Saturday, September 29, 2012

Just Warming Up, Backwards

This past week I posted a warm-up for the students as they came into class. It read:
This is the second line of an 'order of operations' question.
What might the first line be?
=26 - 84 ÷ 3

The students had reached a certain comfort in their Order of Operations ability and had seen most questions that I had already planned on presenting to them. But for this warm-up I thought I might try something that Marian Small often suggests. Give them the answer (or in this case the second line) and not the question.

Examples of this technique are scattered throughout her book,
Good Questions: Great Ways to Differentiate Mathematics Instruction.

I have a copy on my desk for quick inspiration when I'm looking to start building something new.

This was not the first "work backwards" warm-up I've given my students. But some still looked at it for a minute before diving in. But then answers (or questions rather) were being offered with some ease. Very few students (about one per class) actually misjudged the warm-up and simply solved the next two lines.

Things I liked about this warm up:
  • it was open and offered lots of options (thanks Marian)
  • it gave them a new look at and old question. After all, they saw bedmas last year didn't they? (pemdas for those in the US)
  • they were impressed by the number of solutions and quickly understood that there were infinite solutions
  • it didn't take much time and provided lots of discussion including some valuable mistakes

Are there any other points that make this warm-up useful?
What can I do to improve it? Or make them see bedmas without the same old question?

Your comments (or a comment for that matter) are welcomed.
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Tuesday, September 25, 2012

Go Low or Go Home

Last year I played a math game with all the students at my school. The rules were simple:

1) Choose a positive integer.
2) If your number is chosen by someone else that number is out.
3) Lowest number wins.

I gave out little tickets to all the teachers to distribute and collect once students filled them out.

I love this game and first discovered it years ago on the rubiks cube website.
It is discussed here and done here by another teacher.

Ideally, I'd like to write a program (or use google doc forms) and run this competition weekly and get automatic google doc spreadsheets created. However, last year I did it the old fashion way.

The math team helped me place all the individual tickets down on a lengthy roll of yellow paper. This was quite fun as you might be able to tell in these pictures.

And that is why I'm writing today. I recently got these pictures that had been misplaced due to some yearbook photo/camera logistic meltdown. I don't do the yearbook, but I did finally manage to get these pics.

Let me know what you think.

By the way, can you tell the winning number was 10?

Tuesday, September 18, 2012

Faster, Higher and Stronger...for math

Last week my students compared methods of converting centimetres into feet and inches. At one point during the presentations a student made this statement,

"When would we ever use that method, since this way is much faster."

This instinctive lean towards the more efficient method did not go unnoticed. I think it would be fair to say that students are always looking for the faster method. It might also be fair to say that students enjoy the 'easy way', as opposed to the 'hard way'. (But I'm not defining either of those terms.)

They are looking for efficiency. This is a very natural mathematical idea within students.

It is highlighted in this video where a Japanese math teacher (at minute 4:23) brings the students back to the pneumonic HA-KA-SE. Fast, Easy, Accurate.

It is the math teachers equivalent to the Olympic slogan, Faster, Higher, Stronger.

Citius, Altius, Fortius

This led me to question whether this HAKASE is a constant consideration in Japanese schooling. Is it standard? In watching more of the video you can hear the students suggesting that the given method might only be easy and accurate, and not necessarily fast.

The lesson is analyzed here by Dan in his ongoing discussion about what he calls the Ladder of Abstraction.

Is this the math slogan we can always make reference to?
How often will I be able to refer back to these 3 words when a student asks, "Why are we learning this?"

And when a student does point out, "When would we ever use that method, since this way is much faster?" I can just agree, unless someone can point out a faster or easier method.

Gorgeous prime number generator.

It took me a minute to realize what I was looking at but then...ahhhhhh...a beautiful design. Don't miss out on zooming in and out, and speeding it up.

Sunday, September 9, 2012

The Empire Strikes Back

Last week, my students took a few minutes to measure their heights. In partners, they found a couple spots around the classroom that had tape measures attached to the wall. Results were coming in fairly quickly. A few 170's, lots of 160's and several 150's. These are grade 8 students and the tallest of them I believe was 179cm.

Simple. But were there #anyqs?
Actually just one question, but it came up too often to ignore.

Student: Mr. Rowinsky, I'm 163cm tall, how tall is that?

For American readers this Canadian nuance might be new. There are certain things where the metric system is quite natural to us. And by natural I mean the first thing that comes to mind.

Travel distances for example. (Toronto to Montreal is 550 km).
Another is speed limits. (100km/h is a common speed limit).
Track and Field events. (In school they run the 100m, 200m etc.)

But there are two glaring exceptions.
Personal heights and personal weights.

