Concept maps as a tool for evaluation of modern physics contentsin textbooks1

O uso de mapas conceituais como ferramenta de avaliação do conteúdo de física moderna contido noslivros didáticos

Prof. Dr. Luiz Adolfo de MelloPhysics Department, Universidade Federal de Sergipe, 49100-000, São Cristovão-Sergipe, Brazil

ladmello@uol.com.br

(Recebido em dia de mes de ano; aceito em dia de mes de ano)

__________________________________________________________________________________Abstract: CM is a useful tool for evaluating the instructional sequence or conceptual construction of textbooksfor the basic cycle of the course of Physics and Engineering as for high school. We show that CM is to theanalysis of the concepts of the science the analog of structured language is for programming. As an application,CM will be used to evaluate the topic called Modern Physics written for Basic Physics. We use as theoreticalframe the theory of didactic transposition of Chevallard and generalized by Izquierdo-Aymerich, Adúriz-Bravoand others.

It can be observed that most Modern Physics books follow the curriculum structure proposed in the book“Quantum Physics” of the authors R. Eisberg and R. Resnick. But some stand out by offering some alternativeproposals for the instructional sequence of exposure of subject. Thus, we face the question: Can we evaluate andcompare the sequences of didactic presentation of topics in modern physics that led to the development ofSchröndiger Equation. What are the main topics that should be given greater emphasis and which could betaught superficially, or even deleted?

Concept maps built for the contents of Modern Physics show us the most important links between the keyconcepts introduced by the authors. They show how concepts are sometimes anticipated, sometimes postponedand sometimes partitioned to provide support and consistency to topics and chapters which will follow. The CMbuilt for high school textbook could tell us the level o math used in its text. Key Words: Science teacher education, cognitive science and concept mapping.

Resumo: MC é uma ferramenta útil para avaliar a seqüência instrucional ou construção conceitual com que oslivros didáticos são escritos, tanto para o ciclo básico do curso de Física e Engenharia como para o ensino médio.Mostraremos que MC está para a análise dos conceitos da ciência como a linguagem estruturada está para aprogramação. Como uma aplicação, MC será usado para avaliar o tópico denominado Física Moderna escritopara livros de física básica. Utilizamos como referencial teórico a teoria da transposição didática de Chevallard egeneralizada por Izquierdo-Aymerich, Aduriz-Bravo e outros.

Pode-se observar que a maioria dos livros de Física Moderna segue a estrutura curricular proposta no livro"Física Quântica" dos autores R. Eisberg e R. Resnick. Mas alguns se destacam por oferecer algumas propostasalternativas para a sequencia didática de exposição do assunto. Assim, nos deparamos com a pergunta: Será quepodemos avaliar e comparar as sequências de apresentação de temas da física moderna que levaram aodesenvolvimento da equação de Schröndiger. Quais são os principais tópicos que devem ser dadas mais ênfase eque poderiam ser ensinados superficialmente, ou mesmo suprimidos?

Mapas conceituais construídos para o conteúdo de Física Moderna nos mostram as ligações mais importantesentre os principais conceitos introduzidos pelos autores. Eles mostram como os conceitos ora são antecipados,ora adiados e, por vezes divididos para fornecer suporte e consistência aos tópicos e capítulos que se seguirão. OMC construído para um determinado livro poderia nos dizer o nível de matemática utilizado em seu texto.Palavras Chaves: Ensino de Ciências, Mapas Conceituais, Transposição Didática.

1. Introduction

When teaching a QM course in postgraduate we replace the textbooks written for graduation by thepostgraduate level, that is, by textbook containing higher level mathematical content. At the moment

1 With the support of the Professional Master's Degree in Physics Teaching of SBFísica

that the postgraduate course is not intended to the formation of researchers but to teachers the questionarises: what are the textbooks with “higher knowledge” and suitable for the educator? How, in general,is very difficult to separate the physics from its underlying mathematics, there has very few solutionsto this problem. One option is to teach how the contents of a particular topic of knowledge is built ordeveloped. This allows students to obtain a meta-understanding of the subject. Metaphorically, itwould be like we gave them an external reference to the subject. The idea is evaluate physics conceptsin order to teach the QM, or any subject, in a postgraduate level. For this there are textbooks assupport material for this approach.

