To Be or Not to Be a Function?cdn.carnegielearning.com/assets/page-images/To_Be_or_Not...Determine whether Graphs A –L are linear or non-linear graphs. a. Sort the graphs into two

To Be or Not to Be a Function?

Understanding, Not Memorizing

Sami Briceño Texas Lead Manager of School Partnerships

[email protected]

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14 Chapter 1 Quantities and Relationships

1

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8.2 Describing Characteristics of Graphs • 361

Have you ever followed a trail of animal tracks? For expert animal trackers,

there are many more signs to look for instead of just paw prints. Expert trackers

look for rub, like when a deer scrapes velvet off its antlers. They look for

chews—where a twig or section of grass has been eaten. If there is a clean cut on

the plant, it may likely have been caused by an animal with incisors (like a rodent).

If the plants have teeth marks all over them, those plants may likely have been

eaten by a predator.

And of course, trackers look for scat, or droppings. From scat, trackers can tell an

animal’s shape and size and what the animal eats. Tubular scat may come from

raccoons, bears, and skunks. Teardrop-shaped scat may come from an animal in

the cat family. How do you follow clues in mathematics to solve problems?

Every Graph Tells a StoryDescribing Characteristics of Graphs

Key Terms discrete graph

continuous graph

linear graph

collinear points

non-linear graph

Learning GoalsIn this lesson, you will:

Describe characteristics of graphs using mathematical

terminology.

Describe a real-world situation that could be represented

by a given graph.

Incisors are the sharp teeth in humans and

animals!

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✂

A

x8

6

8

4

4

620

2

0

y B

x8

6

8

4

4

62

2

y

00

C

x8

6

8

4

4

62

2

y

00

D

x8

6

8

4

4

62

2

y

00

E

x8

6

8

4

4

62

2

y

00

F

x8

6

8

4

4

62

2

y

00

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✂

G

x8

6

8

4

4

62

2

y

00

H

x8

6

8

4

4

62

2

y

00

I

x8

6

8

4

4

62

2

y

00

J

x8

6

8

4

4

62

2

y

00

K

x8

6

8

4

4

62

2

y

00

L

x8

6

8

4

4

62

2

y

00

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6. Determine if the graphs you cut out are discrete or continuous.

a. Sort the graphs into two groups: those graphs that are discrete and those graphs

that are continuous.

b. Record your findings in the table by writing the letter of each graph.

Discrete Graphs Continuous Graphs

7. Determine if the graphs are increasing, decreasing, both increasing and decreasing,

or neither increasing nor decreasing.

a. Analyze each graph from left to right.

b. Sort the graphs into four groups: those that are increasing, those that are

decreasing, those that are both increasing and decreasing, and those that are

neither increasing nor decreasing.

c. Record your findings in the table by writing the letter of each graph.

Increasing DecreasingBoth Increasing and Decreasing

Neither Increasing nor Decreasing

A lineargraph is a graph that is a line or a series of collinear points. Collinearpoints are

points that lie in the same straight line. A non-lineargraphis a graph that is not a line and

therefore not a series of collinear points.

8. Determine whether Graphs A–L are linear or non-linear graphs.

a. Sort the graphs into two groups: those that are linear and those that are non-linear.

b. Record your findings in the table by writing the letter of each graph.

Linear Graph Non-linear Graph

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8.3 Defining and Recognizing Functions • 373

Key Terms mapping

set

relation

input

output

function

domain

range

scatter plot

vertical line test

Learning GoalsIn this lesson, you will:

Define relation and function.

Determine whether a relation (represented as a

mapping, set of ordered pairs, table, sequence, graph,

equation, or context) is a function.

To Be or Not To Be a Function?Defining and Recognizing Functions

In September 2009, museum volunteers in England began work on restoring the

WITCH machine—regarded as the first modern computer still able to work. This

huge computer, as long as an entire wall in a large room, was built starting in 1949

and was functional until 1957.

WITCH was used to perform mathematical calculations, but instead of typed input,

the computer had to be fed paper tape for inputs. Then, the computer would

produce its output on paper as well. Even though it was so huge, WITCH could only

perform calculations as fast as a human with a modern calculator.

