Transcript Understanding Student Thinking in Algebra Session Equality 2008-2009 Sessions for Algebraic Thinking      What does the equal sign mean? Relational Thinking Understanding the Properties of Arithmetic Conjectures Justification In order.

Understanding Student Thinking in
Algebra
Session Equality
2008-2009
Sessions for Algebraic Thinking
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What does the equal sign mean?
Relational Thinking
Understanding the Properties of Arithmetic
Conjectures
Justification
In order for students to have a good experience with
Algebra, they must first understand what the equal
sign means.
Outcomes
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Establish that students have some very common
misconceptions about the meaning and use of the
equal sign.
Explore strategies to help students develop and
understanding of the meaning and use of the equal
sign.
Develop an understanding of the importance of the
concept of equality in algebraic reasoning.
“Big Ideas” of Algebraic Thinking
Group Norms
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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
Problem-Solving Activity
Number of Chirps/Minute
Once you see a pattern, try to write a formula
Misunderstandings of the equal sign
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Students across the state have a very common
misconception of what the equal sign
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We gave students these numbers and wanted them
to solve
8 + 4= __ + 5
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These are the results of the 234 students we polled
234 students were asked this question 8 + 4=5+__. The
following results will show the common misconception of
what does the equal sign represent
Grade
7
12
17
12 &17
1&2
2
18%
5
45%
2
18%
2
18%
48
26%
79
43%
29
16%
25
14%
22
43%
14
27%
9
18%
6
12%
11 Students
3&4
181 Students
5&6
51 students
Article
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Children’s Understanding of Equality: A Foundation
for Algebra
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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
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Share your thoughts
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Huge outcomes for this article
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Tell the students what the equal sign means
Revisit equality over, over and over
Two tools to develop
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T/F number sentences
Open-ended number sentences
What about equality?
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.
Third-We must deal with misconception-Children
must understand that equality is a relationship rather
than a signal to do something.
Big Ideas of the equal sign
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See handout
Watching students deal with equality
Video Clip 1.1 (Title 1)
•1’40” in length
•Kindergarten student
•Discussion after the video clip
•What does he understand about the equal sign?
•What strategies does he use to compute his
answer?
Watch students use relationships to
solve number sentences
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Video Clip 1.2
3’ 03” in length
1st grader
Listen to how this student explained how he decided
what number went in the box. This is where we want
our students to move.
Then, watch how the teacher moved to bigger
numbers to see if the student really understood the
strategy he shared using smaller numbers
-What strategy did the student use?
The Same but Different
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How did each student solve and what was
the difference between the two students
strategies from the video?
Developing Children’s Conceptions of
Equality
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Video Clip 1.3-Teacher uses T and F number sentences
4th grade classroom
10’ in length
Listen to how this 4th grade teacher took the misconception her
students had and moved them through the process of T/F and open
number sentences
– Pay attention to questions asked by the teacher during discussion
– Pay attention to how the equal sign appears in different places in
the number sentences.
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THIS VIDEO IS A GOOD EXAMPLE OF HOW TEACHERS
BEGAN WITH A MISUNDERSTANDING AND MOVED
STUDENTS TOWARD UNDERSTANDING
Developing a series of true and false
number sentences
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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?
In grade level pairs (K-1, 2-3, 3-4,5-6), generate a sequence of
number sentences that might be used to help students
develop the meaning of the equal sign. This could be a
series of sequences or one-long series. Include the open
number sentence that you hope the students can do at the
end of that series of true and false statements.
Record these on chart paper.
Developing Children’s Conceptions of
Equality Over Time
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Video Clip 1.5
2nd grade students
Listen how the teacher carefully chooses problems
to challenge children’s misconceptions about
equality. Listen to how some of the children are
beginning to use relational thinking.
Pay attention to the sequence of problems Julie uses
Steps to Equality in your classroom
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First, find out who knows and understands
the equal sign by using open-number
sentences
Then, move to true and false number
sentences
Next, go back to open-number sentences
Connecting equality to formal algebra
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Start with a simple example: 2x + 3 = 9
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Using what you know about numbers solve this
Solve as an elementary student would
Solve this problem: 7x – 36 = 48 – 14x
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Using what you know about numbers solve this
Solve as an elementary student would
Accountability Piece
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Assess your students on the “concept of
equality” and get a schedule set up on how
you will continue to deal with the equal sign
Article-”Fostering Relational Thinking While
Negotiating the Meaning of the Equals