Once upon a time, in a classroom not so far away, a young fourth-grade teacher taught her students all about fractions.
The children began by learning the words numerator and denominator (which were practiced by repeating them in their best Arnold Schwarzenegger “The Terminator” voices).
Next, they learned a bunch of definitions that were all startlingly similar, but also very different. Some of these words were “equivalent fraction,” “least common denominator” and “simplest form.”
Next, they memorized set after set of rules and steps about how to add, subtract, multiply and divide fractions. Throughout the day, children were heard calling out things like “Wait, when do I invert and multiply?” or “What do you mean + doesn’t equal ?”
The teacher was weary, but she persevered, and at the end of three weeks it was all worth it — most of the students could do the operations and even answer most of the fractions questions correctly on the test! So they all lived happily ever after, right?
Wait — no they didn’t! As you may have guessed, that was my classroom seven years ago. As we approached the end of the fractions unit, most of my students could do some or most fourth-grade fractions problems.
Despite their success, I knew there was a big problem one day when I had a conversation that went something like this:
Student: “When are we going to work with numbers again in math?”
Me: “Aren’t fractions numbers?”
Class: “No! Fractions aren’t numbers; they’re fractions!”
Somehow in the midst of all of the memorization of steps and vocabulary words, my students had failed to grasp the most fundamental concept about fractions — that a fraction is indeed a number that represents less than a whole.
What had gone wrong? The answer is that, just like AIMS testing, I was more concerned with correct answers than with the process my students were using to learn and understand mathematics.
Arizona’s College and Career Ready Standards have changed this for me, allowing me to re-evaluate and expand the day-to-day instructional methods that I use in my classroom.
My daily instruction is now much more process-based than product-based. It is still important that children strive to get the correct solution, but it is equally as important to me that my students illustrate their thinking and strategies as they work so that we can work together to correct a misconception early on.
For example, instead of introducing fractions with a vocabulary lesson about the terms numerator and denominator and counting shaded parts of a whole, I give students real-life situations where they split brownies among friends.
We then follow this with a deep discussion about how to name the different-sized parts that they created and about why one student could have a different amount of pieces even though they all had the same amount of brownie.
The richness of these conversations, which led students to discover what fractions are and how they can be used in their lives, are far more interactive and productive than listening to a lecture from me about the same topic.
The standards have been essential in giving me the opportunity to deepen student learning and apply my teaching in new ways to make it more relevant and engaging in my students’ lives. I’m excited about the progress my students are making and look forward see them grow even more.