The classroom becomes

a workshop ...

“ ... as learners investigate together.

It becomes a mini- society – a community

of learners engaged in mathematical

activity, discourse and reflection. Learners

must be given the opportunity to act as

mathematicians by allowing, supporting

and challenging their ‘mathematizing’

of particular situations. The community

provides an environment in which individual

mathematical ideas can be expressed and

tested against others’ ideas.…This enables

learners to become clearer and more

confident about what they know and

understand.”

(Fosnot, 2005. p. 10)

1. ANTICIPATE STUDENT THINKING

An important part of planning a lesson is engaging in solving the lesson problem

in a variety of ways. This enables teachers to anticipate student thinking and the

multiple ways they will devise to solve the problem. This also enables teachers to

anticipate and plan the possible questions they may ask to stimulate thinking and

deepen student understanding.

2. LINK TO LEARNING GOALS

Learning goals stem from curriculum expectations. Overall expectations (or a cluster

of specific expectations) inform teachers about the questions to ask and the problems

to pose. By asking questions that connect back to the curriculum, the teacher helps

students centre on these key principles. During the consolidation phase of the

three-part lesson (see pages 7 and 8), students are then better able to make

generalizations and to apply their learning to new problems.

Linking to Learning Goals

Example for the big idea The same object can be described by using different

measurements.

Teacher’s learning goal: To make a connection between length, width, area and

multiplication.

Problem question: A rectangle has an area of 36 cm2. Draw the possible rectangles.

Possible questions:

• As you consider the shapes you made, what are the connections of the length

of the sides to the total area?

• If you know the shape is a rectangle, and you know the total area and the length of

one side, what ways can you think of to figure out the length of the other three sides?

3. POSE OPEN QUESTIONS

Effective questions provide a manageable challenge to students – one that is at their

stage of development. Generally, open questions are effective in supporting learning.

An open question is one that encourages a variety of approaches and responses.

Consider “What is 4 + 6?” (closed question) versus “Is there another way to make 10?”

(open question) or “How many sides does a quadrilateral figure have?” (closed question)

versus “What do you notice about these figures?” (open question). Open questions

help teachers build student self-confidence as they allow learners to respond at their

own stage of development. Open questions intrinsically allow for differentiation.

Responses will reveal individual differences, which may be due to different levels of

understanding or readiness, the strategies to which the students have been exposed

and how each student approaches problems in general. Open questions signal to students

that a range of responses are expected and, more importantly, valued. By contrast,

yes/no questions tend to stunt communication and do not provide us with useful

information. A student may respond correctly but without understanding.

Invitational stems that use plural forms and exploratory language invite reflection.

Huinker and Freckman (2004, p. 256) suggest the following examples:

As you think about… As you consider…

Given what you know about… In what ways…

In regard to the decisions you made… In your planning…

From previous work with students… Take a minute…

When you think about…

4. POSE QUESTIONS THAT ACTUALLY NEED TO BE ANSWERED

Rhetorical questions such as “Doesn’t a square have four sides?” provide students

with an answer without allowing them to engage in their own reasoning.

5. INCORPORATE VERBS THAT ELICT HIGHER LEVELS OF BLOOM’S TAXONOMY

Verbs such as connect, elaborate, evaluate and justify prompt students to communicate

their thinking and understanding, to deepen their understanding and to extend

their learning. Huinker and Freckman (2004, p. 256) provide a list of verbs that elicit

specific cognitive processes to engage thinking:

observe evaluate decide conclude

notice summarize identify infer

remember visualize (“see”) compare relate

contrast differ predict consider

interpret distinguish explain describe

6. POSE QUESTIONS THAT OPEN UP THE CONVERSATION TO INCLUDE OTHERS

The way in which questions are phrased will open up the problem to the big ideas

under study. The teacher asks questions that will lead to group or class discussions

about how the solution relates to prior and new learning. Mathematical conversations

then occur not only between the teacher and the student, but also between students

within the classroom learning community.

7. KEEP QUESTIONS NEUTRAL

Qualifiers such as easy or hard can shut down learning in students. Some students

are fearful of difficult questions; others are unchallenged and bored by easy questions.

Teachers should also be careful about giving verbal and non-verbal clues. Facial expressions,

gestures and tone of voice can send signals, which could stop students from

thinking things through.

8. PROVIDE WAIT TIME

When teachers allow for a wait time of three seconds or more after a question, there

is generally a greater quantity and quality of student responses. When teachers provide

wait time, they find that less confident students will respond more often; many students

simply need more time than is typically given to formulate their thoughts into words.

Strategies like turn and talk, think-pair-share and round robin give students time to

clarify and articulate their thinking. (For strategies to maximize wait time, See A Guide to

Effective Literacy Instruction, Grades 4 to 6 – Volume 1 (Part 2, Appendix). The Guide

offers tips for using these strategies in the “Listening and Learning from my Peers”

section on page 134.)

(This tip list has been drawn from Baroody, 1998, pp. 17–18. See also A guide to effective instruction innmathematics, Kindergarten to Grade 6 – Volume Two: Problem solving and communication, pp. 32–33.)