IGOR’ KONTOROVICH and BORIS KOICHU

A CASE STUDY OF AN EXPERT PROBLEM POSERFOR MATHEMATICS COMPETITIONS

Received: 30 July 2012; Accepted: 8 May 2013

ABSTRACT. This paper is concerned with organizational principles of a pool of familiarproblems of expert problem posers and the ways by which they are utilized for creatingnew problems. The presented case of Leo is part of a multiple-case study with expertproblem posers for mathematics competitions. We present and inductively analyze thedata collected in a reflective interview and in a clinical task-based interview with Leo. Inthe first interview, Leo was asked to share with us the stories behind some problems posedby him in the past. In the second interview, he was asked to pose a new competitionproblem in a thinking-aloud mode. We found that Leo’s pool of familiar problems isorganized in classes according to certain nesting ideas. Furthermore, these nesting ideasserve him in posing problems that, ideally, are perceived by Leo as novel and surprisingnot only to potential solvers, but also to himself. Because of the lack of empirical researchon experts in mathematical problem posing, the findings are discussed in light of researchon experts in problem solving and on novices in mathematical problem posing.

KEY WORDS: experts, mathematical competitions, mathematical problem, nesting ideas,problem posing

INTRODUCTION

Mathematics education research has accumulated an extended body ofknowledge on problem posing by school children and mathematicsteachers who, as a rule, are novices in problem posing (for a review, seeKontorovich, Koichu, Leikin & Berman, 2012). At the same time,empirical evidence on expert problem posers, i.e. mathematicians andmathematics educators, who create new problems for various mathemat-ical and educational needs as an integrative part of their professionalpractice, barely exists. This is in spite of the proven practice of usingresearch on experts as a source of ideas for promoting mathematicalcompetences in novices. For instance, the mathematics educationcommunity has benefitted from studies on how experts in mathematicssolve problems (e.g. Carlson & Bloom, 2005; Silver & Metzger, 1989),learn mathematics (e.g. Wilkerson-Jerde & Wilensky, 2011), discovernew mathematical facts (e.g. Liljedahl, 2009), etc. The current paperargues that research on how experts pose problems may also be beneficial

International Journal of Science and Mathematics Education 2013# National Science Council, Taiwan 2013

and lead to new ideas on how to improve problem-posing skills of schoolchildren and teachers.

This paper is concerned with a particular aspect of expert problemposing, namely with the role of a pool of familiar problems. Silver,Mamona-Downs, Leung & Kenney (1996) referred to this aspect as anunderexplored component of the mathematical knowledge base neededfor successful problem-posing. Pelczer & Gamboa (2009), who studiedproblem-posing among experienced problem solvers (including partici-pants of Olympiad teams), found that even an extended pool of familiarproblems is not sufficient for posing high-quality problems. Thus, weattempt to take one step further and address two research questions in thispaper: (1) According to which organizational principles can a pool of familiarproblems of an expert problem poser be organized? (2) How can theseprinciples be utilized for posing new problems?

We approach these questions in the framework of a larger study(Kontorovich, unpublished dissertation) focused on the community ofproblem posers for mathematics competitions for secondary schoolchildren. The community is particularly interesting because mathematicscompetitions are widely recognized as a valuable source of elegant andsurprising problems for use not only in out-of-school educational settingsbut also in a regular mathematics classroom (e.g. Koichu & Andžāans,2009). Moreover, many competition problems have served as apowerful means for engaging school children in mathematics challengesand for fostering their mathematical thinking and creativity (e.g. Koichu& Andžāans, 2009; Thrasher, 2008). In addition, it is simply intriguingto understand how the masters succeed in coming up with newproblems after so many mathematical gems have been created forcompetitions during the last century.

THEORETICAL BACKGROUND

This section consists of two sub-sections concerned with expertise. Inthe first one, we review self-reflective papers written by expert problemposers and consider their applicability to addressing the above researchquestions. In the second one, we consider some existing theories ofexpert problem solving as a source for studying expert problem posing.We then argue that, while the theories of expert problem solving can beadapted to some extent to exploring expert problem posing, there isroom for introducing new theoretical constructs which would better fitthe specificity of problem posing.

IGOR’ KONTOROVICH AND BORIS KOICHU

What Do We Know about Expert Problem Posing?