Before the 50s there wasn’t great concern about the proper production of textbooks. Until this timeteaching was based mainly on memorizing formulas and texts, so that the textbooks reflect thisideology. On the other hand there was not a very clear distinction between university careers, so therewere very few texts or none for specific careers formation.

After II World War and the advent of the Cold War, the U.S. felt the need to produce a greatteaching program aimed at training of scientists in general. In 1956 a group of professors, high schoolteachers of physics and from the Massachusetts Institute of Technology (MIT), led by JerroldZacharias and Francis Friedman, formed the Physical Science Study Committee (PSSC) to think andpropose ways to renew the physics teaching in introductory courses [1, 2]. Those educators decidedthat appropriate textbooks could stimulate, at least in part, the students’ interest in the subject and getthem to think like a scientist.

Despite the excellent material produced, this first attempt was unsuccessful precisely because theirgoal was being focused on the forming of scientists. But this project has inspired the creation ofvarious projects world apart from that (Harvard; Nuffield; PEF) [3, 4, 5]. The most important teachingproject inspired by the PSSC is the Harvard Project Physics Course or Harvard Project Physics [3, 6].The Harvard Project Physics is based on a humanistic approach of doing science and in this project theconnective pedagogical approach was proposed. Teaching material and therefore its books werewritten with the goal of educating citizens in general and not only scientists.

With the growing volume of information that students must acquire to graduate forced the creationof university careers specialization and with them the necessity of creation of several educationalprojects by universities. As an example we have textbooks for physical scientists (Alonso; Halliday)[7, 8] and others for engineers in general (Searwey; Young) [9, 10].

Many of these authors are allocated in research groups and universities with very differentcharacteristics that will be named here epistemosphere. To some extent these characteristics must insome way be inserted into the material produced by them. Through the use of CM, it’s possible toshow that aim of training and expertise of these influenced the didactic structure of their textbooks.

Although the modern physics content is inserted in most modern textbooks written to high school,there is a debate about the question if it is effectively taught. An obstacle mentioned by many inachieving this goal is the fact that modern physics is all based on sophisticated mathematical andusually cannot be transcribed in terms of math skills of a high school student [11]. Another objectionis that many principles that are grounded in Quantum Mechanics, as of Uncertainty andComplementarity, are very abstract and not intuitive even for a university student.

Thus it would be very interesting if we had a way to evaluate how the authors of textbooks arebridging the knowledge produced in the scientific sphere (epistemosphere) to the sphere of high school(noosphere). Thus, in the following “concept maps” will be used as conceptual tool to evaluate howthe authors of physics textbooks for high school are doing the didactic transposition of the topic ofModern Physics.

2. Didactic Transposition.

In 1980, the mathematician Yves Chevallard [12] generalizes the idea of Didactic Transpositionoriginally formulated by sociologist Michel Verret, in 1975, in a theory and applied it in the analysisof important issues in the field of didactics of mathematics. In the 90's the scientist Nersessian [13],based on Johnson-Laird ideas [14, 15], generalizes the theory of epistemology of science of Kuhn[16], Feyerabend [17] and others creating the cognitive theory of science. In the last decade authors asIzquierdo [18], Izquierdo-Aymerich [19], Harrison [20] and others have applied the ideas of

Nersessian [13] to the study of DT, focusing the DT analysis on the cognitive processes of the humanmind and epistemology of science.

The Merriam-Webster Online Dictionary defines [21]:Science (Scientia the Latin, translated as "knowledge") refers to any known orsystematic practice. In the strict sense, science refers to the system of acquiringknowledge based on the scientific method and to the organized body of knowledgeachieved through such research. (Webster, as cited in Wikipedia).

But2, scientists propose theories, conceptual models and methods to formulate their explanatorygoals, but this is not entirely possible for students at school [19]. According to Izquierdo-Aymerich[19] when we simplify or define, with didactic purposes, what is science or to do science we candescribe it as a way of thinking and acting in order to interpret certain phenomena and to intervenethrough a series of theoretical and practical structured knowledge. As a result of science education isdesirable that students understand that the natural world has certain characteristics that can be modeledtheoretically. Because of this we present to them, making a DT, some reconstructed facts, simplifiedtheoretical models, arguments, metaphors and propositions that were previously selected. Obviously,this is not "to do science" but "teach science," and the reason for this behavior can be found in thedidactics of science and not just in the science [19].