What types of inputs and outputs do modern computers use and produce? How

does a modern computer turn inputs into outputs?

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374 • Chapter 8 Linear Functions

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Problem 1 Analyzing Ordered Pairs

As you learned previously, ordered pairs consist of an x-coordinate and a y-coordinate.

You also learned that a series of ordered pairs on a coordinate plane can represent a

pattern. You can also use a mapping to show ordered pairs. Mappingrepresents two sets

of objects or items. An arrow connects the items together to represent a relationship

between the two items.

1. Write the set of ordered pairs that represent a relationship in each mapping.

a. 1

2

3

4

1

3

5

7

b. 1

2

3

4

5

1

3

5

7

c. 1

2

3

4

5

1

3

5

7

d. 2

4

6

8

7

9

2

20

2. Create a mapping from the set of ordered pairs.

a. {(5, 8), (11, 9), (6, 8), (8, 5)} b. {(3, 4), (9, 8), (3, 7), (4, 20)}

When you write out the ordered pairs for a mapping, you are writing a set of ordered pairs.

A set is a collection of numbers, geometric figures, letters, or other objects that have

some characteristic in common.

“Use brackets, { }, to denote

a set.

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3. Write the set of ordered pairs to represent each table.

a. Input Output

210 220

25 210

0 0

5 10

10 20

b. x y

20 210

10 25

0 0

10 5

20 10

4. Determine whether the relationships in Question 3 are proportional or non-

proportional. Explain your reasoning.

The mappings or ordered pairs shown in Questions 1 through 3 form relations. A relation

is any set of ordered pairs or the mapping between a set of inputs and a set of outputs.

The first coordinate of an ordered pair in a relation is the input,and the second coordinate

is the output. A functionmaps each input to one and only one output. In other words, a

function has no input with more than one output. The domain of a function is the set of all

inputs of the function. The range of a function is the set of all outputs of the function.

1

2

3

4

1

3

5

7

In the mapping shown the domain is {1, 2, 3, 4} and the range

is {1, 3, 5, 7}.

This mapping represents a function because each input, or domain

value, is mapped to only one output, or range value.

Notice the use of set notation when writing

the domain and range.

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5. State why the relation in the example shown is not a function.

6. State the domain and range for each relation in Questions 2 and 3. Then, determine

which relations represent functions. If the relation is not a function, state why not.

1

2

3

5

4

1

3

5

7

In the mapping shown the domain is {1, 2, 3, 4, 5} and the

range is {1, 3, 5, 7}.

This mapping does not represent a function.

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7. Review and analyze Emil’s work.

EmilMy mapping represents

a function.

1

2

4

7

3

5

96

12

Explain why Emil’s mapping is not an example of a function.

Problem 2 Analyzing Contexts

Read each context and decide whether it fits the definition of a function.

Explain your reasoning.

1. Input: Sue writes a thank-you note to her best friend.

Output: Her best friend receives the thank-you note in the mail.

2. Input: A football game is being telecast.

Output: It appears on televisions in millions of homes.

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3. Input: There are four puppies in a litter.

Output: One puppy was adopted by the Smiths, another by the Jacksons, and the

remaining two by the Fullers.

4. Input: The basketball team has numbered uniforms.

Output: Each player wears a uniform with her assigned number.

5. Input: Beverly Hills, California, has the zip code 90210.

Output: There are 34,675 people living in Beverly Hills.

6. Input: A sneak preview of a new movie is being shown in a local theater.

Output: 65 people are in the audience.

7. Input: Tara works at a fast food restaurant on weekdays and a card store

on weekends.

Output: Tara’s job on any one day.

8. Input: Janelle sends a text message to everyone in her contact list on her cell phone.

Output: There are 41 friends and family on Janelle’s contact list.

9. Create your own context problem, and decide whether it represents a function. Trade

with a partner, and solve your partner’s problem. Then, discuss your responses.

Input:

Output:

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Problem 3 Analyzing Sequences

1. Determine if each sequence represents a function.

Explain why or why not. If it is a function, identify its domain

and range.

a. 2, 4, 6, 8, 10, …

b. 1, 0, 1, 0, 1, …

c. 0, 5, 10, 15, 20, …

2. What do you notice about each answer in Question 1? What conclusion can you

make about sequences?

