A case study: How textbooks of a Spanish publisher justify results related to limits from the 70's until today

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A case study: How textbooks of a Spanish publisher justify results related to limits from the 70’s until today

Laura Conejo, Matías Arce, Tomás Ortega

To cite this version:

Laura Conejo, Matías Arce, Tomás Ortega. A case study: How textbooks of a Spanish publisher justify results related to limits from the 70’s until today. CERME 9 - Ninth Congress of the European Society for Research in Mathematics Education, Charles University in Prague, Faculty of Education;

ERME, Feb 2015, Prague, Czech Republic. pp.107-113. �hal-01280553�

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publisher justify results related to limits from the 70’s until today

Laura Conejo, Matías Arce and Tomás Ortega

University of Valladolid, Valladolid, Spain, [email protected]

In this paper we present the evolution of the proof schemes shown in grades 11 and 12 textbooks of a Spanish publisher related to the theorems of limits. In order to analyse this evolution, we use a framework de- veloped from the definitions of proof schemes, preformal proofs and functions of proofs. Firstly, we describe our framework and then we show a case study applying the framework to textbooks (from the 70s until today) of a Spanish publisher. Some results and reflections about the analysis are described at the end as well as conse- quences for further studies.

Keywords: Proof schemes, preformal proofs, textbooks, limits.

INTRODUCTION

The main goal of a mathematical proof is to verify the correctness of mathematical statements. Under the perspective of mathematical education research we agree with Hanna (1995), who claims that mathematical proof promotes understanding. Moreover, Hanna

& Barbeau (2010) think that proofs are bearers of mathematical knowledge in the classroom because proofs embody “methods, tools, strategies and concepts for solving problems” (Rav, 1999, p. 6) which is the essence of mathematics. Textbooks (that is, any book used by teachers and students, during a scholar year, in a teaching and learning process of a certain subject, González, 2002) are important elements in the teaching and learning process. Schubring (1987) claims that „teaching practice is not so much determined by ministerial decrees and official syllabuses as by the textbooks used for teaching“ (p. 41). In addition, analysis of textbooks give us information about the mathematical knowledge that a society considers relevant in a particular historical moment (González, 2002) be-

cause they affect what and how students should learn (García-Rodeja, 1997). Spanish Educational System is an example of the effect described by Schubring (1987) because textbooks at pre-university education are usually the main reference material for teachers and students during the scholar year. Since the 70s, several changes of the Spanish Educational Law have occurred so a large amount of textbooks have been published and they have evolved in how they present mathematics.

The limit of a function is one of the most difficult and important concepts which are introduced at pre-university school in Spain. Research about this concept has shown that its learning involves a lot of difficulties.

There are different studies which investigate how this concept could be taught at pre-university school: in an intuitive way (Henning & Hoffkamp, 2013), exploit- ing graphs of functions (Gunčaga, 2009),... Blázquez, Gatica & Ortega (2007) consider that students will be able to understand the ε-δ definition once they have understood the concept of limit as a tendency and approximation. We also think that proving results related to limits contributes to students’ understanding of the concept, so we are interested in how textbooks present the concept of limit and justify the results related to them. Closer to our study, several research- es have been developed about proving in secondary school textbooks in other countries but due to the limitation of space we do not include a detailed description of them but we will compare them with our own work in the future. For example, Nordströn &

Löfwall (2005), who studied proof in Swedish textbooks, noticed that the frequency of proof items is low but they often exist invisible in the textbooks and this is a bit similar in Spanish textbooks. Ibañes & Ortega (2001) studied the student’s proof schemes in the last courses of Secondary school and they conclude that

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A case study: How textbooks of a Spanish publisher justify results related to limits from the 70’s until today (Laura Conejo, Matías Arce and Tomás Ortega)

108 students have difficulties in understanding proofs.

Dos Santos (2010) noticed that mathematical proofs are disappearing in textbooks, that is, the newer a textbook is, the less number of mathematical proofs are in it. We claim that mathematical proofs is important in the understanding of mathematics, so it must be included in textbooks and for that we want to know how textbooks deal with this element of mathematics.

