GECDSB Mathematics Learning Teams (MLT) Session #1

School Math Learning Teams

AGENDA

MATHVISIONWhatdoesitmeantobegoodatmath?Burningquestions

EARLYMATHLEARNINGCountingandQuantity

JUNIORandINTERMEDIATE MATHArraysandAreasModels

PEDAGOGICALSYSTEMSPlanningaTask

LEADINGMATHINYOURSCHOOLTheroleoftheschoolmathlearning team

TODAY’S LEARNING

Weare:• Extending ourunderstanding oftheGECDSB:

Mathematics VisionandMathematical Proficiencies

• Building anunderstanding ofnumbersense conceptsfromK-10

• Developing anunderstanding ofPedagogical Systems

• Planning amath task

• Learning toleadmath inourschools

MathPedagogy

MathContent

Leadership&

ProfessionalCapital

THE MATHEMATICS JOURNEY• Mathematics journey isunlikemanyothers• Itisnot animplementation ofaprogramorprocess• Itisthe enaction ofthe GECDSBVision

The Work: Ambitious and Necessary

Enact the Vision

“The GECDSB provides mathematics education that engages and empowers students through collaboration, communication, inquiry,

critical thinking and problem-solving, to support each student’s learning and nurture a positive attitude towards mathematics.”

Table Talk…Why is this work both ambitious and necessary?

SCHOOL MATH TEAM

Clicktoaddtext

Session1 OctoberSession2 NovemberSession3 JanuarySession 4 & 5 FebruaryandMarchSession6 April

TableTalk...Whatistherole(possible role)oftheschoolmathteamatyourschool?

Wherearesomeprofessional learningspacesinyourschool?

What does it mean to be good at mathematics?

What does it mean to be good at math?

Enjoy learning mathDevelop persistence and tenacityLearn to use math to solve problemsDevelop logic and reasoning skillsSee the value for mathematics in their worldLearn their facts and mathematical proceduresUnderstand the ‘whys’ of math

GECDSB: A Vision for MathematicsStrategic Competence

Procedural Fluency

Conceptual Understanding

Adaptive Reasoning

Productive Disposition

This is a vision for mathematics that is both ambitious and necessary.

Math Task Force Data

What does it mean to be good at math?

Math as a functional skillMath as applied to a professionMath as a way of thinking and seeing the world

Math Vision: Understanding Proficiency

StrategicCompetenceProceduralFluencyConceptualUnderstandingAdaptiveReasoningProductiveDisposition

Jigsaw: ReadChapter4

Math Task

Clicktoaddtext

HowCloseto100?Instructionscanbefoundathttps://www.youcubed.org/task/how-to-close-100/

Math Task: Consolidation

Howdoyouseethismathtaskconnectedtothedevelopmentof

mathematicalproficiency?

Isthereareaparticularmathematicalproficiencythatcouldbestrengthened throughthistask?

Doesthischangeyourdefinitionofwhatitmeanstobegoodatmath?

Does this change your definition of what it means to be good at math?

The Work: Guided By Our Questions

We have many questions about mathematics education.

Table Talk…

With the learners at your table, brainstorm some of the

questions you have or your staff may have about mathematics education.

Share them with the larger group.

COFFEE BREAK

Early MathematicsIn2007,itwasfound thatmathematicsskillsamongchildreninKindergartenwerethebestpredictoroflaterschoolachievement,regardlessofgenderorsocio-economicstatus(Duncanetal.,2007). Kindergarten Program, 2016

CriticalQuestionHowcaneducators takeadvantageofthemathematicalknowledgeandexperiencethatchildrenhave?

CriticalUnderstandingThepresencealoneofmathematicsinplayisinsufficient forrichlearning tooccurIntentional,purposeful teacherinteractionsarenecessarytoensure thatmathematicallearning ismaximizedduringplay.

Early Mathematics

Aseducatorswemustconstantlyaskourselves:

Whythislearning,forthisstudent,atthistime?

Early Mathematics

Whatmathematical skillsdoyouthinkouryounglearnersneed?

