2012-2013 Algebra Academy Exploring Student’s Mathematical Thinking Probing the Math Needed for Algebra

Professional Development in MathematicsFor District 287 & Member Districts Special Education Staff

Nancy Nutting nancynutting@comcast.net

Mary Peters mkpeters@district287.orgChristina Shidla cachidla@district287.orgScott Swanson saswanson@district 287.org

District 287 Program Facilitor for Professional Learning

Jennifer Nelson jlnelson@district287.org

763-550-7241

A Minnesota ProjectDeveloped by Districts 287 & 916, Metro ECSU, U of M, Hamline,

Normandale CC

Minneapolis/St. PaulRegion 11 Math and Science

Teacher Center

Math Success: It’s In Our Hands

www.region11mathandsciencecenter.com

Surveyed 287 Special Ed Staff

Math Committee RecommendationsTHIS is what we are doing

todayAlgebra Academy held in 2011-12 positive feedback

share with others

Today’s Session . . .

Understanding Equality is Easy:

True or False?

In this session . . .

Identify benchmarks students reach on their way to understanding equality and being successful with algebra

Identify strategies students might use in working with equations

Practice crafting and sequencing equations that support student learning

Understand PLC structure and assessments you will use in your school between sessions

We want to . . .

. . . think more deeply about arithmetic in ways that are consistent with thinking in algebra.

ARITHMETIC merely involves calculation.

ALGEBRA involves seeing relationships.

Thomas Carpenter, et. al., Thinking Mathematically: Integrating Arithmetic and Algebra, 2003

This year, each session combines . . .

Exploring the math behind

what we teach

Learning instructional strategies to

increase skills & understandings

Assessing our students and planning from what we learn

When you “get” something, practice your questioning skills to help others understand

Take care of your needs

Designed for students who . . .

• Have 1 - 1 correspondence• Can do single-digit addition and have

strategies for basic subtraction• Would benefit from understanding

mathematics

This year . . .

Full Day PD Sessions Wednesday, Sept. 19 (Room 321)

EQUALITYWednesday, Nov. 7 (Room 321)

MODELING WORD PROBLEMSWednesday, Dec. 12 (Room 321)

RELATIONAL THINKINGWednesday, Jan. 16 (Room 321)

OPERATIONS & BASIC FACTSWednesday, Feb. 27(Room 321)

FRACTIONS & DECIMALSWednesday, May 1 (Room 321)

EFFORT, PROBLEM SOLVING & REASONING

+ 2 school-based PLCs between sessions

cover

Back at Your Table . . .

Talk about:How would you recognize a student

who is mathematically powerful? On chart paper, create a circle about 6”

in diameter Web the characteristics of a

mathematically powerful student

1

Checking Out Our Own Number Sense

“EQUAL 9” Post-it Note Posters

Use +, —, x and/or ÷ as many times as you like with 3, 4 or 5 of these numbers:

8 5 10 6 2 to EQUAL 9Write each equation on its own post-it note and add

to your group’s poster. Be sure to write the whole equation. “equation” is a # sentence with an equal sign

2

KRYPTO for older students

Draw 5 numbers from a half deck of cards.Draw a 6th number for the target number. Students create table posters or contribute

to a poster on the board.

Try:http://mphgames.com

WARNING: It could be impossible to make the target number!

2

http://illuminations.nctm.org

DealHint

Solve

2

What’s does this equal?

3 + 4 x 5 = ? 10 – 6 + 5 = ?35 or 23? ̶ 1 or 9?

Parenthesis & Exponents PE Please Excuse

Mult & Divide (in order) MD My Dear

Add & Subtract (in order) AS Aunt Sally

MN Benchmark 5.2.2.1 Apply the commutative, associative and distributive properties and order of operations to generate equivalent numerical expressions and to solve problems involving whole numbers.

2

Think about this problem

8 + 4 = + 5

What would students say belongs “in the box”?

What does belong “in the box”?

3

A Provocative Study (mid 90s)

Source: Carpenter, Franke and Levi

Percent Responding

responses

7* 12 17 12&17

other

Gr. 3 & 4

9 49 25 10 7

Gr. 5 & 6

2 76 21 2 0

Gr. 1 & 2

5 58 13 8 16

3

What if the equations were . . .

Does format matter?

8 + = 7 + 58 + 4 = 7 + 8 + 4 = k + 5, what is k? 8 + 4 = 7 + n, what is n?

3

Instruction matters!

