DIMS Exemplar Set of Items
8th-Grade Mathematics
Wylie and Ciofalo (2008) identified a set of common misconceptions that cut across both mathematics
and science, and provided a framework for teachers to use to identify other student misconceptions. The
eight categories of misconceptions are listed and described below. Following this is a sample set of items
taken from the larger set of 8th-grade mathematics DIMS items that illustrate many of these categories.
Categories of Common Student Misconceptions in Mathematics and Science
Failing to recognize the limitation of diagrams, models, and other representations: Diagrams, models,
and other types of representations used in scientific and mathematical discourse and materials can
be inaccurate, incomplete, or poorly scaled, or the limitations are not articulated. While a teacher
may understand limitations, students are more literal. For example, in science, some students may
interpret a commonly-used poster of the solar system to mean that, since the Sun is located on the
left with the planets arranged horizontally, Jupiter is the center of the solar system. In
mathematics, younger students rarely see triangles that do not have a horizontal base and may
describe them as “upside-down triangles.”
Going too far with abstractions, generalizations, and simplifications: Characteristics and descriptions of
mathematical procedures, expressions, or concepts (and scientific groups or concepts) can be too
abstract, general, or simplified. For example, students need to understand when certain
mathematical “truths” no longer universally apply; e.g., addition always results in a larger number
(until negative numbers are involved). In science, students may think that all birds fly, all
bacteria are harmful, and all metal/silvery objects are attracted to magnets.
Confusing language and vocabulary: Words and phrases used in everyday communication can convey a
confusing or completely different meaning in scientific or mathematical communication. For
example, a student meeting the term “continental drift” for the first time may apply what he or
she knows about the meaning of the word, and as a result, may think that the continents literally
float upon the oceans. Similarly, mathematical terms such as “mean,” “plane,” and “point” have
meanings that are different from everyday language.
Accepting “facts”: “Facts” based solely on intuition or faulty reasoning can be misleading, inaccurate, or
incomplete in relation to both scientific and mathematical concepts. For example, students
sometimes incorrectly think that the first digit to the right of the decimal point is the “oneths”
(trying to be symmetrical around the decimal point). In science, students may think that heavier
objects fall faster, deserts are always hot, or gasses have no mass.
Unpublished Work Copyright © 2010 by Educational Testing Service. All Rights Reserved. 1 Applying real-world experiences and perceptions which can contradict scientific learning: Everyday life
and perceptual experiences can be misleading, inaccurate, or incomplete in relation to various
scientific explorations, studies, or concepts. For example, the five human senses seem infallible
to young students, yet human vision differs from certain insects that see in the UV portion of the
spectrum; it is hard to believe that sounds exist that humans cannot hear; touch is deceptive and
so wood feels “warmer” than metal in same room.
Accepting common sayings, beliefs, and myths: Widely heard/communicated sayings, beliefs, or myths
can be misleading, inaccurate, or incomplete in relation to various scientific explorations, studies,
or concepts. For example, some students believe that humans coexisted with/killed the dinosaurs;
meteors are “falling stars,” and camels store water in their humps.
Applying metaphors and analogies that only partially explain a scientific concept: Metaphors and
analogous comparisons of scientific objects or concepts with everyday terms can be misleading,
inaccurate, or incomplete in relation to various scientific concepts. For example, the following
analogies are partly helpful, but each has its limitations: electricity as water flowing; the heart as
a pump; the eye as a camera.
Accepting false equivalencies: Various mathematical procedures or concepts are sometimes viewed
incorrectly as being equivalent. For example, students may try to manipulate fractions using
whole-number reasoning, think that calculations involving time are the same as working with
decimals, or assume that subtraction and division are commutative, like addition and
multiplication.
About the Sample Set of Items Presented
Each item is presented along with the correct answer (answers) and the misconception category
(categories) that is (are) addressed. For relevant answer choices, a longer explanation is provided
regarding possible reasons that students may or may not select those choices. The explanations are tied to
the common misconceptions that certain students bring to the classroom every year, along with suggested
origins for those misconceptions. Also included are some caveats and suggestions for teachers to be
aware of and think about as they teach the various concepts, procedures, and investigations each year.
