1 MATHEMATICS TEACHING-RESEARCH JOURNAL ONLINE V 3, N 4
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Running Head: TEACHING SIGNED NUMBERS
Using the Number Line to Teach Signed Numbers
For Remedial Community College Mathematics
Alice Welt Cunningham, Ph.D.
Hostos Community College
500 Grand Concourse
Bronx, NY 10451
Telephone: 718/518-6629
Fax: 718/518-6706
Email: [email protected]
© Copyright 2009 Alice Welt Cunningham
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2 MATHEMATICS TEACHING-RESEARCH JOURNAL ONLINE V 3, N 4
December 2009
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3 MATHEMATICS TEACHING-RESEARCH JOURNAL ONLINE V 3, N 4
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Abstract
Using the Number Line to Teach Signed Numbers For Remedial Community College Mathematics
This article considers the use of the number line to teach signed numbers in remedial
mathematics classes designed to prepare community college students for the high stakes exit
examination necessary to graduation and college-level work. The strategy involves using one arrow
for each number in the relevant arithmetic operation, thus representing each number’s magnitude
and orientation. Each drawing reflects either the augmentation or cancellation process necessary to
determining the sign of the result, thus showing the sense behind the rules and promoting long-term
retention rather than short-term rule memorization. Such teaching for sense-making fosters
confidence in math, critical thinking skills and general academic confidence, in turn promoting
increased student retention for this academically-vulnerable population.
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4 MATHEMATICS TEACHING-RESEARCH JOURNAL ONLINE V 3, N 4
December 2009
Using the Number Line to Teach Signed Numbers For Remedial Community College Mathematics
Historically, the arithmetic of negative quantities developed slowly and late
(Mukhopadhyay, Swapna, & Schauble, 1990). “Negative numbers have intrigued and confused
some of the greatest mathematicians who have ever lived” (Davis & Maher, 1993, p. 60).
Mathematicians who found them troublesome include the French mathematician Rene Descartes,
who considered negative solutions of equations to be false, and the Swiss mathematician Leonhardt
Euler (Davis & Maher, 1993; Thomaidis & Tzanakis, 2007). Because of such numbers’ lack of
intuitive support, it took over a thousand years for their acceptance in the mathematical community
(Ball, 1993). This paper outlines a strategy, used by the author with success, for teaching signed
numbers in two different levels of remedial mathematics at a community college in a large East
Coast city. The goal of the strategy is to illustrate the sense behind the rules, thus promoting
understanding and long-term retention rather than short-term rule memorization, as well as
appreciation for, and confidence in, mathematics.
The present paper is an outgrowth of a pilot study performed with learning-disabled fifth-
graders by this author while in graduate school several years ago (Cunningham, 2004). Although
variations on the strategy presented here have been used by a few other authors (National Research
Council, 2001; Martin-Gay, 2007; Akst & Bragg, 2009, Cemen, 1993), the ideas herein presented Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research
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5 MATHEMATICS TEACHING-RESEARCH JOURNAL ONLINE V 3, N 4
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are both original to this author and, for reasons that will be explained, presented with what this
author believes to be greater clarity from the standpoint of student understanding.
The strategy in question utilizes a number-line approach for teaching the addition,
subtraction, multiplication, and division of signed numbers. The article first discusses the school
and its students. The article next summarizes the relevant literature. This discussion addresses four
issues: such students’ need for sense-making teaching strategies, particular difficulties inherent in
mastering operations with signed numbers, the use of diagrams to improve the comprehension of
mathematical topics, and the choice of the number line as a diagrammatic tool for teaching such
operations. The article then details, with accompanying diagrams, the proposed strategy for using
the number line to support the teaching of signed numbers. A discussion of student success ensues,
followed by a brief conclusion.
Nature of the Course and Its Students
The community college in question is a two-year public, open-admissions institution that is
part of a city-wide system receiving state and city funding. Of the current total enrollment of 5500
students, approximately 70% are female, 59% are Hispanic, and 30% are black. Close to 90% of
matriculating students enter needing at least one remedial course in reading, writing, or
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6 MATHEMATICS TEACHING-RESEARCH JOURNAL ONLINE V 3, N 4
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mathematics. Graduates take an average of four years to complete their associate’s degree, with a
graduation rate of only 24% after six years. Percentage retention and graduation rates are
significantly lower for black and Hispanic students. Difficulty completing mathematics courses
constitutes a significant contributing factor to low graduation rates (Hostos Community College,
2008).
The two remedial mathematics courses at issue here are each semester-long courses (Basic
Math Skills and Elementary Algebra) designed to prepare entering students to pass the system-wide
algebra mastery test necessary for graduation and college-level study. During the spring 2008
semester, of the 2600 registrants at this community college enrolled in mathematics courses, close
to 53% were enrolled in these two courses. Such students have had little success with mathematics
and have even less confidence in their ability to do it. At least in this author’s experience, they are
anxious to be given step-by-step algorithms into which to substitute numbers in order to pass the
test.
