Conceptual-holistic knowledge for teaching school mathematics

 

The course is addressed to mathematics teachers of any professional level, for improving the quality of the teaching of school mathematics – in both official school institutions and in private tutoring – in regard to the conceptual understanding of mathematics on the basis of scientific criteria targeting content and didactical methods as well.
The course is conceived and taught by Dr. Cătălin Bărboianu, , with the support of the Theoretical Philosophy Department at the University of Bucharest.
Supervisors-reviewers: Prof. Dr. Ilie Pârvu, Prof. Dr. Radu Miculescu.
The course is delivered online and will last 3 months, consisting of 2 lessons weekly (36 lessons in total), in English. Each lesson is followed by a session of questions-answers and comments. After registration, students receive the technical documentation necessary for attending the course, and upon completion they receive a graduating diploma, which ensures their eligibility for the position of PhilScience math tutor.

 

The thematic structure of the course

 

The structure of the course consists of five main modules whose themes are mentioned in brief below (the complete structure, broken down by chapters and subchapters, as well as bibliography, is soon to be published).

Module 1. General notions: Discipline, theory, conceptual framework, object of study, method. Interdisciplinarity and transdisciplinarity. The status of mathematics among the disciplines. Epistemology of mathematics. Pure and applied mathematics. The goals of mathematical knowledge and mathematics education. Educational aspects of philosophy and history of mathematics.
Module 2. Fundamentals of mathematics in teaching methods: Conceptual foundation and theoretical foundation. The primary concepts of mathematics. Numbers, sets. The logical foundation of mathematics. Aspects of the logical and set-theoretic reducibility of mathematics. The symbolism of mathematics. The formalism of mathematics. Reflection of the fundamentals in the curricular content and the teaching methods (with examples regarding the essential notions taught at each grade). The intensional-type understanding of mathematics with respect to its fundamentals.
Module 3. Concepts of philosophy and epistemology of mathematics that draw on the specificity of mathematics as a discipline: Mathematical concepts, entities, mathematical structures, epistemic structures. The meaning of mathematical concepts. Structural equivalence and classes of structures. Invariance and generality. The language of mathematics. The methods of mathematics. The roles of mathematics. Applicability of mathematics. Mathematical modeling. The content of mathematics. The truths and analyticity of mathematics. The epistemic virtues of mathematics. The teaching of Euclidian geometry as general reflection of the axiomatic method, the proof method, and the language of the practice of pure mathematics. The separation of the empirical interpretation from the mathematical formalism. The teaching of algebra as a reflection of the generation of mathematical structures and of the generalization of the elementary operations with numbers. Examples of explanatory lessons.
Module 4. Mathematical understanding and understanding of mathematics. Holistic-type understanding. Distinctions: The epistemological and the psychological concept of understanding. Conceptual meaning, context of understanding, degrees and kinds of understanding. Intelligibility and understanding. Intensional and extensional meaning. Mathematical thinking. The explanatory metaphors in teaching mathematics and the limitations of the interpretative-metaphorical explanations. The conceptual reading of the symbolic expressions. Concrete examples. The relation between the conceptual understanding of mathematics and the personal universe of knowledge.
Module 5. The practice and technical aspects of the teaching with the goal of reaching the conceptual-holistic type of understanding: Explaining the general concept of definition. Exemplifying and interpreting. The motivation of the definition. The “game” of changing axioms. The relation between the formalism of definition and the extensional-type contexts. Extracting the essential aspects of the proofs of the theorems. Revealing the links between the mathematical concepts by following the components of the definitions and proofs. The proofs of the theoretical propositions and problem solving. The application of mathematics in itself. Managing the radical conceptual transition – from finite to infinite, from discrete to continuous. Generalization and extension. The set of reals as a theoretical construct. Approaching mathematical analysis and calculus from an algebraic, topological, and general-theoretical perspective. The elaborated explanation of the essential notions of mathematical analysis and calculus. The conceptual approach of the problems and patterns of solving. Examples of explanatory lessons. Examples of solutions of problems approached conceptually.

 
Before attending this course, we recommend for teachers, as an introduction and for an overview on the object of the course, our book What is Mathematics: School Guide to conceptual understanding of mathematics.

 

Fee and registration

 

The fee of the course is $290 monthly ($870 for the entire course) and can be paid in full, monthly, or bimonthly.
The course for teachers is presently being developed, and we will soon take registrations for the first series of students. We estimate the start of the course for November–December 2020.
You can register for the course by completing the registration form below and sending it to address orders[at]philscience.org (subject "course registration").

 

download the registration form »                                                                           index »