Pose Questions to Elicit Student Thinking

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James Brickwedde, Project for Elementary Mathematics

Many teachers find it easy to pose questions and ask students to describe their strategies; it is more challenging pedagogically to engage students in genuine mathematical inquiry and push them to go beyond what might come easily for them. (Kazemi & Stipek, 2001, p. 60)

The 2022 Minnesota Mathematics Academic Standards (still in the rulemaking process) incorporates the eight instructional practices articulated in Principles to Actions (NCTM, 2014). The Minnesota STEM Teacher Center website new professional development materials for mathematics has clustered these eight practices into two groups. The top three practices in figure one – Establishing goals, Implement tasks, and Build procedural fluency from conceptual understanding – form Cluster One: Planning & Assessment. The remaining five, captured in figure one under the Facilitating meaningful mathematical discourse are grouped in Cluster Two: Instruction. This article focuses on two of the five in this second cluster; the relationship between posing purposeful questions to elicit and use evidence of student thinking. 

Figure One

Many are familiar with the elements of productive discourse practices that have been outlined in Smith & Stein’s 5 practices for orchestrating productive mathematical discourse (2011). These were referenced in last month’s MathBits article on Teacher Noticing.

  • Anticipating possible student thinking
  • Monitoring actual student thinking 
  • Selecting a subset of student thinking to share
  • Sequencing instances of student thinking to frame the discussion
  • Connecting different instances of student thinking to highlight key mathematical ideas 

Posing questions or prompts is a layered process. The timing of questions within a lesson is one factor (Jacobs & Ambrose, 2008; Jacobs & Empson, 2016). The questions used before students begin to work, while students are working, and after answers are given influence the reflective depth of the classroom discourse. The depth of the questions has been found to impact student achievement. While it is important to ask students to explain their thinking in how a task was solved, that is not enough. It is the pressing for details of that thinking that impacts student achievement the most. (Kazemi & Stipek, 2001; Webb, et al., 2014). Eliciting students’ thinking provides the teacher with the ability to attend to the details of the work, interpret the status of that thinking process, and make decisions on how to respond in-the-moment so that the discourse is a more intentional and purposeful process. A teacher’s questioning techniques influences the depth of knowledge students reflect upon as well as the quality of the formative assessment that is available to base next step decision making.

NCTM has articulated four social norms that support mathematical inquiry which many teachers have come to accept within their classrooms.

    1. students describe their thinking
    2. students find multiple ways to solve problems, and they describe their strategies to their classmates and teacher
    3. students can make mistakes, which are a normal part of the learning process; and
    4. students collaborate to find solutions to problems.

Tell me how you solved that? This is a question that allows a student to describe their thinking. What may result, however, is a procedural explanation of the steps taken to solve it. Kazemi & Stipek (2001) talk about high press questioning that deepens the quality of the explanations within the learning process. They describe four mathematical norms that press students to clarify, elaborate, and justify their reasoning for their own reflective purposes as well as for the benefit of the learning community who are listening to the explanations. Mathematical norms capture the set of expectations about what constitutes mathematical thinking. What contrasts the social norms from the mathematical norms is that mathematical norms require…

    1. An explanation consists of a mathematical argument, not simply a procedural description or summary
    2. Mathematical thinking involving understanding relations among multiple strategies
    3. That errors provide opportunities to reconceptualize a problem, explore contradictions in solutions, or pursue alternative strategies; and
    4. Collaborative work that involves individual accountability and reaching consensus through mathematical argumentation. (p.59)

The scenario used with students in Kazemi and Stipek’s study was how eight friends would share 12 brownies equally and with all the brownies used up, but then an additional nine brownies were included with the arrival of the host’s mother and those brownies were shared equally. The question was how much brownie did each student get? An initial prompt of Tell me how you solved the problem, elicits the opening procedural explanation of steps. Pressing further now influences how reflective the students are about their thinking and how intuitive ideas become explicit ideas that can be used to explore the underlying mathematical structure. 

Examples of pressing for details include requiring the students to provide a mathematical explanation that can be used to compare and contrast with other strategies used or justify various mathematical choices that might also have been used. The teacher questions below are largely extracted from the transcripts in Kazemi and Stipek’s article. 

    • Explain why you did it in half? – Gets the student to clarify what is meant about a mathematical word choice and a choice of decision making.
    • What does that mean if there are eight halves? – Having the student extend the meaning for more mathematical precision
    • Where in your [drawings, equation, chart…] does it show what you are claiming? – Allows students to justify their actions by triangulating verbal, graphical, and numerical strategies 
    • I noticed that you [made a particular choice]. Why did you choose that particular option? You could have done [other possible options]. – Allows the student to justify their explicit choice and opens the conversation around possible efficiencies that transfer to other contexts.
    • What can you tell me about their drawings to represent 1/2 plus 1/6 equals 4/6? – Invites listeners in the sharing process to reflect on what was just presented and in doing so through paraphrasing, restate the justification through the use of the visual evidence. 
    • What did they use or do that was different from what you might have done? – Engages the audience to compare similarities and differences among strategies
    • Okay, so you have two different answers. Could you write them down so people can see it? Class, I’d like you to respond to what they’ve written up here. She says it either could equal 6/8 or 1/8. – Invites the presenters along with their peers to explore an error or a point of confusion as a point of problem solving. It pushes all students to engage in mathematical analysis to understand the source of the error/point of confusion. Mistakes are used as opportunities for students to think further.
    • Why don’t you agree with that? or, what did you just hear [student] say that makes sense to you? – Directs students to articulate their mathematical reasoning by comparing and contrasting decision making between solutions

The intent of the layered questions is to move beyond surface explanations to the mathematical reasoning of the students, to explore errors constructively and in full, and to foster reflection upon one’s mathematical choices. As the researchers state, sustained exchanges allow for but do not ensure conceptual thinking. However, the importance is for the teacher not to be satisfied with surface level explanations and to press for the details of the decision-making and have students justify with evidence why their solution does or does not work. It’s in these classrooms where student achievement is documented to be higher than those who stop with the general explanation and move on. 

Inherent in the questions noted above is the teacher and students use and connect mathematical ideas across the various representations that are the products of the students’ work, which is one of the other key instructional practices noted in figure one. We will explore the use of multiple models in another MathBits.  

References

Jacobs, V.R., & Ambrose, R. (2008). Making the most of story problems: Honoring students’ solution approaches helps teachers capitalize on the power of story problems. No more elusive train scenarios! Teaching children mathematics, 15, 260-266.

Jacobs, V.R. and Empson, S.B. (2016). Responding to children’s mathematical thinking in the moment: An emerging framework of teaching moves. ZDM Mathematics Education 48:185–197.

Kazemi, E. & Stipek, D. (2001). Promoting conceptual thinking in four upper-elementary mathematics classrooms, The Elementary School Journal. 102 (1) pp. 59-80.

National Council of Teachers of Mathematics. (2014). Principles to actions. National Council of Teachers of Mathematics: Reston, VA.

Smith, M.S., and Stein, M.K. (2011). 5 Practices for orchestrating productive mathematics discussions. Reston, VA : Thousand Oaks, CA: National Council of Teachers of Mathematics ; Corwin.

Webb, N.M., Franke, M.L., Ing, M., Wong, J., Fernandez, C.H.,  Shin, N., and Turrou, A.C. (2014). Engaging with others’ mathematical ideas: Interrelationships among student participation, teachers’ instructional practices, and learning, International Journal of Educational Research, 63, pp 79-93.