One experience has stayed with me throughout my teaching career.
A very strong mathematics student moved to our school in Grade 10 from another high school. She continued to excel academically and eventually became one of my strongest students in Math 12.
One day, while we were graphing sine functions, she came to me after class and quietly said,
“Mr. Leung, I didn’t understand what we did today.”
I was surprised.
This was not a student who usually struggled with mathematics.
As we talked, I tried to determine where the misunderstanding was. I asked whether she understood the special angle values. She did.
I asked whether she understood how we generated the ordered pairs. She did.
Eventually I discovered the real problem.
She didn’t understand what I meant when I was marking the horizontal axis in fractions of π.
She knew that π represented approximately 3.14, but she didn’t understand what values such as π/6, π/4, π/3, or 5π/6 actually represented. She had learned to perform fraction calculations such as 3/4 + 5/6 successfully, but she had never developed a meaningful understanding of what those fractions represented as quantities.
The difficulty wasn’t trigonometry.
The difficulty was with counting.
I took out a set of pattern blocks and began representing the fractional parts visually. We talked about what it actually meant to count halves, thirds, fourths, and sixths. We related those same fractional quantities to portions of π around the unit circle.
As we worked through the examples, I could literally see the moment when everything connected.
Her eyes became wide and she looked at me and said,
“That’s what that means!”
In that instant, she wasn’t learning a new trigonometry concept.
She was finally understanding a fraction concept that had remained procedural for years.
That experience had a profound impact on me.
It made me wonder how many students successfully learn procedures without ever fully understanding the quantities those procedures represent. It also reinforced my belief that when students first identify what they are counting, many mathematical ideas become much easier to understand.
That conversation became one of the many experiences that eventually led to the development of Let’s Count On Math™.
During my years teaching Grade 12 mathematics, I began noticing that students often struggled with new topics because they saw each one as a completely different collection of symbols, rules, and procedures.
During one trigonometry unit, while we were simplifying trigonometric expressions, I stopped the lesson and asked the class a question that had nothing to do with Grade 12 mathematics.
“Do you remember the old Sesame Street song, ‘One of these things is not like the others’?”
Almost every student smiled.
Then I asked, “Do you remember The Count?”
Now they laughed.
I explained that Sesame Street had taught them two very important mathematical ideas many years earlier.
First, identify which things are alike and which are different.
Then count the things that belong together.
That is also what we do when we simplify mathematical expressions.
Before students can combine anything, they must first ask:
What kind of quantity am I looking at?
We combine x-terms with x-terms.
We combine x²-terms with x²-terms.
We combine matching trigonometric terms with matching trigonometric terms.
The first task is to recognize which quantities belong together. Once that has been established, “The Count” takes over and we count how many of each kind we have.
As we discussed this, I began connecting the same idea to mathematics they had learned earlier in high school.
Fraction arithmetic follows the same structure. We cannot add thirds and fourths directly because they are different kinds of fractional parts. We first make change by rewriting them as equivalent fractions with a common denominator. Once they are the same kind of part, we count the numerators.
The same thinking appears in algebra when combining like terms, in exponent work when counting repeated factors, and in many of the other procedures students had already learned.
The symbols and vocabulary changed, but the underlying mathematical structure was often familiar.
Many students would respond:
“Why didn’t they tell us this earlier? It would have made things so much easier!”
They were not saying that all mathematics had suddenly become easy.
They were saying that it finally made sense.
Those comments stayed with me. They reinforced my belief that students should not have to experience mathematics as hundreds of unrelated topics. When we help them recognize familiar structures beneath new symbols, they can build new learning on ideas they already understand.
That experience became one of the ideas that eventually led to the development of Let’s Count On Math™:
Help students Find the Counting… so they can Find the Meaning.
In June 2026, I was covering a Grade 3 classroom while the teacher went to participate in collaboration time. I had the opportunity to teach a math lesson!
I began with a set of instructional dice that I had designed. Instead of numbers, the faces showed cats, dogs, and ants.
The activity was simple.
Students rolled the dice, sorted them, and counted the results.
Cats counted with cats.
Dogs counted with dogs.
Ants counted with ants.
No one had to explain the rules. The students immediately understood that unlike things cannot simply be counted together.
After they had become comfortable with the activity, I quietly changed one thing.
I put the animal dice away and brought out another set of dice.
This time the faces showed x², x, and 1.
I didn’t change the instructions. I simply asked them to do exactly what they had just been doing.
Without hesitation, the students grouped the x²’s together, the x’s together, and the 1’s together.
Then we moved one step further.
I wrote symbolic algebraic expressions on the board.
Again, I asked the same question:
“What are we counting?”
