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.
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.