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STUDENT LEARNINGOBJECTIVES ANDMATHEMATICS TEACHINGMatthew DeLong a , Dale Winter b & Carolyn A.Yackel ca Department of Mathematics , TaylorUniversity , 236 W. Reade Avenue, Upland, IN,46989, USA E-mail:b Department of Mathematics , University ofMichigan , Ann Arbor, MI, 48109, USA E-mail:c Department of Mathematics , MercerUniversity , 1400 Coleman Avenue, Macon, GA,31207, USA E-mail:Published online: 13 Aug 2007.

To cite this article: Matthew DeLong , Dale Winter & Carolyn A. Yackel (2005)STUDENT LEARNING OBJECTIVES AND MATHEMATICS TEACHING, PRIMUS:Problems, Resources, and Issues in Mathematics Undergraduate Studies, 15:3,226-258, DOI: 10.1080/10511970508984119

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Septemb er 2005 Volume XV Number 3

STUDENT LEARNINGOBJECTIVES AND

MATHEMATICS TEACHING

Matthew DeLong1, Dale Winter'' , Carolyn A. Yackel"

ADDRESS: (1) Depar tment of Mathem atics, Taylor University, 236 W .Read e Avenu e, Upland IN 46989 USA. mt del ong@tay l or u. edu ,(2)Depar tment of Mathematics, University of Michigan , Ann Arbor MI48109 USA. aman i tav@umich . edu, and (3) Department of Mathemat­ics, Mercer University, 1400 Coleman Avenue, Macon GA 31207 USA.yackel _ca@mer cer . edu.

ABSTRACT: The current work is the first art icle in a two-paper series ex­plorin g the role of explicit learning objectives in undergraduate math­emat ics instructi on . A definition of student-learn ing objec t ive (SLO)is int roduced . We give examples of SLOs for to pics from introduct orycollege and university mathematics cour ses. We list pot ential advan­tages of a genera l pro gram of gro unding mathemat ics instructi on ina set of such explicit SLOs. In the second paper in this series, anexplicit, step-by-step algor it hm for creating set s of student learningobject ives will be described and its use illustrated .

KEYWORDS: College teachin g, learning objectives, teacher planning,lesson planning, st udent motivation , subjec t-matter knowledge.

1 PREPARING FOR TEACHING: A CASE STUDYOne of the most commonly exhorted ideas in college teaching is that effec­t ive teaching is the result of careful and thorough prepar ation [21]. Indeed ,in a survey administered to 356 faculty at a var iety of colleges and un iver­sit ies, "inadeq uate planning" was identified as one of the four most seriousexamples of unprofession al behavior in college teachin g [12] . Popular refer­ences for the teaching of mathematics in colleges and universi ti es (e. g. [8,22, 27, 38, 50, 51]) include simil ar encour agement to prepare thoroughly

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for each indi vidual math emat ics lesson . Krantz [38] for exa mple, beg ins hisbook with the impassioned plea:

"I cannot emphasize too strongly the fact that prepara tion is of utmostimportan ce if you are going to deliver a st imulating class . . . You must besufficiently confident that you can field questions on the fly, can modi fy yourlecture (again on t he fly) to suit circumstances, can to lerate a diversion toadd ress a point that has been ra ised." [38, p. 7]

Authoritative writers and scholars of college teaching appear to be inagreement: Effective college teaching is the resul t of careful preparati on.What we believe needs more discussion and illumination is the type of plan ­ning that most help fully cont ributes to deep and meaningful learning. Toillust ra te that it is not necessarily tha t one plan s carefully - but how onecarefully plans- that is impor tant , we begin with the case of Dr. G .

1. 1 The Case of Dr. G .Dr. G was a member of t he postd octoral faculty in th e math em atics depart­ment at a major research uni versity" . Dr. G taught introductory mathe­mati cs courses (in addition to advanced ma thematics courses) severa l t imesduring his postdoct oral career and was regarded as a dedica ted , cooperative,and conscient ious teacher who was recep tive to new ideas about teaching.The following vignette describes some of the events that took place duringone of Dr . G's pre-calculus classes. In this par ticular class the trigo nometricfunctions were introduced using the uni t circle definit ion.

Dr. G arrived at class a few minu tes early. Many of the st udentsin the class were already present quietly reading the campusnewspa per or wait ing pa tiently for class to begin . As more stu­dents arrived, people from the class greeted each other and en­gaged in conversation. Dr. G participated in these conversationsand joked with people in the class.

At the appointed time, Dr. G returned to the fron t of the roo mand called the class to order. Despi te th e fact that the conversa­tions did not immediately cease, Dr. G 's loud voice was easy tohear. Dr. G began with some administ rat ive announcements­pushing back a test deadline- and briefly not ed some of th e

1 Dr. G's class was visit ed twice dur ing the sa me se mester by one of t he a utho rs(D . \V.). In t he mat he mat ics de partment where Dr. G taught t h is was considered normaland a ll instructor s teach ing int roductor y classes were visit ed at leas t twice d uring t hese mester. In addit ion, t he author met wit h Dr. G (both before and after t he visits) todiscuss the progress of t he class, recognize Dr. G's ac hievements, and suggest areas ofimprovement.

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pervasive errors on the latest hom ework assignment. Aft er re­t urn ing the homework, Dr. G told the class that he was pleasedby the improvement that they had shown compared to the lasthomework ass ignment.

Dr. G then began an interactive mini-lecture reminding st udentsof the radian measure of angle (st udied in the previous class) .Dr . G called on students from the class (calling them by theirfirst names) to supply ste ps and results in the examples that heincluded in his mini-lecture.

When he was sat isfied with the review of ra dians, Dr . G beganto lecture on the definition of sine and cosine using the uni tcircle. As he progressed , Dr. G appeared to be looking at theclass to see which st udents were not involved . He seemed tomake a particular effort to address quest ions to students whodid not appear to be paying at te ntion. When not surveying theclass, Dr. G te nded to face the board as he lectured. Althoughthis did not obscure the notes that he wrot e up , nor did it makeit hard to hear him , Dr. G missed several student quest ionswhen he did not see the raised hands in time .

Aft er approximately forty minutes, Dr. G announced that forthe rest of the class period the student s would be working on agroup problem. He identified the problem from the textbook andasked the students to move into their "usua l groups" and beginworking . The st udents took an inordinate amount of time tomove to their "usual gro ups." When they were finall y sea ted to­gether, many of the gro ups began spontaneous conversations onmatters other than t he assigned problem. After a few minutes,Dr. G began going around the room from group to group check­ing in on st udent progress and asking if there were any pointsthat the gro ups were stuck on. All gro ups answered "no," eventhose groups that did not appear to be worki ng on the assignedproblem .

After a few minutes, t hose groups who had started th e problemevident ly finished to their satisfact ion, while the groups whichdid not appear to be working on the problem cont inued to chatand negotiate. The st udents in the gro ups who had finishedalso began to chat . Realizing that the lesson was almost over ,Dr. G announced the answer to the ass igned probl em and askedif anyone had got ten anything else. No one indi cated th at t heyhad and Dr. G dismissed the class.

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T he vignet te given above describes a class th at was held during the firstfour weeks of the semester. Later in the semes ter, Dr. G's class was visited asecond time. During the first part of the visit , th e class was simply observed .Approximately twenty minutes before the end of class, Dr. G left and theobserver conducted "focus gro ups" (more form ally, t he technique is knownas "Small Group Instruct ional Diagnosis" or SGID - see [17, 22]) with thest udents in the class .

After the second visit. t he feedback from the class was summarized anddiscussed wit h Dr. G. T he students uniformly praised Dr. G 's energy andpositive attit ude in class . The students felt t hat Dr. G mad e a real effortto get everyone involved in t he lesson and they believed that this mean tthat he genuinely cared whether or not st udents und erst ood . Student s alsocommented that Dr . G was easy to understand, easy to get a hold of outsideof class , and pu tting a lot of effort into teaching the class .