Also, if the science class is done in accordance with the principles of meaningful learning [22, 23](Ausubel), that is, a well executed didactic transposition [12] (Chevallard), teachers will be engaged inthe task of connecting scientific models to used by the students themselves, using analogies andmetaphors that can help them move from last to the first [24, 25, 26 ,27; As cited in Izquierdo, 18].

The theory developed by Chevallard deals with the process of how the knowledge produced in theresearch environment arrives or is transposed into the classroom environment. To that end it dividesthis process into three stages. The knowledge produced in the research environment, knowledgetranscribed to text books and knowledge as taught in the classroom. But the DT does not occur entirelywithin the classroom or in the "teacher's room." After Chevallard to understand deeply how scientificknowledge is transcribed to the textbooks we have to include in this analysis the external environmentin which this occurs. According to Pietrocolla, he defines the Didactic Transposition as an efficienttool for analyzing the process by which the knowledge produced by scientists (the scholarlyknowledge) becomes that which is contained in the programs and textbooks (the Knowledge to beTaught) and mainly in what actually appears in classrooms (the Knowledge Taught). After the PSSCand Harvard Physics Projects we are left the impression that the Knowledge to be Taught and theKnowledge Taught are little different from those in the laboratories and research groups. This way ofconceiving teaching brings the idea of simplifying the knowledge.

Chevallard [as cited in Brockington; 28] points out that the way the Knowledge to be Taught isderived from the scholarly knowledge is one of the fundamental points in all didactic. Thistransformation occurs within a university environment (the Didactic System) that lies within a smalluniverse that is the external environment (the Educational System).

Chevallard focuses his study on the process of transformation, simplification and selection ofknowledge produced in scientific research to be transcribed to textbooks on factors outside the schoolsystem, in the wider environment, where all three spheres coexist and influence (the scholarlyknowledge, the Knowledge to be Taught and Knowledge Taught).

The ideas and concepts developed by Chevallard [12] were developed in the study of the passage ofthe "knowledge" from the research environment to the high school. In this didactic transposition modelit defines or considers that the research environment is unique. That is, that the knowledge produced inthe research environment is produced in the final form to be transposed directly (consumed) for highschool. But the theory of DT can be applied to higher education [29] since the transformation of thescholarly knowledge begins in this Knowledge sphere (or Epistemosphere). Moreover, thistransformation process has fewer steps than the previous, and therefore it is somewhat simpler. Wewill show here that the DT occurs in cascading from the research environment for the universityenvironment and from this system teaching for high school.

2 The following two paragraphs are a collection of statements that together form a definition of that is the DT from the TCC point of view.

Thus, we focus the analysis on the process undergone by knowledge when to be transcribed fromthe scientific environment to the classroom (high school). With the expansion of the publishing marketwe have a variety of textbooks produced within this Epistemosphere. This created the possibility andthe need to produce new proposals for education. This production generated a certain amount oftextbooks with specific characteristics and objectives. With the spread of courses and graduateprograms it created another substrate between the knowledge produced in research spheres and thebasic university education. We now have five levels of presentation or transposition of knowledge.The level: 1) Research; 2) Postgraduate; 3) Academic; 4) Basic graduation and finally 5) high school.(Note: In the US and in some countries the high school students have the option to choose betweenthree levels of education. We have the Academic Level, Academic plus (or Applied) and with Honors.The level with honors is equivalent to basic university level).

In this series of articles we will analyze and classify the process and the steps that the knowledgeproduced in research spheres suffers to get to the university environment [29] and then to the highschool. We will investigate how the mathematical and epistemological difficulties filter certainconcepts and favor others. As the introduction of Teaching Methods changed the structure and thechronology in which science subjects are covered. Following we will show how the construction ofCM, to the knowledge produced and translated into an appropriate language to a certain level ofstudents, provides us with information on the conceptual construction of this knowledge.

Thus, we are led to ask: Scholarly knowledge is directly transcribed to ‘Know to be Taught’ (toclassroom) or it occurs in stages until high school? How can we 'track' (investigate) the transformationof knowledge as it is being 'simplified' (like say Prof. Alberto Villani: Diluted) until it reach theclassroom?

