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Mathematical Processes. 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 Instruction - PowerPoint PPT Presentation


  • Mathematical Processes

  • *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 processesExemplarsRubrics and Performance IndicatorsBenchmarkingPractice Marking

  • *What it Will Look Like

  • *Mathematical Processes

  • Traffic Light*CommunicationConnectionsMental Math and EstimationVisualizationReasoning and ProofProblem SolvingUse of Technology

  • *Math Processes in our Curriculum

  • Jigsaw*1. Problem Solving2. Reasoning and Proof3. Communication4. Connections/Representation

    20 min in Expert Group20 min in Home Group

  • *Effective InstructionWhat 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

  • *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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  • *Video- 15min./ guided notes

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  • Your Turn*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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  • Material ScaffoldingMath Makes Sense- Step by Step*

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  • Guided Notes*

  • *Building Assessment into InstructionWhat ideas about assessment come to mind from your personal experiences?

    Now, suppose you are told that assessment in the classroom should be designed to help students learn and to help teachers teach. How can assessment do those things?

  • *Building Assessment into InstructionThink about one child in your math classroom. Write down the childs name.Think about the child in relationship to mathematics. Try to visualize this child using and learning mathematics in your classroom.Now imagine that the childs family is moving to another province. What could you tell the new teacher about the child's learning of mathematics?

    What new mathematical ideas is the child developing?What mathematical ideas has the child mastered?What are the childs strengths in Mathematics? Weaknesses?How does the child best learn mathematics?Does the child like mathematics?Does the child like exploring mathematical ideas with others?

    Brainstorm a list of any thoughts that come to mind.As you read each item ask yourself, How do I know this?

  • *Building Assessment into InstructionNationally, increased attention is being given to ways that mathematical learning is assessed. This interest is being fueled by many factors, including:

    New standards for the teaching of mathematics.

    A strengthened belief that instruction and assessment should be more closely linked.

    A new understanding of the way in which students learn

    An increased concern for equity

    Continued pressure for accountability

  • *Building Assessment into Instruction

    Assessment should enhance students learning

    Assessment is a valuable tool for making instructional decisions

  • *What is Assessment ?

    Assessment isthe process of gathering evidence about a students knowledge of, ability to use, and disposition toward mathematics and of making inferences from that evidence for a variety of purposes(NCTM, 1995, p. 3)

  • *What is Assessment ?

    Assessment should not merely be done to students; rather it should also be done for students

    Assessment should become a routine part of the ongoing classroom activity, rather than an interruption(NCTM Standards, 2003, p. 22-23)

  • *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.

  • *ExemplarsA good problem-based task designed to promote learning is also the best type of task for assessment.

    Problem-based tasks may tell us a lot about what students know, but how do we handle this information?

    Often there is only one problem for students to work on in a given period. There is no way to simply count the percent correct and put a mark in the grade book.

    Scoring is comparing students work to criteria or rubrics that describe what we expect the work to be.

    Grading is the result of accumulating scores and other information about a students work for the purpose of summarizing and communicating to others.

  • *Exemplars Rubric and Performance Indicators

    Problem SolvingReasoning and ProofCommunicationConnectionsRepresentationNoviceNo 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 presentArguments are made with no mathematical basis

    No correct reasoning nor justification for reasoning is presentNo awareness of audience or purpose is communicated.

    Little or no communication of an approach is evident.

    Everyday, familiar language is used to communicate ideasNo connections are madeNo attempt is made to construct mathematical representation

  • *Exemplars Rubric and Performance Indicators

    Problem SolvingReasoning and ProofCommunicationConnectionsRepresentationApprenticeA 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 ideasSome 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.

  • *Exemplars Rubric and Performance Indicators

    Problem SolvingReasoning and ProofCommunicationConnectionsRepresentationPractitionerA 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 answerArguments 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 answerA 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 ideasMathematical 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

  • *Exemplars Rubric and Performance Indicators

    Problem SolvingReasoning and ProofCommunicationConnectionsRepresentationExpertAn efficient strategy is chosen and p


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