the connections before coming to class, they are moreready to appreciate these connections in the lecture. They already constructedthat connection themselves before class and this construction becomes strongerduring the lecture. I use a pre-class activity before introducing Taylor polyno-mials for this purpose. In the activity, finding the tangent line is reviewed, andthe connection between the tangent line and the quadratic approximation isexplained to help students understand similar connections to the higher-orderpolynomial approximations discussed during the lecture.

In the activity, students are first asked to find the tangent line to f (x) = ex

at x = 0 and to provide a linear estimate to f (x) at x = 0.1. The activity thenprovides another equivalent definition of the tangent line L(x) as being the linefor which L(0) = f (0) and L′(0) = f ′(0). Using a similar characterization, theactivity then gives the definition of the quadratic approximation. Students usethis definition to calculate a quadratic approximation to f (x) = ex and obtaina quadratic estimate at x = 0.1. This portion provides practice for students toapply definitions. Students then compare the linear and quadratic estimate tothe function value, and give possible reasons for why the quadratic estimate isa better estimate.

At the beginning of the class period, students discuss their answers to thispre-class activity in small groups. Students usually successfully complete theportion about the linear estimate, although some students struggle with find-ing the quadratic estimate. However, when students discuss their answers insmall groups, they usually reach a consensus on the correct answer in theirgroups. When most groups complete their discussions, the pre-class activity isdiscussed as a whole class. After this discussion, the lecture continues with thecubic approximation. There is usually a correct guess for the cubic approx-imation, after which we conjecture the general formula based on the threeapproximations.

3.2. Pre-class Activity: When Does the Inverse of a Matrix Exist?

In linear algebra, my pre-class activities usually focus on developing con-crete examples and non-examples of concepts and/or relations between theconcepts in order to help students “build adequate ‘concept images’ for the‘concept definitions’” [6]. Before introducing the row reduction method offinding the inverse of a matrix, I assign a pre-class activity in which studentscalculate the inverses of various matrices with a calculator and conjecture what

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row/column properties make a matrix non-invertible. In the activity, the matri-ces are grouped into 2 × 2 and 3 × 3 matrices, as it is easier to conjecture therelationship for 2 × 2 matrices and use that conjecture as a starting guess forthe 3 × 3 matrices. The matrices in the activity serve as examples and non-examples of invertible matrices and help students in abstracting the propertiesof these matrices.

Students almost always come up with the conjecture that if one row and/orcolumn is a multiple of another, then the matrix is non-invertible. For 2 ×2 matrices, this is equivalent to the criterion given in the theorem describinghow to find the inverse of a matrix using the row reduction method. However,for 3 × 3 matrices, the most inclusive criterion is not as obvious to see. Yetthere are always students who come up with the conjecture that a row/columnbeing a linear combination of the others makes the matrix non-invertible. Evenbetter, a student might come up with the formulation that when reduced, thematrix does not have a pivot in every row. At that point, I jump up with excite-ment and introduce the theorem that a matrix is invertible if and only if itsrow reduced echelon form is an identity matrix, in which case the row reduc-tion steps applied to the identity matrix produces the inverse of the matrix.Conjecturing a property that later turns out to be a theorem in the book givesstudents good practice and confidence. Also; examples of the 3 × 3 non-invertible matrices where no row/column is a multiple of another come upin later discussions as an example or a counterexample, such as when a stu-dent wants to reduce the non-invertibility to a row/column being a multipleof another. I write down a matrix on the board which themselves claimed wasinvertible and ask them to explain the situation.

3.3. Pre-class Activity: Counting Multiple Colored Marbles

In the discrete mathematics course, I use an activity in which real-life problemsguide students to discover the idea of the inclusion–exclusion principle beforeit is discussed in the classroom. With this approach, the application motivatesthe theory and increases student interest in the topic. In the activity, studentswork on problems involving counting marbles with multiple colors. The firstproblem is about marbles which are colored red or yellow. The problem givesthe numbers of marbles with each color and the total number of marbles, thatis less than the sum of the numbers of marbles with each color. Students arethen asked how this is possible. In the second problem, the situation gets morecomplicated. Students consider marbles with colors red, yellow, or blue. Giventhe total number of marbles and the numbers of marbles with each color, stu-dents then determine whether it is possible to find the number of multicoloredmarbles.