3. Concept Mapping

Concept Mapping is one of the ways to do structured conceptualization. Structuredconceptualization (SC) is any process or sequence of defined steps that results in a conceptualrepresentation [30]. Joseph D. Novak [31] defines in broad manner what conceptual maps are:

Concept maps are graphical tools for organizing and representing knowledge. Theyinclude concepts, usually enclosed in circles or boxes of some type, and relationshipsbetween concepts indicated by a connecting line linking two concepts. Words on theline referred to as linking words or linking phrases, specify the relationship betweenthe two concepts.

Several authors (Novak, Moreira, Pankratius, Akinsanya, da Silva 2007; and references) [32, 33,34, 35, 36, 37 e 38] stand up for the use of concept maps (CM) as potentially useful tools in teaching,learning assessment and analysis of curricular content. We can construct concept maps to graphicallyrepresent an entire discipline, subdiscipline, a specific topic of a discipline and so on.

When the CM is well constructed allows the visualization and perception of how the key conceptsof a particular topic or field of knowledge succeed, intertwine and organize themselves in thestructuring of this knowledge. Several authors perceive the CM utility and created some basic rules forthe construction and standardization of it. [31, 32, 34, 35 e 38].

Despite of these rules the concept maps is a very flexible tool and can be used in several ways. As stated by Moreira [34]:

"There are no fixed general rules for drawing maps of concepts. The important thing isthat the map is a tool to highlight meanings to concepts and relationships betweenconcepts in the context of a body of knowledge, a discipline, and a field of education."

Thus, for specific purposes and as it will be needed here, it is necessary to define well-establishedrules in the construction of CM. It is argued here that CM is for the analyze of the concepts of sciencethe analogue of structured language for programming.

4. Concept Maps and Curriculum Planning

Due to its concise, graphical and hierarchical form of presenting the key concepts to be taught wehave that CM is a powerful tool to make curricular analysis [31, 34]. The hierarchical organization ofconcepts facilitates the visualization of the optimal sequence of presentation of content both by thosewho organized as by the students or the reader in general.

And since the fundamental characteristic of meaningful learning is integration of newknowledge with the previous concepts and propositional structures of the students,proceeding from the more inclusive or more general concepts to more specific information,we have that CM serve to encourage and improve meaningful learning [Novak, 31].

5. Concept Maps and the Analysis of Textbook

Generalizing this idea we use CM to analyze the conceptual structure that textbooks are written. Asstated above due to its concise, graphical and hierarchical way of presenting key concepts, allowsviewing promptly and succinctly the conceptual framework that a particular author used toconcatenate and organize the key concepts that enter in the preparation of its textbook.

A simple analysis of the index of a book or booklet does not allow us to promptly visualize theunderlying structure of the conceptual construction of a body of knowledge. The statistical counting ofthe number of concepts discussed in a text also does not give us a clear idea how these are connectedtogether to form a conceptual framework. We will show below that the usage of CM help usunderstanding this conceptual construction.

We will do below a comparative analysis of four modern physics texts in order to illustrate thisidea. In this article the analysis will be restricted to the content called early days of quantummechanics until the introduction of the Schrödinger equation.

Part 1 - University Level

We use as a reference textbook the book ‘Quantum Physics’ of the authors Eisberg and Resnick[41], written to the academic level of the physics course. Another feature of this book is that it waswritten to form scientists, in particular to train physicist to work with Modern Physics. As this is oneof the first books used in modern physics courses and it is an improved version of 1961 book of thefirst author, entitled Foundations of Modern Physics, it is natural that the most authors of modernbooks have studied with this textbook or at least used it as one of its references [29]. As comparativeexamples we will analyze two other updated textbooks written to training engineers to work withmodern world.

Let’s beginning with Eisberg’s book. It contains the following topics: Cap 1. Thermal Radiation and Planck's PostulateCap 2. Photons - Corpuscular Properties of Matter. Cap 3. Wavelike properties of matter.Cap 4. Bohr's Model for the Atom.Cap 5. Schrödinger Equation.