Signed numbers form a significant part of the curriculum of each of the two courses. The
Basic Math Skills course covers basic arithmetic concepts, including operations with whole
numbers, fractions and decimals, ratios and proportions, percents, operations with signed numbers
and scientific notation, the metric system, and an introduction to word problems and applications.
The progress is somewhat leisurely. The Elementary Algebra course covers operations with real
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7 MATHEMATICS TEACHING-RESEARCH JOURNAL ONLINE V 3, N 4
December 2009
numbers, algebraic expressions, solving linear equations in one variable (including applications and
word problems), exponential expressions, operations with and factoring polynomials, operations
with rational expressions, solving quadratic equations (again including word problems), roots and
radicals, graphing linear equations and finding the equations of lines, and solving systems of linear
equations in two variables. Concepts are covered at a gallop. Because of the importance of the exit
test for which the Elementary Algebra course is designed, the curriculum tends to emphasize rules
and algorithms necessary for success on the test. The reasoning underlying the rules can get lost in
the rush to cover the syllabus.
College Level Remedial Mathematics And the Need to Teach for Understanding
Recent research on mathematical cognition indicates that students who struggle with
mathematics view it as an obstacle course designed to present meaningless tasks, and their teachers
as out to “trick” them (Ginsburg, 1997, p. 25). Although motivation is thought to be the key factor
in the success or failure of education, by the time many students enter high school, disengagement
from course work and serious study is common (Ramaley & Zia, 2005). The consequences of such
disengagement are all the more serious for people from disadvantaged backgrounds, who often do
not get a second chance at education (Ramaley & Zia, 2005).
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8 MATHEMATICS TEACHING-RESEARCH JOURNAL ONLINE V 3, N 4
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Such concerns are particularly important for students of the “net generation” (e.g., Oblinger
& Oblinger, 2005), now entering community colleges. Today, only about 25% of students enrolled
in post-secondary education are traditional students--those who enter college immediately after high
school, who attend full-time, who work, if at all, only part-time, and who are financially dependent
on their parents (Ramaley & Zia, 2005). By contrast, non-traditional students may enter as adults,
attend part-time, and work full-time while enrolled. Such students are more likely to begin their
secondary education in a community college such as this one, where the yield of successful
bachelor’s graduates is low compared to students who begin their post-secondary education at a
four-year institution (Ramaley & Zia, 2005).
Modern scholars of mathematics pedagogy view the development of mathematical
proficiency as “an exploratory, dynamic, evolving discipline rather than as a rigid, absolute, closed
body of laws to be memorized” (Schoenfeld, 1992, p. 335). Conceptual understanding is considered
to be as important as procedural fluency and strategic competence (National Mathematics Advisory
Panel, 2008; National Research Council, 2001). Rather than constituting a mere by-product of
substantive competence, conceptual understanding is thought to propitiate such competence (e.g.,
National Mathematics Advisory Panel, 2008; National Research Council, 2001; Robinson,
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9 MATHEMATICS TEACHING-RESEARCH JOURNAL ONLINE V 3, N 4
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Menchetti, & Torgesen, 2002). Whereas relational understanding (knowing what to do and why) is
more adaptable to new tasks and easier to remember, instrumental understanding (knowing the rules
without reasons) is often fragile and error prone (e.g., Young-Loveridge, 2005; Tall, 2008).
Procedures need to be translated into an overall process that can be manipulated mentally in a
flexible way (Tall, 2008). While memorizing algorithmic procedures may have a short-term
advantage for passing an imminent test, understanding the process is necessary to the long-term
development of sophisticated mathematical thinking (Tall, 2008). Moreover, understanding the
reasoning behind the rules permits reconstruction of such rules in the case of confusion or a
memory lapse (Weber, 2002). Thus, teaching mathematics for sense-making should propitiate not
only students’ generic academic development but success on their crucial exit exam.
