Alcaro, Patricia C., Alice S. Alston, and Nancy Katims. Fract.docx

Alcaro, Patricia C., Alice S. Alston, and Nancy Katims.
“Fractions Attack! Children Thinking and Talking
Mathematically.” Teaching Children
Mathematics 6 (May 2000): 562–67.
Copyright © 2000 by the National Council of Teachers of
Mathematics, Inc. www.nctm.org. All rights reserved. For
personal use only.
This material may not be copied or distributed electronically or
in other formats without written permission from NCTM.
How do we know when children are think-ing mathematically?
How do we establish
a setting that will elicit mathematical thinking?
What do we learn about the children as a result?
This article explores these questions
through two videotaped vignettes in which
fourth-grade students think and talk mathe-
matically while tackling a complex real-life
investigation called Snack Attack (from the
PACKETS Program for Upper Elementary
Mathematics, developed by Educational Testing
Service [1998] with support from the National Sci-
ence Foundation). The investigation addresses
state and national content standards involving pro-
portional reasoning with whole numbers, fractions,
and decimals, as well as the process standards of
problem solving, reasoning, communication, and
connections (NCTM 1989).

The students worked on the investigation dur-
ing several class sessions, beginning with an
introductory activity that set the context. As part
of this activity, the students received a brochure
called Food Matters that contained the following
information:
• The meaning of calorie
• The way in which calories are burned through
exercise
• The fact that the number of calories burned dur-
ing exercise varies according to the type and
duration of the exercise
562 TEACHING CHILDREN MATHEMATICS
Patricia C. Alcaro,
Alice S. Alston, and
Nancy Katims
Pat Alcaro, [email protected], teaches fourth grade at Point
Road School in Little Silver, NJ
07739. Alice Alston, [email protected], teaches mathematics
education at Rutgers Uni-
versity in New Brunswick, NJ 08901. Nancy Katims,
[email protected], is the
director of assessment, research, and evaluation for the
Edmonds School district in Edmonds,
WA 98026. She was formerly the PACKETS project director
with Educational Testing Service.
Fractions
Attack!

Children Thinking and
Talking Mathematically These students were videotaped during
workwith the vignettes.
www.nctm.org
• The number of calories burned in ten minutes of
doing various exercises (shown as a graphic; see
fig. 1)
After the introductory activity, the students
began working in small groups on the investiga-
tion. The children’s task was to develop a method
to figure out how much time people must exercise
to burn off the calories in snacks that they eat. Each
group had a chart listing the five exercises from the
brochure and the number of calories in seven dif-
ferent snacks (see fig. 2).
Students used a variety of mathematical ideas
and strategies to find the exercise times needed to
burn the calories in a snack of their choice. Then
they tested their methods using a second snack.
The groups worked to explain and justify their
solutions in the format of letters. Each group pre-
sented its solutions, approaches, and explanations
to the class. The teacher facilitated lively discus-
sions throughout the activity’s several days as stu-
dents questioned one another’s solutions, com-
pared different approaches, and worked together to
understand the mathematics involved in the differ-
ent solutions.
Transcripts of videotapes of the sessions docu-
mented the students’ mathematical activity and dis-

course. Two vignettes offer particularly interesting
illustrations of students’ emerging understandings,
as well as their confusion and misconceptions, con-
cerning proportional reasoning and fractions.
Understanding Ratio:
Vignette 1
On the second day of working on the problem, one
group (Allan, Keely, Sarah, and Paul) expressed
some disagreement among themselves and asked
for assistance. They had chosen the chocolate
cookie (55 calories) as their snack and discovered
that less than 10 minutes of jumping rope would be
required to burn the 55 calories because jumping
rope used 60 calories in 10 minutes. Keely and
Allan decided that 9 1/2 minutes would be needed
and made a table to explain their thinking. Paul
was not convinced. The teacher enlisted the entire
class in helping the students think through their
dilemma. Keely came to the chalkboard and drew
the table for the class to review (see fig. 3a).
The students knew that 10 minutes of jumping
rope would burn 60 calories, and they assumed that
9 minutes would burn 50 calories. They reasoned
that half of the 10-calorie difference, or 5 calories,
would correspond to half a minute. So 9 1/2 min-
utes would burn 55 calories. Their table showed a
continuation of this logic for 8 and 7 minutes.
The teacher asked the students how they deter-
mined that 7 minutes burned 30 calories and that 8
minutes burned 40 calories. The group replied that
its calculation was based on “10 calories for every
1 minute.” When the teacher asked the students to

begin with 1 minute and show their idea, the chil-
dren constructed another table on the chalkboard,
shown in figure 3b.
The students moaned when they realized that
according to their new table, 6 minutes of jumping
rope burned 60 calories, but that according to the
brochure, 10 minutes was supposed to burn 60
calories. The group’s first table had shown 30 calo-
ries burned at 7 minutes, but this table showed 30
calories burned at 3 minutes. Looking back at the
group’s original table, the class determined that the
only fact they knew for certain was that 10 minutes
of jumping rope burned 60 calories. Allan and
Keely agreed that their other information repre-
sented guesses, and they knew that they were sup-
563MAY 2000
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1 Exercise data from the brochure
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posed to check all mathematical guesses. When
they checked the logic of their two tables, they
realized that their guesses did not work.
Allan then posed an alternative idea. He said,
“Maybe every odd number of minutes burns calo-
ries ending in 5” and began exploring this idea by
setting up a third table, shown in figure 3c. Allan
quickly realized, however, that this notion did not
work either because 10 minutes of exercise
accounted for only 50 calories in this scenario. The
class sat silently for several seconds. Then Allan
excitedly called out, “I think it counts by sixes!”
When the teacher asked him to explain this
idea, Allan thought quietly for some time, then
said that he could not explain it. All the children
appeared to be thinking intensely about these
ideas. Jeff, another member of the class, eagerly
joined the group at the chalkboard and set up a
table in which the calories increased by sixes. The
students in Allan’s group later used this table,
shown in figure 4, in their written solution.
When the teacher asked Allan to explain Jeff’s
table and why he knew it worked, Allan said that
he had first counted by eights and could not get to
60. Then, he had counted by sevens and could not
get to 60 that way either. When he counted by

sixes, the solution worked.
Referring to Jeff’s table, the group found that 9
minutes accounted for only 54 calories. Allan
quickly said that if 1 minute was 6 calories, then
half a minute would be half of 6, or 3 calories. The
whole class agreed that 9 1/2 minutes of jumping
rope would burn 57 calories.
The groups then continued working on their
own solutions, leaving Allan’s group to think
about the time that would be needed to burn the
55 calories in a chocolate cookie. Allan’s group
wrote a final estimate of 9 1/4 minutes on its solu-
tion chart.
The Mathematics
Revealed in Vignette 1
In this vignette, the students were building a solu-
tion based on equal ratios. In the process, they
showed evidence of logical mathematical reason-
ing. Their challenge was to find the pattern that
would allow all the numbers to make sense, given
F
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3 Students’ tables for determining calories burned by jumping
rope
Minutes of Jumping Rope 7 8 9 1/2 10