NumberSense• Counting• QuantityRelationshipsGeometryandMeasurement

arefoundationalskillsthatmustbeinplacetosupportallfuturemathlearning.

Counting and Quantity

Conservation

One-to-oneCorrespondence

Cardinality

StableOrder

OrderIrrelevance

Abstraction

MovementisMagnitude

Subitizing

Unitizing

Case Study

• Whatprinciplesofquantityandcountingdoesthechildunderstand?

• Whatmightbethenextstep(s)forlearningforthischild?

Early Mathematics Connections

Mathematics

Learning

Teams

MLT

Countingand

Quantity

Stable-Order Order Irrelevance

Conservation

One-to-One Correspondence

Abstraction

Movement is MagnitudeSubitizing

Unitizing

Cardinality

Stable-OrderThe list of words used to count must be in a repeatable order.

This “stable list” must be at least as long as the number of items to be counted.

12

3

45

6

7 8 9 10

Order IrrelevanceThe order in which items are counted is irrelevant.

1 2 3 4 5 6

1 2 3 4 5 6

ConservationUnderstanding that the count for a set group of objectsstays the same no matter whether they are spread out or close together.

7 8 9 101 23 4

5 6

ConservationUnderstanding that the count for a set group of objectsstays the same no matter whether they are spread out or close together.

7 8 9 101

23 4

5

6

… the quantity of five large things is the same count as a quantity of five small things or a mixed group of five small and large things.

Abstraction…we can count any collection of objects, whether tangible or not.

1 23 4

51 2 3 4 5

Understanding that each object being counted must be given one count and only one count. It is useful in the early stages for children to actually tag each item being counted and to move an it out of the way as it is counted.

One-to-One Correspondence

123

4

5

Understanding that the last count of a group of objects represents how many are in the group. A child who recounts when asked how many candies are in the set that they just counted, has not understood the cardinality principle.

Cardinality

1 2 3 4 5 6

The ability to 'see' a small amount of objects and know how many there are without counting.

Subitizing

“5”

Understanding that as you move up the counting sequence (or forwards), the quantity increases by one and as you move down (or backwards), the quantity decreases by one or whatever quantity you are going up/down by.

Movement is Magnitude

1 2 3



1 2 3 4



1 2 3 4 5



1 2 3 4

Understanding that in our base ten system objects are grouped into tens once the count exceeds 9 (and into tens of tens when it exceeds 99) and that this is indicated by a 1 in the tens place of a number.

Unitizing

tens ones


Unitizing

tens ones


Unitizing

tens ones


Unitizing

tens ones


Unitizing

tens ones

1 0


Unitizing

tens ones

1 9


Unitizing

tens ones

2 0

PROCEDURAL FLUENCY

STRATEGIC COMPETENCE

ADAPTIVE REASONING

PRODUCTIVE DISPOSITION

CONCEPTUAL UNDERSTANDING

kylep.ca/gecdsbvision

Math Proficiencies


Ability to formulate, represent & solve mathematical problems using an effective strategy


PROCEDURAL FLUENCY

Understanding and using a variety of mathematical procedures

ADAPTIVE REASONINGCapacity for logical thought, reflection,

explanation, and justification

Inclination to see mathematics as useful and valuable.

Ability to understand mathematical concepts, operations, and relationships


@MathletePearcewww.tapintoteenminds.com

LAST TIME



ArrayMultiplication

Array

Array

Array

Array

Array

3 x 2

3 x 2“3 groups of 2”


1 1


1 11 1


1 11 11 1


1 11 11 1

1 11 1

11or

4 x 3



1 1 1


1 11 1

11


1 11 11 1

111


1 11 11 1

111

1 1 1


1 11 11 1

1 11 1

11or

111

1 1 1

11

1 1 1 1

5 x 3 = ?“5 groups of 3”


1 11 11 1

= ?

111

1 11 1

11

Virtual Manipulatives

catalog.mathlearningcenter.org/apps





= ?


= ?

1

11

11


= ?

1

11

11

1 1 1 1 1 1


= ?

