EDUC4105

It is always encouraging to read reports of research undertaken by teachers where they try things and it works. There were some good ideas mentioned like bringing the class together in a group for class discussions, both for student involvement and behaviour management. This could even be useful for class demonstrations of technology. The encouragement of error is something that is becoming increasingly prominent in Mathematics classrooms. It is definitely important in establishing a conjecturing classroom environment.

Encouraging students to use their own language to contribute in discussions was another good idea. Because the teacher declined to interfere with suggestions by students, everyone was more willing to contribute and there was opportunity for students to realise the need for more descriptive Mathematical language.

I also really liked the activity described right at the end of the chapter, where students  had the chance to come up with their own number pattern and name it. This creative activity places emphasis on Mathematics being a human construct. It could also let students feel more connected with the terminology that somebody else has come up with.

The idea of definitions is conducive to the topic of Geometry, paved by the work of Euclid. The language needed for good definitions is subtle and it is a good exercise for students to create definitions of familiar Mathematical objects to illustrate the difficulty involved and also learn about the nature of Mathematics.

The reasoning required of students to compile a definition can serve as a lead in to discussion of the need for axioms and a basis for developing Mathematical theorems and proofs. Jamison refers to university classes where Mathematical jargon is expected (for all, if…then), but for the context of secondary school, in the exercise it would be just as beneficial for students, while still using correct terminology, to use familiar everyday language instead.

I’m not 100% sure about the need for a procedure required for a definition as students will have presumably seen an English dictionary and have a resonable understanding of what a definition is. I think it might be better to place the emphasis on correct, unambiguous definitions rather than nitpick about ‘good’ definitions if this language exercise was applied to a high school context.

The point was usefully made in the reading that teachers can assist students in thinking through problems by modelling the Mathematical processes involved themselves. In this way teachers are able to scaffold problem solving skills by asking the questions needed to solve the problem rather than simply giving the students the answers they want. Eventually students will ideally go through the process themselves, with teacher questions (”what do we know?”, “what are we trying to find?”) used as prompts.

The use of correct and precise Mathematical language is also a good suggestion made, so that students are used to terminology and are not confused with commonplace language more familiar to non-mathematical settings. I think it is just as easy for students to understand what an integer is rather than ‘positive and negative numbers’.

Again the authors point out some common student misconceptions that have been a feature of the required readings so far in the course. These possible student errors are things we as teachers will benefit from being aware of.

In the reading I liked the explanation of the little things we take for granted like 97 pronounced like ‘9t7′, yet 37 isn’t ‘3t7′, but ‘thir t 7′. I also have a new favourite maths title pun: A symbol-minded approach to mathematics. Perfect.

It was good to read the ways that the topic of number could be broken down and presented in different ways to students. The historical background of the mathematical operating symbols used to manipulate numbers was also interesting and beneficial for students to be exposed to. The discussion of how different symbols are pronounced and what each means would be useful too. This is a good guide to see what to try to do with every topic we learn with a view to teach.

I personally like the idea of teaching Mathematics as a language, perhaps not in isolation, but by highlighting differences between the English we are used to using (ordinary English or OE in the reading) and Mathematical English (ME). I think this would greatly enhance student comprehension of word problems and help them feel less intimidated by ME. The real question for mine is not whether to teach the language explicitly, but how to do it. Do we set aside time to treat ME specifically or is there effective methods to incorporate metalanguage in classrooms regularly (or every lesson, as is suggested)?

It was interesting to read that English is one of the most misunderstood languages and also that ME adds further complications to this language as well. I’ve never really thought about whether it would be easier for mathematics to be taught and learned in a foreign language, and by the evidence in the reading it appears to be the case!

The point made about errors in textbooks is a good one for students to be aware of too if some of their learning is done through textbooks. Students should read the questions critically and acknowledge the possibility of typos or incorrect answers in textbook exercises.

I found this article on the Mathematical Association of America (MAA) website. It was the most emotionally involved that I have ever been when reading anything at university. I recommend that every Mathematics educator, let alone us enrolled in the EDUC4150 course reads this article. Please feel free to post any thoughts as comment.

The article is written by Paul Lockhart who is a mathematics teacher in New York.

Lockhart’s Lament

Certainly in the first reading, Language implications for numeracy, Zevenbergen makes an interesting point when he explores the effect of social differences in language use impacting on numeracy. It had not occurred to me that different social classes might classify different items differently based upon experiences.

