Building a guide to analyse mathematics textbooks based on the didactical suitability theory

Céspedes, María José Castillo; Burgos, María; Godino, Juan D.

doi:10.1590/S1678-4634202248238787eng

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ARTICLES • Educ. Pesqui. 48 • 2022 • https://doi.org/10.1590/S1678-4634202248238787eng copy

Building a guide to analyse mathematics textbooks based on the didactical suitability theory ^* * English version by Angela Helen Barnie.

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Abstract

A textbook lesson can be considered as a potential or planned instructional process proposed by the author’s book, which can help the teacher to design and implement an effective instructional process. This permits the application of didactical analysis tools of the onto-semiotic approach to mathematical knowledge and instruction in order to assess the didactical suitability of the instructional process, identify possible conflicts of meaning and potential improvements. In this article, we describe the process of developing a Mathematics Textbook Lesson Analysis Guide using the didactical suitability theory and its operational breakdown into components, subcomponents and indicators. The formulation of the suitability criteria, as rules that permit to guide an informed assessment of the suitability of a teaching and learning process, has led us to carry out a content analysis of the key research on textbook analysis and the consensus adopted in the research community regarding the suitability criteria. Assuming that a mathematics teacher has decided to use a textbook lesson as a resource to help the teaching and learning process of some mathematical content, the analysis guide is presented as a tool that allows the teacher to assess the didactical suitability and facilitate the making of informed decisions about the lesson use in the classroom.

Didactical analysis; Mathematics textbook; Teacher education; Didactical suitability

Components	Sub-components	Indicators
Meanings	Problems	A representative and an articulated sample of problem-situations that allows the contextualization, exercising, amplification, and application of mathematical knowledge, which comes from mathematics itself and from other contexts, is presented. Problem generation situations are proposed (problematization).
	Languages	A wide repertoire of representations (verbal, graphic, material, iconic, symbolic...) is used to model mathematical problems and ideas, analysing the relevance and potential of each type of representation and carrying out translation processes between them. The language level is made appropriate for the students to which it is addressed. The construction, improvement, and use of representations to organize, interpret, and record ideas are promoted.
	Concepts	The fundamental concepts of the subject are presented in a clear and correct way and are adapted to the educational level to which they are addressed. Situations are proposed where students have to generate or negotiate definitions.
	Propositions	The fundamental propositions of the subject are presented in a clear and correct way and are adapted to the educational level to which they are addressed. Situations are proposed where students have to generate or negotiate propositions.
	Procedures	The fundamental procedures of the subject are presented in a clear and correct way and are adapted to the educational level to which they are addressed. Situations are proposed where students have to generate or negotiate procedures.
	Arguments	Propositions and procedures are explained and argued (justified and demonstrated) in a manner appropriate to the educational level at which they are addressed. The justification of mathematical statements and propositions is favoured through various types of reasoning and proving methods.
Relations		Mathematical objects (problems, definitions, propositions, etc.) are related and connected to each other. The diverse meanings of the objects that intervene in the mathematical practices are identified and articulated.
Processes	Communication, argumentation	Situations are promoted where the student has to argue (describe, explain, verify) and formulate conjectures about mathematical relations, investigate and justify them. Situations are proposed that allow the student to communicate using mathematical language to express his/her ideas with precision. Situations are proposed where the student can analyse and evaluate the mathematical thinking and strategies of others.
	Modelling	Situations are presented that allow the student to use mathematical models to represent and understand quantitative relationships (identify, select characteristics of a situation, represent them symbolically, analyse and reason the model, recognize the characteristics of the situation, the precision, and limitations of the model). The use of technology and the use of functions to model patterns of quantitative change are promoted.
	Generalisation	Situations, where students have the opportunity to describe, explain, and make generalizations and conjectures about geometric and numerical patterns, are promoted.
Epistemic conflicts		The contents, problem-situations, and their solutions, concepts, propositions, language, etc. are presented correctly without errors, contradictions, ambiguities.

Components	Indicators
Relations	The experiences (situations, examples, explanations...) proposed allow the assessment of whether the student establishes relationships or connections between mathematical objects and between their corresponding meanings.
Previous knowledge	The textbook contemplates the previous knowledge necessary for the study of the topic. The intended contents are attainable (have a manageable difficulty) in its various components.
Individual differences	Expansion and reinforcement activities are included. Access, achievement and support of all students is provided.
Cognitive Conflicts	Error is valued as a source of learning. Potential cognitive conflicts of the students are foreseen.
Evaluation	Evaluation and self-evaluation instruments are proposed. It is promoted that the results of the evaluations are disseminated and used to take decisions. The various modes of assessment included in the textbook are suitable for evaluating that the students achieve the appropriation of the knowledge, comprehension, and competencies intended (conceptual and propositional understanding; communicative and argumentative competence; procedural fluency; situational understanding; modelling and generalization competence, metacognitive competence). The evaluation takes into account different levels of comprehension and competence.