Calories Burned 30 40 55 60
The group’s first table
(a)
Minutes of Jumping Rope 1 2 3 4 5 6
Calories Burned 10 20 30 40 50 60
The group’s second table
(b)
Minutes of Jumping Rope 1 2 3 4 5 6 7 8 9 10
Calories Burned 5 10 15 20 25 30 35 40 45 50
Allan’s table
(c)
F
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4 Final solution table for calories burned while jumping rope
the one piece of information that they knew was
true—that is, that 10 minutes of jumping rope
burned 60 calories. While solving the problem, the
students demonstrated their ability to use whole-
number multiplication (e.g., “. . . it counts by
sixes”), multiplication of fractions (e.g., figuring

half of 6), and appropriate number sense about
fractions (e.g., estimating 9 1/4 minutes as the
solution because 9 minutes would account for 54
calories and 9 1/2 minutes would use 57).
Understanding the
Meaning of Fractions:
Vignette 2
The next day, Jackie and Takitha asked the class
for help in figuring out how many minutes a per-
son would have to run to burn exactly 10 calories.
When the teacher asked the girls to explain what
they were thinking, Jackie pointed to the table
that her group was constructing and said, “We
needed 10 calories to burn and we needed to find
one-half of 2 1/2. You know in running how it
says for 10 minutes, we burn 80 calories. We kept
going down to see what we’d get” (See fig. 5).
The girls had started with the given informa-
tion, which was that 10 minutes of running burns
80 calories. They then divided each number by 2
to find that 5 minutes of running burns 40 calories
and repeated this procedure to find that 2 1/2 min-
utes of running burns 20 calories. They easily
divided the 20 calories by 2 to arrive at 10 calo-
ries and knew that they should do the same to the
2 1/2 minutes. This requirement, in essence, was
their dilemma.
Jackie and Takitha decided to build a concrete
representation of 2 1/2 using folded strips of paper.
They cut paper strips of equal sizes to represent
whole-number units, which they referred to as
“wholes.” They then folded and cut some of the
strips in half to form “halves.” As they generated

numbers for their solution, they taped the appropri-
ate combinations of paper strips to the board.
The class was now ready to think about Jackie
and Takitha’s problem: “What is half of 2 1/2?”
Again the class engaged in considerable discus-
sion, leading to the following demonstration:
Bonnie. This is 2 1/2, right? So we’re going to take
away one-half [of it]. So we’re going to take one-half
away from the 1/2. That is a fourth. [She took the half
strip from the board, tore it in half, and removed one
of the pieces.] If you take half away from this, . . . and
it equals 1/2. You take half away from this, and it’s
another 1/2. We cut everything in half.
As she spoke, Bonnie tore one whole strip of paper
into two equal parts and removed one part. Then
she repeated this action with the second whole
strip, leaving two half-strips and one fourth-strip
on the chalkboard.
Teacher. What is one-half of 2 1/2?
Class. 1 and 1/4.
Teacher. How did you get 1 and 1/4? I see two
halves and one fourth.
Class. Two halves makes a whole and then
there’s one fourth, so that’s 1 and 1/4.
Teacher. All right then, can you make an equa-
tion up on the board with the 2 1/2?
Class. 2 1/2 – 1/2 = 1 1/4. [As the class spoke,

Jackie wrote the equation on the board.]
Teacher. 2 1/2 – 1/2 = 1 1/4?
The teacher asked the stu-
dents to build the model of 2
1/2 again and think carefully
about what they were doing as
they acted out the problem.
When the children removed
half of the 2 1/2 a second time,
she challenged them to rethink
their equation.
Paul. I mean, 2 1/2 minus half of 2 1/2 equals
1 1/4.
Bonnie. I see.
Teacher. Bonnie, what equation do you see up
there?
Bonnie. 2 1/2 . . . half of everything . . . and this
is what we came up with, right? But what we had
left . . . what we took away from the 2 1/2 . . . was
two halves and a fourth.
Bonnie reconstructed the 2 1/2 using strips of
paper representing two wholes and one half. As she
explained her idea, she ripped each strip in half and
removed one of the pieces.
Lara. Two halves is a whole, Bonnie.
Bonnie. Two halves and a fourth. If you put
these together like that and this like that, you have
2 1/2. But you took half away. So we take 2 1/2

minus one whole and a fourth. ☛
565MAY 2000
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5 Jackie and Takitha’s incomplete table
Minutes of Running 2 1/2 5 10
Calories Burned 10 20 40 80
The students used
folded strips of paper
as concrete
representations
As she continued her explanation, Bonnie
reassembled the strips once more to show the 2 1/2.
This time, as she removed the halves from the two
units, she pieced them together to show that the
remaining amount was indeed one whole and one
quarter. Takitha, standing beside Bonnie as she
modeled the problem, wrote the following equa-

tion on the chalkboard:
2 1/2 – 1 1/4 = 1 1/4
The girls’ written solution, shown in figure 6,
described how they consistently used this strategy
for other snacks and exercises. The girls success-
fully demonstrated their ability to generalize their
solution strategy to other sets
of numbers, with the exception
of a careless calculation error
in figuring the number of min-
utes for biking.
Solution
s generated by
other students in the class also
demonstrated learning from
the discussion about using
paper strips. For example,
Bonnie and Lara’s group also
included a set of paper strips in
the group’s solution. Their explanation and chart
indicated that they used these paper strips to help

them add the fractions 1/2, 1/4, and 1/8 to find
what portion of a minute of running is required to
burn exactly 7 calories.
Mathematics Revealed
in Vignette 2
Just as in the first vignette, the students here were
building on their intuitive understanding of ratio
and proportion, but the essence of the vignette
focused on the meaning of fractions. The students
were dividing a mixed number into two equal parts.
This mathematical concept is difficult for students
to handle, and it is frequently taught as a mechani-
cal rule that they memorize and apply without true
understanding. In this instance, because the stu-
dents decided to use a concrete representation to
help them solve their problem, a meaningful solu-
tion was within their grasp. In the process, they
demonstrated their understanding of such concepts
as “two quarters make a half,” “two halves make a
whole,” and “half of 5 is 2 1/2.” Equally important,
students were able to use the paper strips as tools to
help them solve similar problems.

Closing Observations
In assessing the students’ performance in this activ-
ity, the teacher was struck by several important
observations. For example, she observed that
although the children appeared to understand the
concept of ratio when using a table, they did not
attempt to check whether their initial answers made
sense. If the students in the first vignette had been
left unchallenged, they would have been quite con-
tent to submit their untested tables to the teacher.
When the teacher asked the class to think together
about the entries in the first table, however, the stu-
dents engaged in a thoughtful reasoning process
that resulted in solutions that made sense.
An important insight gained from the second
vignette was the fragile nature of the students’
grasp of particular concepts. Even after asserting
and proving concretely that one-half of 2 1/2 is
1 1/4, the students had difficulty expressing this
action with an equation. In this example, although
it looked as if the students had mastered the con-
cept, their first symbolic expression showed that
their understanding was not yet complete. It is
important for educators to help students connect

concrete, verbal, and symbolic representations in
ways that build meaning and help students develop
precision in their use of mathematical language.
The two vignettes share interesting characteris-
tics, some of which reflect student behaviors and
others, teacher behaviors. Both vignettes illustrate
powerful mathematical thinking on the part of the
students. The students—
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6 Jackie and Takitha’s written explanation
We need to help
students connect
concrete, verbal,

and symbolic
representations
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• posed and solved their own mathematical

questions;
• were undaunted by the challenging nature of
some of the questions;
• persevered through a series of attempts to a suc-
cessful solution;
• used basic calculations and skills, coupled with
logic, reasoning, and higher-order thinking;
• respected one another’s ideas and worked
together to build a solution, showing true coop-
eration rather than competitiveness in their
interactions; and
• were willing to admit when they did not under-
stand something.
The teacher clearly played an important role, also,
by—
• bringing the whole class together to help an
individual group grapple with a challenging
problem;

• using the simple technique of saying “I don’t
understand” or “I’m confused” to encourage
students to explain their thinking more clearly;
• making no assumptions and insisting that the
students clarify their statements every step of
the way; and
• ensuring that all the students understood each
step of the process, asking other students to
explain an idea rather than explain it herself.
Admittedly, for this class, certain concepts
involving fractions were still fragile, but the stu-
dents appeared to be constructing powerful mental
images of how these mathematical processes
work. As teachers make decisions about how to
cover the prescribed curriculum, this type of task
may become increasingly helpful. Such activities
may be used both for instruction and as assess-
ment tools to reveal and document how students
build models of mathematical ideas that help them
make sense of basic skills and procedures, then
use these models in unfamiliar mathematical situ-

ations that call for higher levels of thinking (Lesh
and Lamon 1992).
References
Educational Testing Service (ETS). PACKETS Program for
Upper Elementary Mathematics. Princeton, N.J.: ETS,
1998.
Lesh, Richard, and Susan J. Lamon. “Assessing Authentic
Mathematical Performance.” In Assessment of Authentic
Performance in School Mathematics, edited by Richard
Lesh and Susan J. Lamon, 17–62. Washington, D.C.: Amer-
ican Association for the Advancement of Science, 1992.
National Council of Teachers of Mathematics (NCTM). Cur-
riculum and Evaluation Standards for School Mathematics.
Reston, Va.: NCTM, 1989. ▲
567MAY 2000
POLI 102 Chapter 1 - Quiz 1
Thursday January 21, 2016 (Due in class on Friday, January

