Introduction

Unit 1

Introduction & Ten Principles Thinking Mathematics

Middle School The changing nature of the pool of students taking courses beyond general math requires:

Alternate explanations

Answers to students’ questions; why?

Helping better students go beyond the text

Applied as well as pure mathematics

Statistics as well as proofs

Business & economics as well as science

Daily applications

Zalman Usiskin, U. of Chicago New Needs For Teachers 45 To prepare for the workplace students should learn to respond to questions such as:

Can you plan how to solve this problem?

Can you convince others that your work is correct?

Do you understand well enough to make adaptations in business, to work in changed conditions?

Secretary’s Commission on Achieving Necessary Skills SCANS Recommendation 43 Formula vs. Application

In a study of a high school curriculum that includes rich applied problem situations, students scored somewhat better than comparison students on algebraic procedures and significantly better on conceptual and problem-solving tasks.

Schoen & Ziebarth, 1988 Secondary Curriculum 42 Illustration

Table

Graph

Equation

Verbal Five Modes of Representation 41 40 Note how the teacher introduces the lesson.

In the clips you see, are any of the Ten Principles visible?

Take notes! Observing the Lesson It has been one month since Ichiro’s mother entered the hospital. He has decided to pray with his small brother at the local temple every morning so that she will be well soon. There are 18 ten-yen coins in Ichiro’s wallet and 22 five-yen coins in his smaller brother’s wallet. They have decided to take one coin from each wallet and put them in the offertory box and continue the prayer until either wallet becomes empty. Ichiro and His Brother:

Coins Problem 38

There is no one way to teach higher level skills. Teachers build a repertoire of strategies, constantly choosing, rejecting, changing as the situation requires.

Barak Rosenshine Higher Order Skills 35 If students are encouraged to develop their own procedures for solving problems, they must use understandings they have already constructed. Understandings and procedures remain tightly connected because procedures are built on understandings.

Making Sense: Teaching and Learning Mathematics with Understanding

Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Olivier, & Human Connecting Concept and Procedure 33 Knowledge is organized around core concepts.

Teachers must draw out students’ pre-conceptions and build on or challenge them.

They should help students understand facts and ideas in the context of a conceptual framework.

They should be familiar with the growth of students’ ideas within the discipline.

Suzanne Donovan, NAS

speaking about the report How People Learn

ER&D2000 Winter Institute How People Learn 32 Was there ever a time when everyone computed well?

1931 – Only 20% of 12th graders compute 2.1% of 60

1937 – Less than half of college freshmen divide 175 by .35

1947 – Only 32% of freshmen can pass arithmetical reasoning exam for ROTC.

Larry Sowder

Why I Support Reform:

The Good Old Days That Never Were A Bit of History 31 (Students usually) learn procedures by imitating and practicing rather than by understanding them, and it is hard to go back and try to understand a procedure after you have practiced it many times.

Hatano 1988; Resnick Wearne & Hiebert 1988

In Making Sense: Teaching and Learning Mathematics with Understanding

Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Olivier, & Human Conceptual/Procedural Balance 30 Initial rote learning of a concept can create interference with later meaningful learning.

Students exposed to instrumental instruction (learning rules without reasons) prior to relational instruction (what and why) achieved no more, and most probably achieved less, conceptual understanding than students exposed only to the relational unit.

Interference of Instrumental Instruction in Subsequent Relational Learning

Pesek & Kirshner, 2000 Balance Conceptual & Procedural Learning 29 Understanding: comprehending mathematical concepts, operations, and relations ― knowing what mathematical symbols, diagrams, and procedures mean.

Computing: carrying out mathematical procedures, such as adding, subtracting, multiplying, and dividing numbers flexibly, accurately, efficiently, and appropriately.

Applying: being able to formulate problems mathematically and to devise strategies for solving them using concepts and procedures appropriately.

Helping Children Learn Mathematics

National Research Council Proficiency Strands 26 Mathematical Proficiency

National Research Council 25 I know this looks long and it takes a while to do but when I do it this way my answer is never wrong!

