The Starting Point...
Immerse yourself in the curriculum and supporting documents.
Attain a better understanding of the expectations and the seven mathematical processes by reading about the explanations and rationale in the front matter of the Full-Day Early Learning Kindergarten Program and Grades 1 to 8 Mathematics curriculum. (p11-17)
Look before and beyond the grade you are teaching to see how concepts build upon each other.
Utilize the document provided during our Divisional Meeting which shows the Math Curriculum on a continuum from grade to grade.
There is a wealth of resources that can offer extra insight into the mathe- matics itself and can help to identify and connect the key mathematical concepts.
Some of these include the Guides to Effective Instruction in Mathematics and the works of Dr. Marian Small, Catherine Twomey Fosnot and John A. Van de Walle.
Your professional learning journey will be most effective when you delve into mathematical ideas with colleagues and together inquire about how your understanding impacts your related teaching.
Culture of Classroom Discourse
(Lucy West)
Teacher-facilitated “math talk” significantly increases children’s growth in understanding of mathematical concepts. Knowledgeable educators recognize that although children may have a beginning understanding of mathematical concepts they often lack the language to communicate their ideas. By modelling and fostering math talk throughout the day and across various subject areas, educators can provide the math language that allows students to articulate their ideas.
It is also important to encourage talk among students as they explain, question and discuss their strategies while co-operatively solving problems. In order to facilitate mathematical thinking rather than direct it, knowledgeable educators recognize when student thinking is developing or stalled. If it is developing, the educator observes but leaves the students to work through their thinking (Sarama & Clements, 2009, p. 325). If it is stalled, probing questions can be asked that provoke thinking about alternate ways to perceive the problem.
It is also important to encourage talk among students as they explain, question and discuss their strategies while co-operatively solving problems. In order to facilitate mathematical thinking rather than direct it, knowledgeable educators recognize when student thinking is developing or stalled. If it is developing, the educator observes but leaves the students to work through their thinking (Sarama & Clements, 2009, p. 325). If it is stalled, probing questions can be asked that provoke thinking about alternate ways to perceive the problem.
After students have worked through solving a problem, educators facilitate consolidation time (either with individual students or with small groups or large groups) in order to allow students to talk about their thinking. This consolidation time is sometimes referred to as the third part of the three-part lesson in mathematics. As educators value a variety of strategies and solutions, they guide students to make connections between them, to recognize how the thinking relates to the key mathematical concepts and to make further conjectures and generalizations.
Five productive talk moves ...to create meaningful mathematics discussions.
Gives the educator an opportunity to embed mathematics vocabulary
1. Revoicing – Repeating what students have said and then asking for clarification
"So you’re saying it’s an odd number?"
2. Repeating – Asking students to restate someone else’s reasoning
"Can you repeat what he just said in your own words?"
3. Reasoning – Asking students to apply their own reasoning to someone else’s reasoning
"Do you agree or disagree and why?"
4. Adding on – Prompting students for further participation
"Would someone like to add something more to that?"
5. Waiting – Using wait time "Take your time ... We’ll Wait .."
(Chapin, O’Connor & Anderson, 2009, p.13)
Guidelines for Whole-Class Math-Talk
Explain: “This is my solution/strategy ...” “I think _____ is saying that ...”
• Explain your thinking and show your thinking.
• Rephrase what another student has said.
Agree with reason: “I agree because ...”
• Agree with another student and describe your reason for agreeing.
• Agree with another student and provide an alternate explanation.
Disagree with reason: “I disagree because ...”
• Disagree with another student and explain or show how your thinking/ solution differs.
Build on: “I would like to build on that idea...”
• Build on the thinking of another student through explanation, example, or demonstration.
Go beyond: “This makes me think about ...” “Another way to think about this is ...”
• Extend the ideas of other students by generalizing or linking the idea to another concept.
Wait time:
• Wait to think about what is being said after someone speaks (try five seconds).
The Value of Student Interaction
In the math reform literature, learning math is viewed as a social endeavour.1,2 In this model, the math classroom functions as a community where thinking, talking, agreeing, and disagreeing are encouraged. The teacher provides students with powerful math problems to solve together and students are expected to justify and explain their solutions. The primary goal is to extend one’s own thinking as well as that of others.3
Powerful problems are problems that allow for a range of solutions, or a range of problem-solving strategies. Math problems are powerful when they take students beyond the singular goal of computational mastery into more complex math thinking. Research has firmly established that higher-order questions are correlated with increased student achievement, particularly for conceptual understanding.
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