Viewing “Mathematics for Teaching”as a Form of Applied Mathematics:Implications for the MathematicalPreparation of TeachersAndreas J. Stylianides and Gabriel J. Stylianides

IntroductionThe profound implications of teachers’ mathe-matical knowledge for the quality of the learningopportunities that teachers can offer to theirstudents (e.g., [2]) justifies, at least in part, thegrowing research focus on teachers’ mathematicalknowledge. This research has given rise to thenotion of Mathematics for Teaching (MfT) [3], [4],which describes the body of mathematics that isimportant for teachers to know in order to be ableto successfully manage the mathematical demandsof their professional practice, i.e., teaching mathe-matics to children. This is contrasted, for example,with the mathematics that is important for otherprofessionals such as engineers and physicists

Andreas Stylianides lectures mathematics education atthe University of Cambridge, UK. His email address isas899@cam.ac.uk.

Gabriel Stylianides lectures mathematics education at theUniversity of Oxford, UK. His email address is gabriel.stylianides@education.ox.ac.uk.

The present article draws on an article that we publishedin a general education research journal [1]. Parts of thatarticle are reused herein with permission (license number:2963601258532). The research underpinning the articleswas supported by funds from the Spencer Foundation (grantnumbers 200700100 and 200800104). We thank JamesSandefur and three reviewers for their useful comments onearlier versions of the present article.

DOI: http://dx.doi.org/10.1090/noti1087

whose work also imposes specific mathematicaldemands. Of course there is overlap between themathematics that is important to different work-places. Yet there are also certain mathematicalideas or ways of knowing and knowing how to usethese ideas that are more relevant to one workplacethan to another.

In this article we discuss a conceptualizationwhereby MfT is thought of as a form of appliedmathematics, and we probe the implications ofthis conceptualization for the mathematical prepa-ration of teachers. We also relate our discussionof MfT as a form of applied mathematics to howHyman Bass and Zalman Usiskin used the notion of“applied mathematics” to think about mathematicseducation or the mathematical preparation ofteachers.

Exemplifying Knowledge of Mathematicsfor TeachingWe begin with a classroom scenario that we useto exemplify elements of knowledge of MfT in theparticular domain of proof.

Classroom scenario:A seventh-grade class was reviewing meth-ods for finding a fraction between any twogiven fractions that are not equivalent andcan be located on the positive part of thenumber line. At one point during the lesson,a student in the class, Mark, announced

266 Notices of the AMS Volume 61, Number 3

proudly his discovery of the following gen-eral method for finding a fraction betweenany two such fractions:

To find the numerator of a fractionthat lies between any two givenfractions on the number line, yousimply add the numerators of thetwo given fractions. To find thedenominator, you simply add theirdenominators.

Mark illustrated his method with an example,which is shown in Figure 1. He also clarifiedthat his method gives one out of manypossible fractions between any two given(positive and nonequivalent) fractions.

The other students in the class wereamazed with the method, tested it withlots of examples, and saw that it workedin every case they checked. Then manystudents became convinced that Mark’smethod works for any pair of fractions. Oneof them, Jane, asked the teacher:

Can we use this method every timewe need to find a fraction betweentwo given positive fractions?

The teacher found herself in a difficultsituation: this was the first time that shehad seen the method and was not sure howto respond to Jane’s question.

A major mathematical issue that arose for theteacher in the scenario was whether it wouldbe mathematically appropriate for the studentsin the class to use Mark’s method to find afraction between any two given positive andnonequivalent fractions. The teacher was seeingthis method, which draws on the mediant propertyof positive fractions, for the first time. The teacher’smathematical knowledge could shape the courseof action she would follow in the classroomscenario, which in turn would influence students’opportunities to learn mathematics. Consider,for example, the following two possibilities thatoriginate from two different elements of knowledgethat the teacher could possess.

Possibility 1:

A possible element of the teacher’s math-ematical knowledge (misconception): If ageneral method is found to work for manydifferent cases (a proper subset of all possi-ble cases), then the method can be acceptedas correct.

→ Course of action: The teacher con-siders Mark’s method to be correct and saysto Jane and the rest of the class that theycan use the method every time they need tofind a fraction between two given positiveand nonequivalent fractions.

→Students’ opportunities to learnmathematics: The students add a newmethod to their “toolkit” (which happens tobe correct), but they are led to develop orcontinue to hold the same misconceptionas their teacher.

Possibility 2:

An alternative possible element of theteacher’s mathematical knowledge (soundconception): Unless a general method isproved to work for all possible cases, themethod cannot be accepted as correct.

→ Course of action: The teacher saysto Jane and the rest of the class that eventhough Mark’s method worked in all thecases they checked, there are infinitely manypairs of fractions and so the examination ofsome of these pairs offers no guarantee thatthe method will work for all possible cases.Also, the teacher invites the students to joinher in thinking more about the method tosee whether they can prove that the methodworks for all possible cases.

