EDUC4105

The interactive whiteboard can be beneficial to teaching, with students becoming more enagaged with correct use of the board. This can be due to students becoming more involved in the lesson (depending on the way in which a teacher uses it) especially when ‘hands-on’ applications are use with it. These might include applets from the internet or even the software that came with the whiteboard. In this way, the use of an interactive whiteboard gives another medium through which teachers can reach their students – it can be very visually appealing with the use of colours, as well as the kinaesthetic appeal to students who enjoy a ‘hands-on approach’.

Killen (2003) says that matching the your teaching style to the way in which a student learns can help them achieve their full potential, and the use of the interactive whiteboard can help a teacher practise a variety of teaching styles (as mentioned). The fact that a PC is needed for the use of the interactive whiteboard also means that the internet could be utilised throughout the lesson as an extra resource, however this could be done merely with a PC and projector.

There can be a danger of using the interactive whiteboard purely for student engagement, as this might suggest that a teacher would believe that the use of the interactive whiteboard at any level will increase student engagement. While this might occur to a certain degree, after the ‘novelty’ of the interactive whiteboard wears off the teacher would still need to ensure that the lessons themselves are stimulating; not just relying on the alternative teaching mode to engage students. With this in mind, I don’t think that the interactive whiteboard would best be used as an absolute replacement for the traditional whiteboard, but as an extra tool to be used – it should only be used where it will actually benefit students learning.

This is a lesson using geometer’s sketchpad on the angles formed by a transversal crossing parallel lines. As far as the NSW syllabus is concerned, this lesson would go some way to achieving the outcomes of SGS4.2. I like the way that the lesson is structured, with all of the steps in a logical sequence.

Using dynamic geometry as opposed to sketches of sets of parallel lines means that students can immediately see how the relationships that they might suspect remain constant regardless of the angles involved. It’s also good how the students are not simply told the relationships, the lesson instead scaffolds their progress to reach their own conclusions. I think that using geometers sketchpad might be a benefit as compared to traditional teaching techniques for this lesson.

It took a while to get into, but after digesting what Schlepegrell (2007) was getting at, I realised that she was providing some valuable insight. It was a little bit ‘more of the same’ at first; she called Mathematics’ technical jargon the mathematics register (especially those homonyms which need to be in context before have a mathematical as opposed to colloquial definition; e.g. volume, function, average, mean etc.).  She talked about the fact that we need to develop a student’s mathematics register because you need to learn the language of maths as well as being able to ‘do it’ – just performing the routine procedures doesn’t lead to an understanding whereas understanding the language can help develop an understanding of what the procedures are actually doing.

Schlepegrell talked about the three different parts of the language – the written (English) part, the symbolic (algebra etc.) part and the graphical or diagrammatical part of mathematical language, and that we as teachers need to help students integrate these into a single ‘language’.  She said that it is important for teachers to use oral language (perhaps a fourth part of mathematical language) to help link each of the three different branches of language.

I thought that Schlepegrell’s use of examples was quite helpful in her explanations, such as when she showed that written and symbolic mathematical language seem to achieve a common goal in vastly different ways. Schlepegrell’s example was “the sum of the squares of two consecutive positive even integers is 340″ being equivalent to”a^2 + (a+2)^2=340″. The written version looks at the left hand side of the equation as one thing, whereas the algebraic version looks at it as processes being ‘done’ to ‘a’.

Schlepegrell  included a valuable section at the end about how we as classroom teachers can help the development of the students’ mathematical registers, a main suggestion was talking about language and how it is used throughout the lesson – sounds like good “Quality Teaching” to me – incorporating Metalanguage is not doubt a fantastic way of helping. Interesting, Schlepegrell said that groupwork, while it can be helpful, reduces the teachers timt in talking with individual students and thus the teacher has less time to try and link the three different aspects of mathematical language.

The theme repeated throughout both of Gough’s article is that mathematics is a language in itself, and yet we try to explain it using another language. He (like many of the other authors) says that problems arise when we use words that have relatively straightforward meanings in colloquial use, yet we use these same words to describe quite complex concepts. Further to this, Gough says that when we use such terms as decreasing at a decreasing rate, increasing at a decreasing rate or any combination of such terms (which we already have an idea or picture in our head of the meanings of these words) to describe something as complex as and exponential curve, we are going to run into language problems. He says that part of the problem is actually just having the same (e.g. decreasing at a decreasing) or opposite (increasing at a decreasing) words next to each other, as drawing a picture relevant to each description is conceptually quite difficult. For those of you who have trouble picturing these, try to think of how to describe each curve without looking it up:

I liked the suggestion of Gough’s to discuss the overlap between the ‘language-of-instruction versus subject-language’; quite often we might use our language of instruction (English) and use a word that means one thing, then slip into the subject language and the same word means another thing. I think that it’s important for students to feel free and to be encouraged to discuss this and for students to feel free to question which meaning of a word you are using (which ‘language’ you are using) if they are unsure – the social support of the classroom should be high enough that students feel comfortable in speaking up if they are unsure. This is especially important since, as Gough says, we are likely to forget the problems that we had upon our initial contact with specialist language, so we need to hope that our students will help us with recognising the difficulties that they have.