An example is given in the text where the students take a personal view of a question, which is reflected in their answers. This was good to see as a potential source of confusion for students, but one I think can be easily avoided through careful wording of questions and a clear idea of expected responses.

The list of homonyms and homophones are a useful reference in the second article and are words I think should be clarified and joked about wherever possible (mmmmm… pi). I think a list like this would be good to make verbal references to in a lesson where appropriate. The point made about sides and corners of 2D and 3D shapes was excellent and could easily cause confusion for students.

I can understand the confusion that might be caused by ruler, mass and root, but can anybody help me out with the mathematical meaning of ‘vulgar’?

Graphics calculators are a great mathematical tool for students to use when graphing equations or a table of values. It would be great for students to have access to these calculators if they were curious about what a certain equation looked like on the cartesian plane. Also instead of drawing a diagram to get an understanding of a problem, students using the graphics calculators could type in the equation to be drawn and instantly have a correct visual aid. Drawing a line of best fit for a set of data can be a difficult skill, but using the graphics calculators last week it was a lot quicker and easier using this technology.
The only problem I can see with students having access to these calculators is that the letters of the alphabet are on the keys, so students can write messages on the screen to one another and become easily distracted. Maybe I shouldn’t judge all students by my own standard of behaviour.

I have a feeling that any classroom lesson using graphics calculators needs to be planned so that student are always doing something and there is no time for students to become distracted.

First Reading:

Literacy, Numeracy, Quantitative Reasoning, Mathematical Literacy and Quantitative Literacy.

From reading this material it appears to me that literacy and numeracy depend on context and as such it is difficult to have permanent definitions. This would partly explain the reason for the differences in the wordings. Generally the examples explain the same basic ideas anyway. The important parts of the reading for me were the differences between the terms, rather than the specifics of what is meant by the words.

Second Reading

Doig, B. (2001) Summing up: Australian numeracy performances, practices, programs and possibilities. ACER.

It was all very nice to see the results of the TIMSS and TIMSS-R, but not really that informative or surprising. The differences between the performances of both sexes were negligible and there is concern about the numeracy achievement of indigenous and non-English speaking students. So these trends that we are already aware of continue.

I’m not sure that comparing different year groups is that effective, since surely the achievements of different age groups will differ from year to year. I would have thought that it would be more accurate to focus on the achievement of the same group of students as they progress, although this would be far more time consuming. I confess to be a bit of a sceptic when it comes to this kind of educational research.

I believe that technology should be used as a means to an end and not an end in itself. That is, technology can be a very useful mathematical tool, however it should not just to be used for interest sake or because we can. Any use of technology in classrooms should always have a clear purpose.

I believe it is very good for secondary mathematics students to be able to use and access technology, such as graphics calculators, to gain a better understanding of concepts by using ideas in a different context. I see a lot of benefit for students to play around with the technology and explore ideas for themselves. This is especially useful for the majority of students who are familiar with using computers.

The main benefit of teaching and learning mathematics with technology is that students gain deeper understanding since the concepts are presented from multiple perspectives. The ideas in a topic are not restricted to paper, but can be manipulated and applied in a new context using technology. Teachers are also able to present and demonstrate content in new and interesting ways. Through using technology, students also gain graphing and computational aids for their mathematical toolbox that allows focus on concepts, avoiding any lengthy working out.

It is important to remember that it is the mathematical ideas that are important, not the use of technology itself. Technology should not be used if it is not needed. If we can do the same thing with pen and paper, then why not do it that way? For lessons which incorporate technology, the use of technology should have a clear focus and student activities should be carefully planned. All activities should have the students actually doing something with a view to understand a concept, rather than simply seeing what we can do with the technology. Activities should also take into account different student abilities, so that advanced students can keep exploring ideas, while slower students could input all required data.

A major disadvantage of using technology is that students may not see it as important if it is not part of assessment. Teachers and students must have the focus of using technology to enhance understanding of the assessable content and that it is worthwhile. Alternatively, the use of technology could be included as part of assessment.

There are practical and equity issues involved with teaching and learning mathematics through the use of technology. Schools may only have limited computers for student use, or classes might only be able to use computers at school at certain times. The equity disadvantage is obvious where there might be some students who do not have access to a computer or relevant technology at home, or are unfamiliar with their use compared to other students.

I personally think that technology should be incorporated in teaching and learning secondary mathematics, as long as the technology is used appropriately.

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