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ConstruCtivist Foundations vol. 9, n°3

Radical constructivism

Examining the Role of Re-presentation in mathematical problem solvingan application of Ernst von Glasersfeld’s conceptual analysisVictor V. Cifarelli • University of North Carolina at Charlotte, USA • vvcifare/at/uncc.eduVolkan Sevim • Virginia Commonwealth University, USA • vsevim/at/vcu.edu

> context • The paper utilizes a conceptual analysis to examine the development of abstract conceptual struc-tures in mathematical problem solving. In so doing, we address two questions: 1. How have the ideas of RC in-fluenced our own educational theory? and 2. How has our application of the ideas of RC helped to improve our understanding of the connection between teaching practice and students’ learning processes? > problem • The paper documents how Ernst von Glasersfeld’s view of mental representation can be illustrated in the context of mathematical problem solving and used to explain the development of conceptual structure in mathemati-cal problem solving. We focus on how acts of mental re‑presentation play a vital role in the gradual internaliza-tion and interiorization of solution activity. > method • A conceptual analysis of the actions of a college student solving a set of algebra problems was conducted. We focus on the student’s problem solving actions, particular-ly her emerging and developing reflections about her solution activity. The interview was videotaped and writ-ten transcripts of the solver’s verbal responses were prepared. > Results • The analysis of the solver’s solution activity focused on identifying and describing her cognitive actions in resolving genuinely problematic situations that she faced while solving the tasks. The results of the analysis included a description of the increasingly ab-stract levels of conceptual knowledge demonstrated by the solver. > implications • The results suggest a frame-work for an explanation of problem solving that is activity‑based, and consistent with von Glasersfeld’s radical constructivist view of knowledge. The impact of von Glasersfeld’s ideas in mathematics education is discussed. > Key Words • Mathematics education, mental representation, problem solving, mathematics learning.

introduction

« 1 » in preparing this paper, we ex-amined how Ernst von Glasersfeld’s theory of radical constructivism in mathematics education has contributed to studies of mathematical knowledge. von Glasers-feld described his view of constructivism as radical “because it breaks with conven-tion and develops a theory of knowledge in which knowledge does not reflect an objec-tive, ontological reality but exclusively an ordering and organization of a world con-stituted by experience” (Glasersfeld 1984: 24).

« 2 » in the context of mathematics education, the implications of von Gla-sersfeld’s views presented challenges for researchers. First, there were methodologi-cal issues. The research methods typically

used in the 1970s to study learning were not designed to capture the type of devel-opment that von Glasersfeld referred to. Hence, new research methods that could isolate and illuminate the constructive learning processes, characterized by the construction of knowledge through or-dering and organizing experience, were needed. accordingly, these new research methodologies were developed and came to be viewed as critical tools in our analy-ses of mathematics learning (Cobb 2008; Thompson 2008). For example, the use of individual clinical interviews, teaching ex-periments and conceptual analysis to con-duct research on learning made possible a focus in which researchers could closely observe the actions of learners as learning commenced, and thus provided contexts in which hypotheses about the knowledge

constructed could be formulated and tested (steffe 1991a; steffe & Kieren 1994; Thomp-son 1994, 2008).

« 3 » a second challenge facing re-searchers involved how von Glasersfeld’s philosophical ideas about structure and organizing of one’s experience could be interpreted in the context of mathematical activity. applications of radical construc-tivism in mathematics education are found in early studies of children’s counting and number development (steffe 1991a; steffe & Cobb 1988; steffe 1992). These studies were very instructive for researchers, par-ticularly in suggesting guidelines for how the ideas of radical constructivism could be used to design and conduct particular research studies. For example, Leslie steffe stated that in applying principles of radical constructivism to mathematics education,

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mathematical problem solving victor v. cifarelli & volkan sevim

radical Constructivism

http://www.univie.ac.at/constructivism/journal/9/3/360.cifarelli

researchers must adopt a broad-based view of action that includes both physical and mental actions:

“ The mathematical knowledge of children can be understood in terms of goal-directed action patterns if ‘action’ is taken to refer to mental as well as to physical action. These mental actions constitute operations – interiorized action pat-terns – and the involved schemes are operative rather than sensorimotor.” (steffe 1991a: 179)

« 4 » steffe’s comments suggest the importance of researchers examining goal-directed action patterns of learners, and that our analyses should explain how goal-directed sensori-motor actions are transformed (or interiorized) into mental action patterns, or operations.

« 5 » our research builds on von Gla-sersfeld’s and steffe’s views about learning by examining goal-directed activity in the context of problem solving. Conducting a conceptual analysis (Thompson 2008) of the actions of a college student solving a set of algebra problems, we focus on the student’s problem solving actions, particu-larly her emerging and developing reflec-tions about her solution activity (Cifarelli 1998). We document and explain how in the course of her problem solving, she is able first to internalize and then to interior-ize her solution activity and thereby build conceptual structure from her actions. Central to the analysis is the documenta-tion and explanation of the important role that mental representation, as defined by von Glasersfeld (1987a), played in her con-ceptual development.

« 6 » The paper begins with an over-view of von Glasersfeld’s views on mental representations and the important role they play in ordering and organizing experience, and thus making possible the abstraction of ideas from experience. This overview is fol-lowed by a description of the experimental setting of the conceptual analysis, which includes a list of the operational definitions that we followed. We then present and dis-cuss episodes from the conceptual analysis. Finally, we offer concluding comments on the impact of von Glasersfeld’s work on is-sues related to mathematics teaching.

von Glasersfeld’s views on mental representation and the act of re-presentation« 7 » von Glasersfeld’s theory of radical

constructivism emphasizes that knowledge is constructed by internalizing and interior-izing actions (Cobb 2008; Glasersfeld 1991), and that mental reflections and representa-tions play prominent roles in the abstraction process.

« 8 » noting the various uses of the term representation, von Glasersfeld saw the usefulness of viewing the process of men-tally re-presenting a prior action as an active dynamic process that plays a fundamental role in the construction of conceptual struc-tures. in so doing, he made a useful distinc-tion between two types of representations: those that depict or stand for other objects, and those that involve re-presentation. rep-resentations that depict or stand for other objects, such as iconic and symbolic rep-resentations, relate a pair of objects. For example, the statement “the sketch rep-resents a lily” relates a pair of objects, in which the first object, the sketch, refers to the second object, the lily that it is supposed to depict. in contrast, representations that are formed by the dynamic act of mentally re-presenting to oneself an action or prior experience (Glasersfeld 1991, 1987a) do not stand for or depict other objects. von Gla-sersfeld referred to this latter type of rep-resentation as mental representation. For example, in the statement “Jane re-presents her summer visit to China,” one refers to a primary creation, a mental act of perceptual or imagined construction, for which there is no prior object that serves as original to be copied, replicated, depicted or stood in for. Thus, von Glasersfeld viewed mental representations as internal structures that characterize and direct the reconstruction of prior experiences that function as con-ceptions, or action-based routines “that can be mentally called up and run” (Glasersfeld 1987b: 219).

« 9 » a particular theme that von Gla-sersfeld focused on in his writings was to identify the critical roles that mental re-flections and representations play in the development of abstract conceptual knowl-edge. specifically, von Glasersfeld described reflection as “the ability of the mind to

observe its own operations” (Glasersfeld 1987a: 11) and characterized it in terms of

“ the mysterious capability that allows us to step out of the stream of direct experience, to re-pres-ent a chunk of it, and to look at it as though it were direct experience, while remaining aware of the fact that it is not.” (Glasersfeld 1991: 47)

« 10 » in discussing mental represen-tations as conceptions, he articulated how they function in experiential contexts:

“ Conceptions, then, are produced internally. They are replayed, shelved, or discarded accord-ing to their usefulness and applicability in expe-riential contexts. But no amount of usefulness or reliability can alter their internal, conceptual ori-gin.” (Glasersfeld 1987b: 219)

« 11 » in addition, he discussed the con-nection between mental representations and other more abstract structures:

“ These conceptions may of course exemplify some abstraction - but if they do, they do so by applying the abstraction to quite specific senso-ry-motor material.” (Glasersfeld 1987b: 219)

« 12 » The importance of considering mental representations as dynamic con-ceptual structures is best exemplified in constructivist models of number devel-opment (steffe et al. 1983; steffe & Cobb 1988). These models attribute re-presented sensory-motor action as a major source of the mathematical knowledge constructed by children as they develop counting strategies. specifically, the children’s ability to re-pres-ent to themselves their prior mathematical activity (i.e., to “call it up” and “run through it” in thought) was found to play a crucial role in the mathematical knowledge they constructed.

Experimental setting

« 13 » The purpose of this paper is to il-lustrate aspects of von Glasersfeld’s model of knowing. to this end, we use a method of conceptual analysis (Glasersfeld 1995a) drawing from a case study of a college stu-dent’s problem solving. We want to make clear that our paper does not report findings

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from a study that tries to test hypotheses or build generalized models from a single test subject. By conducting a conceptual analysis of a single case study of problem solving, we offer an illustrative example of conceptual development over the course of a sequence of problem solving activities.

« 14 » We state some additional goals of our analysis. First, we look to isolate and explain episodes that capture the solver’s constructive processes as they build-up conceptual knowledge. Here we note von Glasersfeld’s suggestion that constructive processes can only be inferred “not from the correctness of a response but from the strug-gle that led to it” (Glasersfeld 1987a: 11).

« 15 » Central to this goal is that we ob-serve the problem solver as she experiences what for her are genuine problems. Hence, we define the term problem in order to cap-ture the experiential aspect of what it means when we say that the student “has a prob-lem.” in deciding how to define this term, we noted the work of Gordon Pask, who chal-lenged the view that problems can be neatly specified as well-structured puzzles whose means of solution require a “thought pro-cess that follows well-specified rules” (Pask 1985: 79). instead, he argued that problems seldom come “pre-packaged” (ibid: 79) as parts of the tasks we give to students, but as phenomena that the individual experiences. These phenomena are “problematic situa-tions” (ibid: 80) within which individuals actively formulate particular problems they see fit to address and solve. This view sug-gests a useful distinction between the tasks we may give to students and the problems they actually experience. steffe further spec-ified problems as occurring when a student assimilates a situation, using a concept, and as a result of the assimilation, “establishes a goal but can find no procedure in the con-cept to reach the goal.”1 our analysis thus adopts the view that the problem solver has a problem when he or she assimilates a situ-ation in terms of his or her current concepts, develops goals for action, and sees that his or her goals cannot be achieved with current knowledge.

1 | invited discussant in the symposium The role of inconsistent ideas in Learning Mathemat-ics at the 1988 annual Meeting of aEra, new orleans, La.

« 16 » our second goal is to develop a micro-level theory about conceptual devel-opment that can occur during mathemati-cal problem solving. in order to accomplish this goal, we compare and contrast the student’s actions across a range of problem solving tasks that systematically vary in their underlying mathematical structures. With this analytical focus, we build on von Glasersfeld’s theory of learning (Glasersfeld 1995a) by extending it to the domain of problem solving.

« 17 » The case study involves one par-ticipant, a college student with the pseud-onym Marie, who was a first-year calculus student at a university in the western united states. she was interviewed and videotaped as she solved a set of nine algebra problems (Box 1), consisting of a format that includes an initial task about the depths of two lakes, and eight follow-up tasks that systematically vary the underlying mathematical relation-ships. These tasks were designed by Erna Yackel (1984), and we use these with the fol-lowing goals: 1. to induce problematic situa-tions for Marie; and 2. to compare and con-trast her actions across a range of problem solving situations.

« 18 » While Box 1 refers to the phrase “similar” problems, we are aware that no set of task descriptions can adequately capture the essence of what it means for students to experience problems and further make de-terminations of problem similarity. Consis-tent with von Glasersfeld’s views, we believe that the solver’s problems are his or her own mental constructions based on his or her in-terpretations of the tasks.

operational definitions

« 19 » in conducting a conceptual anal-ysis (Glasersfeld 1995a) of the actions of Marie, we focus on her problem solving ac-tions as she solves a set of algebra problems. in particular, we interpret and model Marie’s own emerging and developing reflections about her mathematical activity.

« 20 » in order to provide context to our conceptual analysis, we note Patrick Thompson’s summary of the different ways that conceptual analysis can be used by mathematics education researchers.

“ in summary, conceptual analysis can be used in four ways: (1) in building models of what stu-dents actually know at some time and what they comprehend in specific situations, (2) in describ-ing ways of knowing that might be propitious for students’ mathematical learning, (3) in describing ways of knowing that might be deleterious to stu-dents’ understanding of important ideas and in describing ways of knowing that might be prob-lematic in specific situations, (4) in analyzing the coherence, or fit, of various ways of understand-ing a body of ideas. Each is described in terms of their meanings, and their meanings can then be inspected in regard to their mutual compatibility and mutual support.” (Thompson 2008: 60)

« 21 » our analysis, with its focus on examining goal-directed activity in the con-text of problem solving, is consistent with Thompson’s first category, which focuses on the mathematics of individual students. This type of analysis has also been referred to as retrospective analyses or the building of sec-ond-order models of students’ mathematics (steffe & & Kieren 1994) without the use of researchers’ own mathematical understand-ings. in contrast, the last three categories could be viewed as thought experiments based on the researchers’ own mathematics.

« 22 » in our conceptual analysis of Ma-rie’s solution activity, we use the following terms and constructs. We view reflection as the learner’s mental distancing of himself or herself from their actions. We discuss the learner’s reflections in the following con-texts: 1. reflections on prior activity, 2. re-flection on potential activity, and 3. reflec-tion on the results of activity. We will make these distinctions in the conceptual analysis.

« 23 » We use the term mental represen-tation to refer to conceptual structures that result from repeated re-presentations of a perceptual or imagined experience without actual sensory-motor material (Glasersfeld 1987a; steffe 1991a). We use the term antici-pation to indicate a realization by the learner prior to carrying out an activity that carry-ing out the activity will lead to a particular result (steffe & Cobb 1988).

« 24 » in the following sections, we dis-cuss episodes of Marie’s solution activity for tasks 1–4, and task 9.

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Results of the conceptual analysis

marie’s problem solving in tasks 1–2: Building an initial conceptual structure« 25 » in constructing a solution to

task 1, Marie’s activity was not at all rou-tine. she initially interpreted the task as an “algebra word problem in two variables” and stated her intention to “assign variables to everything that is important.” symbols representing variables were manipulated in a mechanical fashion as she tried to code and relate all of the information contained in the task statements (Figure 1), without reflecting to the extent necessary to con-sider whether such assignments were rel-evant to the solution of the problem. she proceeded to generate three algebraic equa-tions that she later found to be inappropri-ate.

« 26 » she realized she had a new prob-lem to solve and abandoned her coding strategy in favor of an approach in which she used a diagram as an interpretive tool to aid her conceptualization and elabo-ration of potential relationships. While her use of the diagram indicated a more sense-making approach, she still struggled to complete the task, generating an equa-tion that led to an incorrect solution (Fig-ure 2).2

« 27 » upon discovery of the error in her diagram, Marie reconstructed the al-gebraic expressions and generated a new algebraic equation that led to a correct solution. Thus, by constructing a side-by-side diagram of the lakes, which aided her understanding of the situation and from which she was able to generate an appro-priate equation that related the lengths of the lakes, she was able to solve the task successfully. in particular, she assigned the variable X to signify the length of a quan-tity not directly stated in the problem state-ments (i.e., the overlapping segment when the lakes are positioned side-by-side). Her solution is summarized in Figure 3.

2 | Comments between brackets indicate the researchers’ interpretations of the solver’s non-verbal gestures and actions.

Box 1: set of problem solving tasks

Task 1: Solve the Two Lakes ProblemThe surface of Clear Lake is 35 feet above the surface of Blue Lake. Clear Lake is twice as deep as Blue Lake. The bottom of Clear Lake is 12 feet above the bottom of Blue Lake. How deep are the two lakes?

Task 2: Solve a Similar Problem That Contains Superfluous InformationThe northern edge of the city of Brownsburg is 200 miles north of the north-ern edge of Greenville. The distance between the southern edges is 218 miles. Greenville is three times as long, north to south as Brownsburg. A line drawn due north through the city center of Greenville falls 10 miles east of the city center of Brownsburg. How many miles in length is each city, north to south?