29-late submissions will not be accepted)
Note: This quiz constitutes 10% of the overall grade
True OR False?
1. The two-party competition has helped states’ resurgence.
2. Capacity is the ability of government to respond effectively
to change, to make decisions efficiently and responsively, and
to manage conflict.
3. Government efficiency refers to its accomplishing what it
sets out to do.
4. The key point of this chapter is that the non-national govts
play a central role in the federal system.
5. Federalism is a system in which powers are shared between
national and non-national govts.
6. Over the past thirty years, state governments have become
more dependent on the federal government for revenues.
7. To deal with the complexities of governing, in the 19900,
states became more reactive rather than proactive.
8. As a rule, state governments prefer to increase the so-called
“sin” taxes on alcohol and tobacco, and only reluctantly raise

sales and income taxes.
9. Among the factors that contributed to state resurgence are
reformed constitutions and institutions, and the presence of
active state and local lobbyist organizations at the national
level.
10. States have become increasingly innovative in inventing
ways to enhance their revenue systems.
11. The first Amber Alert system to broadcast information about
abducted children is an example of a federally initiated program
adopted by the states.
12. Interstate cooperation fosters a healthy climate for joint
problem solving among the states.
13. Tension between the national and non-national governments
is minimal in a federal system.
14. Unfunded mandates imposed by federal legislation are
welcomed by the states because they provide much needed
guidance.
15. In the area of economic development, inter-jurisdictional
conflict is a common problem.

16. New ideas and programs implemented at the state level.
17. Unfunded mandates imposed by federal legislation often
create a financial burden for states.
18. Fiscal stress refers to the pressures created when
expenditures are greater than revenues.
19. When states bid against one another for economic
development they often use tax breaks and regulatory relaxation
to attract business and industry.
20. Population trends of the last decade have resulted in
population shifts from the Frostbelt states to Sunbelt states.
21. When political conflicts emerge from deeply held moral
values on issues such as gay rights, abortion, or pornography it
is informally known as culture wars.
22. The unique characteristics of the fifty states, as described in
your text, are diversity, competitiveness and resiliency.
23. The authors of the textbook conclude that revitalized state
and local governments are taking charge and producing results.

24. According to your text, what defines the U.S. federal
system?
a) cooperation and conflict
b) intervention and intrusion
c) capacity and conflict
d) interdependence and autonomy
25. During what time period were the states characterized as
“havens of traditionalism and inactivity”?
a) the 1950s and 1960s
b) the 1960s and 1970s
c) the 1970s and 1980s
d) the 1980s and 1990s
26. All of the following have contributed to the resurgence of
the states except
a) more equitable representation regarding legislative
apportionment.
b) the extension of two-party competition.
c) a reduced focus on lobbying efforts.
d) a restructuring of state institutions.
27. Economic downturns and limits on taxing and spending have
caused states to
a) continue to depend wholly on federal revenue sharing.

b) implement new and innovative revenue-raising strategies.
c) resist efforts to grant local governments flexibility.
d) continue to depend on federal grants-in-aid as their primary
source of revenue.
28. State rainy day funds, legalized gambling through state-run
lotteries and pari-mutuels, and extension of the sales tax to
services are examples of
a) efforts that have generally failed to generate additional
revenue.
b) expenditure equity strategies.
c) tax equity strategies.
d) revenue diversification strategies.
29. New ideas and programs implemented at the state level
a) are the result of the lobbying efforts of the states.
b) are usually an extension of a federal program already in
place.
c) are usually generated by the private sector.
d) are spread rapidly between jurisdictions as states learn from
one another.
30. Increased national-state conflict seems the inevitable result
of
a) more capable state and local governments.

b) the federal government’s efforts to control the state’s ability
to respond to change.
c) the states’ dependency on federal programs and revenue
sources.
d) the state courts’ involvement in national-state issues.
31. Unfunded mandates imposed by federal legislation
a) are welcomed by the states because they provide much
needed guidance.
b) were once a source of considerable irritation to the states but
the Unfunded Mandate Act passed by Congress now provides
ample federal funds to cover mandates.
c) have eased the tension between the federal government and
the states.
d) often create a financial burden for states.
32. Fiscal stress refers to
a) the pressures states and local governments face regarding
interjurisdictional problems.
b) the pressures states and local governments face regarding
interstate conflicts.
c) the pressures created when expenditures are greater than
revenues.
d) the pressures created when revenues are greater than
expenditures.

33. When states bid against one another for economic
development they
a) enhance interstate cooperation.
b) often use tax breaks and regulatory relaxation to attract
business and industry.
c) rely on federal guidelines to ensure business incentives are
equal between competing states.
d) usually work out agreements that allow other states to share
in the economic benefits.
34. What term does your text use to refer to a characteristic of
government that is open and understandable, one in which
officials are accountable to the public?
a) honesty
b) visibility
c) up-frontness
d) transparency
35. According to Census Bureau data, the fastest growing states
during the period of 2000 to 2009 were located in the
a) West.
b) Midwest.
c) Northeast.
d) South.

36. Population trends of the last decade have
a) resulted in population losses for some states, such as Nevada
and Arizona.
b) resulted in unprecedented growth in the Midwest and
Northeast.
c) resulted in population shifts from the Frostbelt states to
Sunbelt states.
d) had little effect on state and local governments.
37. The unofficial region of the United States generally
consisting of the Northeast and the Midwest is known as the
a) Sunbelt.
b) Great Lakes region.
c) Snowbelt.
d) Frostbelt.
38. When political conflicts emerge from deeply held moral
values on issues such as gay rights, abortion, or pornography it
is informally known as
a) culture wars.
b) moral hostility.
c) societal warfare.
d) divided conflict.

39. The unique characteristics of the fifty states, as described in
your text, are
a) openness, independence, and diversity.
b) diversity, competitiveness, and resiliency.
c) openness, accommodation, and interdependence.
d) diversity, independence, and innovation.
40. The authors of the text conclude that revitalized state and
local governments are
a) becoming less proactive in their approach to new problems.
b) taking charge and producing results.
c) becoming less resilient in the face of ongoing challenges.
d) surviving, but having to take a back seat to the national
government.
POLI 102: Hw 2 - Chapter 3: State Constitutions (10% of the
overall grade)
Dr. Jain – Fall 2018 (Posted: Monday 9/23/18, scantron due in
Class on Monday 10/1/18)
Multiple Choice: Chose the Correct Answer
1. State constitutions represent the fundamental law of the state.
Therefore,

a) they must mirror the federal Constitution in their design and
purpose.
b) when an issue arises between the state and the national
government, the state constitution prevails.
c) they are supreme in only those matters specifically delegated
to the state in the U.S. Constitution.
d) only the federal Constitution and statutes have priority over
the state’s constitution.
2. In the U.S. system of dual constitutionalism,
a) the various state constitutions and the national Constitution
stand equal in questions of law.
b) the state constitutions are supreme.
c) the national government has supremacy within those spheres
of authority delegated to it in the U.S. Constitution.
d) the U.S. Constitution is not limited in matters pertaining to
the states.

3. In recent years, revisions to state constitutions have sought to
a) weaken the power of the governor’s office.
b) make state government more responsive to shifting social and
economic forces.
c) increase the power of legislatures substantially.
d) make the documents more uniform in nature and less flexible.
4. The state constitutions of the original thirteen states were
mainly rooted in
a) the U.S. Constitution.
b) their colonial charters.
c) the Magna Carta.
d) the Articles of Confederation.
5. The Massachusetts constitution, although it has been
amended 120 times, is the only one of the original thirteen that
still exists, because
a) its drafter, Thomas Jefferson, extensively researched various

governments before finalizing it.
b) of the complicated procedures required before it can be
replaced.
c) it was rooted in the composite wisdom of the great political
philosophers of the eighteenth century.
d) there is unwillingness on the part of the state to allow it to be
replaced.
6. All the original states’ constitutions granted the most powers
of government to
a) a governor, selected by popular vote.
b) the courts.
c) a governor, selected by the legislature.
d) the legislature.
7. Between 1860 and 1870, considerable constitutional revision
occurred when

a) Jacksonian notions of popular government swept the nation.
b) Populist and Progressive reform movements swept the nation.
c) western states sought to update their constitutions after the
Civil War.
d) the former Confederate states wrote their constitutions to
incorporate certain conditions of readmission to the United
States.
8. The first state constitutions reflected their framers’ fears and
distrust of
a) executive authority.
b) legislative supremacy.
c) bureaucratic supremacy.
d) judicial supremacy.
9. By 1950, most state constitutions were
a) filled with needed details as to the power and authority of

local government.
b) properly rooted in the tradition of the U.S. Constitution to
minimize verbiage.
c) long, inflexible, and overly detailed, thus tying the hands of
state government in its efforts to meet the challenges of the era.
d) modernized by adding much needed details, thus equipping
the states to adapt to the era.
10. The expansion of state constitutions into lengthy documents
resulted from
a) a requirement to spell out designated local government
authority for each major city.
b) U.S. Supreme Court decisions regarding minimum items to
be covered.
c) efforts to protect the state from encroachment by federal
agencies.
d) a perceived need to be specific about what state and local
governments could and could not do.

