Link to Problem:
Developing conceptual explanations including using the problem context to make explanation experientially real
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Model providing a mathematical explanation. Use the context of the problem, not just the numbers.
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Re-voice, and extend an explanation using the problem context. Expect mathematical reasons not “tidying” 19+7=20+6 because 6+1=7 and 19+1=20.
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Questions to scaffold students to extend their explanations to include the problem context and what they did to the numbers mathematically.
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Model and support the use of questions which clarify an explanation. What do you mean by? What did you do in that bit? Can you show us what you mean by? Could you draw a picture of what you are thinking?
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Have the students develop two or more ways to explain a strategy solution which may include using materials.
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Compare explanations and develop the norm of what makes an acceptable explanation. Reinforce what makes it mathematical.
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Launch the problem and have the students read it as a group, discuss, interpret, reinterpret collectively using student voice.
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Shortly after the small groups begin to solve a problem as a large group have them describe their different starting point. Reinforce acceptability of multiple ways. Support them to make connections to other or previous problems.
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Ask the students in small groups to examine their explanations and explore ways to revise, extend and elaborate on sections they think others might not understand.
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Have students examine their explanation, predict the questions they will be asked and prepare explanations.
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Active listening and questioning for sense-making of a mathematical explanation.
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Discuss and role-play active listening. Use inclusive language “show us”, “we want to know”, “tell us”.
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Structure the students explaining and sense making section by section.
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Emphasise need for individual responsibility for sense-making.
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Provide space in explanations for thinking and questioning.
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Affirm models of students actively engaged and questioning to clarify sections or gain further information.
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support and responsibility for the reasoning of all group members: Use core Pasifika values
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Provide students with problem and think-time then discussion and sharing before recording.
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Establish use of one piece of paper and one pen. Explore family structures where everyone participates.
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Establish expectation that students agree on construction of a solution strategy that all members can explain.
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Explore ways to support students indicating need to ask a question during large group sharing. Use no hands up or the use of koosh balls, or pegs, or beany toys.
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Explore ways for the students to support each other using a range of cultural models e.g. all in the same waka paddling together, or a kapa haka group which requires the expert to be responsible to bring the group up to their level of expertise.
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Select a different member of the small group to explain than the recorder.
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During large group sharing change the explainer mid explanation.
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When questions are asked of the small group select different members to respond (not the recorder or explainer).
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Developing justification and mathematical argumentation
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Require that students indicate agreement or disagreement with part of an explanation or a whole explanation.
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Ask the students to provide mathematical reasons for agreeing or disagreeing with an explanation. Vary when this is required so that the students consider situations when the answer is either right or wrong.
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Model and support the use of questions which lead to justification like ‘How do you know it works?’, ‘Can you convince us?’, ‘Why would that tell you to…?’, ‘Why does that work like that?’, ‘So what happens if you go like that?’, ‘Are you sure it’s…?’, ‘So what happens if…?’, ‘What about if you say…does that still work?’, ‘So if we…?’.
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Ask the students to be prepared to justify sections of their solution strategy in response to questions.
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Require that the students analyse their explanations and prepare collaborative responses to sections they are going to need to justify.
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Model ways to justify an explanation
“I know 3 + 4 = 7 because 3 + 3 = 6 and one more is 7”.
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Structure activity which strengthens student ability to respond to challenge
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Encourage the use of ‘so if’, ‘then’, ‘because’ to make justifications. Use this format to validate an explanation
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Expect that group members will support each other when explaining and justifying to a larger group.
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Explicitly use wait time or think time before requiring students to respond to questions or challenge.
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Require that the students prepare ways to re-explain in a different way an explanation to justify it.
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Provide wait time to allow students to prepare questions which lead to justification
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Developing representing as part of exploring and making connections (How can I/we make sense of this for my/ourselves). Communication and justification (How can I explain, show, convince other people)
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Expect the use of a range of representations including acting it out, drawing a picture or diagram, visualising, making a model, using symbols, verbalising or putting into words, using materials.
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Expect the students to explain and justify using the representation as actions on quantities not manipulation of symbols (use context).
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Require that the students compare and contrast representations and evaluate for efficiency.
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Ask students to represent their thinking in different forms in response to questions or for clarification.
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Developing the use of mathematical language
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Expect the use of mathematical language to describe actions while making mathematical explanations
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Expect the use of correct mathematical terms. Ask questions to clarify terms and actions on symbols (using the context).
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Require the use of mathematical words to describe actions. Reword or re-explain mathematical terms and mathematical explanations. Use other examples to illustrate meaning.
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Require students that the students pose questions using appropriate mathematical language.
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Developing generalisations: Representing a mathematical relationship in more general terms. Looking for rules and relationships. Connecting, extending, reconciling.
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Ask the students to consider what steps they are doing over and over again and begin to make predictions about what is changing and what is staying the same.
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Ask the students to consider if the rule or solution strategy they have used will work for other numbers. Consider if they can use the same process for a more general case (e.g. what happens if you multiply any number by 2?).
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Model and support the use of questions which lead to generalisations like ‘Does it always work?’, ‘Can you make connections between…?’, ‘Can you see any patterns?’, ‘How is this the same or different to what was done before?’, ‘Would that work with all numbers?’.
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Observations from lesson:
teacher starts lesson by questioning the norms - and uses the “repeating” talk move - elaborates or revoices if necessary
teacher displays the norms on the TV to remind everyone
warm-up - teacher puts the statement - ½ is smaller than ¼ - groups discuss amongst themselves
teacher asks students to feed back their ideas or theories
teacher asks students to repeat their peers comments
Teacher then presents the problem for the session - teacher gives students a chance to read the question themselves
teacher then asks “what’s the story telling us?”
repeats the information being delivered - Repeat talk moves -
teacher clarifies what the problem is asking us
Teacher gives students 2 minutes to discuss the problem and then will hand out the pencils and books
After students have their books and pencils, teacher makes his way around the groups and listens in on the thinking,
teacher gives a one minute warning to reporting back
teacher asks students to put their pencils away and to listen to the group reporting back has students repeat the information back
after teach groups present - teacher makes connection -3 thirds is a whole
fractions are sharing - connected to divided by
Possible Ideas/thoughts for your programme:
Samuel, you have made so much progress in your maths programme this year, and this was very evident in today’s maths session with your students. Your warm-up created dialogue and the students were able to practice justifying their ideas. Your launch was clear and you were able to demonstrate the talk moves of ‘repeating and revoicing’. Next, I would work on the talk move of Reasoning - Do you agree or disagree and why? You were very aware of the students levels and provided support without “giving them the answer”. It was intuitive of you to make sure the groups moved closer to be more engaged in the conversation during the student’s presentation of ideas. When you do have a group working with you make sure to keep the status the same - have everyone on the floor or at a table when working and make sure to keep the context very clear for the students to avoid confusion. For example, keep reinforcing/repeating with each group how many slices of pizza there were in each pizza, and therefore how many whole pizzas they needed to order. As you move forward with the DMIC model, your work will now be in building a classroom culture where the students are encouraged and praised to ask questions of their peers and seek clarification. Also, make sure to leave yourself enough time to get to the connect and have time to give some generalisations to the students so that they can apply the Big Idea you have covered with them; for example, what if the pizzas were cut into 4 slices, 5 slices etc.
Again, your session was great today, showing so much growth in your delivery of your maths programme, and I can’t wait to see how your team develops your maths programme for Term 3.
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