Task 3: Solve a Similar Problem That Contains Insufficient InformationAn oil storage drum is mounted on a stand. A water storage drum is mount-ed on a stand that is 8 feet taller than the oil drum stand. The water level is 15 feet above the oil level. What is the depth of the oil in the drum? Of the water?

Task 4: Solve a Similar Problem In Which the Question is OmittedAn office building and an adjacent hotel each have a mirrored glass facade on the upper portions. The hotel is 50 feet shorter than the office build-ing. The bottom of the glass facade on the hotel extends 15 feet below the bottom of the facade on the office building. The height of the facade on the office building is twice that on the hotel.

Task 5: Solve a Similar Problem That Contains Inconsistent InformationA mountain climber wishes to know the heights of Mt. Washburn and Mt. McCoy. The information he has is that the top of Mt. Washburn is 2000 feet above the top of Mt. McCoy, and that the base of Mt. Washburn is 180 feet below the base of Mt. McCoy. Mt. McCoy is twice as high as Mt. Washburn. What is the height of each mountain?

Task 6: Solve a Similar Problem That Contains the Same Implicit InformationA freight train and a passenger train are stopped on adjacent tracks. The engine of the freight is 100 yards ahead of the engine of the passenger train. The end of the caboose of the freight train is 30 yards ahead of the end of the caboose of the passenger train. The freight train is twice as long as the passenger train. How long are the trains?

Task 7: Solve a Similar Problem That is a GeneralizationIn constructing a tower of fixed height, a contractor determines that he can use a 35 foot high base, 7 steel tower segments, and no aerial platform. Alternatively, he can construct the tower by using no base, 9 steel tower seg-ments, and a 15 foot high aerial platform. What is the height of the tower he will construct?

Task 8: Solve a Similar Simpler ProblemGreen Lake and Fish Lake have surfaces at the same level. Green Lake is 3 times as deep as Fish Lake. The bottom of Green Lake is 40 feet below the bottom of Fish Lake. How deep are the two lakes?

Task 9: Make Up a Problem That has a Similar Solution Method.

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« 28 » We inferred that she had devel-oped an action pattern from her solution activity while solving task 1, which served as a conceptual structure that enabled her to interpret new tasks as similar, and thus assimilate new situations to her current structure. as we will see in the subsequent episodes, Marie’s use of this action pattern progressed from repeated re-presentations of the actions themselves, which she then transformed into operative knowledge that helped her solve the tasks more effectively and efficiently.

« 29 » in observing her solution activ-ity in task 2, we inferred from her actions that her initial ideas about similarity were somewhat low level in the following sense. While she did indeed see a need to use a similar approach (i.e., draw a diagram and develop appropriate equations), she could not “see” results of carrying out the ap-proach prior to actually carrying out the actions. so, for example, Marie interpreted task 2 as similar, remarking that “the first thing that strikes me is that this problem is a lot like the first one” and constructed a diagram similar to that she constructed to solve task 1 (Figure 4).

« 30 » While she recognized that she would incorporate a solution method simi-lar to what she did to solve task 1, her rec-ognition did not allow her to see and ad-dress a potentially problematic situation due to the added information contained in the task statement. it was only after she carried out her activity that she questioned the additional information contained in the task statement:

“ Marie: a line drawn due north through the city center of Greenville falls 10 miles east of the city center of Brownsburg. How many miles… [reflection]… i’m thinking that this line drawn due north doesn’t seem to have anything to do with the problem… [reflection]… so i’ll just look at the other relationships first.”

« 31 » Marie’s interpretations indicated she could recognize task 2 as requiring so-lution activity similar to that used in solv-ing task 1. she could not see a potential difficulty prior to carrying out her solution activity. Marie went on to construct a cor-rect solution for task 2 (Figure 4).

Marie: I have 4 unknowns and 3 equations. And that’s not good enough for me to solve an algebra problem.

Figure 1: Marie’s coding strategy.

Marie: This is the bottom, this is the surface of Blue Lake, and this is the bottom of Blue Lake. This distance is 12 and this distance is 35. And this whole distance is twice that whole distance. [reflection]Marie: Okay, if I label this whole distance X… I can say… that 12 plus X plus 35, which is the height of Clear Lake, is going to equal twice X. And that’s the relation in one variable I can solve.Marie: And the relation I was missing here is the fact that I’m looking at differences in height, not absolute height.

Figure 2: Marie’s incorrect solution.

Marie: The bottom of Lake… and this lake is 12 feet above the bottom of that lake. So I didn’t draw it that way. I drew it 12 feet below.Marie: That means that my geometrical solution is probably off.Marie: So, the distance between these two is still 35. The distance between these two is 12. Yeah, but X doesn’t mean the same anymore.Marie: So, 35 plus X equals 24 plus 2X. So 35 minus 24 equals… X. So Clear Lake is equal to 35 plus X which is 46. And Blue Lake is equal to 12 plus 11 which is… 23. That’s the solution!

Figure 3: Marie’s solution to Task 1.

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marie’s problem solving in task 3: introducing re-presentation into her actions« 32 » Marie attempted to solve task 3

in the same way as she solved tasks 1 and 2. However, she soon found herself faced with a problematic situation she had not foreseen, which appeared to galvanize her with a new found sense of excitement and wonderment, as she tried to explain the new problem she faced (Figure 5).

« 33 » The suddenness with which Marie was now able to see a potential difficulty, to-

gether with her reflections exploring the na-ture of the problem she faced, suggests that she had attained a level of reflective activity not demonstrated while solving the prior tasks. More precisely, we inferred from her actions that she could now perform a mental “run through” of her potential solution activ-ity and see difficulties that might arise were she to carry out her actions. We assert that Marie’s reflection on her potential solution activity was an act of mental re-presentation (Glasersfeld 1987a; steffe 1991a, 1991b; steffe & Cobb 1988). Her anticipation of the

problematic situation was followed by her hypothesis that the problem might not con-tain enough information, which was later refined to the hypothesis that she needed more information about the heights of the unknowns. While her hypothesis contained uncertainty, it nevertheless helped her to or-ganize and structure her subsequent solution activity as she explored and tested its plau-sibility as an explanatory device. she spent much time pursuing the elusive informa-tion and finally concluded that the problem could not be solved. in summary, we inferred that her ability to anticipate the problematic situation and develop hypotheses express a mental representation that made it possible for Marie to examine her original goals ret-rospectively.

marie’s problem solving in tasks 4 & 9: Further conceptual development« 34 » in what ways did Marie’s reflec-

tions help to evolve her solution activity while solving later tasks? a partial answer to this question is that she became more cau-tious in her activity, spending increased time reflecting on her potential activity. Her reflec-tions on potential solution activity continued to exhibit hypothetical qualities that led to novel conjectures. For example, while solving task 4, Marie quickly noticed the omission of a question from the task statement and was able to hypothesize potential problems for her to solve from the information.

“ Marie: There’s no question! [reflection]… The things they could ask for are things like… [hypoth-esis]… the height of one of the buildings but… [an-ticipation]… there’s not enough information to get that….The only thing we have information about is… [hypothesis]… ah, the relative heights of the two facades. so, if i were… if somebody wanted me to solve any problem, that’s probably what they’re asking for.”

« 35 » Marie’s reasoning about potential problems that could be solved was plausible since, based on her current understanding, these were problems that could conceivably be solved. and at the same time, her reason-ing was provisional in the sense that she saw the need to explore her newly formulated problem further. Marie’s anticipation follow-ing her first hypothesis indicated she had de-

Marie: The first thing that strikes me is that this problem is a lot like the previous one. [recognition] And… I think it would serve me well to start off in this one by just drawing a picture.


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duced a necessary condition of the problem of finding the height of one of the buildings, by performing a mental “run through” of the imagined action of trying to solve the prob-lem. The result of this mental run through was that she rejected her initial hypothesis. similarly, she explored the plausibility of her second hypothesis, concluding that it made more sense to her that the problem of finding the heights of the two facades was a problem that could be solved.

« 36 » unlike in her solution activity for tasks 2–3, where she made explicit compari-son to task 1, her anticipations here express a higher-level cognitive structure in that reflec-tions are now coordinated with her changing view of what is problematic. also, she dem-onstrated this highly abstract activity prior to constructing her diagrams. she generated a solution to task 4 utilizing diagrams to con-struct relationships, in much the same way as she solved earlier tasks (Figure 6).

« 37 » Marie’s solution activity in task 9 was further indication that she had re-or-ganized her conceptual structure at a higher level of abstraction. specifically, she could reflect on a potential solution activity, an-ticipate its results, and evaluate its usefulness for the current situation without the need to carry out the activity with paper-and-pencil. in other words, she could reflect on her po-tential solution activity, determine appropri-ate relationships, and evaluate the efficacy of those relationships for the situation she faced. The task required her to make up a problem, which had a solution method similar to the prior tasks.

“ Marie: okay… i’m thinking of something with different heights. oh,… [anticipation]… book-shelves in a bookcase. no…[reflection]… that’s no good… [anticipation]… How about hot air bal-loons!”

« 38 » Marie ran through potential solu-tion activity for each of the particular situa-tions she considered, and anticipated their results (i.e., that it would not work for “book-shelves” but that she could solve it for “hot air balloons”). in this way, she operated on her re-presentation of the potential solution activity by producing its results and drawing inferences from them. she was able to gen-erate routinely a problem that she viewed as similar (Figure 7).

Marie: I am going to draw a picture. Here is my oil stand. And we have a water stor-age 8 feet taller. And here’s level water. And here’s the oil level. And the water level is15 feet above the oil level [reflection]. So, solve it… [reflection]… the same way….[re‑presentation] [anticipation]… Impossible!! [displays a facial expression suggesting puzzlement]Marie: It strikes me suddenly that there might not be enough information to solve this problem. So I better check that.… [reflection]… I suspect I’m going to need to know the heights of one of these things [points to containers]. But I could be wrong so… I’m going to go over here all the way through.


Marie: Okay. Let’s see if there is anything here that will at least give me information. Okay, the hotel is 50 feet shorter than the office building. So we have distance here which is 50. The facade of the hotel extends 15 feet below the facade of the office building. That distance would be 15. The height of the facade on the office building is twice that on the hotel [reflection]. So I call this distance X, this distance here is 2X. All right! And then I can say that X minus… I’m trying to find a relationship between these two. And I know that… X minus 15 plus 50 is going to equal 2X. So, 35 equals one X. So that would indicate that the facade on the hotel is 35 feet. On the office building is 70 feet.


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discussion

« 39 » in our application of the ideas of radical constructivism to problem solving, we have developed a micro-level theoretical framework through which to examine and explain problem solving activity.

« 40 » We inferred that Marie’s solution activity in solving task 1 left an indelible mark on her experience that she could make reference to in solving subsequent tasks. We interpreted that she had constructed an ini-tial conceptual structure from her problem solving actions and that this initial struc-ture formed a conceptual foundation for her subsequent solutions in tasks 2–9. However, this structure was at a low level of abstrac-tion. in particular, while in solving task 2 she could see the usefulness of applying the

actions she used to solve task 1, she was not able to see the potential problem. This po-tential problem was suggested by the added information contained in the task statement. she needed to carry out her solution activ-ity fully in order to realize that the added information might pose a new problem for her. Hence, her assimilation of task 2 to her current conceptual structure appeared to be an example of von Glasersfeld’s description of the act of recognition:

“ The ability to recognize a thing in one’s expe-riential field, however, does not necessarily bring with it the ability to re-present it spontaneously. our experiential world contains many things which, although we recognize them when we see them, are not available to us when we want to visualize them. There are, for instance, people

whom we would recognize as acquaintances when we meet them, but were we asked to de-scribe them when they are not in our visual field, we would be unable to recall an adequate image of their appearance.” (Glasersfeld 1991: 49–50)

« 41 » Further development of her con-ceptual structure was inferred from her solution activity for task 3. specifically, she could both anticipate the problematic situation and offer hypotheses about its pos-sible causes prior to carrying out her actions with paper-and-pencil. We inferred this to be an act of re-presentation in which she had performed a mental run through of her potential actions and could see and reflect further on the problem she would face. This is consistent with von Glasersfeld’s view of re-presentation (Glasersfeld 1987a).

« 42 » in inferring Marie’s further con-ceptual development in tasks 4 and 9, we found useful olive’s explanation of how ac-tion can be transformed from initial inter-nalized structures into more abstract con-ceptual structures:

“ The activity becomes interiorized through fur-ther abstraction of these internalized re-presenta-tions whereby they are stripped of their contex-tual details.” (olive 2001: 4)

« 43 » Therefore, in tasks 4 and 9, in olive’s terms, Marie had interiorized her solution activity to the extent that she could reflect on her potential activity as a unified set of actions that were now stripped of their prior contextual details. in this way, her reflections became operative (Glasersfeld 1987a).

« 44 » We believe that this operative activity was especially evident while Marie solved task 9. she was able to reflect on mul-tiple potential problem situations and deter-mine the appropriateness of each without further carrying out the actions.

« 45 » We summarize her conceptual development in tables 1 and 2.

« 46 » in addition to the general task-by-task development of her conceptual structure, we characterize her conceptual development in terms of increasingly ab-stract levels of solution activity and indicate where within the tasks each particular level was first demonstrated (table 2).

Marie: Okay, if I were going to draw a picture of the problem I’d have one hot balloon that looks like…[draws balloon]. And a bigger hot air balloon that looks like that. And I’ll make this distance… 3 feet.Marie: I’ll make this distance 2 feet. And I’ll make this height 10 feet high ‘cause that makes this 12 feet and that makes this one twice this one, which is useful.Marie: So, I’ll just say the top of one hot air balloon, HAB being the abbreviation for that, is 3 feet. The bottom of the yellow hot air balloon is 2 feet below the bottom of the green hot air balloon. The yellow balloon is twice the height of the green balloon. Let’s make that a lake. What are the heights of the balloons?


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implications for teaching

« 47 » While von Glasersfeld’s unique contributions that have impacted the ways that researchers view and examine the de-velopment of mathematical knowledge are very specific, his ideas and recommenda-tions for mathematics education practice remain somewhat broad. von Glasersfeld’s view that learning is explained as a con-structive activity, in which learners develop operative thought by reflecting on their ac-tions (Glasersfeld 1984; 1987a; 1989; 1995b; 1996), lies at the center of his recommenda-tions for mathematics education practice.

« 48 » By building on von Glasersfeld’s theory of radical constructivism, in which knowing is viewed exclusively as an order-ing and organization of one’s world consti-tuted by experience (Glasersfeld 1984), we focused on illustrating how one student’s goal-directed activity evolved in the context of problem solving and transformed her mental representations into more abstract

cognitive structures. We believe this con-ceptual analysis has several implications for teaching.

« 49 » First, we view students’ mental representations as constructions that are viable to them within their experiential world; not as misconceptions that lack the characteristics of the experts’ constructions. Therefore, in order for teachers to help change their students’ concepts, they need to listen and interpret what students say and do when solving mathematical problems and build models of their thinking. We ac-knowledge that this is not an easy task to undertake within the day-to-day practice of classroom teaching. We believe more stud-ies are needed that help equip teachers with practical guidelines for such model building practice.

« 50 » at this juncture, we should cau-tion that von Glasersfeld’s views on model building offer a rebuttal to the type of cogni-tive models that are based on the Cartesian epistemology, which requires knowledge

to stand for and mirror an external reality. For example, Cobb, Yackel & Wood (1992) argue persuasively against adopting these models for use in teaching on the grounds that the instructional implications point to inappropriate instructional activities, which have as their primary goals that “students construct mental representations that cor-rectly or accurately mirror mathematical relationships located outside the mind in instructional representations” (ibid: 4). The mental representations that we illustrated in this paper refer to idiosyncratic internal constructions, which do not stand for any objects but are formed by re-presentation of prior experiences.

« 51 » second, as von Glasersfeld put it, teachers need to make sure that students own their problems:

“ to solve a problem intelligently, one must first see it as one’s own problem. That is, one must see it as an obstacle that obstructs one’s progress to-ward a goal.” (Glasersfeld 1995b: 14)

task 1 tasks 2–4, 9

Solves the target task and follow-up

Solves variations of the original task

Emerging StructureEvolving awareness of solution

activityPrimitive Structure

Needs to carry out solution activity with paper and pencil

Develop Abstract StructureCan reflect on potential

activity and see its results

Table 1: Marie’s evolving conceptual structure.

level of abstraction level of Reflection characterization

more abstract

less abstract

Reflective AbstractionTasks 4, 9

Can coordinate and operate on results of a mental “run through” of potential solution activity.

