Earlier this year, the National Mathematics Advisory Panel released 45 recommendations for improving U.S. mathematics education. In particular, the report singled out proficiency with fractions as a “major goal” for K–8 math education, noting that “difficulty with fractions (including decimals and percents) is pervasive and is a major obstacle to further progress in mathematics, including algebra.” Dr. Katherine K. Merseth, director of the Teacher Education Program at the Harvard Graduate School of Education, specializes in the teaching of math and science and was inducted last year into the Massachusetts Hall of Fame for Mathematics Educators. Merseth spoke with Harvard Education Letter contributing writer Mitch Bogen about why fractions are hard to teach and what teachers can do to help students better understand this critical concept.
Why are fractions difficult to learn?
We know from the research on how people learn that we attach new knowledge to already existing knowledge in our heads. So if I’m a third grader, I’ve spent quite a bit of time in first and second grades adding whole numbers: one plus two is three and two plus three is five. And I understand how that works. But whole numbers are very different from fractions. So what a child may do when confronted with a problem like 1/3 + 1/2 is to answer 2/5. And why not?
Here’s another example: If three is less than four—I’ve got three candy bars and you’ve got four, so I’ve got less than you—wouldn’t you think that 1/3 would be less than 1/4? You’re applying this new situation to your knowledge about counting, where 3 is smaller than 4. Now we’re asking the student to take a new thing, which is 1/3, and intuition connects it to what they know from before, which is that three is less than four. It’s absolutely rational. Kids rarely do things randomly. So as a teacher, what do I do? I try as hard as I can to not just put a big red X on the paper.
What is the best way for teachers to respond to these misconceptions?
What often happens is that the teachers don’t take the time, or don’t have the time, to look at individual students’ work, or to listen to the answer and try to figure out: Why did the student answer the way she did? I feel so strongly about this. Why can’t we slow down and listen to the kids? Because they are making sense. It just doesn’t happen always to be the sense that we’d like them to make.
The problem is compounded by the fact that the teachers themselves are often not terribly comfortable with the mathematical concept they are trying to teach. And teachers are also under the gun with the standards and the pacing guides.
I think we could do a better job by educating teachers to listen to their students and then capitalize on the teacher’s understanding of the kid’s understanding. If you ask me, “What is the one thing you’d do to help teachers teach math more effectively?” I would train them to listen and to be able to ask questions that would pursue the thinking of the child. Until we work to understand what a child is thinking, it may remain undetected by the teacher in the child’s head. And there is cognitive research to show this as well, that we hold on to our misperceptions and don’t readily give them up.
What kind of questions can teachers ask to help students think better mathematically?
Let’s say you ask a child what seven times seven is, and the child says, “Fourteen.” If you can hold your breath and not say, “No!” and instead ask the question to which 14 is the right answer, you may be surprised. When the student says, “Fourteen,” you can say, “Oh? Really? Well, then, what is seven plus seven?” And the student often quickly says, “Oh! I meant forty-nine!” Instead of coming down on the child, suggesting that math is simply a matter of right and wrong—and the student is wrong—the best thing is to ask, “Well, let’s explore where you might have been coming from.” It takes tremendous skill as a teacher to be able to stop and say, “What if I asked seven plus seven instead?”
There seems to be a feeling in math—and I was just in some math classes last week where I saw this—where we almost send a message that when you are in math class, you don’t talk. Talking is for language arts. Asking students to draw, to count, to verbalize in math class is critical, because math language is very important and needs to be very precise. We need to help children become comfortable in talking about math. It’s so important to take the time to stop and say: “What does the problem mean? Can you tell me in your words what it means?”
But some of this is very hard to do. In that class I was in last Tuesday, I said to the teacher, “Why don’t you have a student talk to another student? You don’t have to always be the traffic cop.” There are lots of resources in a classroom beyond the teacher. Ask students to talk with other students. Ask students to articulate for the entire class what they are thinking. Ask students to write down four questions that they have about the topic or issue and pass it to the person next to them.
What about the use of manipulatives for teaching fractions? Do they help?
Being able to manipulate things with your hands, tactilely, or to draw a picture to create a visual representation, is another avenue for grasping concepts behind the specific math procedures.
I’m a big fan of teaching fractions with something called pattern blocks, which are blocks of hexagons, trapezoids, rhomboids, and triangles. I can basically demonstrate to a child or an adult all the concepts and operations in fractions with these manipulative materials.
Now, the push-back comes from teachers who legitimately say, “I’ve got a pacing guide. I’ve got to do this in three days, you know. I can’t take the time to take out all those little blocks.” Well, my response is that if you don’t take time now, you’ll have to take time later. Pay now or pay later. Which would you prefer? And which do you think will be less painful for the student?
But manipulative materials don’t work for every child, so I’m also a big advocate for going at the same concept from different points of view: whether it’s manipulatives, whether it’s drawing things on a page, whether it’s counting—use multiple representations. Use different ways to represent the same idea. One might be in symbols, one in a chart, one in a table, one in a graph.
Can you give some examples?
