Make Reasoning a Routine

Thanks to Matt Oldridge for the nudge to write this post.  As an Instructional Coach, I’ve been working hard to respond to teacher feedback, and give teachers exactly what they want, and what they feel like they can use.  As as result, I find myself presenting more and more about Reasoning Routines, and applying them on a daily basis.  I think that in the math education community, there is a thirst for daily arguing, debate, reasoning and proving in the classroom.

So of course, I now introduce teachers to a bunch of stuff that I’ve tried in my classroom, and two of my favourites are and

A more extensive list of Reasoning Routines I’m working on can be found at my RESOURCES page.

I plan on writing a separate post about and visual patterns, but this one is more meant to just talk about the variety of questions we can ask just using some simple visual cues.  And keep in mind my goal to is elicit and respect divergent thinking…that’s why I really value these resources.


The idea is that I show you 4 images, and they all don’t belong for some reason.  I want to use this idea to elicit as much student thinking as possible.

Let’s just look at this set of shapes:

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I can ask a few questions, which all elicit a different level of thinking:

  1. Which one doesn’t belong, and why?
  2. Give me a reason why each one doesn’t belong…
  3. Which one doesn’t belong, and please draw me another one which doesn’t belong for that same reason.

These three simple questions can be used at any point during a block of learning:

  • Access prior knowledge
  • Summarize the work we’ve done so far
  • Summarize and extend the thinking of everyone in the class

The divergent thinking that happens is amazing.  And of course, everyones line starts with…”Well I picked the obvious one…its the___________” except everyone’s obvious blank is different! Its just amazing.

Furthermore, this image can be modified to elicit even further thinking:

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Now the question can become…Draw me a shape so that this WODB still works.  My students know that this means that this new shape has to not belong for some reason, but still has to allow all 3 other shapes to not belong for their own reason.  Holy smokes you should see the thinking and reasoning fly when I pose this, which students and teachers alike.

This site has about 200 growing patterns shown by images that students can use to predict how the pattern is growing.  For example:

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So I can ask my students how many white tiles are in shape #23 for example, but my favourite thing to ask is describe to me how you see the pattern growing.  Then, either together or in small groups, we annotate this description algebraically.  Some really spectacular divergent thinking happens…and again, everyone expects that their way is the “common” way to see it.  Amazing discussion, and amazing source of conversation when we want to talk about simplifying expressions just to make sure that they all say the same thing!  Here are some ways that my students have seen this pattern grow:

Just amazing ways for students to see the same thing differently! So in the end, what we need to do is realize the power of having students reason something to us, to each other, and to themselves.  Number Talks is a great example, and if we do that every day, great! But if you’re looking for other ways to Make Reasoning a Routine in your class, consider,, and keep an eye on my growing list of RESOURCES that I promise I’ll update as soon as I can 🙂

Let’s Walk the Walk

The Toronto Star published this article about how Teacher Education programs are dealing with incoming teacher-candidates in order to make sure that they’ll be well prepared to teach math.  Most of the solutions are centred around a test that candidates will have to take at the start of the program.

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  • Trent – 75 minute test, if the results are not acceptable, they do some courses, and redo the test…
  • OISE – 90 minute test, with software generated remediation suggestions
  • Brock – Refresher online course with instant graded response, with software generated remediation suggestions
  • Ottawa – a 12 week math course which digs deeper into math content…

(all of this is a summary straight from the article)

I’d like to say that its great that there is more focus on helping teacher-candidates with mathematics, but I think that the test kind of negates all that good stuff.  It’s pedagogically backwards isn’t it?

Aside from Ottawa, everything seems to hinge on fairly straight forward question/answer style test, and I think that Prof. Chris Suurtamm (Ottawa) needs to be commended for her and her colleagues’ approach to this.

