Children s understanding of equality a foundation for algebra
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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

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Children s understanding of equality a foundation for algebra

Children’s Understanding of Equality: A Foundation for Algebra

Research by K. Falkner, L. Levi,

and T. Carpenter


The problem
The Problem Algebra

  • 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
Overall, at this Algebraschool:

Answers given to 8 + 4 = + 5


Of the 44 students who got it correct
Of the 44 students who got it correct Algebra

About 3/4ths of them were in second and third grade


Back to 8 4 5
Back to 8 + 4 = + 5 Algebra

  • 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
The basic misconception… Algebra

  • 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
How does this happen? Algebra

  • 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
How to help children understand the concept Algebra

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
The initial reactions were interesting… Algebra

  • 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 5 was back
The next year…8 + 4 = + 5 Algebrawas 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
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
As the year progressed… school year

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
Conclusions? school year

  • 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
Article: school year

  • 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


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