When st udents were asked to ident ify changes that would help them tolearn more, severa l t hemes emerged from the focus groups. A maj ori ty ofst udents (61%) exp ressed the desire that Dr. G go deeper into each con­cept that he taught rather tha n ". . . going through tons of . . . probl ems."Inst ead , 57% of st udents felt that th ey would learn more if Dr. G focusedon fewer problems, carefully identifyin g and explaining key points in each.Echo ing t his, a simila rly lar ge maj ori ty (61%) expressed concern with thelevel of math ematical sophist ication that Dr. G assumed , in part icular hishabit of skipping the steps that he regarded as obvious or trivia l. Finally,all of the st udents (100% of the 28 st udents present ) believed that Dr. Gshould be more careful about covering material in class before ass igning itas hom ework.

We believe that the events described in the vignette indi ca te that Dr. Gwas an ent husiastic and capable college math ematics teacher. He was clea rlyorganized with a plan for how the class would be run. He had prepared hislect ure notes so that he was able to deliver the material in a way that fewwould find deficient . For example, he spoke loudly enough for all to hear ,began with a review of importan t prerequi site material , and developed thecontent that he wanted to teach in a logical fashion . Dr. G was consciousof the fact that it is not always easy for st udents to pay attention duringextended periods of lecturing, and used strategies (e.g. , sca nning the classfor st udents who seem to be dist racted and then calling on them by nam e)to keep st udents involved in what was going OIl. Dr. G appeared to enjoyan excellent rapport wit h the class (e.g ., he knew all of the student s bynam e, he joked with them before the lesson began , and he recogni zed andpr aised the progress that students mad e). Alth ough it was not necessar ily

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an exe mplary dem onst rati on of active learning, Dr. G devoted some of hislesson to an active learning oppo rt unity for t he st udents, perhaps in t hehop e that they would have a chance to delve deep er into som e asp ect of t hemateri al t hat had just been covered in his lecture.

T he vignette does reveal some areas in whi ch Dr. G could pr obablyimprove his t eaching. For example, he did not pay sufficiently close att enti onto the class to see every rai sed hand and thus missed some questions. Wheninitia t ing the gro up exercise, st udents were slow to move to t heir "usualgroups" and , on t he whole, relu ctant to work on t he ass igned probl em whenthey finall y assemb led . Although these are problem s from t he point of viewof classro om managem ent (see [19, 23, 24]) , they are not t he main issu est hat the students seem to see as problematic in t he class.

Before moving on, we would like the read er to consider t he following twoquestions.

1. If you were Dr. G , would you redouble your efforts to become a moreskillfu l controller of classro om events, or would you try to do some­t hing that addressed the problem s rai sed by the st udents?

2. If you decided to address the issues t hat t he st udents raised, whatconcrete change to your teaching pract ices would you make?

Can conventional lesson preparation pract ices adequately address t heissues rai sed by t he st udents? In t he next sect ion, we describe conventionalplanning practi ces, and we make t he case that something radicall y differentis required.

1.2 Preparing for College Mathematics Teaching

Evidence suggests t hat many college mathem ati cs teachers view t he workof preparing for class as a "conte nt organizat ion" task [20, 53, 65, 66]. Thisview of t he nature of preparati on for teaching appears to be particul arlycommon among instructors whose teaching practi ce is dominated by t helecture method [26, 60]. In t his view, prepar ation for te aching involves,

1. select ion of the areas of content that will be demonstrated during t helesson ,

2. making decisions regarding t he treatment (such as rigorous versusheuristic; complete vers us summative ; formal versus informal ; etc.) oft he selected content during t he lecture,

3. selection or invent ion of examples to illu strat e particular aspects oft he conte nt areas t hat have been chosen, and

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4. arran gement of these elements into th e most mathematically coherentsequence tha t is possibl e.

To some exte nt, this characterization echoes som e of the thoughts onlesson planning suggested in Rishel's [51, pp. 17-18] "basic plan for lessonplans."

1. Start by finding out what today 's topic is supposed to be.

2. Peruse the text to see how the aut hor approac hes the to pic - thishelp s to preserve the sa me notation as the book , thereby reducing theamount of confusion in the class.

3. Prepare an intuiti ve explanation (a 'heur ist ic argument') as to whythe to pic is important , useful and relevan t .

4. Next, prepare a few homework-style problems of increasing difficul tyto illust rate to t he st udents the main concepts of the section of thetext .

5. Allot remaining class time to answering questions or doing old hom e­work probl ems."

Several authors [4, 14, 18, 39, 43] have noted that planning for highquality, effective college teaching depend s on the intend ed outcomes of in­st ruction - in particul ar , t he intend ed cha nges in st udents' behavior , knowl­edge and skills. Preparing for and delivering high-quality, effective collegeteaching involves, "... [ident ifying] clea r goa ls and intellectual cha llenge ­making absolutely clea r to the student what has to be understood , at whatlevel and why" [14, p. 461].

We note that while the methods for lesson prepara tion summarizedabove encompass some of these notions of planning for high quali ty instruc­t ion (for example, Rishel st rongly suggests preparing a "heur ist ic argument"to explain to students why the topic is importan t , useful , and relevan t [51,p. 18]) , t here are critical aspects of planning that appear to be absent al­toget her. For example, t here is little to sugges t (especia lly to the noviceinstructor - see [20]) that preparation for teachin g can include formulationof clea r, explicit goa ls (or student learning objectives - see [21]) that describewhat the st udents will know or be able to do as a resul t of inst ruc t ion.

To return br iefly to the case of Dr. G described in Section 1.2, what couldDr. G-already quite a skillful classroom teacher, a highly qualified math­ematician, and a dedicated educa to r- do to address the specific probl emsidentified by his class? We suggest th at th e vignette reveals that alt houghsome ga ins could be mad e through sha rpening Dr. G 's classroo m man age­ment skills, t here is little that actually takes place in "real t ime" during the

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lesson that would address the po ints that the students raised in their focusgroups. Instead, we suggest that a possible solut ion to the problems raisedby this class can be addressed through Dr . G's preparation for teaching .

Dr. G was regarded by both colleag ues and students as a conscientiousteacher who put t ime and genuine effort into preparing for class. T he simpleamount of preparation was not the issue here . Instead , we argue that onesystematic change that Dr. G could have made was to spend part of hispreparation tim e explicitly formulating goals to guide both his instruction,and also to bett er inform students of what they ought to get out of class .It is a fact that in Dr. G's class no students complained abo ut the factthat he sometimes missed some of their questions during class . Nor didany complain that his lectures were poorly organized or lacked examples(two complaints leveled at most inst ruct ors in the same course) . Instead, all(100%) complained that he assigned homework problems before the st udentsfelt prepared for them , and many (61%) com plained that he did not delveinto problems with sufficient depth.

We suggest that a clearly form ulated set of learning objectives couldhave helped Dr. G to locate the examples that could :

• determine which problems do the best job of demonstrating the mat h­ematical points that he wanted the st udents to learn,

• identify the key points wit hin those problems that he must make aparticular effort to bring to students' attention,

• commun icate to the class the particular mathematical points that t heyneeded to know and be able to use in order to successfully tacklehomework pro blems (especially those that are not rou t ine),

• more pru dently select homework problems for the class, and

• become more aware of the mathematical background materi al thatmust be mob ilized in order for st udents to ma ke sense of the newmaterial that they encounter.

Unfortunately, many popular references on college teaching (includ ingthose well-known to college mathematics teachers, such as those of Kran tz[38] and Ris hel [51], and those that speak to college teaching more broadly,such as [43]), do not emphas ize the ident ification and communication ofinst ru cti onal goa ls when dispensing pragmat ic advice on "preparat ion" toprospective and practicing college teachers .

