Mathematical Processes

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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 Instruction
• Content, Task, and Material linking to the
  mathematical processes
• Exemplars
• Rubrics and Performance Indicators
• Benchmarking
• Practice Marking

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What it Will Look Like
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Mathematical Processes
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Traffic Light
Communication   
Connections
Mental Math and Estimation
Visualization
Reasoning and Proof
Problem Solving
Use of Technology
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Math Processes in our Curriculum
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Jigsaw
1. Problem Solving 2. Reasoning and Proof 3.
Communication 4. Connections/Representation 20
min in Expert Group 20 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 Instruction

• What 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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ReflectionHow 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

• What 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?

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Building Assessment into Instruction

• Think 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?

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Building Assessment into Instruction

• Nationally, 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

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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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What is Assessment ?

• Assessment is
• the 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)

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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)

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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

• A 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.

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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 present Note 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 present Note 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 ideas Note 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.

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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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Talk to teachers about teaching
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