DECIMAL FR
.06
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MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
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Vol. 19, No. 7, March 2014
Copyright © 2014 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
RACTIONS:
AN IMPORTANT POINT
Teachers who are skilled at recognizing students’
misconceptions about decimals are better equipped to
make instructional decisions that build on these ideas.
.6
Sherri L. Martinie
MALYUGIN/THINKSTOCK
h
How can a simple dot—the decimal
point—be the source of such frustration for students and teachers? As I
worked through my own frustrations,
I found that my students seemed to
fall into groups in terms of misconceptions that they revealed when talking about and working with decimals.
When asking students to illustrate
their thinking and explain their ideas
using various models, it became
apparent that they lacked an understanding of place value in our base-ten
number system. Furthermore, they did
not recognize that decimal fractions
Vol. 19, No. 7, March 2014
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were a special class of fractions with
base-ten denominators.
Knowledge of rational number is
built on a foundation of prerequisite
knowledge. Decimal representation
is compounded in complexity by the
merging of whole-number knowledge
and common fractions, with very specific kinds of units. As students incorporate new concepts of rational number into their existing knowledge, we
often see systematic and predictable
errors. The conceptual understanding
that students possess results in specific
misconceptions. Such misconceptions
are inherent to learning and unavoidable. The errors that are made provide
important diagnostic information
for teachers. This article will focus
on research on decimal comparison
strategies, the behavior we see from
students, and what it reveals about the
thinking patterns and misconceptions
that students possess as their knowledge develops over time.
MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
421
BACKGROUND
I witnessed students struggling with
decimals throughout middle school
and beyond. Sometimes it occurred
because students failed to connect
existing knowledge to new knowledge. Often, they saw instruction
on decimal fractions as no different
than whole-number knowledge. As
a teacher, I created this environment.
When comparing the numbers to the
right of the decimal point, I would
encourage students to annex zeros
and then compare the numbers as if
they were whole numbers. Trying to
simplify the process for my students
prevented them from creating meaning and masked their true understanding (D’Ambrosio and Kastberg 2012).
Instruction on decimals as an extension of the whole-number system often
occurs with inadequate understanding
of place-value concepts (Fuson 1990).
As a result, various issues begin to percolate. For example, students write too
many digits in a column or they believe
that there is symmetry about the decimal point resulting in a “oneths” place
to mirror a units digit. As with whole
numbers, some students also think that
a decimal fraction with more digits
after the decimal point, the larger the
value of the digits (e.g., 0.271 is viewed
as being larger than 0.81).
Research reveals several potential
reasons for the lack of understanding
of decimal fractions. Students’ prior
knowledge may be predominately
procedural, which may account for
misconceptions applied to decimal
numbers (Hiebert and Wearne 1985).
Often, students inappropriately apply
rules, resulting in the right answer for
the wrong reason, reinforcing an inappropriate use of the rule. The basis for
this thinking then persists, and procedural flaws are not corrected (Hiebert
and Wearne 1985; Steinle and Stacey
2004b). Rule-based instruction produces accurate answers initially, but
students who are instructed using only
procedural methods tend to regress in
performance over time (Woodward,
Howard, and Battle 1997; Steinle and
Stacey 2004a).
Mack (1995) indicates that with
time and direct effort, students can
separate whole-number from rationalnumber constructs and develop a
meaningful understanding of how
fractions and decimals are represented
symbolically. What does this effort
look like? First, it requires a teacher to
be armed with knowledge of how students think about decimals. Second,
it requires that a teacher be equipped
with tools to help reveal student misconceptions and then build on student
thinking in a meaningful way.
WHAT DECIMAL STRATEGIES
REVEALED
When comparing two decimal numbers, students behave in ways that provide insight into their thinking. Four
Table 1 Four misconceptions are labeled, described, and shown.
Misconception
Description
Behavior When Comparing
Longer-is-larger thinking
(numerator focused)
Students know that moving column by column
to the left of the decimal point results in a larger
value. They see the columns to the right as more of
the same and select as larger the number that has
more digits following the decimal point.
They incorrectly select 0.123 > 0.8,
but correctly select 0.812 > 0.3.
Zero-makes-small
thinking
Students select as larger the number with more
decimal places except when a zero is immediately
to the right of the decimal point.