11. Lengthy state constitutions create problems for states, which
include
a) the fear of challenges brought about by the U.S. Supreme
Court.
b) the need for state courts to rule on conflicting provisions.
c) the burden of increased legal costs that are passed on to
business interests.
d) the high costs of revision.
12. The “Jacksonian-era” reform intended to expand
opportunities for public participation in state government has
resulted in the current
a) fragmentation of the executive branch.
b) decline of popular control in government.
c) legislative restrictions placed on local governments.
d) discrimination of racial minorities and women.
13. The Model State Constitution was developed by

a) the U.S. Supreme Court in 1921.
b) the National League of Cities in 1960.
c) the Council of State Governments in 1968.
d) the National Municipal League in 1921.
14. The influential 1955 U.S. Advisory Commission on
Intergovernmental Relations was
a) disbanded before it could complete its deliberations.
b) successful in producing the Model State Constitution, which
would be used by most states to revise their constitutions.
c) charged with actually rewriting several state constitutions.
d) in favor of the states reviewing their constitutions to ensure
they provided for responsible government.
15. Thomas Jefferson believed that each generation should have
the right to choose for itself its own form of government, and
that it would be appropriate for a new constitution to be
considered every

a) five to ten years.
b) nineteen to twenty years.
c) twenty to thirty years. d) thirty to forty
years.
16. The “higher-law tradition” guides efforts to revitalize state
constitutions by
a) advocating that they put forth basic principles and processes
of government and avoid policy choices that are better left to be
handled by legislatures.
b) advocating adoption of a single-model constitution suitable
to all states.
c) locking the legislature into policies that favor the strongest
political and economic interests in the state.
d) spelling out the need for longer, more detailed constitutions
tailored to the particular state’s political culture.
17. The Model State Constitution

a) has twelve basic articles that are embodied, to some extent,
in the various state constitutions of today.
b) has six basic articles that are also found in the U.S.
Constitution.
c) endorses the widespread use of legal phrasing in revised
constitutions.
d) advocates the use of a single constitution suitable to all fifty
states.
18. The major reason for the rebirth of state activism in
protecting civil liberties and rights
a) has been the conservatism of the U.S. Supreme Court.
b) is the desire by the states’ to mirror the rights delineated in
the first eight amendments to the U.S. Constitution.
c) is based solely on the public outcry for guarantees of
individual rights.
d) has been in response to the federal mandates issued by the
national government to the states.

19. All state constitutions currently provide for bicameral
legislatures except for the state of
a) Wyoming.
b) Pennsylvania.
c) South Carolina.
d) Nebraska.
20. In recent years the executive branch of state governments
has been
a) further fragmented.
b) organized to permit the populace to vote for six to eight
different executive officers.
c) organized to permit the governor and lieutenant governor to
be elected separately.
d) centralized so that most power resides in the office of the
governor.
21. The informal method of amending state constitutions rests

upon
a) the introduction of legislative proposals.
b) the use of a direct initiative.
c) the interpretation of constitutional meaning by the various
branches of government.
d) the reliance on challenges in the federal courts.
22. All of the formal procedures for constitutional change in the
states involve two basic steps, that of
a) legislative and executive proposals.
b) constitutional convention or commissions.
c) direct and indirect initiatives.
d) initiation and ratification.
23. The most common manner of making formal changes to state
constitutions is permitted in all fifty states and accounts for 90
percent of all revisions. It is known as
a) the legislative proposal.

b) a direct initiative.
c) a constitutional convention.
d) a constitutional commission.
24. Once the required signatures have been obtained for a
constitutional change to be placed on the ballot, in states with
provisions for the initiative, experience has shown that
a) a majority of the electorate will support the change.
b) there is widespread apathy even among those who signed the
initiative.
c) opposing interests become mobilized and passage is by no
means certain.
d) the provisions of the initiative will have been changed by the
legislature.
25. Direct initiatives are distinguished by the fact that
a) the legislature votes directly on citizens’ proposals.

b) the courts must decide that the proposals are consistent with
federal law before they are adopted.
c) the Secretary of State must certify that sufficient signatures
exist to permit immediate adoption.
d) the proposals are placed directly on the general election
ballot by citizens.
True or False?
26. Only federal law and federal statutes take priority over
state constitutions and state laws.
27. State constitutions are often neglected in secondary school
and college courses in history and political science.
Astonishingly, one national survey discovered that 51 percent of
Americans were not aware that their state had its own
constitution.
28. In more recent years, constitutional revisions have had the
effect of making state government less efficient, effective, and
responsive to shifting social and economic forces.
29. During the colonial era, a constitution of the Five Nations
of the Iroquois called the Great Binding Law existed, but it was

oral and not particularly relevant to the people in the colonies.
Thus, the thirteen colonial charters provided the foundation for
the new state constitutions.
30. Following the War of Independence, the former colonies
drafted their first constitutions in special revolutionary
conventions or in legislative assemblies. With the exception of
Massachusetts, the new states put their constitutions into effect
immediately, without popular ratification.
31. Following the War of Independence, all states rewrote their
constitutions and put them into effect immediately without
popular ratification.
32. According to legend, a patriot hid the Fundamental Orders
of Connecticut in a hollow tree (nicknamed Charter Oak), to
keep the British from seizing the document.
33. The Commonwealth of Virginia has the oldest state
constitution in use, although it has been amended more than one
hundred times.
34. Legislative supremacy remains the norm in most states.
35. The most active period for revising state constitutions
occurred just prior to the Civil War.

36. State constitutions are generally shorter in length than the
U.S. Constitution.
37. The U.S. Constitution contains about 8,700 words, while the
typical state constitution is much longer in length.
38. Before ratifying its new constitution in 1983, Georgia’s
state constitution contained more than 583,000 words.
39. State constitutions are political documents and have been
used to protect the rights of special interests, such as public
utilities, farmers, timber companies, and religious groups.
40. One of the problems with lengthy state constitutions is that
they tend to be plagued by contradictions and meaningless
clauses, which often result in litigation.
41. The constitution of Oregon has been amended nearly 200
times, which is more than any other state.
42. The Kestnbaum Commission devised the Model State
Constitution that many states would follow in revising their own
constitutions.

43. The positive-law tradition is based on detailed provisions
and procedures.
44. All states have substantially revised not only their courts’
organization and procedures but also the election of judges.
45. In general, state constitutions today conform more closely to
the higher-law tradition and the Model State Constitution than
did those of the past.
46. The United States Constitution has been amended twenty-
seven times.
47. The initiative is used more often than legislative proposal in
amending state constitutions.
48. Constitutional conventions are the oldest method for
constitutional change in the states and are used frequently
today.
49. A constitutional commission is often referred to as a study
commission.
50. Since the mid-1960s, most states have adopted new
constitutions or substantially amended their existing ones.

Midterm Exam Fall 2018 - POLI 102: Chapters 4 & 5 - Dr. Jain
Chapter 4: Citizen Participation and Elections
Multiple Choice
1. In a representative democracy, the most common form of
citizen political participation is
a) writing letters to elected officials.
b) running for public office.
c) serving on an advisory committee.
d) voting.
2. Citizens less likely to participate in government are
a) professional workers.
b) middle-aged individuals.
c) younger individuals.
d) white-collar workers.
3. The Voting Rights Act of 1965 has been modified to
a) ensure that physical intimidation is not allowed.
b) limit the use of “white primaries.”
c) make illegal any government action that discourages minority
voting.
d) extend the vote to illegal immigrants.

