A Worked Example for Creating Worked Examples

Abstract

To reduce algebraic misconceptions in middle school, combine worked examples and self-explanation prompts.

Full text

Vol. 21, No. 1, August 2015 ● MATHEMATICS TEACHING IN THE MIDDLE SCHOOL 27
r
Researchers have extensively docu-
mented, and math teachers know
from experience, that algebra is a
“gatekeeper” to more advanced math-
ematical topics. Students must have a
strong understanding of fundamental
algebraic concepts to be successful in
later mathematics courses (Star and
Rittle-Johnson 2009). Unfortunately,
algebraic misconceptions that stu-
dents may form or that deepen during
middle school tend to follow them
throughout their academic careers
(Cangelosi et al. 2013). In addition,
the longer that a student holds a
mathematical misconception, the more
difficult it is to correct (Kilpatrick,
Swafford, and Findell 2009). There-
fore, it is imperative that we, as teach-
ers, attempt to address these algebraic
misconceptions while our students are
still in middle school. One tool com-
monly used to do such a task is the
Creating Worked Examples
To reduce algebraic misconceptions in middle school, combine
worked examples and self-explanation prompts.
Kelly M. McGinn, Karin E. Lange, and Julie L. Booth
combination of worked examples
and self-explanation prompts (see
fig. 1) (Aleven and Koedinger 2002).
This article will describe not only
the benefits of using this strategy but
also how it connects to the Common
Core State Standards for Mathematics
(CCSSI 2010). It will also provide in-
struction on creating worked-example
and self-explanation problem sets for
your own students.
BENEFITS OF WORKED
EXAMPLES AND SELF-
EXPLANATION PROMPTS
A worked example in mathematics is a
problem that has been fully completed
to demonstrate a procedure (Clark,
Nguyen, and Sweller 2011). Worked
examples, in combination with self-
explanation prompts (questions that
encourage students to explain the
problem back to themselves) have
been found to increase algebra learn-
ing (Booth et al. 2015). Students who
receive worked examples make fewer
errors, complete follow-up problems
faster, and require less teacher as-
sistance (Sweller and Cooper 1985;
Carroll 1994). This practice also im-
proves both conceptual and procedural
knowledge by promoting the integra-
tion of new knowledge with what stu-
dents already know, helping students
make their new knowledge explicit,
and focusing students’ attention on
important mathematical principles
(Rittle-Johnson 2006; McEldoon,
Durkin, and Rittle-Johnson 2013).
Furthermore, research has found
that the use of both correct and incor-
rect worked examples can improve
student learning (Booth et al. 2013).
Often, teachers are hesitant to use
incorrect examples because they feel
that exposing students to incorrect
A Worked Example for
5 – 4x + 2
SIMPLIFY
7 – 4x + 2
SIMPLIFY
 

