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Rich Tasks and Productive Struggle Effective Teaching with

Principles to Action Institute, 2015

John SanGiovanni Elementary Mathematics Supervisor Howard County Public School System

[email protected] @JohnSanGiovanni

Table Introductions

• Your name • Your favorite number

5 Howard County Public School System

How to play:

1. Players take turns rolling a number cube

2. After each roll, a player decides which column to place the digit.

3. That player then adds the value to his/her total.

4. The player who is closest to the target (in the last total) without going over the target wins.

Choose 1 to discuss with a partner

• How might you use this game in your mathematics classroom?

• How might you modify this game for your students?

• What would you look for while your students played the game?

Target 1 Tenths Hundredths Total

6 .06

2 .26

7 .33

5 .38

3 .68

2 .70

5 .75

1 .85

Decimals on a Hundred Chart .01 .02 .03 .04 .05 .06 .07 .08 .09 .10

.11 .12 .13 .14 .15 .16 .17 .18 .19 .20

.21 .22 .23 .24 .25 .26 .27 .28 .29 .30

.31 .32 .33 .34 .35 .36 .37 .38 .39 .40

.41 .42 .43 .44 .45 .46 .47 .48 .49 .50

.51 .52 .53 .54 .55 .56 .57 .58 .59 .60

.61 .62 .63 .64 .65 .66 .67 .68 .69 .70

.71 .72 .73 .74 .75 .76 .77 .78 .79 .80

.81 .82 .83 .84 .85 .86 .87 .88 .89 .90

.91 .92 .93 .94 .95 .96 .97 .98 .99 1.00

Outcomes

• Examine productive and unproductive beliefs about teaching and learning mathematics.

• Identify the characteristics of tasks that promote reasoning.

• Consider the role of productive struggle in elementary mathematics.

8

About the Institute: Effective Teaching Practices

1.

1.

2-4. 2-4.

2-4.

2-4.

5.

5.

Thoughts About Teaching and Learning Mathematics

10

• Brainstorm thoughts or beliefs that different stakeholders have about teaching learning mathematics.

• Capture your ideas on your web. • Describe a theme for your web.

Beliefs / Thoughts about Teaching

and Learning Mathematics

teachers

family

Admin / Leadership

students

Community

Media

Thoughts About Teaching and Learning Mathematics

11

• Examine ideas of other groups during the gallery walk. • √ ideas that appeared on your web. • P = Productive belief • U = Unproductive belief

Beliefs / Thoughts about Teaching

and Learning Mathematics

teachers

family

Admin / Leadership

students

Community

Media

UNPRODUCTIVE Beliefs about Teaching and Learning Mathematics

13

PRODUCTIVE Beliefs about Teaching and Learning Mathematics

14

Success for All by…

• Engaging with challenging tasks. • Connecting new learning with prior

knowledge. • Acquiring conceptual and procedural

knowledge. • Constructing knowledge through discourse. • Receiving meaningful and timely feedback • Developing metacognitive awareness.

15

About the Institute: Effective Teaching Practices

1

1

2-4 2-4

2-4

2-4

5

5

Implement Tasks that Promote Reasoning and

Problem Solving

What does rigor look like in mathematics

classrooms?

With a shoulder buddy…

MATHEMATICAL RIGOR

Proc

edur

al F

luen

cy

SYNTHESIS AND EVALUATION Putting together elements & parts to form a

whole, then making value judgments about the method.

EXTENDED THINKING

Requires an investigation; time to think and process multiple conditions of the problem or task.

BLOOM’S TAXONOMY WEBB’S DOK

KNOWLEDGE The recall of specifics and universals, involving little more than bringing to mind the appropriate material.

COMPREHENSION Ability to process knowledge on a low level such that the knowledge can be reproduced or

communicated without a verbatim repetition.

APPLICATION The use of abstractions in concrete situations.

ANALYSIS The breakdown of a situation into its component

parts.

RECALL

Recall of a fact, information, or procedure (e.g., What are 3 critical skill cues for the overhand throw?)

SKILL/CONCEPT

Use of information, conceptual knowledge, procedures, two or more steps, etc.

STRATEGIC THINKING

Requires reasoning, developing a plan or sequence of steps; has some complexity; more than one possible answer

Wyoming School Health and Physical Education Network (2001). Standards, Assessment, and Beyond. Retrieved May 25, 2006, from http://www.uwyo.edu/wyhpenet

http://www.uwyo.edu/wyhpenet

How do tasks and teaching impact mathematics learning?