4. Voter turnout is higher for elections
a) held in off years.
b) when attention is focused on only one race.
c) held during presidential elections.
d) held at the state and local level.
5. The National Voter Registration Act of 1993 enabled voters
to register
a) by mail or electronic mail (e-mail).
b) when they apply for a driver’s license, welfare benefits, or
unemployment compensation.
c) at the Post Office.
d) on the day of the actual election.
6. A closed primary is one in which
a) registered voters with party affiliation may vote in either
primary, but the election is closed to independents.
b) voters must be registered with their party affiliation in order
to vote in their party’s primary.
c) voters may choose either party ballot in secret.
d) registered voters may vote in either party primary, but the
primary is closed to those not registered.
7. In general elections with three or more candidates where no
one gets a majority, the winner is

a) decided in subsequent elections that continue until one
candidate gets a majority.
b) decided in a runoff election.
c) chosen by a legislative caucus.
d) the candidate with a plurality of the votes.
8.State constitutional provisions for popular referenda allow
a. citizens to petition to vote on actions taken by the legislature.
b. legislatures to decide to ask for a vote to endorse a particular
policy.
c. citizens to force officials from office.
d. citizens to vote on bond issues.
9. Today, statewide initiatives for constitutional amendments,
statutes, or both are used
a. in all fifty states.
b. in twenty-four states.
c. in only three states: Florida, Washington, and Colorado.
d. in all states where advance notice of meetings is required and
minutes must be taken.
10. California’s Proposition 13 sought to
a. give citizens tax relief by lowering property tax rates.
b. strengthen affirmative action programs in the state.
c. make marijuana legal for medical uses.

d. dismantle affirmative action programs in the state.
11. The recall mechanism for state officials is available in
a. all but two states: Georgia and South Carolina.
b. eighteen states, but judicial officers are excluded in seven of
these states.
c. twenty-six states, and at the local level in forty-nine states.
d. all states, but it has never been used except in California and
North Dakota.
12. Open meeting laws that serve to “open” the meetings of
government bodies to the public
a. began in 1965 with Florida’s “honesty law.”
b. apply only to local levels of government.
c. are found in all fifty states.
d. do not affect the executive branch of government.
13. Citizen advisory committees serve
a. exclusively as a vehicle to ensure citizen participation in
government.
b. simply as a tool to be manipulated by politicians.
c. a number of purposes at the state level, but they are rarely
used at the local level.
d. a number of political purposes and give citizens an
opportunity to participate in government.

Chapter 4 – True or False?
14. Aside from voting, citizens have few ways to participate in
the political process.
15. Unskilled workers and blue-collar workers participate in
politics at about the same rate as white-collar workers and
professionals.
16. The “suffragists” were women who were actively fighting
for the right to vote.
17. Women gained universal suffrage in the United States with
passage of the Twentieth Amendment in 1920.
18. Presidential elections attract the highest proportion of
eligible voters.
19. The Fifteenth Amendment of the U.S. Constitution (1870)
extended the right to vote to African Americans, but Congress
had to pass the Voting Rights Act of 1965 to get defiant
southern states to allow blacks to vote.
20. Every state that allows for regular absentee voting by mail
stipulates that citizens must have an excuse in order to do so.

21. Passage of the National Voter Registration Act in 1993
allowed for individuals to register to vote at any
U.S. Post Office.
22. Iowa is the only state that does not require voter
registration.
23. The most common type of voting equipment currently used
by states is the punch card devices.
24. When a presidential candidate campaigns for a party
member in a local race in an effort to help that person win the
election, it is called the “coattail effect.”
25. Recall is a procedure that allows citizens to vote an elected
official out of office before his or her term has expired.
26. All fifty states allow for the recall of state officials, but
judicial officers are exempt from recall.
27. The result of Florida’s 1967 “sunshine law” was a surge in
the desire for openness in government and the establishment of
open meeting laws.

28. Citizen advisory committees provide a formal arena for
citizen input.
29. Volunteerism is a constructive participatory activity that can
bring new ideas into government.
Chapter 5 Political Parties, Interest Groups, and Campaigns
Multiple Choice
30. According to your text, probably the most realistic term to
describe the status of political parties over the past thirty years
is
a. stagnant.
b. declining.
c. revitalized.
d. transforming.
31. Ticket splitting usually means
a. voting for both Democrats and Republicans in the same
general election.
b. unwittingly voting for both a Democrat and a Republican in a
nonpartisan election.
c. placing a few Democrats and a few Republicans on the same
third-party ticket.
d. placing an independent on the ticket to achieve balance and

attract nonpartisan voters.
32. Local political parties typically
a. have no problem staffing precinct offices.
b. hire professional staff.
c. are not as professionally organized as state parties.
d. maintain campaign headquarters year-round.
33. The Republican Party, also known as the Grand Old Party
(GOP),
a. is newer than the Democratic Party.
b. emerged from the Jacksonian wing of the Jeffersonian Party.
c. is much older than the Democratic Party.
d. is much larger than the Democratic Party and is associated
with the “blue states.”
34. Interest group membership
a. is a legitimate way for citizens to communicate their
preferences to government and/or seek benefits offered by the
group.
b. ensures that narrow, selfish interests will prevail in our
society, particularly as the number of interest groups increase.
c. is limited to those with business interests.
d. is limited to those who seek some political end.

35. In most states where political parties are strong, interest
group influence tends to be
a. equally strong.
b. overwhelming, given the symbiotic relationship of interest
groups with parties.
c. nonexistent.
d. weak.
36. The most important commodity that lobbyists can provide
legislators is
a. information on the issues under consideration.
b. personal favors.
c. money to buy public policy through corrupt and illegal
activities.
d. their presence in committee hearings, where they monitor
debates.
37. Grassroots lobbying involves
a. lobbyists positioning themselves to greet legislators on the
steps and lawn of the capitol.
b. increasing the number of lobbyists in the state capitol.
c. orchestration of public support in the form of letters, faxes,
and telephone calls.
d. increasing the amount of time the lobbyist spends on the golf
course developing a personal relationship with a legislator.

38. Political action committees (PACs) grew out of
a. a desire to weaken the role of political parties.
b. laws that made direct political contributions by corporations
and labor unions illegal.
c. a concern over the rising influence of interest groups.
d. an effort to discourage public involvement and participation
in politics.
39. Groups that spend money to influence the outcome of
elections but do not contribute directly to candidates are called
a. political action committees.
b. 527 groups.
c. political consulting groups.
d. soft money interest groups.
Chapter 5 - True or False?
40. The condition of today’s American political parties has been
described with words such as decline, decay, and demise, but a
more precise description may be that they are in the process of
transformation.
41. Republicans typically have been considered the party of big
business, and the Democrats the party of workers.

42. State political parties are very centralized organizations.
43. In spite of the fact that two political parties dominate
politics in the states; third parties have achieved limited success
in some states.
44. State political parties are stronger today than at any time in
the nation’s history.
45. Most states currently exhibit substantial two-party
competition.
46. When we speak of divided government in the states, we
typically mean that one party controls the governor’s office and
another party controls the legislature.
47. Individuals join interest groups for no other purpose other
than to influence government.
48. Because so much of local government involves the delivery
of services, local interest groups devote a great deal of their
attention to administrative agencies and departments.
49. Bombarding legislators with mail, e-mail, faxes and

telephone calls are tactics used by grassroots lobbying
organizations.
50. The 527 groups are not connected to candidates, but spend
money to influence the outcome of elections.
Student evaluation of Dr. Jain’s performance as an educator:
1. Strength(s)
2. Suggested improvement(s)
3. Would you recommend me as a professor to other students?

Yes (why?):
No (why not?):
THANK YOU!!!
2

personal use only.
Martinie, Sherri L., and Jennifer M. Bay-Williams.
“Investigating Students’ Conceptual Understanding of Decimal
Fractions Using Multiple
Representations.” Mathematics Teaching in the Middle School 8
(January 2003): 244–47.
http://www.nctm.org
Discussion
WE ADMINISTERED THIS INSTRUMENT TO FORTY-
three sixth graders. The students worked individually and
had as much time as they needed to complete the ques-
tions. An item was scored correct if the student had accu-
rately completed the given representation in a way that
correctly identified the size of the two decimal numbers.
For example, in the first question, students had to label the
number line with a 0 and a 1 and correctly place 0.06 close
to 0 and 0.6 slightly to the right of 1/2.

