2010-11
Algebraic Thinking
SD Counts Cohort Meeting
Platte South Dakota
September 29, 2010
8:30-3:30
House Keeping
Break at 10:00 and 1:45
Lunch at 12:00
Pop machine in break room
Bathrooms
Group Norms
Honor private think time.
Be prepared for sessions, everyone
should have something to contribute.
Listen carefully to take in another
participant’s ideas.
Cell phones off or on vibrate.
Start and end each session on time.
Take care of your needs.
Items to attend too
Website www.mathcounts.org
October meeting dates
Sign-in sheet
Syllabus
Training for next year-visit with principals
-upcoming workshop for Grades preK-2
How is your year going?
Take a few minutes of quiet think time
and think about how this year has
already been better than last year and
why?
What did your e-metric scores look
like?
Be ready to share
Main ideas of Algebraic Thinking in
the Elementary
Equality
Relational Thinking
Understanding Properties
Conjectures
Justification
Algebra Readiness
does not start with pre-algebra.
Is the foundation which begins in the elementary grades and
is supported by sound teaching methods
Begins in Elem→MS→HS
Algebra should build on skills learned at the pre-K level
Must have curriculum and resources to achieve and meet the
standards
Must get kids actively involved so they better understand
what they are doing.
Must be engaging and relevant
PATTERNS, PATTERNS, PATTERNS-STUDENTS MUST
HAVE THE OPPORTUNITY TO WORK WITH NUMBERS
AND SEE PATTERNS BEFORE THEY CAN WORK WITH
AND UNDERSTAND AN EQUATION
What should Algebra look like at each school?
Algebra at the K-2 level..
The patterns and patterning activities in which
young children readily engage serve to form the
basis of their emergent algebraic reasoning.
Whether they are snapping cubes, clapping their
hands or stomping their feet in rhythm, or using
input-output machines, young children are eager
to experience and are capable of identifying and
describing change.
A lot of concrete, hands-on activity.
Linking cube questions what would colors be?
Different problem types
Algebra at the 3-5 level….
The focus at this level is on patterns,
relations, and functions and the ability
to analyze situations using a variety of
representations.
They begin to see how they can use
algebra in the real-world.
A lot of hands-on, concrete activity.
Algebra at the 6-8 level….
Students need multiple opportunities over time
and across contexts to demonstrate fluency,
flexibility, and a deep understanding of algebra
concepts and skills
Much time should be spent examining tables,
charts, graphs, and equations.
Still a lot of concrete moving to abstract toward 8th
grade
As students develop their algebraic reasoning,
they should be engaged in the use of tables,
charts, graphs, and symbolic expressions
containing variable's
Outcomes for this session
Establish that students have some very common
misconceptions about the meaning and use of the
“=” sign.
Explore strategies to help students develop an
understanding of the meaning and use of the
equal sign.
Develop an understanding of the importance of the
concept of equality in algebraic reasoning
Talk about “relational thinking”
Develop as a cohort the “Big Ideas” of algebraic
thinking.
Visit the properties of math that students use
Table Tent Activity
Write your name
Write a 1-digit number below your
name
Problem Solving Activity
Number of Chirps per Minute
Extension activity
-Once you see a pattern, try to write a
formula.
Misunderstanding
Students, across the state have a very
common misconception of what the “=”
sign means.
We gave students these numbers and
wanted them to solve
8 + 4= ? +5
The next slide will show the results
Discuss how your students answered
before we look to see if they answered like
the rest of the students in our state.
234 students were asked this question 8 + 4= ? + 5. The following
results will show the common misconception of what does the “=“
sign represent.
Grade 7
12
17
12 &17
1&2
5
45%
79
43%
14
27%
2
18%
29
16%
9
18%
2
18%
25
14%
6
12%
11 Students
3&4
181 Students
5&6
51 students
2
18%
48
26%
22
43%
Children’s Understand of Equality: A Foundation for
Algebra
Read silently 10 minutes, share small group,
discuss large group
Be looking for an answer to the following questions
What strategies does the author use to help her
students understand the equal sign?