→ Students’ opportunities to learnmathematics: The students are exposed tothe mathematically sound idea that theconfirming evidence offered by some casesis not enough to establish the correctnessof a general method. Also, the studentsengage with their teacher in an explorationthat can potentially lead to the developmentof a proof for the method. If a proof isdeveloped, the students can add the methodto their “toolkit”; otherwise, the class wouldtreat the method as a conjecture.

Prior research (e.g., [5], [6]) shows that manyteachers have the misconception described inpossibility 1, namely, that examination of a propersubset of all the possible cases constitutes a proofof a general method. Prior research shows furtherthat many students of all levels of education havethe same misconception (see [7] for a review ofsome of this research). Possibility 1 illustrateshow a teacher’s misconception can generate orreinforce the same misconception among students,while possibility 2 illustrates how a sound con-ception could allow a teacher to offer to studentslearning opportunities to develop the same soundconception. The sound conception in possibility 2is “fundamental” [2] in the sense that it can applyto the mathematical work of all students (includingyoung children) and contains the rudiments ofmore advanced mathematical issues (notably, whatcounts as evidence in mathematics).

Given the important implications that posses-sion by the teacher of this sound conceptioncould have for students’ opportunities to learn

March 2014 Notices of the AMS 267

I begin with two positive fractions, say 1 2

and 34

.

To find a fraction between these two fractions I do the following:

12

++ 34

= 46

I use the number line to show that my method worked:

46

is between 12

and 34

.

0 1

46

12

34

Figure 1. A student’s illustration of a method forfinding a fraction between two given positive

and nonequivalent fractions.

mathematics, we can consider the conception tobe an element of knowledge of MfT [8]. Note thatwe do not suggest that knowledge of the mediantproperty of positive fractions should also be anelement of knowledge of MfT. Although knowledgeof this property would likely help the teacher dealwith the particular classroom scenario, it is notthe kind of “fundamental” knowledge that couldhave high currency in the work of mathematicsteaching.

Of course the element of mathematical knowl-edge in possibility 2 is important for effectivefunctioning not only in teaching but also in otherworkplaces. Thus, this element is not unique toknowledge of MfT. A related element that is morerelevant to teaching than to other workplacesthat use mathematics is knowledge of an actualproof of Mark’s method that would be not only(a) mathematically valid but also (b) pedagogicallyappropriate for use with seventh-graders. Indeed, aphysicist or an engineer would likely be concernedonly with the mathematical validity of the proof,while a teacher would have to consider also issuessuch as students’ prior knowledge and whether, forexample, an algebraic proof would be accessible tothem. Later we will revisit the classroom scenario,and we will discuss different possible proofs thatcould meet both requirements.

Our previous discussion illustrates the pointthat a teacher’s mathematical knowledge cannotgenerally function in isolation from pedagogi-cal considerations and, by implication, universitymathematics courses for prospective teachers can-not lose sight of the domain of application of thetargeted knowledge (i.e., mathematics teaching).

This should not be interpreted as a suggestionto compromise the mathematical focus of math-ematics courses for prospective teachers. On thecontrary, we are strong believers that the focusin these courses should remain on mathematics.Our interest is in how to best promote prospec-tive teachers’ learning of mathematics in thesecourses, and, in this regard, we suggest that at leastpart of prospective teachers’ learning experiencesshould be contextualized in pedagogical situations,thereby fostering connections with the domain ofapplication of the intended learning.

Conceptualizing Mathematics for Teachingas a Form of Applied MathematicsHyman Bass and Zalman Usiskin both used thenotion of “applied mathematics” in discussing,respectively, “mathematics education” and “teach-ers’ mathematics”. Hung-Hsi Wu also discussedrelevant ideas, though he used the notion of“mathematical engineering” instead of “appliedmathematics”. Below we briefly present the viewsof these researchers and use them to situate ourproposal in this article to view MfT as a form ofapplied mathematics.

Bass [9] suggested that we view mathemat-ics education as a form of applied mathematics:“[Mathematics education] is a domain of profes-sional work that makes fundamental use of highlyspecialized kinds of mathematical knowledge, andin that sense it can… be usefully viewed as akind of applied mathematics” (p. 418).1 Wu [10]expressed a somewhat similar idea when he arguedthat “mathematics education is mathematical engi-neering, in the sense that it is the customization ofbasic mathematical principles to meet the needs ofteachers and students” (p. 1678). In his view, thecustomization of mathematics needs to happenbefore the mathematics can be applied in thework of teaching. It seems to us that Wu’s notionof engineering relates primarily to curriculumdevelopment, whereas Bass’s notion of appliedmathematics is concerned mainly with the practiceof teaching and its mathematical demands.

Zalman Usiskin [11], [12] has also discussedthe notion of applied mathematics, but, contraryto Bass [9], he has done so in the context ofwhat he calls “teachers’ mathematics”. Usiskin [11]discusses eight aspects under “teachers’ mathemat-ics”: “(a) ways of explaining and representing ideasnew to students, (b) alternate definitions and theirconsequences, (c) why concepts arose and how they

1Although this quotation is dated 2005, Bass has discussedthe idea of viewing mathematics education as a form of ap-plied mathematics in public talks since the 1990s. Wu [10]notes, for example, that Bass lectured on this idea in 1996.

