ArticlePDF Available


To reduce algebraic misconceptions in middle school, combine worked examples and self-explanation prompts.
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
g. 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.
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
7 – 4x + 2
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.
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.
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, thats what I do,
what should I do then? If I’m
doing the same thing as this boy in
this problem, whats 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 dont
believe you that its 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, its 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 thats
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).
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
strategy. As explained in more detail
below, the worked example demon-
strates the effort of a fictitious student
(see g. 1). Therefore, the actual stu-
dent may practice “critiquing” another
students 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
Five steps can be used to create a
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-
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.
attention to the target misconcep-
tion or error you chose in step 2.
See gure 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.
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
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.
Kelly M. McGinn, kelly.,
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,,
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@, 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
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.
5. Follow-up practice problem:
Finally, be sure to allow students
to complete a problem similar to
the worked example to practice the
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.
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.
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):
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.
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
See the latest blogs,
and join the discussion!
Standards of
Mathematical Practice
in the Middle Grades
Standards of Mathematical
Practice in the Middle Grades
Look Whos
Join your fellow readers
on MTMS’s blog:
Join your fellow readers
on MTMS’s blog:
... To reduce whole number bias and improve people's understanding of case-fatality rates, Thompson et al. (2021) created a brief online educational intervention that capitalized on a step-by-step worked example (Atkinson et al., 2000;McGinn et al., 2015;Schwonke et al., 2009) and an analogy (Gentner & Hoyos, 2017;Thompson & Opfer, 2010) to a more familiar situation (e.g., apples rotting on trees in an orchard). Given up-to-date information about COVID-19 cases and deaths, those who were randomly assigned to the educational intervention were more likely to correctly recognize that COVID-19 was more fatal than the flu compared to those who did not receive the educational intervention. ...
... The effect of the intervention extended to another health-related math problem-adults in the intervention condition more accurately interpreted changes in COVID-19 case-fatality rates than those in the control group. Thus, techniques primarily used to improve math performance (e.g., McGinn et al., 2015;Thompson & Opfer, 2010) also improved adults' ratio understanding in a health context. However, Thompson and colleagues did not assess whether the benefits of the intervention were durable over time or on a larger number of problems. ...
... They help students to focus on the concept or meaning of a procedure by presenting problems solved correctly, in part, or incorrectly. Worked examples are proven to be beneficial to students' conceptual development when interweaved with problem solving, especially for students who experience difficulties learning mathematics (McGinn et al., 2015). In the ModelME curriculum, the worked examples relate to a concept students are working toward within gameplay or a related skill that connects to a concept built up in gameplay. ...
... The repetitiveness encourages students to notice a particular mathematical concept or make use of a particular strategy. When paired with conversation about how students solved a problem, specifically discussions that highlight efficient strategies and how the strategies work, number strings can support abstraction of mathematical concepts (McGinn et al., 2015). In the ModelME curriculum, the number strings relate to a skill that connects to a concept built up in gameplay. ...
Teaching and learning fraction concepts continues to be increasingly challenging, especially for elementary and middle school mathematics teachers and students in intervention settings. It is critical for educators to implement instruction that proactively considers engagement, access, and conceptual growth for all students. Dream 2B, a web-based universally designed fraction game, has the potential to significantly impact engagement and conceptual understanding of fractions. In addition, it introduces students to Science, Technology, Engineering and Mathematics (STEM) and Information Communication Technology (ICT) careers. This manuscript provides guidance for utilizing Dream 2B as supplemental mathematics instruction. Components of Universal Design for Learning, as well as gameplay Concept/Skill connections, are provided. Guidance for expanding Dream2B into virtual learning environments is also discussed.
... Worked examples relate to a concept that students are working toward within gameplay or a related skill that connects to a concept built up in gameplay. They help students to focus on the concept or meaning of a procedure by presenting problems solved correctly, in part, or incorrectly [56]. Game replays directly connect to a problem students encountered during gameplay in a particular world. ...
Full-text available
People with disabilities are underrepresented in STEM as well as information, communication, and technology (ICT) careers. The underrepresentation of individuals with disabilities in STEM may reflect systemic issues of access. Curricular materials that allow students to demonstrate their current fraction knowledge through multiple means and provide opportunities to share and explain their thinking with others may address issues of access students face in elementary school. In this study, we employed a sequential mixed-methods design to investigate how game-enhanced fraction intervention impacts students’ fraction knowledge, engagement, and STEM interests. Quantitative results revealed statistically significant effects of the program on students’ fraction understanding and engagement but not their STEM interest. Qualitative analyses revealed three themes—(1) Accessible, Enjoyable Learning, (2) Can’t Relate, and (3) Dreaming Bigger—that provided contextual backing for the quantitative results. Implications for future research and development are shared.