Brody, Minnesota Fifth Grader

192

x39

100x30 = 3000

100 x 9 = 900

90 x 30 = 2700

90 x 9 = 810

2 x 30 = 60

2 x 9 = 18

7, 488 A Student Perspective

Statement to Peers When Explaining Strategy 24 According to U.S. mathematicians one element likely to be present in strong mathematics lessons is a variety of solution strategies.

They further noted that the value of such strategies coming from the students. TIMSS Video Study 23 Students who knew both intuitive strategies and a common school algorithm were more accurate when they used their intuitive strategies.

Nunes, Carraher & Schliemann

Street Mathematics, School Mathematics Accuracy and Multiple Strategies 22 Given an estimation problem, a group of 35 mathematicians used 22 different ways to perform the task.

There was no one “correct” way. Mathematicians Use Many Ways 20 We have documented the essential role of the table in the development of functional reasoning and notation.

The first transition is from concrete to pictures of the concrete to symbolic representation. You go from play with things to a pattern to perhaps a numerical expression of that then to a symbolic expression of that and all those together allow you to form a conjecture.

(H. Pollak in Nature and Role) Representations 17 Research on a functional approach with contextual problems (as opposed to rule-based symbol manipulation) shows that students tend to prefer the contextual approach. The challenge is how to order the context so students are gaining progressive conceptual development and a deep level of understanding.

Nature and Role Challenge of Using Situational Problems 14 The majority of student errors on equations can be attributed to difficulties correctly comprehending the meaning of an equation.

It is a misperception that verbal problems are harder than bare equations.

Koedinger & Nathan (2004) Bare Equations vs. Word Problems 12 decomposing and recomposing numbers to simplify calculation

sense of absolute and relative size

using the known to derive the unknown

judging reasonableness of answers

linking numeration, operation and relation symbols

flexible & multiple representations

using friendly strategies

estimating numerical answers

wanting to make sense of numbers

Resnick, Silver, J. Sowder, Trafton Number Sense Indicators 8 There is a need to link formal mathematical understanding to informal reasoning.

Studies show something called an “expert blind spot,” which is the tendency of someone with advanced understanding … to ignore the steps a beginner must take, the first of which is linking to his informal knowledge. How People Learn 6 Children see a distinction between school mathematics and real mathematics.

Unschooled adults and children with little or no schooling are able to function quite well in local markets to buy and sell. People in other occupations also develop intuitive ways of using mathematics for their work without being formally taught.

Nunes, Schliemann, & Carraher Street Mathematics, School Mathematics 4 As used in Thinking Mathematics, intuitive knowledge is what a student knows before being formally taught.

What has been learned informally

Related knowledge that has been learned previously Intuitive Knowledge 3 Divide by Factors

4/324

9/81

3/9

3

4 x 9 x 3 x 3

Write as Primes

(2*2)*(3*3)*3*3 Divide by Primes

2/324

2/162

3/81

3/27

3/9

3

22 x 34 Prime Factorization Methods for 324 49 Multiplicative

72

9 x 8

6 x 12

4 x 18

(4 x 9) + (4 x 9)

. . . Additive

72

70 + 2

60 + 12

50 + 22

25 + 25 + 22

. . . Multiplicative Decomposition Numbers Are Decomposed by Factors 48 After completing a series of workplace or everyday problems with students, we must always remember to help them understand that what we call mathematics comes from generalizing and organizing the common features among the solutions into a coherent structure.

High School Mathematics At Work, MSEB, 1998 Organizing What Students Learn 46 Use ongoing assessment to guide instruction

Teacher: Is 25% of 15 greater than, less than, or equal to 15?

Student: It is less than 15. You subtract 25%-15 =10 and 10 is less than 15.”

Just Because They Got It Right Does It Mean They Know It?

Susan Gay & Margaret Thomas Ongoing Assessment 36

Reasoning: using logic to explain and justify a solution to a problem or to extend from something known to something not yet known.

Engaging: seeing mathematics as sensible, useful, and doable―if you work at it―and being willing to do the work. Proficiency Strands 27 Accept and encourage multiple correct solution strategies.