Re-PresentationTask 3

Can coordinate prior actions to perform a mental “run through” of prior solution activity and anticipate potential problems.

RecognitionTask 2

Can use previously constructed diagrams to aid recognition of potential problems

Table 2: Marie’s solution activity as levels of conceptual structure.

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“ only when students can be led to see as their own a problem in which their approach is mani-festly inadequate will there be any incentive for them to change it.” (ibid: 15)

« 52 » We believe that in such experi-ential situations, where students own their problems, engage in meaningful goal-direct-ed activity, and re-present it in their minds, students will have a better chance of facing occasions in which they notice conceptual discrepancies in their mental representations (Glasersfeld 1996).

« 53 » Finally, as our conceptual analy-sis illustrates, mental representations are originated internally and emerge from self-generated re-presentations of actual or

imagined activity. in other words, they are dynamic constructs that help learners make conceptual advances. Therefore, self-gener-ated questions and hypotheses that emerge in problem solving can help students to place the current problem in a broader per-spective and thus expand its scope. Because students will own the problem in their own ways, this expansion of scope can further help them engage in unexpected generaliz-ing activity that is rooted in their own goals and purposes. to this end, teachers should allow and encourage students to pursue their own solution strategies while guiding them by requesting explanations and offer-ing critical questions.

conclusion

« 54 » We offered an example of how a reflective abstraction took place over the course of a sequence of problem solving ac-tivities. We hope that our conceptual analy-sis makes clear how this process of reflective abstraction, through a series of anticipatory activity and re-presentation of activity, could be applied to describe mathematical learning through problem solving more generally. We believe that our detailed descriptions of one student’s mathematical activity, and our anal-ysis of her goal-directed activity in the con-text of problem solving, support and contrib-ute to von Glasersfeld’s views about learning.

received: 2 February 2014 accepted: 6 May 2014

victoR ciFaRElliis Professor of Mathematics Education and the Coordinator of the Mathematics Education Program in the Department of Mathematics & Statistics at the University of North Carolina at Charlotte. He received his Ph.D. in Mathematics Education from Purdue University. His doctoral research focused on the role of reflective abstraction as a learning process in mathematical problem solving. Dr. Cifarelli has presented his research at AERA, PME, and PME-NA. His articles have appeared in the Journal of Mathematical Behavior, Focus on Learning Problems in Mathematics and the Proceedings of PME and PME-NA. In addition to his research, Dr. Cifarelli has co-directed numerous projects for mathematics teachers under the Eisenhower Professional Development Program and the North Carolina Department of Public Instruction.

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volKan sEvimis Assistant Professor of Mathematics Education in the School of Education at Virginia Commonwealth University. He received his Ph.D. in Curriculum and Instruction with a specialization in Mathematics Education from the University of North Carolina at Charlotte. His doctoral research focused on students’ understanding of quadratic functions. His current research examines learning processes involved in the co-evolution of problem solving and problem posing. Dr. Sevim has presented his research at NCTM, AERA, PME, and PME-NA. Dr. Sevim has taught upper level mathematics courses at a public high school in Charlotte, NC, for five years. In the last four years, he has been teaching mathematics education methods courses to prospective elementary teachers at university level.

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issues around Reflective abstraction in mathematics EducationCatherine UlrichVirginia Tech, USA culrich/at/vt.edu

> upshot • Cifarelli and Sevim’s analysis of Marie’s problem solving activity raises two questions for me. The first regards what Marie is reflectively abstracting: the use of the generic phrase her solu-tion activity finesses a largely unarticu-lated disagreement in the mathematics education community about what the nature of actions are in Piaget’s theory. The second question involves the im-plications that radical constructivist research in mathematics has for day‑to‑day mathematics instruction.

« 1 » Jean Piaget described reflective ab-straction as one of the “motive forces of cog-nitive development” (Piaget 2001: 303) and hence it should be of interest to anyone inter-ested in student learning. However, in partic-ular, “mathematics was the locus classicus for reflecting abstraction” (Campbell 2001: 12) in Piaget’s work. This was no accident. The mathematics education research community has long noted that key distinctions in stu-dent mathematics hinge around the students’ reconstructions of coordinations of mental mathematical actions into mathematical ob-jects that can be acted upon by generalized mathematical operations (cf. dubinsky 1991; sfard 1991). in other words, in a series of re-flective abstractions, students must change coordinated actions into objects of reflection and further action. anderson norton (in

press) has recently gone so far as to say that “mathematics is the objectification of action” and that is what makes mathematics unique. This seems to have been Piaget’s general view of mathematics as well:

“ new degrees of projection are unceasingly con-structed, allowing new reflections, as is demon-strated by the entire history of mathematics, with the successive thematizing that has gone on right through its current phases of development.” (Piaget 2001: 305)

« 2 » However, as norton (in press) goes on to point out, the nature of the ac-tions mathematics education researchers are referring to varies. some focus on formal mathematical processes such as subtrac-tion or division, while others may focus on externalized procedures such as graphing functions, utilizing function notation, etc. i agree with norton that the most productive way for us to interpret action is as mental actions of the student. More specifically, i suggest we focus on the mental actions that construct and coordinate quantitative rela-tionships (e.g., Thompson 1993) in a given mathematical context.

« 3 » in the case of Marie, victor Cifarelli and volkan sevim often refer to Marie’s so-lution activity, and the phrase can be inter-preted throughout as referring to her drawing of a diagram, her translation of quantitative relationships represented in the diagram into algebraic equations, and manipulation of her algebraic notation to arrive at a numerical an-swer. This focus on her solution procedures is particularly evident in §41 and §43. However, i think the authors would agree with me in concluding that these solution procedures are externalizations of her internal construction of quantitative relationships. in particular, in

task 1 we can conclude that Marie construct-ed a linear relationship between differences in heights, or relative heights, as she refers to them in Figure 2. Thus her actions would refer to the constructions of individual quan-tities, such as the relative heights or the con-structions of the additive and multiplicative comparisons between these quantities. The results of the coordination of her actions refer to the overall quantitative relationship that she constructs. Her representations could still consist of her running through the making of a diagram, but i would interpret her focus as being on the quantitative relationships the diagram symbolizes and externalizes for her. Much of the authors’ analysis of the role of re-presentation would still stand up to scrutiny with this alteration of focus, as i show below.

« 4 » using this perspective on the data, i would like to revisit Cifarelli and sevim’s claim in table 3 that Marie’s activity in tasks 4 and 9 are examples of reflective abstrac-tion. While i agree with this claim, i would make a stronger claim, which is that Marie utilized reflective abstractions in each of her solutions described in the article. i believe Marie’s example illustrates not a develop-ment towards reflective abstraction, but a sequence of increasingly higher levels of reflective abstractions that illustrates how Ernst von Glasersfeld’s characterizations of mental representations intersects with Piaget’s most well-developed thinking about reflective abstraction, as found in his final work on the subject, Studies in Reflecting1 Abstraction. i would like to note here that if

1 | For consistency of terminology, i will use the more conventional term reflective where Campbell used the term reflecting in his trans-lation. However, they refer to the same word in French.

open peer commentarieson victor v. cifarelli & volkan sevim’s “Examining the Role of Re-presentation in mathematical problem solving”

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Re-presentations and conceptual structures of What? Kevin c. moore



we could substitute reflected abstraction for reflective abstraction in Cifarelli and sevim’s original claim, it would agree with my analy-sis. Hence this is partially a matter of align-ing terminology.

« 5 » First, let us review Piaget’s charac-terization of reflective abstraction in general:

“ [reflective] abstraction ranges over the coor-dinations of the subject’s actions. These coordi-nations – and the reflecting process itself – may remain unconscious, or may give rise to con-sciousness…[it] involves two inseparable aspects. on the one hand, there is projection (as though by a reflective surface) onto a higher plane of what is drawn from the lower plane (for instance, from the plane of action to the plane of representation). on the other hand, there is reflection, a mental act of reconstructing and reorganizing on the higher plane what has been transferred by projection from the lower one.” (Piaget 2001: 303)

« 6 » utilizing this characterization, an analysis of Marie’s activity reveals a series of reflective abstractions, with different lev-els of projection. That is, the “higher plane” (or elsewhere in Piaget, the “next develop-mental level”) of projection is very gradually getting higher, i.e., Marie has been able to put an increasing distance between herself and her solution activities (cf. Piaget 2001: 306). Here i should note that my use of the term projection refers to the first part of the process of reflective abstraction discussed above. This is in contrast to the “reconstruct-ing and reorganizing” of what has been pro-jected, which robert Campbell translated as reflection. This does not fit exactly with Cifarelli and sevim’s use of reflection as de-fined in §22. Their use of reflection seems to cover both parts of reflecting abstraction, as evidenced by their definition, which implies projection, and their subsequent use of the term in their categories of reflection, which implies reflection in my sense of the word. While i do not find their broader use of the word problematic in the context of their ar-ticle, the distinction will be important in the present discussion.

« 7 » in fact, Cifarelli and sevim’s Levels of reflection in table 3 correspond well with levels of projection that Piaget (2001: 304) outlines. in task 2, Marie could be inter-preted to recognize some of the same quan-titative elements as in task 1, even before

she solves the problem. This is equivalent to a first-level projection. she then constructs the new quantitative relationships in task 2 and is able to compare that to task 1 in order to identify potentially extraneous informa-tion. This is similar to a third-level project in Piaget’s work and presupposes that at some point (either in task 1 or task 2), Marie has attentionally bounded the quantitative rela-tionship from task 1 as a unified whole that can be compared to the relationships in task 2. This would correspond to a second-level projection. Cifarelli and sevim’s account of how Marie moves from recognition to a nascent anticipation of the results of her ac-tions and of her ability to run through her actions using mental representations both serve as a nice elaboration of what comes between Piaget’s third and fourth levels of projections. Finally, when Marie is able to run through the solution activity and evalu-ate the usefulness of the results in task 9 (and arguably task 4), Marie is at the fourth level of projection, which corresponds to re-flected abstraction (cf. Campbell 2001).

« 8 » Therefore, i find that Cifarelli and sevim’s analysis of Marie’s mathematical ac-tivity, particularly with a more psychological interpretation of action, provides an excel-lent illustration and elaboration of levels of reflective abstraction. Most of the other doc-umentations i know of on reflective abstrac-tion, outside of Cifarelli and sevim’s work, focus on the construction of new kinds of quantities or quantitative relationships. Ma-rie’s use of terms such as relative heights, along with her relative comfort in working with diagrams and equations in represent-ing quantitative relationships, implies that Marie is not constructing a completely new type of mathematical structure. However, as opposed to decreasing the value of Cifarelli and sevim’s example, i think this increases its value in terms of implications for instruc-tion.

« 9 » The more far-reaching discussion of objectification of actions that one finds in aPos theory (asiala et al. 1996) or im-plicitly in the work of numerous researchers such as Leslie steffe and John olive’s work on fraction learning (2010) has implications for broader curricular goals and choices. However, the changes they would hope to engender in students’ ways of operating take months or even years to occur. in the mean-

time, it is the micro-scale reflective abstrac-tions that we can hope to engender on a daily basis as educators. in §52, the authors argue that the importance of the general approach of encouraging problem solving and reflec-tion are an implication of their research. i would agree, and further say that this may be the important implication of constructiv-ist research in general on a daily scale.

catherine ulrich is an assistant professor of mathematics education in the School of Education at Virginia Tech. She studies students’ conceptions of addition and subtraction with counting numbers

and integers. She also studies the interplay between the development of additive and multiplicative

reasoning at the middle-school level.

received: 5 June 2014 accepted: 6 June 2014

Re-presentations and conceptual structures of What?Kevin C. MooreUniversity of Georgia, USA kvcmoore/at/uga.edu

> upshot • Education researchers often explain student activity in terms of gen-eral thinking and learning processes, in-cluding those identified by Cifarelli and Sevim. In this commentary, I refocus Ci-farelli and Sevim’s discussion in order to hypothesize the organization of mental actions that comprise and support those learning processes.

« 1 » in their target article, victor Ci-farelli and volkan sevim provide an exam-ple of a student (Marie) developing abstract levels of solution activity over the course of a task-based interview. describing Marie’s activity in solving task 1, the authors claim, “[Marie] developed an action pattern from her solution activity while solving task 1, which served as a conceptual structure that enabled her to interpret new tasks as simi-lar, and thus assimilate new situations to her current structure” (§28). The authors then conclude that Marie’s conceptual structure

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Re-presentations and conceptual structures of What? Kevin c. moore



mathematical modeling carlos W. castillo-Garsow

became more abstract as she reconstructed this conceptual structure in subsequent tasks, eventually enabling her to anticipate or re-present her solution activity indepen-dent of contextual details and without carry-ing out the activity.

« 2 » Marie’s progress during the se-quence of tasks leaves the following ques-tions: What organization of mental actions1 composed her conceptual structure and how did the nature of these mental actions support her development of an increasingly abstract conceptual structure? in this com-mentary, i provide one perspective on the mental actions driving Marie’s actions and conclude with thoughts on relationships between the posed perspective, Cifarelli and sevim’s work, and broader work in math-ematics education.

processes as mental actions« 3 » Cifarelli and sevim attribute Ma-

rie’s development to processes including re-presentation, recognition, reflection, and abstraction, each of which von Glasersfeld considered critical to explaining the nature of knowing and learning. von Glasersfeld also considered each of these processes as sensitive to the particular mental actions and patterns that comprise the processes. For instance, von Glasersfeld described rec-ognition as requiring, “the attentional select-ing, grouping, and coordinating of sensory material that fits the composition program of the item to be recognized” (Glasersfeld 1991: 5). That is, recognition is not a process in and of itself; rather, recognition entails particular mental actions and assimilation based on previous experience. Likewise, von Glasersfeld considered re-presentation to involve the enactment and coordination of particular mental actions, but that re-presentation differs from recognition in that re-presentation involves the enactment and coordinating of mental actions without the available sensory material that first gave rise to the mental actions (Glasersfeld 1991). it is for this reason that von Glasersfeld consid-ered re-presentation to involve a higher level of abstraction than recognition.

1 | For the purpose of this commentary, i use mental actions to refer to both mental operations and mental actions. Mental operations are those mental actions that are reversible.

« 4 » Compatible with and informing von Glasersfeld’s treatment of the aforemen-tioned processes, Jean Piaget characterized thinking and learning in terms of dynamic processes and organizations of mental ac-tions (Piaget 1977). For instance, Piaget de-scribed different kinds of abstraction based on the locus of the entailed schemes of mental actions. two kinds of abstraction are pseudo-empirical abstractions and reflective abstractions. Whereas pseudo-empirical ab-stractions are based on the results of activity (both in the physical and the mental sense), reflective abstractions are based in the coor-dination and re-presentation of the activity itself. Because of these differences, Piaget necessarily provided models of children’s thinking at the level of mental actions in or-der to distinguish between kinds of abstrac-tion when describing children’s thinking and learning.

« 5 » over the past few decades, numer-ous mathematics educators have conducted research in ways that are attentive to Piaget and von Glasersfeld’s sensitivity to mental actions. Leslie steffe’s2 body of work on stu-dents’ fractional and multiplicative reason-ing provides an apropos example (see steffe & olive 2010 for a comprehensive overview). although steffe’s work includes a focus on

2 | steffe has worked with numerous col-leagues in this area, including Paul Cobb, amy Hackenberg, anderson norton, John olive, Erik tillema, Patrick Thompson, and Catherine ul-rich, to name a few. For brevity’s sake, i only refer to steffe.

students’ abstraction with respect to their fractional knowledge, i interpret his primary objective being the characterization of the organization of mental actions entailed at each stage or level of abstraction (e.g., vari-ous levels of units coordination). it is from his descriptions at the level of mental ac-tions and coordination that he has been able to hypothesize abstractions that take place during students’ units coordination devel-opment. in other words, his descriptions of mental actions and coordination enable him to provide models of what it is that students re-present and abstract.

