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Heuristic literacy development and its relation to mathematical achievements of middle school students Author(s): BORIS KOICHU, ABRAHAM BERMAN and MICHAEL MOORE Source: Instructional Science, Vol. 35, No. 2 (March 2007), pp. 99-139 Published by: Springer Stable URL: http://www.jstor.org/stable/41953732 . Accessed: 28/06/2014 16:59 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . Springer is collaborating with JSTOR to digitize, preserve and extend access to Instructional Science. http://www.jstor.org This content downloaded from 46.243.173.84 on Sat, 28 Jun 2014 16:59:08 PM All use subject to JSTOR Terms and Conditions

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Page 1: Heuristic literacy development and its relation to mathematical achievements of middle school students

Heuristic literacy development and its relation to mathematical achievements of middleschool studentsAuthor(s): BORIS KOICHU, ABRAHAM BERMAN and MICHAEL MOORESource: Instructional Science, Vol. 35, No. 2 (March 2007), pp. 99-139Published by: SpringerStable URL: http://www.jstor.org/stable/41953732 .

Accessed: 28/06/2014 16:59

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

Springer is collaborating with JSTOR to digitize, preserve and extend access to Instructional Science.

http://www.jstor.org

This content downloaded from 46.243.173.84 on Sat, 28 Jun 2014 16:59:08 PMAll use subject to JSTOR Terms and Conditions

Page 2: Heuristic literacy development and its relation to mathematical achievements of middle school students

Instructional Science (2007) 35:99-139 © Springer 2006 DOI 10.1 007/s 11251 -006-9004-3

Heuristic literacy development and its relation to mathematical achievements of middle school students

BORIS KOICHU1'*, ABRAHAM BERMAN2 & MICHAEL MOORE Department of Education in Technology and Science , Technion - Israel , Institute of Technology, Haifa , 32000, Israel; 2Department of Mathematics, Technion - Israel, Institute of Technology, Haifa, 32000, Israel (* Author for correspondence, e-mail: [email protected])

Received: 1 July 2005; accepted: 26 May 2006

Abstract. The relationships between heuristic literacy development and mathematical achievements of middle school students were explored during a 5-month classroom experiment in two 8th grade classes (N = 37). By heuristic literacy we refer to an individual's capacity to use heuristic vocabulary in problem-solving discourse and to approach scholastic mathematical problems by using a variety of heuristics. During the experiment the heuristic constituent of curriculum-determined topics in algebra and geometry was gradually revealed and promoted by means of incorporating heuristic vocabulary in classroom discourse and seizing opportunities to use the same heuristics in different mathematical contexts. Students' heuristic literacy development was indi- cated by means of individual thinking-aloud interviews and their mathematical achievements - by means of the Scholastic Aptitude Test. We found that heuristic literacy development and changes in mathematical achievements are correlated yet distributed unequally among the students. In particular, the same students, who progressed with respect to SAT scores, progressed also with respect to their heuristic literacy. Those students, who were weaker with respect to SAT scores at the beginning of the intervention, demonstrated more significant progress regarding both measures.

Keywords: classroom experiment, heuristics, heuristic literacy, mathematical problem solving, mathematical achievements, thinking-aloud interviews

Introduction

Many educators and scholars experimented with the idea to teach heuristic problem-solving strategies as a means for improving students' mathematical achievement (Mayer, 1992; Lester, 1994). In particular, a remarkable effort was made to find which heuristics are involved in successful problem solving and how that knowledge can be used in favor of different categories of students (e.g., Schoenfeld, 1992). In spite of or, perhaps, due to that effort, the 40-year theoretical debate

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about the role of heuristics in mathematical problem solving still continues (e.g., Carlson & Bloom, 2005). Furthermore, the practical questions "How should these [heuristic] strategies be taught? Should they receive explicit attention, and how should they be integrated with the mathematical curriculum?" (NCTM, 2000, p. 54) reappear in the current Principles and Standards for Teaching Mathematics with no clear-cut answers.

We contribute to the discussion of that research problématique by presenting a study focused on a complex relationship between heuris- tic literacy development and changes in mathematical achievements of middle school students. By heuristic literacy we refer to an individ- ual's capacity to use heuristic vocabulary in problem-solving discourse and to approach scholastic mathematical problems by using a variety of heuristics. The underlying hypothesis of the study is as follows. It is well known that classroom experiments dealing with teaching par- ticular heuristics typically have small to moderate mean effect sizes in terms of standardized measures (e.g., Hembree, 1992; Schoenfeld, 1992). We suggest that an expectedly modest mean effect size of a

particular heuristically oriented intervention can is unequally distrib- uted among participating students. Some students can benefit more than others, and the purpose of the study is to investigate presumable within-sample differences in heuristic literacy development in relation to the students' mathematical achievements.

To achieve this goal, a 5-month heuristically oriented intervention was conducted in two Israeli 8th grade classes in cooperation with the teachers who taught mathematics in these classes. The intervention was based on instructional principles extracted from past research about teaching/learning heuristics as well as on our own pedagogical ideas and experience. Heuristic literacy of the students was promoted while teaching traditional mathematical topics that are part of the middle-school curriculum. Two measures were quantified and traced through the stages of the intervention: the students' heuristic literacy development, by means of individual thinking-aloud interviews, and their mathematical achievements, by means of the Scholastic Aptitude Test (SAT). This paper is one of several reports based on this experiment. Here, we address the following research questions:

1. How can heuristic literacy be promoted while teaching the tradi- tional mathematical topics of a middle-school curriculum?

2. How are the changes in heuristic literacy and in mathematical achievement, if any, distributed among the middle school students who took part in the intervention?

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Theoretical framework

Heuristics

We refer to the concept of heuristics as a systematic approach to repre- sentation, analysis and transformation of scholastic mathematical prob- lems 1 that actual (or potential) solvers of those problems use (or can

use) in planning and monitoring their solutions. Some heuristics are narrow and domain-specific (e.g., "Reduce fractions first"), whereas others are universal and cut across many problem-solving domains

(e.g., "Decompose a problem into manageable segments"). In actual

problem solving, a particular heuristics can come as an enduring or as a transient way of thinking, which governs the entire process of con-

structing a solution or triggers just a short-lived problem-solving step. When expressed verbally, as a recommendation to problem solvers or as a unit in the analysis of one's problem-solving behaviors, heuristics is a statement reflecting a generalized and decontextualized piece of

experience of problem solvers. Heuristics at large can be seen as a

cognitive tool used to approach problems, effectiveness of which is never known in advance.

The above conceptualization of heuristics is a synthesis of many definitions produced in the fields of cognition, pedagogy and artificial

intelligence. Selected definitions by some of the contributors to these fields are presented in Table 1.

To recap, heuristics in the literature are treated as (i) rules of thumb or recommendations to problem solvers (Pólya, 1945/1973; Perkins, 1981; Larson, 1983; Schoenfeld, 1985), (ii) useful units in description and analysis of the ways of mathematical thinking (Newell & Simon, 1972; De Bono, 1984; Goldin, 1998), and (iii) cognitive/metacognitive tools in actual mathematical problem solving (Newell & Simon, 1972; De Bono, 1984; Goldin, 1998; Martinez, 1998; Verschaffel, 1999).

In our research we decided to focus on 10 heuristics, which have been chosen as follows. The initial list of heuristics was produced as a

compilation from the above sources. It was then refined on the basis of a preliminary classification study (Koichu, 2003; Koichu et al., 2003a). In this study, the heuristics were classified by their usefulness for experts either in their opinion or in their observed problem-solving. Appendix A consists of these 10 heuristics in descending order of usefulness for the experts, and, accordingly, in descending order of attention in our

experiment. They are described both as problem-solving behaviors and as units in the analysis of the students' thinking-aloud interviews.

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Table 1. Selected definitions of heuristics

Author Definition

Pólya (1945/1973) Heuristics are formulated as questions that good problem solvers should ask themselves at different stages of solving a problem or as general advice (concluded by an exclamation mark!) to problem solvers. The heuristic questions include "what is the unknown?", "what are the data?", "have you seen such a problem before or maybe in a slightly different form?" The heuristic advice includes "draw a figure if possible!", "find the connection between the given and the unknown!"

Newell and Heuristics are treated as strategies that make problem solving Simon (1972) more efficient than random. Examples of heuristics include

"means-ends-analysis", "backward chaining" Perkins (1981) Heuristics strategy is a rule of thumb that often helps in

solving a certain class of problems, but makes no guarantees De Bono (1984) "[The idea of heuristics] includes all those aspects of thinking

that cannot be fitted into mathematical formulations" (p. 10) Schoenfeld (1985) "Heuristic strategies are rules of thumb for successful problem

solving, general suggestions that help an individual to under- stand a problem better or to make progress toward the solu- tion" (p. 23). Examples of heuristics include "draw a figure", "argue by contradiction", "consider a similar problem", "try to establish subgoals"

Martinez (1998) "It [heuristics] is a strategy that is powerful and general, but not absolutely guaranteed to work. Heuristics are crucial be- cause they are the tools by which problems are solved" (p. 606)

Goldin (1998) "[Heuristic process is] the most useful organizational unit and culminating construct, in a [representational] system of plan- ning, monitoring and executive control. Such processes include 'trial and error', 'think of a simpler problem', 'explore special cases', 'draw a diagram', etc." (p. 153).

