1. Deepening Critical Thinking & Reasoning in Mathematics
Amanda Cameron, Kim Lacelle, and Kathy McCord
EOCCC, April 2018

2. Catholic Graduate Links:
An Effective Communicator
CGE2a, CGE2c, CGE2c, CGE2d
A Reflective and Creative Thinker
CGE3b, CGE3c, CGE3d
A Collaborative Contributor
CGE5a, CGE5b, CGE5c, CGE5d, CGE5e, CGE5f, CGE5g
A focus on Well Being This monograph suggests ways that educators can enhance students’ well-being and foster both their sense of self-worth and efficacy while helping them learn in the mathematics classroom
Well being at the center.. RMS has placed well being a the center. Focus on mindset and building a belief in both educators and students that all students can succeed.

3. Categories of the Achievement Chart
The Achievement Chart provides descriptions of four categories of knowledge and skills:
Knowledge and Understanding
Thinking
Communication
Application
I am sure you are familiar with categories of the achievement chart...pg 22 in the Ministry document .Knowledge being the subject specific content acquired in each grade and understanding being the comprehension and significance of that content. Application is the use of the knowledge to make connections . Thinking are usually the multi step questions that encourage students to make a plan.

4. What’s Assessed? GRADE 3 GRADE 6 20% KNOWLEDGE & UNDERSTANDING 13%
APPLICATION
80% %
THINKING
EQAO questions - Open response and multiple choice combined - this is why exposure to multi step thinking questions is so important

5. Knowledge & Understanding
These questions require students to demonstrate:
subject/strand specific content knowledge
the comprehension of its meaning and significance (understanding)
Examples:
What is 6 x 4?
Which of the following is a composite number? 11, 3, 39, 17
These are basic question /answer type questions

6. Application These questions require students to:
select the appropriate “tool” or find the necessary information to solve

7. Categories of the Achievement Chart
What’s the difference between KU and AP?
If you tell the student what procedure to perform then the skill is usually KU.
If the student must decide what procedure to perform then the skill is usually AP.
In AP, the context is integral to the item. 7

8. Thinking These questions require students to:
select and sequence a variety of tools to solve a problem
demonstrate a critical-thinking process
to answer the question, students need to make a plan
A Grade 3 class wins a pizza party for reading the most books in September. There are 23 students in the class and each student will get 2 slices of pizza. If each pizza has 6 slices, how many pizzas should the class buy?
On EQAO - attitudinal data - “I make a plan” perform better on EQAO than those who do not .
Think breaks - usually means multi- step

9. What’s the difference between AP and TH?
Categories of the Achievement Chart
What’s the difference between AP and TH?
Anything to do with the problem-solving process is Thinking (TH).
The selection and fitting of a single mathematical tool is Application (AP).
If the student needs to make a plan, then the item is Thinking (TH).
To re-cap 9

10. Categories of the Achievement Chart Summary
Knowledge/
Understanding
Application
Thinking
- subject specific content
- the comprehension of its meaning and significance
Q. Find the area of this rectangle:
- select the appropriate “tool” or get the necessary information to solve
Q: Create a rectangle with an area
of 12 units .
- select and sequence a variety of tools to solve a problem
- demonstrate a critical-thinking process
- to answer the question, students need to make a plan Q:
Suzie wants to tile the floor of her family’s rectangular playroom The tiles she plans to use are 10 cm by 10 cm squares.
How many of the square tiles will Suzie need to cover the floor of the playroom?
6 units
2 units

11. Worst-Performed MC Items
When we look at the distribution of EQAO worst-performed MC items across skill:
Knowledge and Understanding: 16%
Application: 41%
Thinking: 43%
Resource Link

12. Questions/Strategies to consider...
Are we teaching students how to "calculate the solution" or how to "think their way to the correct response without precise calculation"?
Could the correct response be isolated by estimating?
Could distractors (wrong responses) be eliminated based on the plausibility or reasonableness of the response?
If you give your students these MC questions as an OR (i.e., with the 4 responses removed), do your results change?
If you give your students these MC questions and tell them they have to start thinking about the question for two minutes before you hand out pencils, do your results change?
How often do students have calculation-free think time during independent practice? (Asked to arrive at a reasonable answer without precise calculation)
What is the gradual release model for your students to use thinking strategies (plausibility, estimation) independently by May of grade 6?
Key Strategies for Multiple Choice
Teach students to give themselves think time:
Think through the question piece by piece
Cover the responses
Change unfriendly numbers to friendly numbers and estimate
Make meaning of the question using language
Use plausibility to eliminate responses

13. Problem 9, Popcorn Power: Grade 6
What is the smallest amount of paper needed to make this container?
Show your thinking.
The container of popcorn pictured below is in the shape of a rectangular prism.
EQAO Task Grade (Page 7)
Link to EQAO Exemplars Gr. 6

14. Task 9: Popcorn Power Grade 6 Specific Expectations
Determine, through investigation using a variety of tools (e.g., nets, concrete materials, dynamic geometry software, Polydrons) and strategies, the surface area of rectangular and triangular prisms; – solve problems involving the
Estimation and calculation of the surface area and volume of triangular and rectangular prisms (Sample problem: How many square centimetres of wrapping paper are required to wrap a box that is 10 cm long, 8 cm wide, and 12 cm high?).
Grade 5 Specific Expectations
determine, through investigation using a variety of tools (e.g., concrete materials, dynamic geometry software, grid paper) and strategies (e.g., building arrays), the relationships between the length and width of a rectangle and its area and perimeter, and generalize to develop the formulas [i.e., Area = length x width; Perimeter = (2 x length) + (2 x width)];
solve problems requiring the estimation and calculation of perimeters and areas of rectangles
Grade 4 Specific Expectations
compare, using a variety of tools (e.g., geoboard, patterns blocks, dot paper), two-dimensional shapes that have the same perimeter or the same area (Sample problem: Draw, using grid paper, as many different rectangles with a perimeter of 10 units as you can make on a geoboard.).
Link to Overall Expectations Strand: Measurement

15. Creating a Collaborative Continuum
Great Professional Learning Inquiry K-6
Choose an EQAO task from the samples provided.
Identify the Curriculum Expectation(s) using the Ontario Math Curriculum.
Create a parallel task for 2 grades below, 2 grades above or 1 grade below and 1 grade above.

16. EQAO Blueprint Framework: Grades 4 to 6 Framework: Grades 1-3
This framework provides a detailed description of the EQAO Assessment of Reading, Writing and Mathematics, Primary Division (Grades 1–3), which is conducted each year in Ontario. It also describes how the primary division assessment aligns with the expectations in The Ontario Curriculum.

17. EOCCC Resource

18. Assessment & Documentation
Increasing the efficiency and effectiveness of assessment and documentation using Google Forms.

19. Assessment & Documentation

20. Assessment & Documentation
Using the Explain Everything App to Document and Assess student Knowledge & Understanding, Thinking and Communication.
Listen fors:
-math vocabulary
(gr. appropriate)
-reasoning & proving
-misconceptions/
misunderstandings
-gaps in learning
Look fors:
-understanding of grids/movement

21. Assessment & Documentation