EDUC4105

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.

The key points of these papers are that in learning mathematics students need to be able to understand an array of new terms and symbols and be able to extract meaning from the language used by the classroom teacher. Students need to understand that the language of mathematics is not personal and the context of what is said is important. The language background of students also has an important bearing on how they interpret mathematical language. The importance of language in understanding concepts cannot be underestimated and often is.
This raises the issue of equity in the classroom, with literate students and students who can understand the teacher when their language is implicit, having an advantage over students with literacy problems or who come from a different language background to that used in schools, ie the language of the middle class.
This has implications for how we prepare to teach mathematics. Research in the readings shows that embedding problems in a real world context may lead students from a working class background to not recognize that the task is set in a specific context and thereby become ‘distracted’ from the mathematical task. As a teacher I would have thought real world connections were an important part of an effective teaching strategy, but this needs to be done in conjunction with the literacy needs of the students. Word problems are another area where teachers try to make the maths more realistic. Here its important to recognize that framing maths problems in language places extra demands on students. The problem becomes a multi-layered task.
As a teacher I can see that I should temper my enthusiasm for the real world connection and focus on being explicit with my use of language and encourage students to read for meaning and in the context of a mathematical problem. I also need to be aware of my own use of language and not assume students know the context of how I am speaking or that they share my use of language. Finally, when preparing lessons be vigilant for ambiguity in terms or opportunities for the language to cause students any misconceptions.

My initial reaction to using a graphics calculator for the first time was a mixture of curiosity as to what it could do and a feeling that there were a lot of buttons that I had no idea about. As with any new calculator there are some procedures that have to be relearnt. Shifting from screen to screen presented me with some difficulty. I imagine in a classroom, students could get lost quite easily unless they had very clear insructions and some experience experimenting with them.
I think they would be useful in modelling problems and displaying results. They would also be useful in graphing functions and comparing graphs of similar functions to show how a change in the function effects the shape and position of its gragh. Just how useful could only be determined with some form of assessment and experience using the machines in different topics.

One disadvantage of graphics calculators may be the time it takes for the whole class to complete exercises. If the process is involved this could cut into the time spent on the topic itself. Some students may have learning styles that don’t benefit from visual learning or even worse may misunderstand important concepts. There may also be a limited number available so the whole class may not be able to participate.
If they were readily available, I think they would be useful in giving students another way to learn many ideas.