Identifying Error Patterns and Diagnosing Misconceptions: Part 1

By Dr. Yetunde Zannou, RME Post Doctoral Fellow

Identifying student error patterns, what they do, is the first step in diagnosing student misconceptions, the why behind the errors. Knowing what students do and most importantly why they do it yields invaluable information that teachers can use to guide instruction and bridge gaps in student understanding.

Teachers can identify student error patterns and diagnose misconceptions using several tools such as student work, direct observation, and interviews. Each provides insight into how students think. In addition, certain assessments are designed to gather specific data about student error patterns and misconceptions based on the answers they choose. Classroom evidence and diagnostic assessments complement one another and contribute to the level of confidence a teacher can have in making instructional decisions to meet students’ demonstrated learning needs. This multi-part series will explore two main categories of tools teachers can use to identify error patterns and diagnose misconceptions: classroom evidence and diagnostic assessments.

Using Classroom Evidence to Identify Error Patterns and Diagnose Misconceptions: Student Work

Classroom evidence consists of student work, direct observation data, and interview data. Student work is identified as assignments (e.g., classwork, homework, quizzes, tests, projects, and portfolios) that students submit as evidence of learning stated objectives. Direct observations involve both listening to student responses within small and large group contexts and watching how they solve problems. Interviews probe students’ understanding through questioning about their thinking and can happen spontaneously or can be scheduled. Each type has unique strengths and can be used together to form a robust assessment system.

Student work is commonly used to understand students’ skill and accuracy in performing mathematical procedures, their conceptual understanding, and their ability to apply that understanding in novel situations. In some cases, student work is also used to determine student readiness for new concepts and advanced learning activities. Though student work serves many purposes in the mathematics classroom, the following considerations can help maximize its use in identifying error patterns and diagnosing misconceptions:

Vary problem sets in specific ways to reveal and confirm error patterns. Student work is often used to determine whether or not a student “got it”. As a tool for identifying error patterns and diagnosing misconceptions, activity selection and what specifically you want to know about student understanding take center stage. In other words, if you want to know if students can accurately apply an algorithm, student work might consist of calculations. To identify error patterns and diagnose misconceptions, select problems that are likely to reveal and confirm a variety of specific errors and misconceptions. Choose problems that vary slightly in order to ferret out where students may struggle.

For example, if you want to determine if students can correctly subtract three digit numbers, select problems that: (a) do not require regrouping, (b) require regrouping from the tens or hundreds place, and (c) require regrouping from both the tens and hundreds place. A common misconception that students have with regrouping is treating each digit in a number independently without regard to its position in the minuend or subtrahend. Students with this misconception may subtract the smaller place value digit from the larger place value digit (e.g., To evaluate 742 – 513, the student subtracts 2 from 3 in the ones place because the 2 is smaller than 3) to get around regrouping. Including problems like this and looking for this error pattern can help teachers to see the misconception and teach students about the relationship between the number and place value. Ashlock (2010) provides a wealth of examples to illustrate how slightly varying problem types can help to identify and confirm error patterns in computation.

Maximize your review time by carefully selecting problems. In higher grades especially, student work tends to cover a variety of topics and rarely focuses on a single concept. Balancing conceptual focus and cumulative review can be challenging. When using student work as a diagnostic tool (different from using a diagnostic assessment), less is more! If the goal is to identify gaps and make adjustments, the fewer and more strategic the problem set, the better. Assigning fewer, more strategic problems regularly provides teachers with timely information about emergent proficiencies and struggles when evaluating student work. This information can be gathered rather quickly and used to help teachers to group students accordingly, target common gaps in understanding, and guide instruction in general. In a classroom where student work is used as a diagnostic tool, cumulative assignments can be given periodically.

Use student work to help focus further steps to identify and diagnose learning needs. It can be challenging to track student progress on a single concept or procedure over time through student work alone because assignments rarely revisit the same concept in the manner over an extended period of time. As such, a comprehensive assessment system is the best approach to identify error patterns and diagnose student misconceptions. Student work just may be a good first step! Other tools will be discussed throughout this series such as gathering classroom through direct observations and interviews, and later, diagnostic and progress monitoring assessments. As a first step in an overall assessment program, student work can provide teachers with focus—identify which students you may need to pay close attention to and what to look for in their work, behavior, and responses.


Ashlock, R. (2010). Error patterns in computation: Using error patterns to help each student learn (10th ed.). Boston, MA: Allyn & Bacon.

National Council of Teachers of Mathematics. (1999). Mathematics assessment: A practical handbook for grades 9-12. Reston, VA: Author.

Mastering Explicit Instruction - Part 4

By Dr. Deni Basaraba, RME Assessment Coordinator

We have come to an end on our series on explicit instruction (Please see our previous posts for more information!). Van de Walle (2013) characterizes explicit instruction as highly-structured, teacher-led instruction on a specific strategy. He explains how this approach can help uncover or make overt the thinking strategies that support mathematical problem solving for students with disabilities.