The goal of good teaching, then, is not the memorization of rules or algorithms, but
mathematical “sense-making” (Schoenfeld, 1992, pp. 335, 339, 340, 344). In the words of two
National Science Foundation scholars, “[i]t is important for students to see not only what they need
to know, but also why it is important” (Ramaley & Zia, 2005, p. 8.6). Again, “[l]earner-
constructed, sense-making experiences consistently are found to be the key to improved learning”
(Ramaley & Zia, 2005, p. 8.6). The true meaning of the ability to learn is not just to memorize the
rules of a particular task, but to be able to discern what the rules should be, and to make sense from
that input (Ramaley & Zia, 2005; Young-Loveridge, 2005). Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research
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10 MATHEMATICS TEACHING-RESEARCH JOURNAL ONLINE V 3, N 4
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By contrast, current-day teaching of remedial mathematics at the community college level
often is far removed from sense-making. Signed numbers are a case in point. To quote a recent
textbook expressly designed for teaching remedial mathematics at the college level, “[t]o add two
signed numbers, [i]f they have the same sign, add the absolute values and keep the sign. If they
have different signs, subtract the smaller absolute value from the larger and take the sign of the
number with the larger absolute value” Akst & Bragg, 2009, p. 347). For subtraction, the student is
directed to “[c]hange the operation of subtraction to addition, and change the number being
subtracted to its opposite,” then to “[f]ollow the rule for adding signed numbers” (Akst & Bragg,
2009, p. 355). Yet another recent textbook, designed for the same purpose, is to similar effect
(Martin-Gay, 2007). The students with the most fragile skill sets receive the least teaching for
understanding. Yet, as detailed above, it is precisely those students who most need teaching for
sense-making in order to learn.
Negative Numbers
Published material on the teaching and learning of signed numbers is scarce (Hativa &
Cohen, 1995). Such numbers have two components: a magnitude and an orientation (National
Research Council, 2001; Ball, 1993; see Bruno & Martinon, 1999). Negative numbers can be
viewed as representing both (a) an amount that is the opposite of something (for instance, -5 can Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research
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11 MATHEMATICS TEACHING-RESEARCH JOURNAL ONLINE V 3, N 4
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represent a $5 debt which is the opposite of a $5 credit balance), and (b) a location relative to zero
(for instance, -5 can represent a position that is five units away from zero, Ball, 1993). The
magnitude component leads to the concept of absolute value (Ball, 1993), meaning the distance
between the number and zero (e.g., Martin-Gay, 2007). Thus, comparing signed numbers becomes
complicated. In the words of one scholar, “[t]here is a sense in which -5 is more than -1 and equal
to 5, even though, conventionally, the ‘right’ answer is that -5 is less than both -1 and 5” (Ball,
1993, p. 379), because -5 is further along the negative side of the number line than either 0 or 5.
Simultaneously understanding that -5 is, in one sense more than -1, and, in another sense less than -
1, is central to understanding negative numbers (Ball, 1993). This tension between the magnitude of
negative numbers and their order on the number line underlies both the historical difficulties in
understanding negative numbers and many current students’ confusion over such numbers’ relative
size (Thomaidis & Tzanakis, 1970).
An additional complicating factor is notational. The minus sign appended to a negative
number has three distinct meanings: (a) the operation of subtraction, (b) a negative magnitude, and
(c) inversion, or what Piaget would have called “’reversibility’” (Nunez, 1993, p. 64), meaning that
the prescribed operation must be reversed (Nunez, 1993, pp. 63-64). Specifically, the “! ” sign
means “the opposite of,” or “the additive inverse of” (e.g., Martin-Gay, 2007, p. 32), with the result
that, if a is a number, !(!a) = a. The impact on students of these multiple meanings for the minus Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research
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12 MATHEMATICS TEACHING-RESEARCH JOURNAL ONLINE V 3, N 4
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sign can be seen from one study showing that both young and adult learners who could offset
negative and positive numbers correctly in their heads became confused by the minus signs when
asked to solve the word problems in writing. Those students who were alerted in advance that the
minus sign was being used to denote the type of magnitude (negative) rather than an operation
(subtraction) performed as well on the written problems as on mental ones (Nunez, 1993).
Because of this confusion between negative magnitude and the operation of subtraction,
there have been a number of attempts to denote negative magnitude using a symbol other than the
traditional minus sign, such as an elevated minus sign (Nunez, 1993; Cemen, 1993), or a circumflex
(Ball, 1993). Calculators usually have two types of minus signs, a smaller, slightly raised symbol to
indicate the type of magnitude and a larger, traditional minus symbol to indicate the operation of
subtraction. However, the use of the same notation for the three different meanings (subtraction,
negative magnitude, and inversion) is neither arbitrary nor accidental, because those three situations
obey the same rule with respect to operations: cancellation (Nunez, 1993). Subtracting a number is
the same as adding its inverse (Davis & Maher, 1993; Nunez, 1993; Bruno & Martinon, 1999). For
example, a gain of negative 4 is equal to a loss of positive 4. Rather than negative and positive
numbers representing two separate attributes, one for gains and the other for losses, only one
attribute is involved, the idea of change. The sign of the change alone distinguishes a gain from a
loss. Cancellation becomes automatic, with the sign of the change determining whether the balance Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research
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is increasing or decreasing (Davis & Maher, 1993). In effect, the two operations, addition and
subtraction, have the same meaning, what some scholars have called “double language” (Bruno &
Martinon, 1999, p. 793). “I won 2” is equivalent to “I lost -2.” Using the operation of addition,
“John had 2 and he was given 3; now he has 2 + 3 = 5” is equivalent to saying that “John had 2 and
they took away -3; now he has 2 ! (-3) = 5” (Bruno & Martinon, 1999, p. 793). Analogously, using
the operation of subtraction, “John had 3 and they took [away] 2; now he has 3 ! 2 = 1” is
equivalent to “John had 3, and he was given -2; now he has 3 + (-2) = 1” (Bruno & Martinon, 1999,
p. 793).