Let us use this book to define the rules used here to building the conceptual maps. When the text istoo broad or when each chapter or section contains many topics, we put the main topics in a centralvertical column and its subtopics will appear in a lateral horizontal line. If we need to introduce asubsection we could use a diagonal line, as representing a depth, giving a three-dimensional characterto the CM. Due to the difficulty to putting a subject entirely and clearly in a single CM sometimes weuse linking sentences instead of linking words. When necessary, glossary of abbreviations can appearin brown boxes. In order to highlight a concept referenced in the text, we connect to it a brown boxing.

For example we have the CM-1 of Eisberg’s book, Figure 1. In these the sequence of the chaptersis shown in the center column and the main sections are presented on the sidelines. For our purpose,unlike 'traditional' CM, we will not use cross lines to indicate that a well defined concept within aprevious chapter or section will be used in a later chapter or section to support another concept. This isexpected occur normally in a middle school text, but, as we will see this isn’t necessary in theuniversity level. We will use this strategy only when we need to indicate the cases in which a concept

is used or cited in a previous section and later defined or introduced. When necessary, in secondaryschool textbook, we will introduce equations in yellow boxes and use as connecting word “math”.

As in this text the explanation of each topic is fully contained within the previous chapter and itsconcepts are used in sequence in other chapters, we don’t have crossed lines between secondarycolumns. We will see below a counterexample. In the end of the first chapter there are someapplications of the theory of Planck’s and these are in the same order of importance amongthemselves. Thus, we put them in a depth dimension of equal importance, so that this CM has threedimensions.

It is worth noting that as the emphasis in the textbook is to present experimental facts that result inthe formulation of a particular theory or model we notice that this presents some experimental factsbefore the model or theory that explains these, disregarding the chronological order of their discoveryor formulation. For example, the "wavelike properties of particles" appear before the Bohr model,although chronologically the Bohr model has been proposed before the theorem of De Broglie.Confirming the theory of DT which states that the way to teach science often differ substantially fromthe way of doing science.

As said earlier, the Eisberg book aims to train scientists, mainly to work with high-energy physics.Thus, in the chapter "The Bohr model for the Atom" they do a detailed study of the Rutherfordscattering model, which will be deleted from the majority of textbooks made to the basic or academiccycle.

5.1 Analysis of University Basic Level Textbooks

As an example of textbook that contains Modern Physics (MP) written for engineering course andthat follows almost the same structure as Eisberg’s text, we have the book 'Principles of Physics' of theauthors Serway & Jewett [9]. The MP contents of this book are almost all contained in Chapter 29.

Note that in this collection the Bohr's model is described in Chapter 11 (Mechanics) along with thegravity model, and then it is only used in section 29.7 in the chapter ‘Atomic Physics’. Herewith theauthors condense in this chapter the whole explanation for the atomic structure in the atoms with asingle electron. The analysis of their content tells us that this text should have the same structure as theEisberg text. To check this we do a CM, Figure 2.

In CM-2 (Figure 2) we see that from the ‘blackbody radiation’ passing through the ‘photoelectriceffect’, until "The wavelike properties of particles" the text is simple and succinct and resembles thestructure of Eisberg’s book subtracting the scattering theory of Rutherford. Here we see as the contentwith a high degree of mathematical difficulty is filtered in the DT. In the 'quantum particle' section theauthors change the instructional sequence to take advantage of the concepts of physical optics taughtpreviously and the previous knowledge of differential equations to introduce the concept of quantumparticle and motivate the need to postulate an equation for the wave particle. See CM-2. We verify inthese sections, especially when they needed discuss ‘the physics of a particle in a box’ beforeintroducing the Schrödinger equation, the difficulties encountered by the authors in performing thedidactic transposition of a texts produced for the academic level to a text fitted to the basic cycle of aengineering course. We note in Section 7 to 13 the phenomenon of dilution of the knowledge, that is,as a topic of Modern Physics usually addressed in one section breaks it into several sections.

Figure 1: Concept Maps of the first five chapters of the Eisberg book.

Figure 2: Conceptual Map of the topic Modern Physics of the book of the author Serway (2006).

As a textbook that does not follow the didactic structure of Eisberg we have the University Physicswith Modern Physics, Vol. 2. 12th Edition, Sears and Zemansky’s (Young and Freedman) [41]. Thecontents of this book are distributed in three chapters namely:

Cap38 - Photons, Electrons and Atoms. Cap39 - The Wavelike Nature of Particles. Cap40 - Quantum Mechanics.

