## 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

## Mathematical Proficiencies

The full GECDSB Mathematics Vision is here.

## Paying Attention to Spatial Reasoning

## 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.

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:

## Counting Principles – Counting and Cardinality

A Progression of Counting and Quantity As a former secondary math teacher and intermediate math coach, my new role as K-12 math consultant has led to a wealth of knowledge that I wish I had during my years spent in the classroom. My conversations about student learning needs with intermediate and senior math teachers always seems to come down to gaps in student understanding, however rarely wer...

## 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:

Here are three views of the cube:

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

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:

## Airplane Problem – Trip to Toronto [3 Act Math Task]

### How many seats are there?

From Subitizing and Unitizing to Multiplicative and Algebraic Thinking This 3 act math task was designed specifically to have a very low floor in order to be useful from primary grades and a high ceiling with an opportunity for many extensions so the task can be used in junior and intermediate classrooms. While I believe this task can touch on many different specific expectations at man...

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:

## 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:

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.

## Donut Delight [3 Act Math Task]

### How many doughnuts are there?

Sparking Curiosity to Promote Conceptual Understanding of Multiplication and Division When one of our district math leads, Brennan Jones asked me to brainstorm some ways we could help his staff engage in some professional development around division and incorporate a 3 act math task into the learning, I immediately thought of some contexts where arrays, base ten blocks and area models cou...

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.

## The Progression of Multiplication

Arrays and Area Models to The Standard Algorithm Did you know that the words "array" and "area model" appear in the Grade 1-8 Math Curriculum a combined 22 times? Not only do arrays and area models help to support the development of proportional reasoning when we formally introduce multiplication in primary, but they also help us understand how to develop strategies that lead to building numbe...

## 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:

## 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.

Hope you folks found this professional learning experience useful.

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