Dynamic Mathematics Assessment

Dynamic Mathematics Assessment incorporates elements of three powerful assessment techniques to determine an individual student's mathematics understanding and skill level for any set of mathematics concepts/skills/strategies of interest. By incorporating elements of Concrete-to-Representational-Abstract (CRA) Assessment, Error Pattern Analysis and Flexible Mathematics Interviews, a dynamic evaluation of a student's mathematical understandings is possible.

Dynamic Mathematics Assessment combines principles of CRA Assessment, Error Pattern Analysis, and the Flexible Mathematics Interview to provide teachers with an in-depth yet instructionally relevant picture of any student's mathematical understandings. CRA Assessment includes evaluating a student's ability to demonstrate his/her understanding of mathematics concepts/skills/strategies at the concrete, representational and abstract levels of understanding. Error Pattern Analysis provides teachers insight into how students approach solving mathematics problems using procedures/algorithms and helps them to pinpoint the use of effective and ineffective procedures and possible mathematical misunderstandings or non-understandings. Flexible Mathematics Interviews allow teachers to gain insight into student's mathematical thinking.

A Dynamic Mathematics Assessment begins by having students respond to a set of prompts or items that reflect targeted mathematics concepts/skills/strategies at the Abstract Level of Understanding. Including at least 5-10 items for each target skill is helpful for ensuring reliability: For example, if a target mathematics skill was "division facts 0-9" then the student might respond to a variety of 10 division facts such as:

  8

  4

  6

  9

  2

  6

  4

  3

  8

  2

÷4

÷2

÷3

÷3

÷1

÷3

÷4

÷1

÷2

÷2

After students have responded, then the teacher evaluates their work and determines whether:

1. the student demonstrates mastery of the skill or complete understanding of a concept;
2. the student demonstrates partial mastery of the skill or partial understanding of the concept;
3. the student demonstrates little/no ability or little/no understanding of the concept.

When the student does not demonstrate mastery, then the teacher examines the student's responses for error patterns. Error patterns are consistent mistakes made by a student when responding to the same task (e.g., the student consistently either adds or subtracts instead of dividing; student responses reflect random guessing; the student consistently misses facts where the divisor is not "1" or "2."). The teacher notes any error patterns and determines whether the error pattern indicates conceptual or procedural non-understanding. For example, if the student consistently adds or subtracts instead of dividing or if their responses are random guesses, it is likely that the student does not understand the division process (conceptual non-understanding). Then the teacher "interviews" the student about what they did when they solved the target mathematics concept/skill. Several approaches can be taken when conducting a flexible mathematics interview. Whichever approach is used, the primary purpose is to get a picture of the student's mathematical thinking as it relates to the target concept/skill. When completing a flexible mathematics interview, it is important to incorporate use of concrete materials and drawings to determine whether the student possesses either a concrete or representational/semi-concrete level of understanding. Even though the student initially demonstrates difficulty with the concept/skill at the abstract level of understanding, it is possible that they have a "lower" level of understanding (e.g., concrete or representational/semi-concrete). Determining a student's level of understanding (concrete, representational/semi-concrete, abstract) helps the teacher determine at what level of understanding they must start their instruction if they are to effectively move the student to an abstract level of understanding. Additionally, by providing students with concrete materials and/or allowing them to draw pictures provides them a more tangible way to describe their mathematical understandings, thereby providing the teacher a clearer picture of why their student is having difficulty with the target mathematics concept/skill.

• Target the particular mathematics concepts/skills you are interested in evaluating. It is important to be as specific as possible (e.g., addition sums to 18; addition of fractions with mixed numbers; 2 digit by 2 digit subtraction with regrouping).
• Include at least 5 to 10 items for each target mathematics concept/skill.
• Make sure that the items accurately represent the target concept/skill at the abstract level of understanding (e.g. numbers and symbols).
• Look for error patterns that indicate conceptual and/or procedural non-understanding and make a written record of your observations.
• Incorporate concrete materials and picture drawing when interviewing students about their mathematical thinking.

1. Target the particular mathematics concepts/skills you are interested in evaluating.
2. Develop an abstract level assessment of target concepts/skills that includes at least 5 to 10 items for each target mathematics concept/skill.
3. Evaluate whether students are at a mastery level (90-100% accuracy), instructional level (75%-89%), or frustration level (below 75%) using the abstract level assessment.
4. For concepts/skills that the student is not at a mastery level, conduct an error pattern analysis.
5. Based on your findings, conduct a flexible mathematics interview to get a picture of your student's mathematical thinking and understanding of selected mathematics concepts/skills (include use of concrete materials and/or drawings).
6. Use the information gained to:
   • Select appropriate concepts/skills for instruction, including any prerequisite concepts/skills (e.g., place value).
   • Determine what level of understanding instruction should first incorporate (e.g., concrete, representational/semi-concrete, abstract).
   • Teach conceptual understanding and procedural understanding.

By providing an in-depth picture of a student's mathematical understanding of specific mathematics concepts/skills, a dynamic mathematics assessment helps the teacher better determine how to best teach the student. Students with learning problems often have gaps in mathematical understanding that make it difficult for them to achieve success in mathematics as they proceed through grades K-12. Teachers who are able to gain insight into their students' mathematical thinking as well as their conceptual and procedural knowledge related to the particular mathematic concepts/skills they teach will be better equipped to provide them effective mathematics instruction. By planning instruction that is based on a student's level of proficiency, their level of understanding and their mathematical thinking, a teacher can better pinpoint what prerequisite concepts/skills a student needs to develop, what particular non-successful strategies they are using, and whether they have established conceptual and procedural understanding at the concrete, representational/semi-concrete, or abstract levels. Based on this mathematical learning picture, appropriate instructional goals and instructional practices can be developed for a student.