Reflect:

Stein, M.K. & Lane, S. (1996). Instructional tasks and the development of student capacity to think and reason: An analysis of the relationship between teaching and learning in a reform mathematics project. Educational Research and Evaluation 2(4), 50-80.

So what does THIS mean?

Task Quality Results Implementation

High High High

High or Low Low

High Low Moderate

A

B

C

Low

Research on Tasks

• Not all tasks provide the same opportunities for student thinking and learning (Hiebert et al. 1997; Stein et al. 2009).

• Student learning is greatest in classrooms where the tasks consistently encourage high-level student thinking and reasoning and least in classrooms where the tasks are routinely procedural in nature (Boaler and Staples 2009; Hieber and Wearne 1993; Stein and Lane 1996)

Table Talk

What are characteristics of quality mathematics tasks?

What do high-level tasks look like?

1. Read each task.

1. Sort the tasks by their cognitive level (Lower or Higher Level of Cognitive Demand).

1. Be prepared to defend your reasoning.

What do they have in common?

• Review your sorting. • What do higher-level tasks have in common? • What do lower-level tasks have in common? • Where do you see evidence of procedure,

concept, and application in these tasks?

Levels of Demands Lower-level demands

(memorization): • reproducing previously learned facts, rules, formulas, definitions or committing them to memory • Cannot be solved with a procedure • Have no connection to concepts or meaning that underlie the facts rules, formulas, or definitions

Higher-level demands (procedures with connections):

• use procedure for deeper understanding of concepts • broad procedures connected to ideas instead narrow algorithms • usually represented in different ways • require some degree of cognitive effort; procedures may be used but not mindlessly

Lower-level demands (procedures without connections):

• are algorithmic • require limited cognitive demand • have no connection to the concepts or meaning that underlie the procedure • focus on producing correct answers instead of understanding • require no explanations

Higher-level demands (doing mathematics):

• require complex non-algorithmic thinking • require students to explore and understand the mathematics • demand self-monitoring of one’s cognitive process • require considerable cognitive effort and may involve some level of anxiety b/c solution path isn’t clear

Leinwand, S., D. Brahier, and D. Huinker . Principles to Action. Reston, VA: National Council of Teachers of Mathematics, 2014 (pg 18)

Lower-level demands (memorization):

• reproducing previously learned facts, rules, formulas, definitions or committing them to memory

• cannot be solved with a procedure • have no connection to concepts or meaning

that underlie the facts rules, formulas, or definitions

Leinwand, S., D. Brahier, and D. Huinker . Principles to Action. Reston, VA: National Council of Teachers of Mathematics, 2014

Lower-level demands (procedures without connections):

• are algorithmic • require limited cognitive demand • have no connection to the concepts or

meaning that underlie the procedure • focus on producing correct answers instead of

understanding • require no explanations


Higher-level demands (procedures with connections):

• use procedure for deeper understanding of concepts

• broad procedures connected to ideas instead narrow algorithms

• usually represented in different ways • require some degree of cognitive effort;

procedures may be used but not mindlessly


Higher-level demands (doing mathematics):

• require complex non-algorithmic thinking • require students to explore and understand

the mathematics • demand self-monitoring of one’s cognitive

process • require considerable cognitive effort and may

involve some level of anxiety b/c solution path isn’t clear


Are there any you want to move?

Lower-Level Tasks • D • C • F • G • I • J • O • P • Q • S

Higher-Level Tasks • A • B • E • H • K • L • M • N • R • T

Some Bear Traps

Adjusting..

How can you make it better?

• Select 2 lower-level tasks. • Work with a partner to talk about how you

could make these tasks higher-level. • Record your ideas on the back of the task.

Bumper Cars

• Take 1 of the improved tasks and bump in to someone else.

• Share how you can improve the task. • Trade tasks. • Bump into someone else.

Support Productive Struggle in

Mathematics

More Research on Tasks

• Tasks with high cognitive demands are the most difficult to implement well, and are often transformed into less demanding tasks during instruction (Stein, Grover, and Henningsen 1996; Stigle and Hiebert 2004)

Think-Pair-Share

• How do you know when a student is struggling?

• What do you do?

42

Find the Picture

Which of the following pictures best describes your productive struggle?

A Destructive Struggle

• Leads to frustration. • Makes learning goals feel

hazy and out of reach. • Feels fruitless. • Leaves students feeling

abandoned and on their own.

• Creates a sense of inadequacy.

