Drugs and "Keep, Change, Flip"--Just Say No!

I am a product of 80s schooling.  A popular public service announcement during that time was a man with a perturbed look on his face, arms crossed, asking the viewer, “Is there still anyone out there who still doesn’t understand what doing drugs does?” Holding an egg, he exclaims “this is your brain,” cracking it up against the side of a heated cast iron skillet he says, “This is your brain on drugs” as the sound of the sizzling egg plays in the background. The ad finishes by him asking, “Any questions?”. The motto we rehearsed in school, thank you Nancy Reagan, was “Just Say No!”.

I didn’t like eggs or drugs, so I am not sure the ad had much impact on me--I had already said “No”! However, I did get the gist. If you take drugs you are going to do bad things to your brain like impede mental growth, stunt brain development, lower cognitive functioning, etc. As a thirty-year teacher I often reflect on the question, “Are their current practices in math classrooms, as well as my own, that are deadening minds and stunting development?  Yeah, I know, drugs and mathematics teaching practices, a tough analogy. I believe one such of those practices that we need to “Just Say No!” is tricks to teach mathematics. A prevalent one is the use of the phrase “Keep, Change, Flip” to teach division of fractions. Okay, all I ask is that you hear me out.  

I witness on a regular basis the intellectual residue of students who have been raised on math tricks such as “Keep, Change, Flip” to learn mathematics. I teach mathematics content classes to future elementary, middle, and secondary mathematics teachers. I also teach the lowest college-level developmental mathematics course at our university--a course in which 95% of students have successfully completed Algebra 2 or above in high school. I also have conducted research in 4th through 7th grade classrooms related to understanding of fractions. I often ask these students, including preservice teachers,  to explain the meaning of 3/4÷1/3 or solve a word problem such as, “A cookie recipe calls for 3/4 of a cup of sugar and you have 6 cups of sugar, how many batches, including partial batches, of the cookie recipe, can you make?”.  Questions like these throw many students into a panic. They often reach for a calculator, or simply state, “Is this where I use, ‘Keep, Change, Flip’?” I have also had the opportunity to teach students from South Korea, Taiwan, China, Oman, Germany, Greece, as well as many others. The one constant is that most of the students from these countries have never heard of “Keep, Change, Flip”. Many U.S. students know “Keep, Change, Flip”, but have no idea mathematically why it makes sense to even perform that operation. This is an American-based shortcut that yields correct answers to division number sentences which are prevalent in our assessment-driven culture. However, a shortcut, like the 80s claim about drugs,  that promotes intellectual neuropathy. 

Now there might be a place for “Keep, Change, Flip” in simplifying an algebraic expression such as 

 but, I am becoming more and more convinced that it does not have a place in elementary classrooms. You are welcome to try to convince me otherwise, but at least give me a chance to make my argument.  I myself have done this and I agree that “Keep, Change, Flip” is an easy, efficient way to get an answer to a school-based problem. I am now sitting on the other end witnessing the unintended consequences of this narrow view of mathematics. Our educational goals should push students beyond the “quick high” of just getting a correct answer.

What is the alternative to “Keep, Change, Flip”? Provide students with the tools that they will need to reason about division problems in the real-world, the language necessary to make meaningful interpretations about any division problem and learn an algorithm that connects to whole number division. The book Extending Children’s Mathematics: Fractions and Decimals by Sue Empson and Linda Levi is a great resource for anyone desiring to learn more about a different view of division of fractions. 

Take a moment to think about the solving the problem, Gwen has 6 yards of ribbon to make bows. If each bow takes 3/4 of a yard of ribbon, how many bows can Gwen makes before she runs out of ribbon. In fact, can you create a picture that would represent the situation? Before we formalize the notation that cues dividing fractions it is important that students intuitively experience the actions associated with the process of division. I would venture that in modeling this solution you created 6 whole objects to represent the 6 yards of ribbon, partitioned each whole object into 4 equal pieces (i.e., fourths), and counted sets of 3 of those pieces (i.e., 3 fourths). In general, we want students to connect the process of partitioning a whole unit into n equal parts with the size of a part, 1/n, that results. And, that they can combine unit fractions to make fractions that are multiples of unit fractions (p. 74, Empson & Levi, 2011). In the ribbon problem, the answer of 8 means there are 8 groups of 3/4 of a yard in 6 whole yards.

Students must also develop the language associated with a symbolic division problem. For example, the above problem 6÷3/4 is asking, consistent with the visual model, “How many groups of 3/4 can be created from 6 whole units?” The first number in a division number sentence signifies the whole and the second number, using a measurement perspective, signifies the size of the group. Many of the college level students I teach do not have a language beyond “Keep, Change, Flip”. This is very limiting to their intellectual development because it does not provide them with the tools to use mental arithmetic to solve problems nor the means to estimate or check the reasonableness of their answer. Consider how a student empowered with this language might reason about problems such as 4 1/2÷1/4  or 6÷0.01.  For example, 4 1/2÷1/4 is asking how many 1/4s are in 41/2?, and, 6÷0.01 could be interpreted as, how many pennies in $6?  And, how they might estimate the answer to a problem such as 1/4÷1/3. That is, that the answer is less than one because the whole is smaller than the size of the group. 

If students are to learn an algorithm it should be an algorithm that connects with what they already know about whole number division. That algorithm is called the Common Denominator Algorithm. The visual action of the earlier ribbon problem illustrates this algorithm. The 6 whole units was recomposed into 24 fourths which was used to form groups of 3 fourths resulting in an answer of 8. In other words, the fraction division problem 6÷3/4 is equivalent to the fraction division problem 24/4÷3/4  which is equivalent to the whole number division problem 24÷3 because it is asking, “How many groups of 3 fourths can be made from 24 fourths?”.

Consider another problem introduced earlier 3/4÷1/3. The problem is asking, “How many 1/3s can be created from 3/4?” or making the units explicit, “How many 1 thirds can be created from 3 fourths?” Using what we know about equivalent fractions, an equivalent problem can be created, “How many 4 twelfths can be created from 9 twelfths?”. In other words, the fraction division problem 3/4÷1/3 is equivalent to another fraction division problem  9/12÷4/12 which is equivalent to the whole number division problem 9÷4. 

If we START with, “Keep, Change, Flip”, we are promoting intellectual neuropathy, a numbing of the brain, because our students feel they have all they need to get answers. As a colleague of mine said, “If we choose to share with students ‘Keep, Change, Flip’ we are making the decision that our students no longer need to think about ideas related to division of fractions.” It has been a challenge to develop a deeper understanding of division of fractions because students believe they have all they need if they can get correct answers. I have seen this first-hand in my college level classes. Recently, I asked a class of 50 students why, “Keep, Change, Flip” works. Not one student was able to provide a mathematically valid response (e.g., dividing is the same operation as multiplying by the reciprocal). There is so much more that we want them to understand about division of fractions than getting a correct answer. Let’s, “Just Say No” to “Keep, Change, Flip” and hope that what we replace it with is more effective than those 80s drug ads! I love to engage other teachers in ways to improve mathematics teaching. Feel free to reach out for further dialogue at patricksullivan@missouristate.edu.