An extensive literature search resulted in identification of only two papersdevoted to the principles of expert problem posing for mathematicscompetitions, by Konstantinov (1997) and Sharigin (1991). Both paperscan be classified as self-reflections of the highly reputed problem poserson their successful practices.

These two papers inform the readers about sources of new problems,problem-posing techniques, and the authors’ considerations involved inshaping the quality criteria for the created problems. Sharigin (1991)points out that a new problem usually comes from other problems theposer is familiar with. He presents a list of six techniques that can beused for creating new problems: reformulating, chaining, consideringspecial cases, generalizing, varying the givens, and inquiry. Interest-ingly, roughly the same techniques were found in the arsenals ofnovices in problem posing (e.g. Crespo & Sinclair, 2008; Norman &Bakar, 2011; Silver et al., 1996; Stoyanova, 2005). However, thestudies with novices repeatedly showed that implementing thesetechniques is definitely not enough for posing problems of high quality.Therefore, additional research effort is needed in order to grasp theessence of expert problem-poser performance.

An integral part of expert problem posing is an individual’s perceptionof what a high-quality (competition) problem is. According to Sharigin(1991), such a problem should be well-formulated and have nounnecessary givens; its formulation is supposed to be short and includesome elements of innovation for a particular category of solvers. Theexamples given in Sharigin’s (1991) paper illustrate these criteria, butthey also create an impression that his own masterpieces are much morethan just brief, well-formulated problems, which are new for somecategories of solvers. On the same matter, Konstantinov (1997)skeptically noted that “It is impossible to formulate what a ‘goodproblem’ is. But when the problem is posed it claims for itself (or againstitself)” (p. 168, translated from Russian). Thus, there is room for a deeperinquiry into the expert’s self-imposed criteria for a high-quality(competition) problem.

Experts’ Organization of Knowledge Base

It is well-known that an expert’s knowledge base for mathematicalproblem solving is more than just storage of pieces of information, suchas definitions, facts, and routine procedures. It also includes the ways bywhich this information is represented, stored, organized, and accessed

A CASE STUDY OF AN EXPERT PROBLEM POSER FOR MATHEMATICS

(cf. Schoenfeld, 1992, for an elaborated argument). Another importantpart of the mathematical knowledge base for problem solving, and,apparently, for problem posing, is a set of rules and norms that existin a particular domain regarding legitimate and prototypical connectionsbetween different pieces of mathematical information (Schoenfeld, 1992).This kind of knowledge is constructed through continuous exposure tovarious mathematical problems, elaboration on a part of them, and storage ofproblems in a knowledge base; in this way, a personal pool of familiarproblems is constructed.

A personal pool of familiar problems of an expert problem poser isimmense. According to Miller (1956), when experts operate with a largeamount of information, the first thing they do is to break it down intomeaningful chunks which make the information more accessible. Thenexperts apply their extended arsenal of schemas to the chunked information.Schemas are referred to as organized structures of mental actions forassociating new with already existing information (e.g. Schoenfeld, 1992).They are used to create a personal perception of information, which isencoded and stored in the long-term memory and later recalled and decodedback again (e.g. Chi, Feltovich & Glaser, 1981).

How can chunking and schemas be used to characterize expert’s poolof familiar problems for problem posing? In other words, what kinds offamiliar problems are grouped in the same chunks? Empirical studies inproblem solving have shown that experts group problems together inagreement with the deep versus surface structure theory (e.g. Chi et al.,1981; Schoenfeld & Herrmann, 1982). Briefly, the theory postulates thatexperts tend to identify problems as being similar due to the fundamentalprinciples and strategies that lead to their solutions (deep structure), andnot just according to their surface structure, e.g. similar scientific fields ortopics, usage of the same mathematical terms, etc.

Let us recall that the studies mentioned in the previous paragraphconcern expert problem solvers, while this paper is concerned withexpertise in problem posing. Therefore, the question of how an expertorganizes and takes advantage of his or her pool of familiar problemswhen creating new problems deserves further empirical exploration.

THE CASE OF LEO

The data for the current paper consist of fragments from a case study ofLeo (pseudonym). The case of Leo was chosen for three reasons. First,Leo is a reputed problem-posing expert, whose problems have appeared

IGOR’ KONTOROVICH AND BORIS KOICHU

in such high-level competitions as the Tournament of the Towns,International Mathematical Olympiad (IMO) for university students, andnational-level Olympiads in Israel. Leo is also a coach of the Israeli teamfor IMO for high school students. Second, we found the case particularlyinformative with respect to mathematical, cognitive, and affective aspectsof the process of posing new problems. Third, the problems that Leoposed in our presence during the clinical task-based interview were laterutilized at an Israeli national-level mathematics competition for highschool students. This fact gives us a (rare) opportunity to claim that theproblem-posing processes observed under laboratory conditions had“real-life” implications.