Here are our final thoughts on the elements necessary to carry out explicit instruction in the classroom.

Monitoring the performance of all students closely to verify students’ mastery of the content and help determine whether adjustments to instruction need to be made in response to student errors. Closely monitoring student performance can also help you determine which students are confident enough in their knowledge of the content to respond when signaled and which students are cueing off their peers by waiting for others to respond first.

Providing specific and immediate affirmative and corrective feedback as quickly as possible after the students’ response to help ensure high rates of success and reduce the likelihood of practicing errors. Feedback should be specific to the response students provided and briefly descriptive so that students can use the information provided in the feedback to further inform their learning (e.g., “Yes, we can use the written numeral 2 to show that we have two pencils”).

Delivering instruction at a “perky” pace to optimize instructional time, the amount of content that can be presented, the number of opportunities students have to practice the skill or strategy, and student engagement and on-task behavior. The pacing of content delivery should be sufficiently brisk to keep students engaged while simultaneously providing a reasonable amount of “think-time,” particularly when students are learning new material.

Providing distributed and cumulative practice opportunities to ensure that students have multiple opportunities to practice skills over time. Cumulative practice provides an opportunity to include practice opportunities that address both previously and newly learned content, skills, and strategies, facilitating the integration of newly learned knowledge and skills with previously learned knowledge and skills and supporting retention and fostering automaticity.

Thanks for joining us on our journey with explicit instruction! Please feel free to leave comments below with your thoughts or questions!

Baker, S. K., Gersten, R., & Lee, D. (2002). A synthesis of empirical research on teaching mathematics to low-achieving students. The Elementary School Journal, 103, 51-73.

Carnine, D. W. (1997). Instructional design in mathematics for students with learning disabilities. Journal of Learning Disabilities, 30, 130-141.

Doabler, C. T., Cary, M. S., Jungjohann, K., Clarke, B., Fien, H., Baker, S., Smolkowski, K., & Chard, D. (2012). Enhancing core mathematics instruction for students at risk for mathematics disabilities. Teaching Exceptional Children, 44, 48-57.

Van De Walle, J.A., Karp, K.S., & Bay-Williams, J.M. (2013). Elementary and middle school mathematics: Teaching developmentally, 8th edition. Boston: Pearson.

Witzel, B. S., Mercer, C. D., & Miller, M. D. (2003). Teaching algebra to students with learning difficulties: An investigation of an explicit instruction model. Learning Disabilities Research & Practice, 18, 121-131.

Mastering Explict Instruction - Part 3

By Dr. Deni Basaraba, RME Assessment Coordinator

We have been doing a series on how all students, particularly those who are struggling, benefit from the provision of explicit instruction (Baker, Gersten, & Lee, 2002; Carnine, 1997; Doabler et al., 2012; Witzel, Mercer, & Miller, 2003). Van de Walle (2013) characterizes explicit instruction as highly-structured, teacher-led instruction on a specific strategy. He explains how this approach can help uncover or make overt the thinking strategies that support mathematical problem solving for students with disabilities.

We started a list of the methods and elements of explicit instruction (see the previous two posts!). To continue this series of blogs, here are some additional elements of explicit instruction as well as examples and/or steps to carrying out these recommendations.

Using clear, concise, and consistent language that is not ambiguous, does not introduce irrelevant or unnecessary information into instruction, and is consistent across teacher models, whole group practice, and to elicit student responses so that students know what they are expected to do. Here are some suggestions to help with this element: Train interventionists to explain math content Include math concepts, vocabulary, formulas, procedures, reasoning and methods Use clear language understandable to students

Providing a range of examples and non-examples to help students determine when and when not to apply a skill, strategy, concept, or rule. Including a wide-range of positive examples during instruction provides students with multiple opportunities to practice the using the target strategy or skill while including non-examples can help reduce the possibility that students will use the skill inappropriately or over-generalize it to inappropriate mathematics problems. For example, here are some examples and non-examples of an algebraic expression: Example: Suppose we wanted to write an expression to find the total cost of movie tickets for different groups of people, given that tickets cost $7 each. We can define the variable "p" to represent the number of people in a group. Then the algebraic expression 7 x p or 7p represents the total cost of movie tickets for a group of p people. Non-example:The expression 550 + 20 is not an algebraic expression because it does not contain at least one variable.

Requiring frequent responses by relying minimally on individual student turns (e.g., asking students to raise their hand if they have the correct response) and relying more on the use of questioning to elicit various types of responses such as oral responses (e.g., having pre-identified partners or groups of students such as boys/girls, 1s/2s, apples/oranges respond), written responses, and/or action responses (e.g., thumbs up/thumbs down). Increasing the number of opportunities students have to respond during the lesson promotes a high level of student-teacher interaction, helps keep students engaged with the content, provides multiple opportunities for student practice with and elaboration on the newly learned content, and can help you monitor student understanding.