In this article we will enumerate many of the ways in which collegeinst ru ctors and their st udents can benefit from an approach to instructi onbased on st udent learning objectives. T he specific goals of this first pap erare to :

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1. argue for the importan ce of formulating explicit objectives as a partof instructor planning,

2. review research on the role and effects of explicit learning objectivesin planning and teaching,

3. introduce a definit ion of student- learn ing objective (SLO) , and

4. list potential advantages that are available to instructors who rou­tinely formulate and use exp licit learning objectives.

In the second article in this series, we will present a ste p-by-ste p al­gorit hm that can allow college mathematics instruct ors to harness th eirknowledge of math emati cs to formulate useful sets of st udent learning ob­jecti ves.

2 REVIEW OF RESEARCHON LEARNING OBJECTIVES

For more tha n fifty years , formulation of specific objectives has been ad­vanced as the foundation of prepar ation for teachin g [63]. Alth ough famil­iar to many in the context of K-12 education, the impera tive to groundinstructi on in explicit objectives has only recently begun to penetra te thefield of higher education [44], appa rent ly generating at best limi ted ent hu­siasm among college and uni versity facul ty [44, 48]. Perh ap s as a resul t ,little research has been conducted in college math em atics classrooms on theeffects of setting and following explicit objectives. Rath er more resea rchhas been completed on students ' goa l-setting practices (sometimes referringto goals set by th e st udents th emselves, and at ot her t imes referring to theado ption of vari ous goa ls or goa l orientations by students within a course)and t he effects on such things as st udent performan ce in the course.

This review is divided into two parts: t he first dealing principally withwork from the inst ruct ional point of view, the second dealing with st udiesconducted on student learning and instructional goals. In both parts wehave included resu lts that we see as relevant to college and university math­emat ics teachers who are interest ed in grounding their teaching efforts in afoundation of explicit ly formu lated st udent-learn ing obj ectives.

2.1 Research on Learning Objectives from the Instruc­tional Point of View

Much of th e research that exists on the goal and obj ective set t ing prac­tices of teachers has been conducted with teachers in schoo l settings, and

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has been completed in the context of research into teachers' planning andthinking processes [15,56]. T he resea rch in this area that has imm ediate rel­evance for the college mathemat ics classroom has been reviewed elsewhere(see [21, 56]). In this review we confine ourselves to research on learningobjectives that is not specifically related to bro ader programs of inquiry int oeit her teacher's planning processes or teacher's interactive decision-makingprocesses.

A well-known exa mple of the effect of goal setting in a school scienceclassroom is the teaching of Sist er M. Gertrude Henn essey'' . The classroomconversations and impressive learning out comes achieved by the studentsin her classes have been well known to science educators for at least tenyears. The conversations and outc omes have been docum ent ed in the pro­fessional literature of science educat ion and have genera ted int erest amongresearchers [5, 6]. The st udy of Hewson and Beeth [7] describ ed here is acase study (based on qualitative data) that attempted to answer the ques­t ion: Is the learning that takes place in Sister Gertrude's classroom theproduct of a unique and singular environment or does Sister Gertrude useinstructional practi ces that could be utili zed by ot her teachers to producesimilarly impressive learning outcomes?

Hewson and Beeth noted Sister Gertrude's practice of goal setting asan importan t basis of her instructional practice [7, p. 743]. The resear chersbelieved that Sister Gertrude's practi ce of setting goals in addit ion to theacq uisit ion of scient ific knowledge to be par ti cularly significant and notedseven overarching goa ls that Sister Gertrude formulated for th e course asa whole. These goa ls were formul ated as the following questions, whichst udents were encouraged to ask thro ughout the academic year.

1. Ca n you state your own ideas?

2. Ca n you talk about why you are att racted to your ideas?

3. Are your ideas consiste nt?

4. Do you reali ze the limitations of your own ideas and the possibilityth ey might need to cha nge?

5. Can you try to explain your ideas using physical models?

6. Can you explain the difference between understanding an idea andbelieving in an idea?

7. Can you apply "inte lligible" and "plausible" to your own ideas? [7J

2 A brief introd uction to Sister Gertrude's teachi ng is availa ble on- line t hro ug h t heOhio State University's P-1 2 project. See: ht t p :/ / www .osu .edu/p12

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T he role of these learning goa ls, as seen by Hewson an d Beeth, wasto ". . . establish the cond itions under which st udents. . . learned science"[7, p . 746J. In par ti cular , t hey found tha t t he learning goa ls played animp ortan t role in honing st udents ' ability to reflect crit ically on his or herscientific ideas . to establish local standards or crite ria against which thescient ific val idity of idea.') could be judged or just ified , and in establishingthe importan ce of the st udents ' scient ific ideas in the lea rning tha t went onin the course .

However , alt hough impor tan t , t he resea rchers did not attribute the im­pressive learning outcomes of Sister Gert rude's classroom ent irely to theestablishment and guidance of learning goa ls. Hewson and Beeth not edthree ot her significant features of the Sister 's instructional practi ce. Fi rst ,the st ructure of classroom instruction and learning activit ies were consis­tent wit h her readi ng of research litera ture on children's learning (especiallydea ling wit h conceptual change, children 's concept ions of science and chil­dren 's knowledge construction) . Second, Sister Gert rude 's man agement ofher classroom engages st udents in the learning of som e of the criteria bywhich know ledge in the broader scient ific community is justified . For ex­amp le, Sister Gert ru de asked st udents to work intensively on a particularpro blem in a group of three to five. When this work had progressed toa useful poin t , t he st udents were required to reconcile and integra te theirideas with those of ot her groups. Sometimes gro ups of st udents investigateddistinct related phenomena and later integrated their ideas to create a com­prehensive pict ure of a large-scale scient ific prin ciple. Third , in addit ionto being well founded on established principles of science educat ion, Sist erGertrude's instruct ional practice included empathy for the st udents as theyst ruggled to learn science.

This third factor might lead some read ers to believe that the out comes ofSister Gertru de 's instruction were, in fact , substant ially due to her uniquepersonali ty. However , Beeth and Hewson take pains [7, p. 755] to expresstheir conclusion that the educat ionally impor tan t aspect of this deep em­pathy was th at it led Sist er Gert rude to reserve eno ugh time for studentsto thoroughly explore and talk about their ideas , and tha t t his pract ice is(at least in principle) replica ble by ot her teachers . T he aspect of SisterGert rude's practice that the researchers saw as most probl ematic for ot herteachers was obtaining the administrative and cur ricular freedom that wouldenable them to limit t he number of to pics to be covered and allow st udentssufficient t ime to explore and talk about their ideas [7, p. 753].

The case study of Sist er Gertrude's classes illustrates the role that for­mulati on of exp licit learning goa ls can play in the creation and shaping of

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the conditions under which st udents will learn a technica l subject (generalscience in this case) . This case st udy offers evidence to suggest that the for­mul at ion and use of these over-a rching learning goa ls is an important andintegral com ponent of instructiona l practi ces that produce superior st udentlearning outcomes alt ho ugh, as a single case st udy, does not establish thisbeyond quest ion. This approach did seem to allow Sister Gert ru de to designcurric ula with the students' ideas in mind, to make her expectations known,an d to choose st rategies that support st udent exploration of their own ideas .This case st udy also shows that while goa l setting and use may well be im­por tant components of highly effective instructional practices they are not(in and of themselves) a complete solut ion. This case st udy suggests th at itis a harmonious combinat ion of goals, learning activit ies const ruc ted usingresear ch-based educa t ional princip les, and classroom man agement pr act icesthat cont ribute to achievement of student-learning outcomes .

