Frames, frames, and more framesAuthor(s): MARY FOLSOMSource: The Arithmetic Teacher, Vol. 10, No. 8 (December 1963), pp. 484-485Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41186991 .

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Frames, frames, and more frames

MARYFOLSOM University of Miami, Coral Gables, Florida Professor Folsom is a member of the school of education faculty at the University of Miami. As a member of the Panel on Teacher Training of the Committee on the Undergraduate Program in Mathematics (CUPM), she has been speaking on their State conference programs.

J/ ew deny that the mathematics curricu- lum of the elementary school needed revi- sion. However, the rush to modernize this curriculum has given rise to some rather questionable practices. Experimental pro- grams claim that the changes to be made should result in good, teachable mathe- matics. One could hardly argue with this objective. Behind closed classroom doors, however, the danger always exists that lack of understanding of the mathematics involved can, and often does, lead to some ludicrous errors. The misuse of frames is a case in point.

D+2 = 5, D + D = 6, and D + A = 10 are as common today as the proverbial old shoe. Not only are children being exposed to such equations in the classroom by teachers, but some recent elementary school textbook series also have hopped on the band wagon.

The elementary school teacher finds the use of 5+ □ = 8 easier for the children than 5+X = 8. The children can write the cor- rect numeral in the interior of the frame without rewriting the equation. Conse- quently, some teachers look upon the use of frames only as a gimmick - a novelty which holds the interest of children.

More alert publishers, who are aware of the hazards involved, avoid the use of different-shaped frames in the same equa- tion, and even go so far as to insist that since a number is the replacement for the frame, the numeral may not be written in- side the frame. The equation must be re- written.

484

Unless the teacher understands the algebraic structure behind the use of frames, one is apt to hear a teacher saying, and quite correctly, that given the equa- tion □ + □ = 6 (the same-shaped frame is repeated in the equation), the number re- placements must be the same. The correct solution is 3+3 = 6, and no other solution is possible in the set of whole numbers. Then, falling into the seemingly logical pit- fall, the teacher follows this with, "The pupils saw that in the frame Л + □ = 6 the numerical substitution must be different, and could not be the same numerals as in D+D = 6.lfl

What foolishness is this? If □ and Д are variables, then D + D = 6 is the same as x+x = 6, therefore x = 3. If follows then that different-shaped frames are used in- stead of x and y, and the difference in the shape of the frames allows us to say that the number replacements MAY or MAY NOT be the same. Given the equation □ + A = 6, there are seven possible solu- tions within the set of whole numbers.

□ 1 A 0 a 1 5 2 4

4 2 5 I 6 О

» Elizabeth В. King, "Greater Flexibility in Abetract Thinking Through Frame Arithmetic," The Arithmetic Teacher, X, No. 4 (April, 1963), 185.

The Arithmetic Teacher

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Different-shaped frames may have the same number replacement. This must be true. If it were not true, then the graphic representation of the real numbers satis- fying the equation would be missing the point in the plane which has the coor- dinates (+3, +3).

One might say, with some justification, that an even more fundamental error is made when the equal sign is used incor- rectly. Most elementary school teachers are not familiar with analytic geometry, but they should be expected to know what is implied by equality.

On the same page in the article previ- ously quoted, the following open sen- tences are given:

a) 9X7 = A + D = 72

b) 7+D = A-6 = 4

There are no replacements for □ and Л which make these sentences true, since equality is a reflexive, symmetric, transi- tive relationship. If the result was in- tended to be

a) 9X7 = 63+9 = 72

b) 7+3 = 10-6 = 4

then the equations should have been written

Í9X7 = A [7+D = A ' and ' IA + D = 72 lA-6 = 4

since 9X7^72, 9X7^63+9, 7+3?¿4, and 7+3^10-4.

To tamper with the algebraic nature of frames not only distorts the real meaning and use of such figures as variables, but will inevitably lead the use of frames into disrepute.

Questions they asked- Can kindergarten and first-grade children understand the role of base ten in our num- ber system? Will changing the base help them to gain a better understanding?

In an investigation of the introduction of various radices in six kindergarten and eight first-grade classes, Scott reports that the majority of children in both grades demonstrated some understanding of the use of the base and positional notation in our numeration system. (The extension of the concept of the { base X base} position was not so easily conceptualized.) Al- though it is possible to acquaint children in kindergarten and first grade with some fundamental concepts about the organi- zation of their system of numeration using different bases, the profit of such a pro- gram is not presently known. - From "A Teaching Investigation of the Introduction of Various Radices in Kindergarten and First Grade Arithmetic," by Lloyd Scott, California Journal of Educational Re- search, XIV (January, 1963).

December 1963

Does using other systems of numeration really help children to better understand our decimal system of numeration?

Under the direction of Harold H. Lerch, fourth-grade children used a short story about make-believe people and their make-believe number system as the basis for study of the development and charac- teristics of a number system with base five. Responses of the pupils to the items on the various measuring instruments not only indicated that such study is feasible for fourth-grade children, but also that chil- dren seem to enjoy and appreciate the study. Too, their understanding of the character- istics of the Hindu- Arabic decimal numera- tion system was increased over the experi- mental period, and such a study was not detrimental to other arithmetic skills and understanding. - From uFourth Grade Pu- pils Study a Number System With Base Five, ' ' by Harold H. Lerch, The Journal of Edu- cational Research, LVII (October, 1963).

(Continued on page 490)

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