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THE TOP TEN THINGS I HAVELEARNED ABOUT DISCOVERY-BASED TEACHINGJonathan K. Hodge BS and PhD aa Department of Mathematics , Grand ValleyState University , Allendale, MI, 49401, USA E-mail:Published online: 13 Aug 2007.

To cite this article: Jonathan K. Hodge BS and PhD (2006) THE TOP TEN THINGSI HAVE LEARNED ABOUT DISCOVERY-BASED TEACHING, PRIMUS: Problems,Resources, and Issues in Mathematics Undergraduate Studies, 16:2, 154-161

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June 2006 Volume XVI Number 2

THE TOP TEN THINGS I HAVELEARNED ABOUT

DISCOVERY-BASED TEACHING

Jonathan K. Hodge*

ADDRESS: Department of Mathematics, Grand Valley State University,Allendale MI 49401 USA . hodgejo@gvsu.edu.

ABSTRACT: In recent years, there has been much lively debate about themerits (or lack thereof) of discovery-based approaches to the teachingand learning of mathematics. In this article, I discuss what I havelearned from my own experiments with discovery-based teaching, andoffer suggestions to others who may be interested in implement ing amore discovery-based pedagogy.

KEYWORDS: Discovery-based teaching, teacher change.

INTRODUCTION

During my last semester of graduate school, I gave in to the not-so-subtleurgings of a certain math educator who, for months, had been hou nding meto just once try something ot her than lecturi ng. What I did not know atthe time is that my subsequent experiments with discovery-based learni ngwould ult imately shape and transform my view of teaching and redefinewhat I perceived to be my role as a teacher of mathematics.

It would be far from the truth to say that I have arrived at some completeun derstanding of what it means to teach in a discovery-or iented manner. Ihave only been doing this for a few years and I do not even claim to be thatgood at it . Nevertheless, I have gained some insig hts that I t hink ot hersmight find useful. To those who are toy ing wit h the idea of moving toward

• Many thanks to Tabit ha Mingus for p laying suc h an im port a nt ro le in my journeyas a teacher. Thanks also to Esther Billings, Matt Boelkins , and Mike La ng for t heirhel p ful suggest ions on t h is art icle .

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Hodge Top Ten Things I Have Learned About Discovery-Based Teaching

a more discovery-based pedagogy, I hope that my observations give a senseof what you are getting yourself into. To the seasoned veterans who mayread this article, I hope that my comments in some way resonate with yourexperiences as well.

Before I beg in , a disclaimer is in order: There has been much livelydebate about the merits (or lack thereof) of discovery-based approaches toteaching mathematics. Though I have chosen to enter into these debates inthe past, I will, for the most part, refrain from doing so here . Instead, I willfocus primarily on my own experiences and what I have learned from them.

WHAT IS DISCOVERY-BASED TEACHING?

Good mathematicians start with good definitions and I will not break tra­dition in this regard. I will , however, give the same warning that I give tomy discrete mathematics students on the first day we discuss graph theory:The definition below is by no means "standard" , if by "standard" we meanuniversally agreed upon by experts in the field. Discovery-based teachingmeans many different things to many different people. Any definition ofdiscovery is inherently personal and I will not try to remove my personalbiases here.

DEFINITION. A discovery-based approach to teaching mathemat­ics is one in which the primary goa l is for students to discovermathematical truths and to construct for themselves a deep, work­ing knowledge of these t ruths.

A consequence of this definition is that I view myself as more of a facilita­tor than an all-knowing sage . My aim is not to somehow transfer knowledgefrom myself to my students, but rather to be part of a learning communityin which st udents actively engage the material and take ownership of theirindividual learn ing experiences . If my classroom works the way I want itto, then what we do in class is only a small (but important) part of thelearning process. It is a starting point and by no means the end of the road.Furthermore, much of my success in the classroom amounts not to givingthe right answers, but to asking the right questions.

