You will write a current APA-formatted report responding to each of the case study prompts listed below. Your paper must be at least 1,200 words, and it must include proper headings and subheadings that are aligned with the grading rubric domains.
Access the Iris Center Case Study Unit: Identifying and Addressing Student Errors from the Module 5 Learn material. Read through the Case Study Unit, including all scenarios and the STAR (Strategies and Resources) Sheet.
Case Study Level A, Case 1 – Dalton (p. 3)
Student: Dalton
- Read Dalton’s scenario.
- Read the possible strategies and resources (STAR Sheets pp. 13-29) listed for Identifying and Addressing Student Errors.
What type(s) of errors is evident
How might you determine the reason students make this kind of error and what are some other examples of these types of errors?
What strategies might you employ while addressing these error patterns? - Write a detailed summary of each strategy, including its purpose.
- Describe why each strategy might be used to help Dalton improve.
Case Study Level A, Case 2 – Madison (p. 5)
Student: Madison
- Read Madison’s scenario.
- Describe why each strategy might be used to help Madison improve.
Read the possible strategies and resources (STAR Sheets pp. 13-29) listed for Identifying and Addressing Student Errors.
What type(s) of errors is evident
How might you determine the reason students make this kind of error and what are some other examples of these types of errors?
What strategies might you employ while addressing these error patterns?
Write a detailed summary of each strategy, including its purpose.
Case Study Level B, Case 2 – Elias (p. 9)
Student: Elias
- Read Elias’ scenario.
- Describe why each strategy might be used to help Elias improve.
Read the possible strategies and resources (STAR Sheets pp. 13-29) listed for Identifying and Addressing Student Errors.
What type(s) of errors is evident
How might you determine the reason students make this kind of error and what are some other examples of these types of errors?
What strategies might you employ while addressing these error patterns?
Write a detailed summary of each strategy, including its purpose.
Case Study Level C, Case 1 – Wyatt (p. 11)
Student: Wyatt
- Read Wyatt’s scenario.
- Describe why each strategy might be used to help Wyatt improve.
Read the possible strategies and resources (STAR Sheets pp. 13-29) listed for Identifying and Addressing Student Errors.
What type(s) of errors is evident
How might you determine the reason students make this kind of error and what are some other examples of these types of errors?
What strategies might you employ while addressing these error patterns?
Write a detailed summary of each strategy, including its purpose.
In addition, your assignment must include the following:
- The case study must include a title and reference page formatted to current APA standards. There is no minimum number of references required.
- Each case study must be properly identified with corresponding headings.
- The case study must include professional, positive language.
052621
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Serving: Higher Education Faculty • PD Providers • Practicing Educators
Supporting the preparation of effective educators to improve outcomes for all students, especially struggling learners and those with disabilities
CASE STUDY
UNIT
Mathematics:
Identifying and Addressing
Student Errors
Created by Janice Brown, PhD, Vanderbilt UniversityKim Skow, MEd, Vanderbilt University
iiris.peabody.vanderbilt.edu
The contents of this resource were developed under a grant from
the U.S. Department of Education, #H325E120002. However,
those contents do not necessarily represent the policy of the U.S.
Department of Education, and you should not assume endorse-
ment by the Federal Government. Project Officer, Sarah Allen
Mathematics:
Identifying and Addressing Student Errors
Contents: Page
Credits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ii
Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
STAR Sheets
Collecting Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Identifying Error Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Word Problems: Additional Error Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Determining Reasons for Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Addressing Error Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Case Studies
Level A, Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
Level A, Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Level B, Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Level B, Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Level C, Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Answer Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
TABLE OF CONTENTS
* For an Answer Key to this case study, please email your full name, title, and institutional
affiliation to the IRIS Center at iris@vanderbilt .edu .
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To Cite This
Case Study Unit
Brown J ., Skow K ., & the IRIS Center . (2016) . Mathematics:
Identifying and addressing student errors. Retrieved from http://
iris .peabody .vanderbilt .edu/case_studies/ics_matherr
Content
Contributors
Janice Brown
Kim Skow
Case Study
Developers
Janice Brown
Kim Skow
Editor Jason Miller
Reviewers
Diane Pedrotty Bryant
David Chard
Kimberly Paulsen
Sarah Powell
Paul Riccomini
Graphics Brenda KnightPage 27- Geoboard Credit: Kyle Trevethan
Mathematics:
Identifying and Addressing Student Errors
CREDITS
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Mathematics:
Identifying and Addressing Student Errors
STANDARDS
Licensure and Content Standards
This IRIS Case Study aligns with the following licensure and program standards and topic areas .
Council for the Accreditation of Educator Preparation (CAEP)
CAEP standards for the accreditation of educators are designed to improve the quality and
effectiveness not only of new instructional practitioners but also the evidence-base used to assess those
qualities in the classroom .
• Standard 1: Content and Pedagogical Knowledge
Council for Exceptional Children (CEC)
CEC standards encompass a wide range of ethics, standards, and practices created to help guide
those who have taken on the crucial role of educating students with disabilities .
• Standard 1: Learner Development and Individual Learning Differences
Interstate Teacher Assessment and Support Consortium (InTASC)
InTASC Model Core Teaching Standards are designed to help teachers of all grade levels and content
areas to prepare their students either for college or for employment following graduation .
• Standard 6: Assessment
• Standard 7: Planning for Instruction
National Council for Accreditation of Teacher Education (NCATE)
NCATE standards are intended to serve as professional guidelines for educators . They also overview
the “organizational structures, policies, and procedures” necessary to support them
• Standard 1: Candidate Knowledge, Skills, and Professional Dispositions
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Error analysis is a type of diagnostic assessment that can help a teacher determine what types
of errors a student is making and why . More specifically, it is the process of identifying and
reviewing a student’s errors to determine whether an error pattern exists—that is, whether a
student is making the same type of error consistently . If a pattern does exist, the teacher can
identify a student’s misconceptions or skill deficits and subsequently design and implement
instruction to address that student’s specific needs .
