Reprinted from Barbara Moses, ed., Algebraic Thinking, Grades K–12: Readings from NCTM’s School-Based Journals and Other Publications (Reston,
Va.: National Council of Teachers of Mathematics, 2000), pp. 197–201. Originally appeared in Teaching Children Mathematics 3 (February 1997): 314–18.
© 1997 by NCTM.
Functions from
Kindergarten
through Sixth Grade
Stephen S. Willoughby
O
ne of the most important and pervasive concepts in mathematics is that of a function.
High school and college students often have trouble with the function concept because of the
abrupt and abstract way in which it is introduced.
However, even very young children can understand the concept if it is introduced in a sufficiently
concrete manner and very gradually abstracted
over a long enough period of time. This understanding is demonstrated by their ability to solve
problems involving simple function rules; by the
ease with which they understand and use variables;
by their ability to make and interpret graphs of
functions; and by their understanding of real-world
examples of functions, such as that an individual’s
height can be seen as a function of age.
A brief description follows of one strand of activities that I and other teachers have used to help develop the concept of function from kindergarten
through sixth grade. Of course, the precise level at
which an activity is introduced can vary, but the
sequence and the gradual, long-term organization,
starting from the concrete and moving toward the
abstract over a period of years, is important. In this
Steve Willoughby is in the department of
mathematics at the University of Arizona, Tucson, AZ 85721. He has taught all grades from
kindergarten through graduate school, is a past
president of NCTM, and in 1995 received the
Mathematics Education Trust Lifetime Achievement Award for Leadership.
article I show how we can start with a concrete
representation of a function rule and gradually
make the representations more abstract by going
from physical objects to pictures to “arrow arithmetic” and finally to the standard representation.
Because of the concrete representations to which
students are initially exposed, students’ later use of
the standard representation will seem natural and
easy them.
Kindergarten
The class decorates a large box, into which a child
fits comfortably, so that it looks like a computer. In
the front of the box is cut a slit about 2 cm high
and 13 cm long (see fig. 1). Although children
soon realize that a classmate is in the box, I have
found that they are more interested and thus pay
more attention if the classmate is hidden in the
box than if that person is clearly visible. Their
sense of play and enjoyment of fantasy presumably
are responsible for their heightened attention.
When nobody else is looking, a child is sent into
the box with appropriate materials (sticks, pencils,
chalk, and so on) and a rule, such as “add 2.” The
teacher announces that the box is now a magicnumber machine, later called a function machine,
and suggests that one member of the class put
some ice-cream sticks into the box through the slit.
The class member must announce how many sticks
are being put into the box. Suppose the class
member puts in five sticks. The box appears to
jump up and down a bit and make strange noises,
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these effects being provided by the child inside the
box, and seven sticks come out. Typically, the class
is excited about this result, and several children
start looking around to see who is missing.
doubling the number can be tried. Then the paper
is placed into the machine and comes back shortly
thereafter with the original number crossed off and
a new number written on the paper.
Next, a second class member is asked to put several pencils into the machine. Suppose that two
pencils are put in the machine. The machine repeats its gyrations, and four pencils come out.
The first time I tried this activity, I carefully taped
all the cracks in the box so that class members
could not discover who was in the box. However,
as soon as the first piece of paper went into the
box, I discovered that I had been too diligent. The
box promptly said, “I can’t see!” I recommend leaving some cracks or giving the operator a flashlight.
A third class member is then invited to put some
pieces of chalk into the machine. This time, however, the class member is instructed to tell the class,
in advance, how many pieces of chalk will be inserted, and the class is asked to guess what will
happen. If the class member says, “I’m going to put
eight pieces of chalk into the machine,” the class
should guess that ten will come out. When the prediction is confirmed, students realize that from the
pattern of the first two events, they are able to infer
the machine’s rule and correctly predict the result
of the third event.
This activity can be repeated many times over several days or weeks with different children in the
box and different class members putting things in.
The rule, of course, should be changed each time
but should usually be limited to adding or subtracting a small number, ordinarily 0, 1, 2, or 3.
Some interesting discussions result from this activity. For example, in one class, after the children realized that the rule was “subtract 3,” the next class
member chose to put two pencils into the machine. After a moment of silence, the machine said,
“I need one more!” That is about as good a definition of negative one as we could hope for from a
five-year-old.
