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:

Untitled drawing

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 🙂

Collaboration – in Math Class

So I read my friend Jay Richea’s post and decided to join in the #Peel21st Blog Hop.

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“Here’s the next instalment of #Peel21st Blog Hop. We  thought we would reflect on the 21st Century Competencies released by the Ministry which discusses the core 21st Century skills that are essential elements in modern learning. We thought these competencies could be a great way to prompt a conversation amongst the #peel21st community and afar as well. This month, it’s all about…

Collaboration: Near & Far

So my focus will be on Collaboration in Math Class.

I remember vividly spending most of my first year math undergrad in isolation, and thinking, how is everyone else handling all of this work, and understanding everything? I lived off-campus you see, and I’d head home after lectures.

Second year, I moved on-campus and discovered the marvel of study groups.  People would sit together, figuring out problems, convincing each other that their solution is the one that made the most sense, or is the most elegant. My personal success and confidence in my mathematical ability skyrocketed.  This is a skill that we far too often skip over in secondary math education.  We call it “stealing work” or “cheating”.  But my professors loved it.  They would join us in our debates.

If anyone has the opportunity, check out McMaster’s James Steward Centre for Mathematics – it’s a building with all interior walls painted in chalkboard paint.  We would stand around for hours outside of our professors office, and she or he would join us from time to time, debating about the solutions we’re trying to discover.

So the question is, how do we teach and foster collaboration in our students?  In reading Jo Boaler’s “Mathematical Mindset” recently, I’ve gotten a few ideas of how to not just have collaboration in my class, but to show students that I value it by having criteria clearly laid out for them.  I loved the two suggestions that she provided (from Carlos Cabana):


I love the fact that Carlos is open about the fact that the group needs to lean on each other in order to collaborate, that a single person cannot be successful in all of this, but together the group can.

To add to all of this awesomeness, we now have collaborative tools like GAFE, Formative, and now Desmos Activity Builder specifically for math where we need to collect data from the entire class in order to solve a problem.  Online tools allow us to create activities such as THIS one where we have to use some information from our class in order to get to discovering some math, and we can now even include information from people outside our immediate network if we push the questions further.  The possibilities are endless, but we can focus on the objective if we just keep asking ourselves the questions:

  • Do mathematicians do math in isolation?
  • What do their offices look like?
  • How do students learn best?
  • How can I best support them to believe in themselves and in their own ideas?


Who Makes Me Think – Marian Small

So I’m sure most math educators have heard of Marian Small in some way, but for me, she’s been simply transformational.  She has deeply impacted the way that I not only teach, but interact with my students.

There are so many things that I’ve learned from her, but here are a few that come to mind during my work on a daily basis:

Open Questions

Making sure that students can choose an appropriate entry point into a question.  Things like, instead of saying “Find the equation of the line through the points (2,3) and (5,7)” saying “An increasing line goes through (2,3)…what other point might it go through? What could it’s equation be?”

This question doesn’t only have multiple entry points, but it actually forces students to THINK about so much more than the slope equation and the equation of a line. It also gets students to engage with all of the terminology that we want them to engage with.

When I say the words “What might it be…” instead of “What is…” in my classes, I definitely see students become more relaxed, more at ease, and in turn, more engaged and more willing to take risks.


We often talk about being ready to give feedback to answers we want, or answers we anticipate.  Dr. Small has also talked a lot about being ready and prepared to offer feedback to thinking that you had no idea was coming…generic questions like “Oh…why do you think that is?” or “Oh…why did you think of doing it that way?”.  The other thing that Small does so well is model curiosity for students.  There is a huge importance in the tone in which we ask the question “Why would you do it that way?” It could be either a shut down question, or a feedback opening question…


This is something I haven’t actually heard her talk about, but I just see the way she models it, and having applied it in my class, it does wonders.  Instead of applying complex math terminology from the start, she uses very simple language to start, and then builds up the math vocab.  So when asking a Which One Doesn’t Belong question involving algebraic terms, she won’t say “Which expression doesn’t belong?” but she’ll say “So…like…which one of these guys doesn’t fit?”