For the others, my main questions are:

  1. Are we not just telling teachers that what we value most in math is doing well on a test? Are we really saying…triangulate evidence when you get there…but for now here’s a test…that’s the important part?
  2. If the only teachers that get to teach math did well on a test, are we not just filling the system with more and more people who do not believe in test anxiety because they probably don’t suffer from it? How is that productive to our focus on mental health and well being?

I don’t know what the solutions are, but Ottawa is at least trying something different.  I’d love to know more about the program, and I’ll set out to do that.  Overall, I feel like we need to expose people to good teaching, not testing, to exciting math, not paper and pencil, and to some good, sound, simple approaches to math education which makes students curious enough to want to learn. We have to focus on what is going to be good for the millions of students that these teachers will affect in their future careers, not just what’s easy for us to evaluate.

We, as a system, have to decide…are we going to just talk the talk, or are we going to walk the walk along with the teachers who we are trying to educate to do things differently.  Are we really a system which is going to hide behind “Do as I say, not as I do…” because that really didn’t work for my parents when I was a teen, and I sure don’t think its become any more effective now…

Dan Meyer – Beyond Relevance and Real World

I finally had the opportunity to watch Dan Meyer’s NCTM 2016 talk.  If you haven’t seen it, check it out below, before I share my thoughts…It was a very worthwhile 47 minutes…I personally picked it over MasterChef on a Sunday night…and I love my MasterChef.

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The Desperate Answers

I love discussing this part of math education.  I at one point a couple of years ago spoke to a group about how anything that has students argue and wonder would be engaging, and I then showed Dan’s TED talk – Math Class Needs a Make-Over.  Everyone ooohd and aaaahd, so I was glad.  Then I had a conversation that made me think that people missed the mark.

Someone said, oh I get it, but what if instead of the water tank, we made it more real? Like filling the air in your tire at the gas station? Everyone does that!

In the end, I was really glad that the question came up because I asked the group if that context makes things any more interesting, and everyone, including the person who suggested it, said no.  I’m glad that Dan takes that idea to task.

The Math Dial

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Love the idea – check the volume of the room…tone it down, and then turn it up.  I didn’t really love the name “Math Dial” with the utmost respect to Dan.  I don’t want people to think that if you turn the dial down, the room is less “Mathy”.  I know this isn’t what Dan meant, I just don’t want other teacher’s we’re talking to to feel that we are watering the math down because even estimating, forming arguments, and fighting to back something up is super mathy. So I’d rename it to something like the Formula Dial, or the Algorithm Dial, or the Formal Dial…I dunno, none of these are as fun as the Math Dial, but there it is.

I do love the idea that you really need to see if your students are ready to have the dial turned up.  This is true even in a calculus class.  Students need to have some place to start, some place to throw their experiences in, and some place to argue, before we throw formality into the flow.  Also, the fact that the dial goes in only one direction…key.

The Dial Only Goes UP! It Can’t Go Back…

Oh so true.  The other lens that I look at this through is anxiety.  If we jack up the dial we create anxiety, and things shut down.  It’s super hard to take that anxiety away, but if we start with low floor, where everyone can enter and play, we’re good to go.  We, as professionals need to be able to sense when and how quickly to notch up the structure.

Delete Your Textbook

I love this…I sense that Dan felt out the audience of the median math teacher and recognized that people aren’t super comfortable with the task of creating 3 Act Tasks.  Many teachers I’ve talked to LOOOOOVE using them, but aren’t really jumping on creating their own. They do, however, value the productive struggle and the arguing that’s caused by these tasks.

This piece is a perfect solution for people who want to jump into creating some productive struggle, but not having it be an onerous  task…and I love his example:

Another super simple example I go to is:

Original:  The length of a rectangle is 2cm more than the width.  The area is 35cm^2. What’s the perimeter?

Modified: What’s the perimeter of our rectangle?…Add information as students ask for it.

The Gist of It

Ask for student questions…

Ask for questions about questions…

Start a fight…

Love these three tips…they’ve held true for so long, and I don’t know if I would call anything else more pertinent than these three things in math class…

So once again…thanks Dan 🙂