VILMA MESA CHARACTERIZING PRACTICES ASSOCIATED WITH FUNCTIONS IN MIDDLE SCHOOL TEXTBOOKS: AN...

VILMA MESA

CHARACTERIZING PRACTICES ASSOCIATED WITH FUNCTIONSIN MIDDLE SCHOOL TEXTBOOKS: AN EMPIRICAL APPROACH

ABSTRACT. Exercises and problems about functions found in 24 middle-school textbooksfrom 15 countries were analysed using an adaptation of Balacheff’s theory of conceptionsand Biehler’s notion of the prototypical domain of application of concepts in order todescribe the practices associated with the notion of function. The analysis yielded fivedifferent practices – symbolic rule, ordered pair, social data, physical phenomena, andcontrolling image – that were present both across and within the textbooks analysed. Theexistence of different practices might help to explain the compartmentalised and sometimescontradictory notions that students and teachers have about functions and might shed lighton the processes of designing curriculum and instruction on functions.

KEY WORDS: conceptions, functions, middle school, practices, textbook analysis

1. INTRODUCTION

In spite of the pervasive presence of textbooks in schooling and educativepractices, few research studies have focused their attention on mathemat-ics textbooks in relation to their mathematical content. Part of the reasonmight be that in trying to understand the nature of the mathematics thatstudents learn, how teachers use textbooks is considered much more im-portant. Although I agree with this contention, studying textbooks cannotbe neglected either. Textbooks have many purposes. They ‘expound thebody of acceptable theory’ (Kuhn, 1970, p. 10); they are powerful mediafor teaching and learning (Tanner, 1988, p. 141); they ‘determine whatis school mathematics (in a similar way to syllabuses and examinations)’(Dorfler and McLone, 1986, p. 93); they are essential for ‘effective learn-ing in developing nations’ (Farrell and Heyneman, 1994, p. 6360), and‘together with examinations and assessments, serve an accountability andcontrol function’ (Woodward, 1994, p. 6366).

Because what students learn from textbooks and the practicality of thatlearning are mediated by the school context (teacher, peers, instruction,assignments), the textbook is a source of potential learning. It is one ex-pression of the intended curriculum (the goals and objectives for mathe-matics intended for learning at a national or regional level; Travers andWestbury, 1989, p. 6), and thus an analysis of textbook content becomes a

Educational Studies in Mathematics 56: 255–286, 2004.C© 2004 Kluwer Academic Publishers. Printed in the Netherlands.

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hypothetical enterprise: What would students learn if their mathematicsclasses were to cover all the textbook sections in the order given? Whatwould students learn if they had to solve all the exercises in the textbook?Would they learn the particular mathematical notions that are presentedin the textbook? Would that learning work well in their future work inmathematics? The purpose of the study reported here is to describe howsuch analysis for a particular mathematical notion looks like and to dis-cuss the implications of such knowledge in advancing our understandingof the difficulties inherent to the teaching, learning and understanding ofmathematical notions.

2. TEXTBOOK ANALYSES

Herbst (1995) characterises textbook analyses as either external or internalcritiques. External critiques treat the textbook ‘as a piece of technologyinside the educational system’ (p. 2) ‘a technological product, a container,or a funnel of the mathematics to be learned’ (p. 3). Those analyses ‘refer thetextbook to its external environment, that being the educational system, themathematics of the mathematician, or the process of transposition’ (p. 3).In contrast, internal critiques consider the textbook as an ‘environment forconstruction of knowledge’ (p. 3); the interactions of the elements inside thetextbook (e.g. diagrams, examples and explanations) are seen ‘as a productof the conflict between the temporal and spatial nature of texts’ (p. 3).

The external critiques of textbooks have highlighted, among others, thefollowing issues:

• U.S. textbooks are very different from textbooks from other countriesin many aspects, with U.S. textbooks presenting curriculum that usuallylags several years behind comparable textbooks from other countries,or written less tersely and more fragmented, or having more opportuni-ties for students to experience contextualised problems (Li, 1999, 2000;Schmidt et al., 1996; Schutter and Spreckelmeyer, 1959; Stevenson andBartsch, 1992).

• Teachers use textbooks as the main (or the only) source to assign home-work for their students, with novice teachers relying more on textbooksthan experienced teachers (Ball and Feiman-Nemser, 1988; Burstein,1993; Kuhs and Freeman, 1979).

• A topic that is introduced in several countries is not necessarily treatedin the same manner with the same degree of emphasis (Howson, 1995,p. 66).

• Teachers have a tremendous mediation impact between the textbookand students’ learning, and thus the connection between textbook

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content and students’ achievement is not straightforward (Lumsdaine,1963; Remillard, 2000; Stodolsky, 1989).

Examples of the internal critiques of textbooks are Otte (1986, p. 176),who analysed the relationship between illustrations and explanations inmathematics textbooks, and Herbst (1995), who analysed the number lineas a metaphor for the real numbers in a series of Argentinean mathematicstextbooks.

The work reported in this paper attempts to bridge these two typesof the textbook critiques, looking at textbooks as environments in whichknowledge is constructed but at the same time subject to external forcesthat shape such knowledge. In the next section I present the theoretical andanalytical tools I used in this study.

3. THEORETICAL AND ANALYTICAL FRAMEWORK

I adapted Balacheff’s theory of conceptions (Balacheff and Gaudin, 2003)and Biehler’s (in press) prototypical domains of application – works thataddress the meaning of school mathematics concepts – into a frameworkthat allowed me to investigate the nature of the notion of function that waspresent in the textbooks selected. In the next three sections I briefly discusswhy function was selected as a case for the study and how Balacheff’s andBiehler’s works were adapted for the purposes of this study.

3.1. Function in school mathematics

The notion of function is a most important one for mathematics. It evolvedfrom being a numerical entity (as represented by Babylonian tables) tobecoming an equation (for Leibniz and Euler), an arbitrary correspon-dence between numerical intervals (for Dirichlet) and finally a correspon-dence between any pair of not necessarily numerical sets (Luzin, 1998).This last definition, launched at the beginning of the 20th century byBourbaki, brought ‘a coherence and simplicity of viewpoint which did notexist before and led to discoveries. . . that [made] possible major advancesin mathematics’ (Buck, 1970, p. 237).

The idea of incorporating the study of functions in school mathemat-ics can be traced to Felix Klein, who in 1908 was ‘successful in gettingGermany to include analytic geometry and calculus in the secondary schoolcurriculum, [with] other European countries [following] suit’ (Kilpatrick,1992, p. 135). The trend, though slower, was also present in the UnitedStates (Cooney and Wilson, 1993, p. 137). In the late 1950s, the New Math

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movement made a stronger commitment to the use of function as a unify-ing concept for school mathematics. Mathematicians were convinced thatteachers who were willing to ‘introduce the set definitions for relation andfunction in one or more of their classes’ would find that ‘the results may berewarding’ (May and Van Engen, 1959, p. 110). But they could not foreseethat what had been unifying for mathematics created many problems forschool mathematics.

The incorporation of the set-theoretical definition into school mathe-matics stimulated researchers in mathematics education to investigate theconnection between the ‘unifying’ definition and the difficulties that stu-dents face when attempting to use that definition (e.g. Breidenbach et al.,1992; Vinner, 1983). Such investigations have led to the formulation ofschemas for describing students’ understanding of mathematical concepts(e.g. Sfard, 1991, 1992) with functions being a particular case; similarly,in this study, functions serve as an illustrative case of a more general for-malization, but at the same time fills a gap in investigating the role thattextbooks may play in shaping students’ understanding of functions.

3.2. Conceptions

Following the French tradition, the word knowing (connaissance) is usedhere as a noun to distinguish the students’ personal constructs from knowl-edge (savoir), which refers to intellectual constructs recognised by a socialbody. Although both terms refer to intangible constructs, the knowings areparticular to the individuals that have them. Different situations generatedifferent interactions between the subject (i.e. the cognitive dimension ofa person) and the milieu (only those features of the environment that relateto the knowledge at stake), and in consequence lead to different knowings.The different interactions explain the coexistence of multiple knowings bya subject. Contradictory knowings can coexist, either at different times ina subject’s history or because different situations enact different knowings.In both cases, what is isomorphic to the observer – probably the teacher –is not for the learner.