Research on error analysis is not new: Researchers around the world have been conducting
studies on this topic for decades . Error analysis has been shown to be an effective method for
identifying patterns of mathematical errors for any student, with or without disabilities, who is
struggling in mathematics .
Steps for Conducting an Error Analysis
An error analysis consists of the following steps:
Step 1. Collect data: Ask the student to complete at least 3 to 5 problems of the same type (e .g .,
multi-digit multiplication) .
Step 2. Identify error patterns: Review the student’s solutions, looking for consistent error patterns
(e .g ., errors involving regrouping) .
Step 3. Determine reasons for errors: Find out why the student is making these errors .
Step 4. Use the data to address error patterns: Decide what type of instructional strategy will best
address a student’s skill deficits or misunderstandings .
Benefits of Error AnalysisBenefits of Error Analysis
An error analysis can help a teacher to:
• Identify which steps the student is able to perform correctly (as opposed to simply
marking answers either correct or incorrect, something that might mask what it is that
the student is doing right)
• Determine what type(s) of errors a student is making
• Determine whether an error is a one-time miscalculation or a persistent issue that
indicates an important misunderstanding of a mathematic concept or procedure
• Select an effective instructional approach to address the student’s misconceptions and
to teach the correct concept, strategy, or procedure
Mathematics:
Identifying and Addressing Student Errors
INTRODUCTION
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References
Ashlock, R . B . (2010) . Error patterns in computation (10th ed .) . Boston: Allyn & Bacon .
Ben-Zeev, T . (1998) . Rational errors and the mathematical mind . Review of General Psychology,
2(4), 366–383 .
Cox, L . S . (1975) . Systematic errors in the four vertical algorithms in normal and handicapped
populations . Journal for Research in Mathematics Education, 6(4), 202–220 .
Idris, S . (2011) . Error patterns in addition and subtraction for fractions among form two students .
Journal of Mathematics Education, 4(2), 35–54 .
Kingsdorf, S ., & Krawec, J . (2014) . Error analysis of mathematical word problem solving across
students with and without learning disabilities . Learning Disabilities Research & Practice, 29(2),
66–74 .
Radatz, H . (1979) . Error analysis in mathematics education . Journal for Research in Mathematics
Education, 10(3), 163–172 .
Riccomini, P . J . (2014) . Identifying and using error patterns to inform instruction for students
struggling in mathematics. Webinar slideshow .
Yetkin, E . (2003) . Student difficulties in learning elementary mathematics . ERIC Clearinghouse for
Science, Mathematics, and Environmental Education. Retrieved from http://www .ericdigests .
org/2004-3/learning .html
References for the Following Cases
Ashlock, R . B . (2010) . Error patterns in computation (10th ed .) . Boston: Allyn & Bacon .
Sherman, H . J ., Richardson, L . I ., & Yard, G . J . (2009) . Teaching learners who struggle with
mathematics: Systematic invervention and remediation (2nd ed .) . Upper Saddle River, NJ:
Merrill/Pearson .
Chapin, S . H . (1999) . Middle grades math: Tools for success (course 2): Practice workbook. New
Jersey: Prentice-Hall .
☆
What a STAR Sheet isWhat a STAR Sheet is
A STAR (STrategies And Resources) Sheet provides you with a description of a well-
researched strategy that can help you solve the case studies in this unit .
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Mathematics: Identifying and Addressing Student Errors
Collecting Data
STAR SHEET
About the Strategy
Collecting data involves asking a student to complete a worksheet, test, or progress monitoring
measure containing a number of problems of the same type .
What the Research and Resources Say
• Error analysis data can be collected using formal (e .g ., chapter test, standardized test) or
informal (e .g ., homework, in-class worksheet) measures (Riccomini, 2014) .
• Error analysis is one form of diagnostic assessment . The data collected can help teachers
understand why students are struggling to make progress on certain tasks and align
instruction with the student’s specific needs (National Center on Intensive Intervention, n .d .;
Kingsdorf & Krawec, 2014) .
• To help determine an error pattern, the data collection measure must contain at a minimum
three to five problems of the same type (Special Connections, n .d .) .
Identifying Data Sources
To conduct an error analysis for mathematics, the teacher must first collect data . She can do so by
using a number of materials completed by the student (i .e ., student product) . These include worksheets,
progress monitoring measures, assignments, quizzes, and chapter tests . Homework can also be used,
assuming the teacher is confident that the student completed the assignment independently . Regardless
of the type of student product used, it should contain at a minimum three to five problems of the same
type . This allows a sufficient number of items with which to determine error patterns .
Scoring
To better understand why students are struggling, the teacher should mark each incorrect digit in a
student’s answer, as opposed to simply marking the entire answer incorrect . Evaluating each digit in
the answer allows the teacher to more quickly and clearly identify the student’s error and to determine
whether the student is consistently making this error across a number of problems . For example, take
a moment to examine the worksheet below . By marking the incorrect digits, the teacher can determine
that, although the student seems to understand basic math facts, he is not regrouping the “1” to the
ten’s column in his addition problems .
Note: Marking each incorrect digit might not always reveal the error pattern . Review the STAR Sheets
Identifying Error Patterns, Word Problems: Additional Error Patterns, and Determining Reasons for
Errors to learn about identifying the different types of errors students make .
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TipsTips
• Typically, addition, subtraction, and multiplication problems should be
scored from RIGHT to LEFT . By scoring from right to left, the teacher will
be sure to note incorrect digits in the place value columns . However,
division problems should be scored LEFT to RIGHT .
• If the student is not using a traditional algorithm to arrive at a solution,
but instead using a partial algorithm (e .g ., partial sums, partial products)
then addition, subtraction, multiplication, and division problems should
be scored from LEFT to RIGHT .
References
Kingsdorf, S ., & Krawec, J . (2014) . Error analysis of mathematical word problem solving across
students with and without learning disabilities . Learning Disabilities Research and Practice,
29(2), 66–74 .
National Center on Intensive Intervention . (n .d .) . Informal academic diagnostic assessment:
Using data to guide intensive instruction. Part 3: Miscue and skills analysis . PowerPoint slides .
