Introduction

It is not uncommon for students to make mistakes when solving math problems. Sometimes these are careless errors, and other times they are due to conceptual misunderstandings or skill deficits. When students consistently struggle or perform poorly on mathematics tasks, educators might consider conducting an error analysis. An 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., multidigit 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 by examining the procedures they use to solve problems or by asking them to explain their logic.

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.

Each case study includes multiple STAR Sheets and cases.

About the Strategy

Collecting data for an error analysis involves asking a student to complete a worksheet, test, or progress monitoring measure containing a number of problems of the same type or asking students to explain their thinking and processes.

What the Research and Resources Say

• 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; Hwang & Riccomini, 2021; Lewis, 2016; Lewis et al., 2020; Nelson & Powell, 2018).
• Error analysis data can be collected using formal measures (e.g., chapter test, standardized test) or informal measures (e.g., homework, in-class worksheet, interview) (Riccomini, 2014; Lewis, 2016; Lewis & Fisher, 2018).
• 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.).
• Common error types include using the wrong operation, miscalculation (e.g., basic facts, regrouping), procedural errors (e.g., forgetting to regroup, performing an incorrect operation), and visual-spatial errors (e.g., column alignment, patterns, reading graphs). (Nelson & Powell, 2018; Lin et al., 2025; Rong & Monnen, 2022; Nelson & Powell, 2018).

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 tens column in his addition and multiplication 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 more about identifying the different types of errors students make.

About the Strategy

Identifying error patterns refers to determining the type(s) of errors a student makes when solving mathematical problems. There are three types of errors:

1. Factual errors—errors due to a lack of factual information (e.g., digit misidentification, counting errors)
2. Procedural errors—errors due to the incorrect performance of steps in a mathematical process (e.g., not regrouping, decimal misplacement)
3. Conceptual errors—errors due to misconceptions or a faulty understanding of the underlying principles and ideas connected to the mathematical problem (e.g., misunderstanding of place value, incorrectly applying rules to new 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; Lin et al., 2025).
• Procedural errors are the most common type of error (Riccomini, 2014; Nelson & Powell, 2018).
• Because conceptual and procedural knowledge often overlap, it is difficult to distinguish conceptual errors from procedural errors (Rittle-Johnson, Siegler, & Alibali, 2001; Riccomini, 2014).
• Not every error is the result of a lack of knowledge or a skill deficit. Sometimes, a student makes a mistake as a result of being fatigued or distracted (i.e., careless errors) (Fisher & Frey, 2012).

Common Factual Errors

Factual errors occur when students lack factual information. Review the table below to learn about some of the common factual errors made 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.

Procedural Error	Examples
Regrouping Errors
Forgets to regroup The student forgets to regroup when adding, multiplying, or subtracting.	Example 1: The student adds 7 + 4 correctly but does not regroup one group of 10 to the tens column. 77 + 54 121 Example 2: The student does not regroup one group of 10 from the tens column but instead subtracts the number that is less (3) from the greater number (6) in the ones column. 123 – 76 53 Example 3: After multiplying 2 × 6, the student fails to regroup one group of 10 from the tens column. 56 x 2 102
Regroups across a zero When a problem contains one or more 0s in the minuend (top number), the student is unsure of what to do.	The student subtracts the 0 from the 2 instead of regrouping. 304 – 21 323
Performs the incorrect operation Although able to correctly identify the signs (e.g., addition, minus), students often subtract when they are supposed to add, or vice versa. However, students might also perform other incorrect operations, such as multiplying instead of adding.	Example 1: The student adds instead of subtracts. 234 – 45 279 Example 2: The student multiplies instead of adds. 3 + 2 6
Fraction Errors
Fails to find common denominator when adding and subtracting fractions	The student adds the numerators and then the denominators without finding the common denominator. 3       4  + 1       3  = 4       7
Fails to invert and then multiply when dividing fractions	The student does not invert the 2 to 1/2 before multiplying to get the correct answer of 1/4. 1       2   ÷   2 = 1       2   x   2       1   =   2       2   =   1
Fails to change the denominator in multiplying fractions	The student does not multiply the denominators to get the correct answer. 2       8   x   5       8  = 10       8
Incorrectly converts a mixed number to an improper fraction	To find the numerator, the student adds 2 + 1 + 1 to get 4, instead of following the correct procedure ( 2 × 1 + 1 = 3 ). 1 1       2  = 4       2
Decimal Errors
Does not align decimal points when adding or subtracting The student aligns the numbers without regard to where the decimal is located.	The student does not align the decimal points correctly. In this case, .4 and .2 are in the tenths place and should be aligned. 120.4 + 63.21 75.25
Does not place the decimal in the 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.	As with adding or subtracting, the student aligns the decimal point in the product with the decimal points in the factors. 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. 3.4 x .2 6.8

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. Examine the table below to learn more about some common conceptual errors.

