EDUC4105

In this article, Gough discusses the nature of the technical language of mathematics and how the use of normal English in our instruction can lead to difficulties for students. He makes a good point when he describes how we all feel a sensory version of words, for example, to talk about increasing at a decreasing rate can make us feel a bit whoozy, but to decrease at an increasing rate can have us (and our students) feeling confused or maybe slightly nauseas. Our spatial interpretation of terms can often be at odds with the numerical meaning, making it neccessary for the teacher to be aware that the language they are using may be misinterpreted by their pupils due to the student’s strong intuitive meaning they have of some terms. The real difficulty of teaching maths in terms of language is that we do use everyday language interspersed with mathematical language that has usage of common terms with different meanings. This has further implications for students whose first language isn’t English. Tho only way to know what your students are thinking is to ask them often and encourage them to talk to you and each other about the contradictions between the language they use and the language they are trying to learn. As Gough points out it is easy to forget the confusion of technical terms once they have been learnt and even easier to blame the students for having trouble when they have been subjected to ambiguity and ’slippery words’ from teachers. The recommendations at the end of the article are an excellent checklist to help teachers avoid confusing students and increase students’ understanding of new concepts and terms, while remaining aware of the student who struggles with vague explanations.

This site was rated highly by our expert panel, for more details, read on.

Efficiency
The task is large, but gives a deep understanding of the uses of logarithmic scales. The process link ensures clear understanding and focus on the task. A weakness of the webquest is that there are many tasks and a lot of reaearch for two people.
Link to logs
Technofile
The strengths of this site were good links to next page from the bottom of current page and good use of pictures, fonts and colours. However the tables could have been in excel to do the graphs and the worksheet template could be on word so students could type straight in.
Altitudinists
The task really only required the gathering of information. There wasn’t any synthesis of ideas though some degree of thought and problem solving would have to be done to work out scales to plot different size objects on a sheet of paper. The task also requires the student to connect logarithms to uses in the real world.
From the viewpoint of the NSW Syllabus.
This webquest addresses how to represent a wide ranging set of data and phenomina that changes logarithmically on a easy to plot and read graph. Data is represented on both linear scales and log scales and the advantages of a log scale are discovered by the student. There is good use of technology with great links that explore scales with interesting and relevant examples, from size of bacteria to diatances in space. There are no direct links to outcomes as it is American, so some localising of the aims would be good from a syllabus perspective. Still, a good webquest.

The main thing to come from this paper was Tapson’s observation that confusion in mathematical language often comes from not knowing how to apply words in a mathematical context. Tapson also questions some of our habits in naming shapes ( and parts of shapes) that appear to be inconsistent. I think many of these issues can be overcome with explicit instruction that emphasizes that we use particular words in the context of the mathematical problem at hand and to remember that we are talking in a mathematical mode, not a conversational mode.

This article stressed the need for students to express mathematical ideas clearly to foster their learning of maths. Students should be encouraged to verbalise their thoughts and speak like mathematicians, even if this feels awkward at first. To be able to do this students need a supportive environment and understand that wrong answers are as welcome as right answers. The author suggests that often students may feel that they don’t understand a concept, when really they just don’t know how to communicate their understanding. Before they can do this students need time to reflect on a question before answering, quite often this isn’t the case with many teachers using a rapid question and answer style in the classroom. Many students will feel they can’t do a more challenging question if they can’t answer it quickly. It is probably worth thinking about how we set up a classroom for discussion and how as teachers we mediate classroom discussion. An overly teacher centred style may limit interaction between students and the learning opportunities that come from students listening and talking about their understanding. Finally by listening to student’s discussion closely, we can find many opportunities to assess their understanding during the lesson.

This site uses real time data and collaborative projects with links to both of these to exploit the internet. There are links to educational resources and further links to areas such as podcasts, internet safety, real time data,etc. The Noon day offers a one step process to get expert information in a number of fields.
The layout is easy to follow with links to that follow a logical train of thought. The site has a clearly described set of purposes which it seems to deliver on. Some links for lesson ideas actually were mixed up, but it wasn’t hard to work out how to find the one you were after. The graphics were interesting, but there weren’t any audio clips. The spelling and grammar appeared to be of a high quality.