Even though each of these tasks required some concep-
tual knowledge to represent the answer correctly, stu-
dents’ success with the decimal tasks varied for each rep-
resentation. Many students could accurately show 0.6 and
0.06 in one or two representations but not the others. The
number of students scoring all correct (4) to none correct
(0) are shown in table 1. Only six students (14%) of those
tested were able to represent the decimal numbers in all
four situations. Note that 77 percent of the students
showed some conceptual understanding of decimals by
providing correct responses to one, two, or three of the
tasks, but they were not able to represent the numbers
correctly for all the models.
Students’ success with the different models varied
greatly (see table 2). Students were correct most often
when explaining decimal numbers using the 10 × 10 grid
and using money. Although 58 percent of students an-
swered the place-value question correctly, most compared
the tenths place of each decimal. Only six students (14%)
stated that six-tenths is more than six-hundredths or made
any quantitative comparison of the two decimals.
V O L . 8 , N O . 5 . J A N U A R Y 2 0 0 3 245

Conceptual Understanding
Using Multiple Representations
TABLE 1
Correct Responses on Decimal Questionnaire
NUMBER OF PERCENT OF
NUMBER OF STUDENTS STUDENTS
CORRECT RESPONDING RESPONDING
RESPONSES CORRECTLY CORRECTLY
4 6 14%
3 14 33%
2 12 28%
1 7 16%
0 4 9%
Total 43 100%
TABLE 2
Correct Responses for Each Item on the Decimal
Questionnaire

PERCENT OF
NUMBER OF STUDENTS
STUDENTS RESPONDING
ITEMS ON RESPONDING CORRECTLY
QUESTIONNAIRE CORRECTLY (OUT OF 43 STUDENTS)
Number line 11 26%
10 × 10 grid 28 65%
Money 28 65%
Place value 25 58%
The number line was the most difficult of the four
models. In fact, of the fourteen students who missed
only one representation, eleven missed the number
line. The most common error (made by sixteen of the
thirty-two students who missed this question) was to
label 0 and 1 on the number line, place 0.6 accurately,
then incorrectly place 0.06 or leave it off entirely (see
fig. 2). Notice that the student whose work is shown
in figure 2 considered 0.06 to be halfway between 0
and 0.6, confusing one-tenth the size of 0.6 with one-
half the size of 0.6. Another common error was to label
0.06 on the number line to the left of 0.6 but to place

both decimal numbers inaccurately between 0 and 1
(see fig. 3). Students seemed to understand that 0.06
was smaller than 0.6 but did not indicate the sizes of
the decimals in relation to 0 and 1. Figure 4 shows an-
other common error, which was to identify 0.06 as
larger than 0.6, perhaps with the idea that the longer
decimal is larger, as is true with whole numbers. In
two of the student samples, students used 0.5 and 0.05
as benchmarks to try to identify the correct placement
of 0.6 and 0.06. This approach illustrates students’ at-
tempts to apply what they know about the sizes of
these decimals, specifically, that 0.5 is one-half and 0.6
is slightly larger than one-half.
Follow-up Assessment on Linear Representation
IN THE NUMBER-LINE MODEL, STUDENTS HAD DIFFI-
culty labeling endpoints of 0 and 1 and relating the values
0.6 and 0.06 to the endpoints. Because we could not deter-
mine whether students were struggling with the number
line or with the relative values of the decimals, we de-
signed another assessment that included four number-line
tasks of increasing complexity:
1. Draw a number line that shows the numbers 1 through 5.

2. Draw a number line that shows 2.5.
3. Draw a number line that shows 0.4.
4. Draw a number line that shows 0.4 and 0.04.
What percentage of your students would successfully plot
the numbers for each of these four tasks? Figure 5 shows
the results for the forty-three sixth graders that we tested.
Most students understood the number line in relation to
whole numbers, but many could not place the decimals, espe-
cially those less than 1. Only one in five students was able to
place 0.04 and 0.4 accurately! Recall that on the first test, 26
percent of students were able to label and place 0.6 and 0.06
correctly. This additional task revealed that students’ diffi-
culty with a number-line representation was specific to those
decimals less than 1, in particular, those less than 1/10.
Implications for Teaching and Learning
TO MAKE SENSE OF DECIMALS, STUDENTS NEED
multiple experiences and contexts in which to explore
them. Our assessment instrument using four representa-
tions indicates that students may appear to understand
decimals using some models, but they may lack a pro-
found overall understanding of decimal concepts. In in-

struction, therefore, teachers must include many represen-
246 M A T H E M A T I C S T E A C H I N G I N T H E M I
D D L E S C H O O L
Fig. 2 A student places 0.6 correctly but is unable to place 0.06
correctly.
Fig. 3 This student’s solution recognizes that 0.6 is greater
than
0.06, but the student does not indicate the relative sizes of the
deci-
mal numbers compared with 0 and 1.
Fig. 4 A student places 0.06 to the right of 0.6, apparently
based on
the misconception that the longer decimal is greater in value.
Fig. 5 Results of follow-up assessment using four number-line
tasks
of increasing complexity
0
20

40
60
80
100
Label number
line 1–5
Label number
line with 2.5
Label number
line with 0.4
Label number
line with 0.4
and 0.04
P
er

ce
nt
C
or
re
ct
Follow-up Tasks
93%
83%
57%
20%
Percent of Accuracy
V O L . 8 , N O . 5 . J A N U A R Y 2 0 0 3 247

tations of decimal concepts to broaden and deepen stu-
dents’ understanding.
Teachers should also use multiple representations to
assess students’ understanding. Without the number-line
question in our assessment instruction, we might have
concluded that our students had a sound understanding of
decimals and their relative magnitude. Mistakes can reveal
student misconceptions or overgeneralizations and pro-
vide opportunities for learning, both for the teacher and
students. An instrument that asks students to provide dif-
ferent representations and explanations for a particular
concept can be an eye-opener for teachers and can guide
instructional decisions to enable students to deepen their
understanding of concepts. The purpose of our decimal
questionnaire was to identify student misconceptions and
use that information to guide instructional planning.
Collecting data from students often results in more ques-
tions. In our classrooms, the surprising difficulty of the num-
ber line led to a follow-up inquiry to find out more about what
students could and could not do. The follow-up number-line
questions revealed that students’ number-line difficulties were
specifically related to the size of the numbers, in particular, to
decimals less than 1/10. We might offer several possible ex-

planations for the students’ difficulty with locating numbers
less than 1/10 on a number line. One explanation is that stu-
dents were asked to draw and label all parts of the model, in-
cluding the 0 and the 1 without any visual organizers already
marked for them. This task was also the only
one that called for approximation; students
might not have been able to estimate approxi-
mate positions for the two values, even though
they could illustrate exact representations
(such as the shading required in item 2, fig. 1).
Students might also have been inexperienced
with number lines. Using the number line to
discuss the approximate magnitude of decimal
numbers (as well as fractions and percents) is
an effective tool for developing students’ num-
ber sense (Bay 2001). Given that students
struggle with the number-line model and
knowing that decimals often appear in linear
models in real-life situations, such as on a ther-
mometer or metric ruler, we must recognize
the importance of including linear models in
our teaching of decimal concepts.
Summary