Quantitative structures and abstraction« 6 » a line of research that i find rel-

evant to Cifarelli and sevim’s work, as well as including a focus on mental actions critical to abstraction, is that which characterizes students’ quantitative reasoning (Thompson 1993). Work in this area (Castillo-Garsow 2012; Ellis 2007; Johnson 2012) charac-terizes students’ thinking and learning in terms of their constructing and abstracting measurable attributes (e.g., quantities) and relationships between these attributes. as an example, Patrick Thompson (1994) char-acterized relationships between rate and the abstraction of constructing and coordinat-ing proportionally accumulating quanti-ties. in short, Thompson argued that a stu-dent reflectively abstracted notions of rate through repeatedly (re)constructing and coordinating quantities.

c

QA

c

QA–b

QB=a·QA QB=a·QA=

b

Figure 1: A hypothetical quantitative structure.

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« 7 » returning to Marie, a possible explanation for her construction of an in-creasingly abstract conceptual structure is that she had repeatedly constructed and coordinated particular quantities and re-lationships that resulted in an increasingly abstract quantitative structure (Thompson 1994). Figure 1 provides a hypothetical model of this quantitative structure. The quantitative structure (Figure 1) is com-posed of two lengths, QA and QB, beginning and ending at (potentially) different refer-ence points, thus forming the lengths of b and c to account for the amount by which QA and QB exceed QB and QA, respectively. also, QB is known to be equivalent to a times the length QA. From here, it can be deduced that QB is equivalent to the sum of c and the difference between QA and b (e.g., Figure 1, right, and Marie’s equation of x – 15 + 50 = 2x in task 4).

« 8 » in characterizing Marie’s solution activity on task 1, Cifarelli and sevim de-scribe that she constructed a diagram that “aided her understanding of the situation” (§27). one way to explain how the diagram “aided her understanding of the situation” is that Marie constituted her diagram in terms of quantities and relationships be-tween quantities that she then symbolized using equations and variables, whereas her initial approach to the problem was entirely symbolic and devoid of imagined quantities and relationships. as Marie then progressed to subsequent tasks, during which she seemed to spend more time fo-cusing on her image of the situation and its quantities (§28), she was able to recognize that she was constructing similar quanti-tative structures. That is, reconstructing, recognizing, and re-presenting quantita-tive structures enabled her to construct a sense of invariance or similarity across the tasks, eventually leading to her anticipat-ing the implications of this structure (e.g., solutions, calculations, and the necessity of particular information and values).

« 9 » Echoing von Glasersfeld’s ap-proach to recognition and re-presentation, i find it important to emphasize that the assimilation of a situation to a quantita-tive structure is not a passive absorption of information (Thompson 2013). rather, assimilation to a quantitative structure in-volves constituting the situation such that

it entails that quantitative structure and its implications, with such activity look-ing different at various levels of abstraction (e.g., the extent that the assimilation entails anticipatory actions). Hence, each time Marie came to a new situation, she had to conceive the situation so that it entailed a quantitative structure, and it was through this repeated process that she constructed an increasingly abstract and anticipatory structure.

coordinating foci« 10 » research on students’ mathe-

matical thinking and learning encompasses a number of foci ranging from specifying students’ mental actions to characterizing general activity and learning processes. regardless of the primary foci of research, i contend that inquiry into student think-ing and learning must be sensitive to the particular mental actions at play. if not, we can end up taking structures and processes for granted as opposed to being inherently tied to the organization of mental actions that constitute the structures and processes (see dawkins 2012 for a discussion of this phenomenon).

« 11 » an important area moving for-ward is the determination of systematic ways to coordinate research foci in order to construct more extensive models of stu-dent thinking and learning. For instance, there is a significant body of literature on students’ problem solving processes, with several researchers providing frameworks by which to categorize students’ activity (e.g., Carlson & Bloom 2005; Pólya 1973). an open question thus becomes: Can we create relationships between problem solv-ing frameworks and those constructs that are more fine-grained in terms of men-tal actions and processes of mathematical thought? For instance, there seems to be a natural connection between Carlson and Bloom’s conjecture-imagine-evaluate cycle, students’ quantitative reasoning, and their abstraction of conceptual structures; a stu-dent that has abstracted particular quanti-tative structures so that these structures are anticipatory might be supported in men-tally running through conjecture-imagine-evaluate cycles. as another example, it seems intuitive that students’ cycling back or checking activity involves processes of

re-presentation and reflection. What men-tal actions might support such processes as being more productive than not in the moment of problem solving? The answers to these questions will no doubt further our understanding of student thinking and learning.

Kevin c. moore is an Assistant Professor of Mathematics Education. His research focuses

on student cognition in the area of pre-calculus and calculus ideas. His work includes exploring

students’ quantitative and covariational reasoning in the context of angle measurement, trigonometry,

and the use of multiple coordinate systems.

received: 25 May 2014 accepted: 27 May 2014

mathematical modeling and the nature of problem solvingCarlos W. Castillo‑GarsowEastern Washington University, USA ccastillogarsow/at/ewu.edu

> upshot • Problem solving is an enor-mous field of study, where so‑called “problems” can end up having very little in common. One of the least studied cat-egories of problems is open‑ended math-ematical modeling research. Cifarelli and Sevim’s framework – although not de-veloped for this purpose – may be a use-ful lens for studying the development of mathematical modelers and researchers in applied mathematics.

« 1 » victor Cifarelli and volkan sevim’s target article bears superficial similarity to some of my own work in studying stu-dent’s mathematical development (Castillo-Garsow 2010, 2012, 2013; Castillo-Garsow, Johnson, & Moore 2013). We both study the mathematical development of individual students engaging in structured series of mathematical tasks; however, applications differ. My work focuses on the development of particular mathematical tools for model-ing, while the authors propose a framework (§46, table 3) for how a student might ex-pand the scope of a problem to see other

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problems as similar. The authors’ frame-work is one that i have wanted for a very long time, and i am pleased to see it here.

« 2 » on the other hand, i have always found the phrase “problem solving” to be troublesome. What i would like to do in this commentary is discuss my reservations about the phrase “problem solving” and argue that the results of Cifarelli & sevim have broader applications to critically un-derstudied areas of mathematics education such as open-ended research modeling.

distinguishing research modeling and problem solving« 3 » There is no commonly accepted

definition of “problem solving.” in §15 the authors take the perspective of Leslie steffe, saying that “problem solving” involves a sit-uation, a goal, and that there is “no proce-dure in the concept to reach the goal.” Bor-rowing from John dewey (1910: 9), i will call this last aspect a “perplexity” and refer to this meaning of problem solving as the situation-goal-perplexity meaning of prob-lem solving. This meaning is compatible with the authors’ work, but there are other meanings of problem solving also compat-ible with the authors’ work.

« 4 » For example, Frank Lester defined “problem” to mean “a situation in which an individual or a group is called upon to perform a task for which there is no read-ily accessible algorithm which determines completely the method of solution” Lester (1978: 54). Lester’s definition implies that for a student to engage in “problem solving” there must be an external task, the student has to believe that the task is a problem, and the student believes that there is a solution (a yet unknown, but recognizable stopping point, which Lester also calls the “out-come”). This task-problem-solution mean-ing of “problem solving” is also compatible with the tasks described in the authors’ work, in my own work involving structured tasks, and other literature that focuses on problem solving, including Marilyn Carl-son and irene Bloom (2005) and George Pólya (1973).

« 5 » The task-problem-solution mean-ing differs from the situation-goal-perplex-ity meaning in two ways. The first is the dis-tinction between a “solution” and a “goal.” a goal may encompass finding a solution,

finding multiple solutions, making progress towards a solution, or even a goal entirely unrelated to solutions (such as improved understanding). two examples are the “open-ended” problems such as those de-scribed by Jerry Becker and shigeru shima-da (1997) or the “Model Eliciting activities” described by richard Lesh and Guershon Harel (2003). These tasks do not typically have a single recognizable stopping point.

« 6 » The second distinction is between “task” and “situation.” Lester (1978: 54) de-scribes a task as being externally imposed, and in all the above cited examples, the task is determined by the instructor or research-er, rather than by the student. However, the situation-goal-perplexity meaning encom-passes a much larger class of problems. i will illustrate with an example from Mitchel resnick (1997: 68–74).

« 7 » resnick describes two high school students, ari and Fadhil, who were work-ing with the agent-based modeling program StarLogo at the same time as they were en-rolled in driver education. ari and Fadhil developed a curiosity: they wanted to know what caused traffic jams. using StarLogo, ari and Fadhil developed several simula-tions of drivers on a highway, and explored driver behaviors that contributed to or eliminated traffic jams. although ari and Fadhil did not succeed in controlling their simulated traffic jam, they discovered quite a bit about traffic jam behavior: that traffic jams moved as waves in the direction op-posed to traffic, and that starting all cars at the same initial speed did not prevent a traf-fic jam, so long as the cars were unevenly spaced.

« 8 » Both the case of Marie and the example of ari and Fadhil involve model-ing in the sense of students thinking about quantities, measurements, and relations between quantities. Both involved imagin-ing fictional situations and describing them mathematically. However, ari and Fadhil’s project differs from Marie’s work in that ari and Fadhil were pursing their own curiosi-ties. unlike Marie, ari and Fadhil were en-gaged in mathematical research.

« 9 » This activity is what i mean when i say “research modeling.” Based on ari and Fadhil’s example, i would propose three criteria for identifying research modeling activity:

1 | The problem originates with the student,2 | it is based on their non-mathematical

experience, and3 | The goal of the activity is understanding,

not a solution.This type of experience is compatible with steffe’s situation-goal-perplexity meaning of problem solving, but it differs greatly from the task-problem-solution meaning. in contrast, Marie’s work was contained entirely within the task-problem-solution meaning.

Research modeling in mtBi« 10 » another example of research

modeling comes from my own work with the Mathematical and Theoretical Biol-ogy institute (MtBi) and the institute for strengthening the understanding of Mathematics and sciences (suMs; Cama-cho, Kriss-Zaleta & Wirkus 2013; Castillo-Chavez & Castillo-Garsow 2009; Castillo-Garsow, Castillo-Chavez & Woodley 2013). MtBi is a summer research Experience for undergraduates (rEu) in mathemati-cal biology, and suMs is its partner sum-mer program for high school students. By encouraging students to return as advanced students, peer-mentors, tutors, and instruc-tors, MtBi/suMs serves as a mentorship pipeline in the mathematical sciences that reaches from high school to tenure.

« 11 » MtBi/suMs has been extraordi-narily successful at recruiting underrepre-sented minorities (urMs) and developing them into mathematical researchers (Cas-tillo-Garsow, Castillo-Chavez & Woodley 2013). as of February 2012, 69 u.s. citizen alumni of MtBi have completed a Ph.d. and 54 (79%) were urMs. Most of these degrees have been recent. Based on job posting data, Castillo-Garsow, Castillo-Chavez & Woodley (2013) estimated that MtBi alumni have been awarded between 1.6% and 4.3% of recent Ph.d.s in applied mathematics and 50% of all mathematical biology Ph.ds awarded to u.s. Latin(o/a)s since 2005.

« 12 » MtBi is an 8-week program. The first three and a half weeks consist of lec-tures and homework in population biology. The students study difference and differen-tial equations, statistics, stochastic models, agent-based modeling, and computer simu-lation. The structure of these weeks roughly

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follows Fred Brauer and Carlos Castillo-Chavez (2012), supplemented by guest lec-turers. This portion of the program follows the task-problem-solution model.

« 13 » The second half of the program is student-driven group research projects. in MtBi’s initial prototype year (1996), the projects were assigned as tasks. in sub-sequent years, students were expected to design their own projects, while gradu-ate students and faculty served in advising roles. This portion of the class follows the situation-goal-perplexity model. Because students choose their topic of study, they frequently know more about the situation than the mentors. The mentors contribute mathematical and modeling experience, but rarely situational knowledge (Cama-cho, Kriss-Zaleta & Wirkus 2013; Castillo-Chavez & Castillo-Garsow 2009).

« 14 » one of the most notable aspects of these student projects is their variety. al-though trained as population biologists and mathematical epidemiologists, students have applied these techniques to a broad range of interests including: the three-strikes law (seal et al. 2007), gang recruit-ment (austin et al. 2011), education (Boyd et al. 2000; diaz et al. 2003), immigration (Catron et al. 2010), political third party formation (romero et al. 2011), mental illness (daugherty et al. 2009; dillon et al. 2002), pollution (Burkow et al. 2011), obe-sity (Evangelista et al. 2004), drug use (or-tiz et al. 2002; song et al. 2006), and even MtBi itself (Crisosto et al. 2010). suMs students work with a reduced curriculum in a similar environment, and similarly extend their mathematical biology tools to interests such as traffic, aerodynamics, and education.

« 15 » MtBi highlights a critical dis-tinction between these examples of research modeling (ari and Fadhil, MtBi/suMs) and Marie: the scope of the abstraction. Ma-rie abstracted a scheme of a particular task type and was able to assimilate new tasks to that scheme, as well as generate new tasks of that type. These research modeling students abstracted mathematical schemes that as-similate not tasks, but their own interests. to MtBi students, bulimia and tick-host interactions are the same “type” of problem (both applications of systems of ordinary differential equations). The way they see

their own world has been mathematized. For any curiosity they have, MtBi students will check if the mathematical tools they learned are appropriate to exploring that curiosity.

sequencing tasks for building modeling« 16 » it is in the context of this distinc-

tion between engaging in tasks and explor-ing a curiosity that i want to speak about the potential ramifications of Cifarelli & sevim’s work. Julie Gainsburg (2006) suggested that these two classes of activities have very dif-ferent challenges for students, and that work in K-12 tasks may not adequately prepare students for “adult” modeling. a question i want to explore is: Can sequences of tasks develop problem solving skills that encom-pass not just task-problem-solution prob-lem solving, but also the larger world of situation-goal-perplexity problem solving, including research modeling?

« 17 » anecdotally, the answer appears to be yes, although that “yes” may be quali-fied by yet unexplored factors. MtBi/suMs students begin working with structured tasks and later appear to assimilate a vari-ety of situations to the schemes they have abstracted from these structured tasks. The process could very well be quite similar to the process of recognition, re-presentation, and reflective abstraction that the authors describe in §46.

« 18 » However MtBi/suMs and simi-lar rEus are little studied by mathematics educators, and the reasons for MtBi’s suc-cess are not well understood (Castillo-Gar-sow, Castillo-Chavez & Woodley 2013). to my knowledge, the closest that anyone has come to a radical constructivist study of a mathematical biology rEu is Erick smith, shawn Haarer, and Jere Confrey’s (1997) study of a graduate level mathematical biol-ogy class. so while it is possible that MtBi students follow a similar trajectory of as-similation as Marie, this development has never been observed, because no radical constructivist has been around to assimilate it. Furthermore, it is unclear exactly how the tasks MtBi/suMs students engage in are different from the tasks that Marie engaged in. The resulting abstractions appear to be different, but what are the reasons for those differences?

conclusion« 19 » since its foundation, construc-

tivism has had a history of careful study of students’ understandings by way of precisely designed tasks (e.g., Piaget 1967, 2001). These studies have provided tre-mendous insight into the development of the mathematics of students. But the field has grown enough that it is time for con-structivists to give up some of that con-trol, and apply our lenses to the careful study of the mathematics of researchers, including student researchers. We need to extend our studies beyond the set of task-problem-solution problems to the broader set of situation-goal-perplexity problems by making a large-scale concerted effort to understand problem solving in the relative complement.

« 20 » The large-scale study of math-ematical modeling research programs such as rEus by multiple groups of radi-cal constructivists is critical both to the development of these rEus and to the ad-vancement of the constructivist study of mathematics education. as it stands, we simply do not know how mathematical re-searchers are formed, or how the develop-ment of mathematical researchers might be promoted in the lower grades. i would urge myself, the authors, and all interested read-ers to form collaborations with mathemati-cal scientists and applied mathematicians for the purpose of studying precisely how mathematical researchers are formed using tools such as Cifarelli & sevim’s framework.

carlos castillo-Garsow is an MTBI alumnus. He began his training in mathematical biology and

linguistics before completing a Ph.D. in mathematics with a focus on mathematics education. He now

combines these disciplines to study the conceptual semantics of students’ modeling activities.


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a Fine conceptual analysis needs no “ism”Gerald A. GoldinRutgers University, USA geraldgoldin/at/dimacs.rutgers.edu

> upshot • The key philosophical premise of von Glasersfeld’s radical constructiv-ism is not necessary to the insightful conceptual analysis presented by Cifarel-li and Sevim, which could benefit from abandoning it.