There are three main visual representations for demonstrating the concept of parts to a whole with a fraction. Take the fraction 3/4. For the first example, I might draw one circle and divide it into quarters, and I would shade in three parts of the one whole. Then I would say to the student, “This is three fourths,” A different visual representation would be to take four separate discs and shade in three of them. This is more of a “discrete” representation. The third representation is the number line. Instead of having discrete parts, we draw a line from zero to one and shade it in up to the three-quarters point, as if it were a ruler. This representation is called “continuous.”
When using these representations, teachers need to be very aware that students might interpret them as totally different things. In the first case, the student sees one big glob being cut up into four pieces, but in the second, there are four globs and I’m coloring three of the four discrete pieces. That can be confusing at a young age. So it is important for the teacher to stress that even though these examples look different, they all represent the same numerical quantity.
Still, it’s very important to represent fractions in these multiple ways because some children will go for the continuous model, others will go for the discrete model, and so on. And we want to give them every opportunity to grasp the concept of three-fourths. One of the tricky things about fractions is that they are inherently multi-faceted. And as the students proceed beyond the idea of the fraction as a numerical quantity, or parts to the whole, they’ll encounter even more complexity. For example, a fraction, besides being a number, can represent a relationship between the numerator and the denominator, or it can be seen as a division problem—four divided into three. Imagine being the student who just learned to take three parts out of four: “Now you’re telling me it’s a division problem? Come on!” What does a fraction mean? Well, it can mean many things. It’s an area of mathematics that can be a real minefield for both students and teachers.
Why are fractions important for further math proficiency?
The reality is you could probably get through mathematics, albeit with extreme difficulty, without a total mastery of fractions. But as a student moves up through the grades and comes to an algebraic fraction like x+3 over 7, if that student didn’t grasp earlier that whole notion of ratio, or relationship between parts and the whole, they’ll be hard pressed to understand how to approach that problem. Not impossible, but hard pressed. I think the [National Math] Panel was right to put their finger on the fact that in the upper elementary grades we spend an inordinate amount of time trying to teach kids about fractions, and we need to do it better. And it has to do with helping children and teachers have multiple ways to represent and understand fractions.
It’s a challenge for teachers when older students don’t have a good grasp of fractions. There are no shortcuts, and students need to learn all the core concepts. But one thing that teachers can do to help these older students is to find a context that is relevant to the age group. It’s difficult enough for students to admit they don’t fully understand something they feel they should have learned earlier. Don’t embarrass a seventh grader who is learning fractions by using a picture of three tricycles.
What are the cultural factors that make teaching math a challenge?
There seems to be a social acceptability to saying, “I’m no good at math.” And it goes from parent to child. The child comes home and says, “Oh, Mom, can you help me with my math homework?” And what’s the message? “Oh, I never could do it either, sweetie. Do your best.” What’s the child going to think? That this is important? No. So I’m 150 percent behind changing children’s beliefs and society’s beliefs about mathematics—and about science.
One of my theories about why people are successful or unsuccessful in math, odd as it may sound, is based on their relationship with numbers. Do they feel friendly with numbers? Are numbers a friend that they can play with and do things with as youngsters, either in their minds or on the playground, or are they objects to be afraid of? I think it’s important for adults to make children more aware of the mathematical objects all around us.
One great example of making math accessible and exciting is in Melbourne, Australia, where a public park features a mathematical walking tour. You go from station to station, and a sign at one station might say: “Look at the leaves in this tree. Do you notice that they are in the relationship of 3 to 5 to 8, depending on where they branch out?” Or, “Look at the spirals in this sunflower. Do you see how they unfold in a mathematical relationship?” Discussing objects and phenomena not only for what they are but also for what they represent in mathematics is a good way to make math more accessible.
Another simple strategy is to engage students with the mathematical and statistical aspects of baseball, basketball, soccer, and all the other sports they love. There are almost unlimited examples of ways to help young people become comfortable and friendly with math. Most of them involve taking math out of the books and into the everyday world of children.
It’s also important to explore with students the mathematical aspects of jobs and careers that are out there. This can help answer that question: “So what good is this going to do? How is this going to serve me in the future?”
I venture to say that while our literacy folks are doing a very good job in this regard, a big change in the world in the next 50 years will be the increasing importance of math. And if we don’t focus on math we’ll be left behind. The whole world must be literate in mathematics because it grounds the rationality of so much of what we do: decisions about voting, decisions about spending millions or billions of tax dollars, concepts about risk. And it comes right down to personal quality of life. Can I afford to take out that subprime loan? I think that the problems for many people who are losing their homes in the subprime loan crisis are partially our fault, because they might not have understood interest rates or what the risks were. Why? Because they didn’t grasp the math.
There is a level of mathematics that is very complicated, that I can’t understand, that you can’t understand, that only well-trained mathematicians can understand. That probably does take a special innate ability, but at the level of being a competent citizen, being able to make important decisions, to think for yourself, to have a sense that when this changes how that changes—that is teachable. It’s very teachable. Mathematics is accessible to everyone.
Mitch Bogen is a freelance education journalist based in Somerville, Mass.