Open-Ended Math Questions Reveal Student Thinking

Careful, intentional, and mindful questioning is one of the most powerful tools a skillful teacher possesses (Costa & Kallick, 2000).  Teachers can use open-ended questions during math instruction or assessments to learn how students are problem-solving.

A question is considered open-ended when it is framed in such a way that a variety of responses or approaches are possible (Small, 2009).  As shown in Figure 1, open-ended math questions are designed to uncover student understanding and misunderstandings. The responses are used to inform instruction rather than to make evaluative decisions (Rose & Arline, 2009).

Teachers analyze students’ responses to questions in order to learn how they think.  The responses reveal what students know and how they apply that knowledge.  Teachers then use this information to design instruction that supports student learning.  Additionally, open-ended questions provide opportunities for students to respond and contribute at their respective levels.  This is especially important for struggling students, since they are likely to be passive learners (Lovin, Kyger, & Allsopp, 2004).

Figure 1 illustrates how asking an open-ended question identifies students’ understanding along a continuum.  Teachers use this information to determine where to begin new instruction.

openedendedchart

Figure 1. Continuum of student responses and instructional decisions.

The examples below from Instructional Consultation and Assessment Team (ICAT) Manual Book 3 (Gravois, Gickling, & Rosenfield, 2011, pp. 82-84) illustrates examples of open-ended questions what the responses reveal about students’ understanding of a variety of math concepts.

Specific Assessment: Mathematical Thinking

Good Math Questions Share the Following Features

  • Begin with what the student knows
  • Engage students in the math skills and thinking that you are trying to assess
  • Require more than facts to resolve
  • Are open-ended and have more than one answer

Consider the following types of mathematical open-ended questions:

Number Sense

  1. Jack has 12 pets.  Some are dogs and some are cats.  How many dogs could Jack have, and how many cats could Jack have?
    This question can show you:

    • Can the student count 12 items?
    • Does the student know how to begin?
    • Does the student start with 12 and count back, or does she start with a number and add on to 12? (i.e. 10 or 5)
    • Does the student need counters or can he solve it mentally?
    • Does the student demonstrate knowledge of an operational algorithm?
  1. I made a two-color tower with 6 blocks.  What might my tower look like?
    This question can show you:

    • What factors of 6 does the student use?
    • Does the student know how to begin the problem?
    • Does the student use counters or can she solve the problem mentally?
    • Does the student demonstrate knowledge of an operational algorithm?

 

The following video clips show Marilyn Burn using open-ended questions during math assessments to reveal student thinking and knowledge of concepts.

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(Burn, 1993a)

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(Burn, 1993b)

Note to Readers: Please share how you are using open-ended questions in your classes or ask a question related to this topic by posting a comment at the end of this article.

References

Burn. M. (1993a).  Mathematics: Assessing, understanding Cena. Retrieved from  https://www.youtube.com/watch?v=_ofQ_WnQiZ4

Burn, M. (1993b). Mathematics: Assessing, understanding Jonathan. Retrieved from https://www.youtube.com/watch?v=1puQxclB2aw#t=10

Costa, A. L., & Kallick, B. (Eds.). (2000). Activating and engaging habits of mind. Alexandria, VA:  Association for Supervision and Curriculum Development.

Gravois, T., Gickling, E., & Rosenfield, S. (2011).  ICAT manual book 3. Baltimore, MD: ICAT Publishing.

Lovin, A., Kyger, M., & Allsopp, D. (2004). Differentiation for special needs learners.     Teaching Children Mathematics, 11, 158-167.

Rose, C., & Arline, C. (2009). Uncovering student thinking in mathematics, Grades 6-12. Thousand Oaks, CA: Corwin Press.

Small, M., (2009). Good questions: Great ways to differentiate mathematics instruction. New York, NY: Teachers College Press.

The following resources are available for check out from the
T/TAC William & Mary Library
     goodquest Good Questions for Math Teaching:
Why Ask Them and What to Ask – Grades 5-8
Authors: Lainie Schuster and Nancy Canavan Anderson
Call Number CMT103
 differentmath Good Questions Great Ways to
Differentiate Mathematics Instruction
Author: Marian Small
Call Number CMT116
 youngmath Young Mathematicians at Work:
Constructing Multiplication and Division

Authors: Catherine Twomey Fosnot and Maarten Dolk
Call Number CMT100
 studenthinking Uncovering Student Thinking in Mathematics
Grades 6-12
Authors: Cheryl M. Rose and Carolyn B. Arline
Call Number CMT111
 investmath Investigations, Tasks, and Rubrics to Teach and Assess Math
Grades 1-6

Authors: Pat Lilburn and Alex Ciurak
Call Number CMT128

 

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