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Lessons From MAC Looking Back Looking Forward

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Changing The Didactic Contract Originally focus on Teaching For Understanding – Onset of No Child Left Behind Now Gearing up for Common Core State Standards and National testing – New Focus on Learning for Understanding, Mathematical Practices – Conceptual Knowledge

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Didactic Contract Freds Flat Freds flat has five rooms. The total floor area is 60 sq. meters. Draw a plan of Freds flat. Label each room and show the dimensions (length and width of all rooms).

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Phases of Change PhasesAudience -Resistance- Care about students -Search for Evidence-Desire for student -Ideas for changing success classroom instruction-Uncomfortable with mathematics -Change from normal classroom practice

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Curriculum should be designed to systematically provide students with mathematical experiences that become progressively deeper and broader. When mathematical content and processes recur through the grades, they should be experienced at deeper levels. Learning Across Grade Levels

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The problems posed and the concepts examined within any mathematics area should grow more sophisticated each year of the curriculum until a suitable level of understanding or proficiency is reached. At that point, although formal instruction in the specific content area will end, students should continue to use the understandings and skills they have acquired, thereby maintaining and strengthening both. Is this really something to get excited about?

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Teacher Comments In Perspective Question on mean, medium, mode too vocabulary specific. – Mathematical Knowledge Question could be more specific about what to include so students describe mathematically instead of gray, nice, …. – Class didactic We havent covered it yet. – Knowledge is cumulative Common errors, I noticed that students ½ + ½ + ½ and put 3/6. – Meaningful analysis of student work

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Teacher Comments in Perspective We need to allocate more time for mathematical reasoning. – Genuine concern One of my students raised her hand, isnt this just like what we have been doing? – Yeah! Its hard to give them time to process – it takes too long to teach in depth. – What message are we giving?

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Positive Changes in Learning Tests are getting more difficult over the years. Hugh Burkhardt, Shell Centre Types of problems addition and subtraction Types of problems for multiplication and division Learning around fractions, meaning of fractions

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Positive Changes in Learning Tests are getting more difficult over the years. Hugh Burkhardt, Shell Centre Data/ scale Algebraic thinking/ growing patterns Longer chains of reasoning

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Where are we stuck? Geometry Van Hiele levels Compare/ Contrast attributes Recognizing attributes Nonstandard orientation Composing and Decomposing Shapes

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Level 1 (Visualization) recognize figures by appearance alone, often by comparing them to a know prototype. The properties of a figure are not perceived. Level 2 (Analysis): Students see figures as collections of properties. When describing an object they do not discern which properties are necessary and which are sufficient to describe the object. Level 3 (Abstraction):. At this level, students can create meaningful definitions and give informal arguments to justify their reasoning. Level 4 (Deduction): Students can construct proofs, understand the role of axioms and definitions, and know the meaning of necessary and sufficient conditions. At this level, students should be able to construct proofs.

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Where are we stuck? Percents Reading in content area Data – – Types of Displays – Making versus reading – Purpose of Measures of Center, doing math in context

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Where are we stuck? Transference – Algebra Choosing appropriate strategies and organizing work Measurement – Conversions, links to proportional reasoning – Quantity – how measures change – New Role in CCSSM

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What do we do about it? What are some of the roadblocks to making progress? What do we do next?

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Administer Tasks Examine Student Work Inform Teacher Knowledge Inform Instruction Formative Assessment Cycle MARS Tasks Scoring and Student Works Protocols Tools for Teachers and PD Materials Re-engagement Lessons Common Core Standards

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Tools for Analyzing Student Work Line Graph to see trends in teachers class Analysis by points Tools for Teachers Scoring Questionnaires Grade Level Team meetings Writing Ideas into next years planning Tools for Analyzing Student Work Line Graph to see trends in teachers class Analysis by points Tools for Teachers Scoring Questionnaires Grade Level Team meetings Writing Ideas into next years planning Improve Student Learning

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Re-engagement Give ourselves permission to spend more time on a problem and its discussion. Give students the opportunity to really examine the mathematics and change their ideas through rich dialogue. Promotes sense-making, justification, making conjectures and testing them. Ups the cognitive demand of the task.

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What is the mathematical story of this task? What are the big mathematical ideas in the task? What are the themes that emerge from the student work? What might be underlying causes for problems?

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Data

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Sub Sandwiches Each sandwich needs: 1 sub bun 3 slices of salami Half a tomato Laura makes 6 sub sandwiches. How many slices of Salami does she need? Laura only has 3 tomatoes. How many sandwiches can She make?

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Actual Student Work

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Sub Sandwiches What do you think the student is doing? What do you think the lines represent?

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Sub Sandwiches Another student wrote this. What do you think the student is doing? What do the numbers represent? How is this the same or different from Student As work?

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Original Student Work

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Used as a question What do you think the student is doing? What do you think the lines represent? What do the numbers mean?

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Process of Re-engagement Give students a purpose for re-examining the work or mathematics of a task by creating a dilemma or cognitive conflict. Move students from the process of solving a problem to justification and sense-making. Why did this work? Why doesnt this make sense? Involve them in the discipline of doing mathematics.

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Re-engagement Happens Live The heart of the process is in the discussion, controversy, and convincing of the big mathematical ideas. This is where students have the opportunity to clarify their own thinking, confront their misconceptions to see the errors in logic, use mathematical vocabulary for a purpose, and make generalizations and connections.

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Learning Cycle Strategy to improve the learning cycle Instead of something new, something different, probing more deeply into the mathematics Take advantage of time already invested in thinking about the problem Move students from where they are to more grade level appropriate strategies Accessible to teachers

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Building Lesson Unit How to week out or pare down? How to respond to the not enough time? How to make life more manageable and build in time to develop problem-solving and thinking as well as skills?

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What do students already know? Start unit with assessment task. Use a re-engagement lesson to clear up some misconceptions Take notes on what students understand to week out some material and concentrate on whats new Tie new learning to the task – touchstone experience

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POM or Formative Assessment Lesson Give POM or FAL about ¾ of the way through the unit Check for misunderstandings, misconceptions so that they can be addressed before the end of the unit Work on learning in context and longer chains of reasoning Give students chance to articulate their ideas and talk their way into understanding

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End Unit with Formative Assessment Make it safe for teachers to try lessons – lesson –study like model Instituting Best Practices – Public Learning Records – Structured math talks: Revoicing, restating, agree/disagree, add-on – Wait time – Equity on talk time – sticks or cards

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Hurdles Classroom Management Belief in Students

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End Unit with Formative Assessment Make it safe for teachers to try lessons – lesson –study like model Building students ability to dialog with each other by taking responsibility for their own learning – asking clarifying questions, challenging ideas of others, developing ability to make convincing arguments

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Importance of Feedback Dylan Wiliam, Paul Black – Inside the Black Box In summary, feedback to any pupil should be About the particular qualities of his or her work, with Advice on what he or she can do to improve, and Should avoid comparisons with other pupils.

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Lessons from Drumming Pupils who come to see themselves as unable to learn usually cease to take school seriously. Many become disruptive; others resort to truancy. Such young people are likely to become alienated from society and to become the sources and victims of serious social problems.

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