Children’s Understanding of
Equality: A Foundation for
Algebra
Research by K. Falkner, L. Levi,
and T. Carpenter
The Problem
• Teachers were asked to share the
following problem with their classes:
• 8+4=
+5
• One 6th grade teacher was very
puzzled…then she was astounded by
her classes’ answers.
Overall, at this school:
Answers given to 8 + 4 =
+5
Grade
Number
7
12
17
12 & 17
Other
1
42
0
33
3
0
6
1&2
84
5
45
17
0
17
2
174
11
96
17
24
26
3
208
21
125
42
10
10
4
57
4
5
25
17
6
5
42
3
20
19
0
0
6
145
0
122
20
3
0
Total
752
44
446
143
54
65
percent
100
6
59
19
7
9
Of the 44 students who got it
correct
About 3/4ths of them were in second and third grade
Grade
Number
Percent
1
0
0
1&2
5
11
2
11
25
3
21
48
4
4
9
5
3
7
6
0
0
Back to 8 + 4 =
+5
• How did students (59%) arrive at an
answer of 12?
• How did students (19%) arrive at an
answer of 17?
• How did students (7%) arrive at an
answer of both 12 and 17?
The basic misconception…
• Concerns what the symbol “=“ means
• For a great majority of the students (at
least 85%), it means “time to carry out
a procedure and arrive at an answer”.
• For a small minority of the students
(the 6% who arrived at the correct
answer), it meant “is the same as”, i.e.
• “8 + 4 is the same as + 5”
How does this happen?
• It’s all in how the “=“ is used…
• “Not much variety is evident in how the
equals sign is typically used in the
elementary school. Usually, the
equals sign comes at the end of an
equation and only one number comes
after it…[as a result], the children are
correct to think the equals sign is a
signal to compute” – Falkner, Levi &
Carpenter
How to help children understand
the concept
Falkner (who taught the mixed 1st and 2nd grade class)
chose to help her students understand by using true
and false number sentences, such as the following:
4+5=9
12 – 5 = 9
7=3+4
8 + 2 = 10 + 4
7 + 4 = 15 – 4
8=8
The children used manipulatives
(Unifix cubes and other materials)
to help make models of the
situation
The initial reactions were
interesting…
• All children agreed that 4 + 5 = 9 was true
and that 12 – 5 = 9 was false.
• They were less sure about the others
(read from article, page 234 and 235)
The next year…8 + 4 =
was back
+5
• A few, but not all of the children who
were in the class the previous year got
it correct.
• Most of the new kids (who were in
kindergarten the previous year) either
put 12 or were confused.
• However, this year, more students
were able to explain why the answer
should be 7
Falkner integrated discussion of
equality throughout the school year
• She continued to use the true-false
sentences
• She had students make up their own
true and false sentences
• She wrote open number sentences
where the location of the unknown
varied
• __ = 9 + 5
7 + 8 = __ + 10
7 + __ = 6 + 4
etc
As the year progressed…
More students understood the concept. By springtime,
she was integrating discussions about equality with
discussions about other algebraic concepts
For example, she asked the class to
look at the sentence: a = b + 2.
She asked them: which is larger…a
or b?
For children (or anyone) who
struggles with the concept of
equality, this problem would prove
daunting…so how did a class of 1st
and 2nd graders handle it? (read
page 236)
Conclusions?
• By using the techniques, • This understanding
the teachers were able
is extremely
to help more students
important if students
understand correctly the
are to understand
concept of equality.
how to do even the
most basic
algebraic
manipulations.
• The students also
seemed to enjoy
this…
Article:
• Children’s Understanding of Equality: A
Foundation for Algebra, from Teaching
Children Mathematics, December 1999,
pages 232 - 236
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•
•
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Authors:
Karen P. Falkner
Linda Levi
Thomas P. Carpenter