Retrieved from http://www .intensiveintervention .org/resource/informal-academic-diagnostic-
assessment-using-data-guide-intensive-instruction-part-3
Riccomini, P . J . (2014) . Identifying and using error patterns to inform instruction for students
struggling in mathematics . Webinar series, Region 14 State Support Team .
Special Connections . (n .d .) . Error pattern analysis . Retrieved from http://www .specialconnections .
ku .edu/~specconn/page/instruction/math/pdf/patternanalysis
The University of Chicago School Mathematics Project . (n .d .) . Learning multiple methods for any
mathematical operation: Algorithms. Retrieved from http://everydaymath .uchicago .edu/about/
why-it-works/multiple-methods/
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STAR SHEET
STAR SHEET
Mathematics: Identifying and Addressing Student Errors
Identifying Error Patterns
About the Strategy
Identifying error patterns refers to determining the type(s) of errors made by a student when he or she
is solving mathematical problems .
What the Research and Resources Say
Three to five errors on a particular type of problem constitute an error pattern (Howell, Fox, & Morehead,
1993; Radatz, 1979) .
Typically, student mathematical errors fall into three broad categories: factual, procedural, and conceptual .
Each of these errors is related either to a student’s lack of knowledge or a misunderstanding (Fisher & Frey,
2012; Riccomini, 2014) .
Not every error is the result of a lack of knowledge or skill . Sometimes, a student will make a mistake simply
because he was fatigued or distracted (i .e ., careless errors) (Fisher & Frey, 2012) .
Procedural errors are the most common type of error (Riccomini, 2014) .
Because conceptual and procedural knowledge often overlap, it is difficult to distinguish conceptual errors
from procedural errors (Rittle-Johnson, Siegler, & Alibali, 2001; Riccomini, 2014) .
Types of Errors
1. Factual errors are errors due to a lack of factual information (e .g ., vocabulary, digit identification,
place value identification) .
2. Procedural errors are errors due to the incorrect performance of steps in a mathematical process
(e .g ., regrouping, decimal placement) .
3. Conceptual errors are errors due to misconceptions or a faulty understanding of the underlying
principles and ideas connected to the mathematical problem (e .g ., relationship among numbers,
characteristics, and properties of shapes) .
FYI FYI
Another type of error that a student might make is a careless error . The student fails
to correctly solve a given mathematical problem despite having the necessary skills
or knowledge . This might happen because the student is tired or distracted by activity
elsewhere in the classroom . Although teachers can note the occurrence of such
errors, doing so will do nothing to identify a student’s skill deficits . For many students,
simply pointing out the error is all that is needed to correct it . However, it is important
to note that students with learning disabilities often make careless errors .
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Common Factual Errors
Factual errors occur when students lack factual information (e .g ., vocabulary, digit identification,
place value identification) . Review the table below to learn about some of the common factual errors
committed by students .
Factual Error Examples
Has not mastered basic number facts:
The student does not know basic
mathematics facts and makes errors
when adding, subtracting, multiplying,
or dividing single-digit numbers .
3 + 2 = 7 7 − 4 = 2
2 × 3 = 7 8 ÷ 4 = 3
Misidentifies signs 2 × 3 = 5 (The student identifies the multiplication
sign as an addition sign .)
8 ÷ 4 = 4 (The student identifies the division sign
as a minus sign .)
Misidentifies digits The student identifies a 5 as a 2 .
Makes counting errors 1, 2, 3, 4, 5, 7, 8, 9 (The student skips 6 .)
Does not know mathematical terms
(vocabulary)
The student does not understand the meaning of
terms such as numerator, denominator, greatest
common factor, least common multiple, or
circumference .
Does not know mathematical formulas The student does not know the
formula for calculating the area
of a circle .
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Procedural Error Examples
Regrouping Errors
Forgetting to regroup: The student forgets to regroup
(carry) when adding, multiplying, or subtracting .
77
+ 54
121
The student added 7 + 4 correctly but didn’t
regroup one group of 10 to the tens column .
123
− 76
53
The student does not regroup one group of 10
from the tens column, but instead subtracted the
number that is less (3) from the greater number
(6) in the ones column .
56
× 2
102
After multiplying 2 × 6, the student fails to
regroup one group of 10 from the tens column .
Regrouping across a zero: When a problem contains one
or more 0’s in the minuend (top number), the student is
unsure of what to do .
304
− 21
323
The student subtracted the 0 from the 2 instead
of regrouping .
Performing incorrect operation: Although able to correctly
identify the signs (e .g ., addition, minus), students often
subtract when they are suppose to add, or vice versa .
However, students might also perform other incorrect
operations, such as multiplying instead of adding .
234
− 45
279
The student added instead of subtracting .
3
+ 2
6
The student multiplied instead of adding .
Fraction Errors
Failure to find common denominator when adding and
subtracting fractions
3 1 4
— + — = —
4 3 7
The student added the
numerators and then the
denominators without finding the
common denominator .
Failure to invert and then multiply when dividing fractions
1 1 2 2
— ÷ 2 = — × — = — = 1
2 2 1 2
The student did not invert the 2
to before multiplying to get the
correct answer of .
Failure to change the denominator in multiplying fractions
2 5 10
— × — = —
8 8 8
The student did not multiply the
denominators to get the correct
answer .
Incorrectly converting a mixed number to an improper
fraction
1 4
1— = —
2 2
To find the numerator, the student
added 2 + 1 + 1 to get 4,
instead of following the correct
procedure ( 2 × 1 + 1 = 3 ) .
Common Procedural Errors
Procedural knowledge is an understanding of what steps or procedures are required to solve a
problem . Procedural errors occur when a student incorrectly applies a rule or an algorithm (i .e ., the
formula or step-by-step procedure for solving a problem) . Review the table below to learn more about
some common procedural errors .
1
4
1
2
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Common Conceptual Errors
Conceptual knowledge is an understanding of underlying ideas and principles and a recognition
of when to apply them . It also involves understanding the relationships among ideas and principles .