Productive Struggle • Leads to understanding. • Makes learning goals feel

attainable and effort seem worthwhile.

• Yields results. • Leads students to feelings of

empowerment and efficacy. • Creates a sense of hope.

Which One?

Productive Struggle

What do students look like when they are engaging in productive struggle?

What do Teachers look like when they are facilitating productive struggle?

Productive Struggle: Student Behaviors

• Struggling at times but know that learning comes from confusion and struggle.

• Asking questions to help them understand the task • Persevering in solving problems and realizing that is

ok to say, “I don’t know how to proceed here,” but it is not ok to give up.

• Helping one another without telling their classmates what the answer is or how to solve the problem.

Principles to Action (2014) pg 52

Productive Struggle: Teacher Behaviors

• Anticipating what students might struggle with and being prepared to support them productively.

• Giving students time to struggle, and asking questions that scaffold students’ thinking without stepping in.

• Helping students realize that confusion and errors are a part of learning, by facilitating discussions on mistakes, misconceptions, and struggles.

• Praising students for their efforts in making sense and perseverance.

Principles to Action (2014) pg 52

How Can We Support the Development of Productive Struggle?

• Choose rigorous tasks! • Determine what it looks like in students. • Make it an explicit focus in classrooms. (Tell students

we are doing this!) • Focus on productive behaviors instead of

intelligence. • Communicate with families.

Mindset and Productive Struggle

Fixed Mindset • Understanding,

proficiency, ability are “set”

• You are good at something or you aren’t

Growth Mindset • Understanding,

proficiency, ability are developed regardless of your genes

• You become better at something as you work with it – as you struggle with it

60 Dweck, 2008

Why isn’t there a magic bullet for developing productive struggle?

61

Our personal growth…

62

Session 1 Closing

What are 2 ideas from this session that resonate with you?

63

Disclaimer The National Council of Teachers of Mathematics is a public voice

of mathematics education, providing vision, leadership, and professional development to support teachers in ensuring equitable mathematics learning of the highest quality for all students. NCTM’s Institutes, an official professional development offering of the National Council of Teachers of Mathematics, supports the improvement of pre-K-6 mathematics education by serving as a resource for teachers so as to provide more and better mathematics for all students. It is a forum for the exchange of mathematics ideas, activities, and pedagogical strategies, and for sharing and interpreting research. The Institutes presented by the Council present a variety of viewpoints. The views expressed or implied in the Institutes, unless otherwise noted, should not be interpreted as official positions of the Council.

64

65

Connecting Representations Fraction Equivalency and Comparison



Your name

What is a fraction that describes you?

¼ Howard County Public School System

The Fraction Game

• Make a game board with halves, thirds, fourths, fifths, sixths, eighths, and tenths.

• Take turns rolling fractions with dice to create a fraction.

• Players remove the fraction or an equivalent of what they have rolled.

• The first player to remove all of their pieces wins.

4 Adapted from Illuminations’ The Fraction Game

http://illuminations.nctm.org/Activity.aspx?id=4148

The Fraction Game

• Player 1 rolls a 3 and a 4 to make 3/4.

• Player 1 rolls a 1 and a 3 to make 1/3

• Player rolls a 1 and a 3 to make 1/3 again.

5 Adapted from Illuminations’ The Fraction Game

http://illuminations.nctm.org/Activity.aspx?id=4148





Outcomes for this session

• Examine how representations develop understanding of fraction equivalency and comparison.

• Examine how representations can be applied to problem solving situations.

• Add to our toolbox of classroom ready strategies.

7

Use and Connect Representations

Why Representations?

“Effective mathematics teaching includes a strong focus on using varied mathematical representations.” “Using different representations is like examining the concept through a variety of lenses, with each lens providing a different perspective that makes the (concept) richer and deeper.”

10 10 NCTM (2014) pg 24

So what are these lenses?

Julie shares that

and are equivalent...

Do you agree?

Use 2 different

representations to

justify your thinking.

134

168

How did you represent your understanding?

Julie shares that

and are equivalent...

• Visual

• Physical

• Contextual

• Verbal

• Symbolic

134

168

Deeper Understanding through Representations

When students learn to represent, discuss, and make connections among mathematical ideas in multiple forms, they demonstrate deeper mathematical understanding and enhanced problem-solving abilities.

13

Fuson, Kalchman, and Fransford 2005; Lesh, Post, and Behr 1987

13 NCTM (2014) pg 26

Connections through Representations

14

68


15


16

Developing Understanding Through Representations

Visual representations are of particular importance in the mathematics classroom, helping students to advance their understanding of mathematical concepts and procedures, make sense of problems, and engage in mathematical discourse.