My students have commented on the fact that this encourages them to participate, and they feel at ease.  The worries of math class doesn’t creep up on them, and they don’t feel anxious at all.  This allows me to make sure that many more of my students get the chance to be successful and to be heard.

Anyway, I’m sure I’m not doing justice to everything Dr. Small has taught me, but I want to keep my blogs short.  I’ll probably do another post on her impact in the future.  I was very happy to see Dr. Small speak again, and I got to tell her a very important truth: I use a lot of her ideas when I teach the OISE Math AQ, and the reason is that without her ideas, I don’t think I’d even be the type of teacher that gets the opportunity to teach an AQ.  So thanks again, Dr. Small 🙂


I Taught an AQ…So What???

So I just finished teaching an AQ/ABQ in Intermediate/7 and 8 Math.  The days were long, we had no windows, but wow…what a learning experience.  For those that were involved, I want to extend my gratitude for stretching me and making my think past my current experience.  Here are a few take-aways from the term:

You Can’t Teach an Old Dog New Tricks…Yeah Right!

There was so much experience in the class, people that have been teaching for as long, or longer than I have, and yet everyone walked away with something.  Everyone had some questions, some new conclusions, something that they were excited to try out (including me!)  One thing that resonated with me was that more and more, people were thinking about how to service their students who are yet to come, versus how they have serviced students in the past.  I think a part of this thinking came from me telling a story about a staff meeting where one teacher continued to hold on to an idea that she had.  I knew that the teacher had the best intentions, and in the end it came out that she was scared to let the idea go because she’d be admitting that she had done wrong to so many students in the past.  We talked about how the goal is always to improve and to make things as good as possible for students to come, and eventually she gave herself a pat on the back for accepting the new idea.  Even for me, thinking about things as beneficial for students to come versus detrimental for students in the past has become a conscious effort, and I’ve seen in myself a quicker acceptance to alternative thought.

Open Questions and Parallel Tasks – People LOVE Them!

Come to think of it, people love anything Dr. Marian Small comes up with! I think the most popular take-away this year was open questions and the effect that it has on the un-engaged learner.  (As a side note, I use un-engaged deliberately, because I think dis-engaged implies that the student was never engaged, but un-engaged admits that at one point the student was engaged, and we, as an education system, have done something to un-engage them…not sure who I heard this from, but it’s way too good for me to have come up with.)  So open questions and parallel tasks were really the number one tool according to my class which could be used to re-engage these learners.  I think that people love them because they really give everyone an opportunity to participate.  They allow everyone to “feel the love” in math class.  I remember in my math classes as a kid, I felt the love.  I could answer quickly, I could throw my hand up fast and be confident.  That’s until I got bored, stopped thinking and developed gaps.  Then when I wanted to feel some love, I couldn’t…I wasn’t sure of the answer.  So with open questions, you really give a platform for everyone to participate, and start to build some of their confidence back.

Algebra Tiles – Bring ’em On!

People love algebra tiles (but they have to be taught how to use them!) Oh that day was a roller coaster.  People did not look happy when I brought them out.  Everyone wanted to do my questions using some shortcut they remembered, or some way that they had just taught a class full of kids.  I really had to insist that these little tiles would paint a wonderful picture about algebra, and that we should give them a chance, and one by one, people started seeing the rectangles come out, and started seeing the square get completed, literally.  So…if you’re going to present algebra tiles to teachers, make sure you teach them how to use them.

Technology – We Need It!

I did not expect to be asked so many questions about various tech tools.  I don’t overload the course with tech because I don’t want it to be in the forefront.  I use less in this course than I would with my students, but people wanted more! This is so super exciting to me because it says that teachers are ready to take on this beast, fully knowing that there may be glitches, things may not work out perfectly, and it may be an arena where you aren’t an expert.  But it seems that people are ready none-the-less.  So lets get it! There should be more up-to-date AQ’s about implementing technology in the classroom.  I think that these AQ’s shouldn’t be so much about certain APPs or certain LMSs, because by the end of the course, there’s probably a better app or LMS, but it should be about things like, what makes a good instant response app? What makes a good LMS? What can technology do which pencil and paper can’t???  Hmmm…maybe I’m gonna have a MOOC…stay tuned.

