Day 3

Day 3

Professional Learning for Mathematics

Leaders and Coaches—Not just a 3-part series

1

What’s My Number?• Multiply the number of brothers you

have by 2• Add 3• Multiply by 5• Add the number of sisters you have• Multiply by 10• Add the number of living grandparents• Subtract 150Your number is: ____ ____ ____

# of brothers# of brothers

# of sisters# of sisters

# of grandparents# of grandparents

Inside/Outside Circle• Each person from your board go to one of the

four corners1 min discussion

• Share an interesting aspect from the ‘view & discuss’ or ‘do’ that you participated in.

• How are the Big Ideas impacting your practice?• Which is more comfortable for you: Open or

Parallel Tasks, and why?• How are you using the MATCH Template and/or

PPQT to help you with lesson planning?

Provincial-level EvidenceTrends in Provincial EQAO Math

0

10

20

30

40

50

60

70

80

90

1 2 3 4 5

2004-2009

% a

t P

rovi

nci

al S

tan

dar

d

0.66 in Grade 6

1.00 in Grade 3

1.66 in Combined 9

1.00 in 9 Applied

2.00 in 9 Academic

Point increase per year over the past 3 years

Provincial-level EvidenceProvincial Grade 9 EQAO Data

0

20

40

60

80

100

1 3 5 7 9Semesters 2004-09

% a

t P

rovi

nci

al S

tan

dar

d

Applied

Academic

Rate = 1.6 points per semester

Rate = 2 points per semester

Rate = 1.4 points per semester

Rate = 2.3 points per semesterincreasing

increasing

Board-level Evidence and BIPs

• Halton DSB’s 2009-10 Math GAINS Transition Project paper

• Peterborough Victoria Northumberland Clarington CDSB 2-part roving Math GAINS report

• Greater Essex CDSB roving Math GAINS report

Next Steps – Materials• WINS = Winning with Instructional Navigation

Supports• Grade bands K-1, 2-3, 4-5, 6-7, 7-8, ?-?• Focused on Number Sense• “Thinking Book” for learners and “guide” for

learning facilitators• Could be used at home use, but good for many

other applications• 3-part lessons with 4 parallel questions

addressing the same learning goal• multiple solutions with scaffolding questions

Next Steps – Professional Learning

• Sessions January to June 2010 with possible themes:– WINS– Managing Group Dynamics in Classrooms and

Schools– Strategic planning work sessions

• Access customized provincial-level support through Jeff Irvine, Myrna Ingalls, Demetra Saldaris

• Regular postings on www.edugains.ca

Addressing Questions from Session 2

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• We ask an open question.

• Nobody responds.

• One thing we need to work on are strategies to use in that situation.

Let’s try an example.

Scaffolding Thinking Prompts

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• Let’s start with one of the open questions from last session.


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Open Question:

Create two linear growing patterns that are really similar.

• How are they similar?


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• Could one of your patterns be 1, 4, 9, 16,…? Explain.


Open Question:

Create two linear growing patterns that are really similar. How are they similar?

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• If you were describing your pattern to another student, what information would you give them?


Open Question:


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• Are 2, 5, 8, 11, 14,… and

5, 10, 15, 20,… really similar?

Open Question:



Discuss at your table:

How do the three questions help students with scaffolding their thinking to answer the open question?Open Question:


How does the Big Idea and the Lesson Goal impact the questions

you are asking for scaffolding?

18

• Which two graphs do you think are most alike? Why?

Y = 3x2 – 2 y = -3x2 – 2

Y = 2x2 + 3 y = 3x2 + 2

Another example from last time

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• What would you be looking for to decide if two graphs were alike?

• Do any of the graphs go through the same points?•Do any of the graphs open in the same direction?• Are any of the graphs congruent to other graphs?


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Which one of the Scaffolding Thinking Prompts do you like?

Why?


21

It takes more than 5 English words to describe an algebraic expression that has one term.

What could the algebraic and verbal expressions be?

Algebraic expressions

Scaffolding Thinking Prompts•How many English words does

it take to describe 2n?•Could the algebraic expression

be 2n+3? Why or why not? •How many terms would the

expression “a number squared” take?

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Scaffolding Task:

Choose one of the open questions:

• Decide under what conditions you would use those scaffolding questions.

This picture shows that 4x + 2 = 2 (2x + 1) no matter what x is.

• Explain why.• Now draw another picture that shows another equation that is true no matter what x is.

25

•Graph the 2 lines.

The 1st is 3x + 2y = 6 and

the 2nd is –x + 3y = 17.