Students’ Increase in Understanding of the Meaning of the Equal Sign

(Number of students answering 8 + 4 = ___ + 5 correctly)

Before adjusting instruction

After adjusting instruction

Gr. 1 & 2 5% 66%

Gr. 3 & 4 9% 72%

Gr. 5 & 6 2% 84%NCISLA inBrief: ”Building a Foundation for Learning Algebra,” Fall 2000, http://ncisla.wceruw.org/publications/briefs/fall2000.pdf.

Instruction matters!

7 * (8*) 12 (16) 17 (24) otherSchool

WSept.

07Dec. 07

Sept. 07

Dec. 07

Sept. 07

Dec. 07

Sept. 07

Dec. 07

Grade 2

6% 33% 53% 40% 17% 13% 25% 15%

Grade 3

34% 65% 44% 28% 13% 6% 9% 1%

Grade 4

26% 75% 59% 17% 15% 8% 0% 3%

Grade 5

65% 88% 25% 5% 7% 6% 3% 1%

Minnesota Metro Area School in 2007

(Sept) 8 + 4 = + 5 (Dec) 9 + 7 = + 8 *CORRECT RESPONSE

Metro Area Junior High Data

Responses  n  %

7 268 4012 274 4117 76 11

12 or 17 31 5other 22 3

Junior High Intervention Students n = 671

Same Metro Area High School

December 2, 2004

Geo1 Geo2 Geo3 BCGeoResponses n =

29 n = 31

n = 34 n = 11

65.52% 61.29% 85.29% 72.73% 720.69% 22.58% 8.82% 9.09% 12

0.00% 6.45% 5.88% 0.00% 176.90% 3.23% 0.00% 0.00% 90.00% 0.00% 0.00 0.00% 12 or 176.90% 6.45% 0.00% 18.18% other

Instruction Matters for Students with Special

Needs too!

• Results from 2011-12• Anecdotes from Mary, Chris & Scott

CGI (Cognitively Guided Instruction)

Began as a research project between University of Wisconsin at Madison mathematicians and math educators and Madison area teachers.

Goal: To explore how children understand mathematics and to move instruction from their understanding – what they cognitively know.

Benchmarks in Student Thinking about the Equal Sign

1. BASIC NUMBER SENTENCE SENSE• Children begin to write number sentences

and describe their thinking about the equal sign. They begin to see that numbers or expressions on one side of the equal sign are the same amount as numbers or expressions on the other side.

Adapted from: Carpenter, Franke and Levi. Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School. Heinemann. Portsmouth, NH 2003 www.heinemann.com

4

Begin even with young learners

2 and 3 are the same (amount) as 54 and 1 make the same as 5, etc.

5 is the same (amount) as 3 and 2

5 makes the same as 0 and 5, etc.

4 + 6 = 10 or 10 = 4 + 64 plus 6 is the same (amount) as 1010 is the same (amount) as 4 plus 6

Kevin in Kindergarten

Equation given to student:

• What does Kevin understand about the equal sign?

• If you were this child’s teacher, what problems would you have him work on next?

5

Equality as Balance

Pan Balance Number Balance

Balance

=

14 + 16 17 + =

=

14 + 1617 +

=

=

14 + 16 17 + 13=

8

Check balances on pages 6-12

• What’s my mathematical goal – What do I want students to notice?

• What will you be doing in math during the next few weeks? On pp. 13-15 make a page or two of balance problems to bring out the mathematical ideas in a particular lesson.

• Save p.15 as a master or get template from the Algebra Academy website.

One caution about balances . . .

Balances are a weight metaphor which causes some issues, e.g. work with subtraction or negative numbers.

Watch carefully to see if students are using computation, in which case the equal sign as a balance point may work well. But if students are thinking about the concepts of balancing weights it may cause some misunderstandings.

With older students,consider these two

balances . . .Consider what numbers

create balance in the following situation?

Consider what students might say

about how this equation works on abalance:

What numbers do the circle and the triangle represent if this balance DOES balance?

3 – 5 + 6 = 5 – 1

+ 5 + 5

PAN BALANCES

http://illuminations.nctm.org/ActivityDetail.aspx?ID=26

You Tube Videos

Linear Equations – Balancing the Equationhttp://www.youtube.com/watch?feature=fvwp&NR=1&v=DO-hzLh79uw

Pan Balance with Shapeshttp://www.youtube.com/watch?feature=endscreen&v=vbX83p0xJ9c&NR=1

Equality Benchmarks

2. EXPERIENCE WITH A VARIETY OF EQUATION TYPES

• Children accept as true number sentences that go beyond the form a + b = c. They understand that equations in these forms might be true:

 7 = 3 + 4

2 + 8 = 5 + 5356 + 42 = 354 + 44

y = 3x + 7

Use:Equal 9 Postersor Krypto 4

Equality Benchmarks

3. CALCULATING TO DETERMINE TRUTH (Operational Thinking)

• Children recognize that the equal sign separates two equal values. They carry out calculations to determine that the two sides of an equation are equal or not equal.