17

(Arcavi 2003; Stylianou and Silver 2004)

Equivalency through Representations

18

Establishing a/b = (nxa)/nxb) with Paper Folding

• Fold a piece of paper and shade ½ of it.

• Close it and fold it in half again.

• Open it. What do you notice?

• What would happen if we folded it again?

• How can you show that 3/2 = 12/8 with paper folding?

• Find 3 sets of equivalent fractions you can show with paper folding?

Establishing a/b = (nxa)/nxb)

2 3


3

3x5


2

2x5


23

2x5 3x5

10 15

= =

Equivalency with Cuisenaire

White is ⅓ of light green. Find pairs of colors that show ⅓?

What did you find?

http://nrich.maths.org/4348


• Light green is 1/3 of brown. • Red is 1/3 of dark green. • Yellow is ¼ of (orange + orange). • Purple is ¼ of (orange + dark green). • Red is 1/7 of (orange + yellow).

True or false? Have students

write their own.



• ¼ of (orange + dark green) = ??? • 1/5 of ??? = red • ½ of (orange + brown) = ??? • 1/9 of ??? = white • 1/10 of (orange + orange) = ???

Fill in the Missing Part



• 3 whites/beige = ??? of black • 5 light greens = ??? of (orange + brown) • 3 dark greens = ??? of (2 orange + purple)

Fill in the Missing Fraction


Good Questions for Journal Prompts

• Is 3/4 equivalent to 6/8?

• What are 3 fractions equivalent to 3/5? How do you know?

• Show 2 ways to prove that a fraction is equivalent to another fraction?

• What patterns do you notice in equivalent fractions?

Supporting Fluency Challenges X 0 1 2 3 4 5 6 7 8 9 10 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 7 8 9 10 2 0 2 4 6 8 10 12 14 16 18 20 3 0 3 6 9 12 15 18 21 24 27 30 4 0 4 8 12 16 20 24 28 32 36 40 5 0 5 10 15 20 25 30 35 40 45 50 6 0 6 12 18 24 30 36 42 48 54 60 7 0 7 14 21 28 35 42 49 56 63 70 8 0 8 16 24 32 40 48 56 64 72 80 9 0 9 18 27 36 45 54 63 72 81 90 10 0 10 20 30 40 50 60 70 80 90 100

Use ratio tables to help students identify equivalent fractions after the concept is understood.

Comparison through Representations

32

Think-Pair-Share

1. Fold a piece of paper in fourths.

2. Choose 4 forms of representations to compare

68

46

Applying understanding acquired through representations

34

27

35

89

26

1418

1420

1314

1516

36

1520

56

58

1617

3034

712

920

• Compare the following fractions. • Talk with your table about how you

compared the fractions. • We will share our ideas with the group.

What might these representations look like? Procedural

Conceptual

Find common denominators

Cross multiply (bow-tie,

butterfly, the x)

Compare to 1/2

Same number of pieces

Size of pieces

Think-Pair-Share

How do representations help us compare the fractions without finding common denominators?

36

Are all representations created equally?

• 8/9 or 7/8 • 4/6 or 7/12 • 3/5 or 3/7 • 5/8 or 6/10 • 7/12 or 5/12

• Area Model: circle • Area Model: rectangle • Set Model • Number Line • Measure

Which fraction representation(s) works best?

Representations for Solving Problems

38

Students should be able to approach a problem from several points of view and be encouraged to switch among representations until they are able to understand the situation. (NCTM, 2014)

NCTM (2014) pg 26 Huinker 2013; Stylianou and Silver 2004)

Tangram Values

You should have 7 pieces.

• 2 large triangles (f and g)

• 2 small triangles (c and e)

• 1 medium-sized triangle (a)

• 1 square (d)

• 1 parallelogram (b)

Tangram Values

What fraction of the whole does each piece represent?

Tangram Values

If A, C, E make the whole:

What does C represent?

Tangram Values

If D is the whole:

What does G represent?

Tangram Values

Find as many sets of equivalencies as you can.

A Problem to Ponder…

What are 3 fractions between and ?

49

35

910

Uncomplicating Fractions (2014)

Use pictures, numbers, and/or words to explain your findings.


2 different improper fractions can be written as

a mixed number in the form . What

might the improper fractions be? Use pictures,

numbers, and/or words to explain your findings.