... Worked examples are frequently presented as having been completed by a fictional student and paired with a similar practice problem (e.g., McGinn et al., 2015). Most commonly, worked examples show a correctly worked-out solution. ...
Full-text available
The current meta-analysis quantifies the average effect of worked examples on mathematics performance from elementary grades to postsecondary settings and to assess what moderates this effect. Though thousands of worked examples studies have been conducted to date, a corresponding meta-analysis has yet to be published. Exclusionary coding was conducted on 8033 abstracts from published and grey literature to yield a sample of high quality experimental and quasi-experimental work. This search yielded 43 articles reporting on 55 studies and 181 effect sizes. Using robust variance estimation (RVE) to account for clustered effect sizes, the average effect size of worked examples on mathematics performance outcomes was medium with g = 0.48 and p = 0.01. Moderators assessed included example type (correct vs. incorrect examples alone or in combination with correct examples), pairing with self-explanation prompts, and timing of administration (i.e., practice vs. skill acquisition). The inclusion of self-explanation prompts significantly moderated the effect of examples yielding a negative effect in comparison to worked examples conditions that did not include self-explanation prompts. Worked examples studies that used correct examples alone yielded larger effect sizes than those that used incorrect examples alone or correct examples in combination with incorrect examples. The worked examples effect yields a medium effect on mathematics outcomes whether used for practice or initial skill acquisition. Correct examples are particularly beneficial for learning overall, and pairing examples with self-explanation prompts may not be a fruitful design modification. Theoretical and practical implications are discussed.
... One future research idea is to use worked examples (McGinn et al., 2015) that involve step-bystep instructions on how to calculate final grades and visualizations, like number lines, to help students track their grades relative to total course points (cf. Siegler et al., 2011). ...
... Secara sekilas, metode ini mirip dengan penberian contoh yang telah dikerjakan (worked example) yang telah umum diberikan dalam setiap buku pegangan matematika. Akan tetapi, metode contoh yang keliru tidak hanya menekankan agar siswa memahami konsep seperti halnya worked example (Kay & Edwards, 2012;McGinn et al., 2015;van Loon-Hillen et al., 2012), tetapi penggunaan contoh yang keliru mnekankan agar siswa lebih memahami logika dan alasan dibalik setiap pengerjaan soal terlebih dalam mata pelajaran matematika (Jaeger et al., 2020;McLaren et al., 2016;Melis, 2004;Tsovaltzi et al., 2009). ...
Full-text available
Students errors are actually provide resources for teaching. Using students error, lecturer can emphasized certain concept while teaching, especially those concepts which make students confused. One of the teaching method that use students error is erroneous example. In this article, we developed a certain task using erroenous example method. The task consists of 10 question. Each question is made to tackle students understanding about a concept that is often mistaken by students. For each question, we give a correct/incorrect worked example, then students are asked to find the mistakes (if it is incorrect) or justify the steps (if it is correct). This study is a descriptive qualitative study that aims to describe the task and students response after doing the task. The method of collecting data was using observation, documentation, and interview. The results shows us that erroneous example task can be categorized difficult for students. However, it is also proved that it can be used to supporting students mathematics communication.
There are numerous reasons why students with disabilities struggle in school. A key reason is professionals in the field may not pay enough attention to students’ overwhelmed cognitive capacity. Cognitive load theory explains that all humans have limited capacity at any given time to use their auditory, visual, and tactile inputs (independently or collectively) to acquire new information and store it in long-term memory. When available cognition is overwhelmed – which can be caused by any number of reasons – learning cannot occur. In this article, we introduce the key aspects of cognitive load theory and give specific examples of how special educators can use this information to shape their instruction to support students’ unique needs.
Mathematics difficulties (MD) are widespread. Higher mathematics achievement is associated with college success and greater job opportunities. In this chapter, we address the intersection between cognitive research on the science of learning and the education of students with MD. First, a conceptual framework for mathematics learning is provided, including consideration of individual differences in domain general and domain specific learning processes. Next, learning principles that promote successful learning in mathematics are described, along with examples of how the principles can be translated into educational practice for students with MD. We argue that in addition to providing developmentally appropriate instruction in content knowledge, such as whole number and fraction knowledge, application of more general learning principles validated by research will lead to deeper and more durable learning for struggling students. These learning principles include studying and comparing correct as well as incorrect worked out problem-solutions; using integrated visual and verbal models to reduce splitting attention; interleaving or varying practice with problems of different types; providing frequent cumulative practice or review that is spaced out over time; connecting and integrating concrete and symbolic representations; presenting arithmetic problems in different formats; using physical movements and gestures to promote learning; and incorporating activities with number lines to increase learning of whole numbers and fractions.