In addition to the single attribute issue just discussed, operations with signed numbers
reflect one of the many situations in mathematics where previously learned knowledge causes
confusion that impedes learning. For example, previously learned knowledge suggests that the
operation of subtraction produces less, a result inconsistent with negative numbers, where
subtraction produces more (Tall, 2008). (For other number line studies voicing concern over the
impact of previous learning on students’ expansion of their mathematical understanding, see, e.g.,
Merenluto, 2003; Sirotik & Zazkis, 2007, both arguing that knowledge regarding the discrete nature
of the natural numbers impedes understanding the density of real numbers, which have no “next”
number).
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Despite these difficulties, recent research has shown that even prior to instruction children
have a rudimentary concept of negative numbers (Hativah & Cohen, 1995; Mukhopadhyay,
Resnick, & Schauble, 1990; National Research Council, 2001). For example, children have
exposure to games where points are lost, resulting in scores below zero (National Research Council,
2001), while older but unschooled children have familiarity with buying and selling, owing and
paying (Mukhopdhyay, Resnick, & Schauble, 1990). Thus, it is thought that, building on such
informal knowledge, instruction in negative numbers can begin much earlier than is currently done
(Hativah & Cohen, 1995; National Research Council, 2001; National Mathematics Advisory Panel,
2008), a conclusion surely relevant to overcoming remedial students’ antipathy to such numbers.
The Role of Diagrams in Teaching for Understanding
The 2000 Principles and Standards of the National Council of Teachers of Mathematics cite
“representation” as one of the ten components of the connected body of mathematical knowledge
that students should acquire as they progress through school (NCTM, p. 29) and state that
representations such as diagrams, displays, and symbolic expressions should be treated as essential
elements in supporting students’ understanding of mathematical concepts. As early as 1945, George
Polya, in his book How to Solve it: A New Mathematical Method, named understanding the problem
as the first of the four important elements in his heuristic for problem-solving and suggested
drawing a figure as an important aspect of this understanding (Polya, 1988, pp. 4, 7). Other authors Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research
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15 MATHEMATICS TEACHING-RESEARCH JOURNAL ONLINE V 3, N 4
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speak of visual representations as important both in aiding reflection and for communicating
mathematical ideas (e.g., Singer, 2007; Elia, Gagatsis, & Demetriou, 2007; Diezmann & English,
2001). An increasing body of research supports the view that learning is affected positively by
presenting text and illustrations together (Mayer & Sims, 1994). For instance, one study showed a
better result when visual computer-animated and verbal explanations were presented to college
students simultaneously rather than separately, leading to better science problem-solving transfer
(Mayer & Sims, 1994). Another study showed that the use of diagrams in scientific texts improved
college students’ recall of the conceptual information being presented (Mayer & Gallini, 1990).
One scholar who used meta-analysis to integrate the reports of 487 studies to assess the effects of
diagrams on mathematics problem-solving concluded that, at the 99% confidence level, an
advantage was observed for using diagrams to represent problems (Hembree, 1992). According to
this report, pictures along with text provided the largest performance differences, with formats
using “minimum verbiage” tending toward better performance, especially with slower students
(Hembree, 1992, p. 261).
More recently, some neuropsychological and neuroimaging studies have suggested the
existence of a language-independent spatial representation of numbers in the human brain (e.g.,
Singer, 2007). This topological capacity is thought to substitute global perception for discrete
numerical perception, thus fostering mathematical understanding (Singer, 2007). Admittedly, some Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research
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16 MATHEMATICS TEACHING-RESEARCH JOURNAL ONLINE V 3, N 4
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recent studies claim that diagrams compete for limited working space in short-term memory, thus
interfering with problem-solving (Pike, 2003; Elia, Gagatsis, & Demetriou, 2007). However, net-
generation students, who have come of age using the graphical interfaces of modern technology,
prefer pictorially-grounded teaching methods (Oblinger & Oblinger, 2005).