In cont rast to the harmony of Sist er Gertrude's goa ls with her practice,Stark et at. [58] observed a discontinuity between the stated purposes offacul ty memb ers with specific course goals that they submitted. In their na­t ional st udy, faculty "overwhelmingly" sa id that their prim ary educationalpurpose was to develop effective thinking . They were then asked to submitspeci fic course goa ls. T he overwhelming maj ori ty of the 4000 goa ls thatwere sub mitted related to teaching concepts in their discipli nes, instead ofto deve lop the intellectu al skills that they cla imed to value. Simi lar resultshave been commun icated elsewhere [45, 61].

The first important message that we see here is that processes used byfaculty to prepare for teaching dr aw heavil y on uni versity teachers ' knowl­edge of the sub jects that they teach. Second , that some degree of discontinu­ity between what faculty view as their main educationa l purpose and whatt hey actua lly do when teaching may not be an uncommon phenomenon.We suggest t hat the existence of such discontinuiti es may be due, at leastin part , to the fact that many facul ty memb ers are reluct an t to explicitlyformulate goa ls that describ e their educational purposes [44] and that manyteachers do not use these kinds of guiding principles in a discip lined waywhen planning for instructi on [53, 65, 66]. (Sister Gertrude [7] is an obvi­ous exceptio n.) We suggest that both the pro cess of formulating explicitgoals and the pr acti ce of using th em in a disciplined way when preparingfor teaching may be far more attrac t ive to faculty if the pro cess and prac­t ice are both gro unded in what faculty already spend a lot of time doing:T hinking about the subject mat ter that they teach.

Bol and St rage [9J examined the st udent learning goa ls and assess mentpractices of te n high schoo l biology teachers. As wit h [58], Bol and Strage

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also observed a discontinuity between the goals that the teachers claimed tohold and the goa ls that were actually supported by their assessment s. Theresearchers interv iewed te n high schoo l biology teachers on th eir teachin gphilosophies and intend ed st udents learning outcomes . The researchers alsocollected test and pract ice problems (including tests, qui zzes, textbooksprobl ems, ot her homework probl ems, worksheets and st udy guides) thatthe teachers assigned and rated these according to th e level of processing(basic knowledge, app lication, etc.) and form at.

They found t hat overa ll the teachers stated "achievement goa ls that wereprimarily globa l, higher order obj ectives that emphasized learning skills,moti vation , understanding, and the applicat ions of knowledge" [9, p . 152].However , t heir assessment practices did not support t hese goals. For ex­ample, more th an half (52% of test it ems and 53% of practice problems )required st udents to simply recall basic fact ua l information . Significant ly,Bol and Strage found that the teachers were unaware of this mism atch.These researchers suggest th at an importan t st rategy for meeting the chal­lenges of instructi onal reform is to improve teachers ' understanding of thevalue of aligning their instructional objectives with their assess ment items.Furthermore they suggested that teachers utili ze some sort of blu eprint forclass ifying individu al assess ment items in order to facilitate meeting theirprofessed learning objectives.

We preface the following comments with th e observation that the st udyof Bol and Strage [9] has some obvious limi tat ions (e.g., a sa mple size often , it was conducted in the context of biology teaching) that make anyinferences drawn for college mathemati cs teachin g tentat ive at best . Simplecommo n sense seems to suggest that an individual 's acti ons will be genera llyconsistent wit h his or her overarching philosophy or belief system. That is,an indi vidual will naturally act in accord with his or her professed philoso­phy or beliefs. Bol and St rage's results cast doubts upon the validity of thisnatural assumptio n. For the individual teacher who wants to ensure thathis or her teaching pr actice is representati ve of his or her beliefs about thenature of learning and philosophy of teaching, simp ly formulating beliefs orado pting a philosophy may not be enough. Teaching in a way that is t ru lyreflecti ve of wha t an individu al teacher believes may require the teacher tofollow procedures or approac hes to teaching that are in some sense mor eexaggera ted or unnatural t ha n the pract ices that a teacher might use if leftto his or her own devices. We believe th at one such potenti ally help fulapproach is to ground ins truction in an explicitly [ormulated set o] student ­learn ing objectiv es. Alt hough there is certainly a price to be paid by theindividual teacher (i.e. t ime an d effort) , t he benefit can be a substant ially

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better agreeme nt between what th e teacher thinks he or she is doing andwhat is actually accomplished in th e course.

2.2 Research on Learning Objectives from the Student­Learning Point of View

Much of the research on goa ls and obj ectives from the st udent -learn ingpoin t of view is couched in the st udy of motivation in education. Thereis an immense weal th of published research on goa l set t ing and studentmoti vation, especia lly in genera l psychology and in schoo l (K-12) set t ings.Some excellent start ing points for read ers interested in these issues include:[33,49, 59]. In this review we have focused on those st udies th at

• help instructors to understand how the different goal orientations thatcollege st udents can adopt (or perhaps can be encour aged to adopt)are related to motivation and performance in college-level courses,

• sugges t courses of act ion that instru ctors could take for using goalsetting as a tool to enhance st udent learn ing,

• help instructors to recognize the different goa l st ructures that canbe implemented wit hin a course and the learning outcomes that areassociated wit h these, and

• help ins tructors to understand the various propert ies of learning goals,and the learning outcomes that are associated with these proper ties.

2.2.1 U sing Goal Setting t o Enhance Student Learning

Carter [13] provid es a bri ef summary of the research on the goa l orient ationsof st udents in vari ous classrooms. The independ ently generated consensusof several researchers [1, 28, 46] is that st udents ar e described as adoptingeither a mast ery goal orientation or a performance goal orientation. St u­dents wit h a mastery goa l orientation try to master the material, at te mpt tolearn as much as possible, and view misconceptions as important ste ps to­wards const ruc t ing accurate knowledge. Students wit h a performan ce goalorientation try to outperform their peers , finish their work as quickly aspossible, make high grades, and get teacher app roval. T he research consis­t ent ly shows higher levels of st udent achievement from adopting a masterygoa l orientation than from adopting a performan ce goal orientat ion [1, 47].

Carter identified three ste ps that instructors can take to help st udentsadopt a mastery goa l orientation. The first is to help students set short­term learning goals. The second is to have students assess their own progresstowards their learning goals through a process of self-exa mina t ion of their

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work. T he third is to help st udents reali ze that misconceptions are stepping­sto nes in the learning process and can be importan t steps toward accurateconceptions . Finally, Carter suggested that students draw mental models oftheir conceptions both pri or to and after instructi on as a means to facilitateth e three ste ps toward a mastery goa l orientat ion list ed above.

An impor tan t point we see here is tha t instructors can model the stepssuggested by Carter by themselves, setting out speci fic short -term learninggoa ls, and by designing instruct ion that creates opportunities for progresstowards these goals. Secondly, we suggest in the sequel to this article thatdrawing a specific kin d of mental map can facilit ate the instructors' formu­lation of such short-term learning goa ls.

In a study of 262 college st udents enro lled in psychology courses, Epperand Harju [29] investigated the rela tionship between st udents' goal orien­tation with their academic performan ce as signified by their grade pointaverages. They used Roedel, Schraw, and Plake's [52] Goals Inventory todetermine separate scores for the st udents ' learning and performan ce goals.These goa l orientations are the same as those discussed by Carter , whereinstead of "mastery orientation" these aut hors use the phrase "lea rn inggoals." Unlike Carter, and following after Dweck and Legget t [28], t heseauthors looked at performance goals and learning goals as two completelyindependent factors. T hey determined that students wit h high performan cegoals and high learning goals as well as st udents with high learning goa ls andlow performan ce goa ls had significant ly higher GPAs tha n those ot her goalorientations. Likewise they determined that students with low performan cegoals and low learning goa ls had significantly lower GPAs than those othergoa l orientations. Students with low learning goals and high performancegoa ls had lower , but not significant ly lower , GPA s.