28 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL ● Vol. 21, No. 1, August 2015 Vol. 21, No. 1, August 2015 ● MATHEMATICS TEACHING IN THE MIDDLE SCHOOL 29
procedures may increase misconcep-
tions. However, this is unfounded;
incorrect examples help students rec-
ognize incorrect procedures and think
about the differences between them
and the correct procedures, which
can increase students’ conceptual and
procedural knowledge (Booth et al.
2013).
To maximize the impact of worked
examples, a similar practice prob-
lem can be included, thus allowing
the students to practice their newly
learned concept or skill (Atkinson et
al. 2000). Students who receive alter-
nating worked examples and practice
problems outperform those who
receive all worked examples followed
by all practice problems (Trafton and
Reiser 1993). When students take
time to study a worked-out example,
answer questions that require them
to explain the problem to themselves,
and then practice a similar problem
immediately afterward, they begin to
break down their previously stubborn
misconceptions in ways that help
strengthen their understanding of
algebraic concepts.
TEACHER PERSPECTIVES
Although empirical research has
shown the benefits of providing
middle school students with worked
examples and self-explanation
prompts, reviewing the perspective of
the teacher can be just as valuable (all
names are pseudonyms).
Jane, an eighth-grade algebra
teacher from a public school in the
Midwest, points out the benefit of us-
ing incorrect examples by stating,
The incorrect examples are actually
sometimes the ones that really are
better for showing students. The
incorrect examples are often the
best learning tool. . . . Forcing them
to say, “Well, that’s what I do,
what should I do then? If I’m
doing the same thing as this boy in
this problem, what’s wrong
with that?”
Alyssa, another eighth-grade algebra
teacher from a second school district,
voices another view, finding that there
are benefits to using both types of
examples. She says,
I find that either correct examples or
incorrect examples help kids identify
themselves with somebody else easily.
[With misconceptions], kids can be
really stubborn, and they really don’t
believe you that it’s wrong. To see
a kid look at an incorrect example
and say, “No, but this is correct,” and
kind of have that moment of “Oh, I
really was off, and now I understand
it more” . . . I think they are allowed
to engage with that problem more so
than if it was just a standard practice.
Teachers who have used worked-
example–self-explanation problems
have reported overwhelmingly posi-
tive results. Most important, they see
their students finally mastering con-
cepts that had previously eluded them
because of persistent misconceptions.
By using standards-aligned instruc-
tional strategies that help middle
school students gain access to alge-
braic concepts, teachers are helping
to ensure that all students can achieve
success in higher-level mathematics
courses. Peter, a mathematics teacher
leader, summed it up by saying,
Too often in math class, it is about
just getting the answer, it’s not
about the process. And when
you’re analyzing someone else’s work
. . . you’re dealing with the process,
not just the answer. I think that’s
extremely valuable for students!
Although it is evident that the use of
worked examples and self-explanation
prompts can improve student learn-
ing and help students confront their
misconceptions, it is also important to
ensure that this strategy aligns with the
principles of the Common Core State
Standards for Mathematics (CCSSM).
ALIGNMENT WITH THE
COMMON CORE
Worked-example problems with self-
explanation prompts can be paired
with any CCSSM content standard;
however, this strategy also helps
teachers integrate the Standards for
Mathematical Practice (SMP) into his
or her classroom.
The first standard, make sense
of problems and persevere in solv-
ing them, is a great example (CCSSI
2010). The first line of the standard’s
description is “mathematically profi-
cient students start by explaining to
themselves the meaning of a problem
and looking for entry points to its
solution” (CCSSI 2010, p. 6). The
worked-example instructional strat-
egy helps scaffold understanding so
that students can work through this
process. The use of worked examples
allows students to study a sample en-
try point, whereas the self-explanation
questions prompt students to explain
to themselves the meaning of the
problem. This process not only neces-
sitates students to think through the
problem in a new way but also helps
situate students to be able to under-
stand the different approaches that a
classmate might use.
Worked examples and self-expla-
nation prompts also align with
SMP 2: Reason abstractly and quan-
titatively (CCSSI 2010). Specifically,
the standard emphasizes the impor-
tance of “making sense of quantities
and their relationship in problem situ-
ations” and “attending to the meaning
of quantities, not just how to compute
them” (CCSSI 2010, p. 6). Carefully
worded self-explanation prompts help
students accomplish this goal. For
instance, a student can be explicitly
asked to explain what the y-intercept
represents in a given word problem, a
task that is often left to discussion or
implication with traditional solution-
based assignments.
Finally, SMP 3, Construct viable
arguments and critique the reason-
ing of others (CCSSI 2010), is also
addressed through the use of the
worked-example–self-explanation
strategy. As explained in more detail
below, the worked example demon-
strates the effort of a fictitious student
(see fig. 1). Therefore, the actual stu-
dent may practice “critiquing” another
student’s reasoning in a safer environ-
ment when answering the explanation
prompts. For example, a potential
prompt may read, “Does Natalie’s
price for a pen seem reasonable? Why
or why not?” In addition, the use of
worked examples that are incorrect
gives students the opportunity to
“distinguish correct logic or reasoning
from that which is flawed, and—if
there is a flaw in an argument—ex-
plain what it is” (CCSSI 2010, p. 7).
To ensure that students receive
the full benefits of this strategy as
demonstrated in prior research, it is
important that care is taken to write
problems that specifically address
your own students’ needs. A step-by-
step guide will help to maximize the
benefit of each problem on student
learning.
STEPS FOR CREATING WORKED
EXAMPLES AND SELF-
EXPLANATION PROMPTS
Five steps can be used to create a
worked-example–self-explanation
item.
Step 1
Identify the objective and list a few
common misconceptions associated
with this objective. Similar to plan-
ning a lesson, start by writing the
lesson goal or focus objective. For
example, the objective for our sample
item will be the following: Students
will be able to simplify an expression
by combining like terms.
This is where you must brainstorm.
Think about the mistakes you have
seen students make while solving
problems associated with the objec-
tive in the past. For our sample, the
misconceptions and errors associated
with combining like terms include a
tendency to fail to include the nega-
tive sign as a part of the term and to
combine non-like terms.
Step 2
Choose one misconception or error
for each example. The goal is to focus
students’ attention on one aspect of
the problem at a time; do not over-
whelm them with too many errors or
ideas. Either make an entire activity
sheet focused on one misconception
or error or create a sheet that focuses
on just a few items. For our sample,
we created a worked example focused
on the idea that you must include
the negative sign with the term when
rearranging terms within the expres-
sion.
Step 3
Create the worked example using the
misconception. Write a worked-out
solution to a problem that meets your
objective. Although the worked ex-
ample can be done either correctly or
incorrectly, clearly mark the problem
as correct or incorrect. It is also help-
ful to act as if a fictitious student com-
pleted this example because the actual
student completing the work will
connect with that other student and
realize that a similar misconception
or a similar error is indeed common.
Choose students’ names that reflect
the diversity of the classroom and vary
who completes the incorrect and
Fig. 1 This task combines a worked example with a self-explanation prompt.
Incorrect examples help students recognize
incorrect procedures and think about the
difference between them.

