Research from education sciences has not shaped a unitary concept of mathematical understanding because it has kept the same separation between the conceptual and procedural aspects of teaching mathematics. Both inter- and trans-disciplinary research pursued by mathematicians and philosophers have concluded, even since the 1990s, that, given the epistemological nature of understanding and the special place mathematics does have as an object of study in an intersecting zone of epistemology, philosophy of mathematics, and philosophy of science and adjacently foundations of mathematics and history of mathematics, the necessary contributions for a conceptual and theoretical framework within which to define the complete understanding of mathematics and to obtain the criteria and norms of an adequate teaching to provide this understanding, are expected to come from these domains (click here to download bibliography). These results led to the conclusion that for students to "do mathematics" while understanding it, and for the conceptual and procedural approaches to be unified, the content and methodology of teaching must follow an epistemic-holistic approach, centered around conceptual understanding, methodological justification, and context, which assumes several general principles, of which we mention the following:
  - Referring at any time to the logical and theoretical fundamentals of a mathematical concept reveals important aspects of the nature of mathematics and the links of that concept with other mathematical concepts learned or subsequently to be taught.
  - Explaining a mathematical definition or structure through breaking it down into its constitutive concepts and providing examples is not enough to ensure complete understanding of the concept it designates. It is necessary to go beyond the formalism of the definition to reveal the contexts in which the definition was created and for which it is relevant: 1) the logical context (generalizations, particularizations, and immediate implications, capturing the analyticity by commuting an axiom with a conclusion of the implication and testing it against non-mathematical interpretations); 2) the theoretical context (the mathematical and epistemic structures that are connected through the newly defined notion, the rationale of the mathematician for defining the new notion, examples from applied mathematics); 3) the individual epistemic context (the compatibility of the newly taught notions with student's conceptual framework at that moment, including those specific to other disciplines, like physics); 4) the historical context (mathematics was not given to us all at once, but has been developed over time, and every concept, theory, and procedure has had its own motivation at a certain historical moment).
  - The mathematical methods should not be merely used, but also justified, explained, and described in both their strengths and weaknesses. (There are also weaknesses of mathematics; it should not be presented as a “science of God” or “the absolute science”). .
  - The logical principles that mathematical deductive methods rely on, as well as the methods themselves, must be known when applied. Such principles are embedded in human reason. We use them in daily life, and we are aware of them from early childhood; there is no reason to qualify their presentation in the math class as inappropriate for a certain age or outside the taught discipline
  - Complex symbolism is one of the main factors that make math seem daunting to many students. This is because a symbolic expression is read literally and not conceptually. By visually and graphically decomposing a complex expression into parts that are assigned to the concepts they denote, then verbally clarifying the relationships between those concepts and projecting them back into the symbolic expression, we will reveal the conceptual-epistemic structure of that expression and the instrumental role of symbolism, and we will shift attention from the complexity of the expression to the relationships between its references, thus inducing a "conceptual reading" of it, much "friendlier" and more intelligible than the expression itself.

Teaching under the conceptual approach does not prevail over or affect the goals of the procedural approach. On the contrary, a holistic understanding of mathematical concepts will help in the process of problem solving, especially when difficult problems are encountered; such a vision reveals further connections (multi-steps ahead) between concepts that are not visible when struggling to find available, usable, and applicable procedures, and as such may suggest new, unexpected paths toward the solution, which may then be approached procedurally. As such, a person with a good understanding of the concepts will be a better problem solver.

The principles briefly exposed above are theoretically applicable at any school level, including secondary and high school. However, the application must also take into account the student's own level and universe of knowledge. This conditioning amounts to the means of communication, adaptation through personalized content, and teaching methodology, and also imposing the topics to be taught. (A student who at some moment abandoned the "battle" for understanding mathematics cannot be suddenly plunged into conceptual learning just to keep up with the class material, but must be brought back to the point at which the learning was abandoned – or earlier.).

 

 

index »