Epper and Harju concl uded that instructors could encourage a learn­ing goal orientation by st ructur ing class time so that students are activelyengaged in the learning process. They also suggest that st udents be en­couraged to deve lop a more process-orientated learning style as opposed tofocusing mainly on the outco me. This encourages a point of view where thegoa l is achievement of compete nce in an area and the development of newskills rather than get ting a gra de .

We sugges t t ha t an important first step in st ructur ing active studentengagement wit h mathemati cs is for the instructor to formulate the short­term ob jectives that will underlie the tasks with which the st udents willbe actively engaged . Moreover, if t hese objectives includ e the condit ionsof engagement and the performan ce criteria by which achievement will bejudged, then the objectives can encourage students to focus on th e process

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of mathematical engagement in addit ion to the outcome (e.g ., getting the"right" answer) .

2.2.2 Goal Orientation, Performance and Motivation for Col­lege Students

Har ackiewicz, Barron , and Elliot [36] raised the questi on: What ty pe ofmotiva tional orientat ion is optimal for survival and success at t he collegelevel? In order to address this question, t hey adopted an achievement goalsframework and attempted to determine how different goal orientat ions af­fect performan ce and intrinsic motivation . In t heir article they report ontwo different st udies. In the first st udy, they recrui ted a group of psychol­ogy students to participate in a st udy of pinball playin g. Following [35],they differentiate between purpose goa ls, which are higher level and pro­vide a genera l purpose for engaging in an act ivity, and target goals, whichare task-specific guidelines providing concrete standards or targets. Theydistinguished between mastery goals and performan ce goals. Harackiewicz ,Barron, and Elli ot performed two experiments, in which th ey investigatedthe effect of the interactions of purpose goals and target goa ls with perfor­man ce goals and mastery goa ls on intrinsic moti vat ion. They found that theinteraction of purpose goals with mastery and performance goals dependedsignificant ly on the personali ty, specifically the achievement orientat ion, ofthe students . St udents whose achievement orientation was compa t ible witht he ty pe of purpose goal (maste ry or performance) showed increases in in­trinsic motivation. On the oth er hand, they found th at for target goals,mastery target goa ls enhanced intrinsic motivation relat ive to performan cetarget goals. Here too they found evidence that individual differences inachi evement orientation moderated these effects .

In Harackiewicz, Barron, and Elliot's second study, th ey followed a co­hort of college st udents enro lled in an introductory psychology course overa period of a semest er. The researchers measured individual differencesin achievement orientat ion using a dep artment al survey conducted prior tothe semes te r. They then measured intrinsic interest near the end of th esemes te r and course grades at th e end of the semeste r. They found thata mastery goal orientation stated at the beginning of the semest er impliedhigher intrinsic interest at th e end of the semester, but was unrelated tograde performan ce. On the ot her hand, th ey found that a performance goalorientation stated at the beginning of the semester implied a higher coursegrade, bu t was unrelated to end-of-semester intrinsic interest . Because eachgoal was associated with one positive outcome, but not the other , the au­thors suggested th at st udents who adopted both goa ls were the most likelyto attain both outco mes.

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On e importan t resul t that we see here is that mastery target goa ls wereshown to enhance int rinsic moti vation relative to performan ce target goa ls,regardless of the motivati onal orientation of the st udent . T his is in cont rastwit h purpose goa ls, which seem to require compatibility with motivationalorientation to have their full effect . Since some st udents approach learningwit h differing combinations of a mastery orientation and performan ce ori­entation , havin g an instructor emphasize both ty pes of purpose goals in acourse seems to be importan t . On the other hand, setting mastery targetgoa ls seems to be one way of enhancing student learning broadly, ratherthan for one parti cular ty pe of st udent .

2.2.2 Goal Structures and Attribution of Success

Ames [2] studied the effect of goal structure on achievement attributionsof 88 fifth- and sixth-grade children during mathematica l puzzle solving .She compared an individuali sti c goa l st ruc ture , where the students wereencouraged to get as many puzzles right as they could and improve wit h eachsuccess ive attempt, with a compet it ive goal st ructure, where the st udentswere set against a competitor and encouraged to get more right than thecompet itor . Ames found th at with an individualisti c goa l structure, t hest udents were much more apt to attribute success or failure to effort orlack thereof. Likewise, she found that with a compet it ive goal structure,the students were much more likely to att ribute success or failure to abilityor lack of ability. T he implicit message was th at an individualis ti c goalst ructure is preferabl e, becaus e effort is something that a st udent has directcontrol over, whereas ability is not .

The important resul t that we see here is that instructors can influencestudent attribut ion of success by encouraging individualisti c goal st ruc tures.One possible way to help student s focus more on individual achievement ofoutco mes would be to set forth achievable and measurabl e learning obj ec­t ives for each lesson .

2.2.3 Properties of Learning Goals

Schunk [57] focused on the self-regulated learning process of goa l settingand perceived self-efficacy. In his literature review on goa l setting, he foundthat the effects of goa ls on st udent behavior depend up on three properti esof the goals: thei r specificity, t heir proximity, and their difficulty level.Schunk found that goals incorporating specific performan ce standards aremore likely to enhance learning than general goals. He also found thatproximal goa ls result in greate r student motivation than more distant goa ls.

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Finally, he found that greater effort is expended in order to ach ieve difficultgoals than to achieve easier goals, assuming that students have the requisiteskills to meet the difficult goals. In fact, students who received difficult goalsalo ng with goal attainment informat ion , such as how many of their peerscompleted sim ilar tasks, displayed the highest self-efficacy and skill.

Schunk's findings suggest that instructors could affect student learn ing,motivation, and effort by helping students to identify (or by identifying forthe students) spec ific, prox imate, and difficult yet achievable learning goa ls.In this vein, we will next give a defin it ion and characterization of student­learning objectives that we believe instructors can use to enhance studentlearning, motivation, and effort in their courses.

3 DEFINING THE NOTION OF ASTUDENT-LEARNING OBJECTIVE (SLO)

3.1 Distinguishing B etween Goals and Objectives

As indicated by the st ud ies rev iewed in Section 2, te rms such as "goal,""objective" and "outcome" are often used in educational contexts with atbest an intuitive, but vague, sense of what is intended . Sometimes the termsare used interchangeably, whereas some authors use specific terms such as"learning outcomes," "learning goa l" or "instructional objective" to conveyinformation about the type of educational end that the author is interestedin discussing . (For examples of the ways in which these terms are used, see[3, 25, 31, 34, 37,40,41 , 42, 64].)

Following the convention emerging from a number of scholarly works [11,16, 42] we will use the term "goal" to refer to the large-scale, overarch ingends of education that are generally not ach ieved as a response to any single,discrete instructional action. The "goals" included in the CUPM Sourcebookfor College Mathematics Teaching [54, p . 2], such as:

• mathematics instruction sho uld provide st udents with a sense of thediscipline - a sense of its scope , power, uses and history,

• mathematics instruction should deve lop students ' understanding ofimportant concepts in the appropriate core content, and,

• mathematics inst ruction should help st udents to develop what mightbe called a "mathematical point of view" - a predilection to analyzeand understand , to perceive structure and structural relationships, tosee how things fit together,

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are very much "goals" in the sense we intend here. (As, for example, ar emost of the goals that are formulated and published by authors in collegemathematics educat ion - see [10,22,32, 62] for further examples.)

With "goals" understood as large-sca le, overarching ends of a processof educat ion, instruct ion an d learning, we will use the term "object ive"to refer to th e discrete educa t ional steps that learners take in purs uit ofeducat ional goals. "Objectives" are th erefore much more specific, concrete,lim ited and (usually) connected with specific areas of subject mat ter t hanare goa ls. Nevertheless, achievement of a "goal" will usually be (at least tosome degree) the resul t of successfully acco mplishing a number of objectives.

