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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!

Monday, 26 November 2018

Power of Mindsets In Math

Written by Marsi Quarin-Wright, Instructional Coach
Online Teacher Course by Jo Boaler: youcubed.com

I was recently introduced to the work of Jo Boaler and the website youcubed.org. What caught my interest is where she says math learning begins. Her starting point is fundamental to the vison of FSD and the role relationships play in learning. To me, this is not a typical starting point in understanding math. What furthered my curiosity was watching this approach to math instruction in a classroom.  I listened to the teacher’s positive talk and how engaged students were in the lessons.  The most impressive is listening to how students in her class love math, describe 3-D shapes with excitement and apply proper math vocabulary. I knew I had to investigate more. I went on to the website and found that there was an online course I could take. I signed up!

I was nervous and fearful; what was I doing taking a math course? What if I had to add or multiply something?  To my surprise, as I logged in and found the first sections of the course were on Math Mindset, Mistakes and Persistence, and Teaching for a Growth Mindset. Dr. Boaler quickly dispelled the myths of math, such as:
·         math is only for smart people
·         you either get math or you don’t.

Taken from youcubed.com - Growth Mindset by Jo Boaler: retrieved from https://bhi61nm2cr3mkdgk1dtaov18-wpengine.netdna-ssl.com/wp-content/uploads/2017/05/When-You-Believe-in-Your-Students-They-Do-Better.pdf
She uses brain research to show that everyone is capable of learning math to high levels. I quickly realized that math has a bad reputation and my own worries about taking this class fed into that bad reputation. I quickly understood and turned my worries into wonders and began to see math in a new light.

Over the years I have heard from friends, family, parents and students – “I hate math” or “I’m no good at math”. I think it is vitally important that we become more aware of how these passing comments can affect how students’ view and ultimately learn math. We need to focus on the positives and not create unnecessary anxiety in students by continuing to discuss math in a negative way.  Using a growth mindset can offset many of these fears for our students. We need to use mistakes as a path to a deeper understanding of math. Having students see that persevering through difficult tasks is a way to make our math brain grow.

Where to start? I encourage everyone to take a look at youcubed.org.

In particular, look under Tasks and More à Week of Inspirational Math.

This starts that switch in mindset in math. You can check out the online student class on youcubed.org called, “How to Learn Math: For Students;” its free and another great place to start for everyone. The whole site is really amazing and provides practical, easy to use lessons for all teachers – even for those who think they “aren’t good at math”…

Tuesday, 13 November 2018

Math Tools

By Julie Julian, Instructional Coach
Guided Math & Running Record, CRC Session with Dr.Nicki Newton

Math tools help students scaffold their thinking. All learners should be able to access them at any given moment, regardless of grade level.

“Good problem solvers usually construct a representation of the problems to help them comprehend it “(van Gardener & Montague, 2003), but representing mathematical information visually does not come naturally to most students. Visual representation should be explicitly taught and then practiced using a variety of tools.

Both hands-on manipulatives and virtual tools:

·        Help convey concepts
·        Help visualize mathematical ideas
·        Model number relationships

Classroom Manipulatives

Manipulatives Tip Sheets http://www.edugains.ca/newsite/math/manipulative_use.html
The tip sheets for each of the listed manipulatives include a description of what they are, how they can help students, how many are recommended and sample activities. Unfortunately, many of the weblinks at the bottom of these PDF files aren’t active, but the tip sheets themselves are very helpful.

Geoboards (pdf)
Tangrams (pdf)

Beaded Number Line

Have each student created a 100-bead beaded number line alternating colours every 10 beads This can be used for counting, skip counting, place value, rounding, adding, subtracting, number patterns, multiplication, division, and decimals.

Virtual Manipulatives

There are lots of fantastic interactive virtual math tools and apps for Smartboards, computers, tablets, and phones.

Graph Paper:

Number Frames:

Cuisenaire Rods:



Number Line:

Rekenrek (number rack):

Pattern Shapes:

Wednesday, 7 November 2018

Problem Solving in Math

By Darla Milford, Instructional Coach
Running Record, CRC Session with Dr.Nicki Newton

Real World Connection:

Problem solving in math should not be taught separately from the mathematical skills, processes, and understanding.  In fact, according to Dr. Nicki Newton, everything ‘math’, should be attached to the real world with a story attached to it.   In order to intellectually engage our students in math, it is essential we approach problem solving from a real world context with personalization being key.  

On Dr. Newton’s Guided Math Blog, she quotes Bailey saying students “don’t care how many apples
Bob gave to Suzy. They’re much more interested in things like music, video games, movies, trading
cards, money, and friends” (Bailey, 2002, p. 61) so make it personal!!!

Here are some sites with engaging problems ready to use with your own students:

Cue Words/Keywords
We often fall into the trap of teaching our students to pick out ‘cue words’ or ‘keywords’ within a
problem to figure out the answer.  Some of these words may include ‘how many in all’ referring to
addition or ‘what is the difference’ suggesting subtraction. This strategy is essentially a quick ‘trick’
to help students solve the problem but it is important not to focus entirely on keywords without
considering the contextual information within the problem.  Focusing on words alone takes away
from the mathematical reasoning and understanding that is required in higher level or multi-step
problems students will encounter as they move forward in math.

Here’s an example:

Joe has 8 marbles, which is 2 times more than his brother. How many marbles do they have

If kids focus primarily on key mathematical vocabulary, it may lead to a misunderstanding that the
term ‘times’ means that we multiply 8 X 2, when, in fact, if you read the problem in context you will
figure out that you already know that Joe has 8 marbles but you need to figure how many marbles
Joe’s brother had before adding these numbers together to get our answer.

Joe-8 marbles
Brother-½ of Joe’s 8 marbles = 4
8+4=12 marbles altogether

Open-Ended Problem Solving
Dr. Nicki Newton emphasizes that math is best learned in contextual situations and that it is
important to provide opportunities for kids to demonstrate their mathematical understanding through
the use of ‘Open’ word problems.  ‘Open’ word problems are just that, open-ended, with a variety
of questions that could be asked as well as proven.  She suggested using a ‘3 Reads Protocol’
reading strategy to help make the problem more about the math than the reading  component.

Here’s an ‘Open’ problem example using the ‘3 Reads Protocol’ Reading Strategy.

The Smith Family Vacation Problem

The Smith family went on vacation. They drove 201 km on Monday, 177 km on Tuesday and 99 km
on Wednesday.

Write a story problem that includes the information given with an answer that makes sense.

When using this strategy, which can be used for whole group or guided math instruction there are 3

  1. Begin by reading the problem altogether out loud.

     2) Reread the problem altogether again, this time, unpacking the math.

     3) Read the problem a 3rd time and have students create possible problems linked to the  
information and have them figure out the answer.

Here is an interesting article about open-ended questions and the value in having students write their
own problems to increase engagement and metacognition.  

Dr. Nicki Newton’s authentic approach to teaching math problem solving in real world contexts makes

sense.  When students are connected to their learning and they can see the implications in the world
around them, they become much more invested.  Make sure you check out Dr. Nicki’s blog and if you
have any questions about her material, be in touch with the Instructional Coach in your building!