8 + 4 = ___ + 5

12 124

Equality Benchmarks

4. RELATIONAL THINKING • Children compare the expressions on each side

of the equation and check for truth by identifying relationships among numbers and reasoning instead of actually carrying out the calculations.

 

8 + 4 = ___ + 5

• “7 is the missing number because 5 is one more than 4, so I need a number that is one less than 8.”

4

Why does algebra make sense?

3m + 4 = 13 PP 4 PP 4 3m = 9

3m 3 = 9 3 m = 3What if you only think about the equal sign as a

signal to compute rather than separating two sides that have the same value?

Terry Wyberg, U of M, Region 11 Grant Developer

Middle School Study by Knuth

3 + 4 = 7

a)The arrow points to a symbol. What is the name of the symbol?

b)What does the symbol mean?c)Can the symbol mean anything else? If

yes, please explain.

Source: Knuth, “The Importance of Equal Sign Understanding for the Middle Grades,” MTMS, May 2008

Study by Knuth 3 + 4 = 7

Source: Knuth, “The Importance of Equal Sign Understanding”, MTMS, May 2008

Best Definition Percent of Students (n=375)

Relational

Operational

Other No response

don’t know

Gr. 7 36 52 9 3

Gr. 8 46 45 8 1

Gr. 6 29 58 7 6

What if students were asked . . .

3 + = 7

or 3 + m = 6

The arrow points to a symbol.What does this symbol mean?

4th Grade Bilingual Classroom

• Note equations given to students (LH column).

• Why is each number sentence useful in developing students thinking about equality?

• If you were these children’s teacher, what equations might you use with them next? 16

Usually avoid equal sign in strings

8 + 4 = 12 + 5 = 17 what’s wrong?

but 8 + 4 ≠ 12 + 5and 8 + 4 ≠ 17

Use a “goes to” arrowto track ongoing thinking

8 + 4 12 + 5 17 17

Equation Chains

• An “equation chain” can use multiple equal signs if all the terms surrounding any equal sign are equal to each other.

For example, children might generate many ways to make 10 and write the following “equation chain”:

10 = 6 + 4 = 7 + 3 = 20 – 10 = 100 – 90 = 7 + 2 + 1 

17

How long can you go?

• Consider having students create chains on adding machine tape to encourage flexible thinking about a given quantity and expressions that represent that amount.

75 = 3 x 25 = 100 – 25 = 7 x 10 + 5

Thanks to teachers at Willow Lane, White Bear Lake Area Schools, MN for the adding machine tape idea

Work equation chains with a partner

Start with the day of one of your birthdates.Take a strip of adding machine paper equal

to your height.

Add one or two expressions equal to that number. Move to next strip and add to it.

Use with Calendar Math or Morning Meeting or Number of Day

27 = Whole Numbers +

Misconceptions about Equality

• It is difficult to sort out exactly why misconceptions about the meaning of the equal sign are so pervasive and so persistent. A good guess is that many children see only examples of number sentences with an operation to the left of the equal sign and the answer on the right and they over generalize from those limited examples. Carpenter, Franke and Levi. Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School. Heinemann. Portsmouth, NH 2003 www.heinemann.com p. 22 18

The Clothespin Card

6 + 4

Benchmarks 1, 2, 3 4

18

Generate Equations

6 + 4 = 10

0 + 10 = 10

4 + 6 = 10

1 + 9 = 10

2 + 8 = 10

7 + 3 = 10

3 + 7 = 10

3 + 7 = 10

4 + 6 = 10

5 + 5 = 10

5 + 5 = 10

6 + 4 = 10

7 + 3 = 10

etc. etc.

Algebrafy basic fact work and work with equality misconceptions.Why do you think you have found all the ways to make 10 with 2 numbers?

Introduce TRUE OR FALSE as a way to look at an equation in its entirety.

18

True or False Sentences

Write the number sentence and then show whether it is TRUE or FALSE?