Verschaffel (1999) "[Heuristic methods are] systematic search strategies for problem analysis and transformation" (p. 217).

Heuristic literacy

We define heuristic literacy as an individual's capacity to use heuristic vocabulary in discourse and to approach (not necessarily to solve!) scholastic mathematical problems by using a variety of heuristics. We chose the term heuristic literacy (see also Koichu et al., 2004) because

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of its connotation to other kinds of literacy, such as scientific literacy and mathematical literacy. Generally speaking, scientific literacy and mathematical literacy deal with some habits of mind that people need to develop in order to engage in scientifically laden or mathematically laden situations arising in their professional and social life (e.g., OECD, 1999). Analogously, heuristic literacy deals with some habits of mind that learners need to develop in order to engage in solving mathematical problems arising in their scholastic life, for instance, in a middle school classroom. In the same way that a scientifically or math- ematically literate person is not necessarily a professional scientist or mathematician, a heuristically literate person is not necessarily an ex- pert scholastic problem solver. Rather, a heuristically literate person is prepared to approach a scholastic mathematical problem even when he or she does not have a readily avaliable solution procedure.

The above definition of heuristic literacy includes two interrelated aspects of an individual's problem-solving capacity: the use of the heuristic vocabulary in discourse and of the internalized heuristics in actual problem solving. Accordingly, two branches of research inspired the above definition: research on mathematical discourse (e.g., Sfard, 2001, 2002) and on teaching problem-solving skills (e.g., Schoenfeld, 1979, 1983, 1985, 1987, 1992).

Sfard (2001, 2002) justified the importance of negotiating metadis- course rules and metalevel intimations in a mathematical classroom. According to Sfard (2002),

Metalevel intimations are ideas for discursive decisions induced by interlocutors' tendency to behave in a regular rather than acciden- tal way that is in accord with metadiscoursive rules that seem to regulate discourses (p. 337).

She pointed out that metalevel intimations are vital both for advancing discourses (including discourses with oneself) and for the process of learning. Koichu et al. (2004) argued that heuristic vocabu- lary, shared by students and teachers, can serve as such metalevel intimations in a middle school mathematical classroom.

Schoenfeld (1992), discussing many attempts to make heuristics work, pointed out that there is extensive evidence that promoting heu- ristics as rules of thumb or general advice to problem solvers is not effective. In his own research, Schoenfeld (1979, 1983, 1985, 1987) sug- gested that heuristics can be better received by the students when pre- sented as more detailed, domain-specific and situation-specific constructs. He argued that decomposing universal heuristics into more

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prescriptive procedures is empirically useful, since it enables novice

problem solvers to implement the strategies with adequate domain- specific recourses. At this point, let us note that teaching highly pre- scriptive heuristics can also be ineffective, especially in a scholastic set- ting, not in a lab. Indeed, as Schoenfeld (1992) noted, teaching a great number of domain-specific heuristics in a middle school classroom is sometimes a daunting task for the teachers. Thus, the question of the level of granularity of heuristics' description needed in order to help students to use heuristics in problem solving remains open (e.g., Schoenfeld, 1987). In particular, this enables us to experiment with the granularity of particular heuristics in problem solving (see Appendix A).

In closing, the terms heuristics and its corollary, heuristic literacy, are used here to isolate as a research object planning and monitoring problem-solving tools and capacities from the rest of problem-solving tools and capacities. The term promoting heuristic literacy is used to briefly express the essence of the intended intervention: revealing the heuristic constituent of any given mathematical content in accordance with certain instructional principles, discussed below. The term heuris- tic literacy development is operationally defined below as an integra- tive variable that has emerged from the analysis of thinking-aloud interviews.

Instructional principles of promoting heuristic literacy

Building on previous research, we assume that some heuristics can be taught not just as rules of thumb or prescriptive procedures, but can be gradually constructed as desirable ways of thinking through deliber- ately arranged practices and reflections.2 Particular classroom activi- ties, designed to promote heuristic literacy in our study, can be seen from many constructivist-oriented pedagogical perspectives, such as: advancing classroom discourse and intuitions (e.g., Confrey, 1993; Fischbein, 1999; Sfard, 2001, 2002; Sfard & Kieran, 2001), encouraging reflection and using indirect instruction (e.g., Schoenfeld, 1983, 1987; Confrey, 1990; Leinhardt & Schwarz, 1997; Stillman & Galbraith, 1998), teaching in accordance with the principle of smallest possible help (Selz, 1935, cited in De Leeuw, 1983), teaching for transfer of learning (e.g., Mayer & Wittrock, 1996; Perkins & Grotzer, 1997; Perkins & Salomon, 1988).

More specifically, five instructional principles underlie the interven- tion. In part, these principles were adapted from the aforementioned

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literature, and, in part, they emerged from our own experience and reflective thinking about the intervention:

1. "Non-routine" problems can be used in order to seize opportunities to learn heuristics. The same holds for "routine" tasks given under certain constraints (Episodes 1, 2 and 3 in Appendix B illustrate the implementation of this principle in the intervention).

2. Disposition of a teacher as an experienced problem solver sharing her or his personal struggle with the students can help students to

appreciate heuristic aspects of mathematical problem solving (see Episode 2).

3. The smallest possible help can be provided, if needed, to less suc- cessful problem solvers by more successful peers or by the teacher. In some cases, such help can be worded in terms of the intended heuristics (see Episode 2 and 3).

4. Developing heuristic vocabulary and its use in classroom discourse can help students to become more thoughtful and reflective prob- lem-solvers (see Episodes 1 and 2).

5. Appreciation of heuristics is more likely to occur when the same heuristics appear to be helpful in different problem-solving con- texts, for example, in algebra and geometry (see Episodes 2 and 3).

Method

The intervention

In Brousseau's (1997) terms, the intervention consisted of applying didactical engineering to the given mathematical topics. This means that only the ways of teaching the curriculum-determined topics could be modified in order to reveal their heuristic potential, but not the

topics or the order of the topics. It is important to note that the deci- sion about the term of the intervention during the school year has been made under many logistic constraints on the part of the participating schools. As a result, the curriculum-prescribed topics that fell into that term were "Abridged multiplication formulas", "Identical expressions including fractions", "Quadratic equations", "Quadrilaterals" and

"Pythagorean theorem". The intervention was arranged as follows. Once a week the teachers

of two experimental classes - we will call them Liora and Ada - told us which subject matter they were going to teach during the following week, often indicating specific problems and exercises in the textbook.

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Based on this information, we - the researchers along with the teach- ers - designed a 90-min lesson aimed at promoting particular aspects of heuristic literacy. Then the teachers conducted the lessons in their classes, where non-participant observations were carried out by the first author of this paper. As a rule, the same heuristic-oriented activi- ties were conducted in two classes. Then the lessons were discussed in follow-up conversations with the teachers. Commonly, these conversa- tions were focused on the teachers' reflections, heuristic aspects of the observed lessons, and on planning the future lessons. It is worth men- tioning that the teachers were fairly optimistic regarding our coopera- tion. In Liora's words from the interview conducted after the intervention:

We started to prepare the lessons, only you at first, afterwards together ... At the beginning it worked sometimes better, some- times worse, but after a short period of time there were many very successful activities in succession.

During the first 2 months of the intervention, we put the main effort into developing a heuristic vocabulary that could be shared by students and teachers. The heuristic vocabulary was promoted by means of deliberation and fixation of particular patterns of classroom discourse in the students' reflection on their problem solving (see Episodes 1 and 2 in Appendix B).

Simultaneously, many unconventionally difficult problems were proposed for class work and homework. Some of the tasks included the option of stepwise help provided in terms of the constructed heu- ristic vocabulary. A weaker as well as a stronger student could solve the most difficult problems, using some heuristic intimations designed either by a teacher or peers (see Episode 3). Gradually, problem solving in small groups followed by whole-class discussions became a custom of heuristically oriented lessons.

Many activities that implicitly invited the use of the intended heuristics were incorporated in the second half of the intervention. Along with "non-routine" problems, many conventional tasks and exercises given under certain constraints, were used to promote students' capability to plan their solutions (see Episodes 2 and 3). Let us also note that, unlike some past research (e.g., Schoenfeld, 1979; 1985), pretest or posttest sorts of problems were carefully excluded from the teaching during the experiment in order to avoid direct tutoring.

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Three examples of the classroom activities illustrating the interven- tion are presented in Appendix B; more episodes from the interven- tion can be found in Koichu (2003) and Koichu et al. (2004).

Pretest and posttest

The pretest consisted of two parts: a two-dimensional figure-reasoning test (FRT) and a mathematical achievement test. The FRT consisted of the odd items of the 60-items Raven's Progressive Matrix Test. The mathematical achievement test consisted of the mathematical part of the SAT, translated into Hebrew by Zohar (1990).

