Additional Support for What to Look For 2

To build a sense of number and support students as they move from counting, to additive thinking, and into multiplicative reasoning, the LKDSB has developed recommended instructional practices to support students from K – 8.

These recommended instructional practices are based on the research by Dr. Alex Lawson based out of Lakehead University and the Ontario Mathematics Curriculum. Students will begin with early numeracy skills in K-2 classrooms that build a strong sense of number, counting skills and early additive thinking. Students in Grade 3 and 4 continue to strengthen their additive thinking and begin developing multiplicative reasoning skills. Both additive thinking and multiplicative reasoning skills must continue to be strengthened throughout Grades 5, 6, 7 and 8, as they move from applying this understanding from whole numbers into fractions, decimals and percents.

Accessing the What to Look for 2 Resource

Each school has been provided copies of the What to Look For 2 Book. These copies are intended for use in Grade 2/3 - 5/6 classrooms as Tier 1 instruction.

Targeted Tier 2 intervention support will occur in priority schools in grade 3, 6, 7 and 8 with the support of the math facilitator, in collaboration with the classroom teacher.

Teachers should use the code at the front of the book to access the Pearson resources and files provided by the authors.

In the case for teachers who are not able to open SmartNotebook files with their iPads/TVs, you can access a powerpoint version of the files only available to LKDSB teachers using One Drive with this link.

Laying the Groundwork for Multiplicative Reasoning

Research has shown that the typical "Groups of" instruction, common across North America, does not align with how students naturally construct the idea of multiplication. The research found that "groups of" instruction:

falls short for many students in most word problem situations because it doesn't deal with quantities
encourages counting and addition, rather than multiplication, keeping children stuck in early strategies
might be an unnecessary detour

Research completed with students and analyzed by the research team from Lakehead University has instead developed a unit for teaching multiplicative reasoning that builds on additive thinking, and honours how students naturally begin to connect to multiplication.

As with any change in practice, there is new language, a new order of suggested teaching and new methods of introducing key concepts of multiplicative reasoning, that may be challenging for teachers. The research, as well as our initial work in this space, has confirmed that students who engage in this new way of learning multiplication grasp concepts more naturally than previous teaching methods.

The LKDSB is recommending that all Grade 3 and 4 students engage in this new method of learning multiplication and will support this with the physical resource and math facilitator support throughout the year.

In addition, the LKDSB is recommending that Grade 5 - 8 priority classrooms also engage in this learning through Tier 2 intervention.

The Importance of Additive Thinking

It was confirmed in the research the importance of student additive thinking as a solid base for being able to think multiplicatively.

Both the addition/subtraction, and the multiplication/division continua are built around important phases that students may be working in.

When we look at the addition/subtraction continua, we can see that students move from direct modelling and counting, into counting more efficiently, then develop the ability to work with numbers before they become proficient.

The research once again confirmed the importance of recognizing where a student is currently working in order to support them to become more efficient. In the case of additive thinking, students need to be working with numbers or proficient with addition before they are able to think multiplicatively.

The work that is done with students in the early counting and additive thinking areas of number sense in Kindergarten to Grade 2 set the stage for success when we begin to develop multiplicative reasoning in grade 3 and 4. These strategies are also essential to student success as they work through additive thinking and multiplicative reasoning with whole numbers into fractions, decimals and percents in grade 5 to 8.

Tier 2 intervention, that honours where students are and supports their growth along the continua is necessary in all grades as we work to help students develop from counting, to additive thinking, and into multiplicative reasoning.

Why is Multiplicative Reasoning Important?

Multiplicative reasoning is not just the ability to multiply. It is the ability to use one multiplication fact to derive other multiplication facts. It is understanding what is happening and how the two are related. For example:

Multiplicative reasoning appears in division, area, volume, coding, fractions, place value, early algebra, and so many other areas of the junior and intermediate curriculum.

The researchers asked themselves "Is there a better way to teach multiplication or more broadly, multiplicative reasoning, than what I have been doing?!" and argued the following important points:

when children simply memorize facts and procedures, they miss out on an enormous amount of math that is the foundation for junior, intermediate, and senior level mathematics, as well as a general proficiency with mathematics essential to living
Just memorizing the facts is no longer enough and never was
children should eventually know their facts automatically; however, they should come to know them by constructing a range of increasingly sophisticated strategies and the related key ideas
many children who bypass the territory in between are at a significant disadvantage - particularly struggling students

Important Concepts from What to Look for 2

Although this book is written as a unit of study, it actually will take approximately 2 years to complete. The Ontario Curriculum expectations spreads the exploration and automaticity of learning multiplication facts out over Grade 3 and 4. Throughout the unit of study, teachers are encouraged to pause and provide time and space for students to work with the concepts, develop their understanding and strengthen the automaticity of the multiplication facts being taught. Teachers are asked not to rush the process and encouraged to seek support from the school assigned math facilitator throughout the year.

