Fractions with PSTs: Celebrate Just Because They Came!

Fraction Adventures: Finding Mathematical Brilliance in Everyday Moments

Today was a great day with my preservice teachers (PSTs)! We explored several mathematical concepts, counted pumpkin seeds, investigated stages of place value understanding, and engaged in a Number Talk using our student-made rekenreks. We were busy doing mathematics!

But the highlight of the day? We celebrated attendance!

Yes, you read that correctly. We celebrated simply because students showed up to class!

Now, I know that sounds a little strange. After all, aren't students supposed to come to class? Absolutely! However, I have been intentional about finding creative ways to encourage attendance while helping my PSTs recognize that mathematics, particularly fractions, is embedded in our everyday experiences.

And what better way to explore fractions than by using the people who actually show up?

Why Fractions? Why Attendance?

If you have been following my Fraction Adventures, you know that I am on a mission to help my PSTs develop a deeper conceptual understanding of fractions. For many future elementary teachers, fractions bring back memories of confusion, frustration, and a collection of rules that never quite made sense.

But fractions are more than numerators, denominators, and procedures. They represent quantities, relationships, and ways of making sense of the world.

Research supports the importance of developing fraction understanding. Siegler and colleagues (2012) found that early fraction knowledge predicts later mathematics achievement, even after accounting for other mathematical knowledge and abilities.

Additionally, Lamon (2012) emphasized that fractions have multiple interconnected meanings, including part-whole relationships, measures, quotients, ratios, and operators.

That is why I am intentionally incorporating multiple representations into our daily classroom routines. Every time a student enters our classroom, we explore attendance fractions using three models:

• Set model: Representing students as members of the whole class.

• Linear model: Using a number line to explore fraction magnitude.

• Area model: Partitioning a whole into equal parts using arrays.

My goal is simple: I want my PSTs to see fractions, reason about fractions, and, most importantly, understand fractions.

The Adventure Begins: Three Out of Fourteen!

When class began, only three of my fourteen students were present.

Three!

Now, I could have been frustrated. Instead, I saw a mathematical opportunity.

I asked, "What fraction of our class is here today?"

Of course, they responded, "Three-fourteenths!"

But you know me. Giving the correct answer is only the beginning. I want to know what they understand.

So, the mathematical adventure began!

Representation 1: The Number Line

First, I asked my PSTs to construct and label a number line representing our entire class.

Where would 3/14 be located? What does that point tell us about our attendance? Are we close to having half the class present?

We discussed the importance of partitioning the interval from zero to one into fourteen equal parts.

I wanted them to recognize that 3/14 is not simply two numbers stacked on top of each other. It represents a quantity with a specific magnitude.

Representation 2: The Set Model

Next, we represented our class using a set model.

My PSTs drew thirteen female figures and one male figure to represent our fourteen students.

And, of course, they placed a crown on the male figure!

I laughed!

Apparently, being the only male in our class comes with royal privileges.

But even in this humorous moment, there was mathematics. We discussed how the entire set represented our class and how the students present represented a fraction of that whole.

Representation 3: The Area Model

Now, this is where things became interesting.

When I asked my PSTs to represent 3/14 using an area model, they were not quite as confident.

Instead of immediately showing them how to construct the model, I asked:

"How can we represent fourteen using an array?"

Silence. Thinking. More thinking.

Then I reminded them of our previous Fraction Adventure, when we used Alabama A&M University's Homecoming football score to explore arrays and area models.

Suddenly, the connections began!

"Two times seven!"

"One times fourteen!"

Yes!

We could create a 2 × 7 array or a 1 × 14 array, partitioning the whole into fourteen equal-sized units. Then, we could shade three of those units to represent the students present.

This was exactly what I wanted: my PSTs connecting prior mathematical experiences to new situations.

Wait! Somebody Else Just Walked In!

Just when we were making sense of 3/14, another student entered the classroom.

And you know what that meant.

Our fraction changed!

We stopped and updated all three representations.