PRINCIPLES AND STANDARDS FOR SCHOOL
Mathematics (NCTM 2000) states, “Students
must learn mathematics with understanding,
actively building new knowledge from expe-
rience and prior knowledge” (p. 11). With
decimals, prior knowledge of whole numbers
may cause misunderstandings. For students
to fully understand the similarities and differ-
ences of decimals and whole numbers, instruction must
emphasize conceptual development, including the use of a
variety of decimal representations.
References
Bay, Jennifer M. “Developing Number Sense on the Number
Line.” Mathematics Teaching in the Middle School 6 (April
2001): 448–51.
National Council of Teachers of Mathematics (NCTM). Princi-
ples and Standards for School Mathematics. Reston, Va.: 2000.
Resnick, L. B., P. Nesher, F. Leonard, M. Magone, S. Omanson,
and I. Peled. “Conceptual Bases of Arithmetic Errors: The
Case of Decimal Fractions.” Journal for Research in Mathemat-

ics Education 20 (January 1989): 8–27.
Sackur-Grisvard, C., and F. Leonard. “Intermediate Cognitive
Or-
ganizations in the Process of Learning a Mathematical Con-
cept: The Order of Positive Decimal Numbers.” Cognition and
Instruction 2 (2) (1985): 157–74.
Sowder, Judith. “Place Value as the Key to Teaching Decimal
Op-
erations.” Teaching Children Mathematics 3 (April 1997):
448–53.
Wearne, D., and J. Hiebert. “Constructing and Using Meaning
for
Mathematical Symbols: The Case of Decimal Fractions.” In
Number Concepts and Operations in the Middle School, edited
by James Hiebert and Merlyn Behr, pp. 220–35. Reston, Va.:
National Council of Teachers of Mathematics, 1988. �
Glasgow, Robert, Gay Ragan, Wanda M. Fields, Robert Reys,

and Deanna Wasman. “The Decimal Dilemma.”
Teaching Children Mathematics 7 (October 2000): 89–93.
personal use only.
89OCTOBER 2000
I
f you are aware of the results given in the
media reports about the Third International
Mathematics and Science Study (TIMSS), you
probably know that fourth graders from the United
States (U.S.) scored above the international average
in mathematics and that eighth and twelfth graders
scored below average (Mullis et al. 1997). As an edu-

cator, you are aware of the dangers of looking only at
averages of test scores. Rich information can be
gleaned from the TIMSS data that will help us learn
more about what our students know and are able to
do. The data from a large-scale study, such as the
TIMSS, often raise questions about what the numbers
really mean. This article addresses one such question
that arose from examining part of the third- and
fourth-grade TIMSS data. The process that we used
may be as valuable as the information that we found.
Perhaps this process will help you answer questions
that arise as you reflect on the TIMSS results.

The Dilemma
The TIMSS data are reported in mathematical con-
tent categories (see timss.bc.edu). In the category
of fractions and proportionality, U.S. third graders
performed at the international average, whereas
U.S. fourth graders performed above the interna-
tional average. However, a close look at the
released test items reveals an interesting phenome-
non. U.S. students did not do as well on questions
involving decimals as they did on questions
involving fractions (see fig. 1).
Questions M-5 and J-7 differ primarily in repre-
sentation of the possible responses (decimal and
fractional), yet the performance levels of U.S. stu-
dents were dramatically different for these
two questions. Thirty-two percent of U.S.
fourth graders answered the decimal ques-
tion (M-5) correctly, whereas 80 percent
answered the fraction question (J-7) cor-
rectly. Although this difference between
understanding levels can be seen in the inter-
national scores, the difference in U.S. stu-
dents is more extreme. The results of ques-

tion I-2 seem to further highlight a deficiency in
understanding decimal notation.
The data in figure 1 generate many questions
concerning decimals in third- and fourth-grade
classrooms. Figure 1 suggests growth in under-
standing decimals and fractions from third to
fourth grade, indicating that students are probably
receiving instruction on both topics. The dilemma
Bob Glasgow, Gay Ragan, Wanda Fields, and Deanna Wasman
were doctoral students at the
University of Missouri when this study was done. Glasgow,
[email protected]univ.edu, teaches
and works with preservice teachers at Southwest Baptist
University in Bolivar, MO 65613.
Ragan, [email protected], is currently an assistant professor at
Southwest Missouri
State University, Springfield, MO 65804. Fields,
[email protected], has taught at both the
middle school and high school levels. She has also taught
college algebra courses at a com-
munity college and at the University of Missouri—Columbia.
Reys, [email protected], is on
the faculty at the University of Missouri. Wasman,

[email protected], is teaching in
the Department of Mathematical Sciences at Appalachian State
University, Boone, NC 28608.
The authors gratefully acknowledge Linda Coutts, mathematics
coordinator for Columbia
Public Schools, and Vicki Robb, principal of Russell Boulevard
Elementary School, for their
help in collecting data from teachers and students that are
reported.
Robert Glasgow,
Gay Ragan,
Wanda M. Fields,
Robert Reys, and
Deanna Wasman
Decimal
Dilemma
The

http://www.nctm.org
is this: Why does decimal understanding lag
behind fractional understanding? Is it a result of
when decimals are taught or more a result of how
they are taught in the U.S.? Reflect on the items
and results in figure 1. Would your students per-
form at the U.S. levels? Why is decimal under-
standing weaker than fractional understanding in
U.S. third and fourth graders? How might you col-
lect data to answer these questions?
The Details
We examined the details surrounding the decimal
dilemma by looking at national and state informa-
tion concerning the coverage of decimals in U.S.
third- and fourth-grade classrooms.
The national level
The TIMSS addressed national curricular differ-
ences by asking an agency of each participating
country to report which test items were included in

its country’s intended curriculum by the fourth
grade. For the U.S., the National Research Council
selected persons who were familiar with mathe-
matics curricula across the country to perform a
Test-Curriculum Matching Analysis. The U.S. was
the only country to identify 100 percent of the
questions as being included in the fourth-grade
curriculum. Other countries, such as Japan (identi-
fying 89% of the questions as being covered), Sin-
gapore (74%), and the Netherlands (52%), were
less optimistic about what their curricula included,
even though the fourth graders in all three coun-
tries outperformed the U.S. fourth graders. Even at
the third-grade level, the U.S. curriculum review-
ers found 100 percent of the questions to be
included in the intended curriculum. Again, Japan
(75%), Singapore (51%), and the Netherlands
(23%) were more cautious. The reviewers were
therefore confident that the intended curriculum
for U.S. third and fourth graders would prepare
them to perform well with decimals, but the actual
results on the TIMSS did not support this opti-
mism. Was the intended curriculum misreported, or
is it not being taught?

The state level
We next investigated curriculum expectations at
the state level, since most states have curricular
frameworks that serve as guidelines for local
school districts. An examination of five state cur-
ricular frameworks (Alabama, California, Min-
nesota, Missouri, and New Jersey) found that dec-
imal notation is studied by the time that students
are in the fourth grade in all five states. Each
framework expects that students should correctly
answer question I-2 in figure 1. All but one frame-
work mentioned using shaded regions to represent
decimal notation as found in question M-5. All five
frameworks promote using money to introduce
decimals, as well as looking at decimal equiva-
lences for common fractions. Are these frame-
works representative of elementary school educa-
tion in the U.S.? What does your state or district
framework suggest for decimal instruction in these
grades?
The Discoveries
We then considered the decimal dilemma by exam-
ining a single school district. We examined the cur-

riculum of one local school district and found that
F
IG
U
R
E
1 The TIMSS questions with average percent of students
answering correctly
Grade U.S. International
Question I-2
0.4 is the same as . . . 3rd 21% 21%
a) four c) four hundredths
b) four tenths d) one-fourth 4th 40% 39%
Question M-5
Which number represents the shaded part
of the figure? 3rd 18% 33%
a) 2.8 b) 0.5
c) 0.2 d) 0.02 4th 32% 40%

Question J-7
Part of the figure is shaded. What
fraction of the figure is shaded? 3rd 63% 42%
a) 5/4 b) 4/5
c) 6/9 d) 5/9 4th 80% 61%
in the third grade, decimals were developed within
the context of money; in the fourth grade, students
were expected to be able to read, write, compare,
add, and subtract decimals through the hundredths
place. To see whether the classroom practice
matched the intended curriculum, we surveyed
third- and fourth-grade teachers and interviewed
several students at each grade level.
The district level—teachers
Teachers were asked when they first introduce dec-
imals (see fig. 2) as well as what contexts they use
to teach decimals (see fig. 3). Twenty-eight third-
grade and thirty fourth-grade teachers of mixed-
ability classrooms responded. Teachers who taught
multiple-grade classrooms were excluded.