30 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL ● Vol. 21, No. 1, August 2015 Vol. 21, No. 1, August 2015 ● MATHEMATICS TEACHING IN THE MIDDLE SCHOOL 31
attention to the target misconcep-
tion or error you chose in step 2.
See figure 2b. Although the student
was asked a “what” question, it was
followed up with a question that
prompted the student to explain his
or her reasoning. This is important.
Although it is fine to ask procedural
questions, follow it up with a con-
ceptual question. In fact, research has
found greater learning gains when
students are asked to explain the
concept, rather than the procedure
(Matthews and Rittle-Johnson 2009).
Step 5
Create a practice problem similar
to the worked example. Give this
problem to students to complete on
their own after they have studied
the worked example and answered
the self-explanation prompts, thus
allowing them time to practice. It will
also reinforce the new information
related to their misconception. Note
in figure 2c that the structure of the
“Your Turn” problem is identical to
the worked example. The only aspects
that changed were the numbers and
order of the terms.
ONE MORE EXAMPLE
OF A WORKED EXAMPLE
It is always helpful to see more than
one example when learning a new skill.
Figure 3 illustrates a second worked-
example–self-explanation combination.
Note a few important features:
1. Marked example: Be sure to mark
the example as correct or incorrect.
This example is clearly marked as
correct.
2. Use of fictitious student: Inez
completed this problem.
3. One target misconception or error:
Although many different miscon-
ceptions and errors can be associ-
ated with a particular problem,
be sure to only focus on one at a
time. In this instance, the item
was designed to promote alternate
problem-solving procedures and
counter the misconception that
there is only one way to solve a
problem. With the same example,
one could also target the common
error of not selecting the
Sample Prompts
Although it is acceptable to ask procedural questions, be sure to ask
students to explain and/or justify their reasoning.
1. Why is __________ not included in the answer?
2. What did [student name] __________ as his first step?
]
3. What should [student name] have done to __________?
4. Would it have been OK to write __________? Why or why not?
5. Why did [student name] combine __________ and __________?
6. Why did [student name] first _______ then _______?
7. Is _______ the same expression as _______? Explain.
8. Would [student name] have gotten the same answer if he [or she]
_______ first?
9. Why did [student name] change __________ to __________?
10. Explain why __________ would have been an unreasonable answer.
11. How could [student name] have figured out that his [or her] answer did
not make sense?
12. How did [student name] know that __________ was not equal to
__________?
13. What did the __________ represent in this word problem?
14. How did the __________ in the equation affect the graph?
15. Why did [student name] __________ from both sides of the equation?
Fig. 2 A practice problem can be given to students to complete on their own.
(a) (b) (c)
correct examples. Figure 2a shows our
sample worked example. Note that the
item is clearly marked incorrect, that a
fictitious student completed the item,
and that the fictitious student only
made one error.
Step 4
Write the self-explanation prompt,
focusing on the target misconception
or error. This is the trickiest part of
the process. You will want to write
one or two questions that specifically
ask the student to examine his or her
own misconception through the work
done by the fictitious student. At first,
teachers often have trouble creating
self-explanation prompts; however, it
gets easier with practice. Avoid only
asking such “what” questions as these:
1. What is wrong with the example?
2. What mistake was made?
3. What is the correct answer?
Instead, focus on writing “why”
questions. You want to have students
explain their reasoning, not just state
the procedure. It is important to call
students’ attention to the features of
the problem that you think are impor-
tant; in other words, draw their
Fig. 3 This version shows a second worked-example–self-explanation combination.