A third line lies between them. What might its equation be?

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27

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•Most of the scaffolds we just saw were very problem specific.

• General scaffolds are also helpful.

General & specific scaffolds

Fail Safe Strategies

•Where have you seen something like this before?

•What patterns do you see?

•Have you thought about….?

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•A useful source for general scaffolds is available in the mathematical process package in TIPS on the Edugains website.

General & specific scaffolds

http://www.edu.gov.on.ca/eng/studentsuccess/lms/files/tips4rm/TIPS4RMProcesses.pdf

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•Looking at the parallel task, decide what possible scaffolds might be needed.

Your turn to scaffold

Thinking about feedback• Suppose a student does well on a task.

What kind of feedback do teachers tend to give?

• Does that help the good student move on?• Suppose a student does not do well on a

task. What kind of feedback do teachers tend to give?

• Does that help the weak student to move on?

Looking at Student Work

• In pairs, pick one of the student’s work.

• What feedback would you give students to move them forward?

• Share with another pair.

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•Open questions and parallel tasks are built for instruction.

The focus is not on evaluation, but…

Assessment issues

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•It makes total sense to use parallel tasks to measure communication and/or thinking and maybe (depending) knowledge or application.

Assessment issues

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•to use a previously shared rubric or marking scheme before posing such questions to help students meet success.

It is important …

Example 1Task A:

• Think of a way to represent the pattern with the general term 3n +1.

• How does your representation tell you whether 925 is a term in the pattern?

Task B: • Think of a way to

represent a pattern where each term value is eight times the term number.

• How does your representation tell you whether 925 is a term in the pattern?

Learning goal: Represent the general term of a linear growing pattern

Possible Scoring SchemeThis could be marked as a 6-mark question

• 3 marks for a really good representation of the pattern which means that you are representing the right pattern (which is the most important thing) without errors

• 3 marks for a complete explanation for why 925 is or is not in the pattern based on the representation

39

Task A:

Explain and justify each step you would use in solving the equation without using a calculator.

Another Example

3

5

3

4

2

3x

Task B: Explain and justify each step you would use in solving the equation without using a calculator.

1.5 x – 4.2 = 7.3

40

Task A:

Sketch graphs of y = 2x and y = 2-x.

Tell how the graphs are alike and

different.

Example 2Task B:

Sketch graphs of y = x2 and y = -2x2.

Tell how the graphs are alike and

different. Which of those things could you

have predicted without sketching? Why?

Possible RubricLearning Goal: Examine the effects of the parameters given two graphs of the same type of function

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•It makes sense, even on a “test”, to use open tasks to ensure that students get an opportunity to tell as much as they know in whatever form works for them about an idea they have learned.

Assessment issues

Example 3Choose an equation to solve where the solution is an integer. Solve it in at least three different ways, explaining your thinking for each method.

Suppose you had chosen an equation where the solution turned out to be a fraction. Which of your methods would be more likely to help you? Why?

A possible Rubric

Learning Goal: Different representations of equations are more useful depending on the situation.

45

•Describe a number of examples of measurements or situations that would model direct variation. What is it about them that makes the variation direct?

Another Example

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•Give as many reasons as you can to explain why there are many quadratic relations that pass through the points (0,0) and (2,4).

Another Example

47

•Although assessment of learning should drive what you teach, it should not limit the strategies you use to meet all students’ needs.

Assessment issues

Break out

• Working with Student Samples (Assessment for Learning)

• Developing Scaffolding Questions• Forming Open & Parallel Questions • Creating Consolidating Questions

Move to one of the following areas of interest:

49

•Our “big idea” for this PLMLC series is that if you think more broadly about what you are focusing on in instruction, you are more likely to:

Summarizing big ideas

50

•help students make essential connections

• ensure that students learn what is really important

•ensure that a much broader range of students can meet success

Summarizing big ideas

Work in Boards Group

• Talk about moving forward with the ideas we have been working on

• Facilitators will be happy to join your team if you want a different set of eyes

• Call on another board to share ideas

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•You are welcome to ask any questions or offer any insights on what we’ve discussed in today’s plenary .

Your questions

Next Steps – Professional Learning

• Sessions January to June 2010 with all expenses paid by the ministry - possible themes:– K to Grade 8+ Mathematics Package– Classroom Management Package– Strategic planning work sessions

• Access customized provincial-level support through Jeff Irvine, Myrna Ingalls, Demetra Saldaris

• Regular postings on www.edugains.ca

Documents

Day 3