The Effect of the Number Line
The identification of the real number system with the number line has been described as
“one of the main ideas in mathematics” (Bruno & Martinon, 1999, p. 791). In the words of a recent
report of the National Research Council (2001), the number line:
lets you interpret whole numbers, negative numbers, and fractions all as part of one overall
system. Furthermore, it provides a uniform way to extend the rational number system to
include numbers such as π and 2 that are not rational; it provides a link between arithmetic
and geometry; and it paves the way for analytic geometry, which connects algebra and
geometry (p. 87).
For example, representing fractions and their decimal equivalents as identical points on the line can
help to obviate the student misconception that fractions and decimals constitute two separate sets of
numbers (Pagni, 2004). The use of approximation permits the correct placement on the number line
of irrational numbers as well (National Research Council, 2001). The number line thus serves as a
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dynamical mental structure rather than as a constraining step-by-step algorithm, stimulating the
ability to transfer from one context to another and shortening learning time (Singer, 2007).
Development of such understanding takes time. While the number systems N ! Z ! Q ! R
may be seen by mathematicians as successive number systems represented on the number line
which lies in the complex plane C, each such extension involves a sophisticated learning process for
a novice (Tall, 2008). Operating with whole numbers gives the sense that addition and
multiplication produce a bigger result, while subtraction produces a smaller one, a concept that
conflicts with the behavior of integers, where subtracting a negative number gives more (Tall,
2008).
Moreover, the view of the utility of number lines for teaching mathematics is not universally
shared (Diezman & Lowrie, 2007). For example, one Cypriot study found that first and second-
graders had difficulty calculating addition and subtraction problems on a number line because of
confusion over mathematical information presented implicitly in a diagram (Shiakalli & Gagatsis,
2006), while a Dutch interview of several mathematically adept youngsters found use of the number
line to be constraining (van den Heuvel-Panhuizen, 2008). In particular, a nine-year old girl forced
to use a structured number line with intervals of one to calculate the difference between 59 and 17
found herself reverting to a counting strategy (van den Heuvel-Panhuizen, 2008). These interviews, Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research
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18 MATHEMATICS TEACHING-RESEARCH JOURNAL ONLINE V 3, N 4
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while anecdotal, support the views of at least one researcher, a great proponent of using computer
diagrams in teaching mathematics, who dismisses the number line as a “mismatch” in terms of
manipulatives because its use requires “counting the counts” (Clements & McMillen, 1996, pp.
208-209). In his view, when adding 5 and 4, the student must first locate 5 on the number line and
then count the 6 as 1, the 7 as 2, etc., to reach the result (Clements & McMillen, 1996; cited with
approval, Sarama, 2004).
Others criticize use of the number line as a diagrammatic tool because of confusion caused
by the line’s geometric interpretation of a number as both a point and a vector, as compared to the
line’s arithmetic interpretation as the distance between points (e.g., Elia, Gagatsis, & Demetriou,
2007). This criticism is particularly apposite in the context of research regarding children’s
understanding of measurement, where even older children respond to measurement with a nonzero
origin by reading off whatever number on the ruler corresponds to the end of the object being
measured (National Research Council, 2001). Other difficulties encountered by students include
counting the lines instead of the spaces (Mitchell & Horne, 2008) or failing to understand what part
of the number line constitutes a whole in calculating a fractional part (Hannulu, 2003).
Although at least one study has considered the difficulties that using the number line poses
to algebra students trying to understand quadratic inequalities (Thomaidis & Tzanakis, 2007), none Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research
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of the criticism just voiced involves teaching the arithmetic of negative numbers. By contrast, the
report of the National Research Council (2001) particularly recommends using the number line for
such teaching. For this purpose, the report depicts two different diagrammatic techniques. The first
is a two-arrow approach that uses zero as the starting point and separate directional arrows for the
numbers being added or subtracted. Variations of this technique can be found, for instance, in
Martin-Gay (2007, a remedial college textbook) and Cemen (1993, pre-college math). A second
approach uses a single arrow. The first of two numbers in an operation functions as the starting
point, or origin, of the calculation, and the second number is illustrated by an arrow representing
both that number’s magnitude and its orientation. This latter approach in effect involves a
geometrical interpretation of addition and subtraction as translations to the right or left on the
number line (National Research Council, 2001). This second approach (using the first number as
the starting point of the calculation) is used, for instance, by Akst & Bragg (2009), another remedial
college textbook. However, The National Research Council report (2001) views this approach as
“quite sophisticated” (p. 93). In view of students’ difficulties with negative numbers, and the
difficulties noted above that students experience measuring the length of an object when using a
starting point other than zero, the second approach strikes this author as unnecessarily complex.
