## Building our Students’ Mathematical Intuition

I’ve been asked to share my OAME 2017 presentation on Mathematical Intuitions by a few of my participants.  Instead of just sharing the slides, I thought I would add a bit of the conversations we had, and the purposes behind a few of my slides.  Here is a brief explanation of the 75 minutes we shared together:

I started with an image of the OAME 2017 official graphic and asked everyone what mathematics they saw in the photo:

I was impressed that many of us noticed various things from numbers, to sizes of fonts, to shapes and other geometric features, to measurement concepts to patterns…

I decided to start with an image so I could listen to everyone’s ideas (the group could have simply noticed the numbers visible on the page, or the triangles, but thankfully the group noticed a lot more!).

I then shared a few stories where students have entered into a problem where they have attempted to do a bunch of procedures or calculations without ever doing any thinking, either before or after, to make sure they are making sense of things.

You can read the full stories on these 2 slides here and here.

The bandana problem above is a really interesting one for me because it shows just how likely our previous learning can actually get in the way of students who are attempting to make sense of things.  Most students who learned about how to convert in previous years in a procedural way have difficulty realizing that 1 meter squared is actually 10,000 cm squared!

In an attempt to explain the kinds of mental actions we actually want our students to use when learning and doing mathematics I showed an image shared by Tracy Zager (from her new book Becoming the Math Teacher You Wish You’d Had).  We discussed just how interrelated Logic and Intuition are.  Students who are using their intuition start by making sense of things.  They start by making choices or estimates, which are often based on their previous experiences, and use logic to continue to refine and think through what makes sense.  This process, while often not even realized by those who are confident with their mathematics, is one I believe we need to foster and bring to the forefront of our discussions.

I then shared the puzzle above with the group and asked them to find the value of the question mark.  Most did exactly what I assumed they would do… but none did what the following student did:

Most teachers aimed to find the value of each image (which isn’t as easy as it looks for many elementary teachers), but the student above didn’t.  They instead realized that all of the shapes if you add them up in any direction would equal 94.  This student had never been given a problem like this, so didn’t have any preconceived notions about how to solve it.  They instead, thought about what makes sense.

So, how DO we help our students use their intuition?  Here are a few ideas I shared:

1. Contemplate then Calculate routine (See David Wees for more about this here or here, or purchase Routines for Reasoning by Grace, Amy and Susan)

The two images above show visual representations (thank you Andrew Gael and Fawn Nguyen for your images) where I asked everyone to attempt to think before they did any calculations.  I used Andrew’s picture of the dominoes and asked “will the two sides balance… don’t do any calculations though”.  For Fawn’s Visual Pattern, I asked the group to explain what the 10th image would LOOK LIKE (before I wanted them to figure out how many of each shape would be there, and then find a rule for the nth term).

We shared a few estimation strategies:

and a few “Notice and Wonder” ideas:

However, while I love each of the strategies discussed here (Contemplate then Calculate, Estimation routines, Notice and Wonder) I’m not sure that doing a routine like these, then going about the actual learning of the day is going to be effective!

Instead, we need to make sure that noticing things, estimating, thinking happen all the time.  These need to be a part of every new piece of learning, not just fun or neat warm-ups!

Building our students’ intuition means that we need to provide opportunities for them them to think and make sense of things, and have plenty of opportunities for them to discuss their thinking!

If our goal is for students to think mathematically, and use their logic and intuition regularly, we need to operate by a few simple beliefs:

I ended the presentation with a final thought:

Here is a copy of the presentation if you are interested:

I’d suggest you scroll down to slide 49 and play the quick video of one of my students doing a spatial reasoning puzzle.  It’s one of my favourites because it illustrates visually the thinking processes used when a student is using both their intuition and logic.

To me, there seems to be so much more I need to learn about how to help my students who seem to struggle in math class use their intuition.  Hopefully this conversation is just the beginning of us learning more about the topic!

A few questions I want to leave you with:

• What routines do you have in place that help your students make sense of things, use their intuitions and develop mathematical reasoning?
• Do your students use their intuition in other situations as well (or just during these routines)?
• How can you start to build in opportunities for your students to use their intuition as a regular part of how your class is structured?
• What does it look like when our students who are struggling attempt to use their intuition?  How can we help all of our students develop and use these process regularly?

Special thanks to Tracy Zager’s new book for the inspiration for the presentation.

As always, I would love to continue the conversation here or on Twitter

## Developing Spatial Reasoning through Guided Play

For the last few months, a team of kindergarten teachers and myself have been working together to deepen our understanding of early years mathematics, spatial reasoning, and the importance of guided play as a vehicle to engage our students to think mathematically.  Below is a copy of our slideshow presentation we shared at OAME 2017, and some of the documents we have created over the past few months.

A quick synopsis of our work first:

While our research led us toward Doug Clement’s work about trajectories, and research about spatial reasoning and early mathematics, much of the tasks we actually did with students directly came from the book shown above (Taking Shape) which I can’t recommend enough – if you can, get yourself a copy!

We discussed the quote above to help us realize what actually underpins mathematics success.  More details about how the quote ends here.

We shared research showing just how important early mathematics is, and specifically what the kinds of instruction could / should look like to accomplish this learning.  Duncan et. al., is a widely quoted piece of research that has led many to realize that early math learning needs to be a focus in schools – even more so than early reading!

We played a few games that helped us stretch our spatial reasoning abilities.  The image above was part of our “See It, Build It, Check It” activity (found in Taking Shape).  Everyone saw the image for a minute, then was asked to build it once the image was removed.  What we noticed is just how difficult spatial tasks are for us!

After we had the opportunity to play for a bit, we dug back into the research about spatial reasoning and the jobs typically chosen based on (high school) spatial ability.  Hopefully you noticed something interesting on the graph above!

So, we know how important spatial reasoning is, but the 3 pieces above (taken from Paying Attention to Spatial Reasoning document) might help us realize how important a focus on spatial reasoning is for both our students and us.

In our time together, we learned a lot about the importance of observing our students as they were engaged in the learning.  From the initial choices they made, to how they overcame obstacles, to understanding the mental actions that were happening… Observing students in the moment is far more powerful than collecting correct answers!

See the link at the bottom of the page for our connections to Doug Clements’ work.

We also discussed the specific connections between the mathematics behaviours and the learning that happened beyond.  In our Kindergarten program document, our students’ expectations fall under 4 frames (see above) so we linked the learning we saw to the program document in a way that helps us see the depth and breadth of the kindergarten program (see document linked below).

We then ended our presentation with a synopsis of what we learned throughout our work together.  While the slideshow might be helpful here (I’d love for someone to comment on those slides at the end), the conversations that Sue, Kristi and Kristen had with others have shown me just how valuable it is to spend time learning together.  I couldn’t be prouder to be able to work with such reflective and dedicated teachers!

Our presentation:

Guided Play Slideshow

A few of our handouts:

4 Frames handout

Developmental Trajectories – Adapted from Doug Clements’ work

Spatial Language

@MarkChubb3

@mrs_dt

@mrs_penlington

@kristinWillms