Many students struggle with fraction operations. And just as many adults (parents, teachers, school leaders) have questions about the “new” way of doing math.
It makes sense that some see conceptual math as a solution to a problem that doesn’t exist. It’s easier (at least in the short term) to just teach tricks for fraction operations. But the limits of these strategies are significant.
A fellow educator (I’ll call him Jim) recently asked me about conceptual math. He wanted to know “what the big deal was” with Common Core. Do students really need to do all these drawings? I answered his question by asking him a question.
“What is 1/4 times 2/3?”
“I remember doing this…don’t I need to find a common denominator?”
I don’t mean to cast aspersions on Jim. He’s a technology director who’s made a significant impact on his school. The fact that he’s not proficient in a 5th grade math standard hasn’t held him back in life.
But this exchange highlights several issues about the traditional approach to math. First, what Jim learned in school didn’t stick. He had been taught rote procedures without concepts. When he attempted to recall a procedure years later, he didn’t know which to use.
His approach also shows that he wasn’t considering the size of the numbers. Or what it means to multiply a fraction by a fraction.
Worse still, he didn’t even think about problem-solving to finding the answer. He reached into the memory banks for an algorithm. When he couldn’t find one, he had nowhere else to go. His career success indicates that he has the ability to problem-solve. But he didn’t associate that ability with math.
Fraction Operations Present Unique Challenges
These challenge of connecting concepts to procedures is common to many math standards. But they are particularly evident with fraction operations. One reason is that many students begin fraction operations before they really understand what a fraction is. Another is that fractions are less intuitive than whole numbers. It’s not always clear what is happening when we take away 1/4 from 2/3. Or when we divide 7 by 1/6.
I’ve noticed a trend among schools who use the generic curriculum plans that come with their textbooks. Teachers in middle and upper grades bemoan their students’ struggles with fraction operations. They wonder why elementary teachers never taught fractions. After all, fractions are supposed to be covered in grades 3 through 5.
The reality is that those teachers did teach fraction operations. The problem is that the students never really understood what they were doing. They remembered the tricks long enough to pass the test. By the next school year, their memories of fraction operations were as distant as my friend Jim’s.
Fractions Before Fraction Operations
Many educators would love to just forget all about fractions after grade 5. But fractions provide an essential foundation for advanced math.
Decimals and percents are both types of fractions. Middle school ratio and proportion standards also build upon fractions. Not to mention slope (rise over run). In high school, Algebra, Statistics, and Calculus all involve rational numbers. (A rational number is any number that can be expressed as a fraction).
The first step to mastering fraction concepts is understanding numerators and denominators. A denominator tells us how many pieces a whole is divided into. A numerator tells us how many of those pieces we have. At some point (usually around grade 5), they learn that a fraction can also be thought of as a quotient. The result of a numerator divided by denominator.
Fraction Operations Work Like Whole Number Operations
Once students understand fractions, they can operate with fractions the same way they operate with whole numbers. That is, if students understand how the operations work.
Adding 1/5 to 2/5 brings 3/5. Taking 1/8 away from 1/4 requires breaking the 4th into 8ths — much the same way taking 3 from 62 requires breaking up a ten. Multiplying 4/7 by 1/2 leaves you with 2/7 — half of what you started with.
Students can eventually save time by using algorithms to operate with fractions. Algorithms also allow them to work with more complicated numbers. It’s hard to create a visual model of 29/1256 divided by 15/72.
Sadly, many students learn algorithms without really understanding the meaning of fractions operations. This approach has several drawbacks. For one, when a student can’t attach meaning to an algorithm, they won’t retain it for long. In addition, they get confused about which algorithm to use. (Hence why Jim thinks he needs a common denominator to multiply fractions.)
But there’s an even bigger problem with algorithms sans concepts. Students don’t develop the ability to see the math that is all around them. They don’t recognize math as a language, or a study of space and quantity. Math isn’t beautiful or an art.
Instead, it becomes a collection of “math facts,” random rules, and secret symbols.
Teach Fraction Concepts with The Three Vehicles
I encourage every math teacher to start each year by teaching the three vehicles of conceptual math. The three vehicles are scale models, number sentences, and word problems.
