Roseland Public School Math Professional Learning

GECDSB School Based Learning Sessions (SBL)

Thursday March 2nd, 2017

Thanks for inviting me to learn with you on your school based learning day! The group was energetic and excited to learn about meaningful manipulative use, spatial reasoning, some counting principles, visualizing numerical expressions, some multiplication/division and the importance of concreteness fading!

Here’s a summary of what we explored today throughout all three 100-minute blocks:

The GECDSB Mathematics Strategy

GECDSB Math Strategy


Mathematical Proficiencies

The full GECDSB Mathematics Vision is here.

GECDSB Mathematical Proficiencies


Paying Attention to Spatial Reasoning

Paying Attention to Spatial Reasoning Document


Warm-Up: How Do You See the Dots?

We did a quick Dot-Card warm-up asking you to visualize how you saw a series of dots on the screen. We went around the table and it seemed that everyone had a different perspective.

Jo Boaler Dot Card Warm-Up

Here’s some of the ways you might have visualized the dots:


Counting Principles

We briefly discussed the importance of counting and quantity principles like subitizing and unitizing. A full summary is below:


A Spatial Reasoning Task

Both the primary and junior groups were interested in some tasks that might help students with their spatial reasoning skills.

One that came to mind immediately was from NRich.org:

A Puzzling Cube

Here are the six faces of a cube – in no particular order:

All Six Sides of a Cube

Here are three views of the cube:

Three Views of a Cube

Can you deduce where the faces are in relation to each other and record them on the net of this cube?

A Blank Cube Net

You can use this interactivity to try out your ideas. You will still have to visualize the cube folded up!

Check out the interactive version here.


Then we took some time to explore a 3 act math task with a very low-floor and high-ceiling, called the Airplane Problem:

In the airplane problem, we made predictions, explored subitizing and unitizing.

After the predictions for number of seats, I challenged the group to:

Create any numerical expression that could represent the seat configuration (or array of seats) for the airplane and use your manipulatives to represent your expression concretely.

Then, in number talk fashion, we had each person share what numerical expression(s) they saw. Here’s a few screenshots of what came out:

Airplane Problem 3 Act Task - Sample Number Talk 1

Airplane Problem 3 Act Task - Sample Number Talk 2


Visualizing the Order of Operations

With our intermediate friends, we explored the Airplane Problem and also looked at different ways to write the seat configuration using as many different numerical expressions as possible. Some great conversations were had and it led us to think about some of our order of operations work in the intermediate grades.

Here’s some of where it led:

Airplane Problem and Order of Operations - Distribution

Airplane Problem and Order of Operations - Extending Curriculum Visually

We also took some time to visualize the order of operations in various ways including these:




See the whole playlist of visualizing order of operations videos here.


Extending to Multiplication and Division

Finally, with the junior group, we explored the Donut Delight 3 act math task where we played with multiplication to predict how many donuts there were in the “double hundred dozen” box and then used repeated subtraction to lead to a flexible division algorithm for division when we tried to find how many layers there were.

While we didn’t explore the progression of multiplication explicitly, the following post might be useful to extend some of the thinking from the donut delight task.


Vertical Non-Permanent Surfaces

During the second 100-minute block, some discussion arose from the use of whiteboards – also known as “vertical non-permanent surfaces (VNPS)”. Peter Liljedahl from Simon Fraser University has done extensive research on Vertical Non-Permanent Surfaces (VNPS) including his paper Building Thinking Classrooms. In his research, he has found that standing and writing on non-permanent surfaces like chalkboards or whiteboards is very effective for engagement, time-to-task and time-on-task in mathematics class:

Vertical Non-Permanent Surfaces


Concreteness Fading

We attempted to summarize the use of manipulatives on a continuum called “Concreteness Fading”. While the name suggests that concrete manipulatives fade away over time, it is important to remember that they fade away with one layer of abstraction and then reappear as a new layer is “piled” on.

Applying Concreteness Fading to Progression of Multiplication

Hope you folks found this professional learning experience useful.

I’d be delighted to come back and learn alongside you all again sometime soon!