To tackle the problem of the existence of these contradictory knowings,Balacheff and Gaudin (2003) proposed a formalization of a conception C,as:

a quadruplet (P, R, L, �), in which:

– P is a set of problems;– R is a set of operators;– L is a representation system;– � is a control structure (p.10)


P is described ‘from the observation of students in situations to becharacterised with reference to the related content, the criteria being toquestion the specificity of these situation with respect to this content’(p. 10); in other words those problems for which the given conceptionprovides tools to elaborate a solution. R and L are defined in the ‘classi-cal’ sense, R as elementary operations or actions which when organisedinto procedures solve the problem and L as the tools needed to allow theformulation and use of the operators. The control structure ‘is often leftimplicit. . . [and] it can be identified [as] the organised set of criteria whichallow to decide whether an action is relevant or not, or that a problem issolved’ (p. 11).

However, Balacheff reiterates, that the

characterization of a conception. . . is not more on the subject than to the milieuwith which he or she interacts. On the contrary, it allows a characterisation of thesubject/milieu system: the representation system allows the formulation and theuse of the operators by the active sender (the subject) as well as the reactive receiver(the milieu); the control structure allows to express the means of the subject todecide of the adequacy and validity of an action, as well as the criteria of the milieufor selecting a feedback (p. 11).

As a consequence, if

we call knowing a set of conceptions which have the same content. . . we mayspeak of the domain of validity of a knowing (i.e., the union of the domain ofvalidity of the related conceptions) but at the same time we can acknowledge itscontradictory character if one conception is false from the point of view of another(p. 13–14).

This view implies that variations in the set of problems that learnersface, together with the operators, the representations and the metacogni-tive and verification strategies needed to organise the work, lead to dif-ferent characterisations of the conceptions of function: for example, theproblems that Newton faced, mostly based on physical experiments, incontrast to the problems that Dirichlet faced, the analyses of the conver-gence of the Fourier series, required and used a different set of operators,representations, and control structures, which in turn made it possible forNewton and Dirichlet to operate with two different conceptions of func-tion. The interest in establishing the foundations of mathematics at thebeginning of the 20th century and the appearance of set theory led to adifferent set of problems, operators, representations, and control systems,which resulted in yet another conception of function. Given that by thischaracterisation different conceptions of a notion can coexist, one couldask whether or not it is desirable to have them, or whether there are par-ticular problems that may emerge for the learner when he or she is trying

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to reconcile – if that ever happens – those conceptions, or if there could beinstructional strategies that allow the emergence of multiple conceptionswithin a learning trajectory. These are theoretical problems that may be ad-dressed with this framework, when studying an individual’s learning; thus, atransfer of this theory of conceptions into textbook analysis is problematic.Balacheff (April 2003, personal communication) suggests that becausespheres of practice and conceptions are dual notions, an analysis of thetextbooks from the point of view of the spheres of practice they shape mayin fact speak about the conceptions that such practices may stimulate. Andbecause problems and conceptions are of a dual nature (‘on the one handconceptions need problems as constituents of their characterization, andon the other hand problems get their meaning from the conceptions whichcontribute to their solutions and, indeed, form the nature of this solution’Balacheff and Gaudin, 2003, p. 19), an analysis of the problems is crucialfor using Balacheff’s formalisation of conceptions. Thus the main questionthat I sought to answer was: What are the conceptions of function that maybe stimulated by the solutions to exercises and problems of the 7th- and8th-grade mathematics textbooks in a given sample?

I introduced a second notion, prototypical domains of application, toassist in the characterization of the problems.

3.3. Prototypical domains of application of function

Biehler (in press) stresses that the ‘meaning of a mathematical conceptdiffers in different contexts’ (Section 1, paragraph 1). Regarding the taskof teaching mathematics as a social endeavor, he argues that the teach-ing of a mathematical concept cannot be limited to the meaning giveninside the sphere of academic mathematics. Consequently, it is neces-sary to incorporate the meanings given to the concept in other practicesas well:

As mathematics education, however, has to base its curricular decisions on abroader picture of mathematics than that of academic mathematics, we considerthe reconstruction of meaning, the development of a synthesizing meaning land-scape of a mathematical concept, to be an important task for the didactics ofmathematics that could serve as a theoretical background for curriculum designand implementation (Section 1, paragraph 4).

For Biehler, the three elements are constitutive of the meaning of amathematical concept: the domains of application of the concept (its useinside and outside mathematics), its relation to other concepts and its rolewithin a conceptual structure (a theory), and the tools and representationsavailable for working with the concept (Section 3.2, paragraph 1). Usingthe concept of function as an example, Biehler (in press) identifies the


‘prototypical ways of interpreting functions (prototypical domains of appli-cation) which summarize essential aspects of the meaning(s) of functions.’These are natural laws, causal relations, constructed relations, descriptiverelations and data reductions:

The relation between the quantity and price of a certain article is a constructedrelation: it is imposed by fiat (Davis and Hersh, 1980, p. 70). Using a parabolafor describing the curve of a cannon ball has the character of a physical (natural)law. Contrary to this use, a parabola used in curve fitting may just provide a datasummary of the curvature in a limited interval. Using functions for describingtime dependent processes are different from using functions for expressing causalrelations: time is not a ‘cause’ for a certain movement. . . . In many statisticalapplications, functions are used to describe structure in a set of data that cannot beinterpreted as a natural law (Section 3.7, paragraphs 1 and 2).

Biehler’s characterisation of prototypical domains of application offunction provided the initial characterisation for the uses that can be at-tributed to functions in middle school textbooks. The characterisation wasused to classify the problems available to the students.

4. METHOD

The original sample for the study consisted of 35 secondary mathematicstextbooks from 18 countries chosen from the Third International Mathe-matics and Science Study [TIMSS] database1 according to the followingcriteria: the textbook was intended for secondary grades; the textbook waswritten in English, Spanish, German, French, or Portuguese; and the text-book contained references to functions, linear functions graphing in twocoordinates, graphing in the Cartesian plane, tables, patterns or relations.The final decision of keeping a section was determined by its contentand the evidence of an explicit relation to functions. Thus a section on‘tables’ would be considered if the tables were an aid to construct, use, orinterpret functional relations. Appendix 1 contains the list of the 7th- and8th-grade textbooks that were analysed. Only 24 textbooks from this sam-ple were intended for 7th and 8th grades; these are the focus of this paper.I consulted the curriculum guides for mathematics for the countries whenavailable and noted the instances in which the notion of function appearedin specific objectives. It was not possible to systematise the collection ofsuch information and therefore no further analyses on these documentswere conducted. The information, however, was useful for triangulationpurposes.

Digital copies of the selected chapters and sections were made, ascontinuous access to the books was not possible. All the exercises, hereafter

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tasks, from such sections (N = 2,039; of these only 1,318 tasks were fromthe 7th and 8th grade textbooks) constituted the corpus of data for the study.Each task received a 4-tuple code. The first code, P, identified the use offunction present in the task according to Biehler’s characterization. Thesecond, O, contained all the operations that were needed to solve the task.The third, R, contained all the representations that were needed to solve thetask. Finally, � contained all the activities available for the student to verifythat the operations were appropriate, that a solution was obtained, and thatit was correct (hereafter controls). The development of the categories forcoding each element of the quadruplet was accomplished in four steps.

First, I selected one task from the first section of each textbook to analysein depth (35 tasks). I worked each one, following as much as possible thetextbook presentation that preceded the exercise section, and responded tofour questions (What is the use given to function in the task? What doesthe student need to do to solve the problem? Which representations arenecessary to solve the problem? and How does the student know that he orshe has gotten an answer and that the answer is correct?). The answers tothese questions in narrative form were the base for developing categoriesfor each element of the quadruplet.