1

11

11

1 1 1 1 1 1

1 1 1 11 1 6


= ?

1

11

11

1 1 1 1 1 1

1 1 1 11 11 1 1 11 1

612


= ?

1

11

11

1 1 1 1 1 1

1 1 1 11 11 1 1 11 11 1 1 11 1

61218


= ?

1

11

11

1 1 1 1 1 1

1 1 1 11 11 1 1 11 11 1 1 11 11 1 1 11 1

6121824


= ?

1

11

11

1 1 1 1 1 1

1 1 1 11 11 1 1 11 11 1 1 11 11 1 1 11 11 1 1 11 1

612182430

1

11


1 1 1 1

= ?

1 11 1 1 11 11 1 1 11 1

1 1 1 1 1 1

1 1

1

11


1 1 1 1

= ?

1

11

11 1 1 111 1 1 111 1 1 111 1 1 11

1 1


1 1 1 1

= 25

11 1 1 111 1 1 111 1 1 111 1 1 11


1 1 1 1

= 25

11 1 1 111 1 1 111 1 1 111 1 1 11


1 1 1 1

= 25

11 1 1 111 1 1 111 1 1 111 1 1 11

PERFECTSQUARE


1 1 1 1

= ?

11 1 1 111 1 1 111 1 1 111 1 1 11


1 1 1 1

= 36

11 1 1 111 1 1 111 1 1 111 1 1 11

11111

1 1 1 11 1

What’s the Length of the Pool?

?

1111


?

1111


1 1 1 1

1111


1 1 1 1 1 1 1 1

1111


1 1 1 1 1 1 1 1 1 1 1 1

1111


1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1111

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

What’s the Area of the Pool?

1111

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1


1111

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1


1111

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1


1111

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1


1111

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1


1 1 11 1 11 1 1

111

1 1 1 1

1 1 11 1 11 1 1

111

1 1 1 1

1111

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1


16 16 16 16

1 1 11 1 11 1 1

111

1 1 1 1

1111

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1


16 16 16 1664

1 1 11 1 11 1 1

111

1 1 1 1

1111

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1


16 16 16 1664 square-units

1 1 11 1 11 1 1

111

1 1 1 1

1111

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1


16 161 1 11 1 11 1 1

1 1 11 1 11 1 1

1 1 11 1 11 1 1

1 1 11 1 11 1 1

1 1 11 1 11 1 1

111

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

64 square-units

“Spatial thinking, or reasoning, involves the location and movement of objects and

ourselves, either mentally or physically, in space. It is not a

single ability or process but actually refers to a considerable number of

concepts, tools and processes.”

(National Research Council, 2006)

“The relation between spatial ability and mathematics is so

well established that it no longer makes sense to ask whether

they are related…”

“…moreover, spatial thinking was a better predictor of

mathematics success than either verbal or mathematical skills.”

Activities to Develop Geometric and Spatial Thinking

visualizing diagramming

designing(Davis, Okamoto & Whiteley, 2015)

orientinglocating

perspective taking

slidingrotating

reflecting

modelingexploring symmetry

composing

decomposingscaling

map-making

1

11


1

= ?

11 11

1 1 111 1 111 1 11 1

1 1 1 1 1 1

= 18

1

11


1 11 11

1 1 111 1 111 1 11 1

1 1 1 1 1 1

= 18

1

11


1 11 11

1 1 111 1 111 1 11 1

1 1 1 1 1 1

= 18

1

11

3 x 6“Splitting the Array”

1 11 11

1 1 111 1 111 1 11 1

1 1 1 1 1 1

= 18

1

11

3 x 6

1 11 11

1 1 111 1 111 1 11 1

1 1 1 1 1 1

3 x 4=

= 18

1

11

3 x 6

1 11 11

1 1 111 1 111 1 11 1

1 1 1 1 1 1

3 x 4=

= 18

1

11

3 x 6

1 11 11

1 1 111 1 111 1 11 1

1 1 1 1 1 1

3 x 4= + 3 x 2

= 18

1

11

3 x 6

1 11 11

1 1 111 1 111 1 11 1

1 1 1 1 1 1

3 x 4= + 3 x 212= + 6

= 18

1

11

3 x 6

1 11 11

1 1 111 1 111 1 11 1

1 1 1 1 1 1

3 x 4= + 3 x 212= + 6

3(= +4 2)


=6 x 7 ?


