Transcript Equality Presentation Powerpoint

2012-2013 Algebra Academy
 Exploring Student’s Mathematical
Thinking
 Probing the Math Needed for Algebra
Professional Development in Mathematics
For District 287 & Member Districts Special Education Staff
Nancy Nutting
Mary Peters
Christina Shidla
Scott Swanson
[email protected]
[email protected]
[email protected]
saswanson@district 287.org
District 287 Program Facilitor
for Professional Learning
Jennifer Nelson
[email protected]
763-550-7241
A Minnesota Project
Developed by Districts 287 & 916, Metro ECSU, U of M, Hamline, Normandale CC
Math Success:
It’s In Our Hands
Minneapolis/St. Paul
Region 11 Math and Science
Teacher Center
www.region11mathandsciencecenter.com
Surveyed 287 Special Ed Staff
Math Committee Recommendations
THIS is what we are doing today
Algebra 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
When you
“get”
something,
practice your
questioning
skills to help
others
understand
Learning
instructional
strategies to
increase skills &
understandings
Assessing our
students and
planning from
what we learn
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)
EQUALITY
Wednesday, Nov. 7 (Room 321)
MODELING WORD PROBLEMS
+ 2 school-based
Wednesday, Dec. 12 (Room 321)
PLCs between
RELATIONAL THINKING
sessions
Wednesday, Jan. 16 (Room 321)
OPERATIONS & BASIC FACTS
Wednesday, Feb. 27(Room 321)
FRACTIONS & DECIMALS
Wednesday, May 1 (Room 321)
cover
EFFORT, PROBLEM SOLVING & REASONING
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 9
Write 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
Deal
Hint
Solve
2
What’s does this equal?
3+4x5=?
35 or 23?
Parenthesis & Exponents
Mult & Divide (in order)
Add & Subtract (in order)
10 – 6 + 5 = ?
̶ 1 or 9?
PE
MD
AS
Please Excuse
My Dear
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
2
problems involving whole numbers.
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)
Percent Responding
12&17 other
7*
12
17
Gr.
1&2
Gr.
3&4
5
58
13
8
16
9
49
25
10
7
Gr.
5&6
2
76
21
2
0
responses
3
Source: Carpenter, Franke and Levi
What if the equations were . . .
Does format matter?
8+=7+5
8+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
After
adjusting
adjusting
instruction instruction
Gr. 1 & 2
Gr. 3 & 4
Gr. 5 & 6
5%
9%
2%
66%
72%
84%
NCISLA inBrief: ”Building a Foundation for Learning Algebra,” Fall 2000,
http://ncisla.wceruw.org/publications/briefs/fall2000.pdf.
Instruction matters!
Minnesota Metro Area School in 2007
7 * (8*)
12 (16)
17 (24)
07
07
07
other
(Sept)
School Sept. Dec. Sept. Dec. Sept. Dec. Sept. Dec.
8+4=+5
W
(Dec)
9+7=+8
*CORRECT RESPONSE
Grade 6%
2
07
07
07
07
07
33% 53% 40% 17% 13% 25% 15%
Grade 34% 65% 44% 28% 13% 6%
3
9%
1%
Grade 26% 75% 59% 17% 15% 8%
4
0%
3%
Grade 65% 88% 25% 5%
5
3%
1%
7%
6%
Metro Area Junior High Data
Junior High Intervention Students n = 671
Responses
n
%
7
12
17
12 or 17
other
268
274
76
31
22
40
41
11
5
3
Same Metro Area High School
Geo1
Geo2
Geo3
BCGeo
Responses
n = 29
n = 31 n = 34
n = 11
65.52% 61.29% 85.29% 72.73%
7
20.69% 22.58% 8.82%
9.09%
12
0.00% 6.45% 5.88%
0.00%
17
6.90% 3.23% 0.00%
0.00%
9
0.00% 0.00%
0.00
0.00% 12 or 17
6.90% 6.45% 0.00% 18.18%
other
December 2, 2004
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 5
4 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 + 6
4 plus 6 is the same (amount) as 10
10 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
14 + 16
=
=
=
=
17 + 
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 a
balance:
+5
+5
3–5+6=5–1
What numbers do the circle and the
triangle represent if this balance
DOES balance?