Description of Strategy
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For this reason, the strategy described here starts at zero and uses multiple arrows, one for
each number involved in the computation. In each case, the relevant number and sign is depicted
with its corresponding arrow; the result is demarcated by circling the consequent number. The
theory behind this approach is to reduce the students’ need to focus on both the concept of
magnitude and the concept of orientation at the same time. The drawings are meant to be depicted
with small numbers. That way, in case of a test-induced memory lapse, the relevant rules can be
reconstructed pictorially (see Weber, 2002) and then applied to the larger numbers involved in the
examination question. More importantly, this approach is intended to indicate, without resorting to
jargon-filled absolute value rules of the type quoted earlier in this article, the relevance of
magnitude, or absolute value, to the result. By using one arrow for each number in an operation,
the automatic augmentation or cancellation resulting from a visual comparison of each such
number’s magnitude and orientation immediately becomes apparent. Further to the right on the
number line means both ‘greater in absolute value’ and ‘greater in actual value.’ Further to the left
on the number line means both ‘greater in absolute value’ and ‘less in actual value.’ The pictorial
nature of the number line permits an intuitive understanding of this distinction (see Singer, 2007).
Opposites
The first concept to be covered is that of “opposites” or “additive inverses.” These are
numbers that have the same size or magnitude (i.e., absolute value) but that lie on opposite sides of Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research
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zero, so that they have different actual values. Because the numbers in each such pair are
equidistant from zero, when they are added together, they produce zero (the additive identity, that
number which, when added to any real number, gives the latter number, e.g., Keenan & Dressler,
1990. See Figures 1 and 2).
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Addition
The result of the operation of addition depends on whether the two numbers being added
have the same or different signs.
Same Signs
Adding two numbers (“addends”) with the same sign, whether positive or negative,
increases the amount of the quantity being added and therefore produces a result (“sum”) with the Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research
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December 2009
same sign. Adding two positive numbers must produce a positive result, because the amount of the
positive quantity is increased. Once another positive quantity is added to the initial positive
quantity, the result will be correspondingly further to the right, or positive side, of the number line.
For example, (+5) + (+3) = (+8). Similarly, adding two negative numbers with the same sign must
produce a negative result, because the amount of the negative quantity is being increased. For
example, (-5) + (-3) = (-8). Adding another negative quantity to the initial negative quantity results
in a negative number that is correspondingly further to the left, or negative side, of the number line
(see Figure 3).
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Different Signs
In the case of numbers with different signs, the result of adding such numbers depends on
which number (quantity) has the larger size or magnitude. When the magnitude of the numbers is
compared pictorially (a process equivalent to comparing the two arithmetically by subtraction), the
sign of the larger number determines whether the result will lie on the positive or on the negative
side of the number line. If the positive number is larger, the result will lie on the positive side of the
line, while if the negative number is larger, the result will lie on the negative side of the number Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research
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December 2009
line. For example, (+8) + (-5) = (+3). The addition of the negative number reduces the result to less
than the original number, but the result is still positive (to the right of zero on the number line),
because the negative quantity being added is less than the original positive number. By contrast, (-
8) + (+5) = (-3). The negative quantity being added is greater than the original positive number,
resulting in a number to the left of zero on the number line (see Figure 4). Comparing the quantities
visually obviates the need for memorizing rules regarding comparisons of the respective numbers’
absolute values.
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The same results obtain when the order of the addends is reversed: (-5) + (+8) = (+3), and
(+5) + (-8) = (-3). Addition is commutative, and the same analysis applies regardless of the order in
which the two numbers are added. Pictorially comparing the magnitudes of the two addends results
in automatic cancellation, with the number having the larger magnitude determining whether the
result falls on the positive side of the number line (to the right of zero) or on the negative side of the
number line (to the left of zero).
Subtraction
As can be seen from Figures 5 and 6, subtracting a number is equivalent to the addition of
its “opposite,” because the operation of subtraction reflects taking away the quantity being
subtracted. Taking away a positive amount makes the original number smaller by reducing the net
positive quantity, thus moving the result further to the left on the number line and making the
operation equivalent to subtracting a positive number. Taking away a negative amount makes the
original number larger by reducing the net negative quantity, thus moving the result further to the
right on the number line and making the operation equivalent to adding a positive number. As in the
case of addition, the question is whether the effect of the operation is to increase the total positive
amount, making the result more positive (further to the right on the number line) or to increase the
total negative amount, making the result more negative (further to the left on the number line).
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Different Signs
Because subtracting a number is equivalent to adding its “opposite,” subtracting numbers
with different signs is equivalent to adding numbers with the same sign. Once the number to be
subtracted is changed to its opposite, the sign of that number becomes the same as the sign of the
number from which it is being subtracted. For example, (+5) – (-3) is equivalent to (+5) + (+3), so
the result is (+8), as illustrated for addition in Figure 3 above. A negative quantity, (-3), is being
taken away, making the original positive quantity, (+5), larger: (+8), which is further to the right on
the number line. By the same analysis, (-5) – (+3) = (-5) + (-3) = (-8), because the original negative
quantity, (-5), is being increased by taking away a positive amount, (+3), making the original
negative quantity larger: (-8), which is further to the left on the number line. (For illustrations using
addition, see Figure 3; for correlative examples using subtraction, see Figure 5.)