50 Adapted from Uncomplicating Fractions (2014)

1

What fraction of the large square is shaded…

Is more than ½ of the square shaded? Is more than ¼ of the square shaded?


2 groups used Cuisenaire rods to show that ½ is equivalent to 2/4. But they used different rods. Is this possible? What different rods might they have used?

54

Think-Pair-Share

Jackson shares that

and are

equivalent because

they both are missing

3 pieces. Do you agree?

Use 2 different representations to

justify your thinking.

710

58

Representations for Justification

• Write 2 fractions. • What is a fraction that is between these 2 fractions. • Use pictures, numbers, and/or words to justify your thinking.


These fractions are written in order from least to greatest. What are the missing numbers?

57 Uncomplicating Fractions (2014)

Use pictures, numbers, and/or words to explain your findings.

3

2

3

2

5 8


Use pictures, numbers, and/or words to justify your thinking.


Use pictures, numbers, and/or words to justify your thinking.

Developing Representational Competence

1. Encourage purposeful selection of representations.

2. Engage in dialogue about explicit connections among representations.

3. Alternate the direction of the connections made among representations

(Marshall, Superfine, and Canty (2010)

60

When this happens… What are students doing?

• Using multiple forms of representations to make sense and understanding.

• Describing and justifying their understanding with drawings, diagrams, and other representations.

• Making choices about representations. • Contextualizing mathematical ideas by

connecting them to real-world situations • Considering advantages of using various

representations. 61

When this happens… What are WE doing?

• Selecting tasks that allow students to decide which representations to use

• Allocating substantial instructional time for students to use, discuss, make connections

• Introducing forms of representations. • Focusing students’ attention on the mathematical

ideas • Designing ways to elicit and assess students’

abilities to use representations. 62



63

64

Connecting Representations Place Value and Problem Solving



Your name

What is your favorite number?

5 Howard County Public School System

Knock ‘Em Down

• Players take turns rolling 2 digit dice.

• Players make an addition or subtraction equation to cover one of the “pins”

• The first player to cover all of the “pins” wins 1

2 3

5 4 6

9 10 7 8

Jennifer Throndsen • Teaching Children Mathematics • May 2014, Volume 20, Issue 9, pg 584

Knock ‘Em Down

• Player 1 rolls 7 and 3.

1

2 3

5 4 6

9 10 7 8








Knock ‘Em Down

• Pull a domino. Use the pips to find a sum or difference represented on the board.

1

2 3

5 4 6

9 10 7 8


Outcomes for this session

• Examine how representations can be applied to problem solving situations.

• Examine how representations develop understanding of number, place value, and base ten.

• Add to our toolbox of classroom ready strategies.

9

Use and Connect Mathematical

Representations

Why Representations?

“Effective mathematics teaching includes a strong focus on using varied mathematical representations.” “Using different representations is like examining the concept through a variety of lenses, with each lens providing a different perspective that makes the (concept) richer and deeper.”

11 11 NCTM (2014) pg 24

So what are the lenses?

A 3-digit number has more ones than tens.

What are 3 possible numbers?

Use 2 different representations to justify your thinking.

How did you represent your understanding?

3-digit numbers…

• Visual

• Physical

• Contextual

• Verbal

• Symbolic


Sam created a number using 9 base-ten blocks.

What number did Same create?

Use representations to justify your thinking.


How many different numbers can be made with 9 base ten blocks?

Use representations to justify your solutions.


Use base ten blocks to show each number in 2 different ways.

43 26


When students learn to represent, discuss, and make connections among mathematical ideas in multiple forms, they demonstrate deeper mathematical understanding and enhanced problem-solving abilities.

18

Fuson, Kalchman, and Fransford 2005; Lesh, Post, and Behr 1987

18 NCTM (2014) pg 26

What is 12?

• Think of as many different ways to show 12 that you can.

• We will come together as a group in a few minutes to share our ideas.

Students to naturally connect ideas they already have to

construct a new idea.

Van de Walle, J. (2007). Elementary and Middle School Mathematics: Teaching Developmentally (6th ed). Boston, MA: Pearson

Representations enable….

Students to naturally connect ideas they already have to

construct a new idea.

Van de Walle, J. (2007). Elementary and Middle School Mathematics: Teaching Developmentally (6th ed). Boston, MA: Pearson

Representations enable….

The more ideas used and the more connections made, the better we understand.


24


25

Number Webs

How many expressions (49 + 1) can you write that are equivalent to 50?