We have four more suggestions for mastering explicit instruction! Stay tuned!!

Baker, S. K., Gersten, R., & Lee, D. (2002). A synthesis of empirical research on teaching mathematics to low-achieving students. The Elementary School Journal, 103, 51-73.

Carnine, D. W. (1997). Instructional design in mathematics for students with learning disabilities. Journal of Learning Disabilities, 30, 130-141.

Doabler, C. T., Cary, M. S., Jungjohann, K., Clarke, B., Fien, H., Baker, S., Smolkowski, K., & Chard, D. (2012). Enhancing core mathematics instruction for students at risk for mathematics disabilities. Teaching Exceptional Children, 44, 48-57.

Van De Walle, J.A., Karp, K.S., & Bay-Williams, J.M. (2013). Elementary and middle school mathematics: Teaching developmentally, 8th edition. Boston: Pearson.

Witzel, B. S., Mercer, C. D., & Miller, M. D. (2003). Teaching algebra to students with learning difficulties: An investigation of an explicit instruction model. Learning Disabilities Research & Practice, 18, 121-131.

Mastering Explicit Instruction - Part 2

By Dr. Deni Basaraba, RME Assessment Coordinator

A couple of weeks ago, we discussed how all students, particularly those who are struggling, benefit from the provision of explicit instruction (Baker, Gersten, & Lee, 2002; Carnine, 1997; Doabler et al., 2012; Witzel, Mercer, & Miller, 2003). Van de Walle (2013) characterizes explicit instruction as highly-structured, teacher-led instruction on a specific strategy. He explains how this approach can help uncover or make overt the thinking strategies that support mathematical problem solving for students with disabilities. 

We started a list of the methods and elements of explicit instruction. To continue this series of blogs, here are some additional elements of explicit instruction as well as examples and/or steps to carrying out these recommendations.

Beginning each lesson with a clear statement of the objectives and goals for that lesson so that students know not only what they are going to learn but why it is important and how it relates to other skills and strategies they have already learned. For example: Today, we are going to continue our work with variables and expressions. You are going to learn how parentheses are used in expressions. You need this knowledge in order to solve algebraic equations.

Reviewing prerequisite knowledge and skills prior to starting instruction that includes a review of relevant information not only to ensure that students’ have the prior knowledge and skills needed to learn the skill being taught in the lesson but also to provide students an opportunity to link the new skills with other related skills and increase their sense of success. For example: First, let’s review. Tell your partner what a variable is. (A variable is a symbol that represents a number.) Yes, a variable is a symbol that represents a number. Look at these expressions (5 + x; t – 10). In the first expression, what is the variable? (X) Yes, x is a symbol that represents a number. Everyone, what is the variable in the second expression? (T) Let’s look again at the definition of an expression...

Modeling the desired skill or strategy clearly to demonstrate for students what they will be expected to do so and can see first-hand what a model of proficient performance looks like. When completing these models, think aloud for students so that each step of the strategy or skill you perform is clear and to clarify the decision-making processes needed to complete the procedure or solve the problem. For example: Look at this expression: 3 × ( 4 - 2). When an expression contains more than one operation, parentheses can be used to show which computation should be done first. So, when we have an expression, we first look for parentheses and do the operation inside the parentheses. In this problem, 4 - 2 is inside the parentheses, so I will do that operation first...

Providing guided and supported practice by regulating the difficulty of practice opportunities from easier to more challenging and providing higher levels of guidance and support initially that is gradually decreased as students demonstrate success. For example: Let’s do some problems together. Please stay with me so we can do these items correctly. Write this expression on your paper, but don’t solve it. Do we do the operations inside or outside of the parentheses first...  Write this expression on your paper. Do we do the operations inside or outside of the parentheses first? Inside. Find the value of the expression. (After several examples where the teacher slowly gives less help) Copy this expression and find the value of the expression. Don’t forget . . . parentheses first...

Baker, S. K., Gersten, R., & Lee, D. (2002). A synthesis of empirical research on teaching mathematics to low-achieving students. The Elementary School Journal, 103, 51-73.

Carnine, D. W. (1997). Instructional design in mathematics for students with learning disabilities. Journal of Learning Disabilities, 30, 130-141.

Doabler, C. T., Cary, M. S., Jungjohann, K., Clarke, B., Fien, H., Baker, S., Smolkowski, K., & Chard, D. (2012). Enhancing core mathematics instruction for students at risk for mathematics disabilities. Teaching Exceptional Children, 44, 48-57.

Van De Walle, J.A., Karp, K.S., & Bay-Williams, J.M. (2013). Elementary and middle school mathematics: Teaching developmentally, 8th edition. Boston: Pearson.

Witzel, B. S., Mercer, C. D., & Miller, M. D. (2003). Teaching algebra to students with learning difficulties: An investigation of an explicit instruction model. Learning Disabilities Research & Practice, 18, 121-131.