6 x 7 = ?


6 x 7 = ?


6 x 7 = ?


6 x 7 = ?


6 x 7 = ?


= 426 x 7


= ?6 x 7


= ?6 x 7


= ?6 x 7


= ?6 x 7

5

6


= ?6 x 7

5

6 30


= ?6 x 7

5

6 30

= 6 x 5


= ?6 x 7

5

6 30

= 6 x 5

2


= ?6 x 7

5

6 30

= 6 x 5

2

12

+ 6 x 2


6 x 75

6 30= 6 x 5

2

12+ 6 x 2

= 30 + 12

= 42


= ?6 x 7


= ?6 x 7


= ?6 x 7


=

5

5

2

1

6 x 7


=

5

5

2

1

5 x 5 5 x 2 1 x 5 1 x 2+ + +

6 x 7


6 x 7=

5

5 25

2

10

1

5 x 5 5 x 2 1 x 5 1 x 2+ + +

= 25 10 5 2+ + +

5 2

5 x 14 = ?“5 groups of 14”

11111

1 1 1 1 1 1 1 1 1 1 1 1 1 1

5 x 14 = ?“5 groups of 14”

11111

1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1

5 x 14 = ?“5 groups of 14”

11111

1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1

5 x 14 = ?“5 groups of 14”

11111

1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1

5 x 14 =“5 groups of 14”

11111

1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1

x 105

5 x 14 =“5 groups of 14”

11111

1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1

+x 105 x 45

5 x 14 =“5 groups of 14”

11111

1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1

+x 105 x 45

5 x 14 =“5 groups of 14”

11111

1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1

+(105 )45


BASE 10 BLOCKS

1001,000 10 1

5 x 14 = ?“5 groups of 14”

“5 groups of 10 plus 5 groups of 4”or

1 1 1 1

11111

1 1 1 1 1 1 1 1 1 1 1 1 1 110

5

11111

5 x 14 = ?“5 groups of 14”


1 1 1 1

11111

1 1 1 1 1 1 1 1 1 1 1 1 1 110

5

11111

10

5 x 14 = ?“5 groups of 14”


1 1 1 1

11111

1 1 1 1 1 1 1 1 1 1 1 1 1 110

5

11111

1010

5 x 14 = ?“5 groups of 14”


1 1 1 1

11111

1 1 1 1 1 1 1 1 1 1 1 1 1 110

5

11111

101010

5 x 14 = ?“5 groups of 14”


1 1 1 1

11111

1 1 1 1 1 1 1 1 1 1 1 1 1 110

5

11111

10101010

5 x 14 = ?“5 groups of 14”


1 1 1 1

11111

1 1 1 1 1 1 1 1 1 1 1 1 1 110

5

11111

1010101010

5 x 14 = ?“5 groups of 14”


1 1 1 1

11111

1 1 1 1 1 1 1 1 1 1 1 1 1 110

5

11111

1010101010

1 1 1 1

5 x 14 = ?“5 groups of 14”


1 1 1 1

11111

1 1 1 1 1 1 1 1 1 1 1 1 1 110

5

11111

1010101010

1 1 1 11 1 1 1

5 x 14 = ?“5 groups of 14”


1 1 1 1

11111

1 1 1 1 1 1 1 1 1 1 1 1 1 110

5

11111

1010101010

1 1 1 11 1 1 11 1 1 1

5 x 14 = ?“5 groups of 14”


1 1 1 1

11111

1 1 1 1 1 1 1 1 1 1 1 1 1 110

5

11111

1010101010

1 1 1 11 1 1 11 1 1 11 1 1 1

1 1 1 1

5 x 14 = ?“5 groups of 14”

11111

1 1 1 1 1 1 1 1 1 1 1 1 1 110

5


11111

1010101010

1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1

4 x 12 = ?“4 groups of 12”


1 1

1111

1 1 1 1 1 1 1 1 1 1 1 110

1111

4 x 12 = ?“4 groups of 12”


1 1

1111

1 1 1 1 1 1 1 1 1 1 1 110

1111

1 1101 1101 1101 110


13 x 14 = ?