Second, I used the resulting categories to code the remaining tasks in allthe first sections of each textbook, looking for new categories and refiningthe properties of each. I used the constant comparative method (Glaser andStrauss, 1967) and described the salient features of the categories for anelement and at the same time looked for possible breaks or mismatchesthat could lead to the creation of a new category. For example, to developcategories for operations and controls, I manually compared the narrativesof all the problems looking for common words, activities and processes.When a common activity or process appeared, a short name was assignedand written on an index card, with an abbreviation and a brief description.The continuous comparison of the narratives and the classification of theinstances allowed me to refine the descriptions, constructing terms andsentences that encompassed groups of operations and of controls. Thus, forexample, the operation locate points in a graph was described as follows:

Begin with an ordered pair; the first component is located on the x-axis, and a markis drawn at that point; a perpendicular line through that mark is traced; the secondcomponent is located on the y-axis, and a mark is drawn at that point; a horizontalline is traced through the mark; the point of intersection is the point sought.

The control use check points was described as follows:

There are sentences providing the answers; there are warnings as to what is nota result; the answers to subsequent tasks contradict the answer obtained; there isanother person performing the same activity.


All the descriptions for the operations and the controls were taken fromthe textbooks themselves as they contained thorough descriptions of theprocesses that the students were expected to perform in the tasks and insome cases indications of what to do to make sure that a result was correct.Looking for likely or unlikely results, for example, was a control that wasassigned when there were indicators in the statement of the task (e.g. thestudent obtains a number too big or too small for a given scale in a Cartesianplane, or he or she is getting decimals or negative numbers when whole orpositive numbers are expected, or a set of points in a Cartesian plane arenot aligned on a line) that would help to verify that the procedure or theanswer was appropriate for the task.

As for representations, I began with those given by Balacheff – symbolicand graphical – and added those illustrated in the textbooks: table, picture,arrow diagram, number line, verbal and semi-symbolic. In all, this secondstep involved 518 tasks and resulted in 133 categories.

Because of the large number of categories, the third step consisted ofmerging them within common groups, thus yielding a smaller, more man-ageable number of categories for each element. The final step was to testthe coding system by having other raters use it to code tasks; this processhelped to further refine and validate the categories of the coding system.The coding process required for the raters to have the lessons from wherethe tasks were taken, as one of the assumptions for the codification wasthat students would follow their texts when working with the task and thecodification process accounted for the content that preceded each task. Therange for inter-rater agreement in this stage was 60% to 75%. These resultswere used to produce a refined coding system of 10 codes for uses of func-tion, 36 codes for operations, 9 for representations, and 9 for controls andwas further tested for reliability and validity (inter-rater agreement rangedfrom 80% to 100%). Any given task was assigned a single P code and asmany as needed of the other codes for the other elements of the quadruplet(see Box 1 for an example of such codification).

The codes were reorganised within each element in order to facilitatedata analysis and interpretation. Such reorganization was assisted by theregularities in the data and by the nature of the codes created. One importantcaveat of this reorganization is that it was imposed on data already coded. Adifferent reorganisation of the existing codes might make salient differentaspects of the data. I chose a reorganisation based on frequency of observa-tion of codes; other alternatives – such as content, or complexity – wouldilluminate different aspects of the tasks and the conceptions they may elicit.

In the case of Uses, the collection of all of the contexts and illustra-tions created affinity groups that allowed me to organise the codes intofive categories: physical, that referred to physical phenomena and included

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Box 1. Example of a codification of a task.

cause-and-effect and time relations; social, that referred to human activityand included data reduction and constructed relations; figural, that high-lighted the crucial role of images and patterns in defining mathematicalphenomena and included geometrical, graph-defined, and pattern relations;rule, that highlighted the input–output character of the relation without con-text and included rules and relations that defined proportions; and set ofordered pairs, that referred to relations defined by means of sets and specificassignments.

In the case of operations, the basic criterion for reorganisation wastheir frequency, as determined by configural frequency analysis (CFA, vonEye, 1990, 2000) by country – an analysis that highlights cells in a table


whose frequency is larger (or smaller) than would be expected by chance.Three operations, locating points in a graph, reading points from a graphand finding an element of the range or of the domain of a relationship,were deemed as the most frequent by at least seven countries. In order tocarry out these three operations the relation must be known (either explic-itly or through a graph). The operations produce particular instances ofthe relation (a value or a point in a Cartesian plane) and general featuresmay be obtained by a repeated application of these operations. I called thisgroup of operations manipulate (in the sense of ‘utilize skillfully’, MerriamWebster’s College Dictionary) because they do something with the relation.Six operations – finding the relation between two (sets of) numbers, com-paring without computation, determining the domain and range of therelation, filling a table1, describing the shape of a graph, and determiningwhether a relation is a function – were the most common in at least twocountries and in contrast to the first group, they do not need an explicitdefinition for the relation but attempt to make its features explicit. I calledthis group appreciate (in the sense of ‘grasp the nature, worth quality, orsignificance of’; also, ‘judge with heightened perception or understanding:be fully aware of’, Merriam Webster’s College Dictionary) because theytell something about the relation. Finally, five operations – performing acomputation, conducting an experiment, measuring, listing the elements ofthe relation and giving a definition – were the most common as perceivedby exactly one country. I called this group of operations calculate becausethey do something for the relation. Because each task could have more thanone operation, I characterised each task as having operations in the manip-ulation group only, in the appreciation group only, in both the appreciationand manipulation group, in all three groups, or in no group (other). Thesefive categories were used to characterise all the combinations of operations.

Because there were 70 different combinations of representations, I ex-plored several alternatives for reorganising them. There were some tasksthat used only one representation, but the majority used at least two, whichmade the grouping difficult. A compromise was needed to balance the needfor diversity and the need to highlight particular characteristics of the com-binations. I chose to emphasise the use of the symbolic representation andcreated three groups. The first group, called symbolic only, contained thosecombinations that used either a symbolic or a semi-symbolic representationonly. The second group, called symbolic and other, contained those combi-nations that used the symbolic representation in conjunction with any otherrepresentation (graph, table, picture, number line, arrow diagram, verbal,or numerical); the last group, non-symbolic, contained those combinationsthat did not use a symbolic representation. With this classification, it wasnot possible to make claims about the use of specific representations other

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than symbolic, but the classification highlighted important aspects of theuse of representations in this sample.

The case of the controls was similar to that of the representations inthat there were 74 combinations, but in this case it was possible to regroupthe combinations according to the nature of the activities involved. I de-fined three groups of controls. The first group encompassed activities thatrely on the solution process only to verify that the answer obtained wascorrect as conflicting answers may reveal an inappropriate procedure ora wrong answer. These were double-checking, comparing with previousexamples or exercises and using checkpoints. The second group referred toactivities that require relying on the mathematical content at stake. Thesewere determining whether the relation yielded two different values for agiven number, assuming continuity and using alternative representations.The final group, using a computer or a calculator, looking for likely orunlikely results, and using the given information, encompassed activitiesrelated to the didactical contract (Brousseau, 1997), as these act as indi-cators that a result is not the expected one, for example getting decimalnumbers when integer numbers are expected or a point that can not beplotted in a Cartesian plane (see also Mesa and Herbst, 1997). To suitablygroup the 74 combinations, I characterised each combination according tothe three types of controls. A given combination could have controls ofone type only (process, content, or contract) or a combination of two orthree types. From these seven possibilities, I chose to highlight those com-binations in which the content was important. With this in mind, I createdthree categories: content and other contained all the combinations that hadat least one control of the content type; process-contract, which containedthe combinations with controls of these two types only; and process, whichcontained the combinations with controls of the process type only.

The frequency of occurrence of each possible quadruplet was testedstatistically to identify the most and least frequent configurations of thefour elements using CFA which helped in characterising the practices offunction revealed by the tasks. The reduced sample size did not allow meto carry out statistical analyses by country.

5. RESULTS

That textbooks come in different sizes and shapes has been reported inseveral other studies (Howson, 1995; Li, 1999; Schutter and Spreckelmeyer,1959), and the 24 textbooks in this study illustrated the same variation.Almost every 7th- and 8th-grade textbook contained pictures, drawingsand photographs apparently related to the content but without information


relevant for solving the task (e.g. the picture of a car in a task about distancetravelled per unit of time or the photograph of a plant in a task aboutlength of the leaves of plants). In some countries the textbooks were workbooklets without elaborated explanations for the students. Other textbookscontained text that explained or illustrated related content with task sectionsimmediately after these explanatory sections, yet others had an elaboratedpresentation after all the tasks were presented. These apparently simpledifferences suggest different strategies across countries for organising thecontent but also serves as a cautionary note in interpreting and generalisingthe results.

