Chapter 7: Calculus
PART I A Guide to the Paradigmatical Cases (Episodes) Presented in this Chapter
PART II Data and Analysis
In the following I present the factual and interpretive accounts and conclusions for the 8 Episodes of the table in Part I. In Part III then I synthesise the findings of Part II related to Calculus and briefly discuss the wider cognitive issues that are presented in the overall synthesis of the data analysis in Chapter 10.
(i) Constructing a Meaning of the Concept of Limit: Concept Definition and the Formalism of Mathematical Notation, Concept Image and Visualisation
(ii) The Novel Notion of Continuity: Proof By Definition Or With the Algebra of Limits. A Battle of Ambivalent Preferences and the Cognitive Effect of a Hidden Agenda
(iii) LimS , S lim and the Right to Exchange Limits. The Superiority of Proof Via First Principles and the Convention of Foundational Rigour
(iv) Striving for Meaning and Significance: The New Concept of Fourier Series
(v) The Contrast and the Gap Between the Mechanistic and the Conceptual Approach to the Notion of Derivative
(vi) The Novices' Difficulty with Grounding Intuitive Arguments on Appropriate Theorems. Decontextualised Knowledge, Regression to Quasi-Formal Familiar Modes of Reasoning and the Examples of the Intermediate Value Theorem and the Inverse Function Theorem
(vii) The Gap Between the Novice's Advanced Algorithmic Behaviour and Inadequate Conceptual Understanding. An Example from an Application of the Taylor Series
(viii) The Contrast Between Novice and Expert Approaches to Mathematical Reasoning. The Example of a Convergent Series
PART III A Synthesis of the Findings in the Area of Calculus. Indications for the Cross-Topical Synthesis in Chapter 10