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Wednesday, 5 December 2018

Warming up the Mind for Mathematical Thinking!

Written by Rebecca Forchuk, Principal
DMG Professional Learning Day, November 23, 2018

Students come to math class with different interests and understanding of concepts, but regardless of where they are on the continuum, students are adaptable.  Teachers can enhance learning when we plan for this variability in math.  Exploring instructional strategies that consider this variability was the focus of Dr. Morris Gibson’s staff learning day on November 23rd.  I had the opportunity to learn, alongside  the staff, from their admin team and Wanda Dechant, Designer of Professional Learning, Calgary Regional Consortium.

Kevin, Principal of DMG, pointed out the importance of warm ups in math and made connections to other areas of life: athletes warm up, musicians warm up and so too should mathematicians.  The benefits are:

Bridges transition time for students.
Allows students to shift to mathematical thinking.
Builds confidence through successful math opportunities that are at the right level.
Provides effective, yet easy, differentiation opportunities when using open ended or ‘low floor, high ceiling’ tasks.
Reviews prior learning and provides formative assessment to teachers.
Makes thinking visible; students hear/see the thinking of others
(Adapted from DMG PowerPoint slide 27, Nov 23rd)

There are several warm ups that develop the 7 processes of math, which are foundational to mathematical understanding.    This week we are going to explore one of those warm ups:

Warm Up #1: How Many Dots

Ask students:
 “How many dots do you see?” 
Flash the picture for three seconds on the Smartboard.  Warn students it will be F-A-S-T!  If you need to show it again, do so.  But again, make it quick – three seconds or less.  The purpose of this task is NOT a correct answer. 

Ask students to share how many dots they saw.  Try to get variability in students’ answers.

Here comes the mathematical thinking. Ask:
“How did you see the dots?”  
While students share how they saw the dots, the teacher captures the student’s visual representation on chart paper or Smartboard by doing two things:
  1. Draw lines from dot-to-dot to represent how they saw the dots.
  2. Write down the equation that represents their visualization. 

The picture below demonstrates what your chart paper might look like after students show their thinking:

(Taken from DMG PowerPoint Slide 33, Nov 23rd)

Picture 1: How teacher "saw" the dot card
I participated in this warm up twice: once with Div 1 teachers and again with Div 2 teachers.  Figure 1 shows the diverse ways in which everyone on staff saw the dots on this card.  Each division of teachers came up with entirely different ways to visualize the same dot card! See Picture 1 for the various ways teachers saw the dot card.

Through our discussions after the task, we came to consensus such a warm up can develop the processes of visualization, reasoning, communication and estimation.  This task also develops:
  • Spatial reasoning
  • Positive mathematical mindset
  • Understanding there is a variety of perspectives in math
  • Understanding there are multiple pathways to arrive at an answer.
  • Ability to subitize

At the end of the session, admin tasked teachers with homework: try one of the warm ups from the session before their next PLC meeting. It is clear that students as young as Grade 1 can benefit from “How Many Dots.” It really demonstrates variability when a student visualized the dots as a picture of a fish – what a unique way to see them!  

This proves to be a great low floor, high ceiling task.  Dot cards (just Google “dot cards math” to find an endless number of them) can be simple and range in complexity:

Example 1:

Example 2:

Example 3:

Watch the blog over the next few weeks when we explore more Math Warm Ups that develop number sense, processes of math, and growth mindset based on professional learning at DMG! 

A big thank you to the admin and staff at DMG for inviting me to their session and sharing their learning journey!