50

O’Connell and SanGiovanni (2013)

Developing Quantitative Reasoning with Number Webs


30

Students should be able to approach a problem from several points of view and be encouraged to switch among representations until they are able to understand the situation. (NCTM, 2014)

NCTM (2014) pg 26 Huinker 2013; Stylianou and Silver 2004)

What’s the Question?

Think of an addition or subtraction word problem that would have this answer:

24 tigers

Angela Barlow, Janie Cates Teaching Children Mathematics January 2007, Volume 12, Issue 5, pg 252

Result Unknown Change Unknown Start Unknown

Add to

Take From

Total Unknown Addend Unknown Both Addends

Put Together / Take Apart

Difference Unknown Bigger Unknown Smaller Unknown

Compare


Add to

There were 8 seals in the water. 7 more seals dived in. How many seals are swimming altogether?

There are 8 seals at the zoo. How many more seals does the zoo need to get to have 15 altogether?

There were some seals out of the water. 7 more seals came out. Then there were 11 seals altogether. How many penguins were outside to start with?

Take From




Compare


Add to




Take From The monkey had 13 bananas. He ate 6. How many bananas were left?

The monkey had 9 bananas. He ate some and then he had 3 left. How many bananas did the monkey eat?

The monkey had some bananas. He ate 10. Then there were 6 bananas left. How many bananas did the monkey have to start with?




Compare


Add to









There were 5 brown bears and 2 polar bears climbing on the rocks. How many bears were there altogether?

There were 5 bears at the zoo. 2 of them were polar bears. How many were not polar bears?

There were 5 bears at the zoo. How many could be swimming and how many could be standing?


Compare


Add to









There were 5 brown bears and 2 polar bears climbing on the rocks. How many bears were there altogether?

There were 5 bears at the zoo. 2 of them were polar bears. How many were not polar bears?

There were 5 bears at the zoo. How many could be swimming and how many could be standing?


Compare The zoo has 7 African elephants. The zoo has 2 Asian elephants. How many more African elephants does the zoo have than Asian elephants?

There are 5 more female elephants than male elephants. There are 2 male elephants. How many female elephants are there?

There are 7 elephants at the zoo. That’s 5 more than baby elephants. How many baby elephants are there?

Representing the Problems…

• Equations • Pictures • Models • Words

37

Where does your problem fit?

24 tigers

Thinking vs Getting Answers

The clown gave my little brother 7 red balloons and some green balloons. Altogether my brother got 13 balloons. How many green balloons did he get?

Using “KEY” words

Elliott ran 6 times as far as Andrew. Elliott ran 4 miles. How far did Andrew run?

How many legs do 6 elephants have?

Clement, L. and Bernhard, J. (2005) “A Problem-Solving Alternative to Using Key Words” Mathematics Teaching in the Middle School, March 2005, 10-7 p. 360, NCTM

K-W-S for Problem Solving

The store has 13 cans of tennis balls on the shelf. Each can has 3 balls in it. How many tennis balls does the store have?

KNOW WANT SOLVE

What do I KNOW about the problem?

What do I WANT to find out? How will I SOLVE the problem?

Thinking vs Getting Answers: Solving in Different Ways

There were 17 brown cows in the barn. 8 black cows went in to the barn. How many cows were in the barn?

Picture

Equation Number Line

Estimate


46

pictures, number lines, diagrams, models

counters, blocks, drawings, acting out

equations

written, oral descriptions

Bar Models Joining Problem 1

Oscar has 7 blocks. His father gives him 8 more blocks. Now how many blocks does he have?

7 8

?

7 + 8 = ? 8 + 7 = ?


Mom made 7 plates for dinner. How many more plates does she need to make 12?

7 ?

12

7 + ? = 12 ? + 7 = 12 12 - 7 = ?


Deryn has some points. She scored 5 more points. Then she had 9 points. How many points did she start with?

? 5

9

? + 5 = 9 5 + ? = 9 9 - 5 = ? 9 - ? = 5

Bar Models Separating Problem 1

Oscar has 11 french fries. He eats 7 of them. How many french fries are left?

7 ?

11

11 - 7 = ? 7 + ? = 11 ? + 7 = 11

Bar Models Separating Problem 2

Julie had 27 stickers. She gave some to Scott. Then Julie had 17 left. How many did she give to Scott?

17 ?

27

27 - 17 = ? 17 + ? = 27 ? + 17 = 27

Part-Part-Whole Problem 1

14 boys and 16 girls are in a 2nd grade class. How many students are in the class?

whole

part part 14 16

?

