Careful, intentional, and mindful questioning is one of the most powerful tools a skillful teacher possesses (Costa & Kallick, 2000). Teachers can use openended questions during math instruction or assessments to learn how students are problemsolving.
A question is considered openended when it is framed in such a way that a variety of responses or approaches are possible (Small, 2009). As shown in Figure 1, openended 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, openended 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 openended question identifies students’ understanding along a continuum. Teachers use this information to determine where to begin new instruction.
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. 8284) illustrates examples of openended 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 openended and have more than one answer
Consider the following types of mathematical openended questions:
Number Sense
 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?
 I made a twocolor 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?
Note to Readers: Please share how you are using openended 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, 158167.
Rose, C., & Arline, C. (2009). Uncovering student thinking in mathematics, Grades 612. 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 

Good Questions for Math Teaching: Why Ask Them and What to Ask – Grades 58 Authors: Lainie Schuster and Nancy Canavan Anderson Call Number CMT103 

Good Questions Great Ways to Differentiate Mathematics Instruction Author: Marian Small Call Number CMT116 

Young Mathematicians at Work: Constructing Multiplication and Division Authors: Catherine Twomey Fosnot and Maarten Dolk Call Number CMT100 

Uncovering Student Thinking in Mathematics Grades 612 Authors: Cheryl M. Rose and Carolyn B. Arline Call Number CMT111 

Investigations, Tasks, and Rubrics to Teach and Assess Math Grades 16 Authors: Pat Lilburn and Alex Ciurak Call Number CMT128 
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