5.1. Practices associated with functions in mathematicstextbook exercises

The CFA for the configurations of the four elements of the quadrupletyielded 28 configurations whose frequency differed from chance; 24 ofthem were more frequent than what could be expected by chance (types)and 4 were less frequent than what could be expected by chance (anti-types; the test-wise α protected using a Bonferroni adjustment was α∗ =0.00022). To determine what these 28 configurations were portraying, Igrouped them by use and defined five practices around function as elicitedin the tasks of these middle school textbooks: symbolic rule, ordered pair,social data, physical phenomena and controlling image. The configurationswithin each use followed patterns that helped me to better characterise thosepractices. For example, there were six configurations with a rule use; threeof them were types and three anti-types. In the anti-type configurations,non-symbolic representations were always present, which implies that tasksin which there is a rule use for function, non-symbolic representations arerarely used. I interpreted this result as an indication that using a symbolicrepresentation only or in combination with other representations is thenorm. In other words, non-symbolic representations do not tend to appearwhen dealing with functions defined by rules. The analysis classified abouthalf of the tasks (see Table I). In the following sections I describe each ofthe practices.

5.2. Symbolic rule

In tasks that belong to a symbolic rule practice the use of function is asa rule. When manipulation operations are used alone, the representationneeded is symbolic and when manipulation and appreciate operations areused together, the symbolic representation is not the only representationrequired. The task requires controls based either on the process only or

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Table IFrequency and Percentage of Tasks Exhibiting Each Practice

Practice n N = 1318 (%) N = 635a (%)

Symbolic rule 263 20 41

Ordered pair 185 14 29

Social data 93 7 15

Physical phenomena 53 4 8

Controlling image 41 3 6

Not classified 683 52 –

Note: aThere were 635 tasks that were classified into one ofthe five practices. Percentage points may not add up to 100due to rounding.

on the process together with the didactical contract; content-based con-trols are not required. Thus, within this practice two kinds of tasks ap-pear. In the first kind, the student manipulates the relation given in orderto obtain particular values of the function using the symbolic represen-tation; in the second kind, the student manipulates the relation, uses theresults to tell something about the function, and uses representations thatsupport the symbolic one. In both kinds of tasks, the student may ver-ify his or her solutions by repeating the process or by contrasting theresults with hints given by the setting of the tasks. Tasks that portray func-tion machines enact practices of the first kind whereas tasks in which theCartesian plane is being introduced enact practices of the second kind.The following is an example of a task that exemplifies the symbolic rulepractice:

Consider a function h defined as h (x) = 2x + 1.1.1. Find h (–1), h (0) and h (1)2. Find x such that h (x) = 11.

In this task, there is an input x that is transformed by certain procedure –multiply by 2 and then add 1 – to obtain an output; there is no reference to anexternal context. The student has to obtain particular values of the function(at –1, 0 and 1) and a number, a value for x, such that the function is 11when the transformation is applied to that number. The only representationused is symbolic. The student has to repeat the procedure that was givenin the preceding text (substitute the values into the equation), which, at thesame time, acts as the indication that an answer was obtained. It is unlikelythat the student will determine by himself or herself whether the answeris correct. If he or she gets an indication in this sense, the path to follow


would be to repeat the process. Twenty percent of the tasks suggested asymbolic rule practice.

5.3. Ordered pair

In the tasks that belong to the ordered pair practice, the use of functionis as a set of ordered pairs. The tasks require manipulate or appreciateoperations or a combination of the three kinds (manipulate, appreciate,calculate) and they use any of the possible representations and controls.In this practice the tasks offered the most alternatives for the elements ofthe quadruplet. The tasks may be solved using only the symbolic repre-sentation or using a combination of several other representations (e.g. anarrow diagram, a number line, or a Cartesian plane). Because one com-mon operation deals with determining whether a given relation is a func-tion, the controls tended to be based on the conditions defining a function,which is a content type of control. The following example illustrates suchtask:

The domain of the relation R = {(x, y) | y = x2} is {1, 2, 3, 4}. What is the rangeof R? List the couples of R−1.

In this task, operations of all three types are needed. The student needsto use the relation to find the values of the relation at each point of thedomain, which in turn will determine the range; these correspond to ma-nipulate and appreciate operations. The listing of the elements of the inverserelation (exchange the ordered pairs) corresponds to a calculate operation.In this example, the representation used is symbolic only, and the con-trols available are the procedures themselves; the student might repeat thecalculations in order to verify the solution. Fourteen percent of the taskssuggested an ordered pair practice.

5.4. Social data

In the tasks that belong to the social data practice the use of function is so-cial (as constructed relations or data reduction relations). The task requiresappreciation operations alone or in combination with manipulation opera-tions. The relations tended to be defined through non-symbolic representa-tions (tables, graphs or words) but could ask for a symbolic representation.For verifying correctness of the answers and appropriateness of solutionstudents tend to have examples that used positive integer numbers or thatassumed the continuity of the relations. Thus controls were based either onthe content or on the process with the didactical contract. The following isan example of a task enacting this practice:

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Make a table for the following relation: A car gets 26 miles to a gallon of fuel.Show the relationship of the number of miles driven to the number of gallons offuel used.

The text preceding this task illustrated a similar example in which byusing integer numbers (1, 2, 3, etc.) a pattern is sought so as to describe therelation without symbols. In this case the student is very likely to follow theindicated example, which means that she or he will manipulate the operation(to get the values for 1, 2, 3, etc.) and describe it (an appreciation operation).The context does not preclude the use of fractions of gallons, for example,but it is unlikely that the student will choose fractions or decimals to test ordescribe the relation; therefore, controls are of the didactical contract type.Because the number of miles driven increases as the number of gallonsincreases, the content also acts as a control for the task. Seven percent ofthe tasks promoted a social data practice.

5.5. Physical phenomena

In the tasks that belong to the physical phenomena practice the use offunction is physical (cause–effect or time relationships). The tasks requiremanipulate operations alone or in combination with appreciate operations,or a combination of the three groups, or it requires operations outsidethese groups. The tasks do not use symbolic representations. The controlsare based either on the content in combination with other kinds or on theprocess only. Tasks belonging to this practice usually required the studentsto collect data from experiments (e.g. timing a pendulum). The followingexample illustrates such tasks:

The following table shows the distance travelled by a car after the brake is pressedover a dry road; for example, a car driving at 40 kilometers per hour will need18.6 meters to reach a complete stop. Is there proportionality between the speedand the stopping distance?

Speed (in km/h) Stop distance (in m)

40 18.6

50 26.5

60 35.7

70 46

80 57.5

90 70.7

110 101

130 135.6


This task was presented in a chapter on direct proportion and the stu-dent had at hand two strategies to test the proportionality: either to findthe ratios of corresponding entries or to produce a graph for the givenpairs. The tabular presentation suggests the ratio approach as illustratedin the previous examples. The student might need to repeat the proce-dure for almost all the entries because the ratios (distance/speed) of thesmaller numbers seem to group around 0.5. By finding all the ratios, thestudent shows that there is no (direct) proportionality. In this case, the pro-cess is fundamental to the solution and for establishing the correctness ofthe answers. Four percent of the tasks suggested a physical phenomenapractice.

5.6. Controlling image

In the tasks that belong to the controlling image practice, the use of func-tion is figural (geometrical, graph defined, or pattern relations). The tasksuse operations from the manipulate and the appreciate group or from thecalculate with manipulate and appreciate, or from outside these groups.Non-symbolic representations or symbolic representations in combinationwith other representations are required; tasks use combinations of severalkinds of controls. The few instances in which the symbolic representationis used in combination with other representation correspond to cases inwhich the symbols are not manipulated; they act as labels as in expressionslike A = b × h for the area of a rectangle with base b and height h. Thefollowing is an example of a task enacting this practice:

Angular Height

Draw a semicircle with radius 10 cm. Draw several angles x (0 ≤ x ≤ 180◦) withorigin at the centre of the circle and one side lying on the horizontal radius.