The main goal of this study is to analyse the evolution of mathematical proof in textbooks and if they use other alternative justification procedures. To achieve this goal, we will try to answer some questions: are mathematical proofs replaced by other processes of justification? What kinds of justifications are used?

Are all functions of mathematical proofs considered?

Here we present a case study about the evolution of the treatment of mathematical proof in textbooks belonging to a Spanish publisher since the 70s until the present.

THEORETICAL FRAMEWORK

The aim of our study is to analyse the processes found in textbooks than could be used to convey students about the validity of the mathematical statements formulated in them. We use the concept of personal proof scheme (PPS) defined by Harel & Sowder (1998) because it includes other kinds of justification apart from mathematical formal proofs. Ibañes & Ortega (2001) studied the grade 11 students’ proof schemes and they noticed that students accept any kind of proof schemes as a justification of a mathematical result, and some of these kinds of proof schemes could be found in textbooks. We are conscious that we do not know the intentionality of the publishers/editors.

However, we want to classify the processes shown in textbooks according to the characteristics of personal proof schemes that they exhibit because these processes establish different levels of comprehension of the proofs. For that reason, we have adapted the definition of PPS to textbooks in the following way:

Proof scheme (PS) of textbooks: it consists of what is showed in the textbook which can constitute as- certaining and persuading for a generic reader of this textbook (here, a math student of the grade of the textbook), meaning by ascertaining the process showed in the textbook which could allow the reader to remove her or his own doubts about the truth of an assertion and meaning by persuading the pro- cess showed in the textbook that the reader could

employ to remove other’s doubts about the truth of an assertion.

The processes of persuading and ascertaining are complementary, and both together constitute a PS.

Regarding this definition, we have adapted the categories of classification developed by Harel & Sowder (1998) and Ibañes & Ortega (2001), including a new category using the concept of preformal proof (Van Asch, 1993).

PS0: there are no procedures of justification of the theorem.

Inductive PS of 1 case (IPS1): we convey anyone about the validity of a conjecture by illustrating an example.

Inductive PS of several cases (IPSs): as in the pre- vious case, but now we verify several different ex- amples.

Inductive systematic PS (IsPS): as in the previous cases, but now examples are chosen in a systematic way, out different possible cases.

Transformational PS (TPS): it is done by transfor- mations of elements in a deductive way.

Axiomatic PS (APS): the theorem is proved using ax- ioms, meaning by axioms any primary results and other results which have been deduced previously.

Preformal Proof (PP): a line of reasoning which can be formalised to a formal proof, by in which the es- sential idea is already present. It takes the same character of axiomatic and transformational PS.

Figure 1 shows an example of inductive systematic PS. Something can be classified like an inductive PS if the textbook shows something which suggests that the example could be generalized to any other case.

If not, it could be only considered like an application example of the theorem.

Transformational and axiomatic PS are the closest categories to a formal mathematical proof. Sometimes, we could find that a proof scheme has characteristics of both kinds of proof schemes, so they could be classified in both categories. Anyway, we classify the proof schemes found in textbooks only in the predom-

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inant category to make their first count easier. Figure 2 shows some steps of an example of transformational proof scheme, although it uses some results which make it to be also considered as an axiomatic proof scheme. We classify it as a TPS because the author shows mainly transformations to justify the result.

In the example, the justification ends by calculating the areas described below and comparing.

In the category of axiomatic PS we classify any formal mathematical proof. This kind of proof scheme uses always other theorems that were established before and they use deductive reasoning to prove the theorem. One well-known example of axiomatic PS could be the ε-δ proof to establish the uniqueness of the limit of a function if it exists.

Finally, we show an example of preformal proof.

These kinds of justifications are rarely used but we think that they could be useful at these educational levels because they allow the students to understand a deductive reasoning without doing the abstraction of using a general case. In the example (LOGSE 11, 1998), the author justifies that the limit of a polynomial function at infinity is equal to the limit of the dominant term of the polynomial. The author shows the steps

of the formal proof in a specific function instead in a general polynomial function, so the students only need to substitute by the general function to reach the formal proof.

Let be P(x) = 2x³ – 5x² + 8x – 12. We study the ratio of P(x) by its dominant term 2x³:

P(x)2x3

= 1 −

_2x²

+

_2x2⁸

−

_2x3¹²

^x→±∞

1

Except number 1, when x→±∞ any addend approach to 0. For that reason, the limit is 1.