Full-text available
Superintendents from districts in the Minority Student Achievement Network (MSAN) challenged the Strategic Education Research Partnership (SERP) to identify an approach to narrowing the minority student achievement gap in Algebra 1 without isolating minority students for intervention. SERP partnered with 8 MSAN districts and researchers from 3 universities to design and rigorously test AlgebraByExample, a set of 42 Algebra 1 assignments with interleaved worked examples that target common misconceptions and errors. In a year-long random-assignment study, students who received AlgebraByExample assignments had an average 7 percentage point boost on a posttest containing released items from state assessments, and students in the bottom half of the performance distribution where minority students are disproportionately concentrated had an average 10 percentage point boost on a researcher-designed assessment of conceptual understanding. AlgebraByExample is easily incorporated into any existing curriculum, and naturally serves as a launch point for mathematically rich discussion.
In the 2 experiments reported here, high school students studied worked examples while learning how to translate English expressions into algebraic equations. In Experiment 1, worked examples were used as part of the regular classroom instruction and as a support for homework. In Experiment 2, students in a remedial mathematics class received individual instruction. Students using worked examples outperformed the control group on posttests after completing fewer practice problems; they also made fewer errors per problem and fewer types of errors during acquisition time, completed the work more rapidly, and required less assistance from the teacher.
In a series of two in vivo experiments, we examine whether correct and incorrect examples with prompts for self-explanation can be effective for improving students’ conceptual understanding and procedural skill in Algebra when combined with guided practice. In Experiment 1, students working with the Algebra I Cognitive Tutor were randomly assigned to complete their unit on solving two-step linear equations with the traditional Tutor program (control) or one of three versions which incorporated examples; results indicate that explaining worked examples during guided practice leads to improved conceptual understanding compared with guided practice alone. In Experiment 2, a more comprehensive battery of conceptual and procedural tests was used to determine which type of examples is most beneficial for improving different facets of student learning. Results suggest that incorrect examples, either alone or in combination with correct examples, may be especially beneficial for fostering conceptual understanding.
Self-explanation, or generating explanations to oneself in an attempt to make sense of new information, can promote learning. However, self-explaining takes time, and the learning benefits of this activity need to be rigorously evaluated against alternative uses of this time. In the current study, we compared the effectiveness of self-explanation prompts to the effectiveness of solving additional practice problems (to equate for time on task) and to solving the same number of problems (to equate for problem-solving experience). Participants were 69 children in grades 2-4. Students completed a pre-test, brief intervention session, and a post- and retention test. The intervention focused on solving mathematical equivalence problems such as 3 + 4 + 8 = _ + 8. Students were randomly assigned to one of three intervention conditions: self-explain, additional-practice, or control. Compared to the control condition, self-explanation prompts promoted conceptual and procedural knowledge. Compared to the additional-practice condition, the benefits of self-explanation were more modest and only apparent on some subscales. The findings suggest that self-explanation prompts have some small unique learning benefits, but that greater attention needs to be paid to how much self-explanation offers advantages over alternative uses of time.
The knowledge required to solve algebra manipulation problems and procedures designed to hasten knowledge acquisition were studied in a series of five experiments. It was hypothesized that, as occurs in other domains, algebra problem-solving skill requires a large number of schemas and that schema acquisition is retarded by conventional problem-solving search techniques. Experiment 1, using Year 9, Year 11, and university mathematics students, found that the more experienced students had a better cognitive representation of algebraic equations than less experienced students as measured by their ability to (a) recall equations, and (b) distinguish between perceptually similar equations on the basis of solution mode. Experiments 2 through 5 studied the use of worked examples as a means of facilitating the acquisition of knowledge needed for effective problem solving. It was found that not only did worked examples, as expected, require considerably less time to process than conventional problems, but that subsequent problems similar to the initial ones also were solved more rapidly. Furthermore, decreased solution time was accompanied by a decrease in the number of mathematical errors. Both of these findings were specific to problems identical in structure to the initial ones. It was concluded that for novice problem solvers, general algebra rules are reflected in only a limited number of schemas. Abstraction of general rules from schemas may occur only with considerable practice and exposure to a wider range of schemas.
In 2 experiments, high school students studied worked examples while learning how to translate English expressions into algebraic equations. In Exp 1, worked examples were used as part of the regular classroom instruction and as a support for homework. In Exp 2, students in a remedial mathematics class received individual instruction. Students using worked examples outperformed the control group on posttests after completing fewer practice problems; they also made fewer errors per problem and fewer types of errors during acquisition time, completed the work more rapidly, and required less assistance from the teacher. (PsycINFO Database Record (c) 2012 APA, all rights reserved)