268 Notices of the AMS Volume 61, Number 3

have changed over time, (d) the wide range of ap-plications of the mathematical ideas being taught,(e) alternate ways of approaching problems withand without calculator and computer technology,(f) extensions and generalizations of problems andproofs, (g) how ideas studied in school relate toideas students may encounter in later mathematicsstudy, and (h) responses to questions that learnershave about what they are learning” (p. 3). AlthoughUsiskin’s notion of “teachers’ mathematics” iscertainly not unrelated to the notion of MfT as wedescribed it earlier in this article, the connectionsbetween Usiskin’s “teachers’ mathematics” andpedagogy are mainly curricular and not so muchabout teaching and its mathematical demands, asis the case in MfT. For example, the list (a)–(h) ofaspects of “teachers’ mathematics” includes onlyone item, (h), that specifically refers to teaching.

In thinking about the mathematical preparationof teachers, we were inspired by Bass’s use of “ap-plied mathematics”.2 Yet, given that mathematicseducation in general and the work of mathematicsteaching in particular make use of specializedkinds of knowledge from several other fields inaddition to mathematics (psychology, sociology,etc.), we propose to use the characterization “formof applied mathematics” in reference to the math-ematical component of teachers’ work (i.e., MfT)rather than to mathematics education in general,as Bass used it. Traditionally, the term “appliedmathematics” has been associated with the useof mathematical knowledge in particular domainsof professional work, such as those that relate toengineering and physics. Our use of this term forthe domain of professional work that relates tomathematics teaching aims to extend rather thanchange its traditional meaning.

Our proposal to conceptualize MfT as a formof applied mathematics is partly motivated by thefact that the conceptualization has two importantimplications for the mathematical preparation ofteachers. These implications, which we describenext, are aligned with existing research in this area.

First, the conceptualization implies that themathematical preparation of teachers should takeseriously into account the idea that “there is aspecificity to the mathematics that teachers needto know and know how to use“ ([13], p. 271) as com-pared to the mathematics that other professionalusers of mathematics need to know and know howto use. This point was illustrated earlier when wetalked about mathematical knowledge needed todevelop a proof of Mark’s method that would alsobe accessible to seventh-graders: knowledge of

2Only recently did we learn about Usiskin’s use of “ap-plied mathematics”. We thank the anonymous referee whobrought this work to our attention.

such a “pedagogically situated” proof would becrucial for the teacher in the classroom scenariobut not so much for a physicist or an engineer whomay have an interest in the same method.

Second, the conceptualization implies that themathematical preparation of teachers should aimto “create opportunities for learning subject matterthat would enable teachers not only to know, but tolearn to use what they know in the varied contextsof [their] practice” ([3], p. 99). In other words, theconceptualization underscores the importance of a“pedagogically functional mathematical knowledge”(ibid., p. 95) which can support teachers to suc-cessfully address mathematical issues that arisein their work, such as mathematical evaluationof a novel student method or the generation of a“pedagogically situated” proof, as illustrated in ourdiscussion of the classroom scenario.

To recap, the conceptualization of MfT as aform of applied mathematics necessitates thatmathematics courses for prospective teachersdesign opportunities for prospective teachers tolearn and be able to use mathematical knowledgefrom the perspective of an adult who is specificallypreparing to become a teacher of mathematics.But how might such learning opportunities bedesigned? Next we discuss a special kind ofmathematics task that we call Pedagogy-Relatedmathematics tasks (P-R mathematics tasks), whichwe used in our courses with prospective teachersto support their learning of MfT. We do not suggestthat P-R mathematics tasks are the only or the bestkinds of tasks for promoting learning of MfT. Yet,we claim that P-R mathematics tasks can supportproductive learning opportunities for prospectiveteachers to develop the mathematical knowledgethat they need for their work and that such tasksshould not be overlooked in the mathematicalpreparation of teachers.

Pedagogy-Related Mathematics Tasks:A Vehicle to Promoting Knowledge ofMathematics for TeachingIn this section we discuss the notion of P-Rmathematics tasks as a vehicle to promotingknowledge of MfT. Specifically, we discuss twomain features of P-R mathematics tasks thatcollectively distinguish P-R mathematics tasksfrom other kinds of mathematics tasks.

As an example of a P-R mathematics task,consider the following question in relation to theclassroom scenario described earlier:

What would be a mathematically appropri-ate way in which the teacher could respondto Jane’s question about whether the classcould use Mark’s method every time theyhad to find a fraction between two given,nonequivalent positive fractions?

March 2014 Notices of the AMS 269

Figure 2. An algebraic proof of a general methodfor finding a fraction between two given positive

and nonequivalent fractions.

We will henceforth refer to this task as the FractionsTask. A solution to the Fractions Task would buildon the “course of action” that we discussedearlier under possibility 2, which is the desirablepossibility. According to this course of action, theteacher would engage the class in the discussionof a proof that would not only be valid but alsoaccessible to the group of seventh-graders.