32 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL ● Vol. 21, No. 1, August 2015 Vol. 21, No. 1, August 2015 ● MATHEMATICS TEACHING IN THE MIDDLE SCHOOL 33
Kelly M. McGinn, kelly.
mcginn@temple.edu,
taught seventh-grade and
eighth-grade mathematics
before pursing her PhD in
educational psychology
at Temple University in
Philadelphia. Her work
focuses on understand-
ing how students develop
mathematics concep-
tual understanding and
designing interventions to
improve that learning.
Karin E. Lange, klange@sdb.k12.wi.us,
taught middle school math in Camden,
New Jersey, before pursuing her doc-
torate in math education from Temple
University. She is currently the Director
of Math and Science Curriculum and In-
struction for the School District of Beloit,
Wisconsin. Julie L. Booth, julie.booth@
temple.edu, is an associate professor
of Educational Psychology and Ap-
plied Developmental Science at Temple
University. Her work focuses on under-
standing how students’ prior knowledge
affects their learning in mathematics and
designing interventions to improve that
learning.
appropriate number to multiply
both sides by or not multiplying all
terms in the equation.
4. Explain reasoning: Although it
is sometimes beneficial to ask for
procedural explanations, be sure
to also prompt students to explain
their reasoning. For example, when
students were asked, “Why do you
think Inez multiplied all terms in
the equation by 3 instead of sub-
tracting 6 from both sides?” they
answered in these ways:
• To remove the denominators,
because then it would be more com-
plicated than it needs to be
• So we could get rid of the fractions
• To get rid of the denominator
• To remove the bottom #!
This prompt was meant to highlight
the possibility of additional problem-
solving strategies.
NCTM
Elections
5. Follow-up practice problem:
Finally, be sure to allow students
to complete a problem similar to
the worked example to practice the
skill.
EXTENDED UNDERSTANDING
OF SELF-EXPLANATION
PROMPTS
As mentioned above, writing
self-explanation prompts is the most
difficult part of the process, yet it is
also the most crucial component in
creating a problem that successfully
addresses students’ misconceptions.
See the sidebar (on p. 30) for a few
sample prompts to help you get
started. Recall that although it is ac-
ceptable to ask procedural questions,
do not solely rely on those types of
questions. Be sure to ask students to
explain and/or justify their reasoning.
The simplest way to do this is to ask
“why” questions.
FINAL THOUGHTS
Worked examples paired with self-
explanation prompts show promise
as being a new strategy to accelerate
student understanding and success in
algebra, especially for students who have
held persistent misconceptions over
time. Teachers will find the most success
when they target their students’ individ-
ual needs and misconceptions. By giving
teachers the tools to respond to students
and create their own examples and
prompts, it is hoped that all students
can achieve success in understanding
foundational components of algebra.
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