Carole Fullerton on...Numeracy in Elementary: What Matters Most
Numeracy experiences in the elementary years lay the foundation for so much of what propels students towards success: the ability to think, to reason, to communicate and to persevere.
Numeracy understandings grow naturally from students’ explorations. Their spontaneous play is overwhelmingly numeracy-based: counting, sorting, measuring and comparing make up a large part of what young children choose to do both in and outside of the classroom. As students age and develop in their thinking, they likewise develop more sophisticated questions and are able to pursue richer, more complex problems and situations. Take advantage of their natural curiosity — and set high expectations for mathematical reasoning.
In the primary years, teachers have the opportunity to encourage and take promote this naturally occurring play — to “mathematize” the explorations of number, measurement and pattern that students willingly seek out in their centre time, in their outdoor play and in their interactions with other children. And, because they are play- based, these experiences are authentic and positioned at students’ developmental level by default.
In intermediate, students crave — and thrive on — rich problems, ones with more than one right answer and more than one way to solve them. Sharing and comparing strategies make up an essential part of students’ mathematical conversations, and students grow in their capacity to reason as these conversations take place. It’s crucial to root these numeracy talks in important mathematics (multiplicative thinking, proportional reasoning, algebraic thinking) in order to build up students’ mathematical capacity and competence.
Mathematical Competence
The goal of mathematics education is to develop mathematical competence in all learners. Mathematical competence is the combination of:
conceptual understanding
procedural fluency
strategic competence
adaptive reasoning and
productive dispositions
None of these in isolation is sufficient. Rather, these aspects combine to create a thought-full mathematician who can approach a problem with confidence, draw from a range strategies and assess the best one for the task, and fluently apply the skills necessary to promote and hone conceptual understanding. We all know a student who embodies these aspects. They seem naturally predisposed to mathematical thought, fluently moving between strategies and confidently exploring ideas. By providing quality materials, structuring tasks embedded in the big math ideas, asking good questions and allowing time for productive struggle we can build mathematical competence in all learners.
The Big Math Ideas
The important things to know in elementary mathematics are common across grades. This allows us as teachers to structure tasks and lessons that include all learners in meaningful explorations of numeracy whether we are teaching in a straight grade or in a combined grades classroom, at primary or intermediate / middle.
Number sense is the biggest concept. It encompasses an intuitive sense of number, the ability to estimate reasonably using referents, to describe and make connections to how quantity is used in the world. Number sense includes ideas of place value — that we can think about quantities in groups of ten and some more — and that we can apply these ideas to the operations. Number sense extends to students’ ability to think multiplicatively and proportionally - to compare parts and wholes and to see relationships within and between sets.
Students with number sense understand that addition is the act of joining and subtraction is the act of removal. Students with number sense can apply whole number strategies for the operations. They understand that when we measure, we compare — and that we can measure the attributes of length, height, area, mass, capacity and volume. Likewise, students with number sense at the intermediate level understand that multiplication can be expressed in equal groups of or equal rows of a quantity. They see patterns in the multiples and can use them to make connections to division. They use these ideas to predict and make generalizations; an important concept in algebraic reasoning.
Patterns repeat. When we make a pattern, we sort the elements by their attributes and then build a pattern core over and over again. We can predict what comes next by describing the pattern core. Increasing patterns grow. They grow in predictable ways. This is the foundation for skip counting and multiplicative thinking. Growing patterns (built with materials to start and then formalized in T-charts and in generalized algebraic forms, then graphed) likewise support the understanding of the rate of growth and give our learners the opportunity to connect important concrete, pictorial and abstract concepts that mathematicians know are essential for algebraic reasoning.
Promoting Mathematical Competence in Elementary
Constructing group tasks around these essential Big Math Ideas allows everyone in the classroom community to explore and access what’s important to know. Having students share their thinking in a group debrief promotes communication and allows learners to hear a variety of approaches. We are smarter when we learn together!
Daily, Focused, Uninterrupted Math
Do math every day — and be sure that your students are the ones engaged in thinking. Students should count and sort and build and compare and measure and problem- solve every day. They should grapple with concepts and explore different strategies. They should model and solve problems at their grade level and compare their thinking with their peers.
As we know from our literacy practice, our learners need a broad range of experiences in order to improve — working alone, with a partner, in a whole group, on open and guided tasks. They need exposure to and practice with concepts and skills, strategies and approaches and ways to communicate their thinking. Unless students are actually doing math, counting and sorting and measuring and operating and grappling with problematic situations, they will not develop in their capacity as mathematicians.
We learn to read by reading. We learn to “math” by “math-ing”. Let them loose! Trust their capacity — and set high expectations.
Quality Tools
It is important that students have materials they can use to model important mathematics. The best classroom materials allow for explorations of a range of mathematical concepts. It’s important to have high quality manipulatives like Cuisenaire rods available to ensure that the mathematics is accessible and clear, particularly as the concepts become more complex.
Good Questions
When presenting tasks to the group, be sure to pose mathematical problems worth solving. Ask:
How can you…?
What could it be…?
How many ways…?
What can you find out about…?
What do you notice…?
These prompts allow for a range of answers and multiple strategies. They will include learners across the continuum in exploring important mathematics. Celebrate the diversity of their responses and strategies. Support students by recording their thinking in mathematically appropriate ways and using mathematical terminology to name their ideas.
Productive struggle
As teachers we sometimes worry about students as they struggle to work through a problem. Our inclination is to jump in and “rescue” a learner to save them from feeling incompetent. Know that this space of struggle is actually where the learning happens!
Where possible, resist the temptation to interrupt student thinking with an over-eager prompt or a quick solution. Give students think time — at least 10 seconds — to allow them to process and consider options. Resilience grows from productive struggle: the ability to sit with uncertainty, to calm the mind, make connections to prior knowledge and to find and apply a strategy to a problem. Mathematical competence depends on grit and resilience. If we solve students’ problems for them before they have had the chance to think things through, we promote dependence on us as teachers.
Encourage students to persevere and to make connections by asking:
What are you doing? Why are you doing it? How does it help you?
What does this remind you of?
What else have you done that’s like this?
What tools would help?
Or simply say, “Hm. I wonder…” and walk away, leaving the math in the hands of the learner. It’s a powerful reminder of their own efficacy and of your belief in their capacity. And what’s more important that that?