Lessons From MAC

Looking BackLooking Forward

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

Didactic Contract

Fred’s FlatFred’s flat has five rooms. The total floor area is

60 sq. meters. Draw a plan of Fred’s flat. Label each room and show the dimensions (length and width of all rooms).

Phases of Change

Phases Audience- Resistance - Care about students- Search for Evidence -Desire for student- Ideas for changing success classroom instruction -Uncomfortable with

mathematics-Change from normal classroom practice

“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 LevelsLearning Across Grade Levels

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?Is this really something to get excited about?

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 haven’t covered it yet.” – Knowledge is cumulative

“Common errors, I noticed that students ½ + ½ + ½ and put 3/6.” – Meaningful analysis of student work

Teacher Comments in Perspective

• “We need to allocate more time for mathematical reasoning.” – Genuine concern

• “One of my students raised her hand, “isn’t 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?

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

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

Where are we stuck?

• Geometry• Van Hiele levels• Compare/ Contrast attributes• Recognizing attributes• Nonstandard orientation• Composing and Decomposing Shapes

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.

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

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

What do we do about it?

• What are some of the roadblocks to making progress?

• What do we do next?

Administer Tasks

Examine Student Work

Inform Teacher Knowledge

Inform Instruction

Formative Assessment

Cycle

TOOTHPICK SHAPESTom uses toothpicks to make the shapes in thediagram below.

shape 1

6 toothpicks

shape 2

9 toothpicks

shape 3 shape 4

1. How many toothpicks make sh ape 3?_________________

2. Draw sh ape 4 next to sh ape 3 in th e diagramabove .

5. Tom says, “I need 36 toothp icks to make shape 12.”Tom is not correct. Explain wh y he is n ot correct.How many toothpicks are needed to make sh ape 12?

MARS Tasks

Scoring and Student Works Protocols

Tools for Teachers and PD Materials

Re-engagement Lessons

Common Core

Standards

Tools for Analyzing Student Work•Line Graph to see trends in teacher’s class•Analysis by points•Tools for Teachers•Scoring Questionnaires•Grade Level Team meetings•Writing Ideas into next year’s planning

Tools for Analyzing Student Work•Line Graph to see trends in teacher’s class•Analysis by points•Tools for Teachers•Scoring Questionnaires•Grade Level Team meetings•Writing Ideas into next year’s planning

Improve Student Learning

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.

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?

Data

Sub Sandwiches

Each sandwich needs:1 sub bun3 slices of salamiHalf a tomato

Laura makes 6 sub sandwiches. How many slices of Salami does she need?Laura only has 3 tomatoes. How many sandwiches canShe make?

Actual Student Work

Sub Sandwiches

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

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 StudentA’s work?

Original Student Work

Used as a question

What do you think the student is doing? What do you think the lines represent?What do the numbers mean?

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 doesn’t this make sense? Involve them in the discipline of doing mathematics.

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.

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

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?

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 what’s new

• Tie new learning to the task – touchstone experience

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

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

Hurdles

• Classroom Management• Belief in Students

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

Importance of Feedback

• Dylan Wiliam, Paul Black – Inside the Black Box

In summary, feedback to any pupil should beAbout 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.

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.

Break