They correctly select 0.6 > 0.089
but incorrectly select 0.6 > 0.89.
Shorter-is-larger thinking Students know that moving column by column to
(denominator focused)
the right of the decimal point results in a smaller
portion. They select as larger a number that has
fewer digits following the decimal point.
They incorrectly select 0.2 > 0.56,
but correctly select 0.8 > 0.123.
Money rule
They correctly select 0.89 > 0.6 (by
converting to 0.60), but they may
use shorter-is-larger thinking and incorrectly think that 0.39 > 0.3912.
They may use longer-is-larger
thinking and correctly select
0.3912 > 0.39.
They may also say that
0.39 = 0.3912.
422
Students identify decimal fractions with money and
accurately compare decimals to the hundredths
place. However, when comparing decimals beyond
the hundredths place, they will revert to using one
of the rules above or will select the two values as
equal. Longer-is-larger thinkers can often have their
thinking masked by the money rule.
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Vol. 19, No. 7, March 2014
basic misconceptions can be found
when students compare two decimal
fractions (Moloney and Stacey 1997;
Nesher and Peled 1986; Resnick et al.
1989; Sackur-Grisvard and Leonard
1985; Steinle and Stacey 2004a). The
four misconception categories are
defined in table 1.
“Task experts,” those students who
accurately compare decimal fractions,
often draw on their knowledge of fractional parts and place value. Some use
a “money rule,” as described in table 1.
Others annex zeros as a “coping strategy”: By adding zeros to equalize the
length of the numbers, they use their
whole-number knowledge to compare
the decimal fractions without needing
to draw on a meaningful understanding of decimal fractions.
STUDENT MISCONCEPTIONS
REVEALED
To get a sense of how middle school
students think about decimals and to
identify their misconceptions, I gave
a decimal comparison test designed
by Stacey, Steinle, and Chambers
(1999) to over 350 students in grades
6−8 from two different schools. The
results were consistent with previous
research, and the students fell into
predictable classifications when comparing pairs of decimals (Moloney and
Stacey 1997; Nesher and Peled 1986;
Resnick et al. 1989; Sackur-Grisvard
and Leonard 1985; Steinle and Stacey
2004a). I then asked the students to
represent or explain decimal numbers
to me to get a better sense of the type
of conceptual understanding they possessed. I used an instrument designed
by Martinie and Bay-Williams (2003)
(see fig. 1) and asked them to represent 0.6 and 0.06 in these four ways:
1. Illustrate where these two numbers
would fall on a number line
2. Shade 10 × 10 grids to represent
each number
3. Use money to represent the two
values
4. Use place value to explain your
meaning
Fig. 1 These student tasks highlight multiple representations of decimals.
Compare 0.6 and 0.06 in the following four ways.
a. Draw and label a number line with 0.6 and 0.06. You should add other numbers to your number line, such as 0 and 1,
for example.
b. Use a 10 ×10 grid to shade 0.6 on one grid and 0.06 on the other.
Shade 0.6 of this grid.
Shade 0.06 of this grid.
Explain how you shaded your grids.
a. Use money to explain 0.6 and 0.06.
b. Use place value (what do you know about the place-value columns, where these numbers would be, and what it
means to a number to be in that place).
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MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
423
Table 2 The fact that unacceptable responses are much higher than acceptable
responses shows that additional work is necessary.
Acceptable
Response
Unacceptable
Response
No
Response
Number line
21%
60%
19%
Area model/grids
38%
59%
3%
Money
41%
51%
9%
Place value
25%
49%
26%
Fig. 2 This response is an example of a longer-is-larger thinker.
addressed in the Common Core State
Standards for Mathematics (CCSSI
2010), in Developing Essential Understanding of Rational Number (BarnettClarke et al. 2010), and in Curriculum Focal Points for Prekindergarten
through Grade 8 Mathematics: A Quest
for Coherence (NCTM 2006). The
representational meaning for decimal
fractions written symbolically depends
on an understanding of the base-ten
number system and place value. An
understanding of what this symbolizes in terms of size or value depends
on the existence of number sense and,
in particular, fraction sense developed
through area models and number lines.
STUDENT WORK SAMPLES
Analyzing student work with each
representation reveals much about student thinking, which often correlates
with the classification to which they
were initially assigned. A sample of
work for each representation is given,
along with a brief explanation of what
it reveals and how they connect to the
misconceptions defined in table 1.