The items can be used as presented, or modified according to student needs. For example, an item could
be read to the class or students could be asked to read it; answer choices could be revealed one at a time
and students asked to think about each one; some answer choices could be removed (paying attention to
how answer choices correspond to potential misconceptions) to simplify the item; students could work
alone or in pairs, etc. The primary goal is to provide information that will be useful to inform next
instructional steps.
Unpublished Work Copyright © 2010 by Educational Testing Service. All Rights Reserved. 2 “Properties of Triangles”
Mathematics, Q8 – 39 – 02
Which of the following sides is the base of
triangle RST?
(A) Side RS
(B) Side ST
(C) Side RT
Copyright (c) 2007 Educational Testing Service. All rights reserved.
Correct answer:
(A) Side RS, (B) Side ST, and (C) Side RT
Misconception categories: Diagrams, Models, and Other Representations
Language and Vocabulary
This 8th-grade mathematics question addresses two different types of misconceptions
regarding the concept of bases of triangles.
(A) Side RS and (B) Side ST. Students who do not select these answer choices are failing to
recognize the limitations of a conventional representation. While teachers may understand
those limitations, students are often more literal. Since students likely see triangles with a
base down - the conventional form of a triangle that is ubiquitous in text books and
worksheets – most often, they may not think of the other choices as possible bases. Many
students also harbor the misconception that orientation is a key attribute of geometric
angles, shapes, and figures.
Students who do not select these answer choices may also be confusing language and
vocabulary. In this case, the word “base” used in everyday communication conveys a
meaning that is confusing or different from its use in the mathematical concept of a triangle.
These students generally know that “base” refers to the “bottom side” of something, and
they may not realize that a triangle can be rotated so that any side can be the “base.”
Therefore, they may think of choice (C) as the only correct answer.
Teachers are generally surprised to realize how some of the graphic representations they
have displayed in their classrooms for years or that appear in textbooks and worksheets can
often cause or reinforce common student misconceptions.
Unpublished Work Copyright © 2010 by Educational Testing Service. All Rights Reserved. 3 “Sides of Triangles III”
Mathematics, Q8 – 41 – 01
In the right triangle below, d = 10 and e = 9. What is
the hypotenuse of the triangle?
(A) 10
(B) √19
(C) √181
Copyright (c) 2007 Educational Testing Service. All rights reserved.
Correct answer:
(A) 10
Misconception category: Abstractions, Generalizations, and Simplifications
Diagrams, Models, and Other Representations
This 8th-grade mathematics question addresses two different types of misconceptions
regarding the hypotenuse of right triangles.
(B) √19 and (C) √181. Students who select these answer choices are simplifying the concept
of finding the hypotenuse of a right triangle. These students have observed, read about, or
been told that the Pythagorean theorem states that a2 + b2 = c2, where a and b are the
lengths of the legs and c is the length of the hypotenuse. They are simplifying that knowledge
by thinking that, to find the hypotenuse of any right triangle, just solve for “c,” even if the
length of the hypotenuse is given or if one of the legs of the triangle is labeled “c.”
Students who select these answer choices may also be failing to recognize the limitations of a
conventional representation. While teachers may understand those limitations, students are
often more literal. Since students likely see right triangles labeled “a,” “b,” and “c” with “c”
as the hypotenuse most often in textbooks and worksheets, they may just automatically think
that any side labeled “c” is the hypotenuse.
In both cases, teachers can help these students build a deeper understanding of right
triangles by providing them with multiple representations, contexts, and opportunities with
which to explore and expand their current thinking.
Unpublished Work Copyright © 2010 by Educational Testing Service. All Rights Reserved. 4 “Appropriate Units of Measurement II”
Mathematics, Q8 – 36 – 02
Which of the following standard units of measure is
incorrectly matched?
(A) Length – meter
(B) Mass – ounce
(C) Time – light year
(D) Area – foot2
(E) Volume – cubic centimeter
Copyright (c) 2007 Educational Testing Service. All rights reserved.
