Supporting student inquiry

Anticipate student thinking

Anticipating students’ responses involves developing considered expectations about how students might mathematically interpret a problem, the array of strategies—both correct and incorrect— that they might use to tackle it, and how those strategies and interpretations might relate to the mathematical concepts, representations, procedures, and practices that the teacher would like his or her students to learn.

Smith & Stein 2018, p. 10

Anticipating student thinking is a vital part of effective mathematics instruction. Anticipating involves:

• envisioning likely student strategies, and
• planning questions and responses to these strategies.

Taking the time to anticipate student thinking before you teach gives you a clearer sense of the mathematics that you need to make visible to your students. Understanding what students are likely to do, gives you greater capacity to help them to make connections between different solutions they might see as being disconnected, as well as enabling you to steer their mathematical thinking towards the learning goal of the lesson.

Before you teach

The best way to anticipate likely student thinking is in collaboration with colleagues. Working with others broadens the scope of likely solution strategies, beyond those that a single teacher could come up with. Actually do the task with your colleagues and, as you do, consider the following:

• What are some common misconceptions which may arise during the task?
• How effective or ineffective are the different strategies that you anticipate students will use?
• How might students mathematically represent the strategies that they use?
• What questions or prompts should we pose in response to the different strategies, so that students are moving closer to the mathematical goal of the lesson?

While you are teaching

Anticipating student strategies prepares you to actively observe and interact with students while they are working on the task. As you observe students, focus your attention on what mathematics you notice students actually using. Converse with students to assess what mathematics students do or don’t understand, then use your prepared questions and prompts to advance students’ thinking towards the mathematical goal of the lesson. Having prepared questions and responses provides you with more time to think of purposeful responses when unanticipated student approaches arise.

Have a go

If you can, complete this activity with at least one other colleague: Select a task you will be using with your students. Work on the task individually and compare your solutions.

Discuss:

• What strategies might the students use to solve the task?
• How might they represent the strategies that they use?
• Are there possible misconceptions that may arise?

Plan questions and prompts in response to the different strategies that you have anticipated. Teach the task!

Reflect: How did anticipating student thinking support you in your role as the teacher?

 

References

Smith, M. S. & Stein, M.K. (2018). 5 Practices for Orchestrating Productive Mathematics Discussions. National Council of Teachers of Mathematics.

Observation

Observing students at work is an important part of your role as a teacher. It is important to look at more than what they are doing to establish how engaged they are, or whether they have completed a task. Teacher attention should be focused on the mathematical ideas that students are exploring, the thinking they are engaging in and where they are going next.

To effectively observe students, you need to:

• Know the development of understanding in the content area that you are teaching.
• Probe to better understand students’ thinking (assessing questions).
• Push students towards the mathematical goal of the lesson (advancing questions).
• Help inconsistencies in the students’ thinking to become apparent.
• Encourage deeper inquiry and looking for generalisations and proof.

Observation involves listening and looking

It is always tempting to start questioning students as soon as you begin to observe their mathematical activity. However, first spend time listening to the students as they discuss their thinking in small groups and watching what it is that they are doing. You will be more likely to ask relevant questions because you will hear the mathematical ideas that the students are discussing and see the strategies that students are using.

Teachers can probe deeper into student understanding, by using prepared questions to assess students understanding and to advance them towards the mathematical goal of the lesson (See Anticipate).

Teacher noticing

Teacher observation requires expertise in teacher noticing, which is more specialised than your everyday noticing. You need a clear understanding of the mathematical learning goal that students are exploring, to be able to notice the mathematical significance in what they say and do. Teachers can learn a lot about students’ mathematical thinking by paying attention to:

• How they use representations
• The logic of their reasoning
• The mathematical meaning implied in their gestures

Teachers should record evidence of these kinds of observations, to inform your interpretations of students’ mathematical understandings. Your observations afford you deeper insight into what a student currently understands, and make you more aware of emergent mathematical ideas which may be just out of their reach.

