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Establishing Routines that Support Thinking

Use the vocabulary of thinking as a matter of course in classroom discussion, i.e. pose purposeful questions.

Provide the students with adequate time to reflect on their learning alone and with others, and to think about their answers before being asked to respond.

Provide time for students to explore ideas, try out solutions, and make revisions or even start anew when their initial ideas don’t work out. (Gini-Newman and Case, 2018, Pg. 68).

Asking students what they notice provides a means to encourage mathematical discourse amongst them.

Wondering encourages curiosity and critical thinking, prompting students to raise questions that they can then apply mathematical tools to answer. (Dobie and Anderson, 2021)

Questioning students’ observations and allowing spaces for wondering creates powerful learning opportunities to make connections and develop positive mathematical identities. (Moldavan, Rhodes, Willingham, and Eisenreich, 2024)

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Principles to Action: Mathematics Teaching Practices

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Teamwork: Does Size Matter and How to Select Teammates

Grades K-2: Teams of two

Lacks some diversity, but with the appropriate guidance of the teacher, the polite exchanges between the two students can shift towards collaboration at this age.

Grade 3 and up: Teams of Three

Redundancy: Things that a group of students have in common—language, interests, experiences, knowledge, etc.

Diversity: Things that individual members of the group bring that are not shared with others—different ideas, viewpoints, perspectives, representations, etc.

Benefits of Random Groups

Increase in thinking;

Willingness to collaborate;

Elimination of social barriers;

Increased knowledge mobility;

Increased enthusiasm for mathematics learning;

Reduced social stress.

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Teamwork: Collaborative Thinking and the Productive Struggle Community

Assign roles to the students to ensure that the work is done equitably and to hold them accountable.

Potential roles may include:

The animator or mediator;

The time-keeper;

The note taker.

Collaborative thinking:

Ensure that there is a structure to encourage students to engage with each other’s ideas, e.g., a placemat structure. (Gini-Newman, G. and Case, R., 2018. Pg. 57)

Productive Struggle Community	What it is
Working with partners	-Sharing the work -Taking turns -Listening to other’s ideas
Sharing ideas	-Valuing others ideas
Making mistakes	-Recognizing that mistakes help us learn
Taking risks	-Asking others questions about their ideas
Respecting the ideas of others	-Respectfully critiquing others or giving feedback
What to do when stuck	-Knowing that everyone gets stuck

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Teamwork: Furniture Arrangement in a Thinking Classroom

“Defront” classrooms to promote the mathematical discourse and critical thinking.

Students will begin to collaborate more, and teachers will talk less.

The placement of the tables sends a message of what is expected, and it changes what happens in the classroom.

The placement of the furniture shapes the teachers' intentions.

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Thinking Routines: Notice & Wonder Activities

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What is Included in the Notice & Wonder Activities?

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What Else is Included in the Notice & Wonder Activity?

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Placemat Discussion Structure

Student A

Notice (N): Blocks placed in 7 columns

Wonder (W): Are the colours of the blocks important?

Question (Q): Could the column of blocks represent natural numbers?

Student B

N: There are 30 blocks in total

W: Do the blocks all have the same mass?

Q: Why are the blocks placed that way? Does each column have a value?

Student C

N: Each block appears to be the same, except for their colours

W: Why weren’t cylinders used instead of cubes?

Q: Do the blocks represent the same thing in each column

Collaboration

N: There are blocks in the image

W: Colours important?

Q: What do the blocks/columns represent numerically?

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Adapt the Activity to Align with the Progression of Learning and the Quebec Education Program

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Notice & Wonder Blocks Activity: Manipulatives

As with any other type of representation, manipulatives must be explored and used in a flexible manner to help student understand and represent mathematical concepts and processes or convey their mathematical reasoning or thinking.

Students must understand that manipulatives are not the one and only representation of a concept, but rather one of the many ways in which it can be represented.

A variety of manipulatives should therefore be used to represent a given concept. (Ministère de l’Éducation et de l’Enseignement supérieur. 2022)

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Assessment-Rich Learning & Peer Assessment of Collaborative Learning

Assessment-rich learning means that students are confident and rigorous in their assessments of themselves and of others.

Every task in school is seen as an opportunity to learn.

Assessment-rich learning helps students focus on what they need to think about and do to improve rather than exclusively on what they have done well or poorly.

Encourage students to view failure as a learning opportunity and to assume an active listening role when receiving comments from others.

	Consistently		Moderately		Rarely
1. Willingness to reconsider position	5	4	3	2	1	N/A
2. Willingness to defend opinion	5	4	3	2	1	N/A
3. Respectful of persons who disagree	5	4	3	2	1	N/A
4. Challenges in responsible way	5	4	3	2	1	N/A
5. Works towards establishing consensus	5	4	3	2	1	N/A

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Notice and Wonder: Balloons (Grades 1 or 2)

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Notice & Wonder: Ballons (Grades 1 or 2)

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Notice & Wonder: Balloons (Grades 1 or 2)

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Concrete, Pictoral, Abstract (CPA Approach):�Representations in Mathematics

Using artihmetic reasoning to:

Determine equivalent addition equations
Learn about the commutative property of addition.

Concrete

Pictoral

Abstract

Polypad – Virtual Manipulatives

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Merci!

Thank You!

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References

Bombardier, H., and Ghislaine Cloutier. (2016). Enseigner en classe multiniveau: des stratégies et des outils efficaces. Chenelière Éducation. Québec, Canada. Pgs. 18 to 48.
CDI Spaces. (2023). 4 Steps to Perfecting Classroom Seating Arrangements for Effective Learning. 4 Steps to Perfecting Classroom Seating Arrangements for Effective Learning
Dobie, T. and Eleanor R. Anderson. (2021). Noticing and Wondering to Guide Professional Conversations. Mathematics Teacher Learning & Teaching PK-12. Volume 114, Issue 2. National Council of Teachers of Mathematic (NCTM). Noticing and Wondering to Guide Professional Conversations.pdf
Gini-Newman, G. and Roland Case. (2018). Creating Thinking Classrooms: Leading Educational Change for this Century. Corwin. Carlifornia, USA.
Liljedahl, P. (2021). Building Thinking Classrooms in Mathematics Grades K-12: 14 Teaching Practices for Enhancing Learning. Corwin Mathematics. California, USA.
Ministère de l’Éducation et de l’Enseignement supérieur. (2019). Framework for Interventions to Improve Math Skills. Framework for Interventions to Improve Math Skills.pdf

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