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Vol. 21, No. 1, August 2015 ● MATHEMATICS TEACHING IN THE MIDDLE SCHOOL 27

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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.

REFERENCES

Aleven, Vincent A. W. M. M., and

Kenneth R. Koedinger. 2002. “An

Effective Metacognitive Strategy:

Learning by Doing and Explaining with

a Computer-Based Cognitive Tutor.”

Cognitive Science (26): 147–79.

Atkinson, R. K., S. J. Derry, A. Renkl, and

D. Wortham. 2000. “Learning from

Examples: Instructional Principles

from the Worked ExamplesResearch.”

Review of Educational Research 70 (2):

181–214.

Booth, Julie L., Laura A. Cooper, M.

Suzanne Donovan, Alexandra Huyghe,

Kenneth R. Koedinger, and E. Juliana

Paré-Blagoev. 2015. “Design-Based

Research within the Constraints of

Practice: AlgebraByExample.” Journal

of Education for Students Placed at Risk

(20): 79–100.

Booth, Julie L., Karin E. Lange, Kenneth

R. Koedinger, and Kristie J. Newton.

2013. “Example Problems That Improve

Student Learning in Algebra: Differen-

tiating between Correct and Incorrect

Examples.” Learning and Instruction

(25): 24 –34.

Cangelosi, Richard, Silvia Madrid, Sandra

Cooper, Jo Olson, and Beverly Hartter.

2013. “The Negative Sign and Expo-

nential Expressions: Unveiling Students’

Persistent Errors and Misconceptions.”

The Journal of Mathematical Behavior 32

(1) (March): 69–82.

Carroll, William M. 1994. “Using Worked

Examples as an Instructional Support

in the Algebra Classroom.” Journal of

Educational Psychology 86 (3): 360–67.

Clark, Ruth C., Frank Nguyen, and John

Sweller. 2011. Efﬁciency in Learning:

Evidence-based Guidelines to Manage

Cognitive Load. New York: John Wiley

& Sons.

Common Core State Standards Initiative

(CCSSI). 2010. Common Core State

Standards for Mathematics. Washing-

ton, DC: National Governors Associa-

tion Center for Best Practices and the

Council of Chief State School Ofﬁcers.

http://www.corestandards.org/wp-

content/uploads/Math_Standards.pdf

Kilpatrick, Jeremy, Jane Swafford, and

Bradford Findell, eds. 2009. Adding

It Up: Helping Children Learn Math-

ematics. Washington, DC: National

Academies Press.

Matthews, Percival, and Bethany Rittle-

Johnson. 2009. “In Pursuit of Knowl-

edge: Comparing Self-Explanations,

Concepts, and Procedures as Pedagogi-

cal Tools.” Journal of Experimental Child

Psychology 104 (1) (September): 1–21.

McEldoon, Katherine L, Kelley L. Durkin,

and Bethany Rittle-Johnson. 2013. “Is

Self-Explanation Worth the Time? A

Comparison to Additional Practice.”

British Journal of Educational Psychology

83 (4): 615–32.

Rittle-Johnson, Bethany. 2006. “Promoting

Transfer : Effects of Self-Explanation

and Direct Instruction.” Child Develop-

ment 77 (1): 1–15.

Star, Jon R., and Bethany Rittle-Johnson.

2009. “Making Algebra Work: Instruc-

tional Strategies That Deepen Student

Understanding, within and between

Algebraic Representations.” ERS

Spectrum 27 (2): 11–18.

Sweller, John, and Graham A Cooper.

1985. “The Use of Worked Examples

as a Substitute for Problem Solving

in Learning Algebra.” Cognition and

Instruction 2 (1): 59–89.

Trafton, John Gregory, and Brian J. Reiser.

1993. “The Contributions of Study-

ing Examples and Solving Problems to

Skill Acquisition.” PhD diss., Princeton

University.

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