1. Mathematical Processes

2. 2 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

3. 3 What it Will Look Like

4. 4 Mathematical Processes

5. Traffic Light 5 Communication Connections Mental Math and Estimation Visualization Reasoning and Proof Problem Solving Use of Technology

6. 6 Math Processes in our Curriculum

7. Jigsaw 7 1. Problem Solving 2. Reasoning and Proof 3. Communication 4. Connections/Representation 20 min in Expert Group 20 min in Home Group

8. 8 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

9. 9 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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11. 11 Video- 15min./ guided notes

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14. Your Turn… 14 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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17. REFLECTION… HOW DOES TASK SCAFFOLDING TEACH CONTENT THROUGH THE MATHEMATICAL PROCESSES? 17

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19. Material Scaffolding Math Makes Sense- Step by Step 19

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21. Guided Notes 21

22. 22 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?

23. 23 Building Assessment into Instruction  Think about one child in your math classroom. Write down the child’s 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 child’s 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 child’s 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?”

24. 24 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

25. 25 Building Assessment into Instruction Assessment should enhance students’ learning Assessment is a valuable tool for making instructional decisions

26. 26 What is Assessment ? Assessment is… “the process of gathering evidence about a student’s 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)

27. 27 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)

28. 28 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.

29. 29 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 student’s work for the purpose of summarizing and communicating to others.

30. 30 Exemplars – Rubric and Performance Indicators Problem Solving Reasoning and Proof CommunicationConnectionsRepresentation 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

31. 31 Exemplars – Rubric and Performance Indicators Problem Solving Reasoning and Proof CommunicationConnectionsRepresentation 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.

32. 32 Exemplars – Rubric and Performance Indicators Problem Solving Reasoning and Proof CommunicationConnectionsRepresentation 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

33. 33 Exemplars – Rubric and Performance Indicators Problem Solving Reasoning and Proof CommunicationConnectionsRepresentation 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

34. 34 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.

35. Final Recording Sheet Problem SolvingReasoning and Proof CommunicationConnectionsRepresentationsTotal Novice Apprentice Practitioner Expert Total 35

36. 36 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?)

37. 37 Talk to teachers about teaching…

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