EDUC4105

Mathematish, the use and understanding of mathematical symbols, involves the rules of using such symbols (or the grammar of mathematish as the authors put it) and how it conveys ideas by the way it is used. They point out that our own proficiency in mathematish may make us unaware of our lack of attention to the grammar needed by students who see it as a foreign language. The authors argue that matematical texts are bilingual, that is they have concepts and symbolic language. A lot of mathematish is taken for granted by teachers who are not explicit about the rules or origin of the features of the language, resulting in the knowledge becoming ‘tacit’ by nature, that is, not expressed or reflected on. Many students view of maths as a fragmented group of topics is due to a lack of understanding of the general language. If they knew the grammar, as in any language, they could see that the rules they learn are applicable across all topics. This requires a dialogue, as the authors state, playing the language game, allowing students to learn the rules through social interaction. I thought it interesting that as teachers of maths our tacit knowledge places us apart from most of our students and many of their problems don’t come into our thinking. This is something to be conscious of at all times. It was also interesting to consider the way textbooks amalgamate mathematical representations as if they were the actual objects and our expectation that students should believe such a ‘lie’. If, from an early age , we can teach students that mathematish is seperate from specific applications of it, that is they can see it as a general tool, then they may learn an important insight into the general nature of mathematics.

This paper discusses the idea of a mathematical register that enables us to construct our mathematical knowledge through the interaction of oral and written language, symbolic notation and visual displays. The author also notes the grammatical features of the register as having patterns of technical vocabulary, dense noun phrases and implicit logical relationships. She argues that learning new technical vocabulary may be easier than vocabulary which the student already knows a meaning in another context. Furthermore that although teachers may be aware of the challenge of learning technical vocabulary, many are not aware of the grammatical patterning associated with it. This has ramifications when teaching attributive and identifying processes, especially for students who speak English as a second language, where these relationships may have a different meaning in the grammar of their first language.
The teachers oral language may also adds more complexity, whereby processess are talked about as if they were things and classroom discussion may be conducted from different points of views without the teacher or the student being aware of it. To support students learning they need opportunities to practice their use of the mathematics register. Teachers also need to ‘unpack and explain’ meanings of symbols and explicitly focus on the language, expecting their students to use technical language in the classroom. This is important for students, for example learning algrebra depends on being able to formally talk about patterns and having students verbalise their thinking provides feedback for teachers and students about their understanding. Discussions should also focus on the reasoning students are using rather than their answers. Schleppegrell suggests that these discussions are vital in constructing meaning for learners and students need to be able to handle technical language which is first introduced orally in the context of odinary language before leaping into the technical.

This second article by Gough dicusses the artificial symbolic nature of mathematics, with its reliance on many non verbal ways of representing information. He makes an interesting point when he suggests that we need to be able to describe what we are doing in non-mathematical language as well as mathematical language. This use of students’ own understanding and explanations when dealing with new ideas is important as it connects the new knowledge with established structures that can later be refined to be more in line with conventional definitions. This, Gough asserts, is social constructivism (remember Piaget ?). I also liked the idea for imagining a square root as thinking of a number as a square and a square root as a side of that square, which makes you think geometrically and calls on previous knowledge of squaring something. Finally, though we should encourage students to talk in mathematical English, at times it is appropriate (and neccessary) to talk about examples in normal English to aid discussion and the deeper understanding that comes from such a free flowing discussion.