We answer our research questions based on in-depth analysis of thedata from two video-taped interviews with Leo and an additional audio-taped meeting, in which Leo gave feedback on our analysis of hisproblem posing practice. The first research question is addressed based ona reflective interview conducted in the form of a conversation regardingselected problems created by Leo in the past. The problems to bediscussed in the interview were sent to us by Leo in advance, whichenabled us to prepare well-focused questions about each problem.

The second research question is addressed based on a clinical interviewduring which Leo was asked to think aloud and create problems based ongiven stimuli. Van Someren, Barnard & Sandberg (1994) argued thatthinking-aloud interviews are useful for taking a closer look at experts’ways of thinking and practice. In this paper, we present in detail datacollected in response to a request to pose as many problems as possible inthe context of billiard, i.e. in a situation when a ball is shot, under certainconditions, at the rectangular table.1 The billiard context was chosenbecause of its richness as a problem-posing stimulus combined with thefact that it had already been used in a number of problem-posing studieswith school children and teachers (e.g. Kontorovich et al. 2012; Koichu &Kontorovich, 2013; Silver et al., 1996). Thus, the use of the billiardcontext creates an opportunity to discuss Leo’s performance in light ofnovices’ performances (see “SUMMARY AND CONTRIBUTION” section). The firstinterview lasted approximately 2 hours; a presented fragment from thesecond interview took about half an hour.

The data were analyzed using an inductive approach in order “[…] toallow research findings to emerge from the frequent, dominant orsignificant themes inherent in raw data, without the restraints imposedby structured methodologies” (Thomas, 2006, p. 238). To make theinductive analysis more transparent, we chose to present the findingsbased on the way in which the categories emerged from the data.

A CASE STUDY OF AN EXPERT PROBLEM POSER FOR MATHEMATICS

FINDINGS

Emergence of Organizational Principles of Leo’s Pool of FamiliarProblemsNoting the Existence of a Special Organizational System. Prior to theinterview, Leo sent us a list of 17 of his problems. Roughly speaking, theproblems belonged to the fields of Euclidean, analytical and spatialgeometries, algebra, graph theory, logic, and combinatorics. Twoproblems, which have appeared in an Israeli national-level competitionfor eighth and ninth graders, attracted our attention because they seemedsimilar, shared the same question and could be solved by using the idea ofalgebraic conjugate numbers.

Problem 1: Simplifyffiffiffiffiffiffiffiffiffiffi

ffiffi

2p

− 1pffiffiffiffiffiffiffiffiffiffiffi

ffiffi

2p þ 1

p þffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffi

3p

−ffiffi

2pp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffi

3p þ ffiffi

2pp þ

ffiffiffiffiffiffiffiffiffiffi

2 −ffiffi

3pp

ffiffiffiffiffiffiffiffiffiffiffi

2 þ ffiffi

3pp .

Problem 2: Simplify 13ffiffi

4p þ 3

ffiffi

6p þ 3

ffiffi

9p þ 1

3ffiffi

9p þ 3

ffiffiffiffi

12p þ 3

ffiffiffiffi

16p þ 1

3ffiffiffiffi

16p þ 3

ffiffiffiffi

20p þ 3

ffiffiffiffi

25p .

When we asked Leo to explain how these problems had been created,he chose to focus on the second one and said:

I needed an algebraic problem for a competition. What can be done in algebra so it wouldbe elementary, but still unexpected? I like [algebraic] conjugate numbers since they areunexpected enough. […] Especially when one number is a predecessor of the other, since

then the numerator of 1 is masked [i.e.ffiffiffiffiffiffiffiffiffiffiffi

nþ 1p

−ffiffiffi

np ¼ 1

ffiffiffiffiffiffiffiffi

n þ 1p þ ffiffi

np ]. […] OK, [so

let it be] conjugate numbers. But quadratic conjugate numbers are hackneyed, boring andeverybody knows them. So let’s take a step forward: cubic conjugate numbers. Thisthought gave birth to the second problem [Problem 2].