For our purpose only the chapters 38 and 39 will be analyzed. For best viewing we divided the CMin two parts. Although this sequence be very similar to the Eisberg's sequence, we can see in the CM-3and CM-4, Figure 3 and 4, that conceptually this is completely different. As the text is contained inonly two chapters we put their content into a single one central column. Since the sequence of contentis too long (the phenomenon of dilution of the knowledge) its contents are presented in a very longchain of boxes.

Early on we see that this book was written with a strong emphasis on the point of view of optics.Already in the first section they present a description of the theory of emission and absorption of light(see Box side of CM-3) which use the term photon to describe electromagnetic radiation before theintroduction of this term in the photoelectric effect chapter. They split the chapter between theory ofdiscrete and continuous spectrum of light emission. So they postponed the "blackbody radiation"(which will be taught in the continuous spectrum section) and start with the photoelectric effect (PE).See box in brown in Fig.3. This is possible because the authors are considering that the universitystudents had a first contact with this subject in the high school. But, this wasn’t possible when the firstversion of Eisberg’s book was written.

As we see in CM-3, figure 3, they start addressing the experimental facts which characterize thePE. In sequence they define the Einstein theory to the Photoelectric Theory. Next they use the Theoryof Relativity to introduce the concept of linear momentum of the photon (anticipates De Broglie)which in turn is used to explain the PE, which will be used further ahead to explain the topic"spectrum of atomic lines and energy levels ", and then will be used again in the Compton Effect. Seethe “photon linear momentum” in the third line of fig.3. Here we can see that the authors of moderntextbooks are considering that the most important themes, as Bohr model, are really taught in the highScholl, so that they can use it without defining them. So, in the textbooks writing to form engineers, asin some to form physicist, some physical concepts are presented in a truncated form. It can be seen inthe figure 3.

FIGURE S3Figure 3: Conceptual Map of the chapter 38 of the book of the author Young-Freedman (2008).

In this text we can see that while in the textbooks writing to form physicists the central theme orconcept is the quantization, or the motivation to introduce the Schrödinger equation, the theme that theauthors consider most important or central is the discrete spectrum of the spectral lines. They sacrifice

the chronological order of insertion of the concepts in favor of a more comprehensive explanation ofthe central theme. This can only be seen through its CM.

We can observe that they use the conclusions of the Bohr model to explain the topic "atomicspectrum of lines and energy levels". But the Bohr model will only be elaborated below in the text toexplain the spectrum of the hydrogen atom (see the central brown box in Figure 3). This will also beused to explain the spectrum of X rays.

Until this moment the text only addresses the discrete spectrum of energies. As we can see in thegreen block of boxes, bellow in fig.3, he uses the theme continuous spectrum of energies to introducethe blackbody radiation and Planck's hypothesis. They use the continuous spectrum of X-ray tomotivate and discuss the empirical laws of Stefan-Boltzmann, Wien and Rayleigh.

Figure 4: Conceptual Map of the chapter 39 of the book of the author Young-Freedman (2008).

Thereafter he follows an order of text presentation almost equal to the Eisberg, as we can see in theCM-4, fig.4. Can be noted that the authors use the same strategy of Serway to define the Schrödinger’sequation. That is, they discuss the stationary state of the wave function before introduce theSchrödinger’s equation. Completing this analysis is interesting to note that some textbooks as theHalliday [8] do not address the topic blackbody radiation and introduce energy quantization in the

topic photoelectric effect. That is, we can remove this topic without loss the comprehension of thetext.

Finally, regarding the presentation of atomic models we see that these are sprayed in somechapters. The Bohr model is presented in Sec. 39, "More Matter Waves" where they present severalsolutions of the Schrödinger equation and put the Bohr model as another electron confinementexample. Again we are faced with the problem of DT of the Schrödinger equation. The Raisin Puddingmodel and the Rutherford model are included in the cap. 42 - Atomic Nucleus - where the authors willdiscuss the theme scattering of particles and studies technique for the atomic nucleus.

Part 2 - High School.