Nunes, Carraher & Schliemann found that students who used their own solution strategies were more accurate than those who relied only on the typical school algorithm.

School Math and Street Math

Nunes, Carraher & Schliemann Multiple Solution Strategies 21 Reflecting on the use of physical objects can help students to develop understanding by linking their informal knowledge to school math. However, much depends on how these objects are used.

Teachers must provide opportunities for students to make explicit connections between activities with objects and the math concepts and procedures.

Helping Children Learn Mathematics

National Research Council Concrete Representations 16 Knowledge that is taught in only a single context is less likely to support transfer of knowledge than knowledge that is taught in multiple contexts.

How People Learn: Brain, Mind, Experience and School

National Research Council, 1998 Situational Stories 13 Cooks were presented problems about price, recipes and pharmacies (medicine). Effect of Familiar Context 11 Success rate with problems presented:

In work setting 98.2%

In context 73.7%

Without context 36.8%

Schliemann & Magalhaes, Proportional Reasoning: From shopping to kitchens, laboratories, and hopefully, schools Base Instruction on Situational Problems 10 Students sometimes purposefully suspend efforts to make sense because the problems they are given are so artificial they do not see them as real mathematics.

Boaler, 1998; Nesher & Hershkovitz, 1997

So we must learn to attend to the problems we use. Problems & Making Sense 9 Build a strong number sense through the use of counting, estimation, use of benchmarks and mental computation skills. Number Sense 7 There is substantial evidence that children’s difficulty with school math derives in large

part from their failure to recognize

and apply the relations between

formal rules taught in school and

their independently

developed intuitions.

Lauren Resnick Build from Intuitive Knowledge 5 2 The chess tournament rules state that when only 8 players remain, each person must play each other person one time. If there is a tie at the end of this “round of 8” further playoffs will be set up. How many games must be played in the round of 8?

How many games in a round of 5? 1 High School Mathematics At Work, MSEB, 1998 Credit choices

Lottery

Hospital care

Braking distance

Agriculture

Quality inspection

Telephone, computer rate plans

Etc. Emergency response design

Scheduling elevators

Figuring heating & cooling costs based on weather patterns

Astronomical & flight work

Prescription drugs & health care plans Where Is High Level Math Needed Every Day? 44 One day after they were done with the prayer they looked into each other’s wallet and discovered that the smaller brother’s amount of money was more than Ichiro’s. How many days has it been since they started praying?

Offertory box Coins – Part II 39 When do we study what?

Do we provide enough experience with it?

How do we balance “coverage”

with learning that will

last and transfer? Adjust the Instructional Timeline 37 Require students to explain and justify their mathematical thinking.

“Adaptive reasoning” is one of the 5 components of mathematical proficiency. This includes deductive reasoning, informal explanation and

justification, and inductive

reasoning.

Adding It Up, National Research Council, 2001 Explain and Justify 18 Simplifying Radicals 50 No one strategy works best for all students and all concepts. Use a Variety of Teaching Strategies 34 Conceptual

That’s a little less than 50.

One 4 has a value 100 times greater than the other 4.

Multiplying by 10 would make the sum a whole number

.6 +.8 is more than 1. Procedural

25.6

+23.8

49.4 Balance Conceptual & Procedural Learning 28 23 + 28 + 25 + 24

4 quarters in a dollar.

Give 2 from 28 to 23 and 1 to 24.

Now there are four 25s. Four 25s are

100, so 100. Analogical reasoning, metaphors, and mental and physical representations are tools to think with, often serving as sources of hypotheses, sources of problem-solving operations and techniques and aids to learning and transfer.

L.S. English, 1997 Explain and Justify 19 15 Use manipulatives and other representations to model problems, then link to symbolic notation. 12 x 15

10 x 10 10 x 5

200 50

2 x 10 2 x 5

20 10

220 + 60 = 260

12 x 15 = 250 Practice Problems 51 Adding It Up

National Research Council (2001) Understanding

Computing

Applying

Reasoning

Engaging

Which Thinking Mathematics Principles can you connect to the proficiencies? Five Strands of Mathematical Proficiency 47 Who are we?

