Mathematical Processes

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What We are Learning Today

Mathematical Processes What are they?How do we teach through these processes?How do students learn the content through these processes?

Effective Instruction: Scaffolding InstructionContent, Task, and Material… linking to the mathematical processes

ExemplarsRubrics and Performance Indicators

BenchmarkingPractice Marking

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What it Will Look Like

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Mathematical Processes

Traffic Light

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Communication  Connections

Mental Math and Estimation

Visualization

Reasoning and Proof

Problem Solving

Use of Technology

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Math Processes in our Curriculum

Jigsaw

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1. Problem Solving2. Reasoning and Proof3. Communication4. Connections/Representation

20 min in Expert Group20 min in Home Group

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Effective Instruction

What are Students Doing?

Actively engaging in the learning process

Using existing mathematical knowledge to make sense of the task

Making connections among mathematical concepts

Reasoning and making conjectures about the problem

Communicating their mathematical thinking orally and in writing

Listening and reacting to others’ thinking and solutions to problems

Using a variety of representations, such as pictures, tables, graphs and words for their mathematical thinking

Using mathematical and technological tools, such as physical materials, calculators and computers, along with textbooks, and other instructional materials

Building new mathematical knowledge through problem solving

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Effective InstructionWhat is the Teacher Doing?

Choosing “good” problems – ones that invite exploration of an important mathematical concepts and allow students the chance to solidify and extend their knowledge

Assessing students’ understanding by listening to discussions and asking students to justify their responses

Using questioning techniques to facilitate learning

Encouraging students to explore multiple solutions

Challenging students to think more deeply about the problems they are solving and to make connections with other ideas within mathematics

Creating a variety of opportunities, such as group work and class discussions, for students to communicate mathematically

Modeling appropriate mathematical language and a disposition for solving challenging mathematical problems

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11Video- 15min./ guided notes

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Your Turn…

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In your groups:

•Rewrite the problem using content scaffolding •Using this new rewritten math statement fill in the task scaffolding chart.•As you and your partner discuss this process through Think Aloud- make notes how this discussion supports growth in each of the 5 mathematical processes

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REFLECTION…

HOW DOES TASK SCAFFOLDING TEACH CONTENT THROUGH THE MATHEMATICAL PROCESSES?

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Material ScaffoldingMath Makes Sense- Step by Step

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Guided Notes

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Building Assessment into Instruction

Assessment should enhance students’ learning

Assessment is a valuable tool for making instructional decisions

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Benchmarking

The WNCP 2006 states that assessment in the classroom should be designed to help students learn and to help

teachers teach.

Benchmarking is a way for us to gather data about what the students in our class and within the whole division know, understand and are able to do at any given time

during a school year.

Benchmarking can help us understand what students need to continue their learning and what the teacher needs to do to assist students with their continued

learning.

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Exemplars – Rubric and Performance Indicators

Problem Solving

Reasoning and Proof

Communication

Connections Representation

Novice No strategy is chosen, or a strategy is chosen that will not lead to a solution.

Little or no evidence of engagement in the task is present

Arguments are made with no mathematical basis

No correct reasoning nor justification for reasoning is present

No awareness of audience or purpose is communicated.

Little or no communication of an approach is evident.

Everyday, familiar language is used to communicate ideas

No connections are made

No attempt is made to construct mathematical representation

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Exemplars – Rubric and Performance Indicators

Problem Solving

Reasoning and Proof

Communication

Connections Representation

Apprentice

A partially correct strategy is chosen, or a correct strategy for only solving part of the task is chosen

Evidence of drawing on some relevant previous knowledge is present, showing some relevant engagement in the task.

Arguments are made with some mathematical basis.

Some correct reasoning or justification for reasoning is present with trial and error, or unsystematic trying of several cases.

Some awareness of audience or purpose is communicated, and may take place in the form of paraphrasing or the task

Some communication of an approach is evident through verbal/written accounts and explanations, use of diagrams, or objects, writing, and using mathematical symbols.

Some formal math language is used, and examples are provided to communicate ideas

Some attempt to relate the task to other subjects or to own interests and experiences is made.

Relates to self and experiences.

An attempt is made to construct mathematical representations to record and communicate problem solving.

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Exemplars – Rubric and Performance Indicators

Problem Solving

Reasoning and Proof

Communication

Connections Representation

Practitioner

A correct strategy is chosen based on the mathematical situation in the task.

Planning or monitoring of strategy is evident.

Evidence of solidifying prior knowledge and applying it to the problem-solving situation is presentNote: The Practitioner must achieve a correct answer

Arguments are constructed with adequate mathematical basis.

A systematic approach and/or justification of correct reasoning is present. This may lead to Clarification of the task Exploration of mathematical phenomenon Noting patterns, structures and regularities

Note: The Practitioner must achieve a correct answer

A sense of audience or purpose is communicated

Communication of an approach is evident through a methodical, organized, coherent, sequenced and labeled response

Formal math language is used throughout the solution to share and clarify ideas

Mathematical connections or observations are recognized

Must use math to prove assumption.

Mathematical proof is needed.

Appropriate and accurate mathematical representations are constructed and refined to solve problems or portray solutions

Note: The Practitioner must achieve a correct answer

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Exemplars – Rubric and Performance Indicators

Problem Solving

Reasoning and Proof

Communication Connections

Representation

Expert An efficient strategy is chosen and progress toward a solution is evaluated.

Adjustments in strategy, if necessary, are made along the way, and/or alternative strategies are considered.

Evidence of analyzing the situation in mathematical terms and extending prior knowledge is presentNote: The Expert must achieve a correct answer

Deductive arguments are used to justify decisions and may result in more formal proofs.

Evidence is used to justify and support decisions made and conclusions reached. This may lead to… Testing and accepting or rejecting of a hypotheses or conjecture Explanation of phenomenon Generalizing and extending the solution to other cases

Note: The Expert must achieve a correct answer

A sense of audience and purpose is communicated

Communication at the Practitioner level is achieved and communication of arguments is supported by mathematical properties used

Precise math language and symbolic notation are used to consolidate math thinking and to communicate ideasNote: The Expert must achieve a correct answer

Mathematical connections or observations are used to extend the solution.

Note: The Expert must achieve a correct answer

Abstract or symbolic mathematical representations are constructed to analyze relationships, extend thinking, and clarify or interpret phenomenon

Note: The Expert must achieve a correct answer

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Benchmarking

Work with a partner.

Score each problem using the rubric and performance indicators. (Long sheet of paper.) Score each area of the rubric separately.

Attach the recorded scored rubric (small piece of paper) to the problem and continue with the next problem.

When all your problems have been marked, transfer the information from each scored rubric to the Final Recording Sheet.

Final Recording Sheet

Problem Solving

Reasoning and Proof

Communication Connections Representations Total

Novice

Apprentice

Practitioner

Expert

Total

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Benchmarking

How can the information gathered on the Final Recording Sheet help to inform instruction?

Discuss with your partner (table group) the kinds of experiences/modeling/opportunities/activities that the teacher should provide to take the students to the next level. (IE. What are the strengths of this group of students?

What areas are these students having more difficulties with?)

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