Before I started University Mathematics, I asked my A-Level Maths teacher what it would be like? He replied with the following:
“It’s actually very different and often quite hard. I would say that A-Level Maths is like learning how to drive a car but University Maths is more about how the car works. Think of yourself as going into your Car Mechanics apprenticeship now.”
At the time I thought it was a great analogy but now I realise that it’s completely ridiculous and unfair. Why weren’t we learning how the car worked at School? What about those people who never get to learn how the car works?
Before starting at the new school, I was lucky enough to be reminded of all Jo Boaler’s, Cathy Humphries’ and Carol Dweck’s ideas in Boaler’s, “How to Learn Math” Course. Hence I started by getting every student in every Grade to research fixed and growth mind set (fixed mind set – intelligence fixed, growth mind set – intelligence can be developed). Once they had some idea of the attributes, they wrote up reflections on their current mind set and what aspects of their mind set that they thought they could improve upon. Some of the students then presented their reflections to the class for others to see that they weren’t the only one with certain fixed mind set attributes.
After they’d learnt about mind set, I gave them this questionnaire, collated the results, made some bar charts and then discussed them with each of my classes.
You’re going to love what one of the Grade 6s wrote on his questionnaire:
Just in case you can’t read that, he said, “I don’t compare myself to other people, I compare my self to my improvement.” One word – BOOM!
For example, here’s a bar chart for Grade 8 and 9 on whether they believe making mistakes is a good thing for learning.
Most agreed that the bars would probably be reflected in the centre line if they hadn’t learnt about fixed and growth mind set first.
It was great to also have the opportunity to give a short presentation on mathematics and mind set to parents last week, in which my main message was: “If they’re not struggling, they’re probably not learning,” and “Posing problems is just as important as answering them.”
Some of the Activities that we’ve been doing
1. Number Talks – Of course, after doing Jo Boaler’s course, I wanted to get some Number Talks into the curriculum for grade 6 and 7. For those that don’t know, that’s when you put a question on the board, like 14 x 8 or 23 x 11, ask the students to do it mentally, and then discuss different strategies as a class. This allows two key things:
1) Allows students to see that there’s more than one way to calculate and that they can transform calculations in different ways. i.e. Flexibility of Numbers
2) Start to visualize different strategies to make them more concrete.
As an example, I gave Grade 6 pasta shapes and asked them to represent 18 x 5 in any way they wanted. They started straight away with 9 piles of 10.
2) Four 4s Challenge – Make all of the numbers from 1 – 20 with exactly four 4s.
It’s a standard challenge but it’s great in that it re-iterates the fact that there’s a number of different ways to construct mathematics and, as individuals, we often find different paths (just like with Number Talks). In Grade 7 they were so hooked that they decided to try and obtain 10 different solutions for each number up to 50 – it’s going to be a term long challenge! (As a side note, they were fascinated by the factorial symbol)
3) Understanding all of those Rules they’ve been lumbered with – all of my students have arrived with what I like to call “Rule baggage” i.e. many of them can calculate things but have no idea where the rule came from and why it works. It seems, unfortunately, that in many cases the rules didn’t come from them.
So I decided with all Grades that I’d start with reviewing basic skills and knowledge so that we could iron out all of this rule baggage they carry with them. I encouraged the students to ask their fellow students why something worked during explanations. After two lessons with Grade 9 we already had 6 questions:
1) Why is it okay to flip the second fraction and multiply when dividing fractions?
2) Why do we need to make a common denominator when adding and subtracting fractions?
3) Whys isn’t 1 a prime number?
4) Why is subtracting a negative the same as adding?
5) Why is it that when we multiply two negatives we get a positive answer?
6) Do the prime numbers follow any pattern?
So the only thing to do was to get them into groups researching each question for the third lesson and then get them to present ideas at the end. Here’s some posters that went along with the presentations:
The “two negatives multiplying to make a positive group” struggled so I pointed them towards a previous post I’ve written. I think that they’re starting to appreciate that there’s no point in learning and doing Mathematics if you don’t engage in how the car works. This quotation from one of the girls says it all: “Maths is actually really cool when you understand how things work.”
As a side, re-learning fraction addition and fraction sense making with Grade 8 was brilliant! We used the Fraction addition Geogebra apps I created last year with inspiration from John Golden and David Cox and they were absolutely hooked (we all spent a week making loads of estimation apps which are all currently on www.geogebratube.com).
To start, I asked them to figure out what was going on with this applet-click the picture to go to the applet.
Once they figured out why we needed to make a common denominator, I wanted to get their fraction sense going with this estimation applet. Whilst I don’t encourage competition in my classroom, they challenged me to do better than they could at estimating the addition of two fractions – here’s what happened…
Class Percentage Error: 10% off
My Percentage Error: 0% off
Back of the net!!!!
4) Building curiosity and intrigue
Of course I’ve already had a few 3 ACT Maths tasks on the go, courtesy of Dan Meyer, but in the first few weeks, I wanted to go beyond the classroom walls to get everyone talking about Maths – students, teachers, visitors etc. I gave Grade 7 and Grade 8 homework to find a symbol in the school and investigate it for homework this weekend.
When I was walking past the symbol in the open learning environment, and everyone was talking about it (even some teachers), a Grade 7 student said, “So that’s our homework sir? Is it Art or Maths?” To which my reply was, “What’s the difference?” Of course there are differences but I want my students to understand that Mathematics can be a Science, an Art form, a subject in its own right in which I don’t care whether it applies to anything in real life because it’s fun, etc. If you’ve never come across it before then I won’t say too much other than it’s to do with Eqyptian Mathematics and I can’t wait to discuss and extend it next week with the students.
So hopefully students’ perspectives on mathematics are already changing but still a long way to go!
[Mixed ability classes are interesting and challenging so far – will probably write a post about how it’s going soon].