5.3 Physics Principles and Problems – Glencoe Science [42]

As a textbook written for high school we chose the Physics Principles and Problems [42]. Wechose this because it is widely used in American high schools and is available on the web. The modernphysics topic is presented in four chapters as follows: 27 - Quantum Theory; 28 - The Atom; 29 -Electronic Solid State and 30 - Nuclear Physics. The study realized here will be restricted to Chapters27 and 28. Its CM is show in Figure 5 below. We can see in their CM, Figure 5, his pedagogicalproject is all grounded in three pillars: 1) Introduction to the theme through the experimental problems(connecting word “puzzle” in brown) that afflicted the scientists at that time; 2) presentation of thetheory by describing an experiment rather than the detailed description of the scientific model; 3)finalization through technological applications (boxes in violet).

Due to the fact that physical theories are constituted by theorems, hypotheses, models and lawsformulated in terms of functions or mathematical equations, many physicists defend the idea that aphysical theory can only be transposed if your math can be written in an appropriate language to acertain level of knowledge [43, 44, 45]. So in the case of high school these functions or equationsassume important role in TD. Thus we put these equations or functions in yellow boxes in theGlencoe’s CM.

It notes that what distinguishes partly the academic levels, academic plus and honor in theAmerican high school is the level of mathematical approach. We see that the same is true for bookswritten for the early years of the university course. We have physics books written for “calculus basedphysics courses” and “Fundamentals Physics books” for the algebra based physics courses. Thus,there is consensus that we should create teaching methodologies that develop math skills for studentswishing to pursue research career. These functions and mathematical equations, the problems andquestions raised for developing these skills are called "Physical things" or "objects of physics" by theteaching physics group "Grupo de Relaboração do Ensino de Física" (GREF) [46].

We see here, unlike some European’s "teaching groups", the text sacrifices philosophical factsbehind the modern physics (e.g. Schrödinger's cat) and underlies their motivation in the appeal thatnew technologies have on students. If we take as an example the school of researchers in education ofBarcelona which bases its analysis of TD on the Cognitive Theory of Science [47, 18 e 13] and mentalmodels [48 e 14] we see a trend of American schools to be more pragmatic and base his TD on theinformation and communication technologies (ICT), violet boxes, and in the project-based learningmethodology (PBL), connecting word “puzzle” in brown.

According Pietrocola [28] the characteristic simplicity and operability of a particular scientificcontent are decisive in their admission and permanence in school curricula. That is, a contentcontaining a simple mathematical formulation, compatible with the secondary school, and thatgenerates questions and problems are the preferred by educators. We confirm here, in the Glencoetext, that due to the linear dependence (equation) between kinetic energy (Ec) the work potential (w)and frequency (f) and their generation of exercises (things of physics), that the topic 'photoelectriceffect' is the preferred by high school textbooks. See Figure 5. It can be seen, too, for the topic “Bohrmodel”. See Figure 6.

In fig.5 we see the text for the photoelectric effect contains basically the same conceptual contentthat of the Book of Eisberg. Besides containing the equation that provides the linear dependencebetween the frequency of the light and the kinetic energy of the ejected electron (yellow boxes), they

discuss in detail the experimental facts that contradict the classical model for the electromagneticradiation and confirm the hypothesis of quantization of electrical magnetic energy. See the crossedlines right from its CM. We see in this text that the equation that relates the kinetic energy of theejected electron with the frequency of the incident light is called the Einstein equation (yellow boxes).A striking feature of the didactic transposition to high school and earned the name 'things of physics'.Here we have again the dilution of knowledge phenomenon.

We see in the 'Bohr atom' section, especially in the emphasis given to deduction of the expressionfor the Bohr‘s radius and energy the concern of the authors in preparing students for problem solving.See the boxes 'things of physics' on its CM, Figure 7 and 8. It is observed in the "Matter Waves", fig,6, “The Atom", fig.7 and 8, and "Quantum Model of the Atom" sections, fig.9 and 10, the use ofmodern teaching methodologies as ICT and BLP does not mean that it must necessarily sacrificemathematical rigor of presentation of a theory. See in Fig. 6, 7, 8 and 9, the simultaneous presence ofboxes brown, yellow and violet.