Two aspects of Leo’s story were insightful for us. First, after choosinga field of mathematics for the future problem, Leo turned at once to theidea of conjugate numbers. It looked as if he had applied the notion of“conjugate numbers” as a code name relating to a whole class of problemsfamiliar to him. The class contains problems that involve expressions withroots, which can be simplified using the properties of algebraic conjugatenumbers. The existence of Problem 1 implied that Leo had alreadysuccessfully turned to this class of problems. Thus, it is reasonable toassume that this prior positive experience awarded special status to the ideaof “conjugate numbers” for Leo.

Second, after scanning the class of “conjugate number” problems thatLeo was familiar with, he concluded that all of them were based on theidea of quadratic conjugate numbers. In light of Sharigin’s (1991)perspective on a high-quality competition problem, Leo could assume

IGOR’ KONTOROVICH AND BORIS KOICHU

that, if a solver was familiar with the properties of quadratic conjugatenumbers, she would not be surprised by any new problem employingthese properties. Therefore, Leo decided to enrich the class with aproblem of a relatively different kind—cubic conjugate numbers.

The story behind Problem 2 made us wonder if Leo possessed similarguiding problem-posing ideas in additional mathematical fields and topics,and if so, what their characteristics were. In addition, we decided to explorewhether the triad “choosing an idea that had been fruitful in thepast—scanning a class of familiar problems related to that idea—creating anew problem by modifying the idea,”which was apparent in the above story,also appeared on other occasions in Leo’s problem-posing performance.

Emergence of the Notion of “Nesting Ideas”. During the reflectiveinterview, Leo explained how eight out of seventeen problems from his listhad been created. One of these stories was presented in the previous section;the synopses of three additional stories are presented in Table 1.2 Theproblems' formulations are presented in the left column in Table 1; the rightcolumn includes the code names of the classes used by Leo, the descriptionof commonalities between the problems belonging to the class, and theessence of Leo’s innovation inherent in the posed problem.

Additional examples of classes of problems mentioned by Leo duringthe interview included problems involving telescoping series, theprinciple of homothecy, Euler’s line, etc.

The overall collection of ideas mentioned by Leo during the reflectiveinterview revealed three organizational principles regarding his pool offamiliar problems:

1. Leo groups his pool of familiar problems in classes and gives themnames. The classes’ names are very personal, “Leo-fitted,” and do notalways reveal to an observer the essence of the class (e.g. “clockproblems” or “cutting problems”). The problems in each class areintended to be challenging for the particular category of potentialsolvers Leo is working with.

2. Problem classification is made according to what Leo perceives as acommon idea shared by all the problems in the class.

3. Each common (or grouping) idea is an object of manipulations/modifications/innovations aimed at posing a new problem which, inturn, can be included in some class.

We chose to associate such grouping ideas with nests, each of whichembraces familiar problems (i.e. “eggs”) and serves as a useful frameworkfor “laying” new ones. Therefore, we refer to them as nesting ideas.3

A CASE STUDY OF AN EXPERT PROBLEM POSER FOR MATHEMATICS

TABLE 1

Additional problems created by Leo

Problem Class the problem belongs to

Problem 3: At what time are the clockhands perpendicular?

Clock problems:

(Appeared in an Israeli national-levelcompetition for secondary students in2010).

The class consists of problems aboutanalogical clocks and special positions of theirhands. Probably the best known problem of theclass is: “When do the hour and minute handscoincide after 12 o’clock?”Leo’s innovation in Problem 3 was aquestion about perpendicular position ofthe clock’s hands.

Problem 4: The points A,B,C,D,Eare located on the circle so that thedistance between two neighbouringpoints is constant (see figure below).

Cutting problems:

The broken line ABECD divides thearea of the circle into two areas:below the line (the grey area) andabove the line (the white area).Which area is bigger: the grey orthe white one?

The problems of this class are based on afigure divided into two areas by a curve.The typical question of the class is “Whicharea is larger and why?” and the typicalanswer is that the areas are equal, eventhough they do not look equal. Leo likes thisclass of problems because they are based onquite basic knowledge of Euclidiangeometry and do not require Xknowledge ofrarefied facts.

(Appeared in an Israeli national-levelcompetition for 8th and 9th gradersin 2007).

Leo’s innovation in Problem 4 is that thewhite area is larger than the grey one,although it is not obvious in the picture.

