March/April 2007
An aligned approach to PreK and early
elementary math
Sharon Griffin is an associate professor of
education and psychology at Clark University and author of the Number
Worlds curriculum for teaching number sense in the preK and
elementary years. In this interview with the Harvard Education
Letter, Griffin discusses what cognitive science can teach us
about aligning preK and early elementary curriculum and teaching
methods with the natural development of children’s mathematical
thinking.
What math skills do children need to have
before they start school?
Children need to know the counting words (“one,
two, three …”) and to understand that this reliable
sequence of words only gets meaning by being attached to quantities:
“We’re going shopping and we’re going to get two
boxes of cereal.” “You can have two cookies but not
three.” To link these words to the quantities that give them
meaning is the crucial thing. Then they can use the counting words
by themselves without even needing to see the objects they are counting.
That frees children up to do math in their heads. If children establish
the link between number and quantity in the prekindergarten years,
they will have a solid foundation for future learning.
What kinds of preK experiences do children
need to develop these competencies?
To begin with, they need to learn to say the counting
words in sequence—what I call the counting string. The crucial
thing about the counting words is learning to say them in order.
We can’t leave anything out, and we can’t say anything
twice. It’s a sequential system.
Along with learning the counting words, kids have
to be paying attention to the quantities in their world. The first
thing is to be able to describe quantities in terms of polar differences:
“This block tower is taller than that block tower.”
Taller, shorter; heavier, lighter; nearer, farther; hotter, colder.
If children are not paying attention to height or weight or distance,
then they’re not going to be able to map numbers onto those
dimensions very readily. In settings where there’s rich lan-guage,
all of that vocabulary mapping onto the world of quantity helps
kids understand the quantitative world in a global way. It makes
all these quantitative differences salient, so that when they have
a grasp of numbers, children can start mathematizing those quantities.
All of this happens very naturally in many homes, but most of it
is not taught in school (see Stages
in the Development of Number Sense).
Children also need to become familiar with all
the different ways numbers are represented in our culture. Numbers
can be represented as objects: Two means two things. Another way
is in patterns, like the dot patterns on dice or on playing cards.
That’s already semiabstract, because the quantities are fixed.
This makes it harder to grasp that the five pattern is one more
than the four pattern—you can’t pick one dot up and
put it back like you can with objects. Numbers can also refer to
position. When you’re counting in a game like hopscotch, it’s
not that you’re adding one more object but you’re moving
one more position in space from where you started. A similar representation
is vertical line representation, like a thermometer, where numbers
indicate quantity on a continuous measurement scale. And the fifth
way is dial representations, like a clock, in which quantities increase
as you move around the dial clockwise.
We know that many low-income children
enter school without the kinds of language-rich experiences that
middle-income children have typically had. What other gaps do you
see at school entry and how do they affect the development of children’s
math skills?
More affluent children are exposed to different
forms of representing numbers; lower-income kids much less so. Research
shows that affluent kids are much more likely to have board games
in the home; they may also have dominoes and playing cards. When
they play Sorry, for
example, they learn that numbers indicate position as they move
around the board. To learn that numbers mean position gives you
a solid foundation for understanding the ordinal value of numbers.
A lot of kids come into kindergarten with the
link between number and quantity firmly established, usually the
more affluent kids. A lot of lower-income kids don’t get this
till they’re seven. It’s about a two-year delay. And
many teachers don’t teach the link between numbers and quantity.
They assume all kids have it. Even kindergarten teachers now will
jump right in and write abstract symbols like “4 + 2”
on the board. They’re mapping counting words onto numerals
and more sophisticated algorithms, and kids don’t even have
a sense of what the counting words actually mean. By the time these
children get to second grade, it’s not surprising that many
of them hate math! My research shows that children who start kindergarten
without this understanding and who do not receive remediation fall
farther and farther behind. But with proper, explicit instruction
that gives kids exposure to the ways number is represented and allows
them to figure out the link between number and quantity, they can
catch up with and even surpass their peers.
Once kindergartners have consolidated
this understanding, what should come next?
The business of first grade is to integrate the
world of counting numbers and the world of quantity with the world
of formal symbols. That’s when you introduce numerals and
plus signs and minus signs and equals signs and link them to the
counting words, which children now associate with real quantities.
But teachers tend not to do this in the correct sequence. They’ll
put “5 + 3 = ” on the board and tell children, “Use
your manipulatives to solve this.” But kids may not even know
what the symbols mean, and therefore may not have any idea how to
represent them with objects. Let’s let children solve problems
with real quantities—using cubes or weights or steps along
a line—let’s let them talk about what’s happening
before they start using numerals and symbols. “How many cubes
do you have now? What happened—you got three more? Now how
many do you have altogether? How did you figure that out?”
Let them talk about adding and subtracting, using the counting words
and the language of quantity transactions (“I had five, and
I gave three away. Now I have two.”). Once that’s really
solid, then introduce the symbols. But don’t do it backwards,
which is our typical pattern.
This is the reason for the big push for communication
in the math classroom. We know that talking makes a huge difference.