Comparing the Glencoe text with the Eisberg’s book structure and with the books written to theuniversity basic cycle we can seen that this resembles more the latter than the former. Note that thetextbook of Glencoe program, as many modern books, complement, finalizing or motivate the subjectcontent with experimental activities that most of the times are playful activities on the subject. See redpastel color tone boxes in Figure 6 and 10. This is one of the tools (things) that are part of the teachingmethodology of a DT well performed and has no correspondence with the scientific activity. It isworth mentioning that the conceptual construction of this book allows use it in the honors level. And ifwe subtract the mathematical content of the text it can be used in the Academic level.

Complementing the analysis, we have that in countries where there are not enough places in theuniversity for all candidates, we have the creation of the phenomenon of preparatory courses and theadaptation of texts written for high school with the main purpose of preparing students for thequalifying exams for admission. Unlike textbooks such as Glencoe, it is a class of books written with amore or less standard structure that follow the structure of university books of the 70s. This structure issomething like this: It begins with a historical introduction of the topic, followed by the theory withsimple illustrative examples. Some have applications placed in boxes and ends with questions andexercises for the entrance exam.

6. Conclusion

As previously mentioned, the simple analysis of the contents of the topics of Modern Physics doesnot allow us to visualize the conceptual construction that the author or authors used to write their text.But, through the making, following well-defined criteria, and subsequent analysis of the CM of agiven field of knowledge, we can see how the concepts were introduced, how were partitionedthroughout the text, and how the author or authors have used to support or only introduce otherconcepts.

Without key words or key concepts connection we could not understand how the concepts aresometimes anticipated, sometimes postponed, and sometimes partitioned in order to base the contentsto be taught. If we only had the flowchart of the text we would not have the connections betweenconcepts and not would notice how they interrelate.

The importance of using CM in the textbook analysis is not very evident through the analysis ofEisberg and Serway texts. This importance is emphasized when we make a comparative analysis withthe Sears book. We see in its CM as key concepts were introduced, anticipated and often partitioned.After studying its CM we noted the difficulties that the authors found to write an original text and thatwere built under the optical point of view.

With this analysis we can previously evaluate which textbook should be used for each type ofstudents who enter university. That is, we now have textbooks written for students who have anadequate scientific alphabetization and others for those who superficially seen the Modern Physicscontent.

Figure 5: Conceptual Map of the Chapters 27 and 28 - Quantum Theory – of the book "Glencoe Science - Physics, Principles and Problems".

Figure 6: Conceptual Map to the section “Mater Waves” of the book "Glencoe Science - Physics, Principles andProblems".

When analyzing a CM built to a text of Modern Physics topic for high school, we can see how thiscontent was being diluted and adapting to reach that level. As the authors or agents determinants ofhow the scholarly knowledge is transposed into knowledge to be taught reformulate it to be taught inthe classroom. As the necessity to develop math skills will generate the 'things of physics'. We also seethat when in the educational project of teaching the author’s uses the TICS or project-based learningmethodology that the DT is made, in most cases, from the physics texts for graduate or professionallevel than from scientific articles.

The use of CM in the classroom of a postgraduate course in physics or in science teaching becomesa powerful tool in determining how the didactic transposition occurs. It makes clear how the authors oftextbooks introduced and highlighted some concepts to be collected and evaluated and how otherconcepts are being rejected or filtered. It is a very powerful tool demonstrates that the scienceproduced in the academy is very different from the science to be teaching (TCC).

ACKNOWLEDGEMENT

We are grateful to SBFísica by commitment and administration of national professional master's degree in physics teaching (MNPEF) because this work is the result of lesson preparation for the course of Quantum Mechanics to this master's program. In especial I would like to thank Prof. Marcos Antonio Moreira to discussion and incentives to make this article. I would like to thank Prof. Alberto J. Cañas by incentives given in CCM 2014 - 6th International Conference on Concept Mapping.

Figure 7: Conceptual Map of the section “The Atom” of the book "Glencoe Science - Physics, Principles and Problems".

Figure 8: Conceptual Map of the section “Development of Bohr Model” of the book "Glencoe Science - Physics, Principles and Problems".

Figure 9: Conceptual Map of the section “The Quantum Model of the Atom” of the book "Glencoe Science - Physics, Principles and Problems".

Figure 10: Conceptual Map of the section “Laser” of the book "Glencoe Science - Physics, Principles and Problems".

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