Dan Meyer – We Need More of Him

I think he’s one of my favourites…and I’m sure that I gave a biased account to my class, because they all felt that he was extremely powerful.  I think the big take-aways from him were creating conversation so that math can be used to serve that conversation.  And asking the question first, and then giving information.  I overheard one person say “In textbooks, why don’t they just put it backwards?” Made me really think! A question reads:

The perimeter of a rectangle is 24, the length is 2 more than the width.  What is the area? – Students have so much trouble with this, for many reasons, but I think one main one is organizing the information.  What if the question read:

What’s the area of a rectangle? The perimeter is 24, and the length is 2 more than the width.

I really wonder if that would make a difference??? If you find out, please let me know!

That’s it for now, I’m sure there’ll be a part 2 for the post, but good gosh, I just loved the last 3 weeks so much! Don’t get me wrong, I’m glad its done and I now have the summer to enjoy, but it wasn’t the worst way to spend the first three weeks of summer!

Make Co-ordinates Come Alive with Google Forms

First of all, I want no credit for thinking of this activity.  It was all Jon Orr’s idea, which you can read here, I just adapted it to use Google Forms and Google Sheets.

So I wanted students to graph the f(x)=log(x) function for the first time, but how boring is plotting x and y co-ordinates? Also, I wanted them to make some big picture connections:

  • An x co-ordinate has a matching y co-ordinate.  
  • We can find that y value by using the f(x) rule.
  • If we all use the same f(x) rule for different x co-ordinates, we will come up with some sort of scatter plot.
  • Hopefully we can draw a curve of best fit for this scatter plot, and predict a pattern.
  • We can describe this curve of best fit with some mathematical properties, so that we can forego the scatter plot in the future, and just draw the graph.

I didn’t tell them these ahead of time or course, but that was my goal.

So I started by giving every student an x-value as they came in, ranging from 0.001 to 12, and I also handed someone x=-1, and someone else x=0.

I then sent students a link to the following form on the class site, and instructed them to complete it:

log form

As they completed the form one by one, the screen in front of the room became littered by random points…

First this:

Graph 1

And then this:

Graph 2

Then this:

Graph 3And finally we had this:

Graph 4I’ll explain how I made that happen live in front of the students in Google Sheets below, but first I want to talk about what came out of the students and their questions.

So first I said that I wasn’t happy because I didn’t know where these numbers were coming from and what they meant, so we talked about labelling the graph.  This is what we came up with:

Graph 5 with titles

Then I asked the students to describe to me in math terms what this graph looks like.  They picked out the x-intercept as (1,0) and one other key point at (10,1) which I was impressed by. Then, they asked me if the y-axis was an asymptote.  I said I don’t know (because that’s usually my answer), so they wanted to check.  They decided to plug in x=0.0000001, and add it to the graph (I just had another student submit a second response to the form that I sent, and the point appeared on the graph).Graph 6 is it an asymptote?

So we decided yes, its an asymptote.  They did a similar process to determine that there’s no horizontal asymptote (even though they said it looks like there may be), but we also decided that this graph increases REALLY slowly.

Ok, so we now had key points and features, and we were all happy.  We then talked about the fact that this looks interestingly similar to the exponential graph, and got into inverses the next day.

So…how did I make the magic happen?

When you create a Google Form, a Google Sheet is created to gather responses.  Each field in the Form gets collected in a separate column in the Sheet.  So, when my students responded to the Form, the populated my Sheet.

Before I sent out the Form, I went to the Response Sheet and selected the two columns where the responses to the Form WILL go.

Sheet empty

I then created a Scatter Plot.

Sheet emtpy w scatter

The columns and the scatter plot were obviously blank to begin with, but as soon as students filled the Sheet with responses, the Scatter Plot filled itself.

Sheet with graph full

When I displayed this to the class, I just zoomed in on the empty scatter plot, and as they entered their responses…bam…magic.

My next goal is to make this work for points of intersection for Linear Systems, where one student will have an x-value which will have a y-value which fits on both lines… I’ll let you know how it goes.