We shaded another unit in our area model, moved to the next fraction on our number line, and circled another figure in our set model.

Then another student arrived.

And another.

Each arrival became a reason to celebrate and another opportunity to do mathematics.

I continued asking questions:

"How far are we from having half the class present?"

"How many more students would need to arrive for everyone to be here?"

"Are we closer to zero, one-half, or one whole?"

"How do you know?"

Notice what we were NOT doing.

We were not finding common denominators. We were not converting fractions into decimals. We were not memorizing procedures.

We were reasoning about fraction magnitude, using benchmarks, and making sense of quantities.

And guess what?

By the end of our attendance adventure, nine of our fourteen students had arrived!

Nine-fourteenths!

That was something to celebrate!

We discussed why 9/14 is greater than one-half, how much of the class was still absent, and how our representations helped us make sense of the quantity.

The students could see that 7/14 represented exactly half the class. Since nine students were present, we were two students beyond that halfway point.

That is fraction reasoning!

The Missed Opportunity: Place Value and Fractions

Now, here comes my confession.

As I sat down to write this blog, I realized something.

I missed a wonderful mathematical opportunity!

Earlier in class, we had explored place value concepts. We discussed how students develop an understanding of units, tens, and hundreds.

But I did not explicitly connect our place value discussion to fractions.

Think about it!

A single unit cube represents one-tenth of a tens rod.

That same unit cube represents one-hundredth of a hundreds flat.

And a tens rod represents one-tenth of a hundreds flat.

The value of the unit depends on what we define as the whole.

This is a powerful connection!

Just as our attendance fraction depended on defining the entire class of fourteen students as one whole, our base-ten representations depend on identifying the whole.

What if I had asked my PSTs:

"If the hundreds flat represents one whole, what fraction does one tens rod represent?"

Or:

"If a tens rod represents one whole, what fraction does one unit cube represent?"

Imagine the conversations we could have had about unitizing, partitioning, and the multiplicative relationships embedded in our base-ten system.

We could have explored how fractions are connected to place value rather than treating these as isolated mathematical topics.

Oh, the possibilities!

But that is one of the beautiful things about teaching. Sometimes our best instructional ideas emerge after the lesson has ended.

And that is okay.

There is always another class and another opportunity to make mathematical connections.

My Reflection: Celebrating More Than Attendance

As I reflected on today's class, I realized that we were celebrating much more than attendance.

We were celebrating mathematical thinking.

We were celebrating students making connections between representations.

We were celebrating moments when uncertainty turned into understanding.

And perhaps most importantly, we were creating a classroom environment where mathematics was something to explore rather than something to fear.

My PSTs are learning that fractions are not confined to a textbook or worksheet. Fractions are present in our classrooms, our daily routines, our experiences, and even our attendance records.

As a mathematics teacher educator, I want my future teachers to carry this understanding into their own elementary classrooms.

I want them to look at everyday situations and ask, "Where is the mathematics?"

I want them to recognize that a simple attendance routine can become an opportunity to explore mathematical relationships, encourage discourse, and deepen conceptual understanding.

And I want them to understand that meaningful mathematics instruction does not always require an elaborate lesson or expensive materials.

Sometimes, all you need is a number line, an array, a few stick figures, and students walking through the door.

So yes, we celebrated today—just because they came!

And along the way, we discovered a little more mathematical brilliance.

A Question for My Fellow Educators

How might you transform an everyday classroom routine into an opportunity for students to reason mathematically?

And what mathematical connections have you discovered after a lesson that you wish you had explored during instruction?

I would love to hear your thoughts!

---

References

Lamon, S. J. (2012). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers (3rd ed.). Routledge.

Siegler, R. S., Duncan, G. J., Davis-Kean, P. E., Duckworth, K., Claessens, A., Engel, M., Susperreguy, M. I., & Chen, M. (2012). Early predictors of high school mathematics achievement. Psychological Science, 23(7), 691–697. https://doi.org/10.1177/0956797612440101