Let’s Count On Math™ is a "portable conceptual continuity tool" that gives students a common way to make sense of mathematics and recognize connections across different concepts, strategies, teachers, classrooms, and grades. (Download the summary sheet with working examples at the bottom of the page.)
Over a student’s Kindergarten to Grade 12 school career, mathematics can appear to become an ever-growing collection of concepts, procedures, algorithms, representations, and rules. At the same time, the manipulatives, models, strategies, vocabulary, teachers, and classroom approaches used to explain that mathematics can change from year to year.
Many of those approaches do an excellent job of helping students understand individual mathematical concepts. The challenge is helping students recognize how all those concepts and representations connect both within and through the grades.
Let’s Count On Math™ is an attempt to provide that connection. Rather than introducing another collection of strategies to remember, LCOM gives students a small set of recurring questions they can carry from grade to grade, helping them recognize familiar mathematical structures even as the mathematics, representations, strategies, teachers, and classrooms change.
LCOM can also guide teachers and students by showing when its questions are not the most useful lens. Not every mathematical idea is primarily about quantity or counting. When the questions do not fit naturally, that can help identify what kind of mathematical thinking is actually required and point toward a more appropriate representation or approach.
What are we counting?
Identify the quantities and units involved. Are we counting ones? Tens? Metres? Fifths? Dollars? Negative quantities? xs?
How are we counting?
Are we counting individually, grouping, partitioning, iterating, comparing, scaling, measuring, or relating one quantity to another?
What is this the same as—or analogous to?
Look for mathematical structures and relationships that are already familiar. New mathematics often builds upon ways of thinking students have encountered before.
Sometimes the quantities we want to work with are not yet in a useful or compatible form.
When that happens, ask:
Can we Make Change?
Making Change means regrouping, rewriting, or converting a quantity into an equivalent form without changing its value or meaning.
A ten can become ten ones.
A half can become three sixths.
A dollar can become four quarters.
One hour can become 60 minutes or 3600 seconds.
A metre can become one hundred centimetres.
0.25 can become 25% or 1/4.
The mathematics is different in each situation, but the underlying habit of mind is familiar:
Change how the quantity is represented while preserving what it means.
Very often, "Making Change" is what allows the mathematics to continue, e.g., adding & subtracting decimal numbers, adding & subtracting fractions, BEDMAS, simplifying algebraic expressions, or measurement, time, and currency calculations, etc.
The foundation is surprisingly simple and can even be connected to familiar Sesame Street ideas:
First: Identify like things.
One of these things is not like the others…
Then: Count the things that belong together.
The Count.
Young children learn that cats and dogs are different things. Later, students encounter the same need to distinguish among ones and tens, unlike fractional parts, different measurement units, positive and negative quantities, unlike algebraic terms, different variables, and increasingly sophisticated mathematical expressions.
The representations change. The notation changes. The mathematics becomes more sophisticated.
But some of the underlying ways of thinking remain remarkably consistent.
Vertically Coherent Does Not Always Mean Conceptually Coherent
Most mathematics curricula are carefully designed to be vertically coherent. Ideas are sequenced so that learning in one grade provides a foundation for learning in the next.
But students do not necessarily experience that coherence; especially as they move through the grades and develop uneven understanding of different concepts or skills. From the learner’s perspective, mathematics can still feel like a succession of new topics, rules, representations, and procedures. The connections may exist in the curriculum map without being visible in the student’s understanding.
LCOM is designed to help make those connections explicit—to help students recognize the recurring structures and ways of thinking that remain familiar as the mathematics becomes more sophisticated.
Mathematics educators have developed many effective ways to help students understand mathematics: ten frames, base-ten blocks, number lines, fraction strips, pattern blocks, integer counters, algebra tiles, visual models, rich tasks, discussion, inquiry, and many others.
Let’s Count On Math™ is not intended to replace them.
LCOM is an attempt to connect the ways we already teach for understanding.
A student may encounter different strategies, different representations, different teachers, and different classroom environments from one year to the next. The three questions provide a common conceptual language that can remain familiar through those changes.
The representation can change.
The strategy can change.
The teacher can change.
The classroom can change.
The mathematics is still connected.
The long-term goal is for students themselves to learn to look at unfamiliar mathematics and ask:
What are we counting?
How are we counting?
What is this the same as?
And, when necessary:
Can we Make Change?
The goal is not to add more mathematics for students to remember. It is to help them recognize the structure within the mathematics they are already learning—so that today’s mathematics connects to what they learned yesterday and provides a foundation for what they will learn tomorrow.
Teachers can continue to use their existing resources, instructional approaches, and classroom structures. When planning and teaching, the 3 questions and recurring move help teachers access the meaning within the mathematics. This is the understanding we want students to develop so they know why the steps in an algorithm or process work, and how new mathematics connects to what they already know and understand.
LCOM can also guide teachers by showing when its questions are not the most useful lens. Not every mathematical idea is primarily about quantity or counting. When the questions do not fit naturally, that can help identify what kind of mathematical thinking is actually required and point toward a more appropriate representation or approach.
The questions also help teachers meet students where they are in their mathematical understanding. With 20 to 30 students in a classroom, this can be difficult to achieve. The 3 questions provide common points of focus for questioning, discussion, and diagnosing where a student’s understanding may be breaking down.
From the student perspective, the same 3 questions become a tool for interrogating the mathematics—getting beneath a step, rule, or operation to its underlying meaning. They can also give students a place to begin when they are unsure how to approach a problem.
In talking, thinking, and collaborative classrooms, the 3 questions provide recurring points around which mathematical thinking and conversation can develop. Because the same questions can travel with students from concept to concept and grade to grade, they can also help create greater continuity in how students understand mathematics over time.
Find the Counting. Find the Meaning.
Looking for an overview/summary with some easy examples? Download the LCOM Overview PDF here
YouTube video: Introduction to Let's Count On Math
