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Overview Student Continua of Numeracy Development
Overview Student Continua of Numeracy Development
Dr. Alex Lawson, Professor Emeritus from the Faculty of Education at Lakehead University, wrote What to Look For: Understanding and Developing Student Thinking in Early Numeracy (2015) to support educators in creating a better understanding of students' mathematical development. As part of this project, she created two continua of numeracy development over time, one for addition and subtraction and one for multiplication and division. This site focuses on the addition and subtraction continuum.
Within the continuum, Dr. Lawson and her research team identified and described the mathematical strategies – the actions, words, or written work students use to calculate or solve problems. They also identified the key ideas that students showed or were building as they used specific strategies. When a student solves a problem or does a calculation, you will be able to see and hear the student use a strategy. The student is also demonstrating the use of knowledge that you cannot see, a mathematical key idea. Key ideas are mathematically powerful concepts that children construct as they work with different strategies. They are something that educators must infer from the strategies we see students using (Lawson, pg. 3).
We have adapted the continuum to also include counting principles, which are fundamental skills that students develop prior to developing addition and subtraction strategies.
For more information, we encourage you to:attend the professional learning sessions on Building Foundational Math Skills in the Early Years, check out the book from the TDSB Professional Library, or read the article, The Mathematical Territory Between Direct Modelling and Proficiency.
The Mathematical Territory Between Direct Modelling and Proficiency.pdfInteractive Student Continuum of Numeracy Development: Addition and Subtraction
Interactive Student Continuum of Numeracy Development: Addition and Subtraction
Click on one of the strategies or key ideas below to learn more about how students demonstrate the strategy and what to look for.
Using the Continuum & Online Professional Learning Resource
Using the Continuum & Online Professional Learning Resource
The strategies in the continuum are grouped into five phases of development that become increasingly sophisticated.
Counting Principles (Pre-Addition Strategies)
Direct Modelling and Counting
Counting More Efficiently and Tracking
Working with the Numbers
Proficiency
The counting principles are precursors to addition and subtraction. These include the basic components of counting.
In direct modelling and counting, students fully represent the problem with objects, then count the objects to find a solution (Lawson, pg. 6)
When students transition to counting more efficiently and tracking, they may still use their fingers or marks on the page, but they are using them to track their mental count (Lawson, pg. 6).
In working with the numbers, students are no longer counting or tracking. Instead, they are operating on or with the numbers (Lawson, pg. 6).
When students reach proficiency, they are able to demonstrate conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition.
While the strategies are organized from the earliest strategies on the left to the more efficient and complex strategies on the right, a student's overall progression is not necessarily linear. Students can move forward and backward depending on the problem type and numbers used. Students will also develop different paths towards proficiency. They do not need to develop and apply every strategy.
The key ideas appear as bands across the bottom of the continuum. Each band spans a range of strategies, showing when students first begin to develop the key idea through to when they have a solid understanding of it (Lawson, pg. 3).
Educators can click the links for each strategy on the interactive continuum to learn more about it, and view some suggestions of games and activities to support students in practicing the strategies.
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