3. Do any of the sequences in Question 1 represent proportional relationships? Explain

your reasoning.

Remember a sequence has

a term number and a term value.

Think about the mappings as ordered pairs.

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Problem 4 Analyzing Graphs

A relation can be represented as a graph. Graphs A–L from Lesson 8.2 provide examples

of graphical representations of relations.

A scatterplot is a graph of a collection of ordered pairs that allows an exploration of the

relationship between the points.

1. Determine if these scatter plots represent functions. Explain your reasoning.

a.

x8

6

8

94

4

62

2

73 510

y

9

5

7

3

1

0

Out

put

Input

b.

x8

6

8

94

4

62

2

73 510

y

9

5

7

3

1

0

Out

put

Input

Keep in mind, a function maps each

input to one and only one output.

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c. How could you change the location of point(s) in each of the graphs to create

functions that would represent proportional relationships?

The verticallinetest is a visual method used to determine whether a relation represented

as a graph is a function. To apply the vertical line test, consider all of the vertical lines that

could be drawn on the graph of a relation. If any of the vertical lines intersect the graph of

the relation at more than one point, then the relation is not a function.

Review the scatter plot shown.

x8

6

8

94

4

62

2

73 510

y

9

5

7

3

1

0

In this scatter plot, the relation is not a function. The input value 4

can be mapped to two different outputs, 1 and 4. Those two

outputs are shown as intersections to the vertical line segment

drawn at x 5 4.

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2. Use the vertical line test to determine if each graph represents a function.

Explain your reasoning.

a.

x8

6

8

94

4

62

2

73 510

y

9

5

7

3

1

0

b.

x8

6

8

94

4

62

2

73 510

y

9

5

7

3

1

0

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Problem 5 Analyzing Equations

So far, you have determined whether a set of data points in a scatter plot represents a

function. You can also determine whether an equation is a function.

The given equation can be used to convert yards into feet. Let x

represent the number of yards, and y represent the number of feet.

y 5 3x

To test whether this equation is a function, first, substitute values

for x into the equation, and then determine if any x-value can be

mapped to more than one y-value. If each x-value has exactly one

y-value, then it is a function; otherwise, it is not a function.

In this case, every x-value can be mapped to only one y-value.

Each x-value is multiplied by 3. Some examples of ordered pairs

are (2, 6), (10, 30), and (5, 15). So, this equation is a function.

1. Determine whether each equation is a function. List three ordered pairs that are

solutions to each. Explain your reasoning.

a. y 5 5x 1 3

b. y 5 x2

c. y 5 | x|

So, if two different inputs go to

the same output, that's still a function.

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Problem 6 Function or Not?

1. Sorting Activity

a. Carefully cut out Relations M through X on

the following pages.

b. Refer to Graphs A through L from Lesson 2.

c. Sort Relations A through X into two groups: those that are

functions and those that are not functions.

d. Record your findings in the table by writing the letter of

each relation.

Functions Not Functions

Time to get your graphs from the second lesson

back out.

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M

10

11

12

13

1000

2000

3000

N

10

11

12

13

1000

2000

3000

O

Input Output

22 4

21 1

0 0

1 1

2 4

P

x y

2 24

1 21

0 0

1 1

2 4

Q

The terms of a sequence:

7,10,13,16,19,…

R

The terms of a sequence:

10,30,10,30,10,…

✂

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S

The set of ordered pairs

{(2,3),(2,4),(2,5),

(2,6),(2,7)}

T

The set of ordered pairs

{(2,1),(3,1),(4,1),

(5,1),(6,1)}

U

y5x211

V

This equation is used to

calculate the number of inches

in a foot:

y 5 12x

Let x represent the number of

feet and y represent the number

of inches.

W

Input:

The morning announcements

are read over the school

intercom system during

homeroom period.

Output:

All students report to homeroom

at the start of the school day to

listen to the announcements.

X

Input:

Each student goes through the

cafeteria line.

Output:

Each student selects a lunch

from the menu.

✂

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Talk the Talk

Choose the appropriate description to complete each sentence.

1. A relation is (always, sometimes, never) a function.

2. A function is (always, sometimes, never) a relation.

Be prepared to share your solutions and methods.