In another class, one student wanted to know what
would happen if she put a fellow classmate into
the machine, which was adding 2. The class discussed whether putting Abigail in would produce
three Abigails or just three children. Somebody
claimed that it would not work for people. The
teacher asked why and was told that people would
not fit into the slot.
First and Second Grades
In first grade, we start by repeating the kindergarten activity and then taking a first step toward
abstraction by having the class member write a
number on a small piece of paper. The rule can involve adding or subtracting larger numbers. Even
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Fig. 1. The magic-number machine
Students are able to
infer the machine’s rule
and predict the result of
the third event.
In second grade, I start by repeating the first-grade
activity with pieces of paper. Once students are
comfortable with that activity, I make it more abstract by using pictures of number machines (see
fig. 2) with various pieces of information, such as
the rule, the number going in, or several pairs of
inputs and outputs. I then ask students to supply
the missing information.
If calculators with a built-in constant function are
available, I also use those as function machines at
this level. For example, with the TI-108, clear the calculator, then press + , 7 , = . Then without touching any other keys, press 5 , and the display will
show 12. The calculator is now a “+7” machine.
Hand it to a student with instructions not to touch
any key until told to do so. Tell the student to pick
a number. If the number is 4, tell the student to
enter 4 and then = . The display should show 11.
The student can continue to choose numbers to
enter followed by = until the class can predict what
display will result. Usually students get the rule after
only one or two examples as long as only addition
and subtraction are allowed.
2
+7
?
4
7
9
?
2
5
7
Fig. 2. Students supply the missing information
Third and Fourth Grades
In the third grade, I introduce more complicated
function-machine rules, such as allowing multiplication and division as well as addition and subtraction, and, of course, any numbers the students
wish to use, although they usually stay with
whole numbers.
I also introduce the idea of composite functions,
that is, using the output of one machine as the
input for a second machine. For example, suppose
that the first machine multiplies by 3 and the second machine adds 4. What happens if a 2 is put
into the first machine and the output goes into the
second machine (see fig. 3)? Students quickly conclude that the output of the second machine will
be 10. Next, suppose that some unknown number
is put into the first machine and the output of the
second machine is 16 (see fig. 4). What went into
the first machine? Third graders have no trouble
deciding that 4 went into the first machine. They
simply run the machines backward, deciding that
12 must have gone into the second machine and
therefore must have come out of the first machine.
Therefore, 4 must have gone into the first machine.
The observant reader will notice that the
third-grade students have just solved the linear
equation 3x + 4 = 16. Some years ago when I was
reviewing this activity with a fourth-grade class the
regular teacher greeted me one morning with the
2
+4
X3
?
Fig. 3. Outputs and inputs are combined.
?
+4
X3
16
Fig. 4. Students begin solving for an unknown.
information that she had been a teacher of
first-year algebra the previous year. She suggested
that the work we were doing looked suspiciously
like solving linear equations, with which her ninth
graders had had trouble the previous year. I admitted that a certain similarity was evident and asked
if any of the fourth graders were having trouble.
She said that the fourth graders were having no
trouble at all, “but if my ninth graders were having
trouble with this last year, I don’t think you should
teach it to fourth graders.” The important and general point here is that if a concept is approached
slowly and through the learners’ experience and if
abstract symbolism is delayed until understanding
has occurred, most subjects can be learned and
understood by children. The difficulty that typically
occurs in first-year algebra results from the introduction of a great deal of very abstract material in
too short a time.
As the boxes representing the machines in the children’s work get smaller and are drawn more
quickly so that they look a bit more like circles
than squares, the pictures gradually turn into
“arrow arithmetic.” This revision simply consists of
replacing the machine with a small circle or
square, with the rule inside the circle. For example, the previously mentioned composite function
would be shown as in figure 5, for which I have
introduced letters for variables: x for the original
input, n for the output of the first machine and
input of the second machine, and y for the output
of the second machine. This use of variables does
bother some third-grade students. If substantial difx
X3
n
+4
y
Fig. 5. Drawings of function machines are gradually
replaced by circles.