Students easily solve the first problem and express the idea of theinclusion–exclusion principle for two sets successfully using the set notation.

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Some students might need to recall the set theory notation for this part butthis part is straightforward for almost all students. For the problem with threesets, without being prompted, most students sketch a Venn diagram. Almostall students find that it is not possible to determine the number of multicol-ored marbles using only the sizes of the individual sets and the size of theunion. A student or two might discover that if the size of the triple intersec-tion is known then the number of the multicolored marbles can be determined.However, the successful formulation of the inclusion–exclusion for three setsis usually postponed to the lecture. The two examples in the activity serve as astarting point for students to grasp the general inclusion–exclusion principle.

4. STUDENT REACTION

At all times I used them, a large majority of the students found the pre-class activities to be helpful. At the end of the semester, students submit twopieces of advice to students starting the same course and “completing thepre-class activities” is the most frequent advice given to incoming students.Students also voluntarily elaborate in course evaluations what they like aboutthe activities, as in “I really like having the pre-class activities because theygave me a good introduction to each topic” and “[The instructor] effectivelymotivated me to confidently understand the material by assigning and giv-ing out material/handouts that I could work through.” Students especially likethe group discussion aspect of the activities (both pre-class and in-class) asthey comment that they “like the group-activity parts, it really engrains thematerial,” or:

in this class, we were made to work things out amongst ourselves in smallgroups almost daily, so we were continually given the chance to explain whythe concepts in class worked the way they did.

During courses where pre-class and in-class activities are heavily used,students tend to enjoy the group work in and out of the classroom more.

5. REFLECTION

Having used pre-class activities in a variety of courses for over 4 years, Istrongly believe that the activities are helpful to students in many differentways. The activities uncover students’ pre-existing knowledge related to thenew lecture topic and hence help students in building a well-structured knowl-edge network. When students discuss the activities in small groups, they settledifferences of opinions so that they can start the lecture with similar back-ground knowledge. Additionally, students have daily homework which theyare eager to discuss with each other even before class starts. These various

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group work activities help with building a collaborative atmosphere in theclassroom and with efficient time usage during the class period as most stu-dents have refreshed the related prerequisite material. Even when the activitiesare not explicitly discussed at the beginning of the class period, I lecture withthe assumption that students attempted the problems in the pre-class activityand thought about the ideas which were introduced. This saves class time thatwould have been spent on reviewing prerequisite material and we can spendmore time on building an understanding of the topic. These activities also helpstudents take more active role in the classroom since each student has had achance to think about the problems in the activity at their own pace and hassomething to say about those problems. More generally, students become moreconfident and comfortable in learning mathematics on their own, making con-jectures, making connections between mathematical topics, and reflecting ontheir learning.

When teaching a topic, if I find myself wishing more of my studentsremembered a related previous topic or saw a connection, then I write an activ-ity to make those connections explicit. If a topic is suitable for exploration bystudents on their own, that exploration becomes an activity as well. My aim isto keep these activities short so as not to overwhelm students with assignments,and hence, writing the activities takes little time for me as well. The activi-ties do not have to be written all at once making this method easy to adaptslowly. Each activity might go through minor revisions once or twice, eitherto add more connections to the lecture topic or to make the tasks and prob-lems more clear. The issue of motivating students to complete the activitiesalso deserves some attention. From my experiences, the seniority and mathe-matical maturity levels of the students in the classroom do not seem to have asignificant effect on the motivation. Tying the pre-class work closely to the in-class work increases motivation significantly. In evaluations, students favor thein-class activities building on the pre-class work. Making the problems in theactivities real-life-based and interesting, if possible, seems to help with motiva-tion. Emphasizing previous positive student response also seems to work well.Having students discuss their pre-class work in small groups further providesmotivation by adding peer pressure on each member. Students feel responsibletowards their group members. Assigning points for “thoughtful and relevant”work in the pre-class activity is another option for encouraging students tocomplete the pre-class activities. I assign class participation points by using thegroup sheets collected after the pre-class discussion or by checking off namesin the class roster while roaming in the classroom while students discuss theirwork in groups. Colleagues in mathematics and other disciplines have longused various means to encourage students to complete assigned reading andsimilar ideas can be used for pre-class activities [2, 7, 10, 13]. Finally, pre-class activities can be used in conjunction with reading assignments to increasestudent preparation for class, although sometimes reading may give away theanswers to problems in the pre-class activity.