IGOR’ KONTOROVICH AND BORIS KOICHU

Examples presented in the previous subsection and in Table 1 representthree types of nesting ideas, i.e. three types of reasons for Leo to includefamiliar and newly constructed problems in the same class: (1) deepstructure nesting ideas (see “conjugate numbers” and “clock problems”),(2) surface structure nesting ideas (see “cutting problems”) and (3)nesting ideas based on particularly rich mathematical concepts (see“ellipses”). The first two types of nesting ideas fit the deep vs. surfacestructure theory well (see “THEORETICAL BACKGROUND”). The third type ofnesting ideas refers to concepts with a considerable number ofmathematical properties represented by some problems of the class. Inthis type of nesting ideas, the deep-level connections between problemsare possible, but non-obligatory.

Summing this part up, it can be said that some organizational principlesof Leo’s pool of familiar problems emerged from his reflections.However, Leo’s reflections were of insufficient resolution in order toinform us about how (if at all) the nesting ideas are actually employed atthe time of problem posing (cf. Van Someren et al., 1994, for limitationsof reflections as a research tool). For this reason, a clinical task-basedinterview was conducted.

Leo’s Utilization of Nesting Ideas in the Process of Posing New ProblemsFragment 1—in the Labyrinth of the Known. At the beginning of theinterview, Leo said that he needed a geometry problem for the forthcomingIsraeli national-level competition and would be glad to take advantage of theinterview for making one up. Then he glimpsed at the billiard task and,without trying to understand it in detail, started to recall competition problemsand mathematical facts associated (for him) with the context of the game

Problem 5: Two ellipses share a focus.Prove that the ellipses intersect at twopoints at the most.

Ellipse:

(Appeared in IMO for college studentsin 2008).

The ellipse problems belong to this class.Leo told us that ellipses are one of hisfavorite topics in plain geometry and that hefrequently uses them in his problem posing.This is because ellipses have many interestingproperties, and not many people know them.Therefore, the innovation is realized throughcreating problems using a rarefied property ofan ellipse.Indeed, Problem 5 can be solved using anuncommon definition of ellipse involving apoint and a directix.

A CASE STUDY OF AN EXPERT PROBLEM POSER FOR MATHEMATICS

of billiards. As it will be evident shortly, the problems and factsrecalled by Leo and his eventually posed problems are way beyondtraditional mathematical syllabus for secondary and high-school students.However, this kind problems and facts are not uncommon in a “syllabus” ofprestigious mathematics competitions (see Koichu & Andžaāns, 2009 andGalperin & Zemliakov, 1990, for compatible problems and facts).

[1] There is a triangle with angles of 90°, 60° and 30°. The ball is shot along with themedian from [vertex of the] angle 30°. How many times will the ball hit the bordersbefore entering a pocket? Eight. OK …After a pause of 10 seconds Leo took another direction:[2] You say “billiard.” The principle of billiard is reflection. You can imagine that a ball isreflected, and you can imagine that a table is reflected. That’s good …there is a sense ofsurprise here, it is unexpected …[3] There is another known billiard problem: A ball enters the corner pocket at an angle ofα. How many times would it take it to hit the corner’s rays before starting to come out? [In

order to solve it], you reflect the angle multiple times and get something like πα hits.

[4] OK, let’s think about something less trivial since we are already familiar with theseproblems.

Leo started recalling not problems, but “less known” geometrical factssomehow related to the idea of reflection. In particular, he said: [5]“There are some nice geometry facts about ellipses” and recalled twofacts. Here is one of them:

[6] If points A and B are the foci of an ellipse, then the [marked] angles are equal[see Fig. 14].[7] But we don’t want an ellipse, it would be too much … [Leo meant that the properties

Figure 1. A fact recalled by Leo on the way to creating a problem

IGOR’ KONTOROVICH AND BORIS KOICHU

of an ellipse are beyond the “syllabus” of the competition he had in mind].[8] We can take a particular case of this phenomenon, with a circle …

Next, Leo recalled the following fact: if O is a circumcenter and CC′ then∡ACH=∡OCB (see Fig. 2).