Many instructors who have not practiced discovery-based teaching wantto know exactly what such a classroom should look like. This, unfortunately,is a question with enough answers to fill volumes. In my classroom, I tryto use a variety of techniques to faci litate my students ' learn ing. These in­clude interactive lectures, collaborative gro ups, st udent presentations, boardwork, computer lab activities, games, and challenging homework problems .

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As much as possible, I also try to make tests and quizzes an imp ortan t par tof the learning process and not simply a tool for assigning grades .

Of course, many highly effect ive teachers use only a subset of thesetechni ques , or use other techniques that I haven 't mentioned. For instance,those who pract ice the Moore method focus mainly on problem/ theoremsequences and st udent presentations of solutions to these problems. Otherswho are limi ted by class size or space mig ht use an interact ive lecture format(perhaps enha nced by technology) with an occasional dose of t hink-pair­sha re. In any case, I believe that it is not purely the mechani cs of theclassroom that make a course discovery-based . What matters most is thephil osophical underpinnings that mot ivate the mechan ics, and it is at thislevel that we see many of the differences between discovery-b ased teachingand more t ra dit ional approaches.

With that said , allow me to pro ceed to my (ten) main observat ions.

AND NOW, WITHOUT FURTHER ADO ...

. . . t he to p te n things I have learned about discovery-based teaching:

10. Discovery-based teaching may feel awkward or un­com for t ab le at fir st.

When one moves toward a more st udent-centered , discovery-based ap­proach, a certain element of cont rol is lost. I distinctly remember mourn­ing this loss; to be honest , I probably st ill do from t ime to t ime. It can berather unset tling to not know exactly how a class is going to turn out. It canbe very tempting at ti mes to cut off the discovery process, grab hold of thereins again , and take what Carol Schumacher refers to as the "impermissibleshortcut" - sim ply giving the answers. [4]

9. Change takes time.

I hav e found that my comfort level has gradually increased as I have per­sisted in at tempt ing to make my classes more discovery-oriented. But thecha nge has taken t ime. According to Edwards [3], "[teacher] change is aslow and complicated pro cess." It "happens in phases, or steps, that mayseem very small to an outside observer but feel mor e significant to the in­dividual who executes the cha nge." A number of recent st udies echo t hesesent iments (see, for instance, [1] or [2]), and they certainly resonate with myown personal experiences . One important conclusion that must be dr awn isthat those who experiment for a semester and then give up because things

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didn 't go exactly as planned are unlikely to experience real growt h or havea fighting chance at making discovery work in their classro oms.

8. Personality is important, but only tangentially.

Many opponents of discovery-b ased learning claim that their personali ti esare compatible only wit h lecturing (and by this I do not mean the typ eof lecture described in Item 6 below). I once said the same thing, andwhile I admit that personality traits do and should shap e one's uniqueteaching style, I am no longer convinced that there are certain personali tyty pes that are entirely incompatible wit h discovery-based techniques. Evenif I hyp othetically admit the existence of an instructor whose personali tyabsolutely prohibit s him or her from ever stepping away from the front ofthe roo m, I would st ill argue that such a person could take ste ps to maketheir lectures more interacti ve and more conducive to st udent discovery. AsI menti oned above, this process will likely involve some discomfort , but I donot believe it to be an impossibl e proposition .

7. Discovery-based teaching is a lot of work.

T he fact that st udents are typi cally more involved in classroom act ivit iesdoes not necessarily imply less work for the instructor. In my experience,it is much harder to prepare an activity that accomplishes something usefulthan it is to prepare a clear and informative lecture. I have also founddiscovery-based teaching to be more emotionally taxing, perhaps partlybecau se I am somewhat of a contro l freak . Finally, since I am now askingmore of my students than I used to, I also find myself spending much moret ime answering questions outside of class.