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The benefits of pre-class activities significantly outweigh the amount ofwork needed to implement them. Since the activities ask students to use previ-ous knowledge, including the earlier topics in the course, the activities can actas formative assessment. Although designing a pre-class activity will cost someamount of time, this time will be compensated by not ever needing to preparelecture material reviewing the prerequisite topics, in addition to the pleasure ofhaving active and prepared students in the classroom ready to volunteer infor-mation related to the lecture topic. Also, students’ taking more responsibilityin preparing for class generally leads to them being more responsible in otheraspects of the course. For those who want to implement something similar,I have only one precautionary advice: Start small and expand your pre-classactivity collection over time.

ACKNOWLEDGEMENTS

I would like to thank my colleagues at GVSU and the IBL workshop for manyinspiring conversations on teaching. I would like to especially thank Dr BrianDrake who was a collaborator on a project in which we developed pre-class andin-class activities for a Discrete Mathematics course at GVSU. I also wouldlike to extend a huge thanks to all my students, at GVSU and at other institu-tions, for they motivated me to make my assignments more relevant and moreentertaining.

REFERENCES

1. Barnes, R. and B. Thornton, 1998. Preparing for laboratory work. InB. Black, and N. Stanley, (eds), Teaching and Learning in Changing Times,pp. 28–32. Proceedings of the 7th Annual Teaching Learning Forum, TheUniversity of Western Australia, February 1998. Perth: UWA. Available athttp://lsn.curtin.edu.au/tlf/tlf1998/barnes.html. Accessed 16 August, 2012.

2. Boelkins, M. and T. Ratliff, 2001. How we get our students to read the textbefore class. FOCUS. 21(1): 16–17.

3. Bransford, J. D., A. L. Brown, and R. R. Cocking, (eds), 2000. How PeopleLearn: Brain, Mind, Experience, and School. Washington, D.C.: NationalAcademy Press.

4. Cobb, P., E. Yackel, and T. Woof, 1992. A constructivist alternative tothe representational view of mind in mathematics education. Journal forResearch in Mathematics Education. 23(1): 2–33.

5. Gentry, M. R. 2005. Student success in an inquiry-based laboratory: theEffect of pre-class activities and student preparation. Master’s thesis,Stillwater, OK: Oklahoma State University.

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6. Harel, G. 1997. The Linear Algebra Curriculum Study Group recommen-dations: moving beyond concept definition. In D., Carlson, C. Johnson,D. Lay, D. Porter, A. Watkins, and W. Watkins, (Eds), Resources forTeaching Linear Algebra, MAA Notes, Vol. 42, pp. 107–126. Washington,DC: Mathematical Association of America.

7. Hoeft, M. E. 2012. Why university students don’t read: what professorscan do to increase compliance. International Journal for the Scholarshipof Teaching and Learning. 6(2).

8. PER User’s Guide. http://perusersguide.org/. Accessed 3 March 2013.9. Perkins, D. N. and G. Salomon. 1994. Transfer of learning. In:

International Encyclopedia of Education, pp. 6452–6457. Oxford, UK:Pergamon Press.

10. Shepherd, M. D. 2005. Encouraging students to read mathematics.PRIMUS. 15(2): 124–144.

11. Sundstrom, T. 2006. Mathematical Reasoning: Writing and Proof , SecondEdition. Upper Saddle River, NJ: Prentice Hall.

12. Taber, K. S. 2011. Constructivism as educational theory: contingencyin learning, and optimally guided instruction. In J. Hassaskhah (Ed.),Educational Theory, pp. 39–61, New York: Nova.

13. Weimer, M. (Ed.). 2010. 11 strategies for getting students to read what’sassigned. Madison, WI: Magna Publication. Available at http://www.FacultyFocus.com. Accessed 26 August, 2012.

14. Yamane, D. 2006. Course preparation assignments: a strategy for creatingdiscussion-based courses. Teaching Sociology. 34(3): 236–248.

BIOGRAPHICAL SKETCH

Feryal Alayont is an Associate Professor of Mathematics at Grand Valley StateUniversity. She received her Ph.D. from the University of Minnesota and wasa teaching postdoc at the University of Arizona before coming to GVSU.

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