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.
1) Number and Operations, and 2) Geometry and Measurement. According to the NRC (2009), conceptual development within number and operations should focus on students’ development of the list of counting numbers and the use of counting numbers to describe total objects in a given set. It is recommended that teachers provide students with opportunities to “subitize small collections [of objects], practice counting, compare the magnitude [size] of collections, and use numerals to quantify collections” (Frye et al., 2013). Conceptual development in geometry and measurement should support the idea that geometric shapes have different parts that can be described and include activities that model composition and decomposition of geometric shapes.