Conceptual errors occur when a student holds misconceptions or lacks understanding of the underlying
principles and ideas related to a given mathematical problem (e .g ., the relationship between numbers,
the characteristics and properties of shapes) . Examine the table below to learn more about some
common conceptual errors .
Conceptual Error Examples
Misunderstanding of place value:
The student doesn’t understand
place value and records the
answer so that the numbers are
not in the appropriate place
value position .
67
+ 4
17
The student added all the numbers
together ( 6 + 7 + 4 = 17 ), not
understanding the values of the
ones and tens columns .
10
+ 9
91
The student recorded the answer
with the
numbers reversed, disregarding the
appropriate place value position of
the numbers or digits .
Write the following as a
number:
When expressing a number
beyond two digits, the student
does not have a conceptual
understanding of the place value
position .
a) seventy-six
b) nine hundred seventy-
four
c) six thousand, six
hundred twenty-four
Student answer:
a) 76
b) 90074
c) 600060024
Procedural Error cont Examples cont
Decimal Errors
Not aligning decimal points when adding or
subtracting: The student aligns the numbers
without regard to where the decimal is located .
120 .4
+
63 .21
75 .25
The student did not align the decimal
points to show digits in like places . In
this case, .4 and .2 are in the tenths
place and should be aligned .
Not placing decimal in appropriate place when
multiplying or dividing: The student does not
count and add the number of decimal places in
each factor to determine the number of decimal
places in the product .
Note: This could also be a conceptual error
related to place value.
3 .4
× .2
6 .8
As with adding or subtracting, the
student aligns the decimal point in the
product with the decimal points in the
factors . The student did not count and
add the number of decimal places in
each factor to determine the number
of decimal places in the product
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Conceptual Error cont . Examples cont .
Overgeneralization: Because of lack
of conceptual understanding, the
student incorrectly applies rules or
knowledge to novel situations .
321
−
245
124
Regardless of whether the greater
number is in the minuend (top number)
or subtrahend (bottom number),
the student always subtracts the
number that is less from the greater
number, as is done with single-digit
subtraction .
Put the following
fractions in order
from smallest to
largest .
The student puts fractions in the order
, , , because he doesn’t
understand the relation between the
numerator and its denominator; that
is, larger denominators mean smaller
fractional parts .
Overspecialization: Because of lack of
conceptual understanding, the student
develops an overly narrow definition
of a given concept or of when to
apply a rule or algorithm .
Which of the
triangles below are
right triangles?
The student chooses a because she
only associates a right triangle with
those with the same orientation as a .
a)
b)
c) both
Student answer: a
90˚
12
200
1
351
77
486
12
200
1
351
77
486
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References
Ashlock, R . B . (2010) . Error patterns in computation (10th ed .) . Boston: Allyn & Bacon .
Ben-Hur, M . (2006) . Concept-rich mathematics instruction . Alexandria, VA: ASCD .
Cohen, L . G ., & Spenciner, L . J . (2007) . Assessment of children and youth with special needs (3rd
ed .) . Upper Saddle River, NJ: Pearson .
Educational Research Newsletter and Webinars . (n .d .) . Students’ common errors in working with
fractions . Retrieved from http://www .ernweb .com/educational-research-articles/students-
common-errors-misconceptions-about-fractions/
El Paso Community College . (2009) . Common mistakes: Decimals. Retrieved from http://www .
epcc .edu/CollegeReadiness/Documents/Decimals_0-40
El Paso Community College . (2009) . Common mistakes: Fractions . Retrieved from http://www .
epcc .edu/CollegeReadiness/Documents/Fractions_0-40
Fisher, D ., & Frey, N . (2012) . Making time for feedback . Feedback for Learning, 70(1), 42–46 .
Howell, K . W ., Fox, S ., & Morehead, M . K . (1993) . Curriculum-based evaluation: Teaching and
decision-making. Pacific Grove, CA: Brooks/Cole .
National Council of Teachers of Mathematics . (2000) . Principles and standards for school
mathematics . Reston, VA: Author .
Riccomini, P . J . (2014) . Identifying and using error patterns to inform instruction for students
struggling in mathematics . Webinar series, Region 14 State Support Team .
Radatz, H . (1979) . Error analysis in mathematics education . Journal for Research in Mathematics
Education, 10(3), 163–172 .
Rittle-Johnson, B ., Siegler, R . S ., & Alibali, M . W . ( 2001) . Developing conceptual understanding
and procedural skill in mathematics: An iterative process . Journal of Educational Psychology,
93(2), 346–362 .
Sherman, H . J ., Richardson, L . I ., & Yard, G . J . (2009) . Teaching learners who struggle with
mathematics: Systematic intervention and remediation (2nd ed .) . Upper Saddle River, NJ:
Merrill/Pearson .
Siegler, R ., Carpenter, T ., Fennell, F ., Geary, D ., Lewis, J ., Okamoto, Y ., Thompson, L ., & Wray, J .
(2010) . Developing effective fractions instruction for kindergarten through 8th grade: A practice
guide (NCEE #2010-4039) . Washington, DC: National Center for Education Evaluation and
Regional Assistance, Institute of Education Sciences, U .S . Department of Education . Retrieved
from http://ies .ed .gov/ncee/wwc/pdf/practice_guides/fractions_pg_093010
Special Connections . (n .d .) . Error pattern analysis. Retrieved from http://www .specialconnections .
ku .edu/~specconn/page/instruction/math/pdf/patternanalysis
Yetkin, E . (2003) . Student difficulties in learning elementary mathematics. ERIC Clearinghouse for
Science, Mathematics, and Environmental Education . Retrieved from http://www .ericdigests .
org/2004-3/learning .html
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STAR SHEETSTAR SHEET
Mathematics: Identifying and Addressing Student Errors
Word Problems: Additional Error Patterns
About the Strategy
A word problem presents a hypothetical real-world scenario that requires a student to apply
mathematical knowledge and reasoning to reach a solution .
What the Research and Resources Say
• Students consider computational exercises more difficult when they are expressed as word
problems rather than as number sentences (e .g ., 3 + 2 =) (Sherman, Richardson, & Yard,
2009) .