« 1 » as victor Cifarelli and volkan se-vim note (§1), what makes Ernst von Gla-sersfeld’s constructivism radical is his a priori rejection of any interpretation of knowledge as objective. indeed,

“ Constructivism drops the requirement that knowledge be ‘true’ in the sense that it should match an objective reality. all it requires of knowl-edge is that it be viable, in that it fits into the world of the knower’s experience, the only ‘reality’ acces-sible to human reason.” (Glasersfeld 1996: 310)

« 2 » This premise has drawn intense criticism from many educational researchers, myself included (Goldin 2003a), as having negative consequences for both mathematics education research and practice. First, infer-ences drawn from it limit severely the con-structs that can be used to understand and interpret empirical observations of the kind reported by Cifarelli and sevim. second, by devaluing truth and objectivity and replac-ing validity by viability, it greatly diminishes the influence that important empirical find-ings can have on education policy.

« 3 » However, radical constructiv-ists’ qualitative research and analyses have provided, and continue to provide, some extremely valuable exploratory and descrip-tive findings. These should tend to support policies and practices that are far more pro-gressive and sophisticated than those being implemented now across the united states as test-score-oriented goals for school math-ematics are increasingly mandated. research findings, as their validity and generalizability are established, point toward placing greater emphasis on relational understanding rela-tive to instrumental understanding (skemp

1976), and toward more complex and more qualitative assessment methods.

« 4 » Thus, as i have maintained for some time (Goldin 2003b), it is long over-due for us to question whether the methods, findings, and analyses in this research actu-ally require acceptance of rC’s ultrarelativist epistemology. i would argue that much of the research is equally compatible with scientific perspectives generally taken in mathemat-ics, physics, cognitive science, and empirical psychology. although i am fully aware that many constructivist researchers regard rC as essential to their programs of research, in my view it actually limits the research severely.

« 5 » The present report by Cifarelli and sevim serves as an example for consider-ation. They present an excellent, fine-grained conceptual analysis of problem-solving ac-tivity by one student, Marie, based on von Glasersfeld’s notion of re-presentation, at-tending especially to her reflections (§22) and to her conceptual structures and antici-pation (§23). They do more than provide im-portant empirical evidence of the complex-ity of Marie’s learning and problem solving. They also suggest some valuable theoretical distinctions that may help one to understand their observations, with wider implications if the inferences they suggest are validly gener-alizable to other learners.

« 6 » With this in mind, i read carefully the conceptual analysis of Marie’s problem solving activity, asking throughout whether one needs rC epistemology at all to make the observations or to carry out the analysis. i think the conclusion is straightforward. The choice that Cifarelli and sevim make to study “idiosyncratic internal constructions, which … are formed by re-presentation of prior ex-periences” (§50) was clearly motivated by the claim that they “do not stand for any objects.” But it does not in any way depend on that claim. The study of “re-presentation,” while inspired by von Glasersfeld, need not entail adoption of the view that this is the only pos-sible kind of representing relationship.

« 7 » one does not require rC as one’s motivation to consider Marie’s internal constructions, reconstructions, reflection processes, anticipations, and representa-tions as worthwhile theoretical constructs, or to draw inferences about them from her observed statements, behavior, and produc-tions. one can do quite the same analysis of

Marie’s problem solving without von Gla-sersfeld’s radical tenet. and without rC, there is no need for or value in rejecting con-ventional notions of truth, correctness, cor-respondences between internal and external representations, and so forth. The way is then clear to introducing additional theoretical constructs regarding representation into the analysis (Goldin 1998; Goldin 2003b).

« 8 » For example, one may seek to char-acterize structures in external, real-world objects, configurations, representations, and systems of representation, and allow the ex-amination of dynamically evolving, two-way representing relationships between the inter-nal and the external. Let us elaborate briefly on this.

« 9 » Without the rC tenet, Marie’s own inscriptions – equations, diagrams, numer-als, and written words reproduced pictori-ally in §§25, 26, 27, 29, 32, 36, and 38 – may usefully be considered as real-world objects about which Marie has partial knowledge. in these inscriptions, one may characterize semantic relationships beyond the meanings that Marie herself might attribute to them at the time she created them: relationships that are “there to be discovered,” and that Marie may or may not in fact later discover.

« 10 » in addition, one may consider real-world lakes, altitudes, cities, distances, storage drums, and so forth as objects of the kind that the problems are “about.” These have attributes and structural properties that exist apart from Marie’s experiential world. some of Marie’s re-presentations have to do with her partial representation of those real world properties – again, relationships are “there to be discovered.”

« 11 » External representational systems also include standard mathematical and pedagogical systems, such as our systems of base ten numeration and algebraic nota-tion. These are based, of course, on socially-agreed conventions; but having been created, they incorporate mathematical relationships “outside the mind” – in particular, outside Marie’s mind. nevertheless, some of her mental processes can be usefully examined with respect to such relationships. When major discrepancies occur, the notion of “misconception” rejected in §49 becomes a valuable descriptive construct.

« 12 » Most important, in addition to augmenting the theoretical possibilities for

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Radical constructivism and critical learning theory Karen François



understanding learning and problem solv-ing, abandonment of the “ism” opens the door to achieving conventional, time-tested scientific goals. Mathematical cognition and learning themselves incorporate real-world phenomena, about which we seek objective, scientifically valid knowledge. This is gained by replicating and generalizing exploratory findings, testing and confirming or discon-firming hypotheses, validating inferences drawn, and so forth.

« 13 » in short, the fine conceptual anal-ysis in this article needs no “ism” to justify itself or to motivate its constructs. rather, it may be regarded as a valuable contribution to the exploratory and descriptive stage in the enterprise of conducting objective scien-tific research.

Gerald Goldin, Distinguished Professor of Mathematics Education, received his Ph.D. in physics from Princeton

and studied mathematics education at Penn, He co-chaired the PME Working Group on Representations,

organized New Jersey’s Statewide Systemic Initiative, and directed two research centers at Rutgers. He

publishes on education, mathematics, and physics, and is a Humboldt Prize awardee. His current investigations

include mathematical affect and engagement.


convergences between Radical constructivism and critical learning theoryKaren FrançoisVrije Universiteit Brussel, Belgium karen.francois/at/vub.ac.be

> upshot • The value of Cifarelli & Se-vim’s target article lies in the analysis of how reflective abstraction contributes to the description of mathematical learning through problem solving. The additional value of the article lies in its emphasis of some aspects of the learning process that goes beyond radical constructivist learning theory. I will look for common ground between the humanist philoso-phy of mathematics and radical con-structivism. By doing so, I want to stress

two converging elements: (i) the move away from traditionalist ontological po-sitions and (ii) the central role of the stu-dents’ activity in the learning process.

« 1 » The authors’ examination of the role of re-presentation in mathematical problem solving as an application of Ernst von Glasersfeld’s conceptual analysis serves to connect the philosophical discussion on knowing and learning and the philosophi-cal debate on the ontological status of things (§1). They do not engage in a philosophical examination of Glasersfeld’s view of con-structivism as radical and instead refer to his 1984 paper, in which von Glasersfeld takes a position in the ontological debate (§1). to von Glasersfeld (1984), knowledge does not reflect an ontological reality. instead, it results from an ordering and organization of experiential reality. The authors (§5) fo-cus on the second part of von Glasersfeld’s non-ontological definition of knowledge and do not consider the relation between the ontological assumption and its possible impact on the learning process. They bridge the philosophical discourse and the concep-tual analysis of a goal-directed activity in the context of problem solving by concentrating on the constitution of mathematical knowl-edge by experience, leaving aside the onto-logical issue. The authors go beyond ontolo-gy, and focus on the cognitive actions of the learner in resolving mathematical problems.

« 2 » it was reuben Hersh (1997) who argued for the connection between the phi-losophy of mathematics and the teaching of it. Hersh stated that the ontological position of a mathematics teacher or the ontological position of mathematics as reflected in the curriculum influences the learning process. Humanist philosophy of mathematics takes the teachability of mathematics as a central concern. Hersh (1997: 182) calls his “own slant on humanist philosophy of mathemat-ics” “social-cultural-historic” or just “social-historic.” He tries to link up mathematics with people, society, culture and history. instead of a Platonist ontology, a humanist conception of mathematics brings math-ematics back to Earth as a human activity that is embedded in a historical and cultural environment. it narrows the gap between students and the subject matter so that stu-dents can realize that they are taking part in

the practice of mathematics. students can learn and understand mathematics and they can elaborate on it, going from practical ba-sic skills to theoretical abstract reasoning.

« 3 » The humanist philosophy of math-ematics and of mathematics education is fully adopted by critical learning theories. These theories emphasise the learning of mathematics as a right for all students ir-respective of their social and cultural back-ground. The learning and knowing of math-ematics is seen as a stepping stone for a future career, for the development of society, and as a basis for an informed and critical citizenship. The mathematician and educa-tor ubiratan d’ambrosio – who can be seen as the founding father of ethnomathemat-ics – pleads for educational reform and for more attention to students and teachers as human beings. For d’ambrosio (1990), we have to realize that mathematics and other scientific disciplines are epistemological sys-tems that are embedded in their socio-cul-tural and historical perspective. These bod-ies of knowledge are not finished and thus do not consist of static entities of results and rules. From a humanist philosophical point of view, teachers should accept, understand and access their students’ social and cultural background knowledge as a starting point for the learning and doing of mathemat-ics. Critical voices from the research field of mathematics education (such as Hersh & John-steiner 2011) propose to look at mathematics as the study of mathematical activities, and hence as a complex of social and cultural manifestations, in a move away from traditionalist positions (e.g., against the still dominant Platonic view).

« 4 » For these critical voices, the theo-retical starting point is the recognition of a factual variety of mathematical practices, including “western” academic mathemat-ics. “Western” academic mathematics has a particular setup of concepts and practices of and about the world and thus is, at its ba-sis, a sociocultural complex like any other cultural tradition. This point of departure is consciously and explicitly taken into ac-count when devising curriculum material and educational strategies by researchers from critical research fields in mathematics education. it is of great importance in the learning process that students will under-stand and will take insightful steps toward

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convergences between Radical constructivism and critical learning theory Karen François



Framing our Research in a Radical constructivist tradition Joy W. Whitenack


the solution of a problem. Whatever their diverging ways of problem solving, these ways can be esteemed to be the preferential ways to start with since these ways are most likely based on insights for the students at work. it is thought that solving problems will give self-confidence to the students, which, in the end, is a major issue (Hersh & John-steiner 2011).

« 5 » so far, the humanist philosophy of mathematics has not been adopted by radi-cal constructivism. although radical con-structivism and critical learning theories come from different traditions, there are some points of convergence between them. Based on an analysis of Cifarelli & sevim’s target article, i want to stress two main points of convergence: the move away from traditionalist ontological positions and the central role of the students’ activity in the learning process.

« 6 » The first point is the move away from the traditionalist ontological Platonic view of mathematics as a static and uni-fied body of mind-independent knowledge. radical constructivism not only deals with the process of cognitive construction, “but also with the relationship which the result of any such construct might have to the real-ity of the traditionally presumed ontologi-cal world” (Glasersfeld 2007: 22). Moreover, that relationship is “the very point in which radical constructivism is revolutionary” (ibid). reality is defined as experiential re-ality to stress the confirmation of one’s own construction of the world and not, “in spite of what is often tacitly assumed, knowledge of a world that might have independent on-tological existence” (ibid: 28). This claim is a clear point of convergence with critical learning theories in that both theories reject traditional ontological positions. Mathemat-ics learning has nothing to do with discover-ing a universal, static and ready-made body of mind-independent knowledge. Both theories have in common that mathematics is a product of human activity, an activity that has to be at the center of the learning of mathematics. Learning mathematics is do-ing mathematics, as extensively demonstrat-ed in the target article (§§3, 5, 28, 47, 48, 52).

« 7 » There is a debate about the differ-ences between radical constructivism and social constructivism and how they differ on the ontological status of “society” as another

construct or as a reality (Hersh 2008). How-ever, the outcome of this discussion will be of little or no use for the teacher because the ontological claim of both theories is similar. The most important idea both theories have in common is the emphasis on the learner as an active player in the learning process. This might be the reason why the authors do not deal with ontological aspects and the possible impact of the ontological position of a mathematics teacher or the ontological position of mathematics as reflected in the curriculum on the learning process.

« 8 » The second point of convergence between radical constructivism and critical learning theories concerns the central role of the students’ activity in the learning pro-cess. in §47, Cifarelli & sevim refer to von Glasersfeld’s ideas on learning as a construc-tive activity and the important role of re-flecting on one’s own actions. The students’ constructive activity and the reflection on it is the starting point for the learning of math-ematics. This reflection is a mental activity by which students can critically examine mental re-presentations of prior experiences that function as conceptions. Based on this mental re-presentation in doing mathemati-cal activities in the context of problem solv-ing (as is explained in the article), students can transform mental representations into more abstract cognitive structures. students are learning mathematics by doing math-ematics in their own way, using their own mental representations as viable construc-tions within their own experimental world. The learners can construct their own ways of practicing mathematics.

« 9 » Critical learning theories also present the learner as an active participant in the learning process. More specifically, Critical Mathematics Education (CME), as developed by skovsmose & Borba (2004), put the student’s activity at the center of the learning process, although not from the perspective of the learner as an indi-vidual. CME is concerned with the social and political aspects of the learner and of the learning of mathematics. Therefore the social environment of the learners – their background, the class, the school and, in the end, the whole society – are taken into account as meaningful aspects of the learn-ing process. The background and the fore-ground of the learner are two key concepts

within CME. “Background knowledge” means what the learners bring to the class-room, while the “foreground” is to be un-derstood as “[t]he set of opportunities that the learner’s social context makes accessible to the learner to perceive as his or her possi-bilities for the future” (vithal & skovsmose 1997: 147). CME emphases the political and cultural situation as an important aspect of the background and of the foreground, since they provide – or block – opportuni-ties for the learner. CME is also concerned with the use and function of mathematics in practice and with “the development of critical citizenship” (skovsmose & Borba 2004: 207) since mathematics also has a for-matting power that influences human be-ings and society. This suggests that despite their theoretical differences, these frame-works may have convergent implications for teaching.

« 10 » However, there is also a large gap between radical constructivism and critical learning theories, especially concerning the social dimension of the learning process. alan Bishop (1985) distinguishes five signif-icant levels within the research on the social dimension of mathematics education, go-ing from a macro perspective (culture) to a micro perspective (the individual). The first one is the cultural level, which emphasizes the way in which the history and the de-velopment of mathematics is embedded in culture. The second one is the societal level, which investigates the influences of differ-ent educational institutions in society. The third level is the institutional level, includ-ing the school system and the (hidden) cur-riculum. With the fourth level, we enter the classroom. it is with this pedagogical level that the didactics of mathematics education is concerned. Finally, Bishop (ibid: 4) points to the individual level as a research domain of the sociological study of mathematics education. The focus at this level is on the learner from a social perspective. For critical learning theories, these social dimensions have a direct impact on the individual op-portunities for the learning of mathematics. These five levels are taken into account to set up the learner’s activity that is at the center of the learning process. radical constructiv-ism also takes the activity of the leaner as a starting point but it does not consider the social embedding of it.

convergences between Radical constructivism and critical learning theory Karen François


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Framing our Research in a Radical constructivist tradition Joy W. Whitenack



« 11 » Clearly, there are differences as to how theoretical perspectives deal with the learner and the environment. While most authors emphasize their differences, in my commentary i have emphasized some strik-ing convergences based on the analysis of Cifarelli & sevim’s target article.

Karen François is an Assistant Professor of Philosophy of Science at the Free University Brussels (VUB).

She received her Ph.D. in philosophy focusing on the philosophy of mathematics and mathematics education.

received: 28 May 2014 accepted: 5 June 2014

a case for Framing our Research in a Radical constructivist traditionJoy W. WhitenackVirginia Commonwealth Univ, USA jwwhitenack/at/vcu.edu

> upshot • In this commentary, I address the viability of conducting constructivist teaching experiments to develop models of students’ conceptualizations. I also discuss how this research tradition has been adapted by researchers to conduct classroom teaching experiments. In my concluding remarks, I address the need for researchers to develop models for teacher learning.