Other important aspects that we consider are functions of mathematical proofs which are shown in the text. For that, we consider De Villiers (1990) model functions of proof (verification, explanation, system- atization, discovery and communication). As Ibañes &

Ortega (2001) defend, we also believe that the greatest worth of proofs is their function of explanation, although it is not their only value, and formal proofs could be exchanged for other kind of proofs. Apart from these classifications, we consider other aspects of proof in our broader study, but due to the limitation of space, we will not describe these items.

Figure 2: Example of transformational PS of “the limit at zero of f(x) = x/sin(x) is 1” (translated from the textbook LGE 10, 1980) Figure 1: Example of inductive systematic PS to show the limits of functions defined

by natural number powers (translated from the textbook LOGSE 11, 1998)

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110 METHODOLOGY

Our research problem is related to the teaching and learning process of mathematical proof. We want to know how proofs have been treated in Spanish textbooks since the 70s. In the 70s it was enacted the first Spanish Educational Law which structures all the Spanish Educational System for the first time, they appeared the first common curricula and the govern- ment made control of the published textbooks. After that, there have been two changes in the Educational Law, and also in the curricula. As Dos Santos (2010) claims, we think that proofs are disappearing of Secondary School textbooks. In Spain, textbooks are used by almost every teacher and in almost every classroom of pre-university education, so the analysis of textbooks gives us information about the teaching of mathematics of a special moment. Due to the historical character of our work, we have combined two methods which are appropriate to this kind of work, the method of historical research on education (Ruiz- Berrio, 1976) and the research process on education described by Fox (1969). The combination of both methods gives us a method but here we only show a part of the general study, considering the analysis items of proof schemes and preformal proofs. By using these analysis items, we have classified the justifications which appear in textbooks according to them.

Then, we have compared the different PS or PP used in each textbooks and their evolution along the time.

Although we are analysing textbooks of four different Spanish publishers (selected according to two crite- ria: publishers that have been widely used by Spanish mathematics teachers and those which exist since the 70’s), here we show the analysis of one of them as a case study. We focus on proofs of theorems related to limits which are taught at pre-university levels, that is, the last courses of Secondary School (grades 11 and 12, 16–17 and 17–18-year-old students). The textbooks of this publisher correspond to three different education laws: the first one, LGE, was promulgated in 1970, the second one, LOGSE, in 1990, and the third one, LOE, in 2006, and it is still in force for the considered courses.

The sample is composed by seven textbooks: one corresponding to LGE (grade 10; limits do not appear in grades 11 and 12), four corresponding to LOGSE (we have found two different collections corresponding to different years which differ on the treatment given to limits, and we have the textbooks of grades 11 and 12 of both collections) and two related to LOE (grade 11 and grade 12). We will use the code Law G (Year) for each textbook to simplify the notation, where “Law”

denotes the Law of the textbook, “G”, the grade, and

“Year”, the year of edition. In the Table 1 we show a summary of our sample.

ANALYSIS RESULTS

Each textbook considers a different treatment of the concept of limit and different ways of justifying the results that they show. We have only found some si- militude in the presentation of the concept and its properties between textbooks of the same collection.

For example, LGE 10 (1980) considers an unusual way to present limits: it firstly defines the continuity of a function and it presents limits in the chapter of de- rivability (in fact, it uses a description of limit in the definition of continuity). In the study of limits, the textbook shows the behaviour of functions in points of discontinuity and then, it defines the concept of limit of a function.

Textbooks corresponding to second period (LOGSE) present limits in a different way depending on the collection: the older collection (textbooks from 1998 and 1999) presents limits like a tool to study functions and their graphics; for example, they give a description of limit using an example and then, they give examples and describe the properties of the limit of a function.

