July/August
2008
Investigating student thinking to teach
fractions well
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.
For Further Information
National Mathematics Advisory
Panel. Foundations for Success: The Final Report of the National
Mathematics Advisory Panel, Washington, DC: U.S. Department of Education,
2008. Available online at www.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf
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