Fig. 3 Reverse thinking is explored with a hundred grid.
Place Value
Figure 2 is the response of a longer-islarger thinker, who lacks an understanding of the base-ten number
system. This student sees the placevalue columns as symmetric about the
ones place and fails to acknowledge the
th in the place-value names, resulting
in the belief that to the right of the
decimal point is simply more whole
numbers written in reverse order.
Therefore, 0.06 is 600, and 0.6 is 60.
The results indicate that students’ success with the decimal tasks varied from
representation to representation, and
their work often provided evidence
to support their classification from
the decimal comparison test. Perhaps
the most notable result was the lack
of success demonstrated by the task
experts. Many of these students could
accurately show 0.6 and 0.06 in one
424
or two representations, but not all
four, and some struggled to model
using any representation. Important
ideas about how students think about
decimal fractions emerged from their
work with all four representations, but
they struggled the most with the use
of a number line and the idea of using
place value (see table 2).
All four of these representations are
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Vol. 19, No. 7, March 2014
Area Model/Hundred Grid
Figure 3 demonstrates longer-islarger thinking and, more specifically,
“reverse thinking” on a hundred grid.
This student sees the tenths column
as the units place and the hundreds
column as the tens column, shading
six single squares for 0.6 and 6 groups
of ten for 0.06.
Number Line
Figure 4a’s response demonstrates
zero-makes-small thinking. This
student correctly positions 0.06 before
0.6 on the number line, but then places
0.15 after 0.6. Using longer-is-larger
thinking, this student may simply believe that a longer string of numbers is
bigger. However, this is also an example
of “column overflow,” a perception that
0.6 = 6 tenths; 0.15 = 15 tenths; and,
therefore, that 0.15 > 0.6. Although
this process works with whole numbers,
such as 150 being 15 tens, it is not true
with decimals.
Figure 4b demonstrates longer-islarger, numerator-focused thinking.
This student correctly perceives 0.6
as 6 parts and 0.06 as 6 parts, with no
consideration of the denominator. As
a result, 0.6 and 0.06 are located in
the same position on the number line.
Additional longer-is-larger thinking is shown in figure 4c, in which
0 is followed by 0.6. This student
perceives this as 6 tenths; counting
by tens, he or she finds 16 tenths,
26 tenths, 36 tenths, . . . , 96 tenths
and finally 106 tenths, but here it is
represented by 0.06.
In the shorter-is-larger thinking
shown in figure 4d, a student confuses
the notation of decimals with that
of negative numbers. This student
perceives 0.06 to be smaller than
0.6 because it is farther from 0 on the
left side of the number line.
Fig. 4 Various thinking strategies, using a number line, are drawn.
(a) Zero-makes-small thinking
(b) Numerator-focused thinking
(c) Longer-is-larger thinking
Money
The response that follows demonstrates longer-is-larger thinking:
0.6 = $60 and .06 = $6.00
This student is a reverse thinker who
interprets 0.6 as 6 tens rather than
6 tenths and therefore writes $60
and 0.06 as 6 hundreds rather than
6 hundredths and writes $600. The
response that follows demonstrates
longer-is-larger thinking:
(d) Shorter-is-larger, negative thinking
0.6 would be 6 cents. 0.06 would be
6 cents also!
This numerator-focused thinker
emphasizes the number of parts only;
Vol. 19, No. 7, March 2014
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therefore, both parts appear as 6 cents.
This student draws on knowledge that
when a zero precedes a whole number,
such as in 06, it does not influence the
value of the number and is ignored.
MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
425
WHAT ARE THE IMPLICATIONS
FOR TEACHERS?
Being able to identify student understanding in a lab setting or using retrospective analysis is different from being
able to recognize student understanding in a classroom setting in which
speed of recognition is critical (Steffe
and Thompson 2000). Relating misconceptions to representations helps
reduce lag time and informs teaching
decisions. Questioning strategies and
teacher reflection are the keys to using
the four representations tasks as well
as other decimal tasks to transform
student thinking about decimals.
making they possess. This prompting
may actually cause some students to
rethink their notion of decimals as they
explain the reasoning behind their answers. Other students will persist with
their line of thinking and demonstrate
a misconception that is more resistant.