Conceptual Error	Examples
Misunderstands 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.	Example 1: The student adds all the numbers together ( 6 + 7 + 4 = 17 ), not understanding the values of the ones and tens columns. 67 + 4 17 Example 2: The student records the answer with the numbers reversed, disregarding the appropriate place value position of the numbers or digits. 10 + 9 91 Example 3: When expressing a number beyond two digits, the student does not have a conceptual understanding of the place value position. Write the following as a number: seventy-six nine hundred seventy-four six thousand six hundred twenty-four Student answer: 76 90074 600060024
Overgeneralizes Because of a lack of conceptual understanding, the student incorrectly applies rules or knowledge to novel situations.	Example 1: 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 than the greater number, as is done with single-digit subtraction. 321 – 245 124 Example 2: Put the following fractions in order from smallest to largest. 77       486 1       351 12       200 Not understanding the relation between the numerator and its denominator (i.e., larger denominators mean smaller fractional parts), the student puts the fractions in the following order. 12       200 1       351 77       486
Overspecializes Because of a 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? Associating a right triangle with only those with the same orientation as a, the student chooses a.

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 might not know what strategy to use (Jitendra et al., 2007; Sherman, Richardson, & Yard, 2009; Powell, 2011; Shin & Bryant, 2015; Lien et al., 2020).
• 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 reading difficulties, find word problems challenging (Powell, Fuchs, Fuchs, Cirino, & Fletcher, 2009; Reys, Lindquist, Lambdin, & Smith, 2015; Lien et al., 2020).
• 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.

Example

The word problem to the right illustrates why students might have difficulty solving this type of problem. 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. Additionally, not understanding some of the non-mathematical vocabulary (e.g., received, earned ) might impede the student’s 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.

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

• 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; Nelson & Powell, 2018).
• To help students improve their mathematical performance, teachers must first identify and understand why students make particular errors (Radatz, 1979; Yetkin, 2003; Lin et al., 2025; Lewis, 2016).
• Knowing what a student is thinking when solving a problem can be a rich source of information about what the student does and does not understand (Hunt & Little, 2014; Baldwin & Yun, 2012; Lewis & Fisher, 2018).

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 error, 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 and conceptual errors. For this reason, it can be beneficial to ask a student to talk through the 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 non-verbal 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 a 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 although Cammy conceptually understands what 1/2 of a whole is, she most likely does not know the process for multiplying whole numbers by fractions.

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. Teachers might notice that students who have learning disabilities:

• 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 visual representations or abstract problems
• Struggle with mathematical terminology and written language
• Have visual-spatial deficits, which result in difficulty visualizing mathematical concepts (although this is quite rare)

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

• By conducting an error analysis, the teacher can target specific misunderstandings or missteps, rather than reteaching the entire skill or concept (Fisher & Frey, 2012).
• Students will continue to make procedural errors if they do not receive targeted instruction to address those errors. Simply providing more opportunities to practice solving a given problem is typically not effective (Riccomini, 2014; Lewis & Fisher, 2018; Lin et al., 2025; Nelson & Powell, 2018).
• 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).
• Addressing a student’s conceptual errors might require the use of concrete or visual representations, as well as a great deal of reteaching. Students can often use concrete objects to solve problems that they initially answered incorrectly (Riccomini, 2014; Yetkin, 2003; Lin et al., 2025).
• Without intervention, students have been shown to continue to apply the same error patterns one year later (Cox, 1975).

How to Address Student Errors

After determining the type of error a student is making, the teacher can address the error in one or more of the following ways.

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 reteaching 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 they are doing incorrectly. Simply reteaching 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.

Manipulatives

Manipulatives are objects that students can interact with to gain conceptual understanding (i.e., understand processes or abstract concepts) and to solve problems. Manipulatives can be:

• Physical—examples include base 10 blocks or a geoboard (a small board with nails on which students stretch rubber bands to explore a variety of basic geometry concepts)
• Virtual—examples include clickable dice on an app or integer chips

Manipulatives help a student to represent the mathematical idea they are trying to learn or the problem they are 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 explicitly make the connection between the concrete object and the abstract or 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. 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 other students.

Explicit Instruction

Explicit instruction is a structured instructional method in which educators first provide students with a rationale and clear expectations for learning skills or concepts. Then the educator models, scaffolds, provides opportunities for guided and independent practice as well as engagement, and offers feedback until students have independently mastered the skill or concept.

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.
Scaffolding	The teacher sequences skills that build on each other. The teacher chunks information into smaller parts. The teacher provides prompts. The teacher asks individualized questions to ensure understanding.
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.
Engagement	The teacher provides opportunities for student response. The students engage with peers through group activities.
Feedback	The teacher provides positive and constructive feedback that is informative to students. The teacher modifies activities based on student responses. The teacher provides immediate corrections when possible.

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.

Instructional Tips

• Check for prerequisite skills: Make sure the student has the prerequisite skills needed to solve the problem they have been struggling with. 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 non-examples: 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 non-example of the error pattern to prevent overgeneralization (i.e., incorrectly applying the rule or knowledge to novel situations) and overspecialization (i.e., developing an overly narrow definition of the concept 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.

  Examples and Non-Examples

  Problems 1 and 3 are examples that require regrouping, whereas Problem 2, which does not require regrouping, is a non-example.

  

• 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 failing 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. The student would stop at that point, as opposed to completing the problem, because they know the process from that point forward. The teacher should then continue in the same manner for the remaining problems.

  

  [Stop at this point because you have addressed the error pattern; the student knows how to add fractions]
• Provide ample opportunities for practice: As with modeling, provide a minimum of three to five problems for guided practice, making sure to include a non-example.
• 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 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.

Background

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

Assignment

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.

Background

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’s 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 Strategies

• Collecting data
• Identifying error patterns

Assignment

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.

Background

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

Assignment

1. Read the 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.

Background