For each angle x determine the height y. Draw an approximate graph. Describe thebehaviour of the curve.

This task, which the student has to solve without using trigonometricrelations, uses x and y as labels for angle and height, respectively. Thestudent is not asked to find a relation between the two variables of angle

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and height. To solve the task the student has to collect data by measuringseveral angles and then measuring the corresponding heights. The set ofvalues obtained has to be plotted, and a description of the graph, whichwill be continuous because of how the angles vary, must be obtained.Thus, despite the function not being explicitly exposed, the student usesthe relation to plot the points, discusses it, by describing the graph, anddoes something for it (i.e. collects the data). The picture given in the taskhelps to illustrate that for an angle of 0◦ the height is 0 cm and that for anangle of 90◦ it is 10 cm, which will act as control for the values obtainedfor the height (it must be between 0 and 10). Three percent of the taskspromoted a physical phenomena practice.

5.7. Other results

Over a third of all the analysed tasks (37%) suggested a symbolic rule, anordered pair, or a controlling image practice whereas a small number oftasks (10%) suggested a social data or physical phenomena practice. Theformer three correspond to tasks for which there is no non-mathematicalcontext; in the latter two, the outside-mathematics context is prominent forthe task. The symbolic rule and the ordered-pair practices were present inmost textbooks (71% each), the social data in about half, and the physicalphenomena and controlling image practices in the least number of text-books (33% and 38% respectively). Within textbooks, only 25% exhibitedonly one type of practice – not necessarily symbolic rule or ordered pair –over a half (54%) had two or three practices (usually symbolic rule, or-dered pair, and social data or physical phenomena), and about 20% hadfour or all the practices. These results indicate that textbooks in generaloffer tasks that may enact a variety of practices of function that will be si-multaneously available for students and teachers during the middle schoolyears.

6. DISCUSSION

Figure 1a–e illustrates the different elements highlighted by the codingprocess within each of the practices found. In this section I discuss thedifferences and similarities and propose some interpretations of the re-sults in terms of the possible conceptions of function that such prac-tices may stimulate. The fact that the analysis of this study was morebottom-up than top-down validates these and other results that the litera-ture has highlighted about conceptions of functions in secondary school andbeyond.


6.1. Similarities and differences

The salient characteristics of these practices are the presence of manipu-lation operations and process controls in all of them. The presence of the

Figure 1. (a–e) Graphs of the five practices found in the middle school textbooks analyzed.(Continued on next page.)

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Figure 1. (Continued ). Other refers to operations that were not classified in any of thethree groups, Manipulate, Appreciate, or Calculate and included operations such as givingexamples and counterexamples, carrying out a proof, making a conjecture, or applying atrigonometric relation.

manipulation operations (e.g. find images and pre-images or locate andread points from a graph) in all the practices may be related to the need tofamiliarise students with the notion of dependence or association betweentwo values; for example, for the rule use, mastering the use of the Cartesianplane was signalled as one objective by several countries; such activitiesprovide a relation and require the students to plot points on a Cartesianplane. The easiest way for students to find out whether the results arecorrect is by checking with some external source or by re-doing the calcu-lations. Repeating the operations has the advantage of providing anotheropportunity to rehearse the principles or procedures students are supposedto learn. If it is true that when functions are introduced, textbooks need to


fulfill the demand of familiarising the students with the notion, then givingstudents opportunities to practice the associated procedures is desirable;in controls based on the process students either repeat the solution proce-dures, compare the solution with a given example, or ask for an externalvalidation of the solution. These strategies are also geared to fulfilling sucha familiarisation goal by providing students with another opportunity forpractice.

The salient distinctions are the variety of operations in practices otherthan symbolic rule, the predominance of non-symbolic representations incontext-based practices (social data, physical phenomena, and controllingimage), and the presence of content-based controls in all but the symbolicrule practice.

Of all the practices, ordered pair seems to be the more comprehensive,its elements have aspects of all the categories defined. This could be relatedto the overarching character of the definition. The use of all types of op-erations serves the purpose of showing that the function can be somethingthat is usable and something that one can discuss; the use of several rep-resentations serves the purpose of showing that arbitrary correspondencesare possible, something that cannot be shown when the rule of assignmentis explicitly given (e.g. with an algebraic expression or a Cartesian graph);and the use of several controls also serves the purpose of calling attention tothe processes associated with the definition and to the conditions by whichthe function exists. A large number of tasks required this practice whichencompasses work on the Dirichlet–Bourbaki’s definition of function. Onereason may be an interest on the authors’ part to make sure that the def-inition of function that is accepted by the community is included in thetextbook. Not including the definition could be taken as a symptom of lackof rigour and may devaluate the textbook. A second reason could be that thepush for more ‘down-to-earth’ textbooks, that is, textbooks that distancedthemselves from the New Math or the back-to-basics movements – as re-quested by national commissions (e.g. NCTM, 1980 in the United States)and taken by national organisations (e.g. NCTM, 1989) – may have comelate for most of these authors. Half of the sample of textbooks have copy-rights on or prior to 1990 and even for those published in the early 1990s,the change may not have occurred at all, because as Farrell and Heyneman(1994) have noted, the differences in policies for renewing and adoptingtextbooks have an impact on the rate at which countries assimilate text-book changes, with developing countries changing at a much slower pacethan industrialised countries. One can anticipate different results with morerecent publications, in which set oriented practices may, in fact, be absent.

The tasks that belonged to a social data practice in this sample differ fromthe symbolic rule and the ordered-pair practices. Appreciation operations

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and non-symbolic representations were the more salient characteristics,although manipulation operations are used with appreciation operations.Also, the content and process in combination with the didactical contract actas controls. Formally speaking, there is no difference between the relationscreated from collected data and a definition of function as ordered pair.However, one can speculate that the contextualisation of the task establishesa setting for which the relation has to make sense and this sense makingseems to be the main objective of these tasks. Thus the operations attemptto characterise the relation: to describe it, to produce comparisons withoutusing computations, to find the range and domain for which the relationholds, to fill a table that can be used to better characterise the relation, todescribe a shape in a given graph, or to establish whether a relation is afunction or not. Why symbolic representations play a small role could be aresult of the expected objective of applying abstract ideas or definitions. Itis the case, for example, that in some textbooks, the text preceding one ofthese tasks presented a definition of function as a set of ordered pairs givingdefinitions for domain and range; but these names were not used again inthe tasks, and when they were, the labels appeared without reference toan external context; this is an indication of a separation of the practices inusing the definition:

A relation is a group of ordered pairs. A relation can be shown in a table or agraph. [A graph and a table are provided showing six integral values for numberof gallons (x) and cost in dollars (y) of ethanol fuel.]

The domain of a relation is the set of all the values of x. The range of a relationis the set of all the values of y.

Observe that in these definitions the variables are referred to as x and yand not as number of gallons and cost in dollars. The tasks that followed thedefinition and that dealt with ordered pairs used a symbolic representation,whereas the tasks with a context all required non-symbolic representations.Finally, if the main purpose of these tasks is to make sense of the relation ina given context, then this purpose may justify the use of all the possibilitiesof control but would emphasise controls that depend on the content. Thisis what was found.

One could speculate that the tasks belonging to a physical phenomenapractice could share common characteristics with those belonging to asocial data practice. This is true to some extent. The physical phenomenapractice required operations in the manipulation group mainly and didnot require symbolic representations. One reason could be that time andcause and effect relationships are difficult for students (Monk, 1992); at


these grade levels it may be more important to begin by illustrating therelationship, and therefore more instances to operate with it are given. Also,assuming that the symbolic representation is perceived as more abstract andin consequence more difficult to handle, then it could be expected that oneway to reduce the complexity of the task is to avoid this representation.The preference for giving opportunities to manipulate and make senseof the relation may also explain why controls are based on the contentand on the process. Such tasks act as illustrations of more sophisticatedapplications of function but they may not be presented at a level that mayimpede the students to engage with the activities. Tasks belonging to thispractice required the students to collect data from experiments (e.g. timinga pendulum), which might also explain why the process and the contentwere so frequently used as controls: Students’ unexpected results for thetask could be attributed to the data collection processes or taken as falsifiersof conjectures posed (in the case of the pendulum the conjecture that thesmaller the angle the longer the period, for example, would be falsified bythe unexpected result that the period is the same for different values of theinitial angle). It is interesting to note that of the 136 tasks coded as havinga physical use of the relation, there were only 16 different cause-and-effectrelations (e.g. density of water versus its temperature, Hooke’s law, Ohm’slaw) and 13 different time relations (e.g. speed versus distance, speed versustime, or time versus distance) with several contexts (e.g. people, cars, ortrains) across all textbooks. This implies that there are some paradigmaticways to portray these functional relations and that they are more or lessstandard across countries.