13 x 14 = ?1 110 1 1

10

1

1

1


13 x 14 = ?1 110 1 1

10

1

1

1

100


13 x 14 = ?1 110 1 1

10

1

1

1

100 10 10 10 10


13 x 14 = ?1 110 1 1

10

1

1

1

100 10 10 10 10

10

10

10


13 x 14 = ?1 110 1 1

10

1

1

1

100 10 10 10 10

10

10

10

1 1 1 1

1 1 1 1

1 1 1 1


13 x 14 = 1821 110 1 1

10

1

1

1

100 10 10 10 10

10

10

10

1 1 1 1

1 1 1 1

1 1 1 1


12 x 15 = ?1 110 1 1 1

10

1

1


1 110 1 1 1

10

1

1

100

12 x 15 = ?


1 110 1 1 1

10

1

1

100 10 10 10 10 10

12 x 15 = ?


1 110 1 1 1

10

1

1

100 10 10 10 10 10

10

10

12 x 15 = ?


1 110 1 1 1

10

1

1

100 10 10 10 10 10

10

10

1 1 1 1 1

1 1 1 1 1

12 x 15 = ?


1 110 1 1 1

10

1

1

100 10 10 10 10 10

10

10

1 1 1 1 1

1 1 1 1 1

12 x 15 = 180


17 x 23 = ?


17 x 23 = ?


17 x 23 = ?


17 x 23 = ?


17 x 23 = ?


17 x 23 = 391

22 x 26= ?

22 x 26= ?

22 x 26= ?

22 x 26= ?

22 x 26= ?

22x

26

The “Standard” Algorithm

22x

26


22x

262

1The “Standard” Algorithm

22x

262


22x

262


22x

262

1

12


22x

26

1

213


22x

26

1

213


22x

26

1

213


22x

26

1

0213


22x

26

1

04213


22x

26

1

04213


22x

26

2

1

104

3


22x

26

2

1

104

3


22x

26

1

044213


22x

26

1

044213


+

22x

26

1

044213


+

275


Arrays & Area Models

22x

26

1

044213

+

275



22x

26

1

044213

+

275



22x

26

1

044213

+

275

132



22x

26

1

044213

+

275

13222

6



22x

26

1

044213

+

275

132



22x

26

1

044213

+

275

132



22x

26

1

044213

+

275

132



22x

26

1

044213

+

275

132440



22x

26

1

044213

+

275

13244022

20



22x

26

1

044213

+

275

13244022

20 6



22x

26

1

044213

+

275

13244022

206

572

26


A “Conceptual“ Algorithm

22x

26


22x

26

1

04421 3

+

275



22x

26


22x

26

1

04421 3

+

275



22x

26

12 (6 x 2)


22x

26

1

04421 3

+

275



22x

26

12 (6 x 2)

(6 x 20)


22x

26

1

04421 3

+

275



22x

26

12 (6 x 2)

120 (6 x 20)


22x

26

1

04421 3

+

275



22x

26

12 (6 x 2)

120 (6 x 20)

(20 x 2)


22x

26

1

04421 3

+

275



22x

26

12 (6 x 2)

120 (6 x 20)

40 (20 x 2)


22x

26

1

04421 3

+

275



22x

26

12 (6 x 2)

120 (6 x 20)

40 (20 x 2)

(20 x 20)


22x

26

1

04421 3

+

275



22x

26

12 (6 x 2)

120 (6 x 20)

40 (20 x 2)

400 (20 x 20)


22x

26

1

04421 3

+

275



22x

26

12 (6 x 2)

120 (6 x 20)

40 (20 x 2)

400 (20 x 20)

572

+


22x

26

1

04421 3

+

275

9 x 12 = ?“9 groups of 12”

111111111

1 1 1 1 1 1 1 1 1 1 1 1

9 x 12 = ?“9 groups of 12” 1

11111111

1 1 1 1 1 1 1 1 1 1 1 110 2

5

4

“5 groups of 10plus

4 groups of 2”

or

9 x 12 = ?111111111

1 1 1 1 1 1 1 1 1 1 1 110 2

5

4

?

