Thinking Routines: �Notice and Wonder Activities to Engage Students in Meaningful Mathematical Discourse

By Alexandre Eden�Educational Consultant of Mathematics & Certification�New Frontiers School Board�2024-2025

Establishing Routines that Support Thinking

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

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• 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.

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• 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).

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

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

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• 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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Six Forms of Critical Thinking Tasks

Principles to Action: Mathematics Teaching Practices

Teamwork: Does Size Matter and How to Select Teammates

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• Grade 3 and up: Teams of Three

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

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• 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

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• Increase in thinking;

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• Willingness to collaborate;

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• Elimination of social barriers;

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• Increased knowledge mobility;

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• Increased enthusiasm for mathematics learning;

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• Reduced social stress.

14 Practices Towards Building a Thinking Classroom

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.

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• Potential roles may include:

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• The animator or mediator;

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• The time-keeper;

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• The note taker.

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• Collaborative thinking:

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• 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)

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Teamwork: Furniture Arrangement in a Thinking Classroom

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

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

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

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• The placement of the furniture shapes the teachers' intentions.

Thinking Routines: Notice & Wonder Activities

What is Included in the Notice & Wonder Activities?

What Else is Included in the Notice & Wonder Activity?

Placemat Discussion Structure

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

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.

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

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• 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.

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

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Notice & Wonder Activities : Adapted to our Socio-Cultural Context (Secondary III)

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• The 13-storey mixed-used office building known as Îlot Balmoral, was designed by Canadian architecture firm Provencher_Roy.

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• The building was constructed in the Quartier des Spectacles, a major cultural district, in downtown Montreal, Quebec, Canada.

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• The National Film Board of Canada (NFB) and the Université du Québec à Chicoutimi (UQAC) École des arts numériques, de l’animation et du design(l’école NAD) both use the tower as their base of operations.

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• Provencher_Roy anticipated that the large diagonal cut would serve to distinguish Îlot Balmoral from the functions of more traditional office towers in the district.

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Notice & Wonder : Îlot Balmoral (Point of Intersection)

Notice & Wonder: Îlot Balmoral (Solving a System of Linear Equations Graphically)

Desmos | Graphing Calculator

Notice and Wonder: Îlot Balmoral (Linear Equations)

Line AB (red):

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A(-10, 47)

B(-5, 27)

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y = mx + b

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m = (y₂ – y₁)/(x₂ – x₁)

m = (27 – 47)/(-5 - -10)

m = -20/5

m = -4

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y = -4x + b

47 = -4(-10) + b

47 = 40 + b

b = 47 – 40

b = 7

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y₁ = -4x + 7

Line CD (green):

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C(10, 5)

D(35,0)

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y = mx + b

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m = (y₂ – y₁)/(x₂ – x₁)

m = (0 – 5)/(35 - 10)

m = -5/25

m = -0.2

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y = -0.2x + b

5 = -0.2(10) + b

5 = -2 + b

b = 5 + 2

b = 7

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y₂ = -0.2x + 7

Notice and Wonder: Îlot Balmoral (Solving a System of Linear Equations using the Method of Comparison)

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y₁ = -4x + 7

y₂ = -0.2x +7

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Method of comparison

y₁ = y₂

-4x + 7 = -0.2x + 7

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-4x + 4x + 7 = -0.2x + 4x + 7

7 = 3.8x + 7

7 - 7 = 3.8x + 7 – 7

0 = 3.8x

0/3.8 = 3.8x/3.8

0 = x

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x = 0

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If x = 0, then y₁ = -4(0) + 7

y₁ = 0 + 7

y₁ = 7

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Validation:

If x = 0, and since y₁ = y₂ = 7

Then (7) = -0.2(0) + 7

Thus 7 = 7

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I(0, 7), based on the established reference plane.

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Conclusion:�The coordinates of the point of intersection of lines AB and CD are given by I(0, 7) based on the set reference plane.

Notice & Wonder: Îlot Balmoral (Total height)

Total height (h_tot)

(i.e. from the ground up to the y-intercept of the point of intersection)

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h_tot = 7 + 5 + h_blue

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h_blue = y_CD if x = 30

y_CD = -0.2(30) + 7

y_CD = 1m

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h_tot = 7 + 5 + 1

h_tot = 13m

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

• Using algebraic reasoning to:
• Determine the line equations.
• Solve the system of linear equations.

Concrete

Pictoral

Abstract

Notice & Wonder: Îlot Balmoral (Pythagorean Theorem)

Contextual

Visual

Physical

Notice & Wonder: Geodesics

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• Academo explains that a geodesic is a line [or curve] representing the shortest route between two points.

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• On a standard two-dimensional map of the world, you might assume that the shortest path between two cities is indeed a straight line.

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• However, Earth is not flat, it is essentially a three-dimensional sphere. Thus, the shortest distance travelled between two cities is represented using a geodesic curve.

Notice & Wonder: Geodesics

Notice & Wonder: Geodesics

Notice & Wonder: Geodesics (DESMOS)

Cat Café Logo DESMOS Project

Questions?

Merci!

Thank You!

References

• Academo. (2024). Geodesics on the Earth. Geodesics on the Earth | Academo.org - Free, interactive, education.
• Bombardier, H., and Ghislaine Cloutier. (2016). Enseigner en classe multiniveau: des stratégies et des outils efficaces. Chenelière Éducation. Québec, Canada. Pgs. 18 to 48.
• Canadian Architecture News. (2022). Provencher_Roy Carves Mixed-Use Office Tower In Montréal with a Red-Colored Diagonal Slit. World Architecture Community. Provencher_Roy carves mixed-use office tower in Montréal with a red-colored diagonal slit
• CDI Spaces. (2023). 4 Steps to Perfecting Classroom Seating Arrangements for Effective Learning. 4 Steps to Perfecting Classroom Seating Arrangements for Effective Learning
• Desmos | Graphing Calculator
• 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

References

• Ministère de l’Éducation, du Loisir et Sport (MELS). (2009). Progression of Learning [Secondary School]: Mathematics. Progression of Learning - Mathematics – Secondary
• Moldavan, A., Rhodes, S., Willingham, J. & Heidi Eisenreich. (2024). Notice and Wonder: Exploring Common Mathematical Notions. Mathematics Teacher Learning & Teaching PK-12. Volume 117, Issue 09. NCTM.
• NCTM. (n.d.). Notice and Wonder Lesson Plans 9-12: Fountain 10-12. Final Fountain - High School (2).pdf
• NTCM. (2014). Principles to Action Ensuring Mathematical Success to All. Principles.To.Actions.ebook.pdf
• Reed Furniture. (2022). Classroom Layout Matters: How Flexible Classroom Furniture Can Help Layout Your Classroom. Classroom layout matters: How flexible classroom furniture can help la - Reed Furniture
• Sangiovanni, J., Katt, S. & Kevin Dykema. (2020). Productive Math Struggle: A 6-Point Action Plan for Fostering Perseverance. Corwin Mathematics. California. USA. Pgs. 48 to 72

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