• When they solve word problems, students struggle most with understanding what the problem is
asking them to do . More specifically, students might not recognize the problem type and therefore
do not know what strategy to use to solve it (Jitendra et al ., 2007; Sherman, Richardson, & Yard,
2009; Powell, 2011; Shin & Bryant, 2015) .
• Word problems require a number of skills to solve (e .g ., reading text, comprehending text,
translating the text into a number sentence, determining the correct algorithm to use) . As
a result, many students, especially those with math and/or reading difficulties, find word
problems challenging (Powell, Fuchs, Fuchs, Cirino, & Fletcher, 2009; Reys, Lindquist,
Lambdin, & Smith, 2015) .
• Word problems are especially difficult for students with learning disabilities (Krawec, 2014;
Shin & Bryant, 2015) .
Common Difficulties Associated with Solving Word Problems
A student might solve word problems incorrectly due to factual, procedural, or conceptual errors .
However, a student might encounter additional difficulties when trying to solve word problems, many
of which are associated with reading skill deficits, such as those described below .
Poor vocabulary knowledge: The student does not understand many mathematics terms (e .g .,
difference, factor, denominator) .
Limited reading skills: The student has difficulty reading text with vocabulary and complex sentence
structure . Because of this, the student struggles to understand what is being asked .
Inability to identify relevant information: The student has difficulty determining which pieces of
information are relevant and which are irrelevant to solving the problem .
Lack of prior knowledge: The student has limited experience with the context in which the problem
is embedded . For example, a student unfamiliar with cooking might have difficulty solving a fraction
problem presented within the context of baking a pie .
Inability to translate the information into a mathematical equation: The student has difficulty translating
the information in the word problem into a mathematical equation that they can solve . More
specifically, the student might not be able to put the numbers in the correct order in the equation or
determine the correct operation to use .
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Example
The word problem below illustrates why students might have difficulty solving this type of problem .
Jonathan would like to buy a new 21-speed bicycle. The bike costs $119.76. Jonathan received
$25 for his birthday. He also worked for 3 months last summer and earned $59.50. Find the
difference between what the bike costs and the amount of money Jonathan has.
In addition to solving this word problem incorrectly due to factual, procedural, or conceptual errors,
the student might struggle for reasons related to reading skill deficits .
• Poor vocabulary knowledge—The student might be unfamiliar with the term difference .
• Limited reading skills—The student might struggle with the problem’s final sentence because of
its complex structure . If the student doesn’t understand some of the vocabulary (e .g ., received,
earned), it might impede his or her ability to solve the problem .
• Inability to identify relevant information—The student might attend to irrelevant information,
such as the type of bicycle or the number of months Jonathan worked, and therefore solve the
problem incorrectly .
• Lack of prior knowledge—The student might have limited knowledge about the process of
making purchases .
• Inability to translate information into a mathematical equation—The student might have
difficulty determining which operations to perform with which numbers . This situation might
be made worse in cases involving problems with multiple steps .
References
Jitendra, A . K ., Griffin, C . C ., Haria, P ., Leh, J ., Adams, A ., & Kaduvettoor, A . (2007) . A
comparison of single and multiple strategy instruction on third-grade students’ mathematical
problem solving . Journal of Educational Psychology, 99(1), 115–127 .
Krawec, J . L . (2014) . Problem representation and mathematical problem solving of students of
varying math ability . Journal of Learning Disabilities, 47(2), 103–115 .
Powell, S . R . (2011) . Solving word problems using schemas: A review of the literature . Learning
Disabilities Research & Practice, 26(2), 94–108 .
Powell, S . R ., Fuchs, L . S ., Fuchs, D ., Cirino, P . T ., & Fletcher, J . M . (2009) . Do word-problem
features differentially affect problem difficulty as a function of students’ mathematics difficulty
with and without reading difficulty? Journal of Learning Disabilities 20(10), 1–12
Reys, R ., Lindquist, M . M ., Lambdin, D . V ., & Smith, N . L . (2015) . Helping children learn
mathematics (11th ed .) . Hoboken, NJ: John Wiley & Sons .
Sherman, H . J ., Richardson, L . I ., & Yard, G . J . (2009) . Teaching learners who struggle with
mathematics: Systematic intervention and remediation (2nd ed .) . Upper Saddle River, NJ:
Merrill/Pearson .
Shin, M ., & Bryant, D . P . (2015) . A synthesis of mathematical and cognitive performances of
students with mathematics learning disabilities . Journal of Learning Disabilities, 48(1), 96–112 .
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Mathematics: Identifying and Addressing Student Errors
Determining Reasons for Errors
CASE STUDY
About the Strategy
Determining the reason for errors is the process through which teachers determine why the student is
making a particular type of error .
What the Research and Resources Say
• To help them to improve their mathematical performance, teachers must first identify and
understand why students make particular errors (Radatz, 1979; Yetkin, 2003) .
• Typically, a student’s errors are not random; instead, they are often based on incorrect
algorithms or procedures applied systematically (Cox, 1975; Ben-Zeev, 1998) .
• Knowing what a student is thinking when she is solving a problem can be a rich source of
information about what she does and does not understand (Hunt & Little, 2014; Baldwin &
Yun, 2012) .
Helpful Strategies
Determining exactly why a student is making a particular error is important in that it informs the
teacher’s instructional response . Though it is sometimes obvious why a student is making a certain type
of errors, at other times determining a reason proves more difficult . In these latter instances, the teacher
can use one or more of the following strategies .
Interview the student—It is sometimes unclear why a student is making a particular type of error .
For example, it can be difficult for a teacher to distinguish between procedural or conceptual errors .
For this reason, it can be beneficial to ask a student to talk through his or her process for solving the
problem . Teachers can ask general questions such as “How did you come up with that answer?” or
prompt the student with statements such as “Show me how you got that answer .” Another reason
teachers might want to interview the student is to make sure the student has the prerequisite skills to
solve the problem .