[9] And why does it happen? This is because the reflections of the orthocenter of atriangle [point H on Fig. 2] by its sides lie on the circumscribed circle [point H′ onFig. 2].[10] That’s good, that means that H′ lies on the distance R from O [R is a radius of thecircumscribed circle]. This could be used for creating a problem …

However, Leo did not use this problem and went on to say:

[11] What else is connected to reflections? The minimalism of the figures that areinscribed in something. For instance, an orthic triangle has a minimal perimeter …[Leo referred to the fact that an orthic triangle has the minimal perimeter of all thetriangles that are inscribed in a fixed triangle].[12] I can take a special case of the triangle with given sides and ask to prove thatthe perimeter of the triangle inscribed in it is bigger than some number [a possiblenumber is the perimeter of the orthic triangle]. But it would be a new version of theproblem we know …

Leo ends up with:

[13] Whatever! They all [the recalled problems and facts] are just memories! I should lookfor some new directions since the facts and figures that I already know aren’t interestingany more.

Figure 2. A fact recalled by Leo on the way to creating a problem

A CASE STUDY OF AN EXPERT PROBLEM POSER FOR MATHEMATICS

Remarks on Fragment 1. First, the fragment can be seen as a three-cycle([2–4], [5–10], [11–12]) spiral of recalling associated problem and facts.In each cycle, Leo considered a particular nesting idea, recalled problemsor facts associated with it, and switched to the next one. Every consequentnesting idea carried traces of the previous ones. Leo initially focused hisattention on the word “billiard” from the given task (see [2]) and recalleda problem about it (see [1]). Then Leo connected “billiard” to reflections(see [2]) and recalled a relevant problem (see [3]). The principle ofreflections, which involves the equity of the angles of incidence andreflection, reminded him a mathematical fact concerning equal angles inthe context of his favorite nesting idea of “ellipse” (see the previoussubsection and [6]). The appearance of the “minimal perimeter” nestingidea (see [11–12]) can be explained by the fact that reflection is aprototypical solution for “minimal path” problems.

Generally speaking, the observed spiral enabled us to refine thehypothesis regarding the patterns in Leo’s problem-posing thinking, whichwas formulated on the basis of the reflective interview. Namely, onlythe first two elements of the triad “choosing an idea that had beenfruitful in the past—scanning a class of familiar problems related to thatidea—creating a new problem by modifying the idea” were observed atthis stage.

The second remark is related to the reasons for which Leo decided notto use the recalled material for creating a new problem. His decision isparticularly intriguing in light of the fact that Leo repeatedly asserted thatthe recalled ideas, problems, and facts were suitable and even somewhatsurprising for the potential solvers (see [2], [7] and [10]) (cf. Sharigin,1991, for the criteria of a “good” competition problem). The explanationappeared in [12–13] when Leo revealed that he was trying to surprise notonly the potential solvers of a new problem, but also himself. Moreover,as [12–13] imply, he cannot be surprised easily, and verification ofproblems and facts that he already knows is not the way to createsomething novel for him.

Fragment 2—Towards the Unknown. It so happened that the distancesfrom point O to AB and from point H to AB were almost equal in one ofthe drawings created by Leo, as in Fig. 2. When Leo noted this, he said:

[14] And what happens if the orthocenter and the circumcenter are at the same distancefrom the side? What would it tell us about a type of the triangle? [pause 20 seconds.] OK,it would be possible to say that the length of the altitude (CC ') is three times more than thedistance from O to AB [see Fig. 3], because we know that CH=2OM.[15] Does it make me happy? I’m not sure … What is the geometric meaning of this

IGOR’ KONTOROVICH AND BORIS KOICHU

expression? Apparently, there isn’t one. OK, there is a [numerical] relationship, but there’sno beauty in it …

Leo thought aloud about more ideas but eventually came back to theabove idea of a triangle in which the orthocenter and the circumcenter arelocated at the same distance from a side of the triangle. This idea ishidden in the problem formulation that he eventually wrote down as hisfirst problem posed during the interview:

Problem 65: H is the orthocenter and O is the circumcenter of a triangleABC when HO||AB. Prove that S△OAB ¼ S△OAC þ S△OBC

2 .

Then Leo evaluated the problem:

[16] It is not a masterpiece. It’s a combination of known facts: the Euler line andareas; and even a simple combination. Actually, I knew that the area of AOB is athird of the area of ABC, so I just added something to make a less transparentproblem.

