Children enjoy playing with toys when they can give meanings to their interactions with the toys. It is similar with early encounters with language, number and shape. Although meanings begin as personal and subjective it is soon discovered that they may be shared and this process itself stimulates the growth of meanings. Learning is most successful when the student makes the (unending) journey from personal to public meanings.
Students who think they are ‘hopeless’ or ‘bad’ at mathematics have often not had the experiences needed to allow them even to make personal meanings of mathematical objects or ideas. I shall suggest that such personal meanings are never ‘received’, but can only be ‘made’ by the student themselves in response to appropriate experience.
Meanings begin as personal within individuals but become shared in communities. They therefore always arise within historical and cultural contexts.
At the beginning of the 19C there were several mathematical expressions, or concepts, for which there were no clear, accepted, public meanings. Examples are infinite series, the infinitesimal, terms with an irrational exponent and the derivative of a function. Of course, this situation was compatible with experienced mathematicians developing numerous successful theories and new insights involving these concepts. But those mathematicians were also like students learning – in their pursuit and negotiation of public meanings for what began as private interpretations.
An inspiring example of an educator and mathematician who explicitly struggled with the clarification of the meanings of new concepts is Bernard Bolzano (Prague, 1781 – 1848). This talk will address these issues in the context of Bolzano’s little known early work on the binomial theorem where all four of the concepts mentioned above are explicitly addressed and substantial progress is made in their use and definition. The original German text of Der binomische Lehrsatz …., (1816) is currently difficult to obtain, but there is an English translation in my edition of The Mathematical Works of Bernard Bolzano (OUP, 2004, reprinted 2006).
The mathematics we and our students are learning is full of living concepts and principles which are stable, but not static. They have historical roots and future evolution. The example of Bolzano’s work will be used to demonstrate the inevitable and organic connection of mathematics with other disciplines such as history, philosophy and language.
The overall theme of the talk in relation to mathematics education is not so much about history as an influence on current mathematics but rather to emphasise that all meanings (influential or not) must begin in a personal meaning that is made by the individual in a response to experience – a response that is inseparable from culture and context.
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