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Vol. 94, No. 9 • December 2001 745

What Do Students Really Know about Functions?

Lisa L. Clement

How would your students respond to the follow-ing item?

A caterpillar is crawling around on a piece ofgraph paper, as shown below. If we wished todetermine the creature’s location on the paperwith respect to time, would this location be afunction of time? Why or why not?

Readers might be surprised to learn that 60 percentof the precalculus students who answered this itemapplied the vertical-line test directly to the path ofthe caterpillar to determine whether the caterpil-lar’s location was a function of time, that is, that forevery point in time, the caterpillar was at exactlyone location. What does this response tell us aboutstudents’ understanding of functions, the vertical-line test, and interpreting graphs? Read further tofind out.

The concept of function plays an important rolethroughout the mathematics curriculum. The typi-cal mathematics definition of function from x to y isa correspondence that associates with each elementof x a unique element of y.

The concept of function is central to students’ability to describe relationships of change betweenvariables, explain parameter changes, and inter-pret and analyze graphs. Not surprisingly Princi-ples and Standards for School Mathematics (NCTM2000, p. 296) advocates instructional programsfrom prekindergarten through grade 12 that“enable all students to understand patterns, rela-tions, and functions.” Although the function conceptis a central one in mathematics, many researchstudies of high school and college students haveshown that it is also one of the most difficult forstudents to understand (Tall 1996; Sierpinska1992; Markovits, Eylon, and Bruckheimer 1988;Dreyfus and Eisenberg 1982).

ALIGNMENT OF CONCEPT IMAGEWITH MATHEMATICAL DEFINITIONVinner (1992) uses the term concept image todescribe how students—as well as adults, for that

matter—think about concepts. A person’s conceptimage consists of all the mental pictures that he orshe associates with a given concept. A student’sconcept image can differ greatly from a mathemati-cally acceptable definition; and students’ conceptimages are often very narrow, or they may includeerroneous assumptions.

A student’s concept image of function may, forexample, be limited to the graph of a relation thatpasses the vertical-line test or to a machine thatfurnishes an output when an input is supplied.Vinner identifies several other common aspects ofstudents’ concept images of functions. One is thatstudents believe that a function should be system-atic. “An arbitrary correspondence is not considereda function” (Vinner 1992, p. 200). Other commonaspects of students’ concept images include the following:

• A function should be given by a single rule. Forexample, a function with a split domain is oftenconsidered as two or more functions, dependingon how the domain is split.

• The graph of a function should be continuous.For example, students do not generally considerthe graph of the greatest-integer function to be arepresentation of a function.

• A function should be one-to-one, that is, func-tions have the additional property that for eachelement in the range, exactly one element existsin the domain. For example, f (x) = 12 is often notconsidered a function, since it is not one-to-one(Markovits, Eylon, and Bruckheimer 1988).

Edited by Barbara [email protected] State UniversityCorvallis, OR 97331-4605Karen [email protected] of New HampshireDurham, NH 03824

Lisa Clement, [email protected], teaches at SanDiego State University, San Diego, CA 92182. She is inter-ested in exploring students’ thinking about algebra andrational numbers.

Precalculusstudentsapplied thevertical-linetest directlyto the path of the caterpillar

Copyright © 2001 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM.

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746 MATHEMATICS TEACHER

These overly narrow views impede students’ abilityto determine functionality except in very specificprototypical instances.

Students often believe that a function mustinclude some algebraic formula. However, someimages that students have about the concept offunction—for example, that a function must havean explicit analytical representation—correspondwith historical concepts of functions (Tall 1996).Students may erroneously consider y = !¡x[̀ –̀ 3a function, since it is an algebraic formula; whereasthey might not consider the correspondence thatMary owes $6, John owes $3, and Sue owes $2 to bea function, since no formula “fits it.” Williams (1998)revealed that 26 percent of university students whowere enrolled in a third-quarter calculus coursepersisted with a similar equation-bound view offunctions. The students’ concept images then pre-vented them from identifying some relations thatactually are functions, whereas the studentsaccepted as functions some relations that are not.