The newer collection (textbooks from 2003 and 2004) makes a more traditional presentation of limits: they do not present limits like a tool to study functions but they give formal definitions, the one in terms of sequences and the one in terms of absolute value (but they don’t justify the equivalence between different definitions). Textbooks corresponding to the third period (LOE) give a simpler treatment of limits: firstly

Education Law LGE (1970–90) LOGSE (1990–2006) LOE (2006–14)

Textbook LGE 10 (1980) LOGSE 11 (1998)

LOGSE 12 (1999) LOGSE 11 (2003) LOGSE 12 (2004)

LOE 11 (2008) LOE 12 (2009)

Table 1: Sample of textbooks analysed

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they consider all the definitions of limit of a function (finite limits, infinite limits, limits at a point or limits at the infinite) and then they formulate the arithmetic properties of limits. The differences between grade 11 and 12 are that in grade 12 definitions are formal (it gives the ε-δ definition) instead the intuitive definition of grade 11 (we say that the limit of a function f(x) at the point a is L if the function f(x) approach to L as much as we want, whenever we take a value for x sufficiently close to the value, LOE 11, 2008) and it goes deeper in the calculus of limits than the textbook corresponding to grade 11.

As we have seen before, there are significant differences in how textbooks deal with the concept of limit and the definitions that they use (formal, intuitive, no definition,...) These differences in how to present and define the concept affect the kind of proofs which can be used (informal definitions do not allow the editor to make formal proofs). Regarding the properties of limits, the uniqueness is only formulated in LGE 10 (1980); one-sided limits always appeared in these textbooks (they are necessary in the study of functions) but they are not always used to characterise the limit of a function: LOGSE 11 (1998) and LOGSE 12 (1999) are textbooks that do not link the existence and equality of one-sided limits to the existence of the limit of a function at a point. The arithmetic of limits is formulated in all textbooks but it is never justify except in one textbook (LGE 10, 1980). Indeterminate forms are always presented in the textbooks so, that let us think that textbooks are more oriented to promote me- chanical calculus of limits than understanding of the concept. However, some textbooks give justifications of some cases of indeterminate forms, so we think that this indicates the consideration of the explanation function of proof in some cases.

Table 2 shows a summary of the proof schemes used in textbooks to justify the results corresponding to limits. We notice a big difference on the number of results considered in each textbook. It is due to the different ways in which the textbooks present the properties of limits: some of them consider separately the arithmetic of finite limits at a point, of finite limits at the infinity, of infinity limits at a point and of infinity limits at infinity; other ones consider only the difference between finite and infinity limits; there are textbooks which study the limits of families of basic functions, and consider them like theorems... This diversity in what a textbook considers as a result affects the amount of results presented in textbooks. Regarding the kind of proof schemes used, we notice that there is a significant change after 1999: the amount of results is smaller than in the previous years, but the number of justifications too. The predominant behaviour in all textbooks is not to justify the results (77.7% of results are classified as PS0). Among the different categories of justifications, it depends on the textbooks: LGE 10 (1980) only uses transformational and axiomatic PS;

the textbooks of the older collection of LOGSE use all kind of justifications, but the predominant PS are inductive PS of 1 case in grade 11 (16%) and transformational PS in grade 12 (14.3%); on the contrary, the newer collection does not justify except two inductive PS of several cases in grade 11; finally, textbooks of LOE mainly use inductive PS of 1 case (18.2% in grade 11 and 16.7% in grade 12).

The results justified by textbooks are generally related to limits of any kind of functions (potential, polynomial, rational, logarithmic or exponential functions). Textbooks never justify the properties related to arithmetic of limits (except the addition in LGE 10 (1980)). Moreover, this one is the textbook which uses more axiomatic PS, although it is not the one which shows more results. We specify the results

Textbook Proof schemes Total

results

PS0 IPS1 IPSs IsPS TPS APS PP

LGE 10 (1980) 21 0 0 0 2 3 0 26

LOGSE 11 (1998) 16 4 3 0 1 0 1 25

LOGSE 12 (1999) 35 3 1 1 7 1 1 49

LOGSE 11 (2003) 9 0 2 0 0 0 0 11

LOGSE 12 (2004) 23 0 0 0 0 0 0 23

LOE 11 (2008) 9 2 0 0 0 0 0 11

LOE 12 (2009) 9 2 0 0 1 0 0 12

Table 2: Proof schemes used by textbooks

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112 which are proved in each textbook in Table 3. Due to

the number of inductive PS found in textbooks, we think that authors try to convince students about the validity of the results from intuition and not from mathematical reasoning.

