Posted: September 19th, 2022

ErrorAnalysisCaseStudyAssignmentInstructions xics_matherr

EDUC 530

EDUC 530

**Error Analysis Case Study Assignment Instructions**

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.

· 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 Madison improve.

Case Study Level B, Case 2 – Elias (p. 9)

Student: Elias

· Read Elias’ 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 Elias improve.

Case Study Level C, Case 1 – Wyatt (p. 11)

Student: Wyatt

· Read Wyatt’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 Wyatt improve.

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.

Page 1 of 2

Curran, C., & the IRIS Center. (2003).

Mathematics: Identifying and addressing student errors. Retrieved from https://iris.peabody.vanderbilt.edu/wp-content/uploads/pdf_case_studies/ics_matherr

032422

iris.peabody.vanderbilt.edu or iriscenter.com

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 University

Kim 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

Level A, Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Level B, Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Level B, Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Level C, Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Answer Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

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 https://

iris .peabody .vanderbilt .edu/wp-content/uploads/pdf_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 Knight

Page 15- Geoboard Credit: Kyle Trevethan

Mathematics:

Identifying and Addressing Student Errors

CREDITS

https://iris.peabody.vanderbilt.edu/wp-content/uploads/pdf_case_studies/ics_matherr

https://iris.peabody.vanderbilt.edu/wp-content/uploads/pdf_case_studies/ics_matherr

https://iris.peabody.vanderbilt.edu/wp-content/uploads/pdf_case_studies/ics_matherr

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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/

http://www.specialconnections.ku.edu/~specconn/page/instruction/math/pdf/patternanalysis

http://www.specialconnections.ku.edu/~specconn/page/instruction/math/pdf/patternanalysis

http://everydaymath.uchicago.edu/about/why-it-works/multiple-methods/

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 .

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 .

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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 = — x —-= —-

2 2 1 2

=1

The student did not

invert the 2 to 1/2 before

multiplying to get the

correct answer of 1/4 .

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 41— = —

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 ) .

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

http://www.specialconnections.ku.edu/~specconn/page/instruction/math/pdf/patternanalysis

http://www.specialconnections.ku.edu/~specconn/page/instruction/math/pdf/patternanalysis

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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 1/2 . This seems to indicate that, though Cammy conceptually understands what

1/2 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

http://www.ohioregion14.org/perspectives/?p=1005

19

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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Place an order in 3 easy steps. Takes less than 5 mins.