[Paper Review] Epistemography and algebra

Jean-Philippe Drouhard|ArXiv.org|Oct 26, 2009

Mathematics Education and Teaching Techniques27 references3 citations

TL;DR

This paper introduces epistemography—a novel framework for mapping students' mathematical knowledge through five interrelated systems: the mathematical universe, semio-linguistic representations, instruments, rules of the mathematical game, and identifiers. It proposes that understanding student knowledge requires analyzing these structured components, offering a systematic approach to diagnose and describe mathematical understanding beyond standard assessment.

ABSTRACT

We propose to address the problem of how to know students' knowledge in an entirely new approach called ?epistemography? which is, roughly, an attempt to describe the structure of this knowledge. We claim that what is to be known is made of five tightly interrelated organised systems: the mathematical universe, the system of semio-linguistic representations, the instruments, the rules of the mathematical game, and the identifiers.

Motivation & Objective

• To develop a comprehensive framework for describing the structure of students' mathematical knowledge.
• To address the limitations of traditional assessment in capturing the complexity of student understanding in algebra.
• To identify and systematize the core components that constitute mathematical knowledge in learning contexts.
• To provide educators and researchers with a tool to analyze how students organize and internalize mathematical concepts.
• To integrate semiotic, instrumental, and structural dimensions of mathematical learning into a unified epistemological model.

Proposed method

• Defining 'epistemography' as the systematic description of the structure of knowledge in mathematics education.
• Identifying five tightly interrelated systems: the mathematical universe, semio-linguistic representations, instruments, rules of the mathematical game, and identifiers.
• Analyzing how these systems interact to form a coherent structure of mathematical knowledge.
• Using this framework to map knowledge in algebraic learning, emphasizing the role of representation and instrumental use.
• Applying the model to interpret student responses and learning trajectories in terms of system interplay.
• Positioning epistemography as a tool for both diagnostic assessment and curriculum design in mathematics education.

Experimental results

Research questions

• RQ1 How can the structure of student knowledge in algebra be systematically described beyond surface-level performance?
• RQ2 What are the core components that constitute a student’s mathematical knowledge in a learning context?
• RQ3 How do semio-linguistic representations, instruments, and rules of the mathematical game interact in shaping understanding?
• RQ4 In what ways does the epistemographic framework improve the diagnosis of student misconceptions or knowledge gaps?
• RQ5 How can the five-system model be applied to analyze and support the development of algebraic thinking?

Key findings

• The five-system model—mathematical universe, semio-linguistic representations, instruments, rules of the game, and identifiers—provides a structured lens for analyzing mathematical knowledge.
• Epistemography enables a deeper diagnostic understanding of student knowledge by revealing the interplay between symbolic, instrumental, and conceptual components.
• The framework shows that knowledge is not isolated but organized through tightly interrelated systems, each contributing to mathematical understanding.
• By mapping these systems, educators can identify not just what students know, but how they organize and access that knowledge.
• The model supports a shift from summative assessment to formative diagnosis by making the structure of knowledge visible.
• The approach offers a foundation for designing curricula and instructional tasks that align with the epistemological structure of mathematical learning.