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ficulty seems to occur with the variables in figures
like figure 5, the teacher might choose to delay introduction of letters for variables, possibly until
fourth grade, until students are more comfortable
with the underlying ideas. Notice, however, that
because of the placement of the multiplication
sign, confusing it with “x” will not occur as could
happen if we wrote 3 x + 4 = y.
I introduce more nonlinear functions. For example,
I ask the class to fill in a table with the area of a
square shown as a function of the side of the
square and then to plot the results. (See fig. 6.)
Some students quickly decide that this function
does not produce a straight line, although some
usually argue that a mistake must have been made.
All the third-grade work is reviewed and expanded
on in fourth grade with the connections between
functions, depicted with arrow arithmetic, and tables and graphs emphasized and with more complicated composite functions and applications considered. The idea that certain kinds of functions
will always produce points on a graph that are on
a straight line is also developed along with the fact
that some function rules do not produce points
that are on a straight line. After several linear functions have been graphed, children tend to believe
that all functions will be linear. I start to overcome
this idea by having them graph a person’s height
as a function of age. In many cases, parents of the
children in the class have kept this information,
which we are able to use. Occasionally, even these
data look linear at first glance, but somebody always suggests that the individual could not possibly live up to the graphical prediction by the age
of twenty or forty.
The purpose of new
symbolism should be to
reduce necessary work—
not add mysticism.
Fifth and Sixth Grades
After reviewing the work with functions from previous grades, children are introduced to a standard
algebraic form for writing function rules, for example, 3x + 4 = y. They invariably find this symbolism
much simpler than what they have been using and
complain that they should have been shown this
easy way to do it earlier so as to save work. That
reaction, of course, is exactly what I want. The
purpose of new symbolism should be to reduce
necessary work—not to add mysticism to an otherwise reasonably easy subject. Older students in
eighth or ninth grade, studying the same topics in
the usual abrupt and abstract way, have trouble
because the abstract symbolism is introduced before the concepts are sufficiently clear, not because
the subject is inherently difficult.
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For more realistic problems involving graphs, I use
such examples and tables as shown in figure 7. The
reader will note, of course, that the function discussed in problem 3 is periodic and more nearly resembles a sine or cosine function than anything else.
Elementary school students should see that if the
function were linear, temperatures would rise or fall
forever, which would become very uncomfortable.
If a computer or programmable calculator is available, a compound-function rule can be used to
generate pairs of numbers, and the class can try to
predict outputs given particular inputs. They may
discover that different function rules may produce
the same set of ordered pairs.
Concluding Comments
All children can understand abstract but important
concepts, such as function, if the concepts are developed first from concrete activities and gradually
abstracted over a fairly long period of time. In this
article I have tried to describe some of the many
activities that I have used to help students become
familiar and comfortable with the important and
pervasive concept of a function.
Bibliography
Willoughby, Stephen S., Carl Bereiter, Peter Hilton, and
Joseph H. Rubinstein. Real Math, Levels K–6. La Salle, Ill.:
Open Court Publishing Co., 1978, 1981, 1985, 1987, 1991.
Length of side of square in cm
0
1
2
3
4
5
0.5
1.5 …
Area of square in cm2
Fig. 6. The table of area as a function of the length of a side
1. A bank charges $1.50 per month plus $0.10 per check cashed. Complete the table and graph the results. Did you expect to get a straight line for the graph? Did you get a straight line?
Number of checks cashed
Total cost
0
1
1.50
1.60
2
3
4
5
10
20
2. A barber makes a profit of $9 per haircut after expenses for lotions and such things. The barber also has expenses totaling about $150 per week for things like rent, utilities, and equipment. Do you think income after expenses as a function of the number of haircuts is a linear function? Fill in the following table and graph your results:
Number of haircuts
Income after expenses
0
10
20
–150
–60
30
30
40
50
3. Here are the mean monthly temperatures in Fahrenheit for the city of Albany. January is month 1, February is month 2,
and so on. Do you think that these data will produce a linear graph? Why of why not? Graph the function.
Month
1
2
3
4
5
6
7
8
9
10
11
12
Temperature
21
23
34
47
58
67
71
69
61
51
39
26
You may use whatever city you like. These temperatures are available in The World Almanac and elsewhere.
Fig. 7. Students graph realistic problems.
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