I am a product of 1980s schooling.
If you grew up around the same time, you probably remember the famous anti-drug public service announcement. A stern-looking man holds up an egg and announces:
“This is your brain.”
Then he cracks it into a sizzling frying pan.
“This is your brain on drugs.”
And after allowing us to contemplate the unfortunate fate of the egg, he looks back into the camera and asks:
“Any questions?”
Then there was the other message seemingly everywhere during the decade—courtesy of Nancy Reagan:
“Just Say No!”
I didn't particularly like eggs, and I wasn't interested in drugs, so I'm not sure either campaign deserves much credit for keeping me on the straight and narrow. I had already said no to both.
But I understood the message.
Some choices can have consequences that aren't immediately obvious.
After more than thirty years of teaching mathematics, I find myself asking a similar question about my own profession:
Are there things we do in mathematics classrooms that help students get an answer today but actually make it harder for them to think mathematically tomorrow?
I think there are.
And there is one practice in particular that I think deserves its own “Just Say No” campaign:
Teaching mathematics through tricks.
One of the most common examples?
Keep. Change. Flip. (Or, the “Kentucky Fried Chicken” Method—true story shared by one of my students!)
Okay, before you close the browser or start composing an angry email to the math professor, give me a chance to make my case.
The Problem Isn't That It Works
Let's start with something important:
Keep, Change, Flip works!
If students are asked to calculate
6 ÷ 3/4
they can keep the first number (6), change division to multiplication (6), flip the second number (4/3), and get the correct answer (6 × 4/3 = 24/3 = 8).
That's exactly why the shortcut is so appealing.
It's quick/
It's easy to remember.
It produces correct answers.
And I have taught shortcuts like this myself.
The problem isn't that Keep, Change, Flip doesn't work.
The problem is what students may never learn because it works so well.
I see the consequences regularly.
I teach mathematics content courses for future elementary, middle school, and secondary teachers. I also teach our university's lowest-level college mathematics course. Nearly all of those students have successfully completed Algebra II or a higher mathematics course in high school.
I've also conducted research with students in grades 4 through 7 examining their understanding of fractions.
Across all of those settings, I have asked questions such as:
What does 6 ÷ 3/4 actually mean?
Or, giving the problem some context.
A cookie recipe calls for 3 /4 of a cup of sugar. You have 6 cups of sugar. How many batches of cookies—including partial batches—could you make?
These questions can make otherwise successful mathematics students surprisingly uncomfortable.
Some immediately reach for a calculator.
Some stare at the numbers.
And eventually someone will ask:
“Is this where I use Keep, Change, Flip?”
That question fascinates me.
The student remembers what to do.
But the student may have no idea what the mathematics means.
And those are not the same thing.
An Interesting Difference I've Noticed
Over the years, I've also had the opportunity to teach students educated in South Korea, Taiwan, China, Oman, Germany, Greece, and many other countries.
One thing has always intrigued me.
Many of those students have never heard the phrase “Keep, Change, Flip.”
By contrast, almost all of my U.S.-educated students know the phrase immediately.
But when I ask:
Why can we flip the second fraction?
or
What does dividing by 3/4 actually mean?
the room often becomes much quieter.
That should make us curious.
Our answer-driven system understandably values students getting correct answers. And Keep, Change, Flip is extraordinarily efficient at producing correct answers on pages filled with fraction-division exercises.
But getting answers cannot be the finish line.
Our goal should be to help students become thinkers, not simply answer-getters.
A shortcut can be useful once students understand the mathematics.
But when the shortcut replaces the mathematics, students can become dependent upon it.
They begin to experience mathematics as a collection of mysterious commands:
When you see this, do that.
Move this over there.
Cross these out.
Flip that.
Don't worry about why.
That isn't the mathematical experience I want for students.
So What Should We Do Instead?
If we are going to say NO to Keep, Change, Flip as the starting point for fraction division, we have to replace it with something better.
For me, that means helping students develop three things:
A way to “see” division.
Language for talking about division.
An algorithm that connects to what they are seeing and saying.
One resource that has greatly influenced my thinking is Sue Empson and Linda Levi's Extending Children's Mathematics: Fractions and Decimals.
Let's look at what an alternative approach might actually look like.
First, Let Students See the Division
Consider the contextual problem from before:
Gwen has 6 yards of ribbon to make bows. Each bow requires ¾ of a yard of ribbon. How many bows can Gwen make before she runs out of ribbon?
Before thinking about an algorithm, stop and picture the situation.
You have 6 yards of ribbon.
Each bow uses 3/4 of a yard.
The question is really:
How many groups of 3/4 are contained in 6 whole?
Imagine representing the six yards.
Each yard could be partitioned into fourths.
The 6 whole has been transformed, or equally exchanged, into 24 fourths.
Now instead of asking how many groups of ¾ (3 fourths) are contained in 6, we can ask:
How many groups of 3 fourths can be made from 24 fourths?
That is a question students can see.
24 fourths grouped 3fourths at a time produces 8 groups of 3 fourths. (3 fourths is our unit formed from 3 groups of 1 fourth that are formed by partitioning the whole unit into fourths.)
So:
6 ÷ 3⁄4 is analogous to solving the division problems 24/4 ÷ 3/4 or 24 ÷ 3
The answer 8 isn't merely the result of manipulating symbols.
It means something.
8 groups of 3/4 (3 fourth) of a yard is the same length as 6 yards (24 fourths is 6 ones).
That is mathematics students can reason about. These actions are shown in the figure below.