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Same Signs
By a similar analysis, subtracting a number from another number with the same sign is
equivalent to adding a number with the opposite sign. Once the sign of the number to be subtracted
is changed to its opposite, that number has a sign different from the number from which it is being
subtracted. Therefore, the same rules apply as for adding numbers with different signs.
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Two positive numbers. In the case of subtraction with two positive numbers, whether the
result is positive or negative depends on whether the number being subtracted is smaller or larger
than the original number. For example, (+8) – (+5) = (+8) + (-5) = (+3). The original positive
quantity, (+8), is larger than the positive quantity being taken away, (+5), so the result, (+3), is
smaller than the original number, (+8), but is still positive (to the right of zero on the number line).
By contrast, (+5) – (+8) = (+5) + (-8) = (-3). The positive quantity being subtracted, (+8), is greater
than the original number, (+5), so the result is negative (to the left of zero on the number line. See
Figure 6).
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Two negative numbers. In the case of subtraction with two negative numbers, the original
negative quantity is being reduced by the amount of the negative quantity being taken away, so the
result is larger than the original quantity (further to the right on the number line). However, whether
the result is positive or negative depends on whether the number being subtracted is smaller or
larger, in terms of magnitude, than the original number. For example,
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(-8) – (-5) = (-8) + (+5) = (-3). The result, (-3), is larger than the original number, (-8) (i.e., further
to the right on the number line), but the result is still negative, because the amount of negative being
taken away, (-5), is less than the amount of negative in the original number, (-8). By contrast, (-5)
! (-8) = (-5) + (+8) = (+3). The result, (+3), is not only larger than the original number, (-5), but is
positive, because the amount of negative being taken away, (-8), is greater than the original number,
(-5). (See Figure 7.)
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Multiplication
Just as adding or subtracting a number on the number line may be interpreted as a
translation of the line to the right or the left by the number of units being added or subtracted, so
multiplication and division on the number line may be regarded as transforming the line by
stretching or shrinking it (National Research Council, 2001). However, the strategy proposed here
requires a less subtle approach. Multiplication is repeated addition (e.g., National Research Council,
2001; Martin-Gay, 2007), and division is repeated subtraction. Thus, the number line strategy
detailed above for addition and subtraction (an individual arrow to represent both the magnitude
and the orientation of each component number) serves equally well for multiplication and division.
Same Signs
Multiplication of two numbers with the same sign, whether positive or negative, will always
produce a positive result. For example, (+3) x (+3) = (+9), because (+3) x (+3) = (+3) + (+3) + (+3).
Similarly, the multiplication of two negative numbers will always produce a positive result, because
the negative amount represented by one factor is being taken away the number of times represented
by the other factor. For example, (-3) x (-3) = (+9), because (-3) x (-3) = -(-3) – (-3) – (-3) = (+3) +
(+3) + (+3). Starting at zero on the number line, the quantity (-3) is being taken away 3 times.
Thus, the result must be positive (to the right of zero on the number line). Yet again, use of the
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number line affords a readily understood representation of this often counterintuitive result (see
Figure 8).
Different Signs
By contrast, two factors with different signs will always produce a negative result. In this
case, multiplication is equivalent to adding the negative factor the number of times prescribed by
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the positive factor, so that the negative quantity is increased, producing a negative result. For
example, (+3) x (-3) = (-9), because (+3) x (-3) is equivalent to (-3) + (-3) + (-3), which equals
(-9). Multiplying (-3) x (3) will again produce a negative result, again for the same reason.
Multiplying (-3) x (+3) is equivalent to -(+3) – (+3) – (+3) = (-3) + (-3) + (-3) = (-9). (See Figure
9.) Like addition, multiplication is commutative (e.g., Akst & Bragg, 2009).
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Multiple Factors
The analysis just described extends to multiple factors with different signs.
An even number of the same sign. Multiplication by an even number of positive or negative
numbers will always produce a positive result. For the positive factors, the result is repeated
addition on the positive side of the number line. For the negative factors, the result is the repeated
taking away of a negative factor, again producing a positive result by eliminating the negative
quantity.
An odd number of the same sign. Multiplying an odd number of positive factors will again
produce a positive result, because the result is repeated addition on the positive side of the number
line. However, multiplication by an odd number of negative factors will produce a negative result.