The 30-items FRT was administered for 15 min in accordance with the manual of the Raven's test (Raven et al., 1983).We used Raven's test in our research for several reasons. First, it provides a good indi- cation of general intelligence and allows researchers to describe their samples in comparison with hypothetically true norms, that is, with the test results of representative samples (Raven et al., 1983). Second, Carpenter et al. (1990) explicitly connected Raven's test with heuris- tics in problem solving. They pointed out that the test measures "the common ability to decompose problems into manageable segments and iterate through them, the differential ability to manage the hierar- chy of goals and subgoals generated by this problem decomposition and the differential ability to form high level abstractions" (p. 429).

The 35-item SAT was administered for 30 min (Zohar, 1990). The test consisted of multiple-choice tasks in the fields of algebra, geome- try, series and word problems. Commonly, the SAT is used for selec- tion of high-school students into US colleges (Donlon & Angoff, 1985). In spite of criticism mostly related to gender bias in favor of boys (e.g., Benbow, 1988; Bridgeman & Lewis, 1996; Gray & Shee- han, 1992), many studies about SAT indicate its high internal consis- tency, predictive, construct and content validity (e.g., Young & Kobrin, 2001). Stanley & Benbow (1981-1982) and Zohar (1990) ar- gued that the SAT, given to 11- and 12-year-old students, is likely to activate their mathematical reasoning, and not only previously learned approaches and techniques. In particular, in many tasks it is possible to avoid computations by constructing shortcut solutions. To illustrate this claim, consider the following SAT task:

Let n! = l- 2- 3- ...-(n - 1) • w be the product of all positive inte- gers from 1 to n. Calculate: jfrpjy-

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Consider two different solutions to the task. First, one can calcu- late the nominator, then calculate the denominator, and finally, reduce the fraction: pypy

= 73$^ = '^q0 = 24. Alternatively, one can solve the task by decomposing the given fraction into two manageable segments: jfy = 6, = 4, and subsequent multiplication of the two results: 6- 4 = 24.

To further contrast these solutions, consider middle school students of a mathematical background similar to that of the partici- pants in our research. We offered this task to a group of such students and closely watched how they solved it. These students were asked to solve the task under time constraints and without using a calculator, as in a real test, and then to explain what they did. In most cases, the students saw the definition of «-factorials for the first time in their lives and started from checking the given definition of «-factorials for concrete numbers. They figured out that 6! = 1 • 2 • 3 • 4 • 5 • 6 = 720, 4! = 1 • 2 • 3 • 4 = 24, etc. Then most of the students used the results of their computation in order to handle the task, which led them to the first presented solution, and, eventu- ally, to the fraction '^q0- None of the students was able to simplify the fraction without a calculator in a reasonable time. "7 see the way of solution, but please, don't make me use a long division, it is too long

" was a typical comment. A few students, who reached the answer, decomposed the given fraction into two manageable ones, as it was shown in the second solution to the task. Our point here is that under the given constraints the above computational task constituted a problem, in which there was room for the use of heuristics like the aforementioned ones "Reduce fractions first" or "Decompose a problem into manageable segments". Thus, we argue that the SAT is a relevant instrument for measuring mathematical achievement at the beginning of a study concerning heuristics.

The posttest consisted of a mathematical reasoning test that we developed as an instrument parallel to the SAT. We call it SAT-2. SAT-2 keeps the structure of the mathematical part of the SAT and includes the same types of problems, but not the same ones. SAT-2 rather than SAT was needed at the posttest to avoid a situation where the students could show progress just due to familiarity with the prob- lems. In order to justify a comparison of the scores, the two tests were given under similar conditions to 381 8th graders who did not take part in the experiment, but studied the same mathematics curriculum and had a similar socio-economic and cultural background to the experimental group. The students were randomly divided into four

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groups. Independent supervisors (not the regular teachers) conducted the two tests (70 items) in four different orders of the items - in order to avoid the situation where the last items of SAT as well as SAT-2 would be unattempted. Then each student's SAT and SAT-2 were

separated and scored. The following statistical parameters were calcu- lated:

(i) The Pearson correlation between SAT and SAT-2 scores was 0.74

(p < 0.0001). (ii) The internal consistency of SAT-2 (calculated by Kuder-Richard-

son formula 20) was 0.78. For the sake of comparison, the inter- nal consistency of the SAT was calculated and was found to be 0.77.

(iii) The hypothesis "the means' difference of SAT and SAT-2 equals 1 point (out of 35)" was not rejected ( t = 0.80; df = 380; p = 0.42).

Thus we concluded that SAT-2 may be used as a tool parallel to SAT after the following adjustment: its score plus one is comparable with SAT scores.

Thinking-aloud interviews

Three three-problem sets were used in thinking-aloud interviews con- ducted at the beginning, in the middle and after the intervention. The time of the interview was not limited; usually, it took 30-90 min, depending on the persistence of the interviewees in solving the given problems. In accordance with recommendations by Ericsson & Simon (1993), the students were instructed not to explain their solutions to the interviewer, but to think aloud "on line". The first problem in ev- ery set was a warming-up task aimed at engaging students in thinking aloud. Only at the warming-up part of the interview, which usually spanned 5-7 min, could the interviewer (the first author) help the stu- dents with encouraging notes and hints. For this reason, this part of the interviews was excluded from further analysis of the students' heuristic literacy.

In the main part of the interview, each interviewee was asked to solve two problems and to think aloud. Sometimes, when the student explained his or her problem-solving steps to the interviewer instead of merely verbalizing ongoing thoughts, the interviewer could left the student alone in front of the camera for a short period of time, asking "don't stop thinking aloud". This methodological ploy, adapted from

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Ericsson & Simon (1993), helped in some "difficult" cases: the students usually started a discourse with themselves more naturally without the interviewer and then kept thinking aloud when the inter- viewer came back and situated himself aside. If the student remained silent for more than 15 s, the interviewer prompted him or her in a neutral manner (e.g., "keep talking", "don't be silent"). Thus, in terms of Ericsson & Simon (1993), the resulting interview protocols were rather concurrent than retrospective in nature. This means that what the students spoke out loud during the main part of the inter- view essentially corresponded to their inner problem-solving speech. In turn, this supports our claim that the collected interview data are valid with respect to heuristics the students' actually used while solving the interview problems.

Three word problems concerning whole numbers (IN, 2N and 3N) and three geometry problems concerning quadrilaterals (IQ, 2Q and 3Q) were used in the interviews (see Figure 1). These problems were carefully designed to meet several conditions. First, the interview prob- lems were based on the concepts taught in the classroom not far from the date of the interviews. This was done to assure that the students

1st interview 2nd interview 3 interview Problem IN Problem 2N Problem 3N The sum of the digits of Represent the number The first digit of a three- a two-digit number is 19 as a difference of digit number is 1. If you 14. If you add 46 to this the cubes of two carry the digit 1 to the end

^ number the product of positive integers. Find of the number, you will get g, digits of the new all possible solutions, a new number. It is given I number will be 6. Find that the difference of the S the two-digit number. new and the original 0 number is divisible by 1 1 . ^ A. Find the original Jš number. 1 B. Find all the possible ^ original numbers.

Problem 10 Problem 20 Problem 30 Check the following Check the following Given a quadrilateral statement: If a statement: If a ABCD. Point E bisects

Q quadrilateral has two quadrilateral has two AB, point F bisects CD, ìg congruent opposite sides right angles and two ri7 _ BC + AD J and two congruent congruent diagonals,

ri7 _ ~~ 2 at

•§ opposite angles then it is then it is a rectangle. can y0U say about a Jjl a parallelogram. If you If you think that the quadrilateral ABCD? I? think that the statement statement is correct, Formulate your conjecture jg is correct, prove it. prove it. Otherwise, an(j pr0ve it. Ig Otherwise, disprove it disprove it by § by counterexample or counterexample or by ^ by any other method. any other method.

Figure 1. Interview problems.

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were familiar with all the definitions and theoretical facts needed in order to understand and approach the problems. Technically speaking, the problems could be solved based only on the knowledge available to the middle school students. Second, the interview problems were chosen to enable both stronger and weaker students to start from the point available for each of them, and then to face a situation of challenge. Thus, the problems were designed to stimulate the students to use a variety of heuristics. Third, for the sake of less confident inter- viewees, the option to address the problems' questions partially, for example, by intelligent guessing, was built into some problems. However, full solutions to the interview problems were out of reach for an average 8th grade student. This was evident from piloting the problems with a group of pre-service teachers, who found all the prob- lems extremely difficult. To illustrate the above conditions, consider briefly the mathematical ideas embedded in the interview problems.

At first glance, problems IN and 3N naturally involve a classical solution by means of equations, and the students learned such solu- tions in a classroom. However, composing equations appears ineffec- tive at a second glance. At first glance, problem 2N may be solved by trial and error. Indeed, it is possible to find one pair of positive inte- gers that fits the problem. In order to find all possible solutions, one can use, for instance, symbolic representation and equations.

Problems IQ, 2Q and 3Q looked like problems that had been recently discussed in the classroom, but they were not. For example, a full solution of 3Q presumes discovery and proof of a converse of the theorem of the median of a trapezoid. Problem 3Q was offered in the interview a week after discussing the theorem of the median of a trap- ezoid in both classes. It was expected that the interviewees could try to adapt the learned proof of the direct problem to its converse, but, to our knowledge, this approach was hardly helpful. The geometric problems are even more difficult than the algebraic ones and, poten- tially, can be handled by means of different auxiliary constructions, arguing by contradiction and indirect use of the "prototypical" theo- rems. At this point, let us recall that success or failure in solving the interview problems was not a factor relevant to the purpose of the interviews. Indeed, the interviews were designed as opportunities to elicit as many heuristics as possible from the students' thinking-aloud speech. We deem that if easier problems or out-of-reach problems with no available starting points were given, they might not have fit this purpose.