Armed with the knowledge of
the misconceptions that students
bring with them and the reasons for
them, the teacher can consider asking
questions that can guide students to
change their thinking. In doing so, it is
important to note that most students
in my study generally had at least one
representation that they could use to
Armed with the knowledge of students’
misconceptions, the teacher can
consider asking questions that can
guide students to change their thinking.
A good problem is often a broad
task with a clear purpose that allows
a teacher to formatively assess student understanding and differentiate
instruction based on what the task reveals. Understanding the misconceptions that students bring with them
and what they look like enables the
teacher to plan for them. A teacher
can contemplate the following two
levels of questions as decimal tasks are
being planned and implemented.
1. What questions can I ask to find
out what students are thinking and
to elicit their reasoning?
2. What leading questions can I ask
to guide students to change their
thinking?
As a decimal task is implemented in
the classroom, it is imperative that the
teacher press students to explain their
reasoning. In doing so, he or she is asking students to make explicit the sense
426
demonstrate the meaning of decimals.
However, very few could demonstrate
an understanding using all four representations, including those students
who would be classified “experts” at
using decimal comparison items. This
knowledge points to issues that we can
use in instruction:
1. Students with particular misconceptions prefer certain representations. When asked to compare
pairs of decimals, they will reveal a
misconception (longer-is-larger or
shorter-is-larger behaviors). They
will exhibit this behavior using one
or more representations but often
accurately depict decimal numbers
using a different representation.
2. Although experts accurately identified the larger decimal from a pair of
decimal numbers, they used the representations in a way that indicated
that they held a misconception.
3. By knowing what representa-
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Vol. 19, No. 7, March 2014
tions a student with a particular
misconception is more comfortable with and why, we can (from
a teaching perspective) leverage
uncomfortable representations
as didactic objects to change the
student’s misconception (Thompson 2002). In other words, we ask
students to work with an uncomfortable representation and then
discuss their actions with that
representation. We can highlight
the actions and results that are
incompatible with their previous
understanding and leverage that
reflection on new representation
and new actions to push for a different target understanding.
The remedy for all misconceptions
is not the same. However, for the
purpose of this article, general recommendations can be made once teachers view the work that students do
through the lens of the misconceptions
discussed in this article. These recommendations are based on the assertion
that the symbolic notation of decimals
is to be viewed as a notation for a
particular type of fraction (specifically
with base-ten denominators).
Many students in my study lacked
either the understanding that decimals
are special fractions with base-ten
denominators, a fundamental understanding of the base-ten number
system, or both. It is unlikely that
students will gain proficiency with
decimals if they misunderstand the
fraction-decimal connection and do
not see decimals as being an extension
of the base-ten system. Mistakes reveal
student misconceptions or overgeneralizations and provide an opportunity to
learn, both for the teacher and for the
student (Shaughnessy 2011).
WHAT CAN CLASSROOM
TEACHERS DO?
1. Incorporate multiple representations
in developing decimal concepts. This
SHERRI L. MARTINIE
SHERRI L. MARTINIE
A classroom place-value display incorporates multiple representations of
numbers, including both money and manipulatives.
A number line, displayed in the classroom, is constructed from decimal cards.
SHERRI L. MARTINIE
will deepen and broaden students’
understanding. By identifying representations with which students are
both most and least comfortable, the
four representations can be used for
formative assessment. Deliberately selecting and structuring lessons around
the least-comfortable representations
allows the four representations to be
used for instructional purposes, as
well. Generating class discussions
around common misunderstandings
uncovered in tasks is a powerful tool
(Shaughnessy 2011); building that
task around a particular representation
is a powerful way of targeting which
misunderstandings are uncovered and
discussed (Thompson 2002).
2. Provide explicit instruction and
supporting visuals to help students understand the base-ten number system and
how it is expanded to include decimal
fractions. Students need to work on
place value and special fractions with
dominators of 10, 100, and 1000. They
can use the part-whole representation of fractions, specifically base-ten
blocks or linear arithmetic blocks (see
http://extranet.edfac.unimelb.edu.au/
DSME/decimals/SLIMversion/
teaching/models/lab.shtml), to help
them explore the columns to the right
of the decimal point. As students
“build” numbers, they must symbolically highlight on a place-value chart
the place value of the digits. When
comparing two decimal numbers,
students with misconceptions can be
asked to illustrate them with the base
blocks, compare the numbers, and
explain their reasoning.