But it’s the one area that teachers are inclined to cut because
of time constraints. They want to do a lot of drill and practice
to prepare kids for the state accountability tests, so they don’t
give kids the chance to make sense of the quantity transactions
they are enacting, with or without manipulatives. That’s a
huge tension in elementary schools, and because of that it’s
more and more important that kids have that rich exposure in preschool
and kindergarten, where there’s a little more flexibility
in what teachers do.
Before they even start school, kids have two to
three years of rich experience in the world of language. We need
to build on that experience in school instead of forgetting about
it, and offer continuing opportunities to link formal written symbols
to this rich base of language (see Developmental
Principles for Early Math Instruction).
In your ideal world, schools would take
a developmentally sequenced approach …
… from preschool right up. Absolutely. Everything
builds on that early training. I believe very firmly that the only
way you can teach for understanding is to start your instruction
from where the child is. If you don’t—you can teach
kids tricks and rules and they might learn to use them, but they’re
not going to really understand them. If you don’t understand
how these fundamental concepts are built, there’s no way that
you can really interpret where a child is and there’s no way
you’ll have the resources to gear instruction to fit children’s
needs.
Can you give an example?
At one time or another, almost all children count
on their fingers to solve single-digit math problems. To make them
embarrassed about doing this or to insist they do the math in their
heads deprives them of a wonderful tool for making sense of numbers.
Counting on their fingers doesn’t hinder cognitive development,
it helps it. Children will abandon this tool quite naturally when
they no longer need it.
Another example is telling time. Around grade
2, children reach a new stage of cognitive development in which
they can start to grasp quantity along two dimensions—for
instance, tens and ones, hours and minutes, dollars and cents. At
this point it’s usually pretty easy for them to learn to tell
time. Yet state standards and curriculum guidelines often require
students to master this skill in first grade. These requirements
aren’t appropriate to children’s level of understanding
and expose them to needless frustration.
You have said that today’s curricula
are not well aligned with the developmental sequence of children’s
mathematical thinking.
There are way too many math topics at every level.
In the fall of 2006, the National
Council of Teachers of Mathematics came up with a new set of
curriculum standards called Focal
Points, which narrows it down to three areas for each grade
level, starting in preK. This is a huge step forward. Still, the
amount of time teachers have to teach math is often inadequate.
There are many interruptions in the classroom as children come and
go for special services, and the children who need quality math
instruction the most are often the ones who are pulled out of the
classroom for other services.
Many math curricula emphasize “spiraling.”
But if kids don’t get a concept the first time it is taught,
they are even less likely to get it the second or third time, when
it’s usually taught at a higher level. Spiraling prevents
the teacher from con-solidating knowledge at each level before moving
on, which is a basic principle of effective practice. For instance,
the function of third grade is to consolidate the knowledge gained
from preK through grade 2, in anticipation of the developmental
gains that occur around grade 4.
In Asian schools, teachers take the time to teach
fewer concepts really well. They’ll show students something
like a balance beam, they’ll let them play with it, they’ll
let them predict and explain and argue among themselves. Then they’ll
test their predictions and at the end they’ll say, “How
can we describe this so we’ll know what to do next time?”
Then they start introducing the formal expressions. But we don’t
have the time. We do it the quick-and-easy way for the kids who
understand. We say, “You see this symbol? This is what it
means you do.”
What do teachers need to know in order
to align their math instruction with students’ cognitive development?
Preservice teachers come to my classes and they
think math is about numbers and rules for manipulating numbers.
They don’t say, “Math is about quantity” or “Math
is a set of conceptual relationships between number and quantity.”
If you believe that math is about numbers and the rules for manipulating
them, of course the best thing you can do is to teach kids the numbers
and the rules as early as possible. And when they say “numbers,”
teachers tend not to think of spoken language. They think of numerals,
the written form of numbers. Of course children have to know the
rules and the procedures, but they also have to know when it’s
appropriate to use them. Children need to build that knowledge with
rich experiences. The point is not to teach all areas of mathematics
from prekindergarten on up, but to cover them gradually over the
course of 18 years of schooling. You don’t have to do everything
in every grade. Let’s do the critical things first and do
them well.
For Further Information
S. Griffin. “The Development of Math Competence
in the Preschool and Early School Years: Cognitive Foundations and
Instructional Strategies.” In J.M. Royer, ed., Mathematical
Cognition: Current Perspectives on Cognition, Learning, and Instruction.
Charlotte, NC: Information Age Publishing, 2002.
S. Griffin. “Fostering the Development of
Whole-Number Sense: Teaching Mathematics in the Primary Grades.”
In M.S. Donovan and J.D. Bransford, eds., How Students Learn:
History, Mathematics, and Science in the Classroom. Washington,
DC: National Academies Press, 2005.
S. Griffin. “Teaching Number Sense: The
Cognitive Sciences Offer Insights into How Young Students Can Best
Learn Math.” Educational Leadership February 2004,
vol. 61, no. 5.
National Council of Teachers of Mathematics. “Curriculum
Focal Points for Prekindergarten through Grade 8 Mathematics.”
Available online at http://www.nctm.org/focalpoints/default.asp
Number Worlds http://clarku.edu/numberworlds/
|