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In summary, the three interviews were designed to study the students' heuristic literacy in development, and not to indicate whether they were able to solve six particular problems. Certainly, the interviews were not designed to measure the students' mathematical achievement. On the other hand, SAT and SAT-2 were used in our study to indicate the students' mathematical achievement in development. We have illustrated in the Pretest and posttest section how particular heuristics can help in SAT, but, definitely, this multi- ple-choice standardized instrument was not designed to measure the students' heuristic literacy. In fact, Brody & Benbow (1990) pointed out that "little is known about the factors that contribute to score growth on SAT" (p. 866). Thus, a relationship between these two measures may exist, but is not obvious. This observation adds motivation to our research.

Participants

Two 8th grade classes (N = 37) from two urban Israeli schools took part in the classroom experiment; 12 of the 37 students took part in the interviews. We will refer to the experimental classes as El and E2. There were 24 students, 12 boys and 12 girls, in El, and 13 students, 5 boys and 8 girls, in E2. The students were 11 to 13-year-old at the beginning of the 5-month intervention. Since 7th grade they had taken accelerated mathematics curriculum MOFET3 in which mathematics is taught eight hours a week, 5 h of algebra and 3 h of geometry.

We further describe the research population in terms of the pretest (see Table 2). The difference between the classes with respect to FRT is not significant (df = 35, t = 0.99, p = 0.33). In terms of hypothetically true norms (Raven et al., 1983), the participants' FRT

Table 2. The experimental group at the pretest

FRT SAT

El Mean 23.83a 13.5b SD 2.73 5.34 Range 19-29a 5-27b

E2 Mean 23.00a 12.23b SD 2.27 2.05 Range 19-27a 9-1 5b

a Out of 30. b Out of 35.

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scores mean that from 50% to 25% of a nationally representative sample obtain higher FRT scores than an average El or E2 student.

The difference between the classes with respect to SAT scores was

significant - in favor of El. Specifically, the hypothesis "the difference between El and E2 SAT means equals 1 point" was not rejected (df = 35, t = 0.17, p = 0.43). Moreover, comparison of ranges and standard deviations of the SAT scores (see Table 2) suggests that E2 was more homogeneous with respect to mathematical achievements than El. This finding supports the information we gained from two schools about how the classes had been formed. Two classes were opened for those students, who were interested in the MOFET curric- ulum, but class El accommodated both stronger and weaker students, whereas class E2 accommodated only students who showed relatively low results at MOFET placement tests. The placement tests had been conducted more than a year before the experiment, when the students graduated from their 6th grade. Let us note that we, the researchers, were not involved in the formation of the experimental classes, and took the schools' placement policies as one of the given parameters of our experiment.

Thirty five of the thirty seven students volunteered to take part in three thinking-aloud interviews. Seven and five students were chosen as interviewees from El and E2 classes, respectively. Wiersma (2000) noted that there is no general answer to the question how to specify the number of interviewees at the beginning of qualitative studies and recommended to use a purposeful sampling in educational research instead of a random one. He also noted that although in qualitative studies a satisfactory sample size cannot be specified as an exact number, it may be useful to justify sample selection by quantitative methods.

The students were chosen in accordance with the principle of maxi- mum variation sampling, that is, "[a] selection process that includes units so that differences on specified characteristics are maximized" (Wiersma, 2000, p. 286)

The following factors affected the choice of the interviewees: (i) SAT scores The mean of SAT scores in class El is 13.5 (SD = 5.34), whereas

the mean of SAT scores calculated for the 7 El -students, chosen as interviewees, is 14.42 (SD = 7.91); t = -0.36, df = 29, p = 0.71.

Similarly, the mean of SAT scores in class E2 is 12.04 (SD = 2.05), whereas the mean of SAT scores calculated for the 5

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E2-students, chosen as interviewees, is 12.23 (SD = 2.3); t = -0.15, df = 16, p = 0.44.

The same situation held regarding the two experimental classes as a whole and the entire group of 12 interviewees. The average of SAT scores in the experimental group is 13.05 (SD = 4.47) and for the 12-student sample it is 13.58 (SD = 6.09); t = -0.32, df = 47, p = 0.75.

Therefore, we conclude that seven El students adequately repre- sent class El, five E2 students adequately represent class E2, and the entire 12-student sample adequately represents the experimental group as a whole - with respect to SAT scores.

The additional factors that affected the choice of the interviewees are as follows:

(ii) Proportional representation of boys and girls. There are 5 boys among the 12 interviewees and 17 boys among 37

students in the experimental group. (iii) The teachers' opinions regarding the mathematical background

of the students. The teachers were acquainted with the problems prepared for

the 1st interview and were asked: "Could a student not solve these problems but understand the detailed explanations of the problems' solutions?" A negative answer to the question was a reason for rejection of a candidate. In addition, the teachers' attestations to some of the students being "very strong" or "weak but cooperative" were also taken into account.

Appendix C contains, in particular, information about the intervie- wees that affected their inclusion in the sample at the beginning of the intervention. Pseudonyms of the students are given alphabetically, starting from Alon who had the maximum SAT score and finishing with Lea who had the minimum score - not only among 12 intervie- wees but in the entire 37-student sample. For the reasons listed above, the group of the 12 interviewees is considered a sample representing the experimental group.

Analysis of heuristic literacy development

The videotapes of the interviews were transcribed and analyzed according to the principles of the constant comparison method (Glaser & Strauss, 1967; Dey, 1999). Specifically, the transcripts were segmented into content units and coded in terms of the protocol coding scheme, placed in Appendix A. The codes emerged from the preliminary study dealing with heuristic behaviors of gifted high

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school students (Koichu et al., 2003b). Content units were determined as the largest unbroken parts of the transcript that bear a particular heuristic interpretation. Commonly, content units consisted of several words up to several sentences, which represented a particular step in planning and monitoring of a solution. Comparable segmentation and coding of thinking-aloud protocols can be found in prior research on problem solving (e.g., Schoenfeld, 1985, pp. 301-340; Carlson & Bloom, 2005). Two examples of coding sample protocols are placed in Appendix D.

Thirty six interviews were coded by the first author of the paper. Extended analytical procedure was applied to one protocol (32 con- tent units, 22 min of videotape). First, a mathematics education Ph.D. student, who was unfamiliar with the purpose of the analysis, has been trained how to use the protocol coding scheme and then independently coded the sample protocol. An agreement rate 84% was found, that is, in 27 cases the same heuristic interpretations were assigned by two coders. Five cases of disagreement were resolved by putting the content units into more than one category, when both coders accepted this. Second, in accordance with a tradition borrowed from anthropological research (e.g., Lincoln & Guba, 1985), the findings were presented to the interviewee, Alon. This was done 7 months after completing the intervention. During a 3-h meeting, the student was acquainted with the researchers' interpretation of his heuristic behaviors and asked to reflect on it. The student accepted our interpretation of his videotaped work. Then he was asked to interpret a part of the interview in terms of the coding scheme. Comparing the educators' coding (after the resolution of disagree- ments) with the student's interpretation of his own work, we found a practically full agreement. Let us note that Alon was the most intelli- gent student in the sample (see Appendix C), and that he was unfamiliar with the purpose of the analysis. Thus, we concluded that heuristics indicated as the result of the implemented analytical proce- dure are, in terms of Clement (2000), plausible. In particular, this means that we could determine with some confidence which heuristics the students used in their solutions.

Nevertheless, we argue that reporting exact numbers of heuristics, indicated in one's solutions to the interview problems would be mis- leading with respect to the purpose of the analysis - to trace changes in the students' heuristic literacy. Indeed, transient problem-solving behaviors, captured in terms of the protocol coding scheme, are not fully replicable, thus, direct numerical descriptions of such local behav-

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iors are overwhelming and hardly informative (Goldin, 2000). It should also be taken into account that even though systematic mistakes in coding are unlikely because of the above reported analytical procedure, occasional mistakes could occur. For these reasons, we were compelled to develop a robust criterion of estimating changes in heuristic literacy, which would be insensitive to occasional analytical mistakes, associ- ated with clinical interviewing and the constant comparison method (Erickson & Simon, 1993; Goldin, 2000). In addition, we decided that the intended measure of one's heuristic literacy should be an integra- tive measure, since it was supposed to be confronted with integrative scores of the standardized tests, implemented in our study.

In what follows we introduce such an integrative measure. It is based on consideration of relative heuristic richness of solutions to the problems given in the three interviews. We use the following compari- son criterion: One solution is called heuristically richer than another if the number of different heuristics indicated in the first solution is greater by three or more than in the second solution. Given that 10 different heuristics are considered in our study (see Appendix A), we deem the criterion "... three or more " very demanding, and, in turn, sufficiently robust. It is chosen to decrease chances that changes in students' heuristic literacy during the intervention might be overesti- mated. This criterion is applied to each student individually, for com- parison of her or his solutions by pairs of corresponding problems given in the first, second and third interviews: IN and 2N, IN and 3N, 2N and 3N, 1Q and 2Q, 1Q and 3Q, 2Q and 3Q.