Observe the student—A student might also reveal information through nonverbal means . This can
include gestures, pauses, signs of frustration, and self-talk . The teacher can use information of this type
to identify at what point in the problem-solving task that the student experiences difficulty or frustration .
It can also help the teacher determine which procedure or set of rules a student is applying and why .
Look for exceptions to an error pattern—In addition to looking for error patterns, a teacher should
note instances when the student does not make the same error on the same type of problem . This, too,
can be informative because it might indicate that the student has partial or basic understanding of the
concept in question . For example, Cammy completed a worksheet on multiplying whole numbers by
fractions . She seemed to get most of them wrong; however, she correctly answered the problems in
which the fraction was . This seems to indicate that, though Cammy conceptually understands what of a
whole is, she most likely does not know the process for multiplying whole numbers by fractions .
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Considerations for Students with Learning Disabilities
Approximately 5–8% of students exhibit mathematics learning disabilities . Therefore, it is important
to understand that their unique learning differences might impact their ability to learn and correctly
choose and apply solution strategies to solve mathematics problems . A few characteristics that
teachers might notice with students with learning disabilities is that these students often:
• Have difficulty mastering basic number facts
• Make computational errors even though they might have a strong conceptual understanding
• Have difficulty making the connection between concrete objects and semiabstract (visual
representations) or abstract knowledge or mathematical symbols
• Struggle with mathematical terminology and written language
• Have visual-spatial deficits, which result in difficulty visualizing mathematical concepts (although
this is quite rare)
References
Baldwin, E . E ., & Yun, J . T . (2012) . Mathematics curricula and formative assessments: Toward an
error-based approach to formative data use in mathematics. Santa Barbara, CA: University of
California Educational Evaluation Center .
Ben-Zeev, T . (1998) . Rational errors and the mathematical mind . Review of General Psychology,
2(4), 366–383 .
Cox, L . S . (1975) . Systematic errors in the four vertical algorithms in normal and handicapped
populations . Journal for Research in Mathematics Education, 6(4), 202–220 .
Garnett, K . (n .d .) . Math learning disabilities . Retrieved from http://www .ldonline .org article/
Math_Learning_Disabilities
Hunt, H . H ., & Little, M . E . (2014) . Intensifying interventions for students by identifying and
remediating conceptual understandings in mathematics . Teaching Exceptional Children, 46(6),
187–196 .
PBS, & the WGBH Educational Foundation . (2002) . Difficulties with mathematics. Retrieved from
http://www .pbs .org/wgbh/misunderstoodminds/mathdiffs .html
Radatz, H . (1979) . Error analysis in mathematics education . Journal for Research in Mathematics
Education, 10(3), 163–172 .
Sherman, H . J ., Richardson, L . I ., & Yard, G . J . (2009) . Teaching learners who struggle with
mathematics: Systematic intervention and remediation. Upper Saddle River, NJ: Pearson .
Shin, M ., & Bryant, D . P . (2015) . A synthesis of mathematical and cognitive performances of
students with mathematics learning disabilities . Journal of Learning Disabilities, 48(1), 96–112 .
Special Connections . (n .d .) . Error pattern analysis. Retrieved from http://specialconnections .
ku .edu/~specconn/page/instruction/math/pdf/patternanalysis
Yetkin, E . (2003) . Student difficulties in learning elementary mathematics . ERIC Clearinghouse for
Science, Mathematics, and Environmental Education . Retrieved from http://www .ericdigests .
org/2004-3/learning .html
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STAR SHEET
Mathematics: Identifying and Addressing Student Errors
Addressing Error Patterns
About the Strategy
Addressing error patterns is the process of providing instruction that focuses on a student’s specific
error .
What the Research and Resources Say
• Students will continue to make procedural errors if they do not receive targeted instruction to
addresses those errors . Simply providing more opportunities to practice working a given problem
is typically not effective (Riccomini, 2014) .
• By conducting an error analysis, the teacher can target specific misunderstandings or missteps,
rather than re-teaching the entire skill or concept (Fisher & Frey, 2012) .
• Without intervention, students have been shown to continue to apply the same error patterns one
year later (Cox, 1975) .
• Addressing a student’s conceptual errors might require the use of concrete or visual
representations, as well as a great deal of re-teaching . Students can often use concrete objects to
solve problems that they initially answered incorrectly (Riccomini, 2014; Yetkin, 2003) .
• Simply teaching the formula or the steps to solve a mathematics problem is typically not sufficient
to help students gain conceptual understanding (Sweetland & Fogarty, 2008) .
How To Address Student Errors
After the teacher has determined what types of error(s) a student is making, he or she can address the
error in the following way .
Discuss the error with the student: After the teacher has interviewed the student and examined work
products, the teacher should briefly describe the student’s error and explain that they will work together
to correct it .
Provide effective instruction to address the student’s specific error: The teacher should target the
student’s specific error instead of re-teaching how to work this type of problem in general . For
example, if a student’s error is related to not regrouping during addition, the teacher should focus on
where exactly in the process the student makes the error . The teacher must pinpoint the instruction to
focus on the error and help the student to understand what he is doing incorrectly . Simply re-teaching
the lesson will not ensure that the student understands the error and how to correctly solve the problem .
Use effective strategies: With the type of error in mind, the teacher should select an effective strategy
that will help to correct the student’s misunderstandings or missteps . Below are two effective strategies
that teachers might find helpful to address some—if not all—error patterns .
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Manipulatives
Manipulatives are concrete objects—for example, base-ten blocks, a
geoboard, or integer chips—that a student can use to
develop a conceptual understanding of
mathematic topics . These objects help a student to
represent the mathematical idea she is trying to learn or
the problem she is trying to solve . For example, the teacher might
demonstrate the idea of fractions by using fraction blocks or
fraction strips . It is important that the teacher make explicit the connection
between the concrete object and the abstract or the symbolic concept being
taught . After a student has gained a basic understanding of the mathematical concept, the concrete
objects should be replaced by visual representations such as images of a number line or geoboard
(a small board with nails on which students stretch rubber bands to explore a variety of basic
geometry concepts) . The goal is for the student to eventually understand and apply the concept
with numerals and symbols .
