Summer of Learning 2019 #9 did 8/2(2×2) reinforce that the maths need to be taught/learned different(iated)ly? @mathgarden @cbrownLmath @joboaler
There has been a lot of thinking, arguing and writing about the maths this past week as a question literally divided the internet: 8 / 2 (2×2) = ? – and no, calculators did not universally solve the problem…
It even made the New York Times: https://www.nytimes.com/2019/08/02/science/math-equation-pedmas-bemdas-bedmas.html
And it’s not the first time the maths have proven challenging to the twittersphere:
And yes, I am mindfully putting an S on the end of math. For too many years I have been angry that our (BC) report ca Sd split the Language Arts into reading writing and oral breakouts, but math became “numeracy” – when in reality the maths are more complex than a single definition – as if subtraction multiplication geometry and fractions can be averaged out to a “common” grade that shows math fluency (not to mention word problems which blend those language arts with the math arts). Especially when a “grade” (or performance standard or whatever) assumes that all learners need to know certain skills by a specific grade/age when some very good mathematicians “struggle” with some (literally) elementary concepts. I rematches a Star Trek TNG episode that had Newton Hawking and Einstein playing poker with Data and Einstein being teased about his lack of arithmetic confidence – at the same time I am reading the Einstein biography by Walter Isaacson and coincidently (or not…?) read the part about his schooling where his math marks were quite good (except when/where he was bored or disconnected from his instructor) but got exceptional when he linked with a school that took a look at thinking on a slightly different way, inspired by the 18th century educator Pestalozzi who promoted ‘thinking visually’ in a psychologically ordered sequence – in essence as the learner was ready to tackle more, not because of how old they were.
This is a similar approach to how the language arts have been tackled: differentiated, working on independent levels for comfort and pushing the zones of proximal learning when the ‘conditions” are right. Take it as you want, but based on international test scores, this has led to BC having some of the best “scores”. We also try to find ways to have students “fall in love” with reading…. have fun with writing… find their voice in presentations (even if it’s videoed or visual instead of vocal). And we don’t have everyone using a common text…. So when do we find ways for learners to love maths… to have fun mathing – and not just playing a board game… to work independently and then stretch their thinking and doing….
But in the maths we rely on everyone being on the same page (sometimes/too often learners literally!). Or we platoon students to work with others in homogeneous groups (and confirms to many that “they’re just not good at math”) when really they may need some different access points. Heck, using one of those “stupid” provincial assessments “we” caught a student who had traditionally struggled on math showing that he could excel – when the conditions were challenging enough – his brain loved multi-step problems but rebelled against single step algorithm solving. My son refused to repeat questions – to the extent that he wouldn’t do 9×2 because he already solved 6×3 “and it’s just regrouping 3s” ….
I know that we like to “help” students by drilling times tables into the memory banks – but much like an over reliance on phonics (both of which have its place!) knowing how the letters sound when blended/knowing the answer on the back of a card with a question does not mean that it is understood. Much like a reliance on mnemonic or other memory aids such as pedmas/bodmas/bedmas blah blah blah, if we don’t understand the why, does it matter if we soh cah toa?
Often in education we will want our learners to “think and act like…” (artists, writers, scientists, etc etc) but we do less thinking… less acting…. and more repeating of math algorithms. Is it because we are not confident in how to have math fun? A few years ago I took a throwaway line from Dr Who about Recreational Mathematics and changed how we did the maths: https://technolandy.wordpress.com/recreational-mathematics/
And then I realized I had only started my way down the rabbit 🐇 hole when I discovered @joboaler Mathematical Mindsets
and then Paul Lockharts Mathematicians Lament, both the Book and the must-Read-before-talking-Math-Curriculum-to-me blog: https://www.maa.org/external_archive/devlin/LockhartsLament.pdf
then Pi of a Life by Sunil Singh @mathgarden and Math Recess which He wrote with Dr Chris Brown @cbrownLmath – both books have great math ideas…for mathers of any age! Even inspired me to have a “what is Mr Landy mathing” display next to my “what is Mr Landy reading” display…. my end of year one: count backwards from 100 by 7s – extension: from 1000
I think we need to model math as fun. How the maths are an authentic part of life: why do we need to know fractions and %s? To know if the final sale price of something really is good and why an extra 50% off of something that is already 50% off does not mean it is free!
Now, I’m a librarian who stopped liking math early on – but always liked the maths (and like broccoli – love it as an adult!) and took some of my librarian thinking to encourage math as fun and not just something you did during the one 45-60 minute block each day! I know another librarian who took a project-based-Learning approach to math and made students honestly state that math was their favourite subject. And I think that if we can help learners see math as fun, it will become less socially acceptable to make statements like “some are good at math some aren’t” (because my mom was told she was bad at math and she believed it even though she excelled in music and math thinking… it still makes her mad that she believed what the teacher told her because she didn’t do all the work in the textbook the same as the answer key wanted…).
If we want people to enjoy the maths, we should take advantage of questions such as 8 / 2 (2 + 2) and ensure we all know how to solve it….and why:
Brackets are used to indicate an algorithm to solve first for the rest of the equation
8 / 2 (4) – we can get rid of the brackets 8 / 24 = 1/3! Kidding!!
When two numbers are next to each other (traditionally with brackets) there is to be an assumption of multiplication:
8 / 2 x 4
Like reading English, we then work from left to right: 8/2 is next = 4
Leaving us with 4 x 4 = 16
UNLESS you were taught that brackets include the number attached to it, then we need to reconsider the work after the addition done in brackets:
8 / 2(4) because the brackets work as a sort-of exponent in that any number directly connected to it needs to be solved as part of that sub-algorithm and nobody wanted to use double brackets…. which then means the next part of the equation is:
8/8 = 1
The “why” is very important. Which is why I shake my head when I hear statements such as “math answers are black & white – there is only one right answer”.
Maybe math has one right answer…
But MathS mean there can be many possible answers and multiple strategies to solve questions…. and many ways to do mathematical thinking that goes beyond “by grade/age 7, students will….” let’s have fun and talk about the Why more than “a way How to solve….”
Be mindful of doing the maths different(iated) and make it a favourite subject…something learners “get to do” and not a punishment (if you don’t do x you’ll have an extra page of boring math questions). Just because it’s Tuesday does not mean we all need to be on page 42 doing even questions (odd ones for homework). Maybe Tuesday could be a thematic day doing only math! https://technolandy.wordpress.com/2018/01/10/day-78-of-186-a-day-of-numbers-math-stuff-all-day-long/
Let’s take advantage of the 8 / 2( 2 + 2) phenomenon and think about how and WHY we do the maths as we doM,