The number of the pairs, in which a solution to the second prob- lem is heuristically richer than a solution to the first problem, minus the number of the pairs, in which a solution to the first problem is heuristically richer than a solution to the second problem, is called a student's index of heuristic literacy development (IHL). By this definition, IHL may take on values from -6 up to 6.

Consider an example. One of the interviewees, George, did his best trying to solve problems IQ, 2Q and 3Q, and finally gave up. Five heuristics were indicated in his approach to 1Q, 6 in 2Q and 9 in 3Q. In accordance with the above criterion, his solution of problem 3Q is heuristically richer than the solutions of problems 1Q and 2Q, which contribute 2 point to his IHL. The solution to 2Q contains more heuristics than to Ql, but the difference is not big enough, so none of these two problems is heuristically richer than another, which contributes 0 to the IHL. In Figure 2, a horizontal arrow denotes the fact that one solution is heuristically richer than

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Figure 2. Examples of computation of indexes of development heuristic literacy.

another, the arrow points to the richer solution. For George and Jacob, the numbers of arrows provide the students' IHL, since all the arrows are one-directional. These data regarding all the intervie- wees are provided in Appendix C.

Results

Changes in mathematical achievement

The adjusted SAT-2 scores will be called POST in the forthcoming analysis of the results. To recall, we have justified that SAT scores and POST scores are directly comparable. Descriptive statistics concerning El and E2 students' SAT and POST scores are presented in Table 3. On average, El students improved their results by 4.38 (SD = 4.53), and E2 students by 4.08 (SD = 3.22). The difference between the two mean improvements is not significant (df = 35; / = -0.20; p = 0.84), that is, both classes demonstrated approximately the same progress. This finding supports the above claim that between-classes differences in conducting the intervention were negligible; this claim will be further supported by means of a regression analysis.

Table 3. Pretest and posttest

SAT POST

El Mean 13.5a 17.88a SD 5.34 4.29 Range 5-27 8-28

E2 Mean 12.23a 16.31a SD 2.05 3.22 Range 9-15 10-20

a Out of 35.

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Next, we claim that the students, who were weaker at the pretest, progressed more than their peers who had higher pretest scores. This claim is supported by the following finding: Pearson correlation between pretest scores, on the one hand, and the difference between posttest and pretest scores, on the other hand, is -0.57 (df = 35; p < 0.001). Note that both weaker and stronger students had enough room for improvement (see Table 3).

Changes in heuristic literacy through the stages of the intervention

To recall, the IHL can by definition take on values between -6 and 6. The empirical range of the IHL is between 0 and 5. As we have argued above, a robust criterion was used to indicate improvements in the students' heuristic literacy. By this criterion, 32 improvements out of possible 72 ones are indicated (see Appendix C). That is to say, if the most successful scenario of the experiment (namely, all the students demonstrate progress in comparison with all the previous interviews both in algebraic and in geometry contexts) is denoted as 100%, the experiment resulted in 44% of success.

Improvements are indicated for 11 out of the 12 interviewees. Fourteen improvements (by 8 students) are indicated in algebraic context; 18 improvements (by 9 students) - in geometric context, seven students demonstrated improvements in both contexts. The intervention fits these findings, since equal effort has been made to promote heuristic literacy in geometric and in algebraic contexts. Most of the improvements are indicated in the 3rd interview, at the end of the 5-month teaching experiment. Only six improvements (by six students) are indicated in the 2nd interview in comparison with the 1st one. Apparently, 2.5 months of the treatment, mainly focused on incorporation of heuristic vocabulary, is not enough. Twelve improvements (by eight students) are indicated in comparison of the 3rd and the 2nd interviews. In the classrooms, the period between these interviews was devoted to the implicit treatment of the intended strategies. Thirteen improvements (by 10 students) are related to the entire 5-month intervention and were found in compari- son of the 3rd and the 1st interviews. These findings bridge the intervention, conducted in the classroom, with heuristic literacy development, measured in the clinical interviews. Our point here is that the intervention indeed dealt with promoting the students' heuristic literacy and not just with incorporating problem-solving opportunities into two classrooms.

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The relationship between mathematical achievement and heuristic literacy development

As one can see in Appendix C, the greater values of the IHL belong to students with relatively low SAT scores. Indeed, Pearson correla- tion between IHL and SAT, calculated for 12-student sample is -0.73 ( p < 0.01). In addition, Pearson correlation between IHL, on the one hand, and the difference between posttest and pretest scores, on the other hand, is 0.82 (p < 0.01). These findings support the conclusion that the weaker students obtained the greatest benefits during the experiment. We give this claim merit, taking into account that there was enough room for improvement even for the strongest students in the sample - with respect to two measures (see Table 3 and Appendix C).

Next, a step-wise regression analysis was conducted to explore covariant-free relationships among all the variables, controlled in our study. POST was chosen as the dependent variable. The variables SAT, FRT, IHL (see Appendix C), GENDER (1 for boys; 0 for girls) and El (1 for the students of El; 0 for the students of E2) were considered as potential predictors of the variance in the posttest scores.

A significant ( F = 13.98, df = 9, p < 0.05) regression model (1) with the entered {p < 0.05) variables SAT and IHL accounts for 76% of the variance ( R 2 adjusted = 0.70):

POST = 3.80+0.83 • SAT + 1.51 • IHL

(0.16) (0.58) ( '

The first entered variable was SAT. This variable explained 57% of variance in POST. This finding means: "the better pretest scores, the better posttest scores", which is hardly surprising. The IHL entered second in the model. It accounted for an additional 19% of variance in comparison with the model that included only one predictor, SAT. Thus, the relationship "the greater heuristic literacy develop- ment, the better posttest scores" is established. This supports our previous claims about the intervention. It also contributes to our understanding of content validity of SAT. Let us recall that Brody & Benbow (1990) pointed out that "little is known about the factors that contribute to score growth on SAT" (p. 866). SAT was sensitive to the intervention described. Therefore, promoting heuristic literacy is one of the factors that contribute to relatively fast score growth of SAT.

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The rest of the variables did not meet the entrance criterion p < 0.05.

Exclusion of variable El that distinguishes the students taught by the different teachers supports the statement that learning/teaching processes in two classes during the experiment were similar. Moreover, this provides an additional justification of the decision to consider the 12-student group as a sample that represents two experimental classes as a whole and not to split it into the two samples that represent each of the classes.

Exclusion of GENDER was a surprise. As we have mentioned in the Pretest and posttest section, SAT is considered a gender-sensitive test in favor of boys. This phenomenon was not replicated in our study, which, apparently, points to a particular feature of the sample.

Exclusion of FRT from the significant regression model was also surprising. Let us recall that Carpenter et al. (1990) directly connected Raven's test with heuristic strategies (see a quotation in the Pretest and post test section). The IHL appears to be a better predictor of the posttest scores. We deem that this fact provides an external support for the above procedure of the analysis of the interviews.

In closing, let us recall that there are reasons (see the Participants section) to generalize the above findings from 12-students sample to the experimental group as a whole.

Summary and discussion

This paper is a report on a classroom experiment concerning promot- ing heuristic literacy through teaching a traditional middle-school mathematical curriculum. By heuristic literacy we refer to students' capacity to use heuristic vocabulary in problem-solving discourse and to approach (not necessarily to solve!) mathematical problems by using a variety of heuristics. Following several literature sources, heuristics are conceived here as systematic approaches to representation, analysis and transformation of mathematical problems. Heuristics in our study is a synergetic concept playing different roles: selected heuristics are both subject matter in the intervention and organizational units in the analysis of the students' thinking-aloud protocols corresponding to transient acts of planning and monitoring in mathematical problem solving. To recall, two research questions are addressed in the paper:

1. How can heuristic literacy be promoted while teaching the traditional mathematical topics of a middle-school curriculum?

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2. How are the changes in heuristic literacy and in mathematical achievement, if any, distributed among the middle school students who took part in the intervention?

In response to the first research question, we outlined the intervention in some detail (see also Koichu et al., 2004) and explicitly stated the instructional principles that determined the intervention and emerged from it. On the one hand, the intervention is based on ideas adapted from past research on teaching/learning heuristics. We critically adapted a number of pedagogical perspectives and learned from both positive and negative experiences available from the extended literature (see Theoretical background section). On the other hand, some elements of the intervention essentially deviate from commonly accepted schemes of teaching problem-solving strategies. Specifically, instead of decomposing universal strategies into more prescriptive substrategies in order to enable students to implement them with adequate domain-specific recourses (Schoenfeld, 1979, 1983, 1985, 1987), we concentrated on teaching universal heuristics that had not been characterized (at least, explicitly) in sufficient detail. Instead, they were incorporated in classroom practice both as metalevel intimations (Sfard, 2002) and as efficient ways of thinking (Harel, in press), which could help in different problem-solving situations. The promotion was not necessarily direct; an effort was made to construct didactical situations that readily involve the intended heuristics. An additional important characteristic of the intervention is that many routine problems and exercises prescribed by the curriculum were used to promote heuristic literacy.