The controlling image practice shares some characteristics with the twoprevious practices: operations from different combinations of groups, non-symbolic or symbolic plus other representations, and all types of controls.Tasks in this practice use a more abstract context as compared to the contextspresent in the tasks belonging to the social data or the physical phenomenapractices and differently from the others, they require combinations ofoperations in more than one group. I interpret this as an indication of thesophistication of the tasks belonging to this practice.

The proportion of tasks for the symbolic rule and ordered-pair prac-tices – in which there is no context involved – is almost twice the per-centage of tasks for the practices that involve a context. One explanationmay be that it is easier to set up a larger number of tasks when there isno context with real-life applications requiring more work on the writ-ers’ part. It is difficult to construct tasks that satisfy academic purposesand at the same time resemble the real situation from which they are de-rived. Similarly, tasks involving physical phenomena may be more difficult

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to set up than tasks involving social phenomena because the former mayinvolve situations that are less familiar to the students or more difficultfor teachers to explain. The low percentage of tasks in the controllingimage practice might be a consequence of the separation of subjects (arith-metic, geometry and algebra) found in these texts. If function tends to beconsidered an algebra topic, it is less likely that geometric situations orpatterns involving numerical sequences will be presented in the algebrasection or be treated under a functional perspective. Context seems to beassociated with the use of non-symbolic representations. The use of con-texts related to the student’s world also satisfies a motivational purposeand at the same time provides an interpretation of an arbitrary correspon-dence between sets as a bi-directional dependence relation. Such taskscan be seen as helping the student make sense of the arbitrariness of thecorrespondence.

7. CONCEPTIONS OF FUNCTION

In 1983, Vinner found that high school students who could recall theDirichlet–Bourbaki definition of function also believed that:

• The correspondence defining a function should be systematic and estab-lished by a rule; arbitrary correspondences are not functions;

• The function must have an algebraic expression, formula, or equation;it is a manipulation carried out on the independent variable in order toobtain the dependent variable;

• A function has two representations, a graphical one (either a curve ina Cartesian plane or an arrow diagram) and a symbolic one given byy = f (x);

• A function cannot have more than one rule; a piece-wise function cor-responds to more than one function;

• A domain cannot be a singleton; a rule with one exception isnot a function; the domain must be constituted by contiguousintervals;

• A correspondence not given by a rule is a function if the mathematicalcommunity has so established it;

• The graph of a function is regular and systematic; and• A function must be a one-to-one correspondence.

What Vinner called compartmentalisation, ‘two items of knowledgewhich are incompatible with each other exist in your mind and you are notaware of it’ (Vinner, 1992, p. 201) could be explained as a result of the


different practices associated with the notion functions present in the cur-ricula and enacted by teachers. Norman (1992) found that secondary math-ematics teachers preferred to use the graph of the relation to establishwhether it was a function or not, or to test characteristics such as continuityor differentiability; tended to think of functional situations as involvingonly numerical inputs and outputs; had a concept definition aligned withthe Dirichlet definition of function but were unable to deal with necessaryand sufficient conditions that determine a function; and had difficulty en-visioning physical situations that entail functional relationships. In 1989,Even found that prospective mathematics teachers in their last year ofpreparation viewed functions mainly as equations; thought that graphs offunctions should be ‘nice’ (continuous) and smooth (differentiable); didnot accept that correspondences could be arbitrary; rejected the notionof constant function, such as f (x) = 4; and thought that the domain andrange of a function should be sets of numbers only. Moreover, when askedabout definitions that they would give to their students, these prospec-tive teachers tended not to use modern terms (e.g. relation, mapping andcorrespondence); used the idea of a machine or black box to illustrate atransformation process; and used the vertical line test to characterise func-tions. Even (1989) attributed these results to a lack of ‘rich relationshipsthat characterise conceptual knowledge’ (p. 266). Prospective teachers hadthe knowledge but were unable to connect the different pieces to make itaccessible. One of the reasons for the compartmentalisation or the lack ofrich representations of the notion could be that the practices associated withdifferent uses of function are suggesting particular conceptions of the no-tion. If we imagine that a student could actually answer all the tasks aboutfunction that the sample in this study collected, he or she would be morelikely to enact a conception of function as a rule that he or she could operatewith, the rule would be expressed by an equation or a symbolic expressionand the student would need an external agent to verify the appropriatenessof the solution process and the correctness of the answer in a task relatedto the function. Had the solution been inadequate, the student would haveto repeat the process used or find the appropriate method and start again.This conception actually provides the student with a concrete set of toolsthat he or she can count on for dealing with functions.

Flexibility is the main characteristic that may be inferred from theordered-pair practice. But this characteristic may in itself be problematic;being so comprehensive and general the student may not distinguish whichfeatures of a function are relevant to know and when those features can beused. When the correspondence between the argument of a function andits value is counter-intuitive, the student might not be able to interpret therelation. One advantage of the other practices (symbolic rule, social data,

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physical phenomena, and controlling image) is that the student may, in fact,be more able to figure out what to do and what to expect. Because thesepractices are more constrained, they may offer the student a more secureground for learning, with a rule to use or a context to limit the possibili-ties of the uses, the student can have more opportunities to explore what afunction is in a variety of ways.

8. CONCLUSIONS

Results from research on student learning have highlighted the importanceof using multiple representations for the construction of meaning formathematical notions; research on problem solving and reasoning hasbrought attention to making metacognitive strategies more explicit tothe students; such innovations are very likely to appear in more recenttextbooks, in particular in those that are part of Standards-based curricula;a similar analysis as the one carried out in this study would indicatewhether that is the case or not.

However, learning different conceptions of a mathematical notion seemsan inevitable part of understanding the notion. We could ask whether allthe practices shown to be elicited by the tasks in these textbooks need tobe part of the learning and teaching repertoire associated with functions.Given that such exposure may be inevitable, we might need to begin tostudy, for example, what emphasis should be given to each practice, whatobstacles are associated with particular conceptions of function, and whatacts of understanding (Sierpinska, 1992) are offered to overcome them.Recognising the mediation role that teachers have may be insufficient fordismissing the results from this study. Teachers do things differently whengiven new textbooks to deliver their lessons (Remillard, 2000) and re-searchers are getting more and more involved in producing curricula thatreflect research results on learning and teaching. The awareness that text-book content may in fact determine different, possibly incompatible, ornot useful, conceptions of mathematical notions should serve as a warn-ings for researchers interested in improving mathematical teaching andlearning.

The formulation and shaping of the categories of controls was one ofthe most difficult steps in making the framework operational, because text-books rarely provide clues as to why topics are treated the way they aretreated, or why one procedure is more effective than another. Very fewtextbooks provided explicit indications for the students to control their ac-tivities (e.g. U.S. textbooks) or problems that are solved in more than oneway (e.g. Mexico), or recommendations as to what the answers should look


like (e.g. the English textbooks). Thus strategies that actually teach the stu-dents how to make sure that the procedure or notion used is appropriate,or that the solution obtained is correct, or appropriate for the problem atstake, are reduced to the cleverness of the students who know that in a taskthat gives a formula, they are not required to sketch a graph. It may be thatthese strategies are typically left for the teacher to illustrate, but the factthat these strategies have not made their way into mathematics textbooksfor middle schools, is problematic. In practical terms, this study highlightsthe need for textbook writers to introduce control strategies explicitly intheir mathematics textbooks.