“Seeing” 8 groups of 3 fourths as the same length as 24 fourths or 6 ones.
Then, Give Students the Language
The language we use matters enormously.
For the symbolic expression I want my students to have the language so they can say what the division question is asking.
6 ÷ 3⁄4,
That is, from a measurement perspective of division,
“How many groups of three-fourths are contained in six whole (ones)?”
That simple sentence gives students a tool they can use long after they have forgotten a mnemonic.
Consider another problem :
4 1⁄2 ÷ 1⁄4.
Instead of reaching automatically for an algorithm, a student can ask:
“How many one-fourths are in four and one-half?”
Now mental reasoning becomes possible.
For example, there are four fourths in every whole.
Four wholes contain sixteen fourths, and another half contains two more.
So there are 18 fourths.
Or consider:
6 ÷ 0.01.
The same language tells us:
“How many hundredths are contained in six?”
If we think about money:
How many pennies are in six dollars?
Six hundred.
No Keep, Change, Flip required.
The language gives students access to the mathematics.
It also helps them estimate.
Consider:
1⁄4 ÷ 1⁄3.
Before calculating anything, a student can reason:
“How many groups of one-third fit inside one-fourth?”
Since 1/4 is smaller than 1/3, we know immediately that the answer must be less than one.
That kind of reasoning is nearly impossible when students' only question is:
“Which number do I flip?”
What About an Algorithm?
Students absolutely should learn efficient algorithms.
Understanding mathematics does not mean abandoning efficiency.
But I would argue that the algorithm should grow out of ideas students already understand.
One possibility is sometimes called the common-denominator algorithm.
Return to:
6 ÷ 3⁄4.
Because 6 ones can be equally exchanged for 24 fourths we can rewrite the problem as…
24⁄4 ÷ 3⁄4.
Now the units are the same.
The question becomes:
How many groups of 3 fourths can be made from 24 fourths?
Which is analogous to to the whole number division problem: 24 ÷ 3 = 8.
The quantities are the same, the actions are the same, the only difference is the size of the unit.
Students can see how fraction division connects directly to whole-number division.
Shortcuts Should Follow Understanding
This is really the heart of my concern.
I am not opposed to shortcuts.
Mathematics is filled with shortcuts. In fact, mathematical notation itself is a wonderfully efficient shortcut.
And there may be contexts later in students' mathematical development—such as manipulating complicated rational expressions—where multiplying by a reciprocal is exactly the efficient move we want them to make.
But there is an enormous difference between a shortcut students use because they understand the mathematics and a shortcut students use instead of understanding the mathematics.
I recently asked a class of approximately fifty college students:
“Why does Keep, Change, Flip work?”
Not one student could explain why dividing by a number is equivalent to multiplying by its reciprocal or connect the procedure to the meaning of division.
But almost every student knew the rule.
Think about that.
They had retained the trick.
They had lost the mathematics.
Let’s approach the “why” from two different perspectives: One a little more visual that relies on proportion reasoning and another that is guided by a mathematical principle.
How many 3/4s are there in each whole unit? A visual to help answer this question is shown below.
There are 1 1/3 or 4/3 units of 3/4 in each 1 whole. If we are wanting to know how many 3/4s there are in 6 whole then we multiply by 6, so 6 × 4/3 units of 3/4 or 24/3 units of 3/4 or 8 units of 3/4. Think about the relationship between the numbers 4/3 and 3/4. They are reciprocals because 4/3 × 3/4 = 1.
That leads into the more formal mathematical way. I often tell my students that division is the same as multiplying by the reciprocal of a number. So, keep change, flip is about thinking about division in term of multiplication. That is, the division problem 6 ÷ 3⁄4 can be thought of as 6 × 4/3.
A colleague once described the concern to me this way:
“When we give students Keep, Change, Flip before they understand fraction division, we may unintentionally communicate that there is nothing else left to think about.”
I think that's exactly right.
And it leads to a principle I try to remember in my own teaching:
A shortcut that follows understanding can create efficiency.
A shortcut that replaces understanding creates dependency.
Revisting an earlier point, I am often asked by teachers if there is a time when “Keep, Change, Flip” is helpful to know. I would argue that it becomes very helpful when solving geometry problems involving similar triangles and when simplifying rational expressions involving division. Those are both topics typically taught in high school classrooms. As for elementary classrooms, as well as middle school classrooms, I do not see a need to introduce “Keep, Change, Flip”.
Maybe It's Time to “Just Say No” in Elementary Classrooms
So yes, I am a child of the 1980s.
And apparently those public service announcements stuck with me more than I realized.
I am not quite ready to crack an egg into a frying pan while shouting:
“This is your brain on Keep, Change, Flip!”
Although I admit the visual would probably get some attention at a professional-learning workshop.
But I am ready to ask us to reconsider what we are giving up when we teach tricks before understanding.
There is so much more we want children to know about division than how to produce an answer.
We want them to be able to see the mathematics.
We want them to be able to say what the mathematics means.
And then we want the symbols and procedures they use to represent ideas they actually understand.
So perhaps the message isn't simply:
“Just Say No to Keep, Change, Flip.”
Maybe the better message is:
Say yes to understanding first.
Because our students deserve more than a trick for getting the right answer.
They deserve mathematics that makes sense.
Want to explore these ideas with your teachers?
Looking for professional learning that helps teachers replace procedural shortcuts with meaningful mathematical reasoning? Explore Professional Learning
You can also explore my book See it, Say it, Symbolize it to learn more about superpower understandings such as equal exchanges analogous problems.