Multiplying every factor but one (i.e., multiplication by an even number of negative factors) will
first produce a positive result for the reasons set forth above. For example, given factors with the
signs +, +, -, -, -, multiplication of the first two factors (+ x +) will produce a positive result.
Multiplication of this product by the third factor, (-), will produce a negative result, and
multiplication of that product by the fourth factor, (-), will produce a positive result. Multiplication
of this positive product by the sole remaining negative factor will produce a negative result, just as
in the case of the multiplication of two factors with different signs.
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Division
Division is traditionally defined as multiplication by the reciprocal of the divisor.
a 1 Specifically, = a • (e.g., Keenan & Dressler, 1990, Martin-Gay, 2007). Therefore, the principles b b
set forth above regarding the multiplication of signed numbers are equally applicable to the
operation of division.
Gauging Success of the Number Line Strategy
According to one longitudinal study using the number line to teach arithmetical operations,
the “effectiveness [of this strategy] in school acts as the main argument for its sustainability in
practice” (Singer, 2007, p. 91). This study showed high test results (more than a 60% success rate at
solving open-ended questions as compared to a 20% success rate for similar questions on a national
assessment, Singer, 2007). Additionally, student use of teaching strategies on assessments
constitutes support for attributing improved student performance to the students’ instruction
(Weber, 2002).
By these standards, the number line strategy presented here has assisted the remedial
community college students who are the subject of this article in increasing their understanding of
signed numbers in a way that can be retained in the context of the high stakes exit examination
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necessary to their graduation. Data detailing the school-wide results on the algebra exit test for the
most recent academic year (the author’s first year of college mathematics teaching) appear in
Tables I-III. Table I reflects school-wide results for both semesters (including the author’s results),
while Table II reflects the author’s results. Table III compares these respective sets of results, both
by semester and over the entire academic year.
Table I
School-Wide Algebra Exit-Test Pass Rates (including author’s scores)
Semester Number Tested Number Passing Number Failing Percent Passing Fall 2008 362 200 162 55.2% Spring 2009 312 177 135 56.7% Overall 674 377 297 55.9%
Table II
Author’s Algebra Exit-Test Pass Rates
Semester Number Tested Number Passing Number Failing Percent Passing Fall 2008 12 8 4 83.3% Spring 2009 24 14 10 58.3% Overall 36 24 12 66.6%
Table III
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Semester School-Wide Author Only Fall 2008 55.2% 83.3% Spring 2009 56.7% 58.3% Overall 55.9% 66.6%
The data show the author’s percentage pass rate on the algebra exit test to be higher both
each semester and overall than the school-wide pass rate in each category. These results cannot be
considered to be significant, in that they reflect a test that covers a multitude of subjects. No
separate data yet exist for the author’s number line teaching strategy. However, in-class
assessments over the course of the year reflected student use of this teaching strategy. Moreover,
preliminary results on an anonymous student questionnaire reflect appreciation for the pictorial
comparison approach. Comments include “now I understand how to add a positive and a negative,”
and “the graph basically shows you how many spaces [are] in between [the starting point and the
ending point].” Another student stated about the number line approach:
It’s really helpful because it help[s] students that [are] having problems with
sign[ed] numbers, and even more to do operation[s] +, ! , ÷, * and word problems.
Student use of the strategy on assessments, coupled with the author’s first-year pass rates on
the algebra exit test, provide evidence in support of strategies designed to teach remedial
mathematics for long-term understanding rather than for short-term rule memorization. Specific
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testing of the number line approach and other sense-making strategies used by the author awaits
implementation in coming semesters.
Conclusion
This paper sets forth a strategy for teaching arithmetical operations with signed numbers to
college students taking remedial mathematics classes. The strategy involves the use of a single
arrow for each number in the relevant arithmetic operation, so that both the magnitude and the
orientation of each number is represented by its own arrow, obviating the need for separate mental
juggling of these two defining characteristics of signed numbers. Each number line drawing
immediately reflects either the augmentation or the cancellation process necessary to determining
the sign of the result. Thus, in an area of mathematics that has plagued mathematicians for centuries
and that is rife with rules and jargon, this pictorial strategy shows the sense behind the rules.
To the extent that high pass rates on a compulsory exit examination, together with students’
use of the strategy on in-class assessments, evidence support for this approach, the strategy has
been successful. Enabling students in remedial courses to view the study of mathematics as sense-
making, rather than as yet another onerous remedial barrier to be hurdled, should facilitate not only
success in the course but generic academic confidence, thus fostering increased student retention for
this academically-fragile population. Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research
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Acknowledgements
The author wishes to thank Professor Alexander Vaninsky for his thorough and meticulous
comments on an earlier draft of this paper and Professor Rees Shad for his patience and artistry in
teaching an inept pupil to construct computer number line drawings.
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