Wednesday, August 30, 2006

Another puzzle

Those who enjoyed solving 100 pirates will want to take a look at the 2 Numbers puzzle, on Naught Much. Other people's solutions are in the comments, so be sure not to cheat!


Solution to the priates puzzle

Thank you to those who participated in the puzzle.

The solution in words:
If a number of pirates need only one shoe and of the others half need no shoes and the other half need two shoes, then on average each pirate needs one shoe. So, the answer is 100 shoes.

Algebraic solution, thanks to Mr. Person:
Let x = the number of one legged pirates, and let y = the number of pirates with two legs. If all y pirates wore shoes, we would need 2 y shoes for this group.
But only half wear shoes, so we only need y shoes for this group. And since we only need x shoes for the one-legged (one shoe per pirate), the total number of shoes
required turns out to be equal to x + y, which is simply the number of pirates.

Also, thanks to S. Elsnick who pointed out that some assumptions needed to be made in solving this problem: all of the one-legged pirates always wear one shoe; there are no pirates without at least one leg; no pirates have three legs; the remaining half of the two-legged pirates always wear two shoes.


Monday, August 28, 2006

100 Pirates: a puzzle

Here's a puzzle posed by a student to a colleague of mine. If you can solve it, then please email me:
A certain number of pirates are one-legged. Of the remainder, half never wear shoes. If there are 100 pirates in total, how many shoes will they need?I look forward to your submissions.

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Saturday, August 26, 2006

The most useful course I took at university

One frustration I had with my graduate diploma of education was the fact that we were expected to read a lot of research papers, without being provided with any relevant training. We did not know enough about research methodologies to be able to read the papers critically.

When I went back to uni to do an MEd, I took a course specifically dedicated to reading Maths Ed. research. I was lucky that the course ran with only one enrollment - mine! The university felt guilty for not promoting the course well enough and let the lecturer go ahead and run it. Later on, a DEd student joined in and a Phd student was attending at the lecturer's request.

This was by far the most practical course I have taken in education. Each week we looked at a different methodology - quantitative, case study, ethnography etc...- and each of us gave a report on a relevant paper. What made the course immediately applicable to my teaching practice was the fact that the lecturer allowed us to follow our own interest for the major assignments:
  1. A literature review on the topic of our choice.
  2. A paper aimed at teachers on the same topic.
In my case, I chose introductory algebra. Out of the reading I did, I got a paper published in a teacher's journal and introduced a unit of work for the year 7 classes at my school.

Much academic research happens in the classroom and with the help of teachers. Researchers often interview children to find out the way they think about certain things. All this knowledge is very useful to a teacher. It is not a case of academics telling teachers how to do their job, it is a case of them providing teachers with information which they are best equipped to make use of in their own classrooms.

Unfortunately, the lecturer recently informed me that the faculty has pulled the plug on the course. I still think that teacher education should always include a course on reading research.


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Help! My nerdy humour is not working

I wanted to liven up a very serious revision lesson with my year 10 IT class. So I used some nerdy humour. I jumbled up some sentences on how computers catch a virus and how these can be removed. When rearranged, the paragraph reads:
Comic book guy opened a file from an unknown source. A virus installed itself on his computer. The virus infected his files. Comic book guy panicked. He ran anti-virus software. The files were disinfected. Comic book guy breathed a sigh of relief, and celebrated with a Vegemite sandwich.
Next to this was a cartoon of the comic book guy (from the Simpsons) saying: "No emoticon could describe how I feel".

Needless to say, they thought it was a very lame joke :-(

Hip people out there, can you help me? Or am I beyond help?


Wednesday, August 23, 2006

The 81st carnival of education

Thanks to the very efficient education wonks, this week's canival of education is now up and ready for your reading pleasure.


Tuesday, August 22, 2006

You do the curly whirly and you turn around ...