Feature 1: A mathematical focusP-R mathematics tasks have a mathematical fo-

cus that relates to one or more mathematical ideasthat theory, research, or practice suggested areimportant for teachers to know. The mathematicalfocus is intended to engage prospective teachersin mathematical activity. In the Fractions Task, themathematical focus is the mathematical evalua-tion of Mark’s method, which can be expressedalgebraically as follows:

Given two fractionsab

andcd

(where a, b, c, d > 0

andab<cd

),ab<a+ cb + d <

cd.

Feature 2: A substantial pedagogical contextIn addition to the mathematical focus, a P-R

mathematics task has a substantial pedagogicalcontext that is an integral part of the task andessential for its solution. The pedagogical con-text situates prospective teachers’ mathematicalactivity in a particular teaching scenario and helpsprospective teachers engage with the mathematicsfrom the perspective of a teacher.

In the Fractions Task the pedagogical contextdescribes the teacher’s need to formulate a re-sponse to Jane’s question about whether the classcould use Mark’s method when asked to find a

fraction between two positive and nonequivalentfractions. According to this context, the eventhappened in a seventh-grade class, which allowsthe solvers of the task (prospective teachers) tomake certain assumptions about what the studentsin the class might know or be able to understand.Thus a solution to the task must not only satisfymathematical considerations but also needs totake into account pedagogical considerations. Nextwe discuss four points related to feature 2 of P-Rmathematics tasks.

First, the pedagogical context in which a P-Rmathematics task is situated determines to a greatextent what counts as an acceptable/appropriatesolution to the task, because it provides (orsuggests) a set of conditions a possible solution tothe task needs to satisfy. In the Fractions Task, forexample, an algebraic proof of Mark’s method likethe one in Figure 2, though mathematically valid,would likely not be within the conceptual reachof students in a seventh-grade class. A proof canonly be useful to students if it is understandableto them. It is up to the teacher to decide whether,given students’ prior knowledge and any nationalcurricular expectations, it would be sensible toengage the class in the development of a differentproof that could be more accessible to students.

A potential proof of the inequality ab <

a+cb+d <

cd

that invokes an argument from physics wouldlikely have stronger explanatory power and bemore accessible to students than the algebraicproof. Consider, for example, the distance-timegraph in Figure 3, with the fractions a

b , cd , and a+cb+d

representing respectively the following speeds: theconstant speed for covering distance a in timeb, the constant speed for covering an additionaldistance c in time d, and the average speed forcovering the entire distance (a+ c) in time (b+d).The smaller the fraction the smaller the speed,and so the fact that a

b is smaller than cd implies

that the constant speed for covering distance ais smaller than the constant speed for coveringdistance c; this is illustrated by the smaller slopeof OP as compared to the slope of PQ in the figure.The average speed for covering the entire distance(a+ c) in time (b+d) should be: (1) bigger than a

b ,because, otherwise, a smaller distance than (a+ c)would be covered in time (b + d); and (2) smallerthan c

d , because, otherwise, distance (a+ c) wouldbe covered in less time than (b+d). Thus it followsthat a

b <a+cb+d <

cd .

Another possible argument for the inequalityab <

a+cb+d <

cd that could possibly be more accessible

to students than the algebraic proof might considerthat the fractions a

b and cd represent ratios, say the

ratios of the number of a student’s correct answersin tests 1 and 2 over the number of questions ineach test (ab and c

d , respectively). The fact that ab is

270 Notices of the AMS Volume 61, Number 3

smaller than cd implies that the ratio of the number

of correct answers over the number of questionsin test 2 was bigger than the corresponding ratioin test 1. After applying reasoning similar to theone in the context of the “distance-time” graphhere, one can conclude that the ratio of the totalnumber of the student’s correct answers in thetwo tests over the total number of questions in thetwo tests (i.e., a+cb+d ) has to be bigger than the ratioin test 1 (i.e., ab ) and smaller than the ratio in test2 (i.e., cd ).

Second, it is hard to describe precisely thepedagogical context of a P-R mathematics task:given the complexities of any classroom situation,it is impractical (perhaps impossible) to describeall the parameters of the situation that canbe relevant to the task’s solution. This lack ofspecificity of the pedagogical context is potentiallyuseful for university instructors implementingP-R mathematics tasks with their prospectiveteachers. Specifically, instructors can use theendemic ambiguity surrounding the conditions ofa pedagogical context to vary some of its conditionsin order to engage prospective teachers in relatedmathematical activities. Consider, for example, thepedagogical context of the Fractions Task, whichdoes not specify whether the class would be able toproduce an algebraic proof like the one in Figure 2.An instructor could exploit this ambiguity to engageprospective teachers in the development of otherarguments that are likely to be more accessibleto students, such as the arguments we discussedearlier. Each of these alternative arguments isbased on different assumptions about the levelof students’ knowledge, and it is important tomake this explicit in the proposed mathematicalsolutions to the task.

