Mathematical Model of the Didactic Structure of Physics Knowledge Embodied in Physics Textbooks

Eizo Ohno Faculty of Education, Hokkaido University

Abstract In this study, descriptions in physics textbooks are considered as the sign defined in semiotics. Charles Peirce’s triadic model of the sign is used to analyze each description. Based on the models of experimental inquiry, we formulate three categories into which descriptions in physics textbooks fall. We then construct the basic components of our analysis using these categorized descriptions. The basic components are represented via the notion of classifications, and a one-way relation between classifications is proposed. The didactic structure of physics knowledge embodied in physics textbooks comprises linked classifications. This analysis allows us to use data from teaching plans and students’ activities in actual science lessons. We apply our approach to analyze two physics textbooks used in secondary education and illustrate the organized structure of descriptions in physics textbooks as diagrams. Keywords experimental model, data model, theoretical model, sign, type, token, semiotics Introduction Scholarly knowledge of physics needs to be transformed into a didactic form. The didactic form is referred to as the “physics knowledge to be taught” form and is adapted for teaching physics in specific contexts. This form is further transformed into teaching material, which is then used by students in classrooms. The transformation is a fundamental process in the design of physics curricula. In mathematics education, the entire scheme described above is called “didactic transposition” (Bosch and Gascón, 2006).

Physics textbooks are important teaching materials. In science lessons, teachers teach and students learn physics using textbooks as the teaching materials. Descriptions in physics textbooks can also be interpreted as “physics knowledge to be taught” that is represented by texts, graphs, and figures. Teachers use physics textbooks so that students can interpret the descriptions in them and comprehend the “physics knowledge to be taught” embodied there. In other words, physics textbooks play the following roles: they are teaching materials in science lessons and “physics knowledge to be taught.” Therefore, analyzing the descriptions in physics textbooks as teaching materials will indirectly reveal the underlying structure of physics knowledge embodied in them.

There have been numerous studies on textbook research (Pingel, 2009). Various methods of analysis of textual and illustrative materials in science textbooks have been developed and attempted (Khine, 2013), and semiotics is often employed in the analysis of texts. For example, social semiotics provides an integrated framework for analyzing textual and illustrative

materials in English and science textbooks (Bezemer and Kress, 2010). However, the social semiotic approach does not focus on the use of textbooks in classrooms.

The didactic structures of the “physics knowledge to be taught” form and their visualizations have also been investigated. Drawing concept maps is one of the methods employed to represent the relational structure of our physics knowledge (Novak, 1998). Another method is called noboriori-hyo, a Japanese phrase (Kawakatsu, 2005) that was a drawing method invented to empirically express a series of learning activities as a table connected to the “physics knowledge to be taught” and the intentions of the teacher. Teachers draw noboriori-hyo tables according to their teaching experience and science textbooks.

In this study, we focus on the bilateral character of physics textbooks; teaching materials used in science lessons and the “physics knowledge to be taught” embodied in the textbook descriptions. First, we explain the fundamental notions of our approach. Descriptions in physics textbooks are classified into three categories based on the models of experimental inquiry. Each description is considered as the sign defined in semiotics. We use Peirce's triadic model of the sign. Second, we expound the theoretical framework of our analysis method. A mathematical model is proposed to illustrate the didactic structures of descriptions in physics textbooks. Two physics textbooks used by secondary school students are analyzed as specific examples. The conclusion discusses the characteristics and potential features of our approach. Triadic model of descriptions in physics textbooks In this study, we consider a series of sentences in a physics textbook as a description. A glimpse at the descriptions in physics textbooks reveals the dual nature of the books. Physics textbooks explain scientific experimentations and observations. During science lessons, students execute activities according to the instructions in the textbooks. On the other hand, certain elements of descriptions in physics textbooks embody abstract concepts and universal laws of physics. Teachers design science lessons to elucidate the “physics knowledge to be taught” embodied in these descriptions. By reading these descriptions, students review the knowledge that is gained by studying. In this study, we discuss this dual nature of physics textbooks based on semiotics (Chandler, 2007).

Figure 1. Triadic model of the descriptions in physics textbooks

Representamen

Description (A series of sentences)

Object

“Physics knowledge to be taught”Students’s activities in science lessons.

Interpretant

Teacher’s mental processStudents’ mental process

We analyze the descriptions using Peirce's triadic model of the sign (Liszka, 1996). Peirce defined a sign as consisting of a representamen, an interpretant, and an object. A representamen is the form that the sign takes. An interpretant is the sense made of the sign. A sign in the form of a representamen creates its interpretant, and the sign stands for its object. We regard each description and picture in a physics textbook as a representamen. The representamen stands for an object that is “physics knowledge to be taught,” and students’ actual scientific activities. The interpretant is a teacher’s or a students’ mental process. A teacher interprets the representamens and designs student activities. Students interpret the representamens to comprehend the “physics knowledge to be taught” and to execute learning activities in science lessons. Figure 1 illustrates this triadic model of descriptions in physics textbooks. Categories of descriptions in physics textbooks Mayo (1996), citing Suppes’s (1969) ideas, proposed models of experimental inquiry. Experimental inquiry is delineated in terms of the following types of models: models of primary scientific hypothesis, experimental models and data models. Models of primary scientific hypothesis are also called primary models. They describe how to break down a substantive inquiry into one or more primary questions that take the form of estimating quantities of a theory. Experimental models connecting the primary models to data are concerned with relating primary questions to questions about the particular type of experiment at hand. Data models comprise the generation and modeling of raw data so as to put them into modeled data linked to the experimental models. Data models are also used to check whether generation of actual data satisfies various assumptions of experimental models.