6. A lecture CAN promote discovery.

Proponents of discovery-b ased teachin g are sometimes quick to dismiss thelecture as being ent irely ineffective. I have certainly done so in the pastand occasionally catch myself doing so even now. I think, however , that itis import an t to draw a distinction between the typ e of lecture that simplypresents resul t s to students and that which works as a tool to allow studentsto discover the results on their own . The form er is ty pica lly non-interactiveand revolves around the instructor telling the students what they need toknow. The latter invites st udent interaction and focuses more on asking theright questions (and , equally important, giving students time to answer).

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I think that we must a lso be careful not to equate discovery-based learn­ing and gro up work. While collaborative groups are certainly a useful to olfor facilitating student discovery , they can also be used in ways that arenot helpful at all. For inst an ce, a skill-a nd-drill group activity that simplyrequ ires st udents to mimic a previously demonstrated technique does notconstit ute discovery-b ased learning. It is closer to an in-class homeworkassignment , and not a very good one at that .

5. Discovery takes time.

Another common crit icism of discovery-based teaching is that one cannot"cover" as much material. This is a completely legitimate concern and onethat is especia lly relevant to younger facul ty members or gra duate st udentswho may feel a certain pr essure to adhere st rict ly to departmental contentguidelines or standardized syllabi .

I have no way of getting aro und this complaint , ot her than to ques­tion what is meant when someone says that they have "covered" a certainamount of materia l. If, by this, they mean that they have written all of thenecessary theorems and proofs on the board and that their st udents havecopio usly transferred these scribblings into their notes, then I would whole­hear tedly agree that one cannot cover as muc h mater ial in a discovery-basedclass. This, however , brings me to my next point .

4. Students can learn a significant amount of material ina discovery-based class.

In contrast to much of what I have been to ld, I have actually found adiscovery-based approach to be highly efficient in foste ring st ude nt un der­standing of imp ortan t concepts. As an example, in first-semester calc ulus ,I use a Maple lab to lead st udents to discover all of the important rul es fordifferentiation wit hin a period of about 50 minutes. We ty pically spend thenext hour or two practicing these rul es and discussing why they work theway they do. At the end of three hours, we've covered a significant amountof materi al and have done so in a way that is meaningful and memorableto students.

3. Students do not always respond warmly to discovery­based learning.

T his po int should come as no sur prise to those who have experienced thejoys of reading st udent evaluations at the end of a discovery-b ased course.

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For those who have been deprived of the pleasure, 1 will share some examplesfrom my own experience:

• "Doesn' t seem to teach as much in class. . . we have to learn for our­selves."

• "The only issue that 1 have with th e class so far is the amount ofself-discovery. 1 find it frustrating th at we dive int o things withoutdiscussing and showing examples."

• "He could be more effect ive if he would just teach. We did all of thisgroup work and nobody got anything out of it . Try explaining thematerial first and then do group work ."

• "He was way too interested in self-discovery teaching."

• "Do not assume everyone can come up wit h ideas of their own . 1wish there would have been more teaching rather than come up withanswers on our own."

1 think that these comments speak for themselves; in many ways , theyare ind icative of the sorts of issues that one must confront when askingstudents to st ep outside of th eir comfort zones and think ind ependently.Allow me also to share the thoughts of a few students whose perspectivesare somewhat mor e affirming:

• "I enjoyed his teaching style - 1 learn best when 1 am expected to domore on my own."

• "P roblem presentations and the discussions that resul ted helped meto understand concepts bet ter tha n any other style 1 have had previ­ously."

• "He didn't just lecture. Activities were pointed and thought provok­ing."

• "I really liked the way you would ask the class 'what if. . . ' questionsand let us think about and answer those questions."

While one obviously hopes for comments of the latter vari ety, 1 thinkthat some of the former are inevitable. With that sa id , 1 also think that itis quite possible for an instructor to pos itively shape and influence st udentattitudes toward the discovery pro cess. To do so requires a measure ofempathy and, often, a fair amount of time. We should not be surprisedthat students who have never been asked to think for themselves find itdifficult or uncomfortable when we ask them to do just that . We should beunderstanding of our students' backgrounds and willing to do whatever ittakes to help them understand and appreciate the valu e of a more discovery­oriented approach.