Third, the pedagogical context of a P-R mathe-matics task has the potential to motivate prospec-tive teachers’ engagement in the task by helpingthem see why the mathematical ideas in the taskare, or can be, important for their future work asteachers of mathematics. According to Harel [14],“[s]tudents are most likely to learn when they seea need for what we intend to teach them, whereby ‘need’ is meant intellectual need, as opposed tosocial or economic need” (p. 501; the excerpt in theoriginal was in italics). In the case of prospectiveteachers, a “need” for learning mathematics maybe defined in terms of developing mathemati-cal knowledge that they need for teaching, i.e.,knowledge of MfT. By helping prospective teacherssee a need for the ideas they are being taughtin mathematics courses for prospective teachers,it is more likely that they will get interested inlearning these ideas. This is particularly importantfor mathematical ideas that prospective teacherstend to consider difficult or “advanced” for the

level of the students they will be teaching, such asthe notion of proof for pre-high school students.

Fourth, the design and implementation ofP-R mathematics tasks require some pedagog-ical knowledge by instructors of mathematicscourses for prospective teachers. For example, thedesign of the Fractions Task used knowledge abouta common student misconception in the domain ofproof and considered a possible link between thisstudent misconception and a teacher’s evaluationof, and response to, a novel student method. Thepedagogical demands imposed by the design of P-Rmathematics tasks on the instructors’ knowledgecan create challenges for those who may havelimited background in pedagogy or familiarity withthe school mathematics curriculum. Similar chal-lenges could emerge also for instructors duringthe implementation of P-R mathematics tasks withprospective teachers, especially in relation to thequestion of what kinds of variations an instructorcould make to the pedagogical context of a taskwithout compromising the task’s realistic nature.We return to these issues in the last section of thearticle.

Exemplifying the Use of P-R Mathemat-ics Tasks in a Mathematics Course forProspective Elementary TeachersThis section is organized in two parts. In the firstpart we provide a brief description of major featuresof a mathematics course for prospective elementaryteachers that we developed to promote MfT as aform of applied mathematics. Given the importancewe attribute to P-R mathematics tasks in thinkingabout MfT as a form of applied mathematics, P-Rmathematics tasks had a prominent place in thecourse. Yet P-R mathematics tasks were not theonly kinds of tasks we used in the course. Anotherkind was what we call typical mathematics tasks,which embody only feature 1 of P-R mathematicstasks. Advantages of typical mathematics tasks arethat they allow for a faster pace during universitysessions than P-R mathematics tasks do and donot require any pedagogical knowledge from theinstructor.

In the second part we illustrate the use of P-Rmathematics tasks in the course by discussingthe implementation of a task sequence, whichincludes both a typical and a P-R mathematicstask. We developed this and other task sequencesin a four-year study that used design experimentmethodology [15]. The design experiment includedfive research cycles of implementation, analysis,and refinement of different task sequences in thecourse. In this article we discuss the implementa-tion of one task sequence from the last researchcycle of the design experiment. The last researchcycle was conducted in two sections of the course

March 2014 Notices of the AMS 271

Figure 3. A distance-time graph that can beelaborated on in a potential proof of a general

method for finding a fraction between two givenpositive and nonequivalent fractions.

with a total of thirty-nine prospective elementaryteachers and the second author as the instructor.The data come from the implementation of thetask sequence in one of the sections.

General Description of the Course

This three-credit undergraduate mathematicscourse was prerequisite for admission to themaster’s-level elementary teaching certificationprogram at a large American state university.Contrary to what usually happens with mathemat-ics courses for prospective elementary teachersin North America [16], the course was offeredby the Department of Education rather than bythe Department of Mathematics. Yet this did notmake any difference to the fact that this was amathematics course.3 The students in the coursepursued undergraduate majors in different fieldsof study and tended to have weak mathematicalbackgrounds (for many of them this was the firstmathematics course since high school). Also, giventhat the students were not yet in the teachingcertification program, they had limited or nobackground in pedagogy.

The course was the only mathematics coursespecified in the admission requirements for theteaching certification program. It covered a widerange of mathematical topics in different mathemat-ical domains (arithmetic, algebra, number theory,geometry, and measurement) and was intendedto improve prospective teachers’ understandingof key mathematical concepts and proceduresin those topics. In addition to mathematical top-ics, the course emphasized the following three

3The first two research cycles of the design experiment tookplace at another large American state university where thecourse was offered through the mathematics department.

mathematical practices: (a) reasoning-and-proving(i.e., making mathematical generalizations andformulating arguments for or against these gener-alizations, with particular attention to the use ofdefinitions), (b) problem solving, and (c) makingconnections between different forms of represen-tation (algebraic, pictorial, etc.). With these threepractices, which we treated as strands that under-pinned prospective teachers’ mathematical work,we aimed to also help prospective teachers appre-ciate what it means to “do” mathematics throughengaging them with more authentic mathematicalexperiences, not merely helping them learn (orrelearn) mathematical concepts and procedures.

