NAME _____________________

Beliefs About Teaching and Learning Mathematics

Principles to Actions, Ensuring Mathematical Success for All, (NCTM, 2014) p. 11

Characteristics of Mathematical Tasks

Leinwand, S, D. Brahier, and D. Huinker. Principles to Actions: Ensuring Mathematical Success for All, Reston, VA: NCTM, 2014 (pg. 18)

Productive Struggle Reflection Survey Rate the frequency of each statement in your classroom.

1. I anticipate what students might struggle with during a lesson and I prepare to support them.

1 2 3 4 5 6 7 8 9 10

Never Always

2. I give students time to struggle with tasks and ask questions that scaffold thinking without doing

the work for them.

1 2 3 4 5 6 7 8 9 10

Never Always

3. I help students realize that confusion and errors are a natural part of learning. We talk about

mistakes, misconceptions, and struggles.

1 2 3 4 5 6 7 8 9 10

Never Always

4. I praise students for their efforts more frequently than the correct answers.

1 2 3 4 5 6 7 8 9 10

Never Always

5. My students struggle at times but they know breakthroughs come from struggle.

1 2 3 4 5 6 7 8 9 10

Never Always

6. My students ask questions that are related to their struggles to help them understand.

1 2 3 4 5 6 7 8 9 10

Never Always

7. My students persevere in solving problems. They don’t give up.

1 2 3 4 5 6 7 8 9 10

Never Always

8. My students help one another without telling their classmates what the answer is or how to solve

it.

1 2 3 4 5 6 7 8 9 10

Never Always

Productive Struggle Images

Analyzing Mathematical Tasks: Grades K ─ 2

Briars, 2012

Categorize each of the Grade K-2 tasks below as High Level Cognitive Demand or Low Level Cognitive Demand.1, 2 Then, develop a list of criteria that describe the tasks in each category. Task A Manipulatives or Tools Available: Counters, interlocking cubes

Write and solve a number story for 26 8.

Show how you solved the problem using words, counters, tallies, or pictures. Task B Manipulatives or Tools Available: None

In the number 28 what digit is in the ones place? what digit is in the tens place?

Task C Manipulatives or Tools Available: None Use the graph to answer the questions. 1. How many students picked red as

their favorite color? 2. Did more students like blue or

purple?

3. Which color did only one student choose as a favorite color?

1 Adapted from Smith, Stein, Arbaugh, Brown, and Mossgrove, Characterizing the Cognitive Demand of Mathematical Tasks: A Task Sorting

Activity, Professional Development Guidebook for Perspectives on the Teaching of Mathematics, NCTM, 2004. 2 Since these are K-2 tasks, assume that they would be read or presented orally to students, not just presented in written form.

Analyzing Mathematical Tasks: Grades K ─ 2

Briars, 2012

Task D Manipulatives or Tools Available: counters, interlocking cubes

Some people were riding on a bus. At the bus stop, 7 more people get on. Now there are 13 people on the bus. How many people were on the bus before the bus stop?

Task E Manipulatives or Tools Available: interlocking cubes

Complete each fact family. Add or subtract. The first one is started for you.

1. 6 + 9 = __15_

9 + 6 = _____

15 – 9 = __6__

15 – 6 = _____

2. 8 + 7 = _____

7 + 8 = _____

15 – 7 = _____

15 – 8 = _____

3. 4 + 8 = _____

8 + 4 = _____

12 – 8 = _____

12 – 4 = _____

Task F Manipulatives or Tools Available: None

These puzzles show parts of number grids. Complete the puzzles.

58

Analyzing Mathematical Tasks: Grades K ─ 2

Briars, 2012

Task G Manipulatives or Tools Available: Interlocking cubes (optional)

1. How many legs are on the people in our classroom? Show how you figured it out. 2. How could you tell someone how to figure out the number of legs for any number of

people? Can you think of a rule? For example, how would you finish this sentence: “To find the number of legs in a group of people, you . . . “

3. What if there were 26 legs? How many people would there be?

Task H Manipulatives or Tools Available: None

Complete the addition facts in one minute or less.

Analyzing Mathematical Tasks: Grades K ─ 2

Briars, 2012

Task I Manipulatives or Tools Available: Counters, interlocking cubes

Solve each story problem. 1. Four bees were buzzing around a flower. Three more bees were at the hive. They flew

over to join the others at the flower. Then how many bees were buzzing around the flower?

2. Now there were seven bees buzzing around the flower. Two bees left the flower and flew

back to the hive. How many bees were still buzzing around the flower? You’ve solved two problems about bees. How are the stories different?

Task J Manipulatives or Tools Available: none

Rosa is going to measure her book. First she will measure using crayons. Then she will measure using paper clips. Will it take more crayons or paper clips to measure her book? Why do you think so? The length of Rosa’s book is 2 crayons. How many paper clips long is Rosa’s book? How do you know?