We categorize descriptions based on how strong the interpretant of each description is related to the above three models. In other words, the model that is employed by the teacher and students to interpret a description determines the categorization of the description. The following basic categories are proposed:

Category A: questions, hypotheses, and theoretical models Each section of a physics textbook generally has a primary scientific inquiry. This substantive question is categorized into questions and hypotheses. The questions are addressed and solved by theoretical models for testing hypotheses. The questions are considered significant for students and are reasonably capable of being answered in science classes. Students learn theoretical models and comprehend ideas to explain phenomena using the models.

Category B: experimentation, observation, and empirical examples Students engage in practical investigations and inquiring activities in science classes according to instructions provided in their textbooks. Descriptions in physics textbooks provide illustrative examples from daily life and given experimental data so that students can analyze such phenomena as thought experiments. Descriptions related to experimental models specify the key features of experiments. Students learn choice of experimental model, sample size, experimental variables, and experimental settings. Descriptions related to data models formalize raw data to apply analytical methods. Students understand analytical methods and graphical representation methods to answer questions framed in terms of the experiment.

Category C: review questions, and exercises

We focus on Categories A and B in this study. A description falls into Category A when its interpretant is mainly related to primary models. It falls into Category B when its interpretant is mainly related to the experimental models and/or data models. We analyze descriptions in physics textbooks using the triadic model demonstrated in Figure 1 to categorize them into Categories A and B. Classifying descriptions Let α denote a description in Category A and β a description in Category B. The description α presents general and universal statements containing abstract concepts and universal laws of physics. The object of α is “physics knowledge to be taught.” The description β denotes investigations and inquiring activities in science lessons. The object of β also includes empirical results obtained from those activities.

Students try to interpret a representamen assigned to α in order to acquire knowledge in physics (objet of α). They combine α with some empirical results (object of β) obtained from an experiment they performed in a science lesson. A teacher designs an activity (object of β) in their science lesson so that students interpret a representamen of β successfully and acquire knowledge of physics (object of α). In this situation, we regard α as a representamen, and β as an object. We propose a triadic relation of α, β, and an interpretant in the minds of the teacher and students, as shown in Figure 2.

The relational structure in Figure 2(a) is called classification Cαβ . The symbol ⊨αβ refers to an

Description α

representamen

interpretant

object

αCα

Description γrepresentamen

interpretant

object

γCγ

Description β

representamen

interpretant

object

βCβ

αβ

Cα

Cβ

αβCαβ

Void

Cγ＊

(a) (b) (c) (d)

Figure 2. Classification and semi-classification

interpretant related to the experimental models. The interpretant serves as a bridge between α and β. The bold gray circle shows that β is in Category B. This classification is a fundamental structure in our model. Figure 2(b) is a simplified schematic of Figure 2(a). The classification is a fundamental component in our model. In other words, α is a type and β is a token. Types are abstract and general, whereas tokens are concrete, specific events, states or processes (Wetzel, 2009). Each token is assigned to a type in the classification. This scheme is inspired by the theory of information flow (Barwise and Seligman, 1997) and we use similar notations and diagrams.

Figure 2(c) shows a description γ in Category A to which no description in Category B is assigned. The semicircle in Figure 2(d) is a simplified schematic of Figure 2(c). Tokens of this classification are shown as “Void” in Figure 2. We call the description γ in Figure 2(c) a semi-classificationCγ* . In science lessons, to fill the void of Cγ* , a teacher might incorporate experimental activities. With information on actual lessons, we can change the semi-classification Cγ* to a classification. Our method connects the results of analysis indirectly to actual science lessons through the voids of semi-classifications. Relational structure of descriptions in physics textbooks Relations between classifications and semi-classifications determine the relational structure of descriptions in physics textbooks. Let ρi and ρ j represent descriptions in the same category

(A or B). There exists a one-way relation “ ρi partakes of ρ j ”, which we write ρi → ρ j , when the two classifications meet any of the following criteria:

i) The representamen of the description ρ j includes an equivalent representamen of the

description ρi .

ii) The interpretant of the description ρ j includes an interpretant and object of the description

ρi . The process to interpret the description ρ j requires understanding the description ρi .

iii) The object of the description ρ j includes an object of the description ρi .

The three conditions are not equivalent to “imply” and “entail.” The relation is not yet rigorously defined with exclusive commitment to logic. Some well-defined conditions of the relation are yet to be solved. Accumulating results of analysis might lead to the resolutions of this problem.