ACKNOWLEDGMENTS

This paper is based on the author’s dissertation completed under the di-rection of Jeremy Kilpatrick. Many thanks to Nicolas Balacheff and toJeremy Kilpatrick for their guidance in the completion of this study. Dif-ferent portions of the work have been reported at the Annual Meeting ofthe American Educational Research Association, April 2001; at the 25thconference of the International Group for the Psychology of Mathemat-ics Education, Utrecht, July 2001; and at the 12th International Commit-tee on Mathematics Instruction (ICMI) Study Conference: The Future ofthe Teaching and Learning of Algebra, Melbourne, Australia; December2001.

NOTES

1. The selection of the TIMSS database was tied to other purposes of the large studyfrom which this paper is derived, and which is reported in Mesa (2000). The variety oftextbooks provided size and variability to the sample, which was important to test theapplicability of the framework.TIMSS selected textbooks that altogether would be usedby the majority of students in each country. There were countries that provided morethan three series of textbooks (e.g. United States and Switzerland) and countries thatprovided only one series (e.g. Argentina and Singapore).

APPENDIX: TEXTBOOKS ANALYZED IN THE STUDY

A list of the 24 middle school textbooks used in the study classified bycountry and grade for which they were intended. When two years are listedthe first one corresponds to the copyright year.

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Country Grade Textbook

1. Argentina1 8 Sadovsky, P., Melguizo, M. P., and Waldman, C. (1989).Matematicas 2. Buenos Aires: Santillana.

2.Australia1 8 Lynch, B. J., Parr, R. E., and Keating, H. M. (1980/1988).Maths 8. Melbourne: Longman Cheshire.

3.Australia3 7 Lynch, B. J., Parr, R. E., and Keating, H. M. (1981/1991).Maths 7. Melbourne: Longman Cheshire.

4.Austria1 8 Albrecht, R., Gutschi, H. P., Langgner, D., and Wiltsche, H.(1991). Lebendige Mathematik, Band 4. Vienna: Hoelder-Pichler-Tempsky.

5. Canada1 8 Connely, R. D., Lesage, J., Martin, J. D., O’Shea, T., Charp,J. N. C., Beattie, R. H., Bilous, F., Bober, W. C., Drost, D.R., Hope, J. A., Lee, R., and Tossell, S. (1988). Journeys inMath 8. Scarborough, Ontario: Ginn Publishing Canada.

6. Colombia1 8 Londono, N., Guarın, H., and Bedoya, H. (1993). Dimensionmatematica 8. Bogota: Norma.

7. Colombia2 8 Villegas, M. (1991). Matematica 2000. Bogota: EditorialVoluntad.

8. England1 7/8 School Mathematics Project. (1984/1991). Speed 1: Graphs.Cambridge: Cambridge University Press.

9. England2 7/8 School Mathematics Project. (1984/1991). Graphs 2:Graphs. Cambridge: Cambridge University Press.

10. Hong Kong1 8 Chan, L. K. F., Leung, C. T., and Wise, S. R. (1988/1992).Mathematics for Hong Kong. Hong Kong: Canotta.

11. Ireland1 8 Morris, O. D. (1987/1992). Text &Tests, 1. Dublin: CelticPress.

12. Mexico1 8 Alarcon, J., Lucio, M. G., Parra, B. M., Rivaud, J. J.,Waldegg, G., and Rojo, A. (1991/1994). Matematicas 2.Mexico, DF: Fondo de Cultura Economica.

13. Portugal1 8 Ferreira-Neves, M. A., and Carvalho-Brito, M. L. (1992).Matematica 8. Lisbon: Porto Editora.

14. Singapore1 8 Seng, T., Keong, L. and Yee, L. (1987). New Syllabus: Math-ematics 2. Singapore: Shing Lee.

15. South Africa2 7 Laridon, P. E. J. M., Brown, M., Jawurek, A., Kitto, A.,Stafford, H., Strauss, J., Strimling, L., and Wilson, H.(1990/1991). Classroom Mathematics Standard 5. Isando:Lexicon.

16. Spain1 8 Gil, J., Garcıa, P., Vazquez, C., and Mascaro, J. (1984).Matematicas 8. Madrid: Santillana.

17. Switzerland2 7 Holzherr, E., and Ineichen, R. (1972/1986). Arithmetik undAlgebra 1. Zurich: Sabe.

18. Switzerland4 7 Deller, H., Gebauer, P., and Zinn, J. (1992). Algebra 1.Zurich: Orel Fussli.

19. United States1 7 Bolster, L., Boyer, C., Hamada, R., Leiva, M., Lindquist, M.M., Robitaille, D., Swafford, J., van de Walle, J. (1991).Exploring Mathematics Grade 7. Glenview, IL: Scott Fores-man.


Country Grade Textbook

20. United States2 8 Bolster, L., Boyer, C., Hamada, R., Leiva, M., Lindquist,M. M., Robitaille, D., Swafford, J., and van de Walle, J.(1991). Exploring Mathematics Grade 8. Glenview, IL: ScottForesman.

21. United States3 7 Fennell, F., Ferrini-Mundy, J., Ginsburg, H, Murphy, S., Tate,W. Cavanagh, M., Altieri, M. B., Sammons, K., Long, D.,Sherman, C., and Vogeli, B. (1992). Mathematics 7. Morris-town, NJ: Silver, Burdett & Ginn.

22. United States4 8 Fennell, F., Ferrini-Mundy, J., Ginsburg, H, Murphy, S., Tate,W. Cavanagh, M., Altieri, M. B., Sammons, K., Long, D.,Sherman, C., and Vogeli, B. (1992). Mathematics 7. Morris-town, NJ: Silver, Burdett & Ginn.

23. United States5 7 Eicholz, R., O’Daffer, P., Fleenor, C., Charles, R., Young, S.and Bernett, C. (1993). Mathematics grade 7. Reading, MA:Addison-Wesley.

24. United States6 8 Eicholz, R., O’Daffer, P., Fleenor, C., Charles, R., Young, S.and Bernett, C. (1993). Mathematics grade 8. Reading, MA:Addison-Wesley.

REFERENCES

Balacheff, N. and Gaudin, N.: 2003, ‘Baghera assessment project’ , in S. Soury-Lavergne(ed.), Baghera Assessment Project: Designing an Hybrid and Emergent EducationalSociety. Les Cahiers du Laboratoire Leibniz, # 81, Grenoble, Laboratorie Leibniz-IMAG.(Accessed May 2003, available at www-leibniz.imag.fr/LesCahiers/)

Ball, D.L. and Feiman-Nemser, S. 1988, ‘Using textbooks and teacher’s guides: A dilemmafor beginning teachers and teacher educators’, Curriculum Inquiry 18, 401–423.

Biehler, R.: In press, ‘Reconstruction of meaning as a didactical task: The concept offunction as an example’ , in J. Kilpatrick, C. Hoyles and O. Skovsmose (eds.), Meaningin Mathematics Education, Kluwer, Dordrecht.

Breidenbach, D., Dubinsky, E., Hawks, J. and Nichols, D.: 1992, ‘Development ofthe process conception of function’, Educational Studies in Mathematics 23, 247–285.

Brousseau, G.: 1997, Theory of Didactic Situations in Mathematics, Kluwer AcademicPublishers, Dordrecht, The Netherlands.

Buck, R.C.: 1970; ‘Functions’, in E.G. Begle (ed.), Mathematics Education (Sixty-NinthYearbook of the National Society for the Study of Education), University of ChicagoPress, Chicago, pp. 236–259.

Burstein, L.: (ed.), 1993, The IEA Study of Mathematics 3: Student Growth and ClassroomProcesses, Pergamon Press, Oxford.

Cooney, T.J. and Wilson, M.R.: 1993, ‘Teachers’ thinking about function: Historical andresearch perspectives’, in T.A. Romberg, E. Fennema and T. Carpenter (eds.), IntegratingResearch about the Graphical Representations of Function, Erlbaum, Hillsdale, NJ, pp.131–158.

Davis, P.J. and Hersh, R.: 1980, The Mathematical Experience, Birkhauser Boston, MA.

284 VILMA MESA

Dorfler, W. and McLone, R.R.: 1986, ‘Mathematics as a school subject’, in B. Christiansen,A.G. Howson and M. Otte (eds.), Perspectives in Mathematics Education, Reidel, Dor-drecht, pp. 49–97.