Components	Indicators
Attitudes	Active participation in activities, perseverance, responsibility, etc., is promoted to encourage a mathematical attitude. The argument is favoured in situations of equality; the value of an argument does not depend on who says it. Flexibility is encouraged to explore mathematical ideas and alternative methods for problem solving.
Emotions	Tasks and the contents involved are of interest to students. There are motivating elements: illustrations, humour, poetry, riddles, etc. Logical reasoning, original ideas or useful, practical or realistic work are encouraged and promoted. Specific moments are programmed throughout the sessions so that students can express their emotions towards the proposed situations. Self-esteem is promoted, avoiding rejection, phobia, and fear of mathematics.
Beliefs	Beliefs about mathematics, its teaching, the student’s metacognition, and about the social context in which learning takes place are analysed and considered.
Values	The student is encouraged to value the qualities of aesthetics, accuracy, and usefulness of mathematics in daily and professional life.
Affectivity evaluation	Assessment activities are proposed to evaluate the affective aspects of teaching and learning.

Components	Indicators
Author-student interaction	The author makes an adequate presentation of the topic (clear and well-organized presentation, emphasizes the key concepts, etc.) Situations where consensus is sought based on the best argument are promoted. Diverse rhetorical and argumentative resources are used to involve and capture the students’ attention. The inclusion of students in the dynamics of the lesson is promoted or facilitated.
Students interactions	Tasks are proposed that encourage dialogue, communication and debate among students in which different points of view are explained, justified and questioned using mathematical arguments. Situations in which students have to convince themselves and others of the validity of their statements, conjectures and answers, supported by mathematical arguments.
Autonomy	Sometimes students take responsibility for the study (raise questions and present solutions; explore examples and counter-examples to investigate and conjecture; use a variety of tools to reason, make connections, solve problems, and communicate).
Formative evaluation	Forms of evaluation are included to allow for the systematic and continuous observation of students’ cognitive progress. Assessment is seen as a process in the service of teaching and learning (serves as feedback to students). The use of diverse evaluation techniques is contemplated (problem solving, practical tasks, ...). Self-evaluation, co-evaluation and hetero-evaluation activities are included. Assessment is consistent with the learning goals (tasks similar to learning situations are included).

Components	Indicators
Material Resources	The use of manipulative, audio-visual, and computerized materials is promoted, allowing the introduction of good situations, languages, procedures, arguments adapted to the intended content. Definitions and properties are contextualized and motivated using specific situations, models and visualizations. The bibliographic sources used are made explicit and diverse.
Time	The contents which are more difficult to understand are given sufficient space in time. The timing of the sequencing of activities and content is appropriate.

Components	Indicators
Adaptation to the curriculum	The objectives, contents, their development and evaluation correspond to the curricular guidelines.
Didactical innovation	Innovation based on research and reflective practice.
Socio-professional adaptation	The contents contribute to the students’ socio-professional education.
Education in values	The training of democratic values (respect for diversity, tolerance, integration, cooperation, environmental awareness, pacifism, other values and prejudices) and opportunities for students to question the apparently obvious or given as natural (critical thinking) are contemplated and promoted.
Intra/Interdisciplinary connections	The contents are related to other intra- and interdisciplinary contents (cross-cutting themes, history of mathematics, others).

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E-mail: revedu@usp.br

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[1] María José Castillo Céspedes has a Bachelor’s Degree in Mathematics Teaching from the University of Costa Rica, and is a lecturer from the same university. Master in Didactics of Mathematics from the University of Granada, student of the Doctorate program in Educational Sciences, line of research in Mathematical Education, at the same university.

María Burgos is a Senior lecturer at the University of Granada, in the Department of Didactics of Mathematics, Faculty of Educational Sciences. PhD in Mathematics and PhD in Didactics of Mathematics.

Juan D. Godino is Doctor in Mathematics by the University of Granada, Professor (retired), professor of the Doctorate Program in Educational Sciences at the University of Granada, Spain.

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Building a guide to analyse mathematics textbooks based on the didactical suitability theory * * English version by Angela Helen Barnie.

Abstract

Building a guide to analyse mathematics textbooks based on the didactical suitability theory ^* * English version by Angela Helen Barnie.