Analyzing Mathematical Tasks: Grades K ─ 2

Briars, 2012

Task K Manipulatives or Tools Available: base-ten blocks, tens and ones mat

Write the tens and ones. Circle the number that is greater. The first one is started for you.

1. 56 __5_ tens _6__ones

61 __6_ tens _1__ones

2. 74 ____ tens ____ones

89 ____ tens ____ones

3. 31 ____ tens ____ones

19 ____ tens ____ones

Task L Manipulatives or Tools Available: red and blue crayons or cubes; colored pencils; crayons or markers

I want to put 8 crayons in my box. I want to have a mix of red and blue crayons. How many of each color could I have? How many blues? How many reds? Find as many different ways as you can.

Analyzing Mathematical Tasks: Grades K ─ 2

Briars, 2012

Task M Manipulatives or Tools Available: base-ten blocks, tens and ones mat

Show this many. Subtract this many. Do you need to regroup? (Yes or No)

Solve.

45 8 45 ─ 8 = _____

27 5 27 ─ 5 = _____

32 4 32 ─ 4 = _____

44 8 44 ─ 8 = _____

49 6 49 ─ 6 = _____

34 4 34 ─ 4 = _____

Task N Manipulatives or Tools Available: none

Write a number sentence. Solve. 1. Jesse had 9 marbles. Louisa had 7 marbles. How many marbles did they have in all? 2. Jesse and Louisa had 16 marbles. Louisa took away 7 marbles. How many were left?

Analyzing Mathematical Tasks: Grades K ─ 2

Briars, 2012

Classification of Tasks

Task Low Level High Level

A

B

C

D

E

F

G

H

I

J

K

L

M

N

Criteria

Mathematics Survey

1. To be good in math, you need to ……… because ……..

2. Math is hard when …..

3. Math is easy when …..

4. How can math help you?

5. The best thing about math is …

6. If you have trouble solving a problem in math, what do you do?

Trends Implied by the

Surveys

Positive Attitudes,

Dispositions, Beliefs

Instructional Plans for

Change

Mathematics is a solitary, silent endeavor.

Collaboration and communication contribute

to mathematical understanding.

Structure group tasks; make children’s strategies

public; encourage children to note others’

contributions to their learning

The teacher is in charge of imparting knowledge.

The rewards for developing mathematical expertise are external

and are often postponed until the future.

Mathematics involves learners in constructing

meaning for themselves. The rewards for developing expertise are

intrinsic.

Encourage interaction, revisiting, extending

(Schwartz 1996); involve students through student-authored

problems, mathematics journals, mathematics

“publications:

Problems are solved in a

swift, prescribed manner.

Problems are solved

through flexible use of multiple strategies. The time required to solve

problems depends on the complexity of the

problem.

Encourage strategy

sharing, problem-posing investigations, extended explorations,

mathematics journals (Whitin & Whitin 2000)

Mathematics is unrelated

to other subjects.

Mathematics has real-life

application across the curriculum and in contexts outside school.

Emphasize content-

related problems (e.g., science), problems inspired by children’s

literature, student-authored problems

©Phyllis E. Whitin, NCTM Teaching Children Mathematics, April

2007

FLUENCY INTERVIEW

1. Explain how to use the “count on” strategy for 3 + 9. (S)

2. Jhordan explains that 6 + 7 is 12. Is he correct? Explain how you

know. (A)

3. What strategy did you use to solve 6 + 8? (S)

4. What strategy did you use to solve 9 +3? (E)

5. Jasmine solved 6 + 8 by changing it in her mind to 4 + 10. What did

she do? Does this strategy always work? (F)

6. Which facts do you “just know”? For which facts do you use a

strategy? (E)

©Gina Kling and Jennifer M. Bay-Williams April 2014 issue of Teaching Children

Mathematics

Collect 25 Together

Materials:

One dot cube

35 counters

Players: 2

Object: With a partner, collect 25 counters

How to Play:

1. To start, one player rolls the dot cube. What number did

you roll? Take that many counters.

2. Take turns rolling the dot cube. Take that many counters

and add them to the collection.

3. After each turn, check the total number of counters in

your collection. The game ends when you have 25

counters.

Variations:

a. At the end of the game, determine how many more than

25 counters you have.

b. Play Collect 40 Together.

c. Play with three people.

d. For each turn, write the number rolled and the total

number of counters you have so far.

NCTM Principles to Actions Professional Learning Toolkit: Teaching and Learning

Addition Strings Task

Solve the set of addition problems. Each time you solve a problem, try to use the previous problem to solve the next problem. 7 + 3 = _____ 17 + 3 = _____ 27 + 3 = _____ 37 + 3 = _____ 37 + 5 = _____ After you have solved all of the problems, describe some patterns that you notice in the sequence of equations.

