## A Philosophical analysis of the concepts of undertanding in mathematics education with specificic references to logism.

##### Résumé

This study undertakes to establish a coherent conception of understanding in mathematics education. The study is clarificatory in that the fundamental concern is to elucidate the concept of understandings, and to analyze mathematical understanding with the purpose of identifying levels of understanding attainable in mathematics education.
Chapter one highlights aspects of educational concern with the notion of understanding, both as and educational aim and as it relates to mathematical knowledge. This is followed by the statement of the problems of the study which has two parts: the first part identifies understanding as a vague term which needs explication, the second part shows that the apparent polar nature of mathematical abstraction and mathematical intuition, creates lack of a coherent conception of understanding in mathematics.
In Chapter two, a philosophical analysis of the concept of understanding is offered. Here, understanding is identified as a form of implicit knowledge related to the meaning of an object of understanding. Such Knowledge of meaning is achieved by revealing the significance of that which is to be understood. The idea of significance is used to refer to the state of an object of understanding in as far as it is surmountable through such means as analysis, explanation and interpretation. Such a conception of significance takes care of the fact that a thing could be incomprehensible, either due to its complexity or simplicity.
In Chapter three, mathematical understanding is analyzed under the theory of logicism. Logistic theory places emphasis on the a priori nature and the deductive structure of mathematics, while it downplays those aspects that have to do with extra-logical procedures. Logicism asserts that mathematical propositions are validities, which are tautologous in character; hence every mathematical problem is solved by rigorous deduction.
The final part of chapter three proffers an appraisal of logicism, showing its strengths and weaknesses. Its strength rests on the postulates of logical certainty that goes with its deductive nature. Its weaknesses arise due to the assumption that the power of human reasoning is perfect, and needs no extra-logical aid to realize logical consequences. However, in certain instances, man is psychologically unable to see the logical consequences of mathematical propositions. In such circumstances, man uses certain guides in order to attain the level of logical certainty by deduction. Such guides, apart from being tautological transformations as calculations, may as well include methods of making mathematical discoveries, such as induction and analogy among others. Mathematicians have to discover what to prove deductively, and such end is achieved through extra-logical means in experience.
Chapter four discusses mathematical understanding in education. It is in this chapter that two levels of mathematical understanding are identified; the pre-logical and logical levels of understanding. The pre-logical level of understanding is attained whenever man makes mathematical discoveries using non-deductive methods. Any conclusions reached at this level are probable and are thus merely tentative answers. The logical level of mathematical understanding is attained when such tentative theorems are subjected to rigorous deductive proof, which establishes that the answers are logically correct. Such a process of understanding takes theorems from the realm of inductive probability to that of deductive certainty. This approach is favoured by the spiral curriculum, which may be designed, to develop the same series of concepts or theorems at several occasions in the curriculum, separated by varying intervals of time. Upon such occasions, several approaches are used that develop understanding from the pre-logical level to the logical level of understanding.
Chapter five, which forms the last chapter of this study contains the conclusions, recommendations and proposed problems for further study.