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2. Students need to see the big picture. If they don'tunderstand the point of the activities, they won't par­ticipate meaningfully.

When I first star ted experimenting wit h discovery-based teaching, I mad ethe mistake of replacing all of my lectures with group activit ies, ofte n wit h­out provid ing a sense of purpose or direction for these activit ies. I some­t imes hear colleagues refer to their teaching style as "guided discovery" ,and it took me a while to reali ze how imp ortant the word guided is in thisdescrip ti on.

T here are many ways to convince students that what they are aboutto do is worthwhile. One way is simply to be an excited and ent husiast icsa lesperson. When my st udent s sense that I believe in what we are doing,they are much more will ing to engage the activities in a productive manner.

I have also developed t he practi ce of start ing each class by writ ing focusquest ions on the board. (For inst ance, "W hat is an equivalence relation?What is the correspondence between equivalence relations and par ti ti ons ofset s?" ) T hese quest ions set the stage for our investigati ons and place theday 's activities within the broader context of our course goa ls. Finally, it 'svery important to make sure that students have some sense of closure at theend of the day, whet her this is accomplished through a summary discussion ,a follow-up homework assignment, or some other means.

1. You can't learn much about discovery-based teachingfrom a top ten list - you have to try it for yourself!

This is an obvious point , bu t one wor th mentioning. For the first few yea rsof my teaching career, I was very resistant to making any sort of cha nge tomy teaching style. I had hear d both sides of the debate and had developedenough inte llect ual objections to discovery-based teaching to justify noteven trying it . In retrospect , I'm very glad that I had a change of hear t .T hough I've experienced my share of frustrations and failures over the pastfew years, I am confident that my students are better off now than theywould have been had I just cont inued doing what was comfortable.

I also don 't want to claim that my experience is unique in any sense of theword . I know many colleagues who have undergone similar transformatio nsand I am certainly not the first to have grappled with these issues. I havecome to believe, however , that the pract ice of discovery-based teaching isless common than it ought to be. My fear is that, collecti vely, we may bemissing out on an opportunity to reach a lar ge group of st udents who have

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the potential to be successful in mathematics, but for whom a traditionalapproach is simply not effective .

I believe that a discovery-based pedagogy should be central to any strat­egy for reaching these students. A philosophy of discovery embraces thediversity of learning styles that are present in any given classroom, while atthe same time recognizing that all meaningful learning ultimately requiresownership and engagement on the part of the learner. Adopting such a phi­losophy has made a world of difference in my classroom, and I think that itcan in yours as well.

REFERENCES

1. Bright , George W ., Anita H. Bowman, and Nancy N. Vacc. 1998.Reflect ions about listening and dialogue as evidence of teacher change . InProceedings of the Twentieth Annual Meeting of the North American Chap­ter of the International Group for the Psyc hology of Math ematics Educa­tion (Columbus , OH) (Sarah Berenson, Karen Dawkings, Maria Blanton,Wendy Cou lombe, John Kilb , Karen Norwood, and Lee Stiff, eds .). Vol. 2,ERIC Clearinghouse for Science , Mathematics, and Environmental Educa­tion. 607-612 .

2. Coh en , David K. 1990. A revolution in one classroom: The case ofMrs . Oublier. Educational Evaluation and Policy Analysis. 12: 327-345.

3. Edwards, Barbara S. 2000. The challenges of implementing innova­tion. The Math ematics Teacher . 93(9 ): 777-781.

4. Shumacher, Carol. 2004. Preparing the ground. Presented at the 7t h

Annual Legacy of R. L. Moore conference. Austin TX. March.

BIOGRAPHICAL SKETCH

Jonathan Hodge is an Assistant Professor of Mathematics at Grand ValleyState University in Allendale MI. He received his BS from Calvin Collegein 1998 and his P hD from Western Michigan University in 2002. He is anactive participant in the Legacy of R. L. Moore proj ect and has recent lyco-authored a discovery-based textbook on the mathematics of voting andelections.

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