The students looked at the expressions and began identifying the like terms.
They grouped and counted the x² terms together.
They grouped and counted the x terms together.
They grouped and counted the constants together.
Although these Grade 3 students had never been formally taught algebra, they understood the extension from the algebra dice to the written symbolic expressions because the underlying thinking had not changed. They also recognized that adding x with x did not make it into an x²
They weren’t trying to memorize an algebra rule.
They were simply applying something they already knew:
Like quantities count together.
Watching that lesson unfold reinforced something I had been wondering about for years.
Perhaps many of the foundations of later mathematics can be developed long before students encounter the formal notation. Instead of introducing algebra as a completely new topic, we can begin by helping students recognize, group, and count like quantities using concrete materials and familiar contexts.
The symbols become more sophisticated.
The mathematical thinking remains remarkably similar.
That Grade 3 lesson became another important step in the development of Let’s Count On Math™.
It reminded me that students are often capable of far more than we expect—not because they have learned advanced mathematics, but because they already understand the underlying structure on which that mathematics is built.
Help students Find the Counting … so they can Find the Meaning.
In May 2026, I was filling in as a support principal at a school. I was invited to teach a 40-minute mathematics lesson to a Grade 6/7 class.
Rather than teaching a single curriculum topic, I wanted to see whether students could begin viewing mathematics through a different lens—one that focused on identifying the underlying structure that connects many mathematical ideas.
I began with something very simple.
Using small toy animal figures that I had purchased from Amazon, I asked students to sort and count them.
Dogs counted with dogs.
Cats counted with cats.
Ants counted with ants.
Coloured turtles.
How many legs? How many shells?, etc.
The students quickly recognized that unlike things cannot simply be counted together. Before different quantities can be combined, they first have to become compatible - depending on the question being asked.
That naturally led into the next idea—what I now call Making Change.
Using a simple grocery shopping example, we discussed exchanging one five dollar bill for five one-dollar coins so we can get a shopping cart. The value remained exactly the same, but the quantity and representation changed to make it useful.
I explained that mathematics works the same way.
Fractions are rewritten using common denominators.
Measurements are converted into common units.
Whole numbers are regrouped.
Algebraic expressions are rewritten into equivalent forms.
Very often, making change is what makes counting possible.
From there, we explored several mathematics topics that students would normally think of as completely unrelated.
We talked about BEDMAS, not simply as a rule to memorize, but as a sequence of mathematical changes that eventually produces quantities that can be counted together. Multiplication changes groups into individual quantities before addition combines them.
We looked at adding fractions, where unlike fractional parts must first be rewritten as like parts before they can be counted.
We simplified algebraic expressions, recognizing that x-terms count with x-terms, x²-terms count with x²-terms, and unlike terms cannot simply be combined. We simplified logarithmic expressions even though they had no idea what "logx" meant.
We discussed area and perimeter, emphasizing that they involve counting different kinds of quantities—square units for area and linear units for perimeter.
Throughout the lesson I kept returning to the same questions:
What are we counting?
How are we counting?
What is this analogous to?
By the end of the lesson, students were beginning to recognize that these ideas were not isolated procedures. They were seeing common mathematical structures appearing across topics they had previously thought were unrelated.
After the lesson, the classroom teacher told me she had never seen those mathematical concepts connected together in that way before. She said she had been writing notes as quickly as she could because she wanted to use many of the ideas in her own classroom.
During the lesson, I had also noticed the school’s educational assistant quietly nodding several times. At one point, a student asked her why she kept nodding. She smiled and replied,
“He’s right. I wasn’t taught mathematics this way either, and this totally makes sense.”
The most unexpected feedback came later that day.
One of the students went home and enthusiastically explained the lesson to their parents. The parent was so impressed by both their child’s excitement and their understanding that they sent an email to the school principal expressing their appreciation for the lesson.
That email meant a great deal to me.
Not because someone was complimenting my teaching.
But because a student had gone home wanting to talk about mathematics.
One lesson cannot demonstrate the long-term impact of a conceptual framework. Deep mathematical understanding develops over many years.
However, that afternoon reinforced something I had been observing throughout my career. When students are encouraged to look beyond procedures and recognize the underlying structure of mathematics, they become more engaged, ask better questions, and begin making connections for themselves.
For me, the most encouraging part of the day was not that students understood one lesson. It was that the same way of thinking helped explain ideas from arithmetic, fractions, algebra, measurement, and order of operations—all within a single class.
That experience strengthened my belief that many mathematical concepts are far more connected than we typically teach them to be, and it became another important step in the continuing development of Let’s Count On Math™.
Help students Find the Counting … so they can Find the Meaning.
