EDUC4105

The first reading was yet another reading on the inconsistencies of Mathematical language. It did however avoid the ‘doom and gloom’ by making some recommendations for teachers to deal with the problems.

I really can’t see the use of one-ty one, one-ty two etc. catching on because although it makes sense and may be easier to learn, we simply don’t use it in our lives. We are brought up with the traditional use of eleven and twelve etc. A complete overhaul of how we pronounce numbers would be more trouble than it is worth. Why not completely create uniform of expression and start saying that the winner of a race finished in “oneth” place? It ‘aint broke – it’s just the way things are.

I also completely disagree with the author’s preaching about using zero more to promote it as a number. It seems completely bizarre to me to say that a baby is zero years old. If a child has not yet lived a year, then they have lived months! Or weeks! Zero years is not a useful description and would be taking things to extremes.

Dismounting from my high horse, the second reading got me thinking about triangular roots of numbers and whether or not it would be a useful problem for students to have a go at to better understand why square numbers have developed. Having students apply knowledge of squaring to (equilateral) triangles could be a good way to reinforce understanding of the ways that some particular natural numbers can be arranged into squares (4, 25, etc.) while the operation of squaring can be applied to numbers in general.

The following is the opinions of a small group of university students on a webquest activity. We are all viewing the contents as beginning Mathematics Teachers.

Here is a link to the Chess activity.

Strengths

• Task has good Knowledge Integration with history (cold war)
• Uses research as a teaching strategy
• Relevant links are included for students to read more about different aspects of the task
• Plenty of technology incorporated with links to films and opportunities for students to create part of web pages
• Students have the chance to play chess with one another

Weaknesses

• Irrelevant to maths syllabus
• Research is only of interest to chess fans and requires a degree of background knowledge of chess
• Task 4 is more suited for D&T where design and ease of use is important
• Seems to be a trivial activity aiming at chess promotion
• Outcome is not mathematical and does not require higher-order thinking (more a structured writing task)
• No meaningful motivation for students
• Purely individual work – lost opportunity for collaboration

For anyone interested in learning chess from a grandmaster, here is a blog on the games of Karjakin Sergey.

I think I will jump on the bandwagon from now on and start using “oblong”. It seems like a good idea for a square to be a rectangle, but not an oblong, to better distinguish between the two shapes.

Giving the circle a perimeter rather than a circumference might be a good idea just because the standardising could make composite shapes easier to interpret. Then again maybe circles and ellipses do deserve a reserved title for being curved. Yes the fence is comfortable, thanks for asking.

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’?