We point out that the age of the participants in our study was lower than in many other experiments on learning/teaching problem- solving strategies (e.g., see Hembree, 1992). It is also promising that the promotion of heuristic literacy was done in a relatively short time in real school conditions, and not in a lab. The claim that the interven- tion indeed dealt with promoting heuristic literacy and not just with opening many problem-solving opportunities is further supported by the analysis of the quantitative relationship between two measures investigated in our study. The first measure is the students' heuristic literacy development derived from qualitative analysis of three rounds of clinical interviews with the representative sample of the students. The second measure is the students' mathematical achievements derived from the standardized tests conducted at the beginning and at the end of the experiment with the whole experimental group. These

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two methodologically segregated measures are crucial in addressing the second research question.

In response to the second research question, we make the following claim:

Changes in heuristic literacy and changes in mathematical achieve- ment are correlated yet unequally distributed among the students who took part in the intervention. In particular, the same students, who progressed with respect to SAT scores, progressed also with respect to their heuristic literacy. Those students, who were weaker with respect to SAT scores at the beginning of the intervention, demonstrated more significant progress regarding both measures.

To our knowledge, this claim is new in the literature about problem solving. It also evokes additional questions: How is the discovered rela-

tionship related to the classroom intervention? Why did the weaker stu- dents benefit from the promoting of heuristic literacy more than their stronger peers? A plausible answer can be found in the nature of the intervention. First, let us recall that the intervention dealt with those heuristics that are more typical for successful problem solvers. There- fore, we suggest that the intervention was more novel for weaker prob- lem solvers. They had enriched their cognitive/metacognitive problem solving repertoire by practicing new strategies, whereas the strongest students might have had these strategies in their "natural" heuristic arsenals prior to the experiment, and then they did not acquire really new problem-solving means, except for heuristic vocabulary. Second, many heuristic-oriented activities were designed so that the weaker stu- dents had more chances to contribute to the whole-class discussions than during their regular classes. This might help the weaker or slower students to increase their confidence and, in turn, to upgrade them from attempts to understand the solutions by others to attempts to produce their own solutions. This explanation is consistent with Schoenfeld's (1987) and Sfard & Kieran's (2001) exploration of the role of social context in the development of problem-solving mastery.

We have tried to show that the intervention is a reasonable explana- tion of the observed phenomena. The claim that we are not making in this paper is that the improvements are due only to the intervention, even though 76% of variance in the posttest scores are explained by the variables in our study. To make such a claim, another research design would be needed. It should include more exhaustive effort to control many potentially influencing variables (Cook & Campbell, 1979). Such an experiment, which can perhaps examine several parallel treatments, may be an interesting future research.

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The presented findings may have important pedagogical implica- tions. First, we have shown that the implemented instructional princi- ples of the intervention provide an answer to the questions, quoted in the introduction, from NCTM (2000). Specifically, we present in some detail a way of revealing heuristic aspects of middle-school curriculum without the curriculum being cut or compromised (see also Koichu et al., 2004). Second, we show how routine problems that inevitably constitute a part of any exam-oriented curriculum can be used as vehicles for enhancing the middle school students' heuristic literacy. Third, an important implication of the study is that heuristic literacy can be promoted in favor of weaker students who often remain out of reach of traditional problem-solving activities. Finally, since the mathematical content of the experiment was curriculum-prescribed and was not chosen by the experimenters, we suggest that many addi- tional mathematics topics and curricula may be reorganized likewise in order to reveal their potential in promoting heuristic literacy.

We conclude the paper by suggesting that this study contributes to our understanding of why many attempts to make heuristics work had rather limited success (e.g., Schoenfeld, 1992; Mayer, 1992). Namely, the reported complex relationship between heuristic literacy and mathematical achievement implies that the same problem-solving activities are of different usefulness for different categories of middle school students learning together. It seems to us plausible that the positive effect that many heuristically oriented interventions similar to ours had on "weaker" students of a particular sample, might be overshadowed by lack of the effect of the same treatment on the "stronger" students of that sample. Consequently, the mean effect sizes of such interventions could be found small (Hembree, 1992). At this point, we must note that the reported (new) relationship was found using a new research instrument. Indeed, the qualitative analysis of classic thinking-aloud protocols resulted in constructing an integrative measure of heuristic literacy development that was con- trasted with the integrative scores of the standardized tests. The idea to construct such a measure is in line with the current methodological call to seek for robust and rigorous methods of qualitative analysis (e.g., Goldin, 2000), but it also needs further validation and testing. We suggest, as another venue of future research, to replicate the experiment in other populations (e.g., with students of different ages and backgrounds). It may be interesting to conduct such experiments with different subject matters, perhaps not only in mathematics.

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Acknowledgments

The research of the first author was supported by the S AKTA RASHI Foundation (budget No 765-076-00) and by the Technion Graduate School. The research of the second author was supported by the Fund for Promotion of Research at the Technion. We are grateful to the teachers of MOFET, the association that has enabled us to gather data for this study for their enthusiasm, professionalism and cooperation.

Appendix A: Protocol coding scheme

Heuristic behaviors Descriptions

1. Planning , including... Evaluating of whether or not it is worthwhile to make a particular problem-solving step or several steps before doing it.

la. Thinking forward Decomposing a given problem into a small number of sub problems, each of which can hopefully be handled separately. The direction of the decomposition is from the givens to a goal state.

lb. Thinking from the Decomposing a given problem into a small end to the beginning number of sub problems, each of which can

hopefully be handled separately. The direction of the decomposition is from the goal to the givens.

lc. Arguing by contradiction Assuming that the conclusion is not true and then drawing deductions until arriving at something that is contradictory either to what is given or to what is known to be true.

2. Self-evaluating , including... Evaluating of whether or not it was worthwhile to use a particular problem solving method.

2a. Local self-evaluating Self-evaluating after a particular step of a solution.

2b. Thinking backward Self-evaluating of an entire solution path, including estimating trustworthiness and ele- gance of the solution and results.

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Heuristic behaviors Descriptions

3. Activating a previous Activating knowledge and problem-solving experience , including... methods that are known to the student from his or

her past learning and problem-solving experience. 3a. Recalling related problems Recalling related problems solved in the past and

using the structure of a known problem to help to solve a new one.

3b. Recalling related theorems Recalling related theorems which may help to solve a problem.

4. Creating a model , including... Representing a problem by means of a represen- tational system different from a given represen- tational system. For example, translating a word problem into algebraic equation.

4a. Denoting and labeling Denoting and labeling the elements of a given problem.

4b. Drawing a picture Drawing a figure, a diagram, or a graph. 5. Exploring particular cases, Approaching a given problem by consideration of

including... its particular cases and (generic) examples 5a. Examining extreme or Assigning special or boundary values to problem

boundary values parameters 5b. Partial induction Search for a pattern or hypothesis by examining

relatively small number of problem parameters. 6. Exploring a particular datum Exploring a particular datum in order to highlight

its role in a problem. For example, one can tem- porary ignore a particular datum to see why it is needed.

7. Introducing an auxiliary Approaching a problem by bringing in an auxil- element iary element that was not mentioned among the

givens of the problem. For example, denoting an algebraic expression as a new variable, building auxiliary construction.

8. Finding what is easy to find Search for additional data, which can be derived easily from the given. For example, two angles of a triangle are given, and a student finds the third angle, even though he or she is uncertain how to use it to progress the solution.

9. Exploration of symmetry Using properties of symmetry either in a geo- metrical or an algebraic sense, i.e. exploring the symmetry of a figure as well as symmetry of cases, when it may simplify the solution.

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Heuristic behaviors Descriptions

10. Generalization Approaching a more general problem than a given one, when it may simplify the solution.

Non-heuristic categories 1 1 . Procedural issues Statements and questions initiated by a student

related to the interview procedure. For example, the student asks for the interviewer's reaction to her or his performance, or manifests that an obligation to think aloud is distractive.

12. Explanations, including... Direct responses to the interviewer's requests to explain or clarify what has been done.

12a. Explanations of silence Explanations to the interviewer of what the stu- dent was thinking when s/he was silent.

12b. General explanations Explanations consisting of the student's reflective thinking on mathematical problems-solving in general.

13. Entering and concentrating A student reads a problem for the first time or comes back to a problem's formulation after beginning a solution.

14. Technical performance Computations, use of algebraic transformation, formulas etc., when a student does not plan or monitor his or her solution.

15. Alien statements Statements that seem to be alien to the topic of the interview.

Appendix B: Episodes from the intervention

Episode 1 : Constructing heuristic vocabulary

At the very beginning of the intervention, the students were engaged in the game "Mouse in the maze." They were given the following task:

There is a 4x 4 maze with the only exit from square A4. A piece of cheese is located near the exit (Figure 3). Some between-square doors are open and others are not. A mouse, striving for the cheese, passed the following path: down - to the right - to the right - up - to the right - cheese. Where was the mouse at the beginning of its way?