« 1 » Because radical constructivism is a theory of knowing, research programs address questions that arise from what it means for an individual to know a thing and how the individual comes to know this thing. researchers following this tradition talk of the concepts that individuals build, how they re-present those concepts and, more generally, what the nature and qual-ity of those concepts or structures are. The first- and second-order models that are developed to understand individuals’ un-derstandings are particularly useful. Leslie steffe, Ernst von Glasersfeld, John richards and Paul Cobb’s (1983) counting types work is a case in point. This work, along with the work of others, has helped to shape our

views about children’s conceptual develop-ment (e.g., Harel & Confrey 1994; Fuson 1990; Thompson & Thompson 1994). These efforts are undeniably extremely important; they have provided an empirically ground-ed basis for us to have timely and much-needed conversations about children’s con-ceptual development.

« 2 » victor Cifarelli and volkan se-vim’s work addresses some of important ideas that have framed the field of math-ematics education for over four decades, first with the renewed interest in Jean Piag-et’s work and then through the insightful teaching experiment methodologies that have extended our understanding of chil-dren’s conceptions. These studies have cata-pulted the mathematics education research community into two (and other) important directions, exploring individuals’ under-standing of more advanced mathematical conceptions (e.g., Confrey 1991; Thomp-son & Thompson 1994) and understanding how these studies might be translated into making sense of teaching and learning in the mathematics classroom (e.g., Cobb et al. 1997; Yackel & Cobb 1996; stephan & rasmussen 2002; Whitenack & Knipping 2002; Yackel 2002). i will address both of these research programs in turn in my re-marks.

Radical constructivism as a viable stance« 3 » First, let me highlight a few key

theoretical ideas or assumptions that guide this type of theoretical tradition.1 | in order to understand and know a

thing, it is important that i experience some type of conflict (i.e., perturbation) in order to think about and, if necessary, incorporate this thing into my current thinking. a thing is only problematic if i deem it to be so.

2 | Building new ideas, making new con-nections and expanding my under-standing of a thing, that is, abstracting and/or assimilating ideas, continues throughout my life. i continue to reify these concepts or objects of some kind. This process is important with regard to understanding why i sometimes just do not understand a thing quite yet – and perhaps why there is hope that at some point in my experience i will.

« 4 » i also need to include a more general “rule of thumb” that guides radical constructivists’ thinking when working with children and their teachers. as von Glaser-sfeld (1995: 15) puts it, “whatever a student does or says in the context of solving a prob-lem is what, at this moment, makes sense to the student” (cf. Cobb & Wheatley 1988; steffe & Thompson 2000). and with regard to teaching, he states:

“ it may seem to make no sense to the teacher, but unless the teacher can elicit an explanation or generate a hypothesis as to how the student has arrived at the answer, the chances of modifying the student’s conceptual structures are mini-mal.” (Glasersfeld 1995: 15)

« 5 » These points are critical to how radical constructivists engage in their work. They also align with the theoretical perspec-tive that Cifarelli and sevim draw on to make sense of one student, Maria’s, prob-lem solving processes. and as it goes with radical constructivist studies, Cifarelli and servim make sense of Maria’s current expe-rience – how she reasons about these tasks, what is problematic for her, when it is neces-sary for her to re-present the problem situa-tions and so on.

« 6 » interestingly, Cifarelli and sevim (§7) explain that the “findings that they re-port are not from a study that tries to test hypotheses or build generalized models” – which are some of the aims of conducting constructivist teaching experiments (steffe & Thompson 2000). at the same time, they offer some insight into how the researcher might proceed if his goal was to build an emerging model per se. in fact, the work they report might serve as the necessary groundwork for framing a more extensive and intentional study that explains how students build conceptions and meanings associated with problem solving. steffe and Thompson in fact encourage researchers to engage in exploratory teaching prior to conducting a teaching experiment in or-der to understand the students’ “ways and means of operating” (ibid: 275). so per-haps the constructs outlined in this paper might be a first step in developing a model for problem solving that is tested, refined and replicated – one that offers empirically grounded evidence for the concepts and

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a case for Framing our Research in a Radical constructivist tradition Joy W. Whitenack



Reflective abstraction tracy Goodson-Espy

structures individuals build to solve these types of tasks sensibly.

« 7 » Therefore, these types of analyses continue to be important insofar as they help feed back into the researcher’s work and ultimately provide opportunities to build emerging theories, in this case about problem solving (e.g., steffe & Thompson 2000). (i will say more about the utility of these types of research studies in regular classrooms in the next section.)

« 8 » Cifarelli and sevim’s work is a snapshot of some of the research processes that radical constructivists might draw on to build models that explain individuals’ conceptualizations. Whether or not this work is a basis for building first- or second-order models will require much more, on-going work, as i understand neo-Piagetian research. For instance, the counting types model referred to by the authors was a lon-gitudinal study of children’s activities in which steffe and his colleagues began to test and build first- and second-order mod-els over a several-year period.

« 9 » What happens if we adapt the teaching experiment to answer questions about how students might build increas-ingly sophisticated conceptions in different settings, namely the regular mathematics classroom? My second set of remarks ad-dresses the viability of moving beyond the teaching experiment.

moving beyond radical constructivist traditions« 10 » addressing questions about how

children build these concepts in other set-tings, such as the regular classroom, has motivated researchers to shift their points of view away from considering solely psy-chological stances. in fact, models that de-scribe how children build conceptions in these types of social settings require them to consider other theoretical questions, in-cluding how children’s experiential activity is shaped by the social settings in which they participate or how students participate in and contribute to classroom mathemati-cal discussions, and what is the nature and quality of the individual and collective conceptualizations that they co-construct. Cobb explains this shift in researchers’ fo-cus as follows:

“ The growing trend to go beyond a purely cog-nitive focus is indicated by the increasing number of analyses that question an exclusive preoccupa-tion with the individual learner…in this emerging view, individual students’ activity is seen to be lo-cated both within the classroom microculture and within broader systems of activity that constitute the sociopolitical setting of reform in mathemat-ics education.” (Cobb 2000: 307)

« 11 » to this end, the classroom teach-ing experiment emerged as a plausible methodology for addressing questions that have contributed in part to our understand-ing of mathematics teaching and learn-ing (Cobb 2000). The shift to coordinating these theoretical stances to understand, say, how students collectively develop increas-ingly sophisticated concepts that are taken as shared by students and teachers alike of-fered researchers new methods for describ-ing locally situated mathematical activity in the classroom. This shift of course helps us address questions that were not the focus of earlier research programs. This is not to say that the constructs offered by radical constructivism were no longer useful or necessary. on the contrary, these constructs underpinned understanding how individual children’s conceptions may have become more sophisticated over time. in fact, Cobb and his colleagues continued to conduct in-terviews with individual children to deter-mine the quality and nature of the student’s numerical understandings using tasks often associated with neo-Piagetian research stud-ies.

« 12 » Cobb & Yackel (1996), for in-stance, attempted to coordinate children’s individual and collective conceptual devel-opment as they conducted classroom teach-ing experiments. By taking seriously tenets of radical constructivism and coordinat-ing this theoretical perspective with social learning theory, they began to unpack how one can explain children’s learning per se in classroom settings. and researchers have continued the work in other classroom set-tings – stephan & rasmussen (2002) and Whitenack & Knipping (2002), to name a few. again, these are research programs that explain how students and teachers establish a learning environment in which individual students can build or refine their current understandings. The tasks that the teacher-

researcher used in the teaching experiments are supplanted by developing instructional sequences in collaborations between the teacher and the researchers to understand how students might build more sophisti-cated understandings. and the process of developing and testing tasks, building the sequences and hypothesizing about student learning, as in the teaching experiment, is cyclic. as the sequence is worked out, tested and revised almost daily, working hypoth-eses feed back into and inform instructional design decisions that the researcher and teacher make (Cobb 2000). so practice and theory are thought to go hand in hand in that theoretical findings feed back into the teaching cycle and inform the sequence of activities that are developed and implement-ed in the classroom setting (Cobb). The re-searcher’s focus moves from how individu-als build concepts to how those concepts are understood collectively by the classroom community. Thus, the intentions of the re-searcher mirror the work that the teacher-researcher performed to some extent in the teaching experiments, but has at its setting the regular mathematics classroom.

« 13 » one of the pressing questions is how the regular classroom teacher develops an understanding of this work, and, if you will, builds conceptual entities that in turn feed back into her instructional practice. in other words, how might the researcher support the teacher’s ongoing professional learning as she engages in the classroom teaching experiment? How might the teach-er objectify the activities that she presents and build understandings about those in-structional materials that need to remain invariant, which representations to use and so on? additionally, what happens when the researcher’s work is done? Can the teacher replicate this work somehow? Put more plainly, how might the teacher proceed dur-ing the upcoming school year, assuming the researcher is no longer working collabora-tively with her? to address these questions, is it possible to adapt the classroom teaching experiment to study teacher’s conceptions? rather than primarily focusing on student’s individual conceptualizations as Cifarelli and sevim do, or students’ collective concep-tualizations as Cobb and Yackel do, research-ers could study the teacher’s conceptualiza-tions as they are situated in the mathematics

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classroom. Following this line of reasoning, the researcher could shift his focus to that of considering the teacher’s understandings (of her students’ conceptions, for instance) rath-er than addressing children’s conceptions per se. This research program would align more closely with Cifarelli and sevim’s approach, but their model would need to be adapted in order to take the teacher’s activity as the basis for developing a model of the teacher’s conceptualizations. to clarify my point, let us consider a situation that i encountered in my work with a second grade teacher. dur-ing one of our planning sessions, the teacher insisted that we incorporate a set of take-away subtraction tasks since her students would need to know how to solve these types of problems for the upcoming standardized test. i complied with the teacher’s request, realizing that these types of tasks were con-ceptually less rigorous than the ones that the students had been working through for sev-eral weeks (comparing and missing addend tasks). not surprisingly, students quickly completed the handout with take-away problems that the teacher assumed would take 15–20 minutes. although we discussed this issue at some length later, i wondered if the teacher understood the fundamental dif-ferences behind the two different tasks. in retrospect, was not this situation an oppor-tunity for the teacher to construct new un-derstandings? What questions might i have asked that would have led her to build some new understandings or to refine her current ways of thinking? to apply this research pro-gram to our work with teachers might be a useful next step in supporting them as they make sense of and capitalize on their stu-dents’ activity. and they would have an expe-rientially real basis for making changes in the types of questions, tasks and, more generally, decisions that they make in their daily work.

Joy Whitenack is Associate Professor of Mathematics Education in the Department of Mathematics and Applied Mathematics at Virginia Commonwealth

University. She teaches mathematics and mathematics education courses in the certification

and the mathematics specialist programs. Her research interests include children’s number sense,

instructional design and teacher leadership.


Reflective abstraction as an individual and collective learning mechanismTracy Goodson‑EspyAppalachian State University, USA goodsonespyt/at/appstate.edu

> upshot • Cifarelli and Sevim discuss the development of individual students’ abstract conceptual structures while problem solving, using constructs for analysis that are consistent with von Glasersfeld’s radical constructivism: re‑presentation and reflective abstraction. This commentary discusses the on‑going contributions of reflective abstraction to individual and collective learning.

« 1 » victor Cifarelli and volkan sevim’s case of Marie in §17 provides an example of a student engaging with a meaningful math-ematical task and developing abstract con-ceptual structures from her own problem-solving activity. From a radical constructivist (rC) perspective, Marie is not becoming aware of an external mathematical structure contained in the task; rather she is develop-ing an internal mental structure based on her own goal-setting and problem-solving. This commentary provides information concern-ing the reflective abstraction construct ap-plied in Cifarelli & sevim’s conceptual analy-sis of Marie’s activity and describes how it continues to play a role in spurring advances in mathematics education.

« 2 » a central concern is improving our ability to describe characteristics of stu-dents’ conceptual development and features of classroom culture that support it. re-searchers’ explanations of what they believe to be occurring during students’ learning experiences are based on written artifacts and analyses of classroom activities and individual interviews, as noted in Cifarelli & sevim’s observations in §2 concerning methodology. one of the more robust theo-retical constructs for describing students’ conceptual development is reflective ab-straction. This commentary provides a brief history of reflective abstraction, describes criticisms from the situated cognition per-spective, and describes modern approaches that address them.

a brief history of reflective abstraction in the mathematics education literature and its relationship to cifarelli & sevim’s study« 3 » reflective abstraction may help

explain the way that students construct con-ceptual knowledge. Ernst von Glasersfeld quoted John Locke (1690), “…so i call this reflection, the ideas it affords being such only as the mind gets by reflecting on its own operations within itself ” (Glasersfeld 1991: 45f). Jean Piaget (1970, 1985) explored issues involving a subject’s interactions with external objects and the subject’s internal mental operations. von Glasersfeld (1991, 1995) summarized Piaget’s three types of re-flective abstraction:1 | reflective abstraction;2 | reflected abstraction; and3 | pseudo-empirical abstraction.The first type refers to the subject’s ability to project onto a new level and reorganize a structure created from the subject’s activities and interpretations. The second type, called “reflected abstraction,” is important for un-derstanding higher-level cognitive develop-ment. reflected abstraction’s distinguishing characteristic is that not only is the subject able to project the structures created by his activities onto a new level and reorganize them; the subject is also consciously aware of what has been abstracted. reflected ab-straction thus refers to the subject’s meta-cognitive awareness.

« 4 » in order to encourage these reflec-tive abstractions, one has to place careful emphasis on offering students appropriate mathematical learning tasks. as noted by Cai & Cifarelli (2005) and Cifarelli & Cai (2005), these tasks may be well-structured (schoenfeld 1985), meaning there are clearly defined solution paths, or ill-structured (Kil-patrick 1987), which indicates a more open-ended problem situation. However, even in the case of a well-structured task, rC ad-herents would argue that the solver creates a unique mental structure through her own activity.

« 5 » Cifarelli (1988) defined levels of reflective abstraction attained by college students while solving algebra word prob-lems. The levels defined were: recognition, representation, structural abstraction, and structural awareness. one progresses

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through the levels in terms of whether a solver can:1 | recognize having solved a similar prob-

lem before;2 | re-use previous solution methods;3 | develop novel strategies that the solver

has not used previously;4 | anticipate sources of difficulty/promise

during the solution process when using a method previously applied;

5 | anticipate sources of difficulty/promise when using a new solution method;

6 | mentally run through methods used previously;

7 | mentally run through potential solution methods; and

8 | demonstrate conscious awareness of problem solving activities.

at the highest level, structural awareness, the problem structure created by the solver has become an object of reflection. The stu-dent is able to consider such structures as objects and is able to make judgments with-out resorting to physically or mentally rep-resenting solution methods.

« 6 » These levels seem to be supported in Cai & Cifarelli’s study (2005), which de-scribed college students’ problem-solving activities as hypothesis-driven versus data-driven. Hypothesis-driven reasoning strate-gies indicate that a student is able to formu-late a hypothesis for a possible solution, test it, and make judgments concerning what to do next based on the results, and that the stu-dent can, furthermore, make decisions based on anticipated results of potential hypoth-eses. data-driven activities indicated that a student can only act on data yielded from the results of a specific case. in the case of Marie in §17, we see an example of a student who is able to analyze a problem situation (task 1), develop an initial conceptual structure, and formulate and physically carry out solu-tion activity. What is significant about Ma-rie’s subsequent activity is that as she engages with tasks 2–4 and 9, she deals with varia-tions on the initial task and becomes able to invent and mentally run through potential methods of solution and make decisions based on these mental run-throughs with-out actually performing them (table 2 in the target article). This activity points to a higher level of reflective abstraction.

« 7 » a parallel yet differing approach to reflective abstraction was proposed by

Ed dubinsky. dubinsky argued for the use of Piaget’s reflective abstraction concepts in helping students develop advanced math-ematical thinking and advocated an instruc-tional approach to “induce students to make specific reflective abstractions” (dubinsky 1991: 123). dubinsky’s approach came to be known as aPos theory (actions-processes-objects-schema) and continues to be used in many studies on undergraduate mathemat-ics education that focus on the development of individual student’s conceptual under-standings of advanced mathematical ideas and the development of curricula to support them (arnon et al. 2014).

