# Why can't schoolchildren use a formula to solve math problems that don't contain numerical values ?

Kehchi Magara

Research output: Contribution to journalArticle

### Abstract

Fifth and sixth graders were presented with 2 problems: (1) "Find the area of this triangle [base 8 cm, height 5 cm]," and (2) "How big is triangle B [an isosceles triangle] compared to triangle A [a right triangle]?" In figure accompanying the second question, the bases of the 2 triangles were equal, and triangle B was twice the height of triangle A. However, numerical values for the width of the base and the height were not provided. If the children were able to use the formula, "area of a triangle=base X height-f-2", properly, they should have been able to realize that triangle B had twice the area of triangle A. This is called the "relational operation of a formula". Of the 135 children, 123 (91%) gave the correct answer to Question 1, but only 62 (46%) answered Question 2 correctly. In other words, the children who could calculate area by using the formula could not necessarily operate the variable in the formula. Further analysis suggested that prerequisites for giving the correct answer to Question 2 were: (a) grasping the relative, rather than absolute, difference in the areas of the 2 triangles, and (b) understanding the difference between "the difference in the area of 2 figures" and "their ratio".

Original language English 180-191 12 Japanese Journal of Educational Psychology 57 2 Published - 2009

schoolchild

### Keywords

• Elementary school children
• Formula for calculating area
• Operating variables in a formula
• Understanding formulae

### ASJC Scopus subject areas

• Developmental and Educational Psychology
• Education

### Cite this

In: Japanese Journal of Educational Psychology, Vol. 57, No. 2, 2009, p. 180-191.

Research output: Contribution to journalArticle

@article{ac4bfebb1f874f88b115ddd9036b15b0,
title = "Why can't schoolchildren use a formula to solve math problems that don't contain numerical values ?",
abstract = "Fifth and sixth graders were presented with 2 problems: (1) {"}Find the area of this triangle [base 8 cm, height 5 cm],{"} and (2) {"}How big is triangle B [an isosceles triangle] compared to triangle A [a right triangle]?{"} In figure accompanying the second question, the bases of the 2 triangles were equal, and triangle B was twice the height of triangle A. However, numerical values for the width of the base and the height were not provided. If the children were able to use the formula, {"}area of a triangle=base X height-f-2{"}, properly, they should have been able to realize that triangle B had twice the area of triangle A. This is called the {"}relational operation of a formula{"}. Of the 135 children, 123 (91{\%}) gave the correct answer to Question 1, but only 62 (46{\%}) answered Question 2 correctly. In other words, the children who could calculate area by using the formula could not necessarily operate the variable in the formula. Further analysis suggested that prerequisites for giving the correct answer to Question 2 were: (a) grasping the relative, rather than absolute, difference in the areas of the 2 triangles, and (b) understanding the difference between {"}the difference in the area of 2 figures{"} and {"}their ratio{"}.",
keywords = "Elementary school children, Formula for calculating area, Operating variables in a formula, Understanding formulae",
author = "Kehchi Magara",
year = "2009",
language = "English",
volume = "57",
pages = "180--191",
journal = "Japanese Journal of Educational Psychology",
issn = "0021-5015",
publisher = "Japanese Association of Educational Psychology",
number = "2",

}

TY - JOUR

T1 - Why can't schoolchildren use a formula to solve math problems that don't contain numerical values ?

AU - Magara, Kehchi

PY - 2009

Y1 - 2009

N2 - Fifth and sixth graders were presented with 2 problems: (1) "Find the area of this triangle [base 8 cm, height 5 cm]," and (2) "How big is triangle B [an isosceles triangle] compared to triangle A [a right triangle]?" In figure accompanying the second question, the bases of the 2 triangles were equal, and triangle B was twice the height of triangle A. However, numerical values for the width of the base and the height were not provided. If the children were able to use the formula, "area of a triangle=base X height-f-2", properly, they should have been able to realize that triangle B had twice the area of triangle A. This is called the "relational operation of a formula". Of the 135 children, 123 (91%) gave the correct answer to Question 1, but only 62 (46%) answered Question 2 correctly. In other words, the children who could calculate area by using the formula could not necessarily operate the variable in the formula. Further analysis suggested that prerequisites for giving the correct answer to Question 2 were: (a) grasping the relative, rather than absolute, difference in the areas of the 2 triangles, and (b) understanding the difference between "the difference in the area of 2 figures" and "their ratio".

AB - Fifth and sixth graders were presented with 2 problems: (1) "Find the area of this triangle [base 8 cm, height 5 cm]," and (2) "How big is triangle B [an isosceles triangle] compared to triangle A [a right triangle]?" In figure accompanying the second question, the bases of the 2 triangles were equal, and triangle B was twice the height of triangle A. However, numerical values for the width of the base and the height were not provided. If the children were able to use the formula, "area of a triangle=base X height-f-2", properly, they should have been able to realize that triangle B had twice the area of triangle A. This is called the "relational operation of a formula". Of the 135 children, 123 (91%) gave the correct answer to Question 1, but only 62 (46%) answered Question 2 correctly. In other words, the children who could calculate area by using the formula could not necessarily operate the variable in the formula. Further analysis suggested that prerequisites for giving the correct answer to Question 2 were: (a) grasping the relative, rather than absolute, difference in the areas of the 2 triangles, and (b) understanding the difference between "the difference in the area of 2 figures" and "their ratio".

KW - Elementary school children

KW - Formula for calculating area

KW - Operating variables in a formula

KW - Understanding formulae

UR - http://www.scopus.com/inward/record.url?scp=71749103050&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=71749103050&partnerID=8YFLogxK

M3 - Article

VL - 57

SP - 180

EP - 191

JO - Japanese Journal of Educational Psychology

JF - Japanese Journal of Educational Psychology

SN - 0021-5015

IS - 2

ER -