When it comes to answering the question, How tall are you?, centimetres are not the first choice. And for, How much do you weigh?, pounds will often trump the poor kilogram.

Hence our lesson: Converting from Metric to Imperial

You must unlearn, what you have learned.

Some points to make about this lesson:

1) They were all eager to see how tall they were.
2) Most walked away learning for the first time, how tall they were in centimetres.
3) The question 'How tall am I in feet?' (& inches) only came up after they were done.
4) In all 6 classes this same question came up instinctivly, without prompting.
5) I would like to thank the Empire and also my neighbours to the south for the long lasting influence and cultural dominance that provides me with this mathematical opportunity. I'm serious. No snark intended.

Perplexity (as Dan likes to call it) was high and so too was engagement.

I put 163cm on the board and asked students to convert it to metres.
1.63m appeared on most pages within the 30 second mark.

But how about feet (and inches)?

Any guesses?

We started with the TOO HIGH guesses. Someone called out 11 feet, 100 feet (accompanied by laughs), and 6 feet.

How about TOO LOW? 3 feet, 1 foot, and 5 feet were called out.

So we had our range (5 feet to 6 feet).

"What information can I give you that would help?" I asked the students.

We agreed on two:
1 inch = 2.54 cm
1 foot = 12 inches.

Small groups went to work. And the work that came out led to some great discussion. And this resulted in some lessons for the students as well as the teacher. are a few

1) There is more than one way to get the answer.
2) We can learn from the mistakes we make along the way.
3) Feet is one measurement and inches is another (5.3 feet is not 5'3")

1) There is way more than one way to get the answer
2) I was surprised to see that no students used the "f-word", formula. (No one came up to me and asked for a formula to solve. Instead they went to work to solve it.)
3) 5 feet 3 inches was a common response and students had difficulty getting over that 1 tenth of a foot is not 1 inch. This came to light with a 0.5 feet discussion.

Here are two some student samples to consider.
Which can we consider a candidate for a Michael Pershan entry?

And now back to:

In answer to my first my first blog post where I posed this question, 'When I asked the 130 students "What is Math", what were the 6 most popular words used in their response/definition?'

The top 6 were:
#6 - Life
#5 - Problems
#4 - Shapes
#3 - Equations
#2 - Operations (although I included the quartet of multiplying, dividing, adding and subtracting all in one. There is a 5th operation that was popular "and stuff" as in when you are multiplying and dividing and stuff.)

and finally

#1 - Numbers

honourable mention goes to the following student responses:
"Math is Math" ~ for humour
"Math is Life" ~ for philosophical
"Math is for problems that need to be solved." ~ for truth

I suppose it was ambitious of me to try to get a tonne (metric) of comments to my first ever blog post, although I expected at least one.

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In an effort to cheer myself up, here's a wordle cloud for most popular words my students used to answer 'What is Math?'

Tuesday, September 4, 2012

Day One

In the spirit of sharing and inspired by the likes of Dan, Kate, Fawn and of course Sam, this is my first blog entry.

Day 1 is complete, and I wanted to share the first thing my students did as they entered the classroom. It was a simple enough activity. It included a half sheet of paper with the question, What is Math? written on the top. Students were encouraged to answer the question in the best way they saw fit. Sentences, point form, a paragraph or two. My reason for doing this was to get a sense of how the grade 8 mind sees math after 7 years of elementary school. So now I'll open it up to anyone who happens to read this, in the style of that old favourite TV show called, FAMILY FEUD.

130 students were surveyed, top 6 answers are on the board, here is the question:

When I asked the 130 students "What is Math", what were the 6 most popular words used in their response/definition?

I'll post the answers soon. Please take a moment to guess in the comments.

Secondly, I asked each student to rank from 0 to 10 their ENJOYMENT level and their CONFIDENCE level, when it comes to math class. Results were quickly put together here.

I have lots of questions here in terms of the best way to show the results, what to talk about/emphasize, and how much importance I should put into the results. But primarily I wanted to show that I can take a question and represent the data in different ways (scatterplot is the one that came to mind here). I hope to do similar things in the future and even re-do this question at the end and see if there was a shift (hopefully, up and to the right).

So there you have a couple of things to think about and perhaps comment on. Hopefully the comments will make me a better teacher. That is the goal after all, right?

There are other things floating around that need mentioning. For example, I'm sure I will be getting in on the Khanversation sooner or later. I'll also likely share things that I find way too cool to keep to myself like, what would the planets look like if they replaced our moon.

And finally I hope to tidy up the blog so that it includes all the buttons and links that make your blog look so nice. Yes, your blog. You know who you are.

Thanks for sharing, and thanks for reading.
Oh, am I supposed to sign off with my name? #firstblogproblems
I will this one time.