Resolving theoretical questions: new approaches to reflective abstraction« 8 » Martin simon and colleagues ex-

plain the difficulties inherent in applying the ideas of rC in an instructional setting, spe-cifically defining problems associated with engendering cognitive conflict. They define a learning mechanism as an elaboration of reflective abstraction, we “…attribute devel-opment of a new conception to a process in-volving learners’ goal-directed activity and natural processes of reflection” (simon et al. 2004: 318). The mechanism for conceptual learning defined involves mental activity, activity sequences, learners’ goals, and ef-fects. The learner’s activity is described as being constructive, rather than inductive, resulting from the learner reflecting on a pattern in the activity-effect relationship instead of reflection on a pattern in the out-comes. The work demonstrates how Piaget’s ideas of assimilation and accommodation work in conjunction with reflective abstrac-tion to define a learning mechanism. The paper also specifies how this information can be used for lesson design and defines a four-step process:1 | specifying a student’s current knowledge;2 | specifying the pedagogical goal;3 | identifying an activity sequence; and4 | selecting a task.

« 9 » abstraction has often been reject-ed by researchers who view it as a decontex-tualization process in which one gradually moves away from the empirical aspects of a situation where something was originally learned. Lynn Fuchs and colleagues describe abstractions as, “deleting details across ex-

emplars and avoiding contextual specificity in order to realize generalization” (Fuchs et al. 2003: 294). abstraction is then seen as requiring the process of decontextualiza-tion for generalized learning to take place or for knowledge to be “transferred” in the classic sense, and yet, decontextualization is deemed unacceptable because it separates knowledge from concrete experience (Beach 2003).

« 10 » Paul Cobb (cited in Lobato 2006: 440) argued that situated theorists should not dismiss abstraction on the basis of de-contextualization, rather they should en-gage in developing alternatives. several alternative conceptions of abstraction were discussed in a 2005 american Educational research symposium on “abstraction in Mathematics Learning: Comparing alter-native Emerging Conceptions.” These con-ceptions (among others) were defined as: (a) collective abstraction; (b) abstraction in context; and (c) actor-oriented abstraction.

« 11 » Collective abstraction was defined during a series of classroom design experi-ments targeted at understanding the inter-actions in the immediate social situation of students’ mathematical learning (Cobb as cited in Lobato 2006: 440). Collective ab-straction accounts for the social dimensions of students’ learning, which also provides a context for interpreting students’ individual activities. Cobb describes the social struc-ture as a classroom micro-culture and ana-lyzes students’ mathematical learning in the context of the class norms and practices. He described collective abstraction as occur-ring when members of a community col-lectively use prior group experiences as an explicit object for class discourse, meaning the groups’ previous discussion and activity becomes an object of reflection. individual student mathematical learning is viewed as activity within the classroom culture, involving processes of reorganizing activ-ity and the use of tools and symbols. These ideas have been applied in u.s. K-12 educa-tion and also to research in undergraduate mathematics education (Cappetta & Zoll-man 2013).

« 12 » rina Hershkowitz and colleagues described a model, Abstraction in Context, based on individual student case studies from a large empirical study of a construc-tivist curriculum project. They defined

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three dynamically-nested epistemic actions within this model: constructing, recogniz-ing, and building-with. abstraction is de-scribed from a sociocultural point of view. They base their definition on five principles, the last of which is crucial to explaining the genesis of abstraction as a learning process:1 | abstraction is a chain of actions under-

taken by an individual or group and driven by a context-specific motive;

2 | context is a personal and social con-struct;

3 | abstraction requires theoretical and empirical thought;

4 | abstraction leads from initial entities to a novel structure; and

5 | the structure comes into existence through creation of new internal links within the initial entities (Hershkowitz, schwarz & dreyfus 2001: 14).« 13 » The actor-oriented abstraction

approach is aimed at coordinating indi-vidual and social levels of abstraction. it emerged through research directed at de-scribing an alternative interpretation of educational transfer, called actor-oriented transfer (aot; Lobato 2003). The actor-oriented approach seeks to describe how features of instructional environments, in-cluding curricular materials, social-cultural norms, tool use, and classroom discourse, interact to affect the conceptual attributes to which students pay attention. actor-oriented abstraction includes modifications to the Piagetian construct of reflective ab-straction to address criticisms. aot seeks to describe both individual and social lev-els of abstraction and uses the device of at-tention focusing to do so. at the individual level, a student generalizes from regularities in records of experience in relationship to the student’s goals and activity, which is a structuring process from the perspective of the learner (Lobato 2012). The social level of abstraction is addressed by identifying records of focus, which are the mathematical ideas on which students seem to have fo-cused their attention while interacting with particular representations in the classroom environment (Lobato, Ellis & Muñoz 2003). Joanne Lobato and colleagues seek to un-derstand why these records of focus exist and account for them by what they refer to as focusing phenomena in the classroom, including the teacher’s actions, classroom

discourse, curricular materials, and avail-able tools. decontextualization is addressed in this approach by considering…

“ context from the point of view of the actor (learner) rather than as something inherent in the situation… contextualizing could be seen as a dynamic process rather than as a static feature of situations to be removed.” (Lobato 2006: 440)

Building from individual to classroom learning and implications for teacher education« 14 » While earlier work in reflective

abstraction provided useful tools for de-scribing the conceptual growth of individual students, its weakness was that it was diffi-cult to make recommendations concerning how these ideas could be used to develop curriculum or influence classroom instruc-tion. Earlier approaches to reflective ab-straction focused on the creation of learning tasks designed to induce reflective abstrac-tion and conceptual learning, and focused mainly on the learning activities of the individual student when faced with these tasks. Currently, the learning “situation” is considered not only to be the specific learn-ing tasks, but also the individual classroom culture. Classroom discourse is credited as being crucial to the concepts that individual students develop, and the limitations of cur-ricular materials alone in spurring cognitive dissonance are recognized. one can sum-marize the different approaches to reflective abstraction approaches as encompassing the following components:

� describing an individual student’s cur-rent knowledge of a concept

� describing an individual student’s prob-lem solving actions

� designing learning tasks likely to spur reflective abstraction in individual or group settings

� describing the mechanism of reflective abstraction in the context of tools and language

� describing the mechanism of reflective abstraction in the context of the class-room environment and culture, includ-ing the discourse

� describing reflective abstraction from the perspective of the actor or the ob-server.

Cifarelli & sevim’s work falls into the cat-egory of providing detailed knowledge of an individual student’s current knowledge of a concept and the new knowledge that is created on the basis of the student’s problem solving activity. such studies provide impor-tant insights into the types of mathematical tasks that can engender conceptual develop-ment in students.

tracy Goodson-Espy is Professor of Mathematics Education at Appalachian State University in Boone,

North Carolina. She serves as a Director for the North Carolina Science, Mathematics, and Technology Center. She earned a M.S. in mathematics from

Middle Tennessee State University and an Ed.D. in mathematics education from Vanderbilt University.


Reflecting on a Radical constructivist approach to problem solvingErik S. TillemaIndiana University Purdue University Indianapolis, USA etillema/at/iupui.edu

> upshot • Cifarelli & Sevim outline the distinction between “representation” and “re‑presentation” in von Glasers-feld’s thinking. After making this dis-tinction, they identify how a student’s problem solving activity initially involved recognition, then re‑presentation, and finally reflective abstraction. I use my commentary about the Cifarelli & Sevim article to identify two ways they could extend their current line of research.

direction 1: What did marie reflectively abstract?« 1 » victor Cifarelli and volkan sevim

argue that Marie’s problem solving activity involved a transition from recognition to re-presentation, and subsequently that she made a reflective abstraction. Therefore, i wondered: What could Marie re-present af-ter finishing the problem solving sequence and what reflective abstraction did she

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Reflecting on a Radical constructivist approach to problem solving Erik s. tillema





make? to think about a response to these questions, i note that each of the tasks that Cifarelli & sevim presented had the poten-tial to be solved in (at least) three structur-ally different ways, and that Marie seemed to use each of these three ways at different points in her problem solving activity. Each way is structurally different in that a person conceives of different quantities in the situ-ation as a basis for establishing an algebraic equivalence. i use an analysis of task 1 to il-lustrate the three potential ways that a per-son could conceive of the quantities.

« 2 » The first way to solve each of the problems is to consider the unknown in the problem, x, to be the depth of Blue Lake (the smaller of the two lakes), then to use the stated multiplicative relationship to find an algebraic expression, 2x, for the depth of Clear Lake (the larger of the two lakes), and finally to establish an algebraic equivalence based on finding two equiva-lent ways to express the depth of Clear Lake (the larger of the two lakes), x + 35 – 12 = 2x (Figure 1). see Marie’s solution of task 4 for an example of this way of solving the problem.

« 3 » a second, and closely related, way to solve the problems is to establish the un-known in the problem, x, to be the depth of Blue Lake (the smaller of the two lakes), then to use the stated multiplicative relationship to find an algebraic expression, 2x, for the depth of Clear Lake (the larger of the two lakes), and finally to establish an algebraic equivalence based on finding two ways to quantify the difference between the bottom of Blue Lake and the surface of Clear Lake (a quantity not explicitly stated in the prob-lem), 2x + 12 = x + 35 (Figure 2). see Marie’s solution of task 2 for a potential example of this kind of solution.1

« 4 » The third way to solve the prob-lems is to consider the unknown in the problem, x, to be the difference between the surface of Blue Lake and the bottom of Clear Lake (a quantity not explicitly stated in the problem), then to use this unknown

1 | Here i use the word “potential” because it was not clear to me from the data if she was thinking about task 2 in this way, but one of the algebraic equations she created and solved, 3x + 200 = x + 218, seemed to indicate that she was.

to establish an algebraic expression to es-tablish the depth of Blue Lake, x + 12, and the depth of Clear Lake, x + 35, and finally to use the multiplicative relationship to ex-press the depth of Clear Lake in two differ-ent ways, 2(x + 12) = x + 35 (Figure 3). see Marie’s solution of task 1 for an example of this kind of solution. Based on the problem Marie posed in task 9, my interpretation would be that Marie reflectively abstracted the first way of reasoning. i make this infer-ence because her problem could most read-ily be solved by measuring the height of the taller hot air balloon using the height of the shorter hot air balloon, and the two differ-ences given in the problem (3 ft. and 2 ft., respectively). When working with future students like Marie, it would be interesting for Cifarelli & sevim to see if such students have the potential to reflectively abstract the second or third way of reasoning from their problem solving activity. it would be inter-esting because both ways of reasoning in-volve positing a quantity that is not stated in the problem, as Cifarelli & sevim point out. Moreover, the third way of reasoning has the potential to involve distributive reasoning

Surface of Clear Lake

2x

x Bottom of Clear Lake

Surface of Blue Lake

35 ft

Bottom of Blue Lake12 ft

Figure 1: One way to conceive of Task 1.

12 ft

x

Surface of Clear Lake

2(x+ 12)=x+ 35

Bottom of Clear Lake


35 ft

Bottom of Blue Lake

Figure 3: A third way to conceive of Task 1.

Surface of Clear LakeSurface of Clear Lake

2x

Bottom of Clear Lake


35 ft

Bottom of Blue Lake Bottom of Blue Lake

x12 ft

Figure 2: A second way to conceive of Task 1.

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and reasoning with complex fraction rela-tionships (e.g., the equations for task 7 are 9 / 7(x + 15) = x + 35 or x + 15 = 7 / 9(x + 35)). The additional kinds of reasoning involved in the second and third way of structuring the quantities are important to algebraic problem solving, and so seem worthy of fur-ther investigation with students.

direction 2: What role does the teacher play in problem solving?« 5 » a second direction for further

investigation would be to address more explicitly the role that the teacher plays in facilitating problem solving experiences for students. to suggest how Cifarelli & sevim might do this, i use Ernst von Glaserfeld’s (1987, 1991, 1995a) discussion of symbol formation and Patrick Thompson’s (2000) model for communication2 (see also tillema 2010). i note that this discussion is compat-ible with Cifarelli & sevim’s distinction be-tween representation and re-presentation.

« 6 » von Glasersfeld (1995a: 131) conceptualized a symbol as involving bi-directional relationships among a sound or graphic image, a concept, and a person’s re-presentations (Figure 4). The bi-directional relationships in Figure 4 denote that when a person is a sophisticated symbol user, any of the three call up the other two (e.g., a sound/graphic image calls up both a con-cept and a person’s re-presentations). one of these bi-directional relationships, the relationship between a sound or graphic image and a concept, is built up when a person isolates hearing a sound or seeing a graphic item and coordinates it with an-

2 | Thompson presented the model already in 1999 at the Panel for the PME-na XXi working group on representations, see http://bit.ly/12sBts5

other aspect of an experience. For example, a young child might first isolate the sound produced when his or her parent says the word “apple,” and coordinate this sound image with the experience of biting into an apple. a person’s apple experiences serve as the basis for the construction of concepts, where a concept is a program of operations (Glasersfeld 1991). in the apple example, the child’s concept could consist of his or her use of a unitizing operation applied to the sensation of biting the apple.

« 7 » The other bi-directional links in Figure 1 involve a person’s re-presentations – the capacity to replay and examine prior chunks of experience, while maintaining awareness that this replay is of a past ex-perience. The bi-directional link between, for example, a sound/graphic image and a re-presentation simply means that a person can use a written word such as “apple” to call up prior apple experiences such as an experience of going to an apple orchard to pick apples as a child, or any other experi-ences that have been associated with the particular sound/graphic item.

« 8 » von Glasersfeld (1991, 1995a) identifies that when a person can use a spo-ken word or graphic item as a symbol, then the spoken word or graphic item functions as a placeholder. it holds the place for actu-ally having to implement activity that the person may initially have used to establish the concept and associated re-presentations. This placeholder function of symbols can be illustrated through a comparison of two stu-dents – one working to establish a difference meaning of subtraction and the other a stu-dent who can symbolize a difference mean-ing of subtraction. a student working to es-tablish a difference meaning of subtraction might need to count the number of inches between two people’s heights and may have yet to associate this activity with the graphic item “–”; in contrast, a student who has es-tablished the graphic item “–” as a symbol may simply use it to denote any time she is comparing two quantities using subtraction (e.g., two heights) without actually having to produce any activity – the student can use a graphic item such as h1 – h2 to mean the potential to make a comparison of two people’s heights without actually having to implement activity that would produce the result of this comparison.

« 9 » Given this definition of symbols, communication, then, entails two or more people engaged in reciprocal assimilation of words/graphic items produced by the other person (Thompson 1999). Each person in-volved in communicating forms a model of the meaning the other person intends. For example, person a may present a problem in written form to person B; person B may assimilate the problem, and in response produce some symbolic activity for the situ-ation (e.g., algebraic symbols, as Marie did in task 1); person a may then assimilate person B’s activity using the model they have developed of person B and respond to person B based on what they think might be a fruitful direction for person B to pursue.

« 10 » This model for communica-tion seems compatible with what Cifarelli & sevim outline about their orientation to problem solving. situating their work on problem solving within a model for com-munication could help them to investigate questions such as the following:

� What types of questions asked by the teacher were effective in supporting a student’s problem solving activities?

� What sequencing of tasks was effective in helping students establish particular ways of knowing in the context of their problem solving activities?

� What types of tools were useful for aid-ing in student’s problem solving activi-ties?

Pursuing answers to these types of questions could position Cifarelli & sevim to make more explicit how von Glasersfeld’s theory could help mathematics educators make decisions about individual and classroom instruction.

Erik tillema is a newly-tenured faculty member at Indiana University Purdue University Indianapolis.

Recently, he has been the PI on two grants aimed at investigating how middle-grade students establish

power meanings of multiplication. He has published much of his work in this area. Prior to being a faculty

member, he taught middle and high school for 4 years.


Sound/ graphic image

Re-presentations Concept

Figure 4: The bi‑directional relationships in symbol formation and use.

http://bit.ly/12sBts5

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authors’ Response: Radical constructivist conceptual analyses in mathematical problem solving and their implications for teaching volkan sevim & victor v. cifarelli



authors’ Response volkan sevim & victor v. cifarelli

authors’ Response: Radical constructivist conceptual analyses in mathematical problem solving and their implications for teachingVolkan Sevim & Victor V. Cifarelli> upshot • In this response to the open peer commentaries on our target article, we address two emerging themes: the need to explicate further the nature of learning processes from a radical con-structivist perspective, and the need to investigate further the implications of our research for classroom teaching.

