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CASE STUDY UNIT

Mathematics: Identifying and Addressing

Student Errors

Created by Janice Brown, PhD, Vanderbilt UniversityKim Skow, MEd, Vanderbilt University

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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 [email protected] .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 .pdf

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 .pdf

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 SHEETSTAR 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 .pdf

El Paso Community College . (2009) . Common mistakes: Fractions . Retrieved from http://www . epcc .edu/CollegeReadiness/Documents/Fractions_0-40 .pdf

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 .pdf

Special Connections . (n .d .) . Error pattern analysis. Retrieved from http://www .specialconnections . ku .edu/~specconn/page/instruction/math/pdf/patternanalysis .pdf

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 …

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