It was interesting that Gough talked about whether or not students from a NESB ‘think’ in their ‘mother tongue’, and that we should speak with the students themselves about this to find out. The importance of this is that there might not be equivalent words in their first language to those that we use to try and explain the language that is mathematics. The suggested remedy is to work out what other words we can use that might be applicable to explaining the concept that do have equivalent meanings in the students first language.

I liked how Gough gave a section on his recommendations for combating language problem, and thought it interesting that, especially in his third recommendation he seems to be talking about teaching well to combat the language problem.He suggests using background knowledge, using multiple teaching methods (for students from different learning styles), and emphasising the importance of good quality working out and writing (such as differentiating x from x – and not calculus differentiating either, I mean it in the verb, not the noun sense!!).

It was interesting that it was pointed out that ‘ means feet or degrees, and ” means inches or minutes depending on the context. now this is not so much a problem that us Aussies run into, but it could be a problem where imperial measurement is used.

Finally, I liked the idea that a lot of terms that we use in maths seemingly make little sense, such as squaring numbers. further to this, we don’t always (often?) explain such terms in terms of geometrically squaring a number and then rooting as finding the side that we had to start with – I think that such an explanation would make a lot more sense to students. In future, I will definitely be trying to use the language to my benefit when explaining such concepts as squaring and rooting.

After reading halfway through the first page of the second article for this week, I was not surprised to see that it had the same author as this week’s first reading. It had the same feel about it, the same style. I think that further to this, all of the articles in the course to date are starting to show similar ideas. This has meant that instead of needing to completely think about new ideas and concepts each week, I have been able to focus on the detail that each author is attempting to convey in regards to the problems that we have in using English to teach mathematics. I see this as a good thing, as we can start to see similar themes and ideas that are repeated throughout the course booklet.

One thing that all of the authors seem to be doing is providing solid examples of where we might have problems in teaching mathematics using the English language. Most of these examples are able to be generalised, so I think that I am getting a good feel for where students will have problems. I like that we have had not just say junior school examples but also some from the stage 6 syllabus. The fact that most examples are actually Australian is excellent – a lot of the courses that we do seem to use materials from America, which use quite different schooling systems and I guess that it just feels less relevant.

Speaking of relevant, I think that a recurring idea throughout all of the readings is that we need to teach with Quality. Most of the authors seem to agree that we need to ensure that students feel free to talk about the lesson, that we shouldn’t be too rigid and ‘hands-upish’ – if that makes sense… we need to ensure that our students feel comfortable that their thoughts, opinions and questions are and will be valued. I will try to include the elements of Quality Teaching in my writings of future articles – I think that it is good to see that even though not all of these articles are from a NSW origin, the ideas Quality Teaching are the same everywhere – even if they are not explicitly pointed out.

Well this posting was going to be about Gough’s articles – I got a little carried away, so I guess that I’ll talk about his articles in my next post

Adious

Nick.

Here Ramirez gives practical examples and remedies of where students may have difficulty with mathematical language in the classroom. An important point raised was one of oversimplifying key words and their definitions – only recently I had a student to whom I oversimplified a key word – ‘of’. I said that it usually means times in maths, when working out say 15% of $80. I was careful to say that it was different to 15% off $80 (one letter makes a huge difference!!), but when we looked at 8 as a percentage of 80, for example; he had difficulty with the language.

The problem turned out to be that I had said “‘of’ usually means times in maths” , so he thought that he had to multiply 8 by 80. Before I made this connection I tried to liken this question to ’3 out of 4′. I realised that the problem that he was having was my oversimplifying the key word ‘of’, and I made a mental note not to make too much of a generalisation when explaining ideas. It was quite an eye-opener when I saw that Ramirez had written about similar problems occurring; I now realise that we need to be careful of not only what we say to our students but that fact that the students may have preconceived ideas of what certain key-words mean in maths because of their past teachers.