Remarks on Fragment 2. In this part, Leo relied on two things from theprevious part: (a) following the fact illustrated by Fig. 3, he focused onthe orthocenter and the circumcenter of a triangle; and (b) he deliberatelygenerated questions without knowing their answer, as a way to surprisehimself by discovering a previously unknown and aesthetically appealingmathematics (see [15]). In light of that, Leo’s attempt to mask a familiarfact by reformulation (see [16]) or by considering a special case (see [8])could not lead him to the creation of a problem that would satisfy his self-attributed quality criteria.

Figure 3. A drawing related to the question, which was new to Leo

A CASE STUDY OF AN EXPERT PROBLEM POSER FOR MATHEMATICS

Fragment 3—Towards Even More Unknown. During the next 5 min, Leowas not very talkative and refused to think-aloud since “Not everythought can be verbalized.” He ended up with the following problem:

Problem 7: There are three equal and tangent (or kissing) circles α1,α2,α3.The circle β is tangent to all the three in points B1,B2,B3. P is a point onthe circle β which is different from B1, B2, B3. Three lines constructedthrough P and Bi intersect with the circle αi at some new point Ai, wheni = 1,2,3. Prove that the triangle A1A2A3 is a regular triangle [see Fig. 4].

Leo mentioned two sources when explaining a posteriori how theproblem occurred to him. First, Leo told us that he recalled a problemabout reflection of a point through 100 kissing circles that he had posed inthe past. Second, Leo recalled a proof of Euler’s Line Theorem by meansof homothecy (once more, see Elder, 2006, for mathematical details andnote that homothecy was mentioned as one of Leo’s nesting ideas in theframework of his reflective interview). The idea of homothecy as aparticular type of reflection is hidden in the way of constructing pointsA1,A2,A3 in the problem formulation.

When evaluating the problem, Leo said that it is decent because:

[17] The problem is based on an idea of homothecy, like in Euler’s line, but it is notenough to reveal this fact in order to solve it. It is not transparent. I love it, though I amsubjective, of course …

Figure 4. The drawing for Problem 7

IGOR’ KONTOROVICH AND BORIS KOICHU

Remarks on Fragment 3. In the above fragment, Leo created a problembased on the combination of two nesting ideas: “kissing circles” and“homothecy.”His excitement about Problem 7 seems to be related to the factthat the resulting combination was new for him and thus required making anexploration without being sure that it would lead to a worthwhile result.

In addition, a contrast between Leo’s reflections on Problem 6 andProblem 7 implies that he was satisfied with the posed problems up to thepoint that he could stop the problem-posing process only when he felt thatthe resulting problem involved mathematics that was novel and surprisingalso to himself and not only to the intended problem solvers.

Leo, however, decided to use both problems. Two weeks after theinterview, Problem 6 and Problem 7 were offered to students: the formerone at a preparatory stage and the latter one—at the competition itself.Anecdotally, we know that for some students Problem 7 appeared to beeasier than Leo had intended: They used vectors or complex numbersinstead of classic methods of Euclidian geometry.

SUMMARY AND CONTRIBUTION

This paper reports the first empirical (i.e. not only theoretical or self-reflective) study aimed at revealing the nature of expertise in the field ofmathematical problem posing. Obviously, the limited scope of our studyprescribes modesty in drawing conclusions. The main contribution of thepresented case study is that it offers grounded suggestions regarding someof the mechanisms involved in expert problem posing. Hopefully, thesemechanisms can be tested in future studies with additional problem-posing experts and, probably also, with learners. Specifically, threecharacteristics of Leo’s problem-posing practice require further empiricaltesting. In what follows, the first two characteristics are discussed in lightof the existing body of knowledge on expertise in mathematical problemsolving, and the third one—in light of findings from two studies withschool children and mathematics teachers. Two particular studies werechosen because their participants were offered the “billiard task” as aproblem-posing stimulus and because the resulting problems werecharacterized in terms compatible with those used in our study.

First, in line with research on expertise in problem solving (e.g.Schoenfeld, 1992), we notice that Leo possesses an advanced, well-structured, and easily accessible knowledge base for problem posing. Animportant component of his knowledge base is a pool of familiar

A CASE STUDY OF AN EXPERT PROBLEM POSER FOR MATHEMATICS

problems. We found that the pool is organized according to a diverse setof grouping principles, which we referred to as nesting ideas.