This article explores one theme related to under-standing functions—developing a concept imagethat is well aligned with the mathematical defini-tion. This exploration includes sharing assessmentitems that teachers can use to make sense of theirstudents’ thinking surrounding this theme, describ-ing students’ responses to these items, and inter-preting those responses. Most of the assessmentitems were selected from items used in previousresearch studies. Twenty-eight such items weregiven at the end of the school year to thirty-fiveprecalculus students who attended a public highschool known for its academic excellence. Five ofthe thirty-five students were then interviewed.These students were selected to represent studentswho scored in the high (two students), middle (twostudents), and low (two students) ranges on thepaper-and-pencil assessment. Readers may want tolook at the five items included in figure 1 beforereading further, so that they can reflect on theitems and view the percent of students whoanswered each item correctly.

STUDENTS’ RESPONSES TO ASSESSMENT ITEMSFigure 1 presents items that are related to under-standing students’ concept images of functions. Oneitem asked students to furnish their own definitionof a function. This item allowed the author toassess students’ understanding of function. Onlyfour of the thirty-five precalculus students couldgive a definition that was similar to the mathemati-cal definition of function, which recognizes thatevery element in the domain must be mapped toexactly one element in the range. One studentwrote, “A function is a relation between two vari-ables in which one variable is dependent upon the

Assessment Items to Determine Alignment of Students’ Concept Image of Functions versus the Mathematical Definition

(The percent of students answering item correctly is listed in parentheses.)

1. Given the following graphs of f (x) and g(x), graph f (x) + g(x). (26%)

(Source: Lisa L. Clement, Alba Thompson, and Patrick Thompson)

2. Circle the graphs that indicate that y is a function of x on the Cartesian coordinate system. Assume that the y-axis is the vertical axis. (40%)

(Source: “Horizontal and Vertical Growth of the Students’ Conception of Function,” by Keith Schwindgendorf, Julie Hawks, and Jennifer Beineke. In The Concept of Function: Aspects of Epistemology and Pedagogy, edited by Guershon Harel and Ed Dubinsky, pp. 133–52. Washington, D.C.: Mathematical Association of America, 1992.)

3. Which of the following indicate that y is a function of x? Circle those that arefunctions. (51.4%) a) y = x2 – 4 d) x2 + y2 = 25b) y = x/2 e) y = ex

c) xy = 8 1 if x ∈ rationalsf ) y = –1 otherwise

g) If we let x = club member’s name and y = amount owed, is y a function of x?

Club Members’ Dues StatusName Owed Name Owed Sue $17 Iris 6John 6 Eve 12Sam 27 Henry 14Ellen 0 Louis 6

(Source for items 3(a) – 3( f ) is “Teachers’ Mathematical Knowledge of the Con-cept of Function,” by Alexander Norman, in The Concept of Function: Aspects ofEpistemology and Pedagogy, edited by Guershon Harel and Ed Dubinsky, pp.215–32 [Washington D.C.: Mathematical Association of America, 1992]. Sourcefor item 3(g) is “The Nature of the Process Conception of Function,” by EdDubinsky and Guershon Harel, on pages 85–106 of the same work.)

4. Define in your own words, the mathematical concept of function. (11.4%)(Source: Lisa L. Clement, Alba Thompson, and Patrick Thompson)

5. A caterpillar is crawling around on a piece of graph paper, as shown below. If wewished to determine the creature’s location on the paper with respect to time,would this location be a function of time? Why or why not? Can time be described as a function of its location? Explain. (Two of the five students who were interviewed answered correctly.)(Source: Lisa L. Clement, Alba Thompson, and Patrick Thompson)

Fig. 1

a. d.

b. e.

c. f.

g.

f (x)

g(x)

Page 23 of 25

Vol. 94, No. 9 • December 2001 747

value of the other variable. For each independentvariable, there must be only one value for thedependent variable.” More typical, however, waseither an image of a machine—that is, when num-bers are entered, numbers are produced, or animage of a graph that passes the vertical-line test.One student wrote, “A group of points which, whengraphed, can have a vertical line drawn throughthem and touch the graph exactly once.” Theresponses of twenty of the thirty-five students, or57 percent, were similar to one of the latter tworesponses.