One aspect of the course’s approach to promoteMfT, which is the most relevant to our purposesin this article (other features are discussed in[17] and [18]), was the use of both typical andP-R mathematics tasks in carefully designed tasksequences. A common task sequence began witha typical mathematics task that set the stage fora P-R mathematics task. The typical mathematicstask allowed prospective teachers to work on amathematical idea from an adult’s standpoint.The P-R mathematics task, which followed thetypical task, introduced a pedagogical context thatprospective teachers had to consider in their math-ematical work, thus engaging prospective teachersin mathematical work from a teacher’s standpoint.To address feature 2 of P-R mathematics tasksconcerning situating prospective teachers’ mathe-matical work in a pedagogical context, we used awide range of classroom scenarios based on actualclassroom records: videos or written descriptionsof classroom episodes in elementary classrooms,artifacts of elementary students’ mathematicalwork, excerpts from elementary mathematics text-books, etc. When actual classroom records wereunavailable, we used fictional records that werenevertheless realistic.

An Example of a Task Sequence and ItsImplementation in the Course

The task sequence aimed to promote prospectiveteachers’ knowledge about a possible relationbetween the notions of area and perimeter ofrectangles. Central to this exploration were alsoideas of mathematical generalization and proof bycounterexample. The sequence included a typicalmathematics task (question 1) followed by a P-Rmathematics task (question 2):

Imagine that one of your students comesto class very excited. She tells you thatshe has figured out a theory that younever told the class. She explains that shehas discovered that as the perimeter of arectangle increases, the area also increases.

272 Notices of the AMS Volume 61, Number 3

She shows you the work in Figure 4 belowto prove what she is doing:

Figure 4.

1. Evaluate mathematically the studentstatement (underlined).

2. How would you respond to thisstudent?

The task sequence was an adaptation of tasksthat were originally developed by Ball [19] andsubsequently used by Ma [2].

Although question 1 refers to a student state-ment, it is a typical mathematics task, becausethe prompt asks prospective teachers to mathe-matically evaluate the statement without askingor expecting them to take into account the factthat the statement was produced by a student.Question 2, on the other hand, is a P-R mathematicstask, because it introduces a pedagogical consider-ation that prospective teachers need to take intoaccount in their mathematical work. Although notexplicitly mentioned in the task, it was understoodin the teacher education class that answers toprompts like the one in question 2 should not focuson pedagogical issues (e.g., “I’d teach again thenotion of perimeter and area…”), but should ratherfocus on the underlying mathematical issues byappropriately considering the relevant pedagogicalcontext.

The mathematical focus of this P-R mathematicstask is to evaluate mathematically the underlinedstatement, which is essentially what the prospectiveteachers were asked to do in question 1. Thepedagogical context of the task concerns theteacher’s responsibility to respond to the studentwho produced the statement. An appropriateresponse to question 1 could say that the statementis false and provide a counterexample to refuteit. However, an appropriate response to question2 would need to go beyond that. Considerationalso of the pedagogical context suggests that itwould be useful for the student’s learning if theteacher not only refuted the student’s statement (byproviding a counterexample) but also helped herunderstand why the statement is false and explorethe conditions under which the statement would betrue. This additional work, though pedagogicallysituated, is deeply mathematical in nature and is

the kind of work that we argue deserves moreattention in mathematics courses for prospectiveteachers.

The prospective teachers worked on the twoquestions first individually and then in groupsof four or five. Later on there was a whole-class discussion, which began with the instructor(Stylianides) asking representatives from differentsmall groups to report their work on the task,beginning with question 1 (all prospective teachernames are pseudonyms).

Andria: We [the members of her small group]said that it [the student statement]was mathematically sound, becauseas you increase the size of the figure,the area is going to increase as well.

Tiffany : We [the members of her small group]agreed. We thought the same, be-cause as the sides are gettingbigger…[inaudible]

Stylianides: Does anybody disagree? [No groupexpressed a disagreement.]

Evans: I agree. [Evans was in a different smallgroup than Andria and Tiffany.]

Stylianides: And how would you respond to thestudent?

Melissa: I think it’s true, but they haven’tproved it for all numbers, so it’s notreally a proof.

Andria: I think that you don’t have to tryevery number [she seems to refer toevery possible case in the domain ofthe statement] to be able to prove it,because if the student can explainwhy it works like we just did, likeif you increase the length, then thearea increases. [pause] [. . . ]

Meredith: I’d say that it’s an interesting idea,and I’d see if they can explain why itworks.

As illustrated by the above transcript, all smallgroups thought that the student statement wastrue, but at the same time they seemed to realizethat the evidence that the student provided for herclaim was not a proof (see, e.g., Melissa’s comment).As a result, the prospective teachers started tothink about how to prove the statement and howto respond to the student. For example, Andria andMeredith pointed out that the student needed toprove/explain why the area of a rectangle increasesas its perimeter increases, with Andria appearing tobelieve that she already had a proof. However, theinstructor knew that the statement was false. As therepresentative of the mathematical community inthe classroom, he probed the prospective teachersto check more cases to see whether they couldcome up with an example in which the student

March 2014 Notices of the AMS 273

statement was false. After a few minutes, all smallgroups found at least one counterexample tothe statement and concluded that the statementwas false. We note that earlier in the course theprospective teachers had opportunities to discussthe idea that one counterexample suffices to refutea general mathematical statement.

