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 = + 5Grade 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 = + 5was back

• 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, the teachers were able to help more students understand correctly the concept of equality.

• This understanding is extremely important if students are to understand 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

• Authors:• Karen P. Falkner• Linda Levi• Thomas P. Carpenter