Regarding the functions of proofs, the kind of results that are justified and the kind of PS used let us think that the predominant function that is showed in textbooks is communication. We also consider the function of verification in the justifications classified like axiomatic or transformational PS. The function of explanation could be noticed in some textbooks which select examples that contribute to explain why the result is true, but we do not notice this function especially in proofs but in the application examples of theorems. Indeed, this function appears in results related to calculus of limits (indeterminate forms or the arithmetic of limits).

FINAL REMARKS AND QUESTIONS FOR FUTURE RESEARCH

As we have seen, there is not a significant fall of the number of mathematical proofs in textbooks since the 70’s in this publisher. That is because there are only three axiomatic PS and two transformational PS in the first book and there are four axiomatic PS and

eleven transformational PS in all of them. However, we have noticed that the number of justifications have dropped along the time. It can be due to diversity in how textbooks deal with limits: for example, the newer collection of the LOGSE period does not show formal definitions, so they do not justify. Textbooks make few references to the justifications procedures used, only in some cases they specify that they are justifying.

We think that more justifications must be included in textbooks and it should be indicated that this is a justification and what kind it is. We conjecture that all textbooks (except LGE 10, 1980) have the intention of teaching students to calculate limits but they are not so interested in the understanding of the concept.

We will go deeply in this aspect in future research.

The analysis shown in this paper gives a first insight in this topic, but there are a lot of open questions. For example, we have said that the differences of justifications founded in textbooks probably depend on the way that these textbooks introduce the concept of limit, so it is necessary to study how it affects to proofs.

Other future questions to answer are: Which is the best way (from a didactical point of view) to present and develop this concept? Is it better to consider intuitive definitions of the concept than a formal one?

We think that a good way to present limits is to adopt a rigorous (but not formal) definition of limit in the

Textbooks Proof schemes

LGE 10 (1980) TPS: equivalent infinitesimals, limit at zero of f(x) = x/sin(x)

APS: uniqueness of limit, addition of finite limits at a point, limit at infinity of logarithmic function

LOGSE 11 (1998) IPS1: limits of types →k/→∞ and →k/→0, limits at infinity of polynomial functions, limit of exponential functions

IPSs: limits at infinity of f(x) = xⁿ, g(x) = x^-n, h(x) = x^1/n TPS: limit at zero of f(x) = x/sin(x)

PP: limits at infinity of rational functions

LOGSE 12 (1999) IPS1: limit at infinity of f(x) = x²+k, limits of exponential and logarithmic functions IPSs: limit at infinity of f(x) = 1/xⁿ

IsPS: limit at infinity of f(x) = xⁿ

TPS: limits at a point of rational functions (→0/→0), limit at zero of f(x) = sin(x)/x, addition of infinities, equivalent infinities, equivalent infinitesimals, limits of (1 + f(x))^1/f(x) and f(x)^g(x) (when they contain indeterminate forms, →1^→∞)

APS: limits at infinity of rational functions PP: limits at infinity of polynomial functions

LOGSE 11 (2003) IPSs: limits at infinity of polynomial and exponential functions

LOE 11 (2008) IPS1: equivalence of existence and equality of one-sided limits and the definition of limit, limits at infinity of rational functions

LOE 12 (2009) IPS1: equivalence of existence and equality of one-sided limits and the definition of limit, limits at infinity of rational functions

TPS: limit at zero of f(x) = sin(x)/x Table 3: Results related to the proof schemes used

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sense defined by Blázquez, Gatica & Ortega (2007) and then, introduce the formal definitions. An example of these authors’ definition is: the limit of a function f(x) at a point a is L if, for every approximation K of L, K≠L, there exist a punctured neighborhood of a such that the images of all its points are closer to L than K. We also recommend using preformal proofs, as a way to introduce formal reasoning, using specific functions that allow students to understand the mathematical reasoning without abstraction. This kind of reasoning is also preferred by students as it is claimed in González (2012). Finally, this case study is not enough to know how textbooks deal with the concept of limit and the justifications, but it let us realize of the difficulties of this kind of analysis (for example, how to compare among proofs when results are organizing or formulated in different ways), and that will help us in our future work.

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