Let αi and α j be descriptions in Category A, and βi and β j be descriptions in Category B.

Descriptions αi and βi compose a classification Cαiβi. Descriptions α j and β j compose a

classification Cα jβ j. If αi →α j and βi → β j , then there exists a one-way relation “Cαiβi

partakes Cα jβ j” which we write as Cαiβi

⇒Cα jβ j(see Figure 3). If only αi →α j is satisfied,

then there exists a one-way relation “Cαiβi partakes the type of Cα jβ j

” which we write as

Cαiβi→Cα jβ j

. This means a situation in which students learn about αi and α j using

completely different teaching materials. If only βi → β j is satisfied, then there exists a one-

way relation “Cαiβi partakes the token of Cα jβ j

” which we write as Cαiβi→BCα jβ j

. This means

a situation in which students learn entirely different contents by performing one-way related activities referred to by βi and β j .

Specific examples Two physics textbooks A and B used in secondary education are analyzed as specific examples. Textbook A is the Japanese textbook Butsuri Kiso (Takagi, et al., 2012). It is the first book of basic physics for students aged 15 to 17, depending on their school curriculum. In this textbook, Chapters 1 and 2 are entitled “Motion of Objects” and “Force and Motion” respectively. We analyzed these two chapters and obtained classifications by the triadic model of descriptions. Textbook B is General Certificate of Secondary Education (GCSE) Physics developed for a concept-led physics course (Whitehouse et al., 2011). Students aged 14 to 16 take this physics course. A part of Chapter P4 “Explaining motion” is analyzed.

Figure 4 shows the relational structure of descriptions related to a topic “F=ma” in textbook A as a diagram of the one-way relation described above. Textbook A describes three experimental activities in this topic. The diagram in Figures 5 is a rough sketch of relational structure of descriptions that reflects some structure of physics knowledge to be taught embodied in Chapter 2 of this physics textbook. The sketch omits the details of the classifications. Small blue dot marks represent the descriptions in Category A. Small red dot marks represent the descriptions in Category B. Figure 6 shows structural differences between textbooks A and B. The right

Figure 3. One-way relation Cαiβi⇒Cα jβ j

Description αirepresentamen

interpretant

object

αiCαi

Description βirepresentamen

interpretant

object

βiCβi

αiβi

Description αjrepresentamen

interpretant

object

αjCαj

Description βjrepresentamen

interpretant

object

βjCβj

αjβj C C

Cαi

Cβi

αiβiαiβi

Cαj

Cβj

αjβjαjβj

(a) (b)

A

B

figure outlines the structure of physics knowledge embodied in Chapter P4 of textbook B obtained in the same way.

Figure 4. Relational structure of the classifications of Textbook A

Figure 5. Rough sketch of relational structure of Chapter 2 in Textbook A

We did not use any data from actual teaching plans or students' activities. The data forms the interpretants and objects in Figures 4 and 5. The analyses assume common teaching plans and activities based on textbooks and experience. If we obtain data from teachers’ intentional instruction and students’ specific behaviors, such information will modify the interpretants and objects. The modification will cause the diagrams in Figures 4 and 5 to change. Conclusion and discussion We have proposed a mathematical model of the didactic structure of physics knowledge embedded in physics textbooks. In this study, we created Categories A and B based on the models of experimental inquiry. Category A is related to the primary models and Category B is established in relation to the experimental models and/or data models. The descriptions in both categories were considered as the sign and analyzed using Peirce’s triadic model of the sign. The descriptions were considered as the representamen. The objects mean “physics knowledge to be taught,” teaching plans, teaching materials, and students’ activities in science lessons. The descriptions as the representamen stand for the objects through the interpretants (the teacher’s and students’ mental processes). The basic components called classifications were created from the categorized descriptions. One-way relations between classifications were proposed. The relational structure of classifications was demonstrated by a diagrammatic expression.

The classification is the basic constituent of our analysis. It is considered as an expression of the data structure for representing the descriptions in physics textbooks. We can create a database on physics textbooks by combining the classifications and the data from the one-way relations between them. Such a database would be helpful in handling a large amount of diverse data and

Figure 6. Structural differences between Textbooks A and B

visualizing the relational structure of classifications. We can expand the data structure to add information on the teacher’s intention and students’ activities in science lessons explicitly through data from interpretants and objects. The database will be associated with rich and large data from science lessons. The usage of this data is a key question for future research. Acknowledgement This work was supported by JSPS KAKENHI Grant Number 26350180. References Bosch, M. and Gascón, J. (2006). Twenty-Five Years of the Didactic Transposition, ICMI Bulletin 58, 51-65.

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Takagi, K., ... et al. (2012). Butsuri Kiso (Basic Physics), Keirin-Kan, Osaka.

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Affiliation and address information Eizo Ohno Faculty of Education Hokkaido University Nishi 7, Kita 11, Kita-ku, Sapporo, Hokkaido, 060-0811 JAPAN e-mail: eohno@edu.hokudai.ac.jp