Although these notions may initially help stu-dents think about characteristics of some functions,students who continue to hold only these images offunctions may either become overly restrictive ornot restrictive enough in deciding whether a partic-ular expression represents a function. For example,students who think only of the vertical-line testwhen determining functionality may believe thatexpressions that they cannot view in graphical formare not functions. These students seem to makethese decisions without ever considering a formaldefinition of function.

When students had to determine whether givenrelations were functions, they succeeded withitems that are considered prototypes of functionsand nonfunctions, as in items 2 and 3. Forinstance, in questions that asked whether y was afunction of x, the majority of students correctly cir-cled the graph of y = x2 and correctly did not circlethe graph of x = y2. These graphs are often amongthe most common examples and nonexamplesshown to students.

Students ran into more difficulty when the func-tion given was not a prototype. For example, moststudents did not consider a graph of a noncontinu-ous (greatest-integer) function and a table showingclub members’ names and the amount of moneythat each owed to be functions, even though theyare both functions. In this situation, students werenot using the actual mathematical definition of afunction to determine functionality but were usingtheir own concept image. The first set of studentsmay have a concept image that includes the condi-tion of continuity, whereas the second group of stu-dents may have a concept image that includes thecondition of analytic representation.

During the interviews, students explained theirresponses to item 2, in which they had to determinewhether certain graphs of relations were functionswith respect to x. All five of the students who wereinterviewed said that they used the vertical-linetest to determine whether each item was a func-tion, and three of them stated something similar tothe student who said, “If you have one certain valuefor x, it has to have one and only one y-value.”Pointing to (b), the student indicated, “This one

doesn’t work because this one x has two y-values.” Students tended to recognize functions most often

when the functions were familiar to them. That is,prototypes of functions were a greater factor indetermining functionality than even the vertical-line test. One student used the vertical-line test butmodified its use on the basis of her experience:“Well, item (a) is a function because it is a parabolaand it is always given as an example of a function.”Regarding the singleton point, she said, “I don’tthink a point can be a function.” Finally, given ahorizontal line, she stated, “I guess by my definitionit would be, but I don’t think it is.” That student’sconcept image seemed to involve a vertical-line testtempered by examples of functions that she hadencountered in the past. Another student indicatedthat his concept image of function required that itpass both the vertical-line test and a condition ofcontinuity. He stated that the greatest-integer func-tion was not a function, since the pieces were notconnected. The author asked, “So to be a function,does the graph have to be continuous?” He replied,“I think so; every one I have ever seen has been.”

As mentioned at the beginning of the article,three students who were interviewed about thecaterpillar question in item 5 looked at the path ofthe caterpillar as a graph and determined that itwas not a function on the basis of the vertical-linetest. The other two students, however, drew location-versus-time graphs and determined that the cater-pillar’s location was indeed a function with respectto time. A student explained, “As time progresses,the reason that it is a function is that you can’t gobackwards in time, so even though the caterpillarcircles around, that’s at a different time. . . . ”

CONCLUSIONSThe concept of function is central to understandingmathematics, yet students’ understanding of func-tions appears either to be too narrowly focused or toinclude erroneous assumptions. These results sug-gest at least two implications for teaching andassessment.

First, although time is devoted to teaching func-tions (high school courses often even have the wordfunction in the title), perhaps this time should bespent discussing functions in different ways. Someof the students’ overly narrow views of, or erro-neous assumptions about, functions may be causedat least in part by what students see in textbooksas the prototypes of functions. Teachers also mayoften furnish students with examples of functionsthat fit the types of functions most often seen intextbooks. As a teacher, I certainly gave prototypi-cal examples to my students as a way, I thought, tohelp them better understand the concept of func-tions. I was thus surprised when I found that theassessment items that I gave to students indicated

A graph of a greatest-integer function was not considered a function by most students

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748 MATHEMATICS TEACHER

a large disparity between a mathematical definitionof function and students’ concept images of func-tions. When providing examples to my students, Inever made explicit to them—or better yet, neverhad them make explicit for themselves—theaspects of the definition of function that they wereattending to when they made determinations offunctionality. We never discussed deeply the defini-tion of function, the different ways to representfunctions, and the connections between the two.