Name

Where Do you Teach?

What grades? Other Content Areas?

Teaching Experience?

Something interesting? Housekeeping

Parking Lot

Cell Phones

Breaks

Respect As you enter the room, place your belongings at a seat and then move about the room adding information on the various pieces of chart paper. DECOMPOSITION CAROUSEL HANDOUT

TOURNAMENT PROBLEM SOLTUIONS

1. Make a Diagram or Chart

ABCDEFGH

A–XXXXXXX7

B-XXXXXX6

C–XXXXX5

D–XXXX4

E–XXX3

F-XX2

G-X1

H-

Total: 28 games in the round of 8

2: Develop a Pattern

Teams Games Games each team adds

1 0

2 1 +1

3 3 +2

4 6 +3

5 10 +4

6 15 +5

7 21+6

8 28+ 7

3: CONNECTING DIAGRAM This is best shown in different colors. Show numbers of additional pairings for each succeeding team in a table.

A

H B

G C

F D

E

4. Geometric Solution (staircase model) and rule

7

7

8

7

Use linking tiles to model the number of games to be played. Start with Team A playing the 7 other teams, team B now having 6 more teams to play, etc. You will lend up with a triangular shape with 7 on each smooth edge.

Make an identical “staircase,” turn it upside down and join the two to create a rectangle. The number of squares in the rectangle (representing games) can now be determined by:

*multiplying 7 x 8

Each square in the staircase represents one game; the rectangle has twice as many games because you made and joined two identical triangular shapes. So if you divide by 2, you get the actual number of games for 8 teams. If t = the number of teams, then = number of games in any round where each team must play every other team. Tournament Principle #7 Principle #8 Principle #9 Principle #10 Principle #1 Principle #2 Principle #3 Principle #4 Principle #5 Principle #6 Please solve this problem.

Be specific in naming

variables and vocabulary.

Share out. Activity #3 Activity #1 Activity #2 Activity #7 ACTIVITY 7: THE EFFECTIVE TEACHING FRAMEWORK AND THINKING MATHEMATICS

1. The framework

Explain that because many teachers now work in environments where evaluation and sometimes pay are tied to evaluation system, they need to know that the research-based Thinking Math Principles are connected to the most commonly used frameworks. Otherwise, some may be reluctant to use them.

2. What parts of the framework do the Ten Principles relate to?

Distribute copies of or have participants find the Framework for Excellence in Teaching in the Resources section of their Research Binder. Split into pairs or small groups. (There could be pairs in a class of 18 or 6 groups in larger classes). Try not to have anyone responsible for more than 3 cards. Distribute small envelopes with two or three cards in each. (There are 18 cards.) Each card has a more detailed explanation of a Standard element.

For example:

Knowledge of Content and the structure of the Discipline 1a. Teacher displays solid knowledge of the important concepts in the discipline and how these relate to one another and to 21st Century skills.

Each group will discuss which of the Ten Principles could be a match. Some elements may match more than one Principle. Record the matches on the work sheet. Each group reports their thinking back to the whole group. BREAK LUNCH

12:00-1:00 Thank you for the great day!

See you September 21 at 8:30!!! Day 2 Welcome Back! 4.OA.1

Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations. Multiplication as Comparison Thinking Mathematics 6-8 Introduction to Common Core Standards Number and Algebra in 6-8 …the “immaculate progression” in standards contrasts with the spectacular variation of student readiness in real classrooms.

Standards map stations through which students are led from wherever they start.

LEARNING TRAJECTORIES IN MATHEMATICS

Daro, Mosher & Corcoran Standards and Progression Grades 6 – 8 Progressions Understanding the

Place Value Structure 31 5.NBT.5

Fluently multiply multi-digit whole numbers using the standard algorithm. Standard Algorithm Rectangular Arrangement K.CC.4b.

Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted.