The prospective teachers’ counterexamples tothe student statement made them experience a“cognitive conflict”, because at the initial stages oftheir engagement with the task sequence, they didnot expect that a statement that looked so “obvious”to them would turn out to be false. This unexpecteddiscovery motivated prospective teachers’ furtherwork on question 2. The instructor decided to givethe prospective teachers more time to think intheir small groups about question 2. The transcriptbelow is from the whole-class discussion thatfollowed the small-group work.

Natasha: We said that the way that they [thestudents] are doing it, where they’rejust increasing the length of oneside, it’s always going to work forthem; but if they try examples wherethey change the length on both sides,that’s the only way it’s going to provethat it doesn’t work all the time. Soyou should try examples by changingboth sides.

Stylianides: [referring to the class] What doyou think about Natasha’s response?Does it make sense? [The class nodsin agreement.] So what else? Whatelse do you think about this?

Evans: You can kind of ask them to re-structure the proof so that it wouldwork.

Stylianides: What do you mean by “restructurethe proof”?

Evans: Like once they figure out that itdoesn’t work for all cases, they couldsay it’s still like. . . if they saw it and ifthey revise it like the wording or justadd a statement in there that if theycan come up with a mathematicallycorrect statement. . .

Stylianides: Anything else? [No response fromthe class.] I think both ideas [men-tioned earlier] are really important.So when you have something [a state-ment] that doesn’t work, then it’sclear that this student would be in-terested to know more. For example,why it doesn’t work or under whatconditions does it work, because, ob-viously, some of the examples thatthe student checked worked. . . .

Natasha and Evans raised two related issues thatthe teacher in the scenario of the P-R mathematicstask could address when responding to the student:(a) why the statement is false (instead of simplyshowing that the statement is false with a coun-terexample) and (b) the conditions under which thestatement would be true. Based on our planningfor the implementation of the task, the instructorwould raise these issues anyway, because, as weexplained earlier, we considered it mathematicallysufficient but pedagogically inconsiderate for ateacher to offer only a counterexample to the stu-dent’s statement. We considered such a responseinadequate in light of the pedagogical context ofthe task.

It is noteworthy that the two issues were raisednot by the instructor but by two prospectiveteachers, Natasha and Evans, who (like the otherprospective teachers in the class) had no teachingexperience. It is difficult to say what provoked thecontributions of these prospective teachers, butwe speculate that the pedagogical context of theP-R mathematics task played a role in this. It is alsopossible that the P-R mathematics tasks we usedearlier in the course had helped the prospectiveteachers begin to develop pedagogical sensibilities.

Following a summary of the two issues inEvans’s contribution, the instructor engaged theprospective teachers in an examination of theconditions under which the student statementwould be true. Specifically, he asked the prospectiveteachers to investigate what happens to the areaof a rectangle in each of the following cases wherethe perimeter of the rectangle increases:

1. One of the two dimensions (length or width)is increased and the other dimension is keptconstant.

2. Both dimensions are increased.3. One of the two dimensions is increased and

the other dimension is decreased, so thatthe amount of increase in one dimension islarger than the amount of decrease in theother.

The prospective teachers produced algebraic andpictorial proofs to show that in the first twocases the area always increases and examples toshow that in the third case the area can increase,decrease, or stay the same.

To conclude, our discussion of the implemen-tation of the task sequence exemplified the ideathat the application of mathematical knowledgein pedagogical contexts can be different from itsapplication in similar but purely mathematicalcontexts. Although the typical mathematics taskand the P-R mathematics task in the sequence weredealing with the same mathematical ideas, thepedagogical context in which the P-R mathematics

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task was embedded shaped what could count as anappropriate mathematical solution to it, therebysupporting the generation of rich mathematicalactivity in a realistic pedagogical context.

Concluding RemarksThe design, implementation, and solution of P-Rmathematics tasks suggest that instructors ofuniversity mathematics courses for prospectiveteachers who may want to use these tasks in theircourses need to have not only good mathematicalknowledge but also some knowledge of school-related pedagogy (including familiarity with theschool mathematics curriculum). For example:What arguments for a true generalization couldbe accessible to students of different ages? Inaddition to offering a counterexample, what othermathematical investigations would be pertinent ina pedagogical context where a student proposed afalse generalization?

In countries such as the United States wheremathematics courses for prospective teachers aretypically offered by mathematics departments [16],it may be unrealistic to require or expect that in-structors of these courses have good knowledge ofpedagogy in addition to their robust mathematicalknowledge [20]. However, if certain knowledge ofpedagogy is recognized to be useful for teachingMfT to prospective teachers, ways need to be foundto support these instructors in their work.

One way could be through textbooks intendedfor use in mathematics courses for prospectiveteachers. Textbooks that would be consistentwith the conceptualization of MfT as a formof applied mathematics discussed in this articlewould not only include P-R mathematics tasks(alongside typical and possibly other kinds ofmathematics tasks) but would also include thefollowing: (a) the rationale for the design of P-Rmathematics tasks and associated target learninggoals for prospective teachers, (b) suggestions forimplementing the tasks with prospective teachers,and (c) comments about how the tasks relate toschool mathematics (e.g., an elaboration on thepedagogical context of the tasks and how thiscontext can shape mathematical solutions).