5 + 7 = 10

Benchmarks 1, 2, 3 4

“Play” True or False

• The “combinations that make 10 activity” was only the context for another goal – introducing the true/false strategy for evaluating equations and checking out their understanding of equality.

10 = 6 + 4 T or F?Math people can do things forwards and backwards as long as they tell the truth! 18

Vocabulary Issues

Establish the words true or false connected to students.

e.g. Today everyone is wearing the color red. True or False?

¿Es verdad o falso?

“Grown-up” vs “Kid” Hmong

Tseeb Tsi Tseebtrue not true

Yog Tsi Yogcorrect not correct

Order from Easiest to Hardest

select one quadrant• If you wanted to expose your students to

equations beyond a + b = c which equations would be easiest and which would be hardest for your students?

• Use label: T or F?Benchmarks 1, 2, 3

19

Open Number Sentences-Make it True

Write the number sentence with a box or a variable representing some missing information. What goes in the box (or what does the letter have to be) to make the sentence (equation) true?

10 = 6 + n

+ 7 = 10Benchmarks 1, 2, 3 4

Using True-False SentencesUsing Open Number

Sentences

On p. 20 create some

true/false sentences or

open number sentences you could use next week with your students

20

One Side is the Same as the Other

Which symbol makes the sentence true?

Benchmarks 1 & 2

21

One Side is the Same as the Other

• Insert the or into each equation – circle or highlight all the true equations

• To what mathematical ideas was I trying to draw attention? What do you notice?

• What mathematical conversations do I want my students to have? Benchmarks 1, 2, 3, 4

22-27

One Side is the Same as the Other

• What is your mathematical focus for your students (or adults with whom you work) next week or in the near future?

• What theme will you use for crafting or equations on pp. 28-29? (save p. 30 as a master or get template at Algebra Academy website)

• What kind of conversation do you want to occur? Benchmarks 1, 2, 3 4

28-29

Fitting in True/False, Open Number and =/≠ Work

True or False? Why?

• Daily board work as students come in to class• Math Circles/Morning Meeting/Daily Oral Math• Do a few before/after lessons using white

boards• Quick transition after specialist or lunch

Exploring Student Thinking

Between our Professional Development sessions, engage in 2 PLC sessions, facilitated by a staff member at your school or form a professional partnership.

PLC #1 centers on the Baseline Assessment

PLC #2 centers on Instructional Strategiesand Interviews with Students

Putting PD into practice with YOUR students

2 PLC sessions + Next PD Session

PLC 1

PLC 2

Bring to Next PD SessionNov. 7

Throughout our ten-year study,

whenever we found an effective school or

an effective department within a school,

without exception that school or department

has been a part of a collaborative

professional learning community.

- Milbury McLaughlin

Need for a Collaborative Culture

based on the work of Nancy Love, ICTM Pre Session Conference, Oct. 17, 2002

State & DistrictEvidenc

eBuilding Evidence

Classroom specific evidence such as

student work samples, observations, surveys,

interviewsImprove

d Student Learning

Improvement in

Curriculum, Instruction

and Assessment

Drilling Down through Data to Improve Student Learning

Baseline Assessment/PLC#1

• How well do your students understand equality?• Give the Baseline Assessment, preferably THIS

WEEK, to your students without any direct instruction. You may read items to students. Manipulative may be available.

• We are looking for a baseline – what do they actually understand right now?

• Summarize your results on the recording sheets and bring students’ work and recording sheets to your 1ST PLC meeting. (Also save for Nov. 7th)

Baseline Assessment Form B

5 problems to give to students:1) True or false?

2)

3)

4) (grades PK-3?)

5) Two teams have the same number of players. There are 8 girls and 4 boys on one team; if 5 boys are on team two, how many are girls?

Check out recording sheets

Baseline Assessment Form A

5 easier problems to give to students:

1) 2 + 3 =2) 10 = 6 + 4 True or false?

3) 8 + 4 = + 54) 3 + = 5 + 25) Two teams have the same number of players. There are 8 girls and 4 boys on one team; if 5 boys are on team two, how many are girls?

Check out recording sheets

If your students are less verbal

. . . read problems to them, as needed

Then collect papers Discuss some of the problems with a student

or a group of students Record their ideas and bring to PLC/next

session

What do you think?

• In one Minnesota District in 2006…

• 59% of grade 4-5-6 students correctly put “7” in the equation in response to:

2. What number can you put in the box

to make this true? 8 + 4 = + 5

Consider this similar problem…

4. The Cardinals and Blue Jays soccer teams have the same number of players on each team. There are 8 girls and 4 boys on the Cardinals. If there are 5 boys on the Blue Jays, how many girls are there on the team? Explain you thinking.