The intended curriculum of the district appears
to be the curriculum implemented, since all teach-
ers gave a particular time when decimals are first
introduced at their grade level. Knowing that deci-
mals are introduced predominantly toward the end
of the school year, after the time when the TIMSS
was administered, provides insight as to why the
U.S. student performance on decimals was low. It
is interesting to note the varying responses in fig-
ure 2. If the instructional schedule varies within a
school district, it is certainly expected to vary
across the country. When do you introduce deci-
mals in your classroom? When do other teachers in
your district first introduce decimals?
The contexts in which these teachers use deci-
mals seem consistent with the conclusions reached
by examining state curricular frameworks (see fig.
3). The predominant way to teach decimals is to
use money, but the use of pictorial representations
and fraction equivalences seems consistent with
the level of preparation required for the TIMSS
questions discussed in this article. How do you pre-
sent decimals to your students? Do you think that
other teachers in your district teach decimals in

ways significantly different from yours?
Teachers predicted what percent of their stu-
dents would answer each of the three TIMSS
items correctly (see fig. 4). Not surprisingly,
teachers thought that their students would do best
on question J-7, which dealt with fractions. Teach-
ers’ predictions for the items did not differ greatly
from the overall U.S. results shown in figure 1.
More surprising was the wide range of responses.
For example, for question M-5, fourth-grade
teachers’ predictions ranged from 0 percent to 95
percent. Out of thirty teachers, fourteen predicted
20 percent or less and four predicted 80 percent or
higher. Similar ranges were found at both grades
for all questions. What percent of your students
would answer each question correctly? Would the
predictions of teachers in your district have such a
wide range?
The building level—students
Finally, in an effort to gain some insight into what
students might have been thinking when they
answered the TIMSS questions, we talked with stu-

dents. We conducted interviews with fifteen third
graders and twenty fourth graders from the same
school district in which the teacher survey was
conducted. In the United States, the TIMSS was
administered in April 1995. Accordingly, we con-
ducted our interviews in late March.
We initially presented the students with the
three TIMSS questions (I-2, M-5, and J-7). Ques-
tion I-2 was presented first without the multiple-
choice responses. Students were asked to name
“0.4” and were encouraged to say it in more than
one way. If they did not say “four tenths” (zero of
fifteen third graders and one of twenty fourth
graders did), they were given the multiple choices.
Responses indicated a very low level of under-
standing of the decimal-notation terminology.
Nearly all the students of both grades initially said
“zero point four,” and only one of fifteen third
91OCTOBER 2000
F
IG

U
R
E
2 When teachers introduced decimals
F
IG
U
R
E
3 Contexts that teachers used to teach decimals
0
5
10
15
20

25
30
35
40
45
1st
quarter
2nd
quarter
3rd
quarter
4th
quarter
Not at all
%

o
f T
e
a
ch
e
rs
R
e
sp
o
n
d
in
g
0

10
20
30
40
50
60
70
80
90
100
Money
%
o

f
T
e
a
ch
e
rs
R
e
sp
o
n
d
in
g
Pictorial Metric
system
Equivalence

to fractions
Base 10
blocks
Other
graders and three of the remaining nineteen fourth
graders chose “four tenths” when given choices.
When we asked about their experiences with these
kinds of numbers, most of the third graders
claimed little exposure to decimals. Most did not
even recall having heard the term decimal before
the interview. Some students said
that they had seen these types of
numbers when studying money. The
only third grader who was able to
choose the correct response claimed
knowledge from outside the school
setting. The student said, “I’ve seen

on basketball games there’s like a
zero point four seconds left on the
clock, zero point four, and they say
there’s four tenths of a second left
on the clock.”
On question M-5, students were
given the multiple choices immedi-
ately. Three of fifteen third graders and eight of
twenty fourth graders chose the correct answer.
These numbers are somewhat deceiving, because
when asked to explain their answers, it was clear
that of the eleven students who answered correctly,
only five (one third grader and four fourth graders)
were able to explain why 0.2 was the correct
answer. For example, one fourth grader answered
M-5 as follows:
(c) 0.2 because if there was a number here [in place of the
0], it would probably be a 9. And I know it is not (b), 0.5,
because that is half of ten.
The answer, even though correct, indicates that
the student’s understanding of decimals is limited.

Students had great difficulty relating the decimal
notation to the picture. The predominant incorrect
answer was “2.8” (given by twelve of fifteen third
graders and six of twenty fourth graders). This
response is consistent with the most common
incorrect answer chosen when the TIMSS was
administered. When students were asked to explain
their reasoning, they said things like, “Maybe it is
two point eight because there are two shaded and
there are eight that are not.” Other students tried to
use their knowledge about fractions to help them.
One student responded in this way:
Well, we did this before so our teacher had us do this and I
was real good at it and she explained it real well, so there
was only two shaded parts and the rest was eight. I put two
eighths and then two slash eight. Two goes with the shaded
part and eight with the unshaded part.
Fourth graders did claim to have studied deci-
mals more than third graders. Some of them
reported to have studied decimals in terms of
money and explored the relationship of fractions to
decimals on a number line. Very few students

reported using pictures, such as the one in question
M-5, to study decimals.
Question J-7, which deals with fractional nota-
tion, yielded results very different from those gen-
erated by the decimal questions. Four of fifteen
third graders and seventeen of twenty fourth
graders were able to answer correctly without
being given the multiple choices. When given the
multiple choices, three more third graders and one
more fourth grader answered correctly. Most stu-
dents, even if answering incorrectly, showed some
understanding of fractional notation. Nearly all the
students at both grade levels reported studying
fractions in school. Their performance levels at
both grade levels indicate clear differences
between students’ fractional and decimal thinking.
The Discussion
Our interviews with students document that most
students are not as familiar with decimals as with
fractions. Our survey of teachers supports this
F
IG

U
R
E
4 Teachers predicted what percent of their students would
answer each question correctly.
Question Grade Average Teacher Prediction
I-2 3 22%
4 34%
M-5 3 25%
4 33%
J-7 3 54%
4 71%
What percent of your
students would

answer each question
correctly?
observation by indicating that teachers have lower
expectations of their students for responses on dec-
imal questions than on fractional questions.
Although district curricula and state frameworks
address decimal notation, decimals take a back seat
to fractional notation in third and fourth grades. We
also found that decimals may not be addressed
until late in the year in many classrooms. When
decimals are studied, students may rarely see fig-
ures like the one used in question M-5. Perhaps
teachers should use pictorial representations more
frequently when discussing decimals and should
attempt to teach decimals in conjunction with
teaching fractions. What do you think?
Have we gained new insight into the decimal
dilemma? We obviously still have a great deal to
think about. The data from the TIMSS only begin to

tell the story of what students know and understand.
Our examination shows the complexity of trying to
expand on a few of the TIMSS items. This examina-
tion included looking at the expectations for students
at national, state, local, and classroom levels. We
hope that our questions throughout this article, as
well as the structure of our explorations, will invigo-
rate your thinking about what your students are
learning. Whether it concerns the decimal dilemma
or other questions that emerge from the TIMSS, we
challenge you to investigate the situation in your
classroom, make a plan, and attack the problem. We
encourage you to examine the TIMSS data in detail
and then explore related information at the national,
state, district, and building levels. Find out what
other teachers in your district are doing. Most impor-
tant, try the TIMSS questions with your own stu-
dents. The results may surprise you.
Bibliography
“California Department of Education Academic Performance
Index.” www.cde.ca.gov/board/mcs–intro.html. World
Wide Web.

“ENC Standards and Frameworks.” www.enc.org/reform/
fworks/states.htm. World Wide Web.
Minnesota Department of Children, Families, and Learning.
Minnesota K–12 Mathematics Frameworks. Saint Paul,
Minn.: SciMath, 1998.
Missouri Department of Elementary and Secondary Education.
Missouri’s Framework for Curriculum Development in Math-
ematics K–12. Jefferson City, Mo.: Missouri DESE, 1996.
Mullis, Ina V.S., Michael O. Martin, Albert E. Beaton, Eugenio
J. Gonzalez, Dana L. Kelly, and Teresa A. Smith. Mathe-
matics Achievement in the Primary School Years: IEA’s
Third International Mathematics and Science Study. Chest-
nut Hill, Mass.: International Association for the Evaluation
of Educational Achievement (IEA), 1997.
Rosenstein, Joseph G., Janet H. Caldwell, and Warren D.
Crown. New Jersey Curriculum Framework. Trenton, N.J.:
New Jersey Mathematics Coalition and New Jersey Depart-
ment of Education, 1996.
“Third International Mathematics and Science Study.”
timss.bc.edu. World Wide Web. ▲

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