It is important that the teacher’s instruction match the needs of the student . Teachers should keep in
mind that some students will need concrete objects to understand a concept, whereas others will be
able to understand the concept using visual representations . Additionally, some students will require
the support of concrete objects longer than will other students .
FYIFYI
Recall that students with learning disabilities sometimes have visual-spatial deficits,
which makes it difficult for them to learn concepts using visual representations . For
these students, teachers should teach concepts using concrete materials accompanied
by strong, precise verbal descriptions or explanations .
Keep in MindKeep in Mind
The type of instruction a teacher uses to correct conceptual errors will likely differ
from that used to address factual or procedural errors . Simply teaching a student the
formula or the steps to solve a mathematics problem will not help the student gain
conceptual understanding .
Geoboard
Credit: Kyle Trevethan
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Explicit, systematic instruction
Explicit, systematic instruction involves teaching a specific skill or concept in a highly structured
environment using clear, direct language and incorporating the components listed in the table below .
Components of Explicit Instruction
Modeling • The teacher models thinking aloud to demonstrate the completion of
a few sample problems .
• The teacher leads the student through more sample problems .
• The teacher points out difficult aspects of the problems .
Guided Practice • The student completes problems with the help of either teacher or
peer guidance .
• The teacher monitors the student’s work .
• The teacher offers positive corrective feedback .
Independent
Practice
• The student completes the problems independently .
• The teacher checks the student’s performance on independent
work .
Adapted from Bender (2009), pp. 31–32
Reassess student skills: After providing instruction to correct the student’s error(s), the teacher should
conduct a formal or informal assessment to make sure that the student has mastered the skill or concept
in question .
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Instructional Tips
• Check for prerequisite skills: Make sure the student has the prerequisite skills needed to solve
the problem with which he has been struggling . For example, if the student is making errors
while adding two-digit numbers, the teacher needs to make sure the student knows basic math
facts . If the student lacks the necessary pre-skills, the teacher should begin instruction at that
point .
• Model examples and nonexamples: Be sure to model the completion of a minimum of three
to five problems of the kind the student is struggling with . Add at least one nonexample of
the error pattern to prevent overgeneralization (incorrectly applying the rule or knowledge
to novel situations) and overspecialization (developing an overly narrow definition of the
concept of or when to apply a rule or procedure) . For example, in the case of a student who
does not regroup when subtracting, a teacher modeling how to solve this type of problem
should also include problems that do not require regrouping .
• Pinpoint error: During modeling and guided practice, focus only on the place in the problem
where the student makes an error . It is not necessary to work through the entire problem . For
example, if the student’s error pattern is that she fails to find the common denominator when
adding and subtracting fractions, the teacher would only model the process and explain the
underlying conceptual knowledge of finding the common denominator . She would stop at that
point, as opposed to completing the problem because the student knows the process from that
point forward . The teacher should then continue in same manner for the remaining problems .
• Provide ample opportunities for practice: As with modeling, provide a minimum of three to
five problems for guided practice, making sure to include a nonexample .
• Start with simple problems: During modeling and guided practice, begin with simple
problems and gradually progress to more difficult ones as the student gains an understanding
of the error and how to correctly complete the problem .
• Move the error around: When possible, move the error around so that it does not always
occur in the same place . For example, if the student’s error is not regrouping when
multiplying, the teacher should include examples that require regrouping in the ones and tens
column, instead of always requiring the regrouping to occur in the ones column .
1 1 — + —
4 2
1 2 — + —
4 4
[Stop at this point because you have addressed the error pattern; the student
knows how to add fractions.]
Problems 1 and 3 are examples that require regrouping, whereas problem 2, which
does not require regrouping, is a nonexample .
121 231 376
− 17 − 120 − 229
1 . 2 . 3 .
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References
Colarussso, R ., & O’Rourke, C . (2004) . Special education for all teachers (3rd ed .) . Dubuque, IA:
Kendall Hunt .
Cox, L . S . (1975) . Systematic errors in the four vertical algorithms in normal and handicapped
populations . Journal for Research in Mathematics Education, 6(4), 202–220 .
Fisher, D ., & Frey, N . (2012) . Making time for feedback . Feedback for Learning, 70(1), 42–46 .
Riccomini, P . J . (2014) . Identifying and using error patterns to inform instruction for students
struggling in mathematics. Webinar series, Region 14 State Support Team . Retrieved from
http://www .ohioregion14 .org/perspectives/?p=1005
Sweetland, J ., & Fogarty, M . (2008) . Prove it! Engaging teachers as learners to enhance
conceptual understanding . Teaching Children Mathematics, 68–73 . Retrieved from http://www .
uen .org/utahstandardsacademy/math/downloads/level-2/5-2-ProveIt
Yetkin, E . (2003) . Student difficulties in learning elementary mathematics. ERIC Clearinghouse for
Science, Mathematics, and Environmental Education . Retrieved from http://www .ericdigests .
org/2004-3/learning .html
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Background
Student: Dalton
Age: 12
Grade: 7th
Scenario
Mrs . Moreno, a seventh-grade math teacher, is concerned about Dalton’s performance . Because
Dalton has done well in her class up to this point, she believes that he has strong foundational
mathematics skills . However, since beginning the lessons on multiplying decimals, Dalton has
performed poorly on his independent classroom assignments . Mrs . Moreno decides to conduct an
error analysis on his last homework assignment to determine what type of error he is making .
Possible Strategies
• Collecting Data
• Identifying Error Patterns
! ! AssignmentAssignment
1 . Read the Introduction.
2 . Read the STAR Sheets for the possible strategies listed above .
3 . Score Dalton’s classroom assignment below . For ease of scoring, an answer key has been provided .
4 . Examine the scored worksheet and determine Dalton’s error pattern .