Most of the students obtained the correct answer (Bl) quickly and confidently. From the follow up discussion, it appeared that in

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Figure 3. Mouse in the maze.

answering the question the students used one of two different strate- gies. The first strategy, in one student's words, was this: "/ drew all the steps from the end. Instead of "right" -"left", instead of "up"- "down" and so on" Another student articulated the second strategy: "/ started from any point ... from C2... and drew the mouse's way step by step. When I missed the cheese for two squares, 1 moved the entire path one square up and one square left. Then I got B1 ." The teacher encouraged the students to give names to the above strategies. The students suggested: "From the end to the beginning" or "To change a direction" for the first strategy, and "To guess and fix" or "To think forward" for the second one. The teacher listed on the board all the students' suggestions and then modified the game.

Now the students were asked to play in "Mouse in the maze" under different constraints. For example, a new path of the mouse was offered only verbally, and the students were asked to find where the mouse was at the beginning of its way without using pencil and paper. As was evident from the classroom discussions following every round of the game, the constraint "no writing" leads students to refining their previous strategies. For example, at some point the students noted that the strategy "From the end to the beginning" is difficult to implement without writing, since its use includes memoriz- ing the whole path of the mouse. They also observed that "Thinking forward" works even without writing, but having a drawing of the maze on the board is beneficial.

At the next stage, the drawing of the maze was erased from the board in order to make the students rely only on the memorized images of the maze. This additional constraint triggered a new strategy, which was eventually called "Reject what is impossible" or "Neutralize." For in- stance, one of the students observed that the fragment "up-up-up" in a path of the mouse eliminates rows A, B, and C from the list of possible answers. Besides, he noted that fragments like "to the left-to the right" neutralize each other, which was important in overcoming difficulties re-

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lated to memorizing of the longer paths under the given constraints. Afterwards, at the same lesson, the teachers asked their students to recall situations when they used strategies called "Thinking forward", "From the end to the beginning" and "Rejection of possibilities" in mathemati- cal problem solving. The introduced strategies' names helped students to communicate their ideas in many mathematical contexts.

Episode 2: Utilizing heuristic vocabulary in reflective problem-solving discourse

In this episode, "routine" algebraic tasks are utilized to promote the students' capability to think forward and to think from the end to the beginning (see Appendix A for descriptions of these heuristics). The lesson presented here was on algebraic transformations of polynomials.

At the beginning of the lesson, the teacher seized the opportunity to think "on line" in front of her students. When Ada checked the home- work, she found that many students did not solve the following task:

Factor: 30a:3 - 15x2 - 14x2 + Ix - 4x + 2. The students asked Ada to explain it. Ada started to solve the

problem on the blackboard talking thoughtfully and trying different approaches. At that moment, the observer was sure that Ada did not know how to solve this problem and was really thinking out loud in front of her class:

Let's try this: 30x3 - 1 5x2 - '4x2 + Ix - Ax + 2 = 30x3 - 29x2+ 3.x 2

What's next? Let's think forward: I can group the first two... and the second two, but it won't give me something in common. Will it? Perhaps, we can group 30x3 + 3x and -29x2 + 2... no ... the same thing ... Let's start from the beginning ... Oh, I see! 30 and 15, 14 and 7, 4 and 2... Did you try to group the pairs without doing addi- tion of similar terms?"

From this point, the student independently completed the exercise. Then Dan, one of the students who were noticeably impressed by Ada's problem-solving skills, asked the teacher:

Dan: How did you understand what to do? Ada: I tried different things; I was trying to think forward, to plan the solution one step ahead of my writing.

Later on, the students practiced how to transform a product of binomials into a canonic polynomial. They solved several tasks like the following one: (x - 1 ) • (x + 2) = x2 + 2x - x - 2 - x2 + x - 2.

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Then the teacher asked the students to solve a similar task without writing the intermediate steps (e.g.,(x - 2) • (x + 5) = x2 + 3x - 10). Under the constraint "no writing", the routine task appeared to be a problem to the students. The challenge was to keep in memory an intermediate result, and then to manipulate it mentally. In order to bypass this difficulty, the students were encouraged to consider the structure of a final answer before diving into technical details and to organize computations smartly so that the resulting polynomial would emerge gradually, step by step. For instance, one could think first of how to obtain an element with x2, then elements with x and then the x-free part of the resulting polynomial. In the above example, there is only one combination leading to x2,x ■ x - x2, and one can write x2 as a part of the answer. Now, how can one obtain x? There are only two combinations, 5x-2x, and one can write 2>x as a part of the an- swer. Similarly, only one combination contributes to a x-free part in the resulting polynomial, -2- 5 = -10, and thus the answer x2 + 3x-10 is found. In the follow up discussion a few students connected the game "Mouse in the maze" (see Episode 1) to the above algebra task. They pointed out that the constraint "no writing" turns even routine problems into interesting ones.

Episode 3: Heuristic similarities among different problems and the smallest possible help

When the students learned Pythagorean Theorem and practiced it in many problems on right-angle triangles, we designed a lesson based on not straightforward implementations of the theorem. The follow- ing problem was offered at the beginning of the lesson:

Let MNPK be an isosceles trapezoid (see Figure 4a). Prove that d2 - ab + c 2

The students were asked to work in small groups on planning solu- tion to this problem. In 15 min, the teacher collected the students' suggestions and wrote them on the board. To recap, the following plan was created:

- The equation to be proved reminds the equation of the Pythagorean Theorem (at least, in part of d2 = ). The idea appears to insert diago- nal MP into a right-angle triangle. It can be done using auxiliary construction:P//_LMÃ" (see Figure 4b)

- The last step in solution will be applying Pythagorean Theorem to the triangle MPH:*/2 = MH2 + PH2. Thus, our next goal is to express MH and PH using a, b, and c.

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Figure 4. For the problem on isosceles trapezoid.

- On finding MH. MH is a part of MK, greater than "a" and smaller than "6". To find MH precisely, it is worthwhile to have "a" and "b" together on the picture. For this reason, it is worth to build the second altitude of the trapezoid:NQJ_MK (see Figure 4c). Now MH = MQ + QH = MQ + a, and the new goal is to find MQ.

- We still did not use the given that MNPK is an isosceles trapezoid; from here it is possible to prove that MQ = HK. Now the new goal is to show that MQ = HK but this is easy since MQ + HK = b-a, and MQ = MK = (b-a)/ 2.

Of course, the classroom discussion of the problem was not as straight- forward as the resulting plan. For example, the students had difficult time thinking of how to find MQ. At that point, the teacher suggested the students to read again the problem's formulation. This was enough in order to trigger the understanding that the problem's given that the trapezoid is isosceles is still unused, and in turn, that MQ and HK can be found simultaneously. When the plan was understood, the students were asked to complete the solution independently. This work took the rest of the first half of a 90-min lesson, and after a 5-min break the class was continued. The students were given the following problem:

The sides of a triangle are 9, 10, and 17. Find the altitude to the side 17. At the beginning of small-group discussions the latter problem

appeared to be unrelated to the former one. The deep-level connec- tions between the problems became visible when the students drew an appropriate picture (Figure 5) and obtained two right-angle triangles. The students observed:

- As in the first problem, there are two right-angle triangles with the given hypotenuses.

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Figure 5. For the problem on the altitude of the triangle.

- It is enough to find MH or HK, and then to apply the Pythagorean Theorem to a right-angle triangle in order to solve the problem.

- The sum MH + HK is given, but not MH and HK. - The given triangle is not isosceles, and thus similarity with the

previous problem is limited(MH ^ HK).

Based on these observations, two students were able to complete the solution, whereas the others needed some assistance. After a brief consultation with the two students who solved the problem on their own, the teacher offered the following question:

Liora: You have two right-angle triangles including the unknown PH, and the idea is to apply the Pythagorean Theorem to one of them. Which right-angle triangle do you choose?

From the whole-class discussion, it became clear that some of the stu- dents would prefer MHP and the others would prefer PHK (e.g., for the reason that "9 is a nice number" or "10 is a nice number"), but most of the students could not make a choice in favor of one of the triangles. They also noted that in the previous problem about a trape- zoid the choice of a triangle was clear because of the equation that had to be proved. At this point, the following idea was expressed by one of the students: " Let's not make a choice, let's work with two tri- angles simultaneously ." It led the students to using the Pythagorean Theorem twice, and, in turn, to a system of two equations with two unknowns (PH = h, MH = x, KH = 17-x; x2 + h2 = 102 and (17 - x)2 + h2 - 92), which completed the planning stage. Implemen- tation of the plan evoked additional difficulties to those less proficient in algebraic transformations. The students, who solved the problem first, were asked to help the teacher and to advise their peers with the smallest possible help. The students, who solved the problems with assistance from their peers, also became advisers, and formulated their own heuristic intimations, depending on needs of the classmates they were helping.