3.Use language that highlights the
nature of the values. For example,
explicitly using the term “special fractions” when talking about denominators of 10, 100, or 1000 will draw
attention to the fact these base-ten
numbers can be easily translated from
fraction to decimal. Use terms, such as
“friendly fractions,” to refer to those
that can be converted fairly easily to a
Several cards and numbers are written for the same place; see, for example,
0.3, 0.30, 0.300.
base-ten special fraction. For example,
because 3/4 = 75/100 = 0.75, 3/4 is a
“friendly fraction.”
4. Incorporate money in a way that
highlights the fractional aspects of the
decimal notation. The fundamental
fact about equivalent fractions immediately tells us that a fraction with
denominator 10, such as 6/10, can be
considered a fraction with a denominator 100, in this case 60/100. In the
classroom, this can be supported by
the conversation that 6 dimes equals
60 cents and 6 dimes combined with
12 cents makes 72 cents. Modeling
and talking about fraction and decimal fraction relationships will enable
students to compare decimals using
the meaning of a decimal as a fraction.
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5. Use the area model to help students
make sense of number lines. Given that
students begin to work with fractions
using the part-whole area model, this
makes sense. Transform fraction strips
into a number line, draw the number
line the length of a set of fraction
strips, and then use the strips to mark
and label tick marks on the line.
A favorite task involves creating
a classroom number line with ends
labeled 0 and 1. As students enter the
room, they are given a decimal square
(these squares can be either made or
purchased). The cards include squares
shaded by tenths, hundredths, and
thousandths. The student is to record
the fraction and decimal for the shaded part and then place the card and
MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
427
the symbolic notation on the number
line. Students then discuss the placement of the cards. They also notice
how the shading of the cards and the
symbolic notation progress across the
number line. At some places on the
number line, several cards and numbers will be written; see, for example,
0.3, 0.30, 0.300.
Once students are proficient with
this activity, the endpoints of the
number line can be changed to 0 and
2 and students given two cards, which
they are to add. They can record the
decimal fraction for the shaded part
fractions using multiple representations. Prior to this work, I approached
teaching decimals in a conventional
manner, using rules and procedures
that lacked the potential for sense
making. Instruction did not include a
meaningful interpretation of symbols
and notation for decimals and at times
seemed to avoid what math is really
about (D’Ambrosio and Kastberg
2012). I found that when my students
did not understand a process, their
reaction was to memorize it.
Students need support as they
come to terms with the complexity of
Research indicates that how students
come to understand decimals results in
specific errors in thinking that are innate
and an inevitable part of learning.
on each card as well as the resulting
addition problem and solution. Students use blank decimal square cards
(they may need one or two cards,
depending on their solution) to shade
the solution. This new decimal square
card(s), along with the addition problem and solution, would be attached
to the new number line. Finally, the
endpoints of the number line can be
changed to −1 and 1. Students are
given decimal square cards that are
designated negative or positive; they
then record the value on a piece of paper and attach the decimal square card
and symbolic notation (as positive or
negative) at the appropriate place on
the number line. This helps students
who are victims of negative thinking.
TRANFORMATION IN TEACHING
After diving deep into the concept
of decimals with students, I conclude
that the key to students’ understanding is merging place value with
428
decimals. That support comes from
the preparation and implementation
of tasks by the classroom teacher.
Research indicates that the manner in
which students come to understand
decimals results in specific errors
in thinking that are innate and an
inevitable part of learning. However,
the types of “rules” that students apply
provide important diagnostic information for teachers and can direct
instruction in powerful ways.
CCSSM Practices in Action
SMP 1: Make sense of problems and
persevere in solving them.
SMP 2: Construct viable arguments
and critique the reasoning of others.
SMP 6: Attend to precision.
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MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
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Copyright © 2014 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
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Any thoughts on this article? Send an
e-mail to mtms@nctm.org.—Ed.
Sherri L. Martinie,
martinie@ksu.edu, is an
assistant professor at
Kansas State University.
She is interested in the
development of student understanding of
decimal fractions. Her work in this area
stems from her experience as a classroom
teacher: seven years at the elementary level,
five years at the middle school level, and
seven years at the high school level, and her
current work with preservice teachers.
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