Most Maths teachers would be familiar with the following algorithm for converting mixed numbers into improper fractions. Improper fractions are those where the numerator (top number) is greater than the denominator (bottom number). Thanks to Mr. Person for the illustration.

Basically, you get the numerator of the improper fraction by multiplying the whole number by the denominator and adding the result to the numerator. Fabulous and very sensible! I am being sarcastic, in case you cannot tell :-)

Every year I revise this topic with my year 8s and they all know some part of this algorithm by heart and usually forget another. They fumble through "you times by the bottom and plus by the top". The poor souls get to repeat this, until they use proper verbs (add to, not plus by)!
Not one of them can explain why this works, or is interested in knowing why this works.

This year, I have the fortune of having a year 7 class instead of my usual year 8s. They were happy thinking: one and a quarter is the same as 4 quarters plus another quarter. This can be written as 5 quarters. After they all understood this, I thought I would show them the obligatory shortcut illustrated above. They had seen it in primary school but could not fully remember it.

For the first time, I was the first teacher they had who called it the "curly whirly", and they were not interested. They wanted to think about it logically, turning the whole number into a fraction. So, this leaves us with one good use for the algorithm:
Oh, the curly whirly
Oh, the curly whirly

May your Maths classes be filled with song!

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It's hanging on the fridge!

I began this semester with a new year 9 class. Two weeks into the semester, one of the students invited me to a talk she was giving in her English class entitled "why I despise Maths"! The student certainly displayed a negative attitude in class, though she did achieve a high mark on the test.

At parent-teacher interviews, I learned that she enjoyed creative subjects. I suggested that, since we were learning "linear graphs", she could try out the virtual beadloom software. Today she announced to me that she had made a beadloom and enjoyed it. I asked her to show it to me, but she said "It's at home, hanging on the fridge!" She then turned to her neighbour and began to describe the activity.

As you can imagine, I was so delighted. A piece of Maths work is on display at a year 9 student's house!


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Wednesday, August 16, 2006

The 80th Carnival of Education

If you're after news, opinions and all things education, then check out this week's Carnival of Education at the Education Wonks.

Saturday, August 05, 2006

What is wrong with rote learning?

The Victorian curriculum seems to be built on one of two assumptions: either people do not have memories, or else their memories need to be left unused. To my knowledge, the students are never asked to memorise poems or mathematical definitions.

A year level coordinator once told me about an English teacher who made the students repeat a poem until they could recite it by heart. That took place during an excursion. When they were back at school, many students commented that they had never realised one could remember something if one repeated it over and over! These were 15 year olds who had not developed strategies for memorisation. I once read about Western hostages in Beirut, and how they kept themselves sane by reciting their favourite poems.

The other day I was describing a project on the Pythagorean Theorem to my nephew, a French educated 17-year old from the middle east. I first asked him if he knew the theorem. He thought for a couple of seconds and said: "Dans un triangle carré, Le carré de l'hypoténuse est égal à la somme des carrés des deux autres côtés" (In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides).

Did the fact that he knew it by heart mean that he understood little of it? Not at all. It gave him the necessary vocabulary to describe his understanding. As we walked on, I described the geometric proofs that my students had to describe as part of their project. He had never seen those proofs before, but we could discuss them abstractly, without having the pictures in front of us. I could use words like "somme" (sum), "surface" (area) and "longueur" (length), with which he was entirely comfortable.

I then mentioned the concept of a proof by induction, something taught in the first year of a science degree at a typical Australian university. He said, "yes, I know what that is. Induction is the opposite of deduction in that you begin with a particular case and generalise. In deduction, you apply a general rule to a particular case."

Let me temper all this by stating that I am not a "back to basics" teacher. I teach algorithms and shortcuts only when absolutely necessary. I do believe that mathematics needs a context, and that understanding is paramount. My complaint is that we seem to have thrown the baby out with the bath water. We often speak as though learning by rote is a poor alternative to learning with understanding. I think that we can use some rote learning to support understanding. Let me know what you think.