3.2 A Preliminary D efinition of "Learning Objectives"

Farrell and Farmer [30] give a broad definit ion of the term, "inst ruct ionalobject ive." According to Farrell and Farmer [30, p. 196], an inst ructionalobj ective is "... a statement that describes a desired student outcome ofinstruction in terms of observable performan ce under given condit ions."

DeLong and Winter [21] have noted that this defini tion differs from thestatements of teaching objectives given by novice college mathem atics in­structors in that this defini tion requires that an instructi onal objective mustdescribe (i) t he outcomes of instructi on in te rms of st ude nts' behavior orperforman ce, and (ii) that the stated outcomes must be pot entially observ­able.

Inst ru ct ional obj ectives, then , consist of three components:

1. a description of the observable student beh avior th at will result frominst ruct ion,

2. a characterization, description or definition of the conditions underwhich students will exhibit t his behavior, and ,

3. a characteriza tio n, descript ion or defini tion of the performan ce stan­dards that will be judged to represent success for the student .

3.3 A Characterization of Student-Learning Objectives(SLOs)

We will use the term student-learni ng objective (SLO) to refer to instruc­tio nal objectives (in the sense of [30]) that satisfy the criteria listed below.

1. Specificit y: A student -learn ing objective must be specific and re­fer to a student acco mplishment that is plau sibly achievable by themaj ority of students in one class meeting or lesson . This is not to

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suggest that instructional obj ectives that require a longer time framefor accomplishment are unimportant , merely that we will not use theterm student-learning obj ective (SLO) in this art icle to refer to in­structional obj ectives that involve a time fram e tha t is longer than asingle class meeting.

2. Premeditation: A student-learning objec t ive is formulated beforeclassroom instruction actually takes place. Much of this activity willlikely take place during the pre-active or planning phase of instruction(as most of the decisions that ultimately affect instructi on are madet hen) but does not necessarily have to take place when an instructor isformally engaged in a process completely dir ected towards planning forthe instruction and learning that will be at tempted during a parti cularlesson .

3. Deliberateness: The instructor mus t deliberately set aside someportion of the class meeting for th e specific purpose of providing stu­dents with a realisti c opportunity to achieve the stated obj ective.Classroom teachin g can be a highly dynamic pro cess and it is en­ti rely possibl e that the events that take place in an actual class willinvolve significant deviati ons from th e agenda that the instructor hasplanned . One possible consequence is that in reality, the finit e dura­t ion of a single class meeting may not allow all planned SLOs to beoffered to the students. In this art icle, whether or not an objective isact ua lly accomplished (or opportunities to accomplish the objectiveoffered to students) does not disqualify the object ive from being anSLO. As long as the instructor 's plans for the class meet ing includeopportunit ies that will offer the students a realistic chance to accom­plish the set obj ectiv es, t he objectives can be referred to as SLOs.

4. Cognitive indivisibility: When formulating SLOs, an instructor isconfronted with the pr oblem of unambiguously specifying the condi­t ions and procedures that will constit ute the SLO , and yet at thesa me time preserving the intention of the SLO. That is, the instructoris t ryi ng to expose as many of the detailed specificat ions (the "trees")that make up the SLO without obscur ing the intent behind the SLO(the "forest") . We call learning obj ectives that achieve this balanc ecogni tive ly indivisi ble to indicate that further subdivision into smaller ,more specific learning objectives would seriously obscure the intent ofthe original learning objective.

For example in a pre-calculus class that includ ed regression analysis,we suggest that the following instructional obj ective could be regarded as

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cognit ively indivisibl e:

(I) "Given a tabl e of data showing values of two related qu an t iti es and anindica t ion of which qua ntity is best suited to the role of t he ind epen­dent var iabl e, st udents are able to determine which typ e of fun cti on(constant, linear , power, exponent ial or polynomial of degree at mostfour) will do the best jo b of representing the major trends in the data ."

It is certainly true that this objective could be decom posed int o a col­lection of subo rdinate objectives, each following t he pat t ern (wit h "linear"replaced by power , expo nentia l, quadrati c, etc .) :

(II) "Given a tabl e of data showing values of two related quanti ti es and anindicati on of which qu anti ty is best suited to t he role of t he indepen­dent va riable, st uden ts are able to determine whether or not a linearfunction will do a reasonabl e job of representing the major t rend inthe data ."

Despi te t he fact that objective (I) could be split into a collect ion ofsubordinate learning objectives (each resembling obj ective (II)) , we arguethat objective (I) can be regarded as cognit ively indi visibl e. On our read ing,ob jective (I) has the intenti on of st udents being able to take any set of dataand find a plau sibl e best- fit curve for the data . It implies that t he st udentshave not only ga ined t he subordinate skills of judging the goodness of fit ofeach individual ty pe of regression that t hey have learned to perform, bu t canalso discriminate amo ngst t he vari ous form s of regression to select t he onethat do es t he best job of representing the trend in the data. A collect ion oflearning objecti ves modeled afte r obj ective (II) does not necessarily revea lt his intent - that st udents are able to discriminate between t he variousoptions t ha t are ava ilable to t hem and det ermine which will do t he bestoverall job .

5. Compatibility: In the mod els of st ruc tur ing a course, instructionalplanning and instructional ana lysis discussed earlie r, all featured alevel that referred to t he overa rching goals or t hemes of a course. Thefinal requirement for an instructional obj ective to be referr ed to as anSLO (at least in t he context of t his article) is that it is compatiblewit h the highest and most overarching level of course goals.

Student-l earning objectives are well-sui ted to describ ing instruction andlearning at t he second lowest of Schoenfeld 's [55] levels of teaching analy­sis, and most relevant to the task of planning one individual lesson wit hina mathemati cs course with a predetermined curriculum. T his is perhap sthe most common form of teacher planning done by college and universitymathematics instructors [21].

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3.4 Examples of Student-Learning Objectives (SLOs)

The purpose of this section is to list some concrete examples of SLOs fromto pics ofte n taught in college or uni versity math ematics or statist ics courses .

3.4.1 Example: 8LOs for an introductory lesson o n ExponentialGrowth

T he followin g SLOs were formulated to guide development of a class thatintroduced th e concept and definition of exponent ia l growt h, exponent ialregression , and the (approximate) numerical solut ion of exponent ia l equa­t ions . P lease note that these SLOs are not intended to be "exemplary"in any sense. They are included as a concrete exa mple for the read er 'sreference.

1. Given an explicit formula for an exponent ia l function, Y = A . B X, inwhich A is a number and B is a posit ive number , and a specific valueof x , st udents are able to calculate y correct ly.

2. Given the x- and y-coordinates of two points that are not vertica lly(Y2 =1= Yl , Yl =1= 0) or hori zontally (X2 =1= xd aligned, st udents cancalc ulate the growth factor:

(Y2) X22. X1

B = -Yl

3. Given a table of data, stude nts are able to check to see whether ornot the data can be represented (at least approx imately) using anexponent ial function . T he st udents are able to do this by ca lculat ingthe growth factor using several different pai rs of points from the tableand checking to see whether or not all of the growt h factors obtainedare approximately the same or not .

4. Given a tab le of data that can be approximated reasonab ly well byan exponent ial fun ct ion , st udents are able to ente r the data into theircalculators an d obtain an equa tion for the exponent ial funct ion usingexponent ia l regression.

5. Given a solvable exponential equat ion of the form A . B X = C andthe dimensions of an appropriate viewing window, st udents are ableto use the "INT ERSEC T" feature of a graphing calculato r to find anapproximate solution of the equation (i.e. t hey are able to find thenumerical value of x) .

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3.4.2 Example: SLOs for a lesson on The Concept ofComposi tion of Function s

T he followin g set of SLOs was formulated to guide a class that introducedthe concept of creating new fun ctions from old using the opera tion of com­position from graphical, num erical and symbolic points of view.