Answer Key
1) 7 .488 2) 3 .065 3) 0 .5976 4) .00084 5) .5040 6) 2 .6724
7) .006084 8) 7 .602 9) .00183 10) 4 .6098 11) $39 .00 12) 732 .48 cm
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Mathematics: Identifying and Addressing Student Errors
Level A • Case 1
CASE STUDY
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Background
Student: Madison
Age: 8
Grade: 2nd
Scenario
Madison is a bright and energetic third-grader with a specific learning disability in math . Her class just
finished a chapter on money, and her teacher, Ms . Brooks, was pleased with Madison’s performance .
Ms . Brooks believes that Madison’s success was largely due to the fact that she used play money to
teach concepts related to money . As is noted in Madison’s individualized education program (IEP),
she more easily grasps concepts when using concrete objects (i .e ., manipulatives such as play coins
and dollar bills) . In an attempt to build on this success, Ms . Brooks again used concrete objects—in
this case, cardboard clocks with movable hands—to teach the chapter on telling time . The class is now
halfway through that chapter, and to Ms . Brooks’ disappointment, Madison seems to be struggling
with this concept . Consequently, Ms . Brooks decides to conduct an error analysis on Madison’s most
recent quiz .
Possible Stragegies
• Collecting Data
• Identifying Error Patterns
! ! AssignmentAssignment
1 . Read the Introduction .
2 . Read the STAR Sheets for the possible strategies listed above .
3 . Score Madison’s quiz below by marking each incorrect response .
4 . Examine the scored quiz and determine Madison’s error pattern .
Answer Key
1) 3:00 2) 9:25 3) 7:15 4) 5)
6) 7) 8) 9)
10)
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Mathematics: Identifying and Addressing Student Errors
Level A • Case 2
CASE STUDY
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Mathematics: Identifying and Addressing Student Errors
Level B • Case 1
CASE STUDY
Background
Student: Shayla
Age: 10
Grade: 5th
Scenario
Shayla and her family just moved to a new school district . Her math class is currently learning how to add and
subtract fractions with unlike denominators . Shayla’s math teacher, Mr . Holden, is concerned because Shayla is
performing poorly on assignments and quizzes . Before he can provide instruction to target Shayla’s skill deficits
or conceptual misunderstandings, he needs to determine why she is having difficulty . For this reason, he decides
to conduct an error analysis to discover what type of errors she is making .
Possible Strategies
• Collecting Data
• Identifying Error Patterns
• Word Problems: Additional Error Patterns
! ! AssignmentAssignment
1 . Read Introduction .
2 . Read the STAR Sheets for the possible strategies listed above .
3 . Score Shayla’s assignment below by marking each incorrect digit .
4 . Examine the scored assignment and discuss at least three possible reasons for Shayla’s error pattern .
4
8
3
18
6
12
1
10
5
6
7
8
3
4
1
4
7
16
2
6
5
8
3
6
Answer Key
1) 2) 3) 4) 5)
6) 7) 8) 0 9) 10)
11) 12) 13)
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Mathematics: Identifying and Addressing Student Errors
Level B • Case 2
CASE STUDY
Background
Student: Elías
Age: 7
Grade: 2nd
Scenario
A special education teacher at Bordeaux Elementary School, Mrs . Gustafson has been providing
intensive intervention to Elías, who has a learning disability, and collecting progress monitoring data
for the past six weeks . His data indicate that he is not making adequate progress to meet his end-of-
year goals . Mrs . Gustafson has decided that she needs to conduct a diagnostic assessment to identify
areas of difficulty and to determine specific instructional needs . As part of the diagnostic assessment,
Mrs . Gustafson conducts an error analysis using Elías’ progress monitoring data .
Possible Activities
• Collecting Data
• Identifying Error Patterns
• Determining Reasons for Errors
! ! AssignmentAssignment
1 . Read the Introduction .
2 . Read the STAR Sheets for the possible strategies listed above .
3 . Score Elías’ progress monitoring probe below by marking each incorrect digit .
4 . When Mrs . Gustafson scores the probe, she finds two possible explanations . One is that Elías is
making a conceptual error, and the other is that he doesn’t understand or is not applying the correct
procedure .
a . Assume that his error pattern is procedural . Describe Elías’ possible procedural error
pattern .
b . Assume that his error pattern is conceptual . Describe Elías’ possible conceptual error
pattern .
5 . Because the instructional adaptations Mrs . Gustafson will make will depend on Elías’ error pattern,
she must be sure of the reasons for his errors . Explain at least one strategy Mrs . Gustafson could use
to determine Elías’ error type .
Answer Key
1) 40 2) 87 3) 45 4) 22 5) 42
6) 34 7) 5 8) 122 9) 5 10) 80
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For illustrative purposes, only 10 of the 25 problems are
shown .
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Mathematics: Identifying and Addressing Student Errors
Level C • Case 1
CASE STUDY
Background
Student: Wyatt
Age: 12
Grade: 6th
Scenario
Mr . Goldberg has been teaching a unit on fractions . He was pleased that all of his students seemed to
quickly master adding and subtracting two fractions . However, when he began teaching the students
how to multiply fractions, a small number of them did not readily learn the content . But after a quick
mini-lesson, it appears that all but three students seem to understand how to solve the problems .
One of these students, Wyatt, seems to be really struggling . Mr . Goldberg determines that he needs
to collect some data to help him decide what type of error Wyatt is making so that he can provide
appropriate instruction to help Wyatt be successful . To do so, he decides to evaluate Wyatt’s most
recent independent classroom assignment .
! ! AssignmentAssignment
1 . Read the Introduction.
2 . Read the STAR Sheets .
3 . Score Wyatt’s classroom assignment below by marking each incorrect digit .
4 . Review Wyatt’s scored assignment sheet .
a . Describe Wyatt’s error pattern .
b . Discuss any exceptions to this error pattern . What might these indicate?
5 . Based on Wyatt’s error pattern, which of the two strategies described in the Addressing Error Patterns
STAR Sheet would you recommend that Mr . Goldberg use to remediate this error? Explain your
response .
1
8
2
9
14
48
12
25
21
56
12
121
24
108
48
48
2
6
1
3
1
4
2
12
6
12
Answer Key
1) 2) 3) 4) 5)
6) 7) 8) or 1 9) or 10)
11) 12)
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