An observation chart may be useful to keep track of what you notice and consider relevant to the lesson’s mathematical learning goal.

 

References

Imm, K. L., Fosnot, C. T., Dolk, M., Jacob, B., & Stylianou, D. (2012). Learning to support young mathematicians at work: An early algebra resource for professional development. Heinemann.

Mason, J. (2021). Learning about noticing, by, and through, noticing. ZDM Mathematics Education, 53(1), 231-243.

Questioning

Posing purposeful questions involves asking questions that deepen students' understanding of mathematics while providing information about their mathematical thinking.

National Council of Teachers of Mathematics, 2014

Why ask students questions?

• To encourage them to explain their thinking
• To find out what they understand
• To focus their thinking
• To deepen their understanding

When teachers ask students questions, it is an opportunity to encourage them to think deeply about the mathematics they are working on. To ask purposeful questions, you need to understand the mathematical content, as well as plan in advance what questions will best engage students in deep mathematical thinking.

Purposeful questioning can also be scary to students: questioning can generate cognitive disequilibrium when it pushes students beyond their existing understanding, causing confusion and discomfort. But when you use questioning purposefully, you give students chance to grapple with unfamiliar mathematical ideas and new insights emerge. Cognitive reorganisation of the student’s current mathematical understandings occurs and these new connections are added, and new understandings emerge.

reSolve tasks use questioning in a number of different ways and for different purposes throughout the Explore and Connect phases of a task. The only question posed in the Launch is the one which sets students on task. During the Summarise phase, the student may be asked to reflect on what they have learnt during the lesson.

Questioning in the Explore phase

Assessing questions are intended to make a student’s current thinking visible, ensuring that the teacher understands what the student did and why he or she did it. Advancing questions are intended to move students beyond where they currently are, toward the goals of the lesson.
Smith & Stein 2018, p.44

As students explore a task, you can use assessing questions to…

• gather information.
• probe student understanding.
• make the mathematics more visible.
• get students to reflect on and justify their reasoning.

Assessing questions focus on the mathematics that students have produced, as well as what they understand about that mathematics. It is important that teachers actively listen to the students' ideas to establish what they actually understand, in order to help them advance this understanding.

An advancing question is a question you ask before you walk away from a student. It focuses and extends students’ existing mathematical thinking towards the mathematical goal of the lesson.

Questioning in the Connect phase

In the Connect phase, your questions should be crafted around student solutions, to make the mathematics visible to students during the whole class discussion. Teachers intentionally select student solutions which are central to the key mathematical learning goals of the lesson, then use targeted, purposeful questioning to help steer the class’s attention towards seeing explicit connections between the student solutions and the lesson’s mathematical goals.

Questions in the Connect phase can…

• ask students why they used a specific strategy, rather than asking what they did.
• emphasise specific connections between the student solutions and the lesson’s mathematical goals.
• make the mathematics visible to the presenter and the listeners.

 

References

Huinker, D., & Bill, V. (2017). Taking action: Implementing effective mathematics teaching practices in K-grade 5. Reston, Virginia: National Council of Teachers of Mathematics.

Imm, K. L., Fosnot, C. T., Dolk, M., Jacob, B., & Stylianou, D. (2012). Learning to support young mathematicians at work: An early algebra resource for professional development. Heinemann.

National Council of Teachers of Mathematics (2014). Principles to actions: Ensuring mathematics success for all. Reston, VA: National Council of Teachers of Mathematics.

Smith, M. S., Bill, V., & Sherin, M. G. (2020). The five practices in practice: Successfully orchestrating mathematics discussions in your elementary classroom. Reston, VA: National Council of Teachers of Mathematics.

Smith, M. S. & Stein, M.K. (2018). 5 Practices for Orchestrating Productive Mathematics Discussions. National Council of Teachers of Mathematics.