« 1 » We wish to thank all of the com-mentators for their thoughtful analytical comments and recommendations. in re-sponse to their commentaries we will re-iterate the major points of each paper, and while reflecting on our own ideas in light of their useful comments, we will seek to ad-dress some of the questions that they raise. We organize our response in terms of what we believe to be two emerging themes from all the commentaries: the need to explicate further the nature of the learning processes such as reflective abstraction and problem solving from the perspective of Ernst von Glasersfeld’s radical constructivism (rC), and the need to study further the implica-tions of our research for classroom teaching.

learning processes and radical constructivism« 2 » catherine ulrich’s paper closely

examines our discussion of Marie’s cogni-tive development and offers a well-written analysis of how Piaget’s characterization of reflective abstraction with different levels of projection can be applied in our study. We agree with ulrich’s characterization of Marie’s activities of drawing, translating and ma-nipulating as externalizations of her internal constructions of the involved quantitative relationships (§3), and we value her sugges-tion that it is more useful for us research-

ers to focus on students’ solution activities as mental actions and not as sets of external procedures. Considering Marie’s solution activity as a series of mental constructions and reflective abstractions with different levels of projections, ulrich states that our claim regarding Marie’s movement from recognition towards reflective abstraction can be restructured as a sequence of increas-ingly higher levels of reflective abstractions (§4). according to ulrich, because Marie seems to be engaged in reflective abstraction in each task (either in the form of projection of mental action from a lower level of activ-ity to a higher level of representation, or in the form of reorganizing or reconstructing the projected mental actions or concep-tions at the higher level), Marie’s reflective abstraction in task 9 resembles a “reflected abstraction” (Piaget et al. 1977: 303).

« 3 » ulrich’s reference to reflected ab-straction reminds us that Piaget addressed how reflective abstractions involve the growth of awareness. Piaget referred to the result of a reflective abstraction as a “re-flected” abstraction once it has become con-scious; and learners do this independently of its level (Piaget et al 1977: 303). While we agree with ulrich that Marie demonstrated increased awareness in task 9, von Gla-sersfeld discussed the difficulty of inferring awareness since “one can be quite aware of what one is cognitively operating on, with-out being aware of the operations one is carrying out” (Glasersfeld 1991: 61). Thus in terms of Marie’s growth of awareness, we noted that her act of re-presentation while solving task 3 resulted in her hypothesis that she needed more information about the heights of the unknowns. at this point in the interview, we maintain that she had some awareness about the relationships involved in the task. on the other hand, as ulrich has commented (§7), Marie’s level of aware-ness in task 9 had increased to the extent that the only checking she needed to do was put numbers to the relationships she knew would work.

« 4 » in sum, we agree with ulrich that interpreting Marie’s actions as levels of pro-jections and reorganizations of actions, as given in Piaget’s theory, can be useful in our search for understanding how students learn through problem solving. in addition to ulrich’s suggestions about re-examining

the levels of reflective abstraction engaged in each task, we also find useful her obser-vation that our analysis of reflective abstrac-tion differs from other studies that focus on the construction of new kinds of quantities or quantitative relationships (§8). specifical-ly, ulrich’s comment that Marie did not ap-pear to construct a new type of mathemati-cal structure suggests that our approach is but one way to think about and study struc-turing activity in problem solving.

« 5 » While we appreciate the alterna-tive analysis stated by Kevin c. moore (§6, 7), we believe that it represents a focus different from our analysis, appearing to be an exam-ple of the analyses cited by ulrich that focus on quantitative relationships (§8). neverthe-less, moore’s comments are helpful because they suggest to us the need to be aware of the possibility of other kinds of conceptual structures as we proceed. in addition, with regard to his suggestion in §11 to examine the extent to which the construction of ab-stract conceptual structures follow the cyclic sequence that was hypothesized by Marilyn Carlson and irene Bloom (2005), we believe that there is some compatibility between this research and our studies on problem solving and problem posing, which found the solver’s novel solutions to be gener-ated recursively through varying degrees of sense-making, problem posing and reflec-tion on solution activity (Cifarelli & sevim in press; Cifarelli & Cai 2005).

« 6 » another commentary focusing on our conceptual analysis of the learning processes involved in problem solving is carlos castillo-Garsow’s. While we commend castillo-Garsow in urging all interested re-searchers to form collaborations and study how student researchers in mathematics are formed using our framework that explains progression through levels of conceptual structure within the context of problem solving, we differ slightly in our view of the kind of problem solving activity that Marie was engaged in. castillo-Garsow’s commen-tary encourages us all to think about how we use the terms “problem” and “problem solv-ing.” Briefly, he refers to two types of prob-lem solving, “situation-goal-perplexity” and “task-problem-solution,” citing the works of Leslie steffe, John dewey and Frank Lester. to expand the definition of the first type, situation-goal-perplexity, we would add

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Jean Lave’s characterization of problems as dilemmas as another example of where problems and problem solving are defined within situations that arise when one’s way of “seeing” or operating is challenged (Lave 1988). These dilemmas are personally con-structed in the sense that they arise from an individual’s own experiences within a par-ticular social situation and thus constitute a unique view of the problem components. in contrast to this type of problem solving, where the solvers pose their own problems to pursue their curiosities and to further their understandings, the second type in-volves tasks that are given to them external-ly. Citing Lester, castillo-Garsow suggests that in the task-problem-solution activity, a task with objective qualities is described explic-itly and given to a student, while at the same time the student believes that this external task “is a problem, and the student believes that there is a solution (a yet unknown, but recognizable stopping point, which Lester calls the ‘outcome’)” (§4). The purpose of this type of activity is to arrive at the solu-tion, or the recognizable stopping point. We value this distinction by castillo-Garsow, and we see the usefulness of this way of concep-tualizing mathematical research that all stu-dents can engage in at all levels.

« 7 » We differ with castillo-Garsow’s as-sessment of the kind of problem solving ac-tivity that Marie was engaged in, because we believe that Marie’s solution activity might be more compatible with the situation-goal-perplexity meaning of problem solving (§3). in addition, we believe that castillo-Garsow’s suggestion that Marie’s experience is contained entirely within task-problem-solution and that her activity does not in-clude examples of her pursuing her own curiosities (§§8f), is, at best, accurate only for certain parts of her solution activity. For instance, in our target article we cited ex-amples in which Marie self-generated hypo-thetical situations to explore, which served to frame her subsequent actions (§§33, 34, 37). These situations went beyond the given tasks and thus were experienced by Marie as open-ended problems. to investigate fur-ther how students might engage in given tasks in such “open-ended ways,” and how problem solving and problem posing might co-evolve to further students’ understand-ings, see Cifarelli & sevim (in press).

« 8 » Perhaps the commentary that is most different in its approach to our paper is that of Gerald Goldin. in his commentary, Goldin offers criticisms of the use of rC as a theoretical framework within conceptual analyses of students’ problem solving activ-ity such as ours. Chief among these criti-cisms is Goldin’s question of whether or not the basic tenet of von Glasersfeld’s rC is essential to constructivist programs of re-search (§4) and his assertion that adherence to the fundamental premises of von Glasers-feld’s theory limits the constructs that can be used to understand fully our empirical observations (§§2–4). Thus he argues that our conceptual analysis of Marie’s problem solving activity does not need rC’s funda-mental premises on knowledge and know-ing (§§6–8), and provides several sugges-tions and recommendations for extending our analysis (§§11f).

« 9 » Goldin rightfully acknowledges that radical constructivists’ qualitative research and analyses have yielded “valuable explor-atory and descriptive findings” that “should tend to support policies and practices that are far more progressive and sophisticated than those currently implemented […]” (§3). However, he claims that it is long over-due for researchers to question whether the methods, findings and analyses in construc-tivist studies actually require acceptance of rC’s “ultrarelativist epistemology” (§4). Fur-thermore, according to Goldin, our choice to study internal structures, which are formed by re-presentations of prior experiences, does not depend on the claim that re-pre-sentations do not stand for any objects. We differ with Goldin on these points.

« 10 » First, as we claimed in the target paper, the dynamic process of mentally re-presenting a prior action plays a fundamen-tal role in the construction of conceptual structures, and this kind of representation does not involve any form of interaction between a learner and an independently ex-isting object. We agree with von Glasersfeld (1974) that all objects are results of an indi-vidual’s coordinatory activity, and that his or her constructs do not correspond to or re-flect structures that “exist” prior to his or her constitutive activity. von Glasersfeld wrote:

“ We call this school of constructivism ‘radi-cal’ because it holds that the knower’s percep-

tual (and conceptual) activity is not merely one of selecting or transforming cognitive structures by means of some form of interaction with ‘ex-isting’ structures, but rather a constitutive activ-ity which, alone, is responsible for every type or kind of structure an organism comes to ‘know’.” (Glasersfeld 1974: 5).

« 11 » second, our method, analysis and findings do require rC’s epistemology because, as teacher-researchers, we adopt the learner’s point of view, to see things as they see them. The examination of dynamic constructs of learning such as the learner’s interpretations, their goal planning and for-mation, how they carry out, monitor and assess their problem solving, how these feed into their overall conceptual understand-ings, require a constructivist model build-ing process that cannot be accomplished by analyzing internal representations that stand for external objects. it is the contex-tual mathematical reality of Marie, as con-structed by us, the observers, that we are concerned with, not her construction of ordinary experiential realities. Goldin as-serts that:

“ without the rC tenet, Marie’s own inscriptions – equations, diagrams, numerals, and written words reproduced pictorially in §§25, 26, 27, 29, 32, 36, and 38 – may usefully be considered as real-world objects, about which Marie has partial knowledge” (§9).

We, by necessity, must construct not only Marie’s inscriptions, but also Marie’s mean-ing for the inscriptions and explicitly ac-knowledge that these are our meanings, not Marie’s meanings. Then, what we would need to do is to establish our own meanings for Marie’s inscriptions other than our con-struction of Marie’s meanings and compare and contrast the two. Citing Piaget’s episte-mology, von Glasersfeld discussed this ne-cessity to actually construct what might be “out there”:

“ taken seriously, a statement to the effect that the child constructs his universe and then expe-riences it as though it were external to himself, would be rather shocking. We would all like to be hard scientists, and such an ‘as though’ threatens to pull the rug from under our feet.” (Glasers-feld 1974: 2)

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in the same way, we construct Marie’s in-scriptions and meanings and then experi-ence them as though they were external to us.

« 12 » While we cannot speak for other constructivists, our analysis is based on two important implications drawn from prin-ciples of rC:

� The learner’s meaning-making in a problem solving situation is inherently subjective; and

� The learner’s conceptual structures re-main stable for them until their expe-riences suggest that they are no longer useful and must be revised.« 13 » These considerations characterize

important though under-analyzed ques-tions about the nature of problem solving: How do solvers make sense of the problems they face? and what happens when solvers realize that current understandings do not work? These questions are related to von Glasersfeld’s notion of viability (Glasersfeld 1987b), that one’s knowledge remains viable until one’s experiences suggest that one’s knowledge has outlived its usefulness. We have based our analysis on these principles of rC, not to reject any objective reality or to take any ontological stance, but solely for the purposes of focusing better on the cognitive actions of learners.

« 14 » Lastly, Goldin argues that many of the constructivist research programs are compatible with scientific perspectives gen-erally taken in mathematics, physics, cogni-tive science and empirical psychology (§4). Given the large number of constructivist studies conducted over the past 40 years, it is difficult to assess this claim. However, Patrick Thompson (2014) recently listed the major areas of impact made by constructiv-ist researchers including: the development of teaching experiments (Cobb 2000; Cobb & steffe 1983; steffe & Thompson 2000) and design experiments (Cobb et al. 2003) as vi-tal and vibrant methodologies in mathemat-ics education theory development; the con-ceptual analysis of mathematical thinking and mathematical ideas as a prominent and widely used analytic tool (Behr et al. 1997; Glasersfeld 1995a; Lobato et al. 2012; smith, disessa & roschelle 1993; Thompson 2000); and the development of clear and operation-alized implications for the design of instruc-tion (Confrey 1990; Forman 1996; simon

1995; steffe & d’ambrosio 1995; Thompson 2002) and assessment (Carlson, oehrtman & Engelke 2010; Kersting et al. 2012). it is difficult to see all the compatibilities be-tween constructivism and the varied scien-tific perspectives Goldin cites. For example, according to Lieven verschaffel and Brian Greer, despite the cognitive-psychological theoretical studies on learning and problem solving over the past 30 years, the general findings do not appear particularly appli-cable to current mathematics education re-search because they do not place “cognitive functioning in a broader perspective that takes into account aspects such as affect, motivation, attitudes, beliefs, and intuitions, as well as social and cultural factors” (ver-schaffel & Greer 2000: 62f).

« 15 » in sum, as Karen François states in her commentary, we hope to…

“ bridge the philosophical discourse and the conceptual analysis of a goal-directed activity in the context of problem solving by concentrating on the constitution of mathematical knowledge by experience, leaving aside the ontological is-sue.” (§1)

Because we focus strictly on “the learner as an active player in the learning process” (François §6), it is irrelevant for us to consid-er the ontological status of the mathematics.

implications for classroom teaching« 16 » With regard to the need to study

further the implications of our research for classroom teaching, Joy Whitenack’s com-mentary suggests that radical constructivist studies focus more on applying von Gla-sersfeld’s theoretical constructs to examine groups of students’ collective learning pro-cesses and classroom teachers’ professional learning. Whitenack’s commentary raises two important points regarding our research program, which focuses on individual stu-dents’ cognitive development within prob-lem solving. While acknowledging its signif-icance as a basis for “developing a model for problem solving that is tested, refined and replicated” (§6), Whitenack suggests that our study can be further extended and adapted to study students’ collective learning pro-cesses through the use of the classroom teaching experiment. in addition to exam-

ining students’ collective conceptions that they co-construct in the classroom, White-nack also suggests that radical constructivist research programs examine teachers’ own conceptions and learning processes as part of their professional development. We fully agree with Whitenack on the importance and value of these two research directions.

« 17 » similarly, focusing on the role of reflective abstraction as an individual and collective learning mechanism, tracy Good-son-Espy summarizes how researchers have applied individual analyses of abstraction to studying classroom learning processes (§§11f) and thus provides valuable insights into how we might revisit the nature of the construct of reflective abstraction used in our study. For example, Goodson-Espy cites Paul Cobb’s definition of collective ab-straction as one such application in which students’ mathematics learning is studied within the context of the classroom norms and practices (§11), occurring when mem-bers of a community collectively use prior group experiences as an explicit object for class discourse. in this approach, individual student mathematical learning is examined within the classroom culture, involving pro-cesses of reorganization of activity and the use of tools and symbols.

« 18 » Goodson-Espy identifies records of focus as another promising approach through which to examine the emergence of abstraction in classroom settings (§13). This approach addresses the mathematical ideas on which students appear to focus their at-tention. The research in this area (Lobato 2012) seeks to understand how these re-cords emerge from focusing phenomena in the classroom, drawing from the teacher’s actions, classroom discourse, curricular ma-terials and tools available to students.

« 19 » in his commentary, Erik tillema of-fers two clear research directions for us to consider. First, tillema illustrates three struc-turally different ways of conceiving of the tasks that we gave to Marie, and suggests that we might investigate the ways solvers might employ these different mathematical ap-proaches where solvers must pose a quantity not stated in the task and where important algebraic problem solving activity involv-ing distributive reasoning and reasoning with fractions are present. second, tillema offers excellent suggestions about how to

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study further the role of the teacher in sup-porting students’ problem solving activities such as the one we present in our analysis. according to tillema, von Glasersfeld’s dis-cussion of symbol formation and Thomp-son’s model for communication could be used as frameworks to examine further how students’ problem solving activities can be supported better, both individually and col-lectively (§6). We value both of these sugges-tions and think that our research program on students’ conceptual development within problem solving could benefit from them.

« 20 » Finally, we believe that François’ commentary offers an insightful look into some points of convergence between tenets of rC and those of critical learning theories. as she states, in both theories, traditional ontological positions are abandoned and mathematics is considered to be a product of human activity. Moreover, students’ activity plays a central role in the learning process within these theories:

“ students are learning mathematics by doing mathematics in their own way, using their own mental representations as viable constructions within their own experiential world. The learners can construct their own ways of practicing math-ematics.” (§8)

We find François’ summary of alan Bishop’s social dimensions of the learning process valuable in that she articulates clearly how the integration of the social embedding of students’ activity can potentially narrow the gap between these two theories (§10). Thus, in order to understand effective teaching practices better, especially in the complex setting of actual classrooms, in addition to focusing on the central role of the learner’s individual activity, we may also consider the social dimensions of this activity at various levels, including that of the classroom, insti-tution, society and culture. Both Goodson-Espy’s and Whitenack’s commentaries support these suggestions.

received: 27 June 2014 accepted: 3 July 2014

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