Answer __________ Write a number sentence to show how you solved this problem.

of the same grade 4-5-6 students got this problem right.92%

Context matters!

Teaching is the constant in classrooms; learning is the variable.

Learning should be the constant; teaching the variable.

Lee Jenkins, The Ten Root Causes of Educational Frustration, DataNotGuesswork Seminar, Sept. 19, 2001

Instructional Strategies/PLC#2

Try some of the instructional strategies or games suggested today:Krypto/Equal 9 posters/Equal #

Increasing the variety of equation formatsUsing “is the same as” language with “equals”Using = and ≠ or balances

Using true/false or open number sentences

Using equation chains or clothespin cards

Come to the 2nd PLC meeting ready to share what is happening with your students & how you use these strategies with your students.

Why is “Equality” so important?

• When students are limited in the kinds of equations with which they work, the conversation can only be about the right answer.

• When students can think about the equal sign differently, the conversations can focus on understanding important math concepts and skills.

• Listening to students’ math conversations makes us smarter teachers--we learn what students know and what they don’t know. “Math Talk” is essential!

PLCs and Teacher Knowledge Matter

In the 7-county Metro Area, those schools which had not made AYP and whose staff members did attend the gr. 3-5 Math Academies . . .

had greater MCA scoresthan schools who did not attend

Region 11 Evaluation Report and Metro ECSU communication, 2010

Assessing Student Learning

We need multiple indicatorsNo one kind of information tells the whole story.

You need to match the evidence to the

audience What evidence will they trust?

Thomas R. Guskey, conference address, MSDC, May 15, 2002

Interview Protocol/PLC #2

Select up to 3 students for a teaching interview with each of the students.

Seeing (hearing & watching) is believing!

Check out your Interview Protocol.Bring your recording sheets and student

work to the 2nd PLC meeting.

Summative Assessment/SD Day

• Parallel to the Baseline• Give to your students before 11/7• Bring Baseline AND Summative

recording sheets and student work to next session on 11/7

Get all assessments online at the Academy Website

MN Mathematics Standards Numbering System

78

3. 2. 1. 4GRADE LEVEL

STRANDSTANDARD

BENCHMARK

K-8, 9-11 1 Number2 Algebra 3 Geom/Meas4 Data

May have several per strand

May have several per standard; includes examples

Supporting Standards & IEP Goals

Using the Standards by Progression Document, identify a standard and some benchmarks that could be supported with one of today’s activities

ALGEBRA

NUMBER

DATA/PROBABILITY

GEOMETRY/MEAS.

MATHEMATICS & SCIENCE FRAMEWORKS FOR MN STANDARDS

STEM TEACHER RESOURCE CENTERwww.scimathmn.org/stemtc

A New Resource for Teachers to Identify Resources to Help Students Achieve

Minnesota Standards in Mathematics or Science

Launched 6/30/2011 – work in progress – your feedback is appreciated!

OVERVIEWSummarizes Big Ideas and Essential Understandings of the Standard/BenchmarksCorrelations to NCTM StandardsCorrelations to national Common Core StandardsMISCONCEPTIONSPoints out frequent misconceptions and common errors students have about these math ideasVIGNETTEA snapshot into a classroom featuring an acitivity and teacher/student dialogueGives a picture of what standard/benchmarks meanRESOURCESTeaching notes and ideas for learningResource listsKey vocabulary and ideas for working with vocab.Professional development/PLC resourcesReferences

ASSESSMENTSample tasks to assess this standard/benchmarksMay be used formatively or summatively

DIFFERENTIATIONIdeas for struggling learners and English LearnersIdeas to extend the learning for kids who “get it”Resource lists

PARENTS/ADMINISTRATORSChecklists and resources for parents, coaches, administrators

Scavenger Hunt

• Select a mathematics standard and cluster of benchmarks on which to focus

• Explore the 7 tabsScavenger

Hunt

Next time (Wednesday, Nov. 7)

• Bring both Baseline & Summative Assessment recording sheets/student work.

• We’ll review your experiences with equality, equations and talking about the equal sign

• We’ll focus on Modeling Word ProblemsIs every word problem a unique experience for students? How can graphic organizers help students understand, categorize and solve word problems?

Feedback please . . .

• Please fill out the sheet with shapes to help us plan for next time

• Check in with your school/district team about PLC meetings

• Talk about what you can try out from today’s session with your students

• Identify an article to read