Mathematics is a language, and it uses its symbols to represent mathematical concepts. The language of uses symbols, notations, and diagrams, rather than only using words, and these all have a part in how we understand mathematics not only as a static description of something in our world, but as how we can manipulate the mathematical statements as part of the dynamic nature of our mathematical descriptions.
Mathematics is essentially a set of manipulable representations, and to develop proficiency in mathematics it is essential to develop an proficiency in the language of mathematics.
Mathematics takes place in the mind of the observer, and not in books or on paper, and these notations and representations are to help us developing our understanding, and consequently to internalise the concepts.
This language of mathematics is large and complex, and can be understood better by dividing it into parts, using a sequenced approach to understanding the base before building on these basis. This is the purpose of curricula, and it is evident that the lack of a suitable basis of knowledge prevents further development on more advanced concepts.
We take the view that treating mathematics as “difficult” or “hard” makes no sense as a general statement. At one level mathematics is always difficult, since the learning of mathematics means developing an understanding of new conceptualisations which you have not previously encountered, and in which you have to have a solid basis in the background mathematics to support you. If you do not have this basis, then the new mathematics will naturally be “hard” but if you do have this basis well-established, then it may be a natural extension of what you already know. Thus, the notion of “difficult” when applied to new mathematics is relative to what you know already, and your ability to develop a new understanding of new elements of the mathematical language.
Mathematics is an abstract language, in which the notations and symbols and diagrams become increasingly abstract as they build on other elements, and it is not possible to understand a notation or symbol or diagram without understanding the basis of this.
We argue strongly against the progression of learners in the mathematics curriculum having achieved a low pass rate or 30-50%, and that 90-100% should be the norm. This is then coupled with the complementary proposal that mathematics progression should be on individual trajectories and not on class trajectories, where classes are structured as supporting environments rather than instructional environment.
However, enough about our positions, let us get on with the work of developing a universal thesaurus of mathematical symbolisms.
The goal of Programme TN is to develop an authentic and peer-reviewed source of all mathematical terms and notations, as an aid to all mathematics scholars. The goal is specifically to render these terms and notations accessible to teachers and learners at all levels.
A secondary goal is to help people to understand the interconnectedness of mathematics through exploration of this structured repository.
This is perhaps the largest of all of the programmes in MUMA at present.
The outcome of this programme is a structured repository of mathematical terms, notations, symbols, and diagrams, with sufficient information to serve as an online aid for the benefit of the various users.
The structure will be that of a thesaurus of terms, to enable a range of relationships to be included, such as general-specific relationships, and related terms, as well as synonyms and alternate forms.
We have been compiling a list of all mathematical terms and notations for many years, and these have been structured into a database with searchable access.
These will form the core of the repository, and from this we will continue to improve the structural relationships between the terms, and aim for consistent and accurate definitions for each.
One of the major challenges is that many of the mathematics symbols do not have ease of entry or description, since they do not exist on the computer keyboard, and thus usage will be made of various tools and standards, such as MathML, LaTeX for diagrams, and MathGL (the mathematics graphical language) for diagrams.
For every term and notation, examples will be provided sufficient to understand the term, and every term will be linked to underlying bases, and to alternative which can be used, since mathematical notation is not consistent, with some symbols and representations used for more than one mathematical concept, and a single concept may be represented in various ways. As one important example, consider the number 1,234. Now ask yourself if you read this as “one thousand two hundred and thirty-four” or as “one comma two three four” as a decimal fraction.
The thesaurus will be organised by major topics in mathematics, which are mostly taught as a unit, such as Numbers, Arithmetic, Trigonometry, Analytical Geometry, Statistics, etc… It will also be organised in terms of curricula, linking specific curricula back to the thesaurus in terms of what should be known when within the proposed curriculum statements.
One important outcome of this repository is to explore how mathematics is highly connected, and how each of these topics draws on others.