Show how you might represent student reasoning with a drawing or on a number line so that students could visually see the relationships among the quantities.

Find the Sums

Sort and Group Basic Facts

Fluency Station: Skill: One-digit subtraction

CCSSM: 1.OA.C.6 and 2.OA.B2

One Less The goal of this game is to find two cards where the number on one card is one less than the number on the other card. Number of Players: Two Players Materials:

Numeral cards 1 – 10 (four of each)

How to Play: Players sit side by side. The cards are shuffled and placed facedown in a stack.

Player 1 takes ten cards and places them faceup in a line between both players.

Player 1 looks for two cards with a difference of one. When he finds two such

cards, he says, “I can take these cards because ____ is one less than ____.”

Player 2 adds two more cards to the faceup line. (The line always has ten cards.)

She hunts for two cards with a difference of one.

If there are no cards in the line with a difference of one, another card is added

to the line.

Players alternate turns until all the facedown cards have been added to the line

and all possible “one-less” combinations have been made.

Variations: “Two Less” and “Three Less” are played in a similar manner. Teacher Question:

Can you prove to me that one less than ___ is ____? NOTE: Counters or a 0 to 10 number line might be very helpful so that children can visualize what they are being asked to do.

0 1 2 3 4 5 6 7 8 9 10

Mastering Basic Math Skills: Games for Kindergarten through Second Grade by Bonnie Adama Britt ©National Council of Teachers of Mathematics 2014

Fluency Station: Skill: One-digit subtraction; Variation – Two-digit subtraction

CCSSM: 1.OA.C.6 and 2.OA.B2, Variation – 2.NBT.B.5

More: A Game of Differences The goal of this game is to practice finding differences using randomly selected numbers. Number of Players: Two Players Materials:

Numeral cards 1 – 10 (four of each)

Counters

How to Play: The cards are shuffled and put facedown in a stack. Each player turns over one

card. Players decide who has the greater number, and then figure out how much

“more” that player has. The player who has “more” takes the quantity of

counters that equals the difference between the two players’ numbers.

Example: Player 1 turns over a 3. Player 2 turns over a 9. Player 2 has the greater number, which is 6 more than Player 1’s, so Player 2 takes 6 counters. Play continues until all the cards have been drawn. Players count their counters, and the player with more counters wins the game. Variation: Each player turns over two cards and makes the greatest two-digit number possible. The players then determine who has the greater number, and using pencil and paper, figure out the difference between the two numbers. The player who has “more” gives himself points equal to the difference between the two numbers. When all the cards in the facedown stack are gone, the player with more points wins the game. Example: Player 1 draws a 6 and 7, and makes 76. Player 2 draws a 1 and a 3, and makes 31. Player 1 has the greater number, in this case, 45 more, so Player 1 scores 45 points for this round. Teacher Questions: What have you noticed while playing this game? Convince me that ___ is that much more than ____. When you take a lot of counters in a turn, what do you notice about the two numbers drawn? What do you notice about the amount of counters you get when both numbers are close together?

Mastering Basic Math Skills: Games for Kindergarten through Second Grade by Bonnie Adama Britt ©National Council of Teachers of Mathematics 2014

Spin, Circle, and Solve!

1.Spin the Strategy Spinner

2.Circle all of the problems on the page that you can solve using

that strategy. (Don’t solve them yet.)

3.Solve the problems you circled.

©Putting Essential Understanding of Addition and Subtraction into Practice Pre-K -2, National Council of Teachers of

Mathematics, 2014

Spin, Circle, and Solve Strategy Spinner

©Putting Essential Understanding of Addition and Subtraction into Practice Pre-K -2,

National Council of Teachers of Mathematics, 2014

Ring Facts

©Putting Essential Understanding of Addition and Subtraction into Practice Pre-K -2,

National Council of Teachers of Mathematics, 2014

Collect 20 Together Materials: One dot cube 30 counters Players: 2 Object: With a partner, collect 20 counters How to Play:

1. To start, one player rolls the dot cube. What number did you roll? Take that many counters.

2. Take turns rolling the dot cube. Take that many counters and add them to the collection.

3. After each turn, check the total number of counters in your collection. The game ends when you have 20 counters.

Variations: a. At the end of the game, determine how many more than 20

counters you have. b. Play Collect 40 Together. c. Play with three people. d. For each turn, write the number rolled and the total number of

counters you have so far.

Levels of Classroom Discourse

Hufferd-Ackles, K., Fuson, K. C., & Sherin, M. G. (2004). Describing levels and components of a math-talk learning

community. Journal for Research in Mathematics Education, 35(2), 81–116.