Even, R.: 1989, ‘Prospective Secondary Mathematics Teachers’ Knowledge and Under-standing about Mathematical Functions’, Unpublished doctoral dissertation, MichiganState University, East Lansing.

Farrell, J.P. and Heyneman, S.P.: 1994, ‘Planning for textbook development in developingcountries’, in T. Husen and T.N. Postlethwaite (eds.), International Encyclopedia ofEducation (2nd ed., Vol. 2), BPC Wheatons, Exeter, pp. 6360–6366.

Glaser, B.G. and Strauss, A.L.: 1967, The Discovery of Grounded Theory, Aldine de Gruyter,Hawthorne, NY.

Herbst, P.: 1995, ‘The Construction of the Real Number System in Textbooks: A Contributionto the Analysis of Discursive Practices in Mathematics’, unpublished master’s thesis,University of Georgia, Athens.

Howson, G.: 1995, Mathematics Textbooks: A Comparative Study of Grade 8 Texts, PacificEducational Press, Vancouver.

Kilpatrick, J.: 1992, ‘America is likewise bestirring herself: A century of mathematicseducation as viewed from the United States, in I. Wirszup and R. Streit (eds.), Develop-ments in School Mathematics Education Around the World (Vol. 3), National Council ofTeachers of Mathematics, Reston, VA, pp. 133–145.

Kuhs, T.M. and Freeman, D.J.: April, 1979, ‘The Potential Influence of Textbookson Teachers’ Selection of Content for Elementary School Mathematics, ’ paper pre-sented at the annual meeting of the American Educational Research Association,San Francisco.

Kuhn, T.S.: 1970, The Structure of Scientific Revolutions (2nd ed.), University of ChicagoPress, Chicago.

Li, Y.: 1999, An analysis of algebra content, content organization and presentation, and to-be-solved problems in eighth-grade mathematics textbooks from Hong Kong, MainlandChina, Singapore, and the United States, unpublished doctoral dissertation, Universityof Pittsburgh.

Li, Y.: 2000, ‘A comparison of problems that follow selected content presentations inAmerican and Chinese mathematics textbooks’, Journal for Research in MathematicsEducation 31, 234–241.

Lumsdaine, A.A.: 1963, ‘Instruments and media of instruction’, in N.L. Gage (ed.), Hand-book of Research on Teaching, Rand McNally, Chicago, pp. 583–682.

Luzin, N.: 1998, ‘The evolution of. . . function: Part 2’, American Mathematical Monthly105, 263–270.

May, K.O. and van Engen, H.: 1959, ‘Relations and functions’, in National Councilof Teachers of Mathematics (ed.), The 24th Yearbook: The Growth of MathematicalIdeas Grades K-12, National Council of Teachers of Mathematics, Washington, DC,pp. 65–110.

Mesa, V.: 2000, ‘Conceptions of function promoted by seventh- and eighth-grade text-books from eighteen countries’, unpublished doctoral dissertation, University of Georgia,Athens, GA.

Mesa, V. and Herbst, P.: 1997, ‘A problem solving session designed to explore the efficacy ofsimiles of learning and teaching mathematics’, in J. Dossey, J. Swafford, M. Parmantieand A. Dossey (eds.), Proceedings of the 19th Annual Meeting of the North Ameri-can Chapter of the International Group for the Psychology of Mathematics Education,Bloomington/Normal, Illinois State University, pp. 331–338.


Monk, S.: 1992, ‘Students’ understanding of a function given by a physical model’, in G.Harel and E. Dubinsky (eds.), The Concept of Function: Aspects of Epistemology andPedagogy (Vol. 25, MAA Notes), Mathematical Association of America, Washington,DC, pp. 175–193.

National Council of Teachers of Mathematics: 1980, An Agenda for Action: Recommenda-tions for School Mathematics of the l980s, National Council of Teachers of Mathematics,RestonS, VA.

National Council of Teachers of Mathematics: 1989, Curriculum and Evaluation Standardsfor School Mathematics, National Council of Teachers of Mathematics, Reston, VA.

Norman, A.: 1992, ‘Teachers’ mathematical knowledge of the concept of function’, in G.Harel and E. Dubinsky (eds.), The Concept of Function: Aspects of Epistemology andPedagogy ( Vol. 25, MAA Notes), Mathematical Association of America, Washington,DC, pp. 215–232.

Otte, M.: 1986, ‘What is a text?’ in B. Christiansen, A.G. Howson and M. Otte (eds.),Perspectives in Mathematics Education, Reidel, Dordrecht, pp. 173–204.

Remillard, J.T.: 2000, ‘Can curriculum materials support teachers’ learning? Two fourth-grade teachers’ use of a new mathematics text’, Elementary School Journal 100, 331–350.

Schmidt, W.H., McKnight, C.C., Valverde, G., Houang, R.T. and Wiley, D.E.: 1996, ManyVisions, Many Aims, Vol. 1: A Cross-National Investigation of Curricular Intentions inSchool Mathematics, Kluwer, Dordrecht.

Schutter, C.H. and Spreckelmeyer, R.L.: 1959, Teaching the Third R: A ComparativeStudy of American and European Textbooks in Arithmetic, Council for Basic Education,Washington, DC.

Sfard, A.: 1991, ‘On the dual nature of mathematical conceptions: Reflections on processesand objects on different sides of the same coin’, Educational Studies in Mathematics 22,1–36.

Sfard, A.: 1992, ‘Operational origins of mathematical objects and the quandary of reifica-tion: The case of function’, in G. Harel and E. Dubinsky (eds.), The Concept of Function:Aspects of Epistemology and Pedagogy (Vol. 25, MAA Notes), Mathematical Associationof America, Washington, DC, pp. 59–84.

Sierpinska, A.: 1992, ‘On understanding the notion of function’, in G. Harel and E. Dubinsky(eds.), The Concept of Function: Aspects of Epistemology and Pedagogy ( Vol. 25, MAANotes), Mathematical Association of America, Washington, DC, pp. 25–58.

Stevenson, H.W. and Bartsch, K.: 1992, ‘An analysis of Japanese and American textbooks inmathematics’, in R. Leetsma and H. Walberg (eds.), Japanese Educational Productivity,Center for Japanese Studies, University of Michigan, Ann Arbor.

Stodolsky, S.S.: 1989, ‘Is teaching really by the book?’, in P. W. Jackson and S. Horoutunian-Gordon (eds.), From Socrates to Software: The Teacher as Text and the Text as Teacher(Eighty-Ninth Yearbook of the National Society for the Study of Education, Part I),University of Chicago Press, Chicago, pp. 159–184.

Tanner, D.: 1988, ‘The textbook controversies’, in L.N. Tanner (ed.), Critical Issues inCurriculum (Eighty-Seventh Yearbook of the National Society for the Study of Education,Part I), National Society for the Study of Education, Chicago, pp. 122–147.

Travers, K.J. and Westbury, I.: 1989, The IEA Study of Mathematics 1: Analysis of Mathe-matics Curricula, Pergamon Press, Oxford.

Vinner, S.: 1983, ‘Concept definition, concept image, and the notion of function’, In-ternational Journal of Mathematics Education in Science and Technology 14, 293–305.

286 VILMA MESA

Vinner, S.: 1992, ‘The function concept as a prototype for problems in mathematical learn-ing, in G. Harel and E. Dubinsky (eds.), The Concept of Function: Aspects of Episte-mology and Pedagogy (Vol. 25, MAA Notes), Mathematical Association of America,Washington, DC, pp. 195–213.

von Eye, A.: 1990, Introduction to Configural Frequency Analysis: The Search for Typesand Antitypes in Cross-Classifications, Cambridge University Press, Cambridge.

von Eye, A.: 2000, ‘Configural Frequency Analysis: A Program for 32-bit Windows Oper-ating Systems’, (Version 2000), unpublished manuscript, East Lansing, Michigan StateUniversity.

Woodward, A.: 1994, ‘Textbooks’, in T. Husen and T.N. Postlethwaite (eds.),International Encyclopedia of Education (2nd ed., Vol. 2), BPC Wheatons, Exeter,pp. 6366–6371.

VILMA MESA

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