Part-Part-Whole Problem 2

Some markers and 12 crayons were in the art bin. There are 31 pieces in the art bin. How many markers were in the bin?

whole

part part 12 ?

31

How are they connected?

54

Focus on the Question The children were collecting seashells at the beach. The list

shows the number of shells collected by each child. Number of Shells Katie – 5 Bridget – 4 Jackie – 9 Allison – 12 Kim – 10

Turn and share:

Katie and Kim put their shells in a box. How many did they have altogether? Tell how you figured out the answer.

Monday


http://www.clipartguide.com/_pages/1386-0803-2514-4114.html



Turn and share:

Allison wanted to collect 16 shells. How many more will she need to collect? Tell how you figured out the answer.

Tuesday





Turn and share:

The children wanted to collect 50 shells altogether. Did they collect enough? How do you know?

Wednesday





Turn and share:

Write a question you could ask using the data.

Thursday



Leveraging Physical Representations

Kim is getting new fish for her aquarium. Her favorite fish is the clownfish. There were 2 different types of clownfish at the pet store. Her aquarium can hold 10 fish. She wants to fill her aquarium with both types of clownfish.

What are the different ways she can fill her aquarium with two different clownfish?

Leveraging Physical Representations

Use a bead counter to model the possibilities.

Write a number sentence for each possibility.

Developing Understanding Through Representations

Visual representations are of particular importance in the mathematics classroom, helping students to advance their understanding of mathematical concepts and procedures, make sense of problems, and engage in mathematical discourse.

61

(Arcavi 2003; Stylianou and Silver 2004)

C -> R -> A Concrete -> Representational -> Abstract

62

Arrow Math

1. Give a starting number.

1. Provide arrows to

find a new number. 1. Arrows indicate +/- 1

or +/- 10 on a hundred chart.

+10 -1 +10

+1 -10 -10

NumberSENSE (1997)

Numbers to 100 (With Hundred Charts)

Make It, Build It, Find It • Roll numbers with place value dice • Draw a picture of it • Put a counter on the hundred chart Four in A Row • Roll numbers with place value dice • Put a counter on the hundred chart • First to get 4 in a row wins

Mystery Number

301 302 303 304 305 306 307 308 309 310

311 312 313 314 315 316 317 318 319 320

321 322 323 324 325 326 327 328 329 330

331 332 333 334 335 336 337 338 339 340

341 342 343 344 345 346 347 348 349 350

351 352 353 354 355 356 357 358 359 360

361 362 363 364 365 366 367 368 369 370

371 372 373 374 375 376 377 378 379 380

381 382 383 384 385 386 387 388 389 390

391 392 393 394 395 396 397 398 399 400

1. A secret number is selected.

1. Players ask yes/no

questions but cannot ask what the number is. (Is it more than 50? Does it have a 7 in the ones place?)

1. Players keep track of

clues and take turns until someone finds the secret number.

Developing Representational Competence

1. Encourage purposeful selection of representations.

2. Engage in dialogue about explicit connections among representations.

3. Alternate the direction of the connections made among representations (C-R-A, A-R-C)

(Marshall, Superfine, and Canty (2010)

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When this happens… What are students doing?

• Using multiple forms of representations to make sense and understanding.

• Describing and justifying their understanding with drawings, diagrams, and other representations.

• Making choices about representations. • Contextualizing mathematical ideas by

connecting them to real-world situations • Considering advantages of using various

representations. 78

When this happens… What are WE doing?

• Selecting tasks that allow students to decide which representations to use

• Allocating substantial instructional time for students to use, discuss, make connections

• Introducing forms of representations. • Focusing students’ attention on the mathematical

ideas • Designing ways to elicit and assess students’

abilities to use representations. 79



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Eliciting and Using Evidence of Student Thinking

Effective Teaching with

Principles to Action Institute, 2015


[email protected]

Evidence of Student Thinking

Eliminate It!

O’Connell and SanGiovanni, Heinemann (2013)

21 63

17 84

Eliminate It!


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14

12

28

Eliminate It!


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38

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Outcomes

• Identify the purpose and strategies for formative assessment

• Develop the process of eliciting and using student thinking

• Connect the effective teaching practices that have been examined during the institute

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Think-Pair-Share: How are instruction and assessment related?