“9 groups of 12”


4 groups of 2”

or

9 x 12 = ?111111111

1 1 1 1 1 1 1 1 1 1 1 110 2

5

4

? ?



4 groups of 2”

or

9 x 12 = ?111111111

1 1 1 1 1 1 1 1 1 1 1 110 2

5

4

?

?

?



4 groups of 2”

or

9 x 12 = ?111111111

1 1 1 1 1 1 1 1 1 1 1 110 2

5

4

?

?

?

?



4 groups of 2”

or

9 x 12 = ?111111111

1 1 1 1 1 1 1 1 1 1 1 110 2

5

4

?

?

?

?



4 groups of 2”

or

9 x 12 = ?111111111

1 1 1 1 1 1 1 1 1 1 1 110 2

5

4

50

40

10

8



4 groups of 2”

or

9 x 12 = ?

111111111

1 1 1 1 1 1 1 1 1 1 1 110 2

5

4

50

40

10

8



4 groups of 2”

or

9 x 12

= (5 + 4)(10 + 2)

= (5 + 4)(10 + 2)

= (5 + 4)(10 + 2)

= (5 + 4)(10 + 2)

= (5 + 4)(10 + 2)

= (5 + 4)(10 + 2)

First

= (5 + 4)(10 + 2)

First

Outside

= (5 + 4)(10 + 2)

First

Outside

Inside

= (5 + 4)(10 + 2)

First

Outside

InsideLast

= (5 + 4)(10 + 2)

First

Outside

InsideLast“FOIL”

= (5 + 4)(10 + 2)

First

Outside

InsideLast

“FOIL”

“BEDMAS”

= (5 + 4)(10 + 2)

First

Outside

InsideLast

“FOIL”

“BEDMAS”

9 x 12

111111111

1 1 1 1 1 1 1 1 1 1 1 110 2

5

4

50

40

10

8

= (5 + 4)(10 + 2)

= (5 + 4)(10 + 2)

First

Outside

InsideLast

“FOIL”

What’s My Area Equation?


x x x x


x x x x111


x x x x111


x x x x111

Area = 3( )4x


x x x x111

x x x xx x x xx x x x

Area = 3( )4x= 12x units2

xx x x

x

Area = 2x ( )3x

xx x x

x

x2

Area = 2x ( )3x

x2xx x x

x

x2

Area = 2x ( )3x

x2x2xx x x

x

x2

Area = 2x ( )3x

xx x x

x

x2 x2 x2

x2 x2 x2

Area = 2x ( )3x

xx x x

x

x2 x2 x2

x2 x2 x2

Area = 2x ( )3x

xx x x

x

x2 x2 x2

x2 x2 x2

Area = 2x ( )3x

xx x x

x

x2 x2 x2

x2 x2 x2

6x2Area = 2x ( )3x =

What’s My Area Equation?x

1 1

1

1

x

1

x x

Area = (x + 3)(3x + 2)x

1 1

1

1

x

1

x x

x1 1

1

1

x

1

x x

x2 x2 x2

x x xx x xx x x

x x

1

11

1

11

Area = (x + 3)(3x + 2)

x1 1

1

1

x

1

x x

x2 x2 x2

x x xx x xx x x

x x

1

11

1

11

Area = (x + 3)(3x + 2) = 3x2 + 11x + 6

Concreteness Fading

1 2 3Enactive

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Concreteness Fading

1 2 3Enactive

Concrete

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22x

26

1

04004

21 3

+

275

PROCEDURAL FLUENCY


ADAPTIVE REASONING



kylep.ca/gecdsbvision

Math Proficiencies


Ability to formulate, represent & solve mathematical problems using an effective strategy


PROCEDURAL FLUENCY

Understanding and using a variety of mathematical procedures

ADAPTIVE REASONINGCapacity for logical thought, reflection,

explanation, and justification

Inclination to see mathematics as useful and valuable.

