Numberphile youtube channel
Take it from the guys at Numberphile to make pi interesting again.
A million digits of pi on rolled paper — unravelled across a runway. Each digit is 1/15th of an inch. The site claimes the length is just over a mile. Are they correct? What kind of questions would a grade 7 or 8 class ask after viewing this video ?
Prendrais-tu coupon A ou B ?
I stole this from John Stevens’ math site “Would You Rather?” and (very liberally) transcribed it into French for my school board. Basically it’s a great tool to start a dialog and really work math content on a concrete level. Students are given the choice between two options and they have to justify their answer mathematically.
I’m starting to translate a few from John’s site into french and I’d like to start thinking of a few of my own.
Here’s the info for his site :
WouldYouRatherMath.com
Twitter @Jstevens009
Here’s a question from Marion Small www.onetwoinfinity.ca on algebraic reasoning :
Notice this pattern:
4 x 5 = 4 x 4 + 4
5 x 6 = 5 x 5 + 5
6 x 7 = 6 x 6 + 6
How can you say the pattern in words? using visuals? using algebra?
Often we try to solve the problem by finding the pattern or inputting the formula. How often do we take the time to really understand what these equations and expressions mean? How many students that are in grade 10, 11 and 12 that don’t realize the difference between an expression and an equation? Sometimes by showing a different way to the same problem can help clarify understanding.
Scroll down for two examples on how you can use algebra and manipulatives on how you could represent your answer.
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Algebraically, you represent n for any number… and n(n + 1) = n^2 + n, which you can see after expanding the left side of the equation. This is how we usually expect to find the answer to this question.
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What happens if we use manipulatives to show the same equation ?
Math Bites reminds of Bill Nye the Science Guy or Sesame Street, but with Math and taught by Winnie from Wonder Years. The hyperkinetic scenes are fun and make a point about explaining basic concepts. Danica, former child actress, is also an acclaimed mathematician who’s put a lot of focus encouraging girls to love and appreciate math. She’s now recently hosting a series of videos for the Nerdist channel on youtube. Worth a visit.
The start of the week isn’t always easy. I need to sometimes give myself an asymptotic high five, where the fun never ends.
Then I get so annoyed when nobody understands my awesome math jokes! I mean, asymptotes, right? Right?
I think the point of the exercise in this image is for the student to decide whether or not the question is correct and to explain why they think so. He or she has to analyse each step and justify his or her conclusions. This is a lot more work than a question which asks you to find the answer in the simplest way possible.
Now don’t get me wrong: I also think it’s important for students to know basic skills as well like how to subtract numbers the traditional way. It’s a very effective strategy because it’s quick and simple to use. But we don’t take enough time to teach the processes behind those steps, such as reasoning and problem solving. Give that student the question, 427 – 316, and if he memorized the traditional way of subtracting he’ll probably get the answer right.
But change that question into a problem solving question such as “find the error on a number line” and most students who have only learned the traditional strategies might find it more difficult because they only see one way of solving subtractions.
Often, the traditional way of teaching meant teaching the most effective way to get to the answer. It would be often modelled as follows:
Do we see a pattern here? I’m sure this method was pretty familiar for who took math a generation ago and beyond. The teacher is doing most of the work by directly telling the students what to learn, as though they were reading a living, breathing instruction manual, and it’s expected that the students follow these instructions and complete the work. No one questions if there are any other methods because it’s again expected that the teacher is ultimately giving his or her students the correct and only steps on how to find the solution. It’s also expected that the students mimic and memorize the model the teacher has established.
Let’s look at the type of questions Math teachers often use to model their strategies in the traditional way of teaching. These are the “What” type of questions – which mostly answers direct questions: “What is 427 – 311?” This direct method of teaching is great for those who find it easy to follow instructions verbally and written down step by step. And has that method worked well for everyone in the past?
It’s comparable to an instruction manual from Ikea. The steps are drawn as pictures and they can be followed gradually. What’s the common complaint from Ikea manuals? There are no words or labels for each piece, so there can be difficulty interpreting each step, or which piece to use. People get frustrated and pieces get broken or missing – which ends with them swearing off buying DIY furniture ever again. My question for you: Have you ever felt frustrated when you couldn’t understand a particular step? Do you always have to depend on the instructions or can you find an alternative method to solve your problem? Does it matter to solve it exactly like how the instructions describe? Sometimes a futon or a bed doesn’t need all the pieces or look exactly like the store model to be an effective piece of furniture.
We often expect Math to have a universal truth: For every math question, there must be a given answer. It’s a very rigid way of thinking because it gives a quick answer to one type of problem without having to reflect too much on the question because we have faith in the process. We don’t also take enough time to ask the “Why” questions: “Why is 427 – 311?” “Is there a different way to see this answer?” “Why is subtracting the traditional way often the most effective strategy?”
This type of questioning goes further than just knowing how to subtract. We are now growing up in a society that is changing socially, technologically, financially at an exponential rate. In the last 30 years, we grew up with several generations of technological devices that have already outgrown society: For music devices, we went through vinyl records, 8-tracks, cassettes, compact discs, to digital media (mp3s). When I was a child of the 80s, I was able to reprogram my VCR clock, work a remote control and understand how to program software on pre-Windows computers. Children today are able to work on tablets, SMART phones, and video games; they can browse the Internet, and create apps, all at a very young age.
The types of jobs are also rapidly changing – we are training students for jobs that haven’t been invented yet until after they have graduated. We are also teaching for jobs that will become obsolete. Yes, students need to learn basic numeracy and literacy skills so that they can be used as tools to help them solve problems. But they should also be encouraged to learn other processes such as problem solving skills and reasoning so they can learn to adapt in an every changing world.
Parents like the one in this image lived in a world where life decisions were black and white. You followed a straight forward path – you had a problem: here’s the solution. Most people had jobs with simple tasks – such as inserting parts in a car factory. With the advent of the 21st Century, society has been fortunate to have many options offered to us – but not without many more difficulties as well.
As well, our society is also now competing financially with other countries that are growing and flourishing. Workers in car factories get laid off because robots can do the work 100 times better and faster. It’s also much cheaper to hire labour in countries with lower GDP if we want more t-shirts, more TVs and more tablets available in our homes. Students need to be trained to think critically as opposed to just basic skills. We also want to teach students to decide which strategy would be the most effective for them to use. If we’re only teaching students to follow instructions and not to think critically, there could be serious consequences in the long run. If we’re only teaching students to memorize the formula as opposed to understanding it, are we training them responsibly?
Here’s a yummy formula to remember volume of cylinders!
Have a Good Weekend!
I know it’s a little late (or way too early), but I found these photos today and I wonder what kind of math question can we create from these photos?
Numberphile is a british channel on youtube where they essentially talk about the joy of numbers. For number lovers, in which the name implies, this is an interesting discussions in numeracy. They bring up a lot of advanced math and talk about it in terms that most laymans can better understand.
A lot of the content seems intended for those in grade 12 high school or university-level math, but they do bring up some good questions such as “Why can’t we divide by 0?” and “How to order 43 chicken McNuggets?” where they wonder can they buy exactly 43 chicken McNuggets based on the boxes the chain sells.
My favourite Numberphile episode is where they find the ratio of pi using real pies. The video, which is clocked at 3 minutes 14 seconds, uses hundreds of pies to form a large circle and its diameter and even the host seemed a bit surprised at the results.
Mathworkz 