1. Given a pair of functions (defined by formulas ) for which it is mathe­maticall y possib le to form a composition, st udents can find formulasfor both of the usual compos it ions (i.e. f (g(x )) and g(J(x))) .

2. Given two functi ons, f (x ) and g(x ) defined by formulas or graphs,st udents are able to determin e whether or not it is possible to forma composit ion such as g(J (:r )). The st udents are able to do this bydetermining the ra nge of f (:r ) an d the domain of g(x ) and checkingthat the ran ge of f(x) will have a non-empty intersect ion wit h thedomain of g(x ).

3. Given a pai r of functions that ar e defined by gra phs and for which itis mathemat ically possible to form a composit ion, st udents can sketchgraphs that accurately represent each of the usual compos itions (i.e.f (g(x )) and g(J (x ))).

4. Given a pair of functions that are defined by tables an d for which itis mathem ati cally poss ible to form a composit ion, and a stateme ntdescribing which compos it ion (i.e. f (g(:1: )) and g(J (x ))) is required ,st udents can write down a table that includes each value of the com­posite function that it is possible to dedu ce from the f (x ) and g(x)tables that have been given.

4 ADVANTAGES OF EXPLICITSTUDENT-LEARNING OBJECTIVES INCOLLEGE MATHEMATICS TEACHING

Earlier we suggested a handful of potent ial advantages that could have beenrea lized by Dr . G had he incorporated the formulation of a set of clearl ystated st udent learn ing objec t ives (SLOs) as par t of his lesson planning pro­cess. In this sect ion we build on th ose suggestions by providing a large num­ber of potenti al advantages that can be realized by the pr actice of regu larlyformulating (and recordin g) explicit SLOs in introductory undergraduatecollege math ematics courses.

Our aim here is not to formula te a list in which every item is intendedto apply equally to all instructors or all college and universi ty teaching

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environments but to create a resource from which the majority of collegeand university mathematics inst ru ctors will be ab le to find a few reasonsfor using SLOs that make sense to them, given their persona l ph ilosophy ofteaching and their specific institutional circumstances.

4.1 Potential Benefits for Mathematics Departments

(a) Explicitly stated and recorded SLOs can form the bas is for judg­ing the eq u iva le n ce of courses at different un ivers ities (for example, inthe context of awarding cred it towards a degree or waiving prerequisitesfor transfer students) . They can also serve as the basis for articulationagreements between departments or universiti es that work together.

(b) Sets of expl icit SLOs can provide client disciplines with ac­curate information on the learning outcomes of a course. (Forexample, calculus courses that serve as prerequisites for courses in otherdisciplines.)

(c) Describing the content of a course by a list of exp lici t SLOs(as opposed to list ing sections from a textbook) can make the courset extbook-independent. In departments that cha nge textbooks fre­quently, the practice of describing a course using SLOs can help to en­courage more cont inuity in the course from one semester to the next , an davoid the t ime-consuming rev isions of syllabi an d course material ot herwisenecess itated by text changes.

4 .2 Potential Benefits for Mathematics Courses

(d ) Formulating a set of explicit SLOs may give a much more realisticimpression of how much material can reasonably b e included in alesson or co u r se . By having to carefully think about the spec ific st udentlearning outcomes that are desired and then estimating how much t ime andeffort will be required to ach ieve these outcomes, instructors may save a lotof problems such as having to rush at the end of lessons or towards the endof a course in order to cover every thing that was included in the syllabus .

(e) Explicit SLOs - especially ones that are stable over time - can helpto maintain course and program integrity and quality. T hey canprovide a clear standard of what st udent accomplishments are required inorder to pass a course. Stable and explicit set s SLOs used as the founda tionof assess ment in a course can serve as both a guard against and a rebut talto accusations of grade inflation .

(f) T he SLOs from an entire course can form a stable description ofwhat that parti cular course attempts to deliver in terms of learning out-

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comes for the st udents. If a course is substant ia lly revised , then these SLOsca n guide the course r evision, thro ugh eit her the ident ification of de­sired learning outcomes that t he course does not address, or by ident ifyin girrelevant learn ing outco mes that are not needed .

(g) In a large mult i-secti on course in which many instructo r teach classesthat are supposed to deliver equivalent student learni ng opport un it ies, acourse-wide set of explicit SLOs can help to ensure a greater d egreeof uniformity in learning outcomes from sectio n to sectio n withou tnecessarily demanding a that all instructors follow a st rict ly regimented andcent ra lized set of pedagogical plan s and st rategies.

(h) W hen co-t eaching a course , or passing a course from one instruc­tor to another at the end of a semester, a set of ex p lici t 8LOs can helpto e n su re more continuity in the course from instructor to instructor.

4.3 Potential Benefits for Individual Mathematics In­st r uct or s

4. 3. 1 Enhancing the Quality of Classroom Instruction

(i) Formulating a set of explicit SLOs can h elp the instructor to setpriorities using course r esources. (For example, class t ime , lab t ime orequipment .) Ex plicit SLOs can also be used to det ermine which learningoutcomes will be pursued most vigorously in the course . T his is opposed,for example, to the pract ice of ass igning equal importan ce to every thingin the course and then running into sit uations in which the instructor willhave to rush to cover the most importan t material in the course (or omit italtoget her) .

(j) Formula ting explicit st udent learni ng objec t ives pr ior to instructionmay allow the instructor to r eflect on the specific p edagogical st r a te­gies that will offer the greatest number of students the b est op­portunity to learn. This may allow the instructor to ant icipate the prob­lems that constra ints (such as the amount of instructional t ime ava ila ble orthe physical set-up of the classroom) will present for different pedagogicalstrategies. Simply thinking about what materia l is to be covered may en­cour age an instructor to think purely in terms of presenting the material tothe student s thro ugh a lecture. T his may limi t t he inst ructo r by deprivinghim or her of an opportunity to explore and pract ice alte rnat ive pedagogica ltechniques.

(k ) A set of explicit SLOs - especially if formulated by the instructorhim or herself - can contribute to the clarity of the instructor's pre­sentations and learning activities in the classroom. T hrough the

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pro cess of formulating and art iculating the SLOs the inst ructor will havehad to reflect on what it mean s to underst and the mathematics that thest udents will learn, and then organize these ideas into some kind of coher­ent framework. Having created this mental model for what it means forst udents to understand the materi al , t he instructor may be better able tocommunica te the relevant aspects of the subject matter - and the connec­tions between them - to the students in the class.

(1) A set of explicit SLOs can cont ribute to the instructo r 's ability tomotivate the material and learning activities to the students in theclass. In the case where the instructor has actually formulated the SLOs himor herself, t he instructor will have had to go through a pr ocess of mentallysift ing and weighing the different learning outcomes that the class couldpossibl y st rive for , and determine not only which are the most important ,but perhap s also develop some rational criteria for these choices that canbe communicated to st udents .

(m) Formulating explicit SLOs may encour age instructors to focusmore clearly on what they want the students to actually learnand gain from their class, as opposed , say, to focusing on what areas ofmaterial t hat they will cover in th e course . That is, formulating the SLOsmay help to make inst ruct ors more learner-focused in their approaches tocurriculum , course and lesson planning.

(n) The process of formulating explicit learning objectives can help aninstructor to think about the links between old material (perhaps whatthe st udents are assumed to know when th ey ente r the course, or materialthat was previously tau ght in the cause) and the new material t ha t will betaught . By taking advantage of this opport unity, instructors may be ableto explicitly and clearly remind students of the material that theyhave previously learned, and link new material to old in explicit,robust and concrete ways.