I I

I I

A A

A A

1 2

3 4

INSTRUCTION ASSESSMENT

Observations

Interviews

Conversations

Extended response tasks

Journals and reflections

Open-Ended Questions

Exit tickets Reports and projects

Portfolios

Summative Assessment Selected Response

Items

Formative Assessment is Like Looking for Icebergs

INFORMative Assessment (2012)

Formative Assessment is like GPS.

1. Where are we going? 2. Where are we? 3. How are we getting there?

Why Use Formative Assessment?

• To inform instruction and provide feedback to students (and you) on their learning.

The Big Picture • Where are you? –

formative assessment

• Where are you going? – standards

• How are you going to get there? – progressions

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What Beliefs Might We Hold about Mathematics Assessment?

At your table: Choose two belief statements and: 1. Decide whether the statement is a

productive or unproductive belief. 2. Explain your reasoning. 3. Describe the point of view of someone you

know about the same belief statement. How might have that belief statement developed?

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Unproductive Belief Statements

• The primary purpose of assessment is accountability for students through report card marks or grades.

• Assessment in the classroom is an interruption of the instructional process.

• Only multiple-choice and other “objective” paper-and-pencil tests can measure mathematical knowledge reliably and accurately.

• A single assessment can be used to make important decisions about students and teachers.

• Assessment is something that is done to students. • Stopping teaching to review and take practice tests improves

students’ performance on high-stakes tests.

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Productive Beliefs Statements

• The primary purpose of assessment is to inform and improve the teaching and learning of mathematics.

• Assessment is an ongoing process that is embedded in instruction to support student learning and make adjustments to instruction.

• Mathematical understanding and processes can be measured through the use of a variety of assessment strategies and tasks.

• Multiple data sources are needed to provide an accurate picture of teacher and student performance.

• Assessment is a process that should help students become better judges of their own work, assist them in recognizing high-quality work when they produce it, and support them in using evidence to advance their own learning.

• Ongoing review and distributed practice within effective instruction are productive test preparation strategies.

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How Do We Know What Students Know?

Using the student work as your only

evidence, what do you believe

the student understands?

25

Investigating a Task/Question

Let’s consider… • What counts as evidence…

– Prerequisite/progression of concept

– Representations – Concepts/conceptual

understanding – Possible procedures/symbols

• How you might collect evidence?

Task/Question to investigate:

• Interpreting their thinking – Do your students understand? – How do you know?

• Making In-The-Moment Decisions – How is their work similar/different than what you

expected? – How do you reflect on what you expected? – How might you adapt what you expected? – Does anything surprise you?

• What might you change?

What Counts As Evidence

What Might Count As Evidence?

Compare. Explain how they could be compared without finding common denominators.

56

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Making Use of Evidence: What Do We Notice?

Making Use of Evidence: What Do We Notice:

Interpreting Student Thinking

Fractions on a Number Line The point shows ¾ on each number line. Write the missing endpoint for each number line in the box.

SanGiovanni, J. (2013) NCTM Winter Institute, Orlando FL

Fractions on a Number Line

How might our students find the missing endpoint? What strategies might they use? What misconceptions might they have?

Interpreting Student Thinking: What’s Happening Here?

1. 2.

3. 4.

Interpreting Student Thinking: What’s Happening Here?

Making In-the-moment Decisions

Interpreting Student Thinking

Students are solving ¾ + ½ • What might you expect to see? • How might you react?

IMAP • Integrating Mathematics and Pedagogy San Diego State University • 2002

http://www.sci.sdsu.edu/CRMSE/IMAP/vid_frac_add.html

Reflecting on Evidence of Student Learning to Inform Next Steps

What Happens Next?

A student is solving ⅚ + ⅖… • What might you expect to see? • What would you plan to do next for

this student?

IMAP • Integrating Mathematics and Pedagogy San Diego State University • 2002

Putting It All Together (The Session)

What do you anticipate students might do, say, and represent?

What are some possible student misconceptions? What kinds of student evidence do you expect?

When could ¼ be bigger than ½? Use words and pictures to explain your thinking.

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Putting It All Together (The Institute)

Institute Closing: Personal reflection on practices

• Which is the easiest for you to implement?

• Which is most challenging?

• How might you use your strength to develop you challenge?

Institute Closing: Content Debrief

What might you Start as a result of the Institute?

What might you Stop as a result of the Institute?

What might you Keep as a result of the Institute?



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Rich Tasks and Productive Struggle - nctm.org · Rich Tasks and Productive Struggle Effective Teaching with Principles to Action Institute, 2015 ... someone else. • Share how you

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