How should students be thinking about the equal
sign?
Are there just some things you “just” tell the
students
Read Article
Group 1-Beginning to The Next Year
Group 2-Beginning of The Next Year-end
Skip-about the Mathematics
Share articles
Key ideas from your section???
Huge outcomes for this article were..
This teacher does a good job of teaching
equality to students
Revisit equality; over, and over, and over.
2 tools to develop equality
True and false number senses
Open-ended number sentences
Why know about equality as a
relationship?
First-In order to think about relationships expressed by
number sentences, children must understand that
equality is a relationship that expresses the idea that
two expressions hold the same value.
Second-Because students do not understand equality
as a relationship, this becomes a major stumbling
block for students when they move from arithmetic
to algebra.
Must deal with misconception-Children must
understand that equality is a relationship rather
than a signal to do something.
Article Reading
Balancing Act: The Truth behind the
Equals Sign
Read individually and be ready to share
Equality Handout
Ask “purpose” questions before each
section
Watch video clips
1.1, 1.2, 1.3, 1.4, 1.6
Steps to Equality
What are the steps in teaching
equality to our students?
Where do we start?
Where do we move to?
What do we revisit?
Video 1.4
Questions
-What misconceptions about the
equal sign do children in this episode
express?
-How do the children who understand
the equal sign help the other children
to develop their understanding?
Watch video clip 1.6
Questions to answer
How does each child solve this problem?
What do their strategies tell you about
their understanding of equality?
“Big Ideas” of equality
In your groups create a list of “big ideas”
about “equality” and put on the wall
What are the words that you have read or we
have talked about that you would like listed
on your sheets of paper.
Let us compare and add if we need to add
any to our list.
Developing a series of true and
false number sentences
Read page 14-15-16-17 in book.
“Developing Children’s Conceptions
of Equality”
Groups to create a series of true and
false number sentences (Grade level)
Where do we start?
Where do we go?
What do we come back to?
Article: Fostering Relational Thinking
while Negotiating the Meaning of the
Equals Sign
This article will again talk about the
meaning of the equal sign, but also talk
about how relational thinking can be use
to help solve problems.
So what is relational thinking?
Relational Thinking
Take out handout “Intro to Relational
Thinking”
Solve independently
USE ANYTHING YOU KNOW
ABOUT NUMBERS TO SOLVE these
number sentences.
Find at least 2 ways, if you finish early 3
ways
Definition of “relational thinking”
Solving problems by looking at
expressions/equations completely, rather
than as procedures to be carried out step
by step
Must have understanding of equality before
you develop “relational thinking”
“solving problem”= instead of procedure
“add on, subtracting”=relational thinking
Relational Thinking
Take out Emma’s Problem Handout
Video clip 2.1
Decide why each number sentence
posed might be particularly useful in
developing relational thinking?
If you were this child’s teacher, what
problems would you have her work on
next?
Video 2.3
Take out Relational Thinking handout
Question
The number sentences posed led to the
number sentences 4 x 7 = ?. Design a
similar sequence of number sentences to
lead to the number sentence 6 x 8 = ? What
implications might such series of number
sentences have for learning multiplication
and division facts?
Video 2.4
Question
Describe each strategy. What strategy
might each of these children use to
decide whether 84 + 27 = 74 + 37 and
56 + 38 = 59 + 36
Big Idea’s of Relational Thinking
See handout, are there any you would
like to add?
Course Expectation
Read and reflect a professional book
(I have some you can choose from)
-At our last meeting we will take time
to reflect on the book you read.
-How/has this book impacted you as
a classroom teacher.
Next meeting date
Snacks for next month
Evaluation
Read Chapters 4-5-6 Jot down key
ideas for each chapter
Next meeting-CHANGED TO
OCTOBER 25TH