Correct answer:
(C) Time ─ light year
Misconception categories: Language and Vocabulary
This 8th-grade mathematics question addresses a common misconception regarding the
concepts of time and distance.
(C) Students who do not select this answer choice may be confusing language and vocabulary.
In this case, the term “year” used in everyday communication conveys a meaning that is
confusing or different from its use as a mathematical and scientific measure of distance.
These students know that “year” is a measure of time and refers to the time it takes Earth to
revolve around the sun ─ a calendar year or approximately 365 days. Therefore, they may
think of choice (C) as a correct match.
“Light year,” used as a measure of distance in astronomy, is the distance light travels in a
vacuum in one year, or about 5.88 trillion miles. This term is often misused as a unit of time
by the public in speaking, writing, and reporting. Students need a deeper understanding of
the terms and their contexts and the flexibility of mind in order to switch back and forth
between them.
Teachers need to be alert for words and terms used in everyday language that can convey a
confusing or completely different meaning in mathematical communication and endeavor. Unpublished Work Copyright © 2010 by Educational Testing Service. All Rights Reserved. 5 “Theoretical and Experimental Probability I”
Mathematics, Q8 – 65 – 01
Samuel tosses a penny 7 times. On each toss,
"tails" is the outcome. What is the probability that
Samuel's 8th toss will also be tails?
(A) P(Tails) < ½
(B) P(Tails) = ½
(C) P(Tails) > ½
Copyright (c) 2007 Educational Testing Service. All rights reserved.
Correct answer:
(B) P (Tails) = ½
Misconception category: Commonly Accepted “Facts”
This 8th-grade mathematics question addresses a common misconception regarding the
concepts of simple probability and independent events.
(A) P (Tails) < ½ and (C) P (Tails) > ½. Students who select these answer choices are
accepting a “fact” that has been told to them or that is based on intuition or faulty
reasoning. These students think that previous outcomes influence the probability of an
independent event. Even though they may logically know and understand the probability of
getting “tails” as the outcome of tossing a fair, 2-sided coin, they often have a stronger
emotional response that is based solely on previous outcomes. In the first case, they may
think that there have been 7 “tails” in a row, so a “heads is due.” In the second case, they
may think that there have been 7 “tails” in a row, so another “tails is more likely.”
In either case, these students do not understand enough about independent events and
random patterns. Teachers must be aware that students (and many adults) do not grasp the
significance of a random pattern – they can neither recognize one when they see it, nor can
they construct one if asked to do so. They also tend to see meaningful patterns where there
are none. Therefore, these students are unduly swayed by previous outcomes of independent
events and give them more meaning than they warrant.
Unpublished Work Copyright © 2010 by Educational Testing Service. All Rights Reserved. 6 “Algebraic Expressions V”
Mathematics, Q8 – 28 – 05
If f + g = 8, what is f + g + h?
(A) 9
(B) 12
(C) 15
(D) 16
(E) 8 + h
(F) You cannot tell without more information.
Copyright (c) 2007 Educational Testing Service. All rights reserved.
Correct answer:
(E) 8 + h
Misconception category: Equivalencies
This 8th-grade mathematics question addresses a common misconception regarding the
concept of algebraic variables.
(B) 12; (C) 15; and (D) 16. Students who select these answer choices are giving variables
numerical values instead of keeping them as unknowns or generalized numbers. They are
incorrectly viewing variables as being equivalent to other variables or to some pattern formed
by numbers or by other variables. By assuming that “f” and “g” equal certain amounts, these
students then go on to equate “h” with that assumption. By assuming that variables are
equivalent letters in the alphabet, students then equate “h” with the number “8.” Either
way, by making these false assumptions, students cannot correctly solve problems that
include variables.
Teachers must be aware of the tendency of certain students to treat as equal various
mathematical procedures and concepts. In this case, students need to understand that
variables are symbols used to stand for any one of a set of numbers or entities. They differ
from constants which stand for a particular thing.
Unpublished Work Copyright © 2010 by Educational Testing Service. All Rights Reserved. 7