To conclude, the conceptualization of MfT asa form of applied mathematics highlights theidea that prospective teachers’ learning of MfTshould not happen in isolation from the context inwhich teachers will need to apply this knowledge.P-R mathematics tasks can provide a vehiclethrough which prospective teachers’ learning ofmathematics can be connected to the teachingpractice. On the one hand, these tasks havemathematics at the core of prospective teachers’activity; this is a necessity given that they are

meant for use in mathematics courses. On theother hand, they situate this mathematical activityin a substantial pedagogical context that shapesand influences the activity, thereby ensuring thatprospective teachers’ learning of mathematics doesnot lose sight of its domain of application.

References

[1] G. J. Stylianides and A. J. Stylianides, Mathematicsfor teaching: A form of applied mathematics, Teachingand Teacher Education 26 (2010), pp. 161–172.

[2] L. Ma, Knowing and Teaching Elementary Mathematics,Erlbaum, Mahwah, NJ, 1999.

[3] D. L. Ball and H. Bass, Interweaving content and ped-agogy in teaching and learning to teach: Knowing andusing mathematics, in Multiple Perspectives on Math-ematics Teaching and Learning (J. Boaler, ed.), AblexPublishing, Westport, CT, 2000, pp. 83–104.

[4] , Making mathematics reasonable in school, inA Research Companion to Principles and Standardsfor School Mathematics (J. Kilpatrick, W. G. Martin,and D. Schifter, eds.), National Council of Teachers ofMathematics, Reston, VA, 2003, pp. 27–44.

[5] M. Goulding, T. Rowland, and P. Barber, Does itmatter? Primary teacher trainees’ subject knowledgein mathematics, British Educational Research Journal28 (5) (2002), pp. 689–704.

[6] W. G. Martin and G. Harel, Proof frames of pre-service elementary teachers, Journal for Research inMathematics Education 20 (1989), pp. 41–51.

[7] G. Harel and L. Sowder, Toward comprehensive per-spectives on the learning and teaching of proof, inSecond Handbook of Research on Mathematics Teach-ing and Learning (F. K. Lester, ed.), Information AgePublishing, Greenwich, CT, 2007, pp. 805–842.

[8] A. J. Stylianides and D. L. Ball, Understandingand describing mathematical knowledge for teach-ing: Knowledge about proof for engaging students inthe activity of proving, Journal of Mathematics TeacherEducation 11 (2008), pp. 307–332.

[9] H. Bass, Mathematics, mathematicians, and mathemat-ics education, Bulletin of the American MathematicalSociety 42 (2005), pp. 417–430.

[10] H. Wu, How mathematicians can contribute to K–12mathematics education, Proceedings of the Interna-tional Congress of Mathematicians, Madrid, 2006,Volume III, European Mathematical Society, Zürich,2006, pp. 1676–1688.

[11] Z. Usiskin, Teachers’ Mathematics: A Collection ofContent Deserving To Be a Field , National Summiton the Mathematical Education of Teachers, 2001,(Retrieved January 6, 2012, from http://cbmsweb.org/NationalSummit/wg_speakers.htm.)

[12] Z. Usiskin, A. Peressini, E. A. Marchisotto, andD. Stanley, Mathematics for High School Teachers:An Advanced Perspective, Prentice Hall, Upper SaddleRiver, NJ, 2003.

[13] J. Adler and Z. Davis, Opening another blackbox: Research mathematics for teaching in mathe-matics teacher education, Journal for Research inMathematics Education 37 (2006), pp. 270–296.

[14] G. Harel, Two dual assertions: The first on learningand the second on teaching (or vice versa), AmericanMathematical Monthly 105 (1998), pp. 497–507.

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[15] P. Cobb, J. Confrey, A. Disessa, R. Lehrer, andL. Schauble, Design experiments in educational re-search, Educational Researcher 32 (1) (2003), pp.9–13.

[16] B. Davis and E. Simmt, Mathematics-for-teaching: Anongoing investigation of the mathematics that teach-ers (need to) know, Educational Studies in Mathematics61 (2006), pp. 293–319.

[17] G. J. Stylianides and A. J. Stylianides, Facilitatingthe transition from empirical arguments to proof, Jour-nal for Research in Mathematics Education 40 (2009),pp. 314–352.

[18] A. J. Stylianides and G. J. Stylianides, Impactingpositively on students’ mathematical problem solvingbeliefs: An instructional intervention of short duration,Journal of Mathematical Behavior 33 (2014), pp. 8–29.

[19] D. L. Ball, Knowledge and reasoning in mathemati-cal pedagogy: Examining what prospective teachersbring to teacher education, unpublished doctoral dis-sertation, Michigan State University, East Lansing, MI,1988.

[20] J. O. Masingila, D. E. Olanoff, and D. K. Kwaka, Whoteaches mathematics content courses for prospectiveelementary teachers in the United States? Results ofa national survey, Journal of Mathematics TeacherEducation 15 (2012), pp. 347–358.

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