This sight provides resources to link activities from the curriculum in a social context. The authors of connecting maths to our lives, provide activities that link content to individual experience, critical analysis and exploration of social issues.
The layout of the web site is easy to follow, finding specific examples of maths topics in relation to various social topics can be selected at the top of the menu, with associated links to issues grouped together.
The sight is has links to teachers lessons from various countries. Not all images have ALT tags.
Graphics are well related to the topics though there aren’t any audio clips or video.

This paper suggests that an understanding of the rules of mathematics empowers students to take the leap from procedural maths to a conceptual undersatnding of mathematics. The fact that mathematical language is devoid of emotion puts it at odds with our every day spoken language. For many students I think this tends to make it difficult to stay focused on the precise language used in a mathematical argument, leading to a superficial or incorrect understanding of what is being explained. If students can be taught to master this language and use it as a tool, nothing more, nothing less, then I think a door opens up for them and they can gain deep insights into mathematical concepts, rather than feel confused and shutout of the club. My experience as a student was that such language was the domain of the ‘teacher’ and no attempt was made to show us how to use it to enable us to learn on another level. This is something to keep in mind when we deliver our version of essential maths knowledge. I also liked the link the author made to analytical thinking in general and how important this is for our students to understand complex social and political issues in today’s world. As a tool for life , the skill in pulling apart a theorem in a logical and structured manner has the potential to make all our maths students better citizens who have the skills to question anything they are told, and not be fooled by rhetoric.

This paper details many of the problems highlited in ‘Language and Mathematics’, (last week). As teachers, it is important to be aware of ambigueties in the English language and as the authors state, build up a library of commonly misunderstood and misused terms to be discussed with our students.
Modeling our own thought processes when teaching is an excellent way to encourage students to think about their own cognition when problem solving. I know for me as a student that I find it particularly helpful in understanding a new concept or process when a lecturer models their own thoughts in trying to find a way through a problem. It gives you a logical stucture to organise your own thoughts. This directs the ‘internal chatter’ that we all have into the right direction, helping students make connections and pushing them to think about what it is they are trying to do.
This paper also raised the importance of classroom dicussions being a,forum where everything is open to question and any mysteries can be undrerstood collectively.
The final thing I’d like to comment on is encouraging students with a primary language other than English to discuss new concepts in their primary language first. This is the first time I have read this suggestion and on reflection feels like a good strategy considering the complex nature of English, the precise nature of mathematical English and the layers these students have to negotiate to form an understanding. Suggestions such as these bring equity to these students by letting them use their strengths to process new and difficult concepts.

This is one of the best articles I have read to date on language in mathematics. Booker spells out in emphatic terms the importance of being numerate in today’s world. The construction of our mathematical knowledge is facilitated by the communication of ideas which largley depends on our ability to use language to express those ideas. Booker suggests that teachers should be careful to match the language not only to the age of the students , but also to the ‘concepts and processes’ being taught. Prviously I hadn’t considered the different ways we represent numbers, ie in the context of materials, language and symbols. This article highlites the importance of reading and writing numbers using place value and the ability of students to be able to rename numbers during computations, ie to reassign the numbers place. I also became aware that a lot of the language linked to ‘pseudo-conceptional’ understanding of maths is engrained in my own experience. Using words and phrases like ‘carry’ and ‘move the decimal point ‘ are signs of a procedural understanding whereas we should be trying to link concepts with ‘basic facts’ and let the processes come from these deeper understandings. Makes me wish I could start again from scratch and go through school minus the rote learning. The difficult thing about this as a secondary teacher is our students have probably already formed many strategies in their thinking long before we see them. I suppose you can just try and build some deep basic number facts while waiding through the curriculum.

This analyisis of ordinary English and mathematical English provides some valuable insights into the complex issue of how the use of the English language shapes our understanding of mathematical concepts. I found it helpful to note the author’s observation that native speakers of English skim read when reading text to minimise having to read redundancies. When this same technique, which we use intuitively, is used to read mathematical language, some important parts of the language can be missed or misinterpreted. As a teacher it is important to teach students to read mathematical language as though it was a foreign language to the one we use in contexts outside of maths.
I know from my own experience when reading a longish proof I have the tendency to start to skim read, resulting in having to reread something several times to follow the logic of what is being said. Also the structure of our language provides many opportunities to introduce errors into a maths problem. This is something to be conscious of when using statements to frame a question. By helping students to focus on each word and what that means mathematically in the context of the question is probably one of the best skills we can teach students.