Precursor Standards You have been given student work from kindergarten to sixth grade which relates to key concepts of multiplication. Try to put this work in the order students would develop their understanding of multiplication. 3 Fewer standards does not mean children will be learning less.

There are fewer because learning is more carefully sequenced and focused. Fewer is no substitute for focused standards.

These standards stress

Conceptual understanding of key ideas

Continually return to organizing principles of the discipline

…and respect what is known about how children learn Goals: Fewer, Higher, Clearer 4.NF.4

Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. Multiplication Standards 4.NBT.5

Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. Multiplication Strategies Heather has two times as many books as Barbara and four times as many as Kip. Together the three girls have 49 books. How many more books does Heather have than Barbara? Multiplicative Comparison Problems 3.NBT.3

Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations. Multiplying by 10 3.OA.7

Fluently multiply and divide within 100, using strategies such as the relationship between multiplica-tion and division or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers. Multiplication Standards 3.0A.5

Apply properties of operations as strategies to multiply and divide.

Commutative property of multiplication, Associative property of multiplication, Distributive Property Properties of Multiplication 3.OA.1

Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. Multiplication Meaning 2.OA.4

Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends. Sum of Equal Addends 1.OA.6

Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten; decomposing a number leading to a ten; using the relationship between addition and subtraction; and creating equivalent but easier or known sums. Precursor Standard Grades 6 – 8 Progressions 2

Toward greater focus and coherence.

What do you think is most important for the classroom in this section of the standards document? Focused Reading -2.6 π Unit 2-Activity 2 Unit 2-Activity 1 Unit 2-Activity 5 CVMI TRAINING

Math as a Second Language

Functions and Algebra

Contact Wayne Harner at Mayerson Academy for further information. http://vimeo.com/29567534 http://vimeo.com/29484921 http://vimeo.com/29568179

http://vimeo.com/29568179 Unit 2-Activity 6 http://vimeo.com/album/1702025/video/29568008 Unit 2-Activity 3 Unit 2-Activity 3 http://www.sei2003.com/APEC/files/video/hfull.mp4 Unit 2-Activity 2

Progressions

Video on Progressions π 10 Principles and TES (Danielson Framework) Activity #2

The Ten Principles of Teaching Mathematics

Please complete the Ten Principles of Teaching

Mathematics Packets in partners. Activity 4

Manipulation is not Enough Activity #4

Connection Between Principles and Practices Activity #6

Decomposition A Prerequisite Skill The Effective Teaching

Framework and Thinking Mathematics What Knowledge Enabled the Thinking Activity #5 Do You Have a Window Seat? BREAK LUNCH Standards for Mathematical Practice BREAK BREAK Unit 3-Activity 5 Reflection We will now divide into two groups.

Half of you will work with the 10 principles.

Half of you will work with the 8 Mathematical Practices.

Where do you see these practices emerging in the work that we have done to date? Common Core Implementation

Focus on Teaching Deeply

Key Shifts:

Focus

Coherence

Rigor

Have a great Year! District News AFT designed and vetted learning modules

Common Core

Teaching Deeply

Research Based

PLCs to be offered monthly (2hr) –open to everyone Thinking Math PLCs StaffNet

http://staffnet.cps-k12.org/staffnet/

Common Core State Standards

Domain/Cluster/Standard

CCSS Site

Hyperlink

Resources Coming! Pacing Guides Math Manager – Sheila Radtke Radtkes@cpsboe.k12.oh.us

Pacing Guides/StaffNet

Learning Teams

District News and Thinking Math PLCs Welcome to the 2012/13 School Year August 21, 2012 Elementary Math Professional Development Learning Team 1

Math Content Learning Team 2

Literacy Through Math In lieu of 2nd monthly principal staff meeting- 1 hour 4 meeting- 1.5 hours

Schools select 1 date within range September 24th October 29-31 October 22nd November 27-30 January 28th January 16-18 February 25th February 20-22 March 25th April 22nd Total 6 hours Total 6 hours Learning Teams At this time, please work in

groups to match the 8 Mathematical

Practices with 10 Teaching Principles.

Please make a list and be prepared

to explain and justify your thinking.

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