Second, one of the characteristics of expert problem solving is looking fordeep-structure similarities between a problem-to-be-solved and the familiarproblems. For Leo, grouping problems by deep-structure nesting ideas wasonly one of three grouping principles in his possession. It was observed that,when posing a new problem, Leo utilized two additional types of grouping:by surface-structure and by a particularly rich mathematical conceptemployed in several problems. Moreover, we observed that, when Leofocused on deep-structure connections, he could hardly come up with aproblem that would be satisfactory for him, since the fact that he knew itssolution in advance was a stumbling block.

The third characteristics stems from contrasting our findings with thefindings of Silver et al. (1996) on mathematics teachers and of Kontorovichet al. (2012) on high-achieving high school students. In both studies, thesubjects posed problems, which were directly related to the billiard contextand whose formulations and solutions were in the spirit of standard textbookgeometry problems or curricula-based problems mentioning the billiardtable. As a matter of fact, the resulting problems were only slight variationsof the problems familiar to the posers. In contrast, Leo put substantial effortinto creating a problem unrelated to the billiard context, the solution to whichwould be surprising to him and not only to the potential solvers. It isparticularly interesting to note that, in order to surprise the potential solvers,he could just use his immense arsenal of nesting ideas whereas, in order tosurprise himself, he had to modify them. He was not content with varying hisfamiliar problems and used them only as starting points. In an attempt tomove away from his familiar problems, Leo deliberately left his “comfortzone” and placed himself in a situation of searching for the unknown.

Why did an expert like Leo, and novices participating in the twoaforementioned studies, act so differently when given a similar problem-posing stimulus? Using the terminology developed in our prior study(Kontorovich et al., 2012), we argue that one of the underlying explanationsmay be related to the differences in posers’ considerations of aptness, i.e.“the poser’s comprehensions of explicit and implicit requirements of aproblem-posing task within a particular context” (p. 153). In the case of thenovices, the task was perceived as a necessary framework with relativelyrigid boundaries which the posers needed to fit. All the novices posedproblems related to the billiard context so that their relevance to the taskwould be out of the question for them or for the potential evaluators. As aresult, this type of considerations might hinder establishing a personal

IGOR’ KONTOROVICH AND BORIS KOICHU

connection with the task and lead to posing uninteresting problems, even forthe posers (cf. Crespo & Sinclair, 2008).

In contrast, the “billiard task” was only suggestatory for Leo, butcreating a “good” problem was self-mandatory. Creating such problemsas those presented in this paper can be seen as an act of significantknowledge acquisition by the expert. This perspective is in line withEricsson’s (2006), who wrote that experts tend to engage themselves indeliberate practices in order to extend their already well-developedknowledge base and to sharpen their professional skills.

In closing, we would like to note that the current study was concerned witha cognitive aspect of expert problem posing. However, posing new problemsin Leo’s case was accompanied by a positive emotional package consisting ofthe excitement of scientific exploration, the thrill of discovery, and the sense ofownership of the result. The affective dimension of problem posing was not atthe focus of the current study and therefore leaves room for further research(Kontorovich & Koichu, 2012). In line with the recent stream in mathematicseducation research (e.g. Furinghetti &Morselli, 2009), it can be suggested thatthe next step would be a confluence research, which would shed light on theinterplay of the cognitive and the affective in expert problem posing.

ACKNOWLEDGMENT

The research of the first-named author was supported by the GraduateSchool of the Technion. We are grateful to an anonymous participant inour study for his cooperation and to Royi Lachmy for his valuablecomments on an early draft of this paper.

NOTES

1 The exact formulation of the task can be found in several sources, includingKontorovich et al. (2012) and in Koichu & Kontorovich (2013). We do not present it herebecause, as will be evident shortly, Leo took into consideration only the general context of“billiard” while disregarding the details of the task.

2 These stories are also presented in Kontorovich & Koichu (2012), but in that paperthey are analyzed from another (affective) perspective.

3 Note that this metaphor, as any other one, has its limitations. For instance, Leo’snests of ideas embrace problems created by Leo as well as problems created by others.

4 Leo’s sketches were of a poor graphic quality, and we chose to reproduce them usingGeoGebra software.

5 We use continuous numeration of the problems posed by Leo for the ease of referringto them.

A CASE STUDY OF AN EXPERT PROBLEM POSER FOR MATHEMATICS

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Department of Education in Technology and ScienceTechnion—Israel Institute of TechnologyHaifa, IsraelE-mail: ik@tx.technion.ac.il

A CASE STUDY OF AN EXPERT PROBLEM POSER FOR MATHEMATICS