Five Practices for Orchestrating Productive Classroom

Discussions (Smith and Stein, 2011)

1. Anticipating student responses prior to the lesson

2. Monitoring students’ work on and engagement with

the tasks

3. Selecting particular students to present

mathematical work

4. Sequencing students’ responses in a specific order

for discussion

5. Connecting different students’ responses and

connecting the student responses to key

mathematical ideas.

5 Practices for Orchestrating Productive Mathematics Discussions, by

Margaret S. Smith and Mary Kay Stein (2011)

The Case of Amanda Smith and the Donuts Task Principles to Actions Professional Learning Toolkit: Teaching and Learning    

The Donuts Task 1. Dion chooses 3 chocolate donuts and 4 vanilla donuts. Draw a picture and write an

equation to show Dion’s donuts. 2. Tamika has 4 vanilla donuts and 3 chocolate donuts. Draw a picture and write an

equation to show Tamika’s donuts. 3. Tamika claims that she has more donuts than Dion. Who has more donuts, Dion or

Tamika? Draw a picture and write an equation to show how you know who has more donuts.

Extension: Tamika changes her mind and she gets 3 chocolate, 2 vanilla, and 2 sprinkle donuts. Draw a picture and write an equation to show Tamika’s donuts. Tamika claims that she has more donuts than Dion because she has three kinds of donuts. What do you think about Tamika’s claim? Who has more donuts and how do you know?

Pose Purposeful Questions

What are teachers doing? What are students doing?

National Council of Teachers of Mathematics. (2014). Principles to Actions: Ensuring

mathematical success for all. Reston, VA: Author.

Framework for Mathematics Teaching Questions

Question Type Description Examples

Gathering Information Students recall facts, definitions, or procedures.

What do we know about Darren’s donuts?

When you write an equation, what does the equal sign tell you?

Probing Thinking Students explain, elaborate, or clarify their

thinking, including articulating the steps in

solution methods or the completion of a task.

Can you tell me how your picture and number sentence helped you solve the problem?

Can you tell me how you figured out how many donuts

Darren has and how many donuts Tamara has?

Making the Mathematics Visible

Students discuss mathematical structures and make connections

among mathematical ideas and relationships.

What does your number sentence show about the donut problem?

How do you know that both Tamara and Darren have 7

donuts?

Encouraging

Reflection and Justification

Students reveal deeper

understanding of their reasoning and actions, including making an

argument for the validity of their work.

How do you know that the sum of two odd numbers will

always be even? What is another number sentence you can write to tell about

Tamara’s donuts?

National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA:

Author.

Beliefs About Mathematics Assessment

Select 2 statements about assessment. Write P if it is a productive belief or U if it is an unproductive belief. Be sure to explain your reasoning.

The primary purpose of assessment is accountability for

students through report card marks or grades.

Only multiple-choice and other “objective” paper-and-pencil tests

can measure mathematical knowledge reliably and accurately.

Assessment is an ongoing process that is embedded in

instruction to support student

learning and make adjustments to instruction.

Assessment in the classroom is an interruption of the instructional

process.

Mathematical understanding and processes can be measured

through the use of a variety of assessment strategies and tasks.

A single assessment can be used to make important decisions about

students and teachers.

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.

Assessment is an ongoing process

that is embedded in instruction to support student learning and make

adjustments to instruction.

Assessment is something that is done to students.

Stopping teaching to review and take practice tests improves

students’ performance on high-stakes tests.

Multiple data sources are needed to provide an accurate picture of

teacher and student performance.

Ongoing review and distributed practice within effective instruction

are productive test preparation strategies.

National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring

mathematical success for all. Reston, VA: Author

Beliefs About Mathematics Assessment

Beliefs About Mathematics Assessment

Unproductive Beliefs Productive Beliefs

The primary purpose of

assessment is accountability for

students through report card

marks or grades.

The primary purpose of

assessment is to inform and

improve the teaching and learning

of mathematics

Assessment in the classroom is an

interruption of the instructional process.

Assessment is an ongoing process

that is embedded in instruction to support student learning and

make adjustments to instruction.

Only multiple-choice and other

“objective” paper-and-pencil tests can measure mathematical

knowledge reliably and accurately.

Mathematical understanding and

processes can be measured through the use of a variety of

assessment strategies and tasks.

A single assessment can be used to make important decisions about

students and teachers.

Multiple data sources are needed to provide an accurate picture of

teacher and student performance.

Assessment is something that is

done to students.

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.

Stopping teaching to review and

take practice tests improves

students’ performance on high-stakes tests.

Ongoing review and distributed

practice within effective

instruction are productive test preparation strategies.

National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring

mathematical success for all. Reston, VA: Author.