I call them vehicles because once students learn to drive them, they can cover a lot more ground the rest of the year. John Van de Walle popularized the idea that there are five ways to represent a mathematical idea. The first vehicle, Scale Models, aligns with two of his representations: manipulatives and visual models. The second vehicle, Number Sentences, aligns with symbolic representations. And the third vehicle, Word Problems, aligns with real world situations.
His fifth representation, oral language, always felt a bit out of place. Of course, it’s a valuable representation. But when discussing math orally, you are usually still discussing one of the other operations. Regardless, students have ample opportunity for discussion in a Three Vehicles workshop lesson.
Vehicle 1: Scale Models
Whenever I begin working on fractions with a new group, I ask everyone to draw me pictures. “Show me what 1/2 looks like. Show me 3/4.” Many teachers are shocked to see their students struggle to draw these common fractions.
When conducting workshops, I’m regularly surprised at how many math teachers have difficulty with these tasks. Like our friend Jim, these adults are often highly successful teachers. They are unable to create these models
through no fault of their own. It’s simply that many weren’t taught to model with mathematics when they were in school.
Visual models can be used both as an instructional and an assessment tool. If a student can tell you that 4 times 1/5 is 4/5, ask them to draw it. If they can’t draw the picture, they don’t understand the concept.
There is more than one right way to use visual models, but there are some general rules. To be a visual model, it should be drawn to scale – otherwise it’s just a drawing. Also, the model should clearly show starting values and operations, not just a result. If someone else can turn your visual model into another representation, it’s probably good.
Vehicle 2: Number Sentences
A number sentence is a complete mathematical statement. Each consists of two expressions on either side of a comparison symbol, such as an equal sign. Equations and inequalities are both examples of number sentences. While Stephen Colbert has come out against them, experienced educators recognize their value.
Number sentences are another great way to teach conceptual math, including fraction operations. Manipulating expressions and equations can help students see how math standards are interconnected.
Students should be taught the conventions of working with equations. For example, they need to be shown that each step goes beneath the step before it. Teachers should also teach students to only use one equal sign per row. Then, they can manipulate number sentences to discover new concepts and connections.
One of my favorite hands-on math lessons involves number sentence proofs. Students are given a series of number sentences. Instead of simply ‘solving for x,’ they must determine whether the sentence is true or false. With a closed number sentence (no variables), the sentence is simply true or false. Open number sentences can be ‘always true,’ ‘sometimes true,’ or ‘never true.’
This activity forces students to explore the concepts behind fraction operations. Even students who know the algorithms still need to explain why their solution works.
Vehicle 3: Word Problems
Word problems can help students see the connections between math and the real world. The right word problem at the right time can make abstract math more concrete and easy to understand.
I remember a student I taught who struggled in math, but loved Pokemon cards. I had asked him to solve a problem on the board: “If you take away 1/3 of 120, how much is left?” He stood and stared at the problem, not knowing where to begin.
I asked him to imagine he had Pokemon character that started with 120 health points. If the character lost 1/3 of its points how many were left. He instantly responded, “80 points.”
One of my favorite word problems involves a jug containing 25 cups of lemonade. If we pour the lemonade out as 1/2 cup samples, how many samples can we pour? The beauty here is that is provides a real world example of division by a fraction.
Fraction division is a tough concept because most students divide into a set number of groups. Dividing into “1/2 of a group” is hard to visualize. (Which is why many students make the mistake of dividing by two instead). But groups of 1/2 make more sense. Students can also see the connection between division and repeated subtraction.
Unfortunately, not all word problems simplify complex ideas. Many students and teachers view word problems as complex and scary. This is often because word problems demand students understand the concepts. They can’t perform an algorithm if they don’t know where to put the numbers from the story.
But just as often, students get lost in all the steps required to solve a word problem. Graphic organizers can be an essential support in this case.
Creativity, Collaboration, and Conceptual Learning
Giving students the opportunity to learn fraction operations through hands-on activities can increase both student achievement and engagement. To learn more, find tips and ask questions in our Facebook group.
Diving into these hands-on activities is not always easy. Our Hands-On Fractions QuickStart Guide is a free download that includes everything you need to start teaching fraction operations through inquiry. It includes lesson plans and reproducibles you can try out in class tomorrow.