Make Reasoning A Routine

I have written before about how talking changes everything in math classrooms. Kids will surprise you with the power of their thinking. They will conjecture, and wonder. They will back up their reasoning with mathematics.

Reasoning shouldn't be an event in math classrooms. Reasoning should be a routine, or even- the default state of the classroom. I have a strong belief in defining terms in plain language, so I will define reasoning in math classrooms as, "providing mathematical reasons to support our answers, verbally, or on paper." If reasoning is pushed further, we get into the concept of providing airtight "proof", and generalizing for all cases. In K-12 education, you will some breakthroughs with proof, and a growing ability to generalize (we particularly see that in the progression of algebraic concepts and thought from K-12). The ability to generalize develops the more kids are given the chance to reason through their ideas, and, of course, as their mathematical toolkit develops over time (learning all four operations, and then powers and roots, fraction sense and arithmetic, algebraic reasoning and working with equations, and so on.)

In this picture, I stood beside two kids while they explored various trapezoids. They were very close to finding something out about the area of all trapezoids. With a bit of a push, they could have gone from examining specific cases, to generalizing for all cases. They were tantalizingly close. I didn't get to see if they got there in the end. I do know this: letting those kids play with trapezoids is a lot more interesting than just throwing out "the" trapezoid area formula and having them do exercises. There is time for that, later.

There is a lot of stuff out there that is user friendly, repeatable on a day to day basis, and supports getting kids to share their reasoning. With the exception of number talks, these are all things that have sprung up from the dynamic and amazing #MTBoS. I suspect lots has been written about using these things as routines, so this is just a brief summary and survey.

Here are a few things you can do.

Number Talks. Depending on how you choose to do your number talks, you may be more focused on specific strategies for mental math, than reasoning. but number talks are portable, short, and get kids talking.

Estimation180. We used one involving a piece of a pie with some of our adult learners, and I was thinking it could go to fractions, or well, pi, if you wanted to actually take the picture and carry out some calculations. I think these are mostly good for developing that horse sense, or intuition about things like quantities. How many? How much? Why do you think so?

Fraction Talks. This site (and Twitter, @FractionTalks), has a wonderfully diverse selection of pictures that can inspire reasoning about fractions. It is so, so important that kids don't see fractions as a strange and new species of number, when they first encounter them. Kids should be able to reason with linear, area, volume, and set models of fractions.

In this picture, a teacher is sharing her reasoning about the lovely "Quarter the Cross" tasks.

Flags make lovely fraction provocations:

Hat tip to @Madame_JB for this one!

WouldYouRather Math.  This one presents two options, and has kids chose which one is best. The instructions include the lovely formulation, "justify your reasoning with mathematics." I love it. The best part about it- you could make your own. You just need two options that inspire interesting reasoning. I am fond of pizza tasks- what's better, at what price, a medium or large, that sort of thing. 

I am tagging @MathManAnusic to talk about Which One Doesn't Belong and Visual Patterns.

Given what's out there, it's pretty easy these days to make reasoning a daily routine, and you should. You could pick any one of these websites, find something matching the curriculum you are working on, and use it as part of your routine. 

Get kids talking, exploring interesting tasks, and let them amaze you with the power of their reasoning. 

Letter to the Editor of a Major National Newspaper on the Ontario Revised Math, Which Wasn't Good Enough To Print, Because It's Not Negative At All, In Its 200 Words, And Refuses To Indulge In The "We're Terrible" Narrative About Ontario Education

What’s missing from the Globe and Mail’s articles on the new Ontario math strategy is the voices of actual math teachers-who teach every day, in classrooms from Windsor to Moose Factory.

The pervading media narrative about math education is far too negative.

My personal perspective is that the Ontario curriculum is strong, with an appropriate balance of skills and concepts. What is needed is not a complete course correction. With subtle tweaks, and reshaping of the curriculum to focus on bigger mathematical ideas, and making sure skills don’t go overlooked, I think it could be even better.

We already have world-class teachers, who with the new focus on math, are getting even better. I have seen massive breakthroughs these past two years. I have seen teachers learning more new and interesting math content than ever before, and they are excited by it! .

$60 million is nothing to sneer at. Together, we can go from good to great, if we keep working together. Change and improvement are not sudden. There is no “silver bullet”, only careful, patient work.  

Math is...What Parents and Teachers Make It

My OAME 2016 Ignite speech, "Math is...What Parents and Teachers Make It," is here, in PDF form.

I think a lot about what math "is", and "is not". I am involved in professional learning for practicing teachers, I am a teacher myself, and, I have two young children.So it is that I have conversations like this, in my house.

I thought about just letting that drop but naw, teachers need to teach, so we explored it, with both boys. That's a pretty decent provisional definition of infinity for a 5 year old, but can we push his thinking further? I asked him "what's the biggest number you know?" He started with 12. I said that if I add one, I get 13. The play continued. I got Callum to write a 1 and a string of zeroes down, and showed him that you can always add another zero. He doesn't really know place value yet, but I was thinking that playing with ever bigger numbers would help him. 

This is still a work in progress. It's also just one example of how you can model math talk at home. I really do think parents have enormous power to expose their kids to everyday math. Tackling infinity is not for everyone. The new resource Inspiring Your Child To Learn and Love Math has all kinds of practical activities parents can do with their kids, from cooking, to shopping, to all kinds of games and fun things you can do to get kids seeing that math is a normal, interesting, inspiring, and fun part of their lives. 

We shape the very idea of "mathematics" in kids' heads. That's a great responsibility we have! pic.twitter.com/t7qwZm9bZS

— Matthew Oldridge (@MatthewOldridge) May 10, 2016

And from the teacher''s side, I think that tweet says it all. We have such an enormous power to shape what mathematics IS, in students' minds. Is it disconnected topics, strands, and facts? Or is it a powerfully connected and interesting body of knowledge? Is it a way of thinking you only do in math class? Is it a subject where you can do powerful critical and creative thinking?

Do we expose our kids to both the awe-inspiring mathematical world, and, math used in the real world?

I hope so.

Here are some of Melissa Dean's (@Dean_of_math) amazing students' responses.

From Kindergarten kids: 

1 + 2

10 - 1

Minus

Adding

Playing thinking games

Dividing

Numbers

Sorting

Shapes

Work

Patterns

Writing

From Primary Kids:

The language to explain certain happenings.

Fun for me. Math is all around us. 

I love to do math because, first, it makes your mind smarter. Next, it just makes me happy. Lastly, I love my teacher every day in math.  

From Junior kids: 

Important, since, well, it’s used everywhere for everything, and so it’s basically necessary in our lives. 

Awesome and it helps you learn. Math is also cool.

Everywhere. This is actually true, considering it covers all strands of life. For example in gym, strategic games, and science! You also use it when you don’t know it. That’s how math is everywhere. 

And from this grade 6 philosopher: 

Everything. Everywhere you go, everyone you meet, all have some connection with math. It’s logic, common sense, and thinking out of the box, not only the seemingly tedious arithmetic and problem solving. Math has grown into the world, and so the world wouldn’t be the world without it.

From Intermediate kids:

A way of solving equations/problem. Math is everything, everything is math.

A combination of numbers and symbols that is useful for every day life and can help you in the future (finding a job.)

Needed in life. Without Math, we’re idiots! Yeah!

And from a grade 8 philosopher:

A metaphor of life. Asks you to solve the problems it creates. It’s simple. Its just us creating ways to explain things we don’t fully understand.