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Appendix C: Interviewees

Name ¡ Class j FRT SAT POST i Interview problems j IHF 1 i 1 1 I

i i 1 ! 1A 2A I 3A I Alón : El : 29 : 27 ! 28 1 I ! 0

! ! l ! IG 2G ! 3G I j ! ! I i ? I

I

I

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! 1A 2A î

3A ¡

Ben ! El ! 27 23 ! 25 ! - i- 1 2 II II 1G LiG^ j j i i I ¡j ► i

i í ! 1A 2A i 3A Charlie E2 i 26 15 | 19 i ? -► i 1

1 ! ¡ IG 2G i 3G ! i i i j

I 1A 2A i 3A Dalit El ! 27 i 14 15 i 1 1

! ! : IG 2G ! 3G j ¡ i ► i

1

S

i

1A 2A I 3A Elena E2 ¡ 24 ¡ 14 20 _ _ - ¡ » , 4

i ¡ IG 2G i 3G j j I > I

¡|;| 1A j 2A j 3A 5 Fruma El f 23 f 13 f 16 i i i I 2

! ! ! ! 1G ! 2G 3G i j j I I i

I

¡~2Ã

!~3a I George E2 i 22 13 17 ! 2

! IG j 2G 3G ^ i i ; s

^

1A 2A ¡ 3A Hava El 22 11 21 ' - ■» i 2

! 1G 2G 3G j ►

!

Ta TA *~3A llana S E2 25 10 15 i ► 3

S 1G ! 2G 3G ^ i i ^ ^

i TÃ "~2Ã 3a Jacob ! E2 20 I 10 20 ► 5

! ! 1G 2G 3G . 1 S » T ¡ ¡ TÃ ^2Ã "~3Ã

Kalanit ï El i 23 ! 8 20 L ► ======t 5 S 1G 2G 3G

ï ' l ! 1A 2A 3A î

Lea j El 25 5 15 - E. | 5 !

1G 2G ^ ;3G t I

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Appendix D: Analysis of sample interviews

In the presented episodes, Alon and Dalit solve Problem 3N (see Figure 5). In the excerpts, content units are separated by numbers of different heuristic categories in the protocol coding scheme (see Appendix A). The codes are denoted as follows:

(3a): This symbol means that the content unit before it is coded as 3a. In the protocol coding scheme, No. 3a is related to "Recalling related problems" (see Appendix A for description of this category).

(13 + 4): This symbol means that the content unit before it is simultaneously coded as 13 and 4. In the protocol coding scheme, No. 13 is related to "Entering and con- centrating" and No. 4 to "Selecting representation" (see Appendix A for descriptions of these categories).

Alon solves Problem 3N

Notes Content units and codes

Reading aloud quickly The first digit of a three-digit number is 1. If you carry the digit 1 to the end of the number, you will get a new number. It is given that the difference of the new and the original number is divisible by ll.A. Find the original number. B. Find all the possible original numbers (13).

Speaking thoughtfully, with 1- It means ... [Just a] second. The number ... What 3 s pauses, denoted as is written here is that an initial number begins

with 1 and the second number is ended by 1 ? (1 3) OK. Then the original number is 100 and some- thing, or between 100 and 199, and we know that the difference is divisible by 1 1 (4 + 13)

Speaking and writing So ... the given number is 100+1 Ox + j>, and after transformation ... it is 1 + 100x+ 10j> (4)

Speaking thoughtfully We know that first [number] minus the second [one] is divisible by ''(13).

Speaking slowly and writing Then we have that 99 + . . . No ... - 90jc - 9 y is divisible by 11 (4)

Speaking slowly, sometimes Then ... the number is divisible by 11 (13), cri- looking at the page terion of divisibility by 1 1 ... sum of digits divides

... (3a). But I have no ... How can I get from here that the sum of digits is divisible by 1 1 1(1 + 2).

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Notes Content units and codes

Then 99 - 90* - 9y ... we know that it is divisible by 1 1 ( 1 3 + 6) 99 is divisible by 1 1 , then -90*-9 must be divisible by 1 1 (8 + lb), since if sum or difference of two numbers is divisible by the same number, they also are divisible by this number (la + 2a), [mathematically speaking, the state- ment is incorrect]). Then we know that -90x-9 must be divisible by 1 1 (1 3) . They both . . . both are divisible by 9, aha ... to reduce 9 (8), let's say that -'0x-y must divide by 11 (14 + lb ), since it's indeed is divisible by 9 [points to the expression - 1 Ojc- it is -90x-9y. If x and y are equal, then . . . indeed what we will have

Speaking slowly and writing -10*-*, it is -11*, and it ... it divides by 11 (lb + 14).

Speaking thoughtfully, some- So I can say that ... the three-digit number is a times looking at the page number 100... greater than 100 and smaller than

200, and numbers of units and tens are equal (2b) ... Speaking very quickly and They are 111, 122, 133, 144, 155, 166, 177, 188, looking at the interviewer 199 (12a + 14 + 10). [Alon addresses ques-

tion B, which includes addressing question A ] Dalit solves Problem 3N Reading aloud quickly, with The first digit of a three-digit number is 1. If short pauses you carry the digit 1 to the end of the number,

you will get a new number. It is given that the difference of the new and the original number is divisible by ll.A. Find the original number. B. Find all the possible original numbers (13).

Speaking thoughtfully, with So first, I have ... the three-digit number which 1-3 s pauses, denoted as "..." begins with 1, so this is 1 times ... I call a number of

tens X, so '0x times ... and I will call a number of hundreds (13 + 4) ... No, sorry, this should be backward (2a)... 1 is [a number of] hundreds, so this is 100, and this is 10* and this is y [writes 100 + 10*+ (4) If you carry out the digit 1 to the end ofthe number, so it will be y ... (13 + 4). Justa sec, tens and units remain at the same place? Only 1 is moved? OK. (2a + 13) So ... 100*+ 10j^+ 1. It is given that the difference of the new and the original number is divisible by 1 1... OK ... (13)

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Notes Content units and codes

Writes and talks slowly The new number lOOx -h 1 Oy -hl - 1 00 H- 10x+ y = ' IOjc + 1 ly - 99 . . . (14) OK, llQx+lly -99 is divisible by 11 (2a + 13). OK,' i°x+

1^-" _ iox + y - 9. I've divided it by 11.

Talks to Int. What does it mean - to find all the possible numbers? (11)

Int. answers the question To find concrete numbers ... An answer should be a number, a three-digit number.

Dalit is speaking thoughtfully, So I should just find x and y (lb). But there is with 1-3 s pauses a little problem. I have an equation with two

variables (lb + 3). I know just -9, so I know that the first digit is 1 , the number of units is 1 (2a + 13). This means that I should know numbers of hundreds and tens (lb). I don't know how to solve it [Pause 12 s] I obtained an equation with two variables, so I cannot know ...

Int. Dalit: So is there no solution to this problem? Oh, just a second, just a second [points to the expression 10x + ̂ -9], no, I think that it's impossible (2). We have done some exercises in which a sum of the digits was given, I thought that perhaps, I could use it here, but no ... (3)

Int.: Do you want to read the problem again, or to check what you have done?

Reading aloud The first digit of a three-digit number is 1. If you carry the digit 1 to the end of the number, you will get a new number. It is given that the difference of the new and the original number is divisible by ll.A. Find the original number. B. Find all the possible original numbers (13)

Speaking thoughtfully Perhaps, I should not divide it by 11. So ... (1 + 2) The new number is... [Pause 20 s].

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Notes Content units and codes

Int.: Check this, please. You forgot parenthesis... [Int. points to Dalit's writing 100* + lOy + 1 - 100 +10 x+y = 1 10* + 1 'y - 99].

Dalit talks to Int. OK. I'll write it from the beginning. Writes and talks 100* + 1 Ov + 1 - (100 + lO.r- y) =

90x + 9y - 99 (14). So now it is divisible by 9. It is not divisible by 11. 0 f2 + 13). I don't know. I don't know.

Two excerpts are of comparable lengths and reflect the same mathe- matical approach. Namely, the students tried to solve the problem by translating the word problem into algebraic expression(s). In heuristic terms, both Alon and Dalit used the following heuristics: "Planning" (1); "Self-evaluating" (2); "Activating a previous experience" (3) and "Creating a model" (4). The difference in two solutions is partic- ularly apparent when both students achieved the expression 90x + 9y-99 and tried to use it. At that point, Alon utilized two new heuristics "Exploring a particular datum" (6) and "Finding what is easy to find" (8), whereas Dalit did not.

We point out that the analysis undertaken essentially reduces the com- plexity of the protocols. Indeed, the analysis is not sensitive to order in which heuristics were called into play or to numbers of content units to which the fragments were segmented. It is only sensitive to the overall num- bers of different heuristics indicated in the solutions. Let us also note that integrative Indices of Heuristic Literacy (IHL) are computed for each stu- dent individually, that is, direct comparison between Alon and Dalit's solu- tions of Problem 3N is not a necessary part of the analytical procedure. It is presented here anyway to clarify segmenting and coding the protocols.

Notes

1 . By scholastic mathematical problems we refer to mathematical problem in a scholas- tic setting. The scholastic setting dealt with in this paper is a middle school setting. Mathematical problem is a problem that requires use of mathematical concepts and principles (Kilpatrick, 1982). A problem is a task for which the solution method is not known in advance by the person(s) engaged in it (e.g. NCTM, 2000). Note that a particular task can or cannot be a problem depending on the persons engaged in it and also on the constraints under which the task is given (Kilpatrick, 1985).

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2. Harel (in press) coined the term way of thinking to allude to characters of a mental act of problem solving, such as problem-solving strategies, beliefs about mathemat- ics and proof schemes.

3. MOFET is an acronym of "Mathematics", "Physics" and "Culture" in Hebrew. The word mofet also means in Hebrew "shining example".

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