Understanding Student Thinking in
Algebra
Session Equality
2008-2009

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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.

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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

 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

Number of Chirps/Minute
Once you see a pattern, try to write a formula

Misunderstandings of the equal sign
 Students across the state have a very common misconception of what the equal sign
 We gave students these numbers and wanted them to solve
8 + 4= __ + 5
 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
1 & 2
11 Students
3 & 4
181 Students
5 & 6
51 students
7
2
18%
48
26%
22
43%
12
5
45%
79
43%
14
27%
17
2
18%
29
16%
9
18%
12 &17
2
18%
25
14%
6
12%

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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
– Share your thoughts

 Tell the students what the equal sign means
 Revisit equality over, over and over
 Two tools to develop
– 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.

 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
1 st 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?

 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
4 th grade classroom
10’ in length
Listen to how this 4 th 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.
– 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 14-15-1617 in book. “Developing Children’s
Conceptions of Equality”
Groups to create a series of true and false number sentences
(Grade level)
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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
2 nd 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
 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
 Start with a simple example: 2x + 3 = 9
– Using what you know about numbers solve this
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– Solve as an elementary student would
Solve this problem: 7x – 36 = 48 – 14x
– Using what you know about numbers solve this
– Solve as an elementary student would

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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