Ability to understand mathematical concepts, operations, and relationships


Leading Mathematics Learning: Building Confidence

What’s your next best move?

In Math ClassTable Talk…What would we see, hear and feel in an exemplary math class?

Write one idea per sticky note.

Identify the groups with a heading/theme/category.

Pedagogical System

Non-threateningClassroom

Environment

InstructionalTask

ToolsandRepresentations

ClassroomDiscourse

In Math Class

Table Talk…

Fold the chart paper into quarters to reflect the 4 aspects of the pedagogical system.

Do your ideas fit/match these categories?

What other ideas can you add?

InstructionalTask

Non-threateningClassroom

Environment

Tools andRepresentations

ClassroomDiscourse

Pedagogical System: Understanding Task

What is a Math Task?Any problem or set of problems that focuses students' attention on a particular mathematical idea and/or provides an opportunity to develop or use a particular mathematical habit of mind.

High or Low Cognitive DemandThe cognitive demand of a task is the level of cognitive engagement needed to complete the task (Stein et al. 2009).

A task by itself is not rich;it is what we do with the task and how it connects to the pedagogical

system that makes it rich.

Understanding Task

Bump It UpBumpupataskasanAssessment forLearning•UsetheTaskCardsatyourtable

•Grade,Topic,OverallExpectationandaTask

•UsetheMathematicsCurriculumandfindthespecificexpectations

• Re-writethetask

Task: Assessment for Learning

TomSchimmer’s (GradingFromtheInsideOut,2016)premiseisthatallassessmentpracticesshouldbeputthroughtwofilters:

1.Isitaccurate?2.Doesitpromoteconfidence/optimisminstudents?

“Schoolisnolongeraboutthecompletionofaseriesofactivities,butratherthepursuitofproficiencyasasetofoutcomesthatstudentsachievethroughtheinstructional experience”

UnderstandingMathTasksGrade2Topic: Counting

OverallExpectations: read,represent, compare,andorderwholenumbers to100,anduseconcretematerials torepresent fractions andmoneyamounts to100¢

SpecificExpectation(s): countforwardby1’s,2’s,5’s,10’s,and25’sto200,usingnumber lines andhundredscharts,starting frommultiples of1,2,5,and10(e.g.,countby5’sfrom15;countby25’sfrom125)

TaskCountby2s

Task: Assessment for Learning

Whatwasyourtask?

Whatdidyounoticeabout thecurriculum?

What/Howcanthetaskbemodified, refined,extended tosupport ALLstudents?CCoonnssiiddeerr- studentswithpersistent learningchallenges- students identified asgifted

Howcouldyouusethis learning toleadmath learning inyourschool?

Understanding Task: Consolidation

UnderstandingMathTasks

If you deny students the opportunity toengage in this activity – to pose their ownproblems, to make their own conjectures anddiscoveries, to be wrong, to be creativelyfrustrated, to have an inspiration, to cobbletogether their own explanations and proofs –you deny them mathematics itself.

Paul Lockhart, A Mathematician’s Lament, 2009

Mathematics Learning

UnderstandingMathTasks

Quotationsaboutmathematicsbymathematicians.• Choose1thatconnectswithyourthinking• Explainyourchoiceandyourthinking

BuildingConfidenceinourNextBestMove…Whatisyournextbestmove?

Leading Math Learning in Our Schools

Bringstudentworkbased onamathematics task.ConsiderCurriculum expectationAssessment forLearning

Wewillanalyzethetaskinthecontext ofpedagogical system.

For Next Time…

Whatwasone(ormore)keylearning(s)fromtoday?

Whatwasonethingyouwouldhavechangedintheday?

Whatquestionsdoyoustillhave?

Whatlearningwouldyouliketoseefornextsession?

Feedback

GECDSB Mathematics Learning Teams (MLT) Session #1

Education