For the 2025-2026 school year, Grade 5 teachers are also asked to complete the lessons Tier 1.

New Language and New Way of Introducing Multiplication

Researchers worked with, recorded, and re-watched videos of children solving multiplication problems prior to being taught multiplication. They found the following:

before instruction, Gr 1 & 2 students who solved the problems mentally, consistently started with the intensive quantity and then iterated it
this happened regardless of the order of the wording and the number of sizes

Let's look at the research question: You have 3 packages of gum. Inside each package there are 4 pieces. How many pieces of gum do you have?

The What to Look For 2 book incorporates teaching multiplication with this idea of Intensive Quantity and Iterations. However, when working with students, we refer to these in more child friendly language.

Intensive Quantity - "Per" - we refer to this as the "per amount"

Iterations - Stamping - we ask students how many times the "per" was stamped

The units of study include learning about the intensive quantity, building fluency, solving word problems, introducing division and working with the array.

The Revised Multiplication/Division Continua

As the researchers spent time teaching using this new method, they also developed a deeper understanding of the strategies students were using. This new revised continua is different than the previous version and aligns with the suggested new method of teaching multiplication.

On this revised continua, students move from precursors of adding, to direct modelling and counting, into counting and adding with tracking, then develop the ability to work with multiplicative relationships before they can become proficient at solving and calculating with meaning.

The research once again confirmed the importance of recognizing where a student is currently working in order to support them to become more efficient. One of the big things to notice is the shaded green line. Students to the left of the green line are still developing and solidifying their additive thinking strategies. Students using strategies that sit on or straddle the green line are beginning to develop their multiplicative thinking and students who are working with strategies to the right of the green line are now working with multiplicative relationships.

The strategies "with trial and error" and "with commuting" help to explain how students may be working with the other strategies and do not stand alone. Strategies with dashed outlines are beyond the scope of the work in the What to Look for 2 book, but are evident in student observations.

Teachers can learn more about each strategy by clicking on the links on the PDF below or by visiting the multiplication or division area of this website.

Final continum MD_APRIL 2025.pdf

Tier 2 Reminders for Grade 3 - 8 students in Counting or Early Addition Strategies

In order to help move students to multiplicative reasoning, we need to ensure that they have understandings that help them get over the green line. For students who are working in the counting or early addition phase, students should be completing Tier 2 intervention in addition strategies to help them solidify their understanding and use targeted supports to move them to more efficient strategies. For support in this area, teachers can access information in the Addition/Subtraction area of this website.

Tier 1 Reminders for Students Developing Multiplicative Reasoning

In order to help move students to multiplicative reasoning, we need to ensure that they have understandings that help them get over the green line. Ontario curriculum expectations in Grade 3 and 4 are focused on the development of mathematical reasoning and automaticity of multiplication facts.

Teachers should use the What to Look for 2 book to explicitly teach multiplicative reasoning to students as part of the 60 minutes per week or number sense and are encouraged to seek support from their assigned math facilitator as required.

Important Tier 1 teaching moves in Grade 3 and 4, outlined in What to Look For 2, to support movement over the green line include:

Creating and working with the Intensive Quantity
Working on Fact Fluency
Exploring the Commutative Property
Teaching a working model of multiplication

By the end of Grade 3 - students will have worked to understand and automatize 10x, 2x, 5x, 1x and 0x facts

By the end of Grade 4 - students will have worked to understand and automatize 9x, 3x, 4x, 6x, 7x, 8x, 11x and 12x

Tier 2 Reminders for Grade 5-8 Students Developing Multiplicative Reasoning

Teachers should use the What to Look for 2 Intervention supports and may seek guidance from their assigned math facilitator as required to prepare Tier 2 intervention for students who require support understanding and automatizing their 0x0 to 12x12 multiplication facts.

Tier 2 intervention, to support movement over the green line should include:

Creating and working with the Intensive Quantity
Working on Fact Fluency
Exploring the Commutative Property
Teaching a working model of multiplication

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