Second, assessment instruments—and ways ofassessing students’ thinking—should be changed tobetter determine what students understand aboutthe underlying concept of function. The items infigure 1 furnish some ways to assess students’ con-cept image of functions, and the students’ responsesin this article indicate a range of possible studentresponses to the items. The items may also furnishan occasion for teachers to have mathematical con-versations about functions with students. Forexample, the interviews with students shed muchmore light on how students were thinking than thepaper-and-pencil assessment alone. Teachers canuse the items to assess students before or after theyhave completed a unit, or they can use a variety ofthe items as questions for generating class discus-sion. The items are appropriate to use in coursesranging from algebra through calculus.

Although I have discussed in this article only onemajor theme related to understanding the conceptof function—developing a concept image that is wellaligned with the mathematical definition—teacherscan begin to help their students understand func-

tions by assessing in multiple ways their currentunderstandings and building from there. Best ofluck in your journey to investigate your students’thinking about functions.

REFERENCESDreyfus, Tommy, and Theodore Eisenberg. “Intuitive

Functional Concepts: A Baseline Study on Intu-itions.” Journal for Research in Mathematics Educa-tion 13 (November 1982): 360–80.

Dubinsky, Ed, and Guershon Harel. “The Nature ofthe Process Conception of Function.” In The Conceptof Function: Aspects of Epistemology and Pedagogy,edited by Guershon Harel and Ed Dubinsky, pp.85–106. Washington, D.C.: Mathematical Associa-tion of America, 1992.

Markovits, Zvia, Bat Sheva Eylon, and Maxim Bruckheimer. “Difficulties Students Have with theFunction Concept.” In The Ideas of Algebra, K–12,1988 Yearbook of the National Council of Teachers ofMathematics (NCTM), edited by Arthur Coxford, pp.43–60. Reston, Va.: NCTM, 1988.

National Council of Teachers of Mathematics (NCTM).Principles and Standards for School Mathematics.Reston, Va.: NCTM, 2000.

Norman, Alexander. “Teachers’ Mathematical Knowl-edge of the Concept of Function.” In The Concept ofFunction: Aspects of Epistemology and Pedagogy,edited by Guershon Harel and Ed Dubinsky, pp.215–32. Washington, D.C.: Mathematical Associa-tion of America, 1992.

Schwindgendorf, Keith, Julie Hawks, and JenniferBeineke. “Horizontal and Vertical Growth of the Stu-dents’ Conception of Function.” In The Concept ofFunction: Aspects of Epistemology and Pedagogy,edited by Guershon Harel and Ed Dubinsky, pp.133–52. Washington, D.C.: Mathematical Associa-tion of America, 1992.

Sierpinska, Anna. “On Understanding the Notion ofFunction.” In The Concept of Function: Aspects ofEpistemology and Pedagogy, edited by GuershonHarel and Ed Dubinsky, pp. 25–58. Washington,D.C.: Mathematical Association of America, 1992.

Tall, David. “Functions and Calculus.” In Internation-al Handbook of Mathematics Education, Part 1, vol.4, edited by Alan J. Bishop, Ken Clements, ChristineKeitel, Jeremy Kilpatrick, and Colette Laborde, pp.289–325. Dordrecht, Netherlands: Kluwer AcademicPublishers, 1996.

Vinner, Shlomo. “The Function Concept as a Prototypefor Problems in Mathematics Learning.” In The Con-cept of Function: Aspects of Epistemology and Peda-gogy, edited by Guershon Harel and Ed Dubinsky,pp. 195–214. Washington, D.C.: Mathematical Asso-ciation of America, 1992.

Williams, Carol G. “Using Concept Maps to AssessConceptual Knowledge of Functions.” Journal forResearch in Mathematics Education 29 (July 1998):414–41.

The author would like to thank Vicki Jacobs, RandyPhilipp, and Barbara Edwards for their commentsand suggestions on earlier drafts of this article. ¿

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The interviewsshed much

more light onhow students

were thinking

than did thepaper-and-

pencil assessment

alone

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To Be or Not to Be a Function?cdn.carnegielearning.com/assets/page-images/To_Be_or_Not...Determine whether Graphs A –L are linear or non-linear graphs. a. Sort the graphs into two