(0) An explicit ly formulated set of SLOs can help to establish theinstructor's credibility with students by demonstrating to the studentsthat the instructor has a clea r grasp of the important elements of the subjectmatter. This can be further reinforced by a harmonious concordance ofclassro om instruction , learning act ivit ies, hom ework assignments and teststhat are guided and orchestrated by the SLOs.

4.3.2 Assessment

(p) Expli citl y formulated (and record ed) SLOs provide a definitive set ofstandards for relevan ce and appropriateness that the instructor can use asa guide for select ing learning activit ies, hom ework assignments, projects,

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and for formulating and scoring course assessments, such as testsand exams.

(q) If t he instructor regularly attempts to assess the quali ty of st udentlearning in the class roo m while the lesson is in progress, t hen a set of explic­itl y stated SLOs provides the instructor with a state ment of the skill andknowledge development that he or she is t rying to detect while assess ing st u­dent learning. T he explicit ly stated SLOs can serve as a starting pointfor selecting particular classroom assessment techniques (or CATs,see [4]) that are well suited to collect ing information that will actually beuseful to the instruct or.

4 .3.3 Documentation and Systematic Improvement of Teaching

(r) When documenting teaching accomplishments (for examplein a teaching por tfolio), an explicit list of SLOs can do an excellentjob of describing your course to the reader. Clea r communicationin such a context might be a real boon if t he portfolio is connected wit h ajo b application , promotion and/or tenure decision or an instit utiona l review.Furthermore, st ude nt teachers and educat ion maj ors (who may intend to be­come K-12 teachers) can use list s of SLOs to describe their st udent -teachingexperiences to potenti al employers and their own teachers.

(s) T he long-term pract ice of writ ing explicit SLOs and using them asthe bas is for instructi on can help college mathematics t eachers to r e­fine and focus their collections of ideas and feelings about teachingand learning (sometimes assemb led into a "state ment of teaching philos­ophy" ).

(t) When trying to fix things that didn't work out while teach­ing, the instructor can go back to the SLOs to determin e exactly what wasachieved, and what st ill remains. T his mean s that the instructor's sub­sequent attempts to teach may be more efficient as he or she willhave a clea rer idea of what the st udents have already mastered, an d theob jectives that st ill remain to be accomplished .

4.3.4 Potential Benefits to Students

(u) An explicit set of SLOs (t hat is communicated to the st udents) canhelp st udents to judge the effectiveness and relevance of their ownlearning and st udy habi ts and act ivit ies.

(v) If SLOs are mad e available to st udents the st udents may be betterab le to approach learning in the way that seems most natural to them,given their preferred learning style. Similarly, formulating explicit SLOs

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and communicating these clearly to st udents may h elp the st u d e n t s totake more r esponsibility for their own learning because they areaware of exactly what they are respons ib le for.

(w ) Explicit SLOs can help st udents prepare for quizzes , t ests andex a m in a t ions by providing them wit h an explicit statement of the skillsand knowledge that they are expected to have mast ered (and , if well written,convey a great deal of useful informat ion to the student about what levelsof skill and mastery are requi red).

(x) An explicit list of SLOs communicated to students pri or to an exami­nat ion can make students ' r eview time (and the time that an inst ructorspe nds work ing wit h students, especially in one-on-one context common tooffice hour s) much more efficien t . St udents are abl e to assess their skilland knowledge levels against the requirements of the SLOs, identify thoseareas that t hey are confident in, and develop a detailed and high ly specificpicture of the required skills and knowledge that they have yet to acquire .

(y) Basing instruction on an explicitly formul ated set of SLOs may h elpthe st u den t s to more readily transfer the mathematical informa­tion and t echniques that they are learning to new and novel sit u ­ations . Cognit ive science suggests that an important feature of instructionthat promotes transfer is the identificati on of the so-called "cognit ive ele­ments" of a concept [11], and the explicit communicatio n of t hese elementsto st udents. Formulat ing SLOs may help the instructor to distill t he "cog­nitive eleme nts" of the material that the st udents will have to learn, as theprocess of formulating SLOs is, in many ways, analogous to the cogn it iveanalyses that are used to identify the cognit ive elements of a concept.

(z) Dur ing the initial meetings of the course , students can use ex p lic­itly st a ted SLOs - especia lly reasonabl y detailed descriptions of the SLOst hat will be pursued at different po ints in the course - to decide whetheror not they are appropriately qualified to take the course , and whetheror not the course will deliver the knowledge and skills that they need todevelop and master.

C ONCLUSION

In this pap er we have defined a notion of st udent-learning obj ective (orSLO ). In order to be regarded as an SLO, a stated learni ng objective mustdescribe an observable outcome of instruction, list ing both the condit ionsunder which the outcome should be observed and the performan ce standardsthat will be used to assess success or fai lure.

We have compiled a substant ive list of pot ential benefits that can be

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realized by depar tments, courses, instructors, and st udents through the dis­ciplined and systematic use of the SLO concept in the design , practice andassess ment of mathemati cs teaching . Fur thermore, we have argued (t hroughanalysis of an illustrati ve vignette) that some of th e problems experiencedby otherw ise profic ient and committed teachers can be addressed throughthe explicit formulation of sets of SLOs and the disciplined use of these toguide teaching decisions.

We have also reviewed relevan t literature to indicate the potenti al bene­fits that can be realized by instructors and st udents by goa l setting practi cesin general, and some practical suggestions for the im plementation of learn­ing goals in courses. However , t he lit erature review also revealed that suchpr acti ces are seldo m employed in education, ofte n leading to cont radict ionsbetween the professed intentions of teachers, what actua lly goes on in class­rooms and the nature of the problems that appear on tes ts. Throu gh theresults described in th e lit erature review, we have argued that:

• substant ia l benefits can accrue from th e formulation and disciplineduse of explicit st udent- learn ing objectives in teaching,

• formul ation and disciplined use of explicit learning objectives in teach­ing is cur rent ly an underu til ized st rategy in teaching, and

• the standard operating procedures of teachers (and par ticularl y collegeand un iversity faculty) are strongly rooted in and influenced by thepar ticular s of the subject mat ter that they teach.

In the next art icle in this series we will describe (and illustrate the useof) a step-by-step algorit hm for helping college and uni versi ty mathemat­ics facul ty to use their subject matter knowledge to create well formul ated ,help ful st udent-learn ing objectives, an d begin to realize some of the educa ­t ional and administ rative benefits that can accrue from this strategy.

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Septemb er 2005 Volume XV Number 3

66. Zah orik, J . A. 1975. Teacher 's planning models. Educational Lead­ership. 33(2) : 134-139.

BIOGRAPHICAL SKETCHES

Matt DeLong is an associate professor of mathemati cs at Taylor University.He received his doctorate from t he University of Michigan under the dir ec­tion of J. S. Milne. In addit ion to thinking about teaching and math ematics,Mat t thoroughly enjoys spending time with his wife, son and dau ghter , di­rect ing his church choir , and acting in community th eater productions.

Dale Winter is an assistan t professor at th e University of Michigan wherehe helps to direct the introductory program . He has also tau ght at Har­vard University, Duke University, Bowling Green State University and theUniversity of Auckland. He received his doctorate from the University ofMichigan under the direction of Jo el Smoller. His dissertation focused onmathematical methods in genera l relati vit y and cosmology. In addit ion tohis professional interests in math emat ics and education , he enjoys the novelsof Primo Levi, mili tary history, marine biology and evolut iona ry psychology.

Carolyn Yackel is an assistant professor at Mercer University. She receivedher doctorate from the University of Michigan under the directi on of MelHochster. Her main are of mathematical interest is commutat ive algebra ,in addit ion to which she enjoys working wit h students and thinking aboutteaching. Her hobbies incl ude gardening , cooking, kni t t ing, crochet ing andmaki ng Temari balls, which can be used to illustrate the du ality relat ionsbetween P latonic solids .

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