This example really goes to show that we need to not only be careful of our use of and explanations of language in the classroom, but also we need to be aware of any preconceived ideas of use and definitions of such language that students may have. We need to be aware of the metalanguage that we use in the classroom.

This article discusses the practical problems and solutions to using Mathematical Language in the classroom, especially for those students who are unfamiliar with using such language. Such practical suggestion included changing the classroom layout  one more conducive to conversation, which should foster communication via mathematical language with the teacher and peers in the classroom.This may even mean asking students to leave their desks and gather around the front of the room (not necessarily on the carpet, especially at high school levels) to ensure that students can’t just put their heads down to avoid participation in a conversation. The idea is also to make students feel more comfortable in a less formal situation than sitting at their desks which should encourage them to participate.

I also liked the suggestion that if students are closer to one another, the teacher will not have to repeat what a student say. the importance behind this was that teachers would quite often rephrase the statement in this situation, which means that students are not required to become as familiar with and competent at using mathematical language.

Other suggestions include ‘getting the ethos right’, by not making everyone put their hands up before they wish to talk, and ensuring that students feel comfortable with making a mistake – it is O.K. to be wrong. This is all about making students feel comfortable within their learning environment.

Some of the ideas in the article are good and practical suggestions, but for those familiar with the NSW Quality Teaching model, it will sound very familiar. I guess that it is a good read to get some practical suggestions at implementing the Quality Teaching model in their classroom.

This week we’ll look at various maths web sites and their usefulness to teachers.

National Council of Teachers of Mathematics

This site is very much a commercial site – you need to pay for access to most of its resources. The ‘preview’ articles and resources do seem useful; there are articles from their Journals such as ‘Teaching Mathematics in the Middle School’ and ‘Mathematics Teacher’. One such article is about teaching algebra, saying that we need to ensure that we don’t give students the impression that there is only one method to solve given algebra problems. The benefit of a site such as this is that it’s very professionally laid out and designed. I probably can’t really recommend this site however, as you need to pay to be a user… I think that if you look hard enough, there will be free resources out there.

The Math Forum

This site is hosted by Drexel University, and it contains a couple of options for purchasing problems, but on the whole there is a lot of free information for students, where they can ask questions; teachers, where they can ask other teachers questions such as ‘how do I explain this concept’. There is also a Maths tools page, which is focussed on technological resources.

The Teachers section had a good way of explaining the multiplication of negative numbers, and they’ve given about four different explanations – excellent if students don’t quite understand the first, second or third methods!! This is just one such example, there are many such explanations available. The students can also ask similar questions and get similar results if they don’t understand a certain concept.

The Math forum doesn’t have a particularly great interface – it’s certainly nothing like the web sites and forums that we might be used to seeing today, however it has been around since 1992 (which might explain the interface) which means that there is a lot of information archived.

To Be Continued

Given that a big problem with interactive whiteboards is the cost, this is an interesting alternative using a ‘wii remote” which I believe has a rrp of about $70. If this really does work, it might be an alternative to the ~$5000 plus projector that you might normally have to spend on an interactive whiteboard. For anyone interested this ‘wiimote whiteboard’ concept is still being developed, and an online community exists including teachers and members of the IT community – if you have some programming skills you might want to help out. The forum also has non-experienced users and seems to have a wealth of information so if you’re interested, check it out.

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The article by Padula, Lam and Schmidtke was very enlightening when describing some of the problems with using English to teach Mathematics. The fact that we have 1120 ways to spell only forty different sounds that are in the English language does make it difficult for kids to become fluent in literacy, which is really a prerequisite for learning Mathematics.

In a literacy class a couple of weeks ago, the tutor made a comment that ‘ghoti’ spells fish:

• gh as in enough
• o as in women
• ti as in direction

I thought that was quite amusing, but it really highlights the problems that kids might have in our classroom.

It was also mentioned that mathematical writing is very dense compared to other written subjects as we use many symbols to condense our texts, even further enhancing difficulties with understanding. I know that when I first arrived at Uni and was introduced to symbols such as∀ (for all), ⇒ (implies), ⇔ (if and only if) amongst many others, I was at first daunted. Not only do you need to be able to decode the symbols into English, you also need to think about what they actually mean. Many of the lecturers simply assume that this sort of ‘language’ is common knowledge, but the new-to-university students have considerable problems for at least the first few weeks AND THESE ARE RELATIVELY ‘HIGHLY EDUCATED’ PEOPLE (to have been accepted into university). Imagine the problems that some of our younger students have.

The other problem that I found with the lecturers using such symbols was that it meant they could fit quite a bit more information into their notes, so we probably got more information because of this mathematical English than we might otherwise have gotten.