PAN BALANCES
http://illuminations.nctm.org/ActivityDetail.aspx?ID=26
You Tube Videos
Linear Equations – Balancing the Equation
http://www.youtube.com/watch?feature=fvw
p&NR=1&v=DO-hzLh79uw
Pan Balance with Shapes
http://www.youtube.com/watch?feature=end
screen&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:
Use:
Equal 9
Posters
or Krypto
7=3+4
2+8=5+5
356 + 42 = 354 + 44
y = 3x + 7
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
12
4
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
̶ 4
3m
3m  3
m
= 13
̶ 4
= 9
= 93
= 3
What 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
Best Definition  Percent of Students (n=375)
response
Relational Operational Other No
don’t know
Gr. 6
29
58
7
6
Gr. 7
36
52
9
3
Gr. 8
46
45
8
1
Source: Knuth, “The Importance of Equal Sign Understanding”, MTMS, May 2008
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
but 8 + 4 ≠ 12 + 5
and 8 + 4 ≠ 17
what’s wrong?
Use a “goes to” arrow
to 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.
27 =
Whole
Numbers +
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
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
Benchmarks 1, 2, 3  4
6+4
18
Generate Equations
6 + 4 = 10
4 + 6 = 10
7 + 3 = 10
3 + 7 = 10
5 + 5 = 10
etc.
0 + 10 = 10
1 + 9 = 10
2 + 8 = 10
3 + 7 = 10
4 + 6 = 10
5 + 5 = 10
6 + 4 = 10
7 + 3 = 10
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.
¿Es verdad o falso?
“Grown-up” vs “Kid” Hmong
e.g. Today
everyone is
wearing the
color red. True
or False?
Tseeb
true
Tsi Tseeb
not true
Yog
correct
Tsi Yog
not correct
Order from Easiest to Hardest
19
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
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 = 10
Benchmarks 1, 2, 3  4
Using True-False Sentences
Using 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
22-27
• 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
One Side is the Same as the Other
28-29
• 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
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 Strategies
and Interviews with Students
Putting PD into practice with YOUR students
2 PLC sessions + Next PD Session
PLC
2
PLC 1
Bring to
Next PD
Session
Nov. 7
Need for a Collaborative Culture
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
Drilling Down through Data to Improve
Student Learning
State &
District
Evidence
Building
Evidence
Improved
Student
Learning
Classroom specific
evidence such as
student work samples,
observations, surveys, interviews
Improvement in
Curriculum,
Instruction and
Assessment
based on the work of Nancy Love, ICTM Pre Session Conference, Oct. 17, 2002
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)
Check out
recording sheets
(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?
Baseline Assessment Form A
5 easier problems to give to students:
2+3=
2) 10 = 6 + 4 True or false?
1)
3) 8 + 4 =
4) 3 +
+5
Check out
recording sheets
=5+2
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?
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.
92%
of the same grade 4-5-6 students got this
problem right. 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 formats
Using “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 scores
than schools who did not attend
Region 11 Evaluation Report and Metro ECSU communication, 2010
Assessing Student Learning
We need multiple indicators
No 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
3.
GRADE
LEVEL
K-8, 9-11
2.
STRAND
1.
4
STANDARD BENCHMARK
1 Number
May have
2 Algebra
several
3 Geom/Meas per strand
4 Data
May have
several per
standard;
includes
examples
78
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
NUMBER
ALGEBRA
GEOMETRY/MEAS.
DATA/PROBABILITY
MATHEMATICS & SCIENCE
FRAMEWORKS FOR MN STANDARDS
STEM TEACHER RESOURCE CENTER
www.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!
OVERVIEW
 Summarizes Big Ideas and Essential Understandings of the
Standard/Benchmarks
 Correlations to NCTM Standards
 Correlations to national Common Core Standards
MISCONCEPTIONS
 Points out frequent misconceptions and common errors students have
about these math ideas
VIGNETTE
 A snapshot into a classroom featuring an acitivity and teacher/student
dialogue
 Gives a picture of what standard/benchmarks mean
RESOURCES
 Teaching notes and ideas for learning
 Resource lists
 Key vocabulary and ideas for working with vocab.
 Professional development/PLC resources
 References
ASSESSMENT
 Sample tasks to assess this standard/benchmarks
 May be used formatively or summatively
DIFFERENTIATION
 Ideas for struggling learners and English Learners
 Ideas to extend the learning for kids who “get it”
 Resource lists
PARENTS/ADMINISTRATORS
 Checklists and resources for parents, coaches, administrators
Scavenger Hunt
• Select a mathematics standard and cluster
of benchmarks on which to focus
• Explore the 7 tabs
Scavenger
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 Problems
Is 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