amt 60 (4)28

Variable is generally acknowledged as adifficult concept for students to grasp, but onethat can be developed meaningfully over time(White & Mitchelmore, 1993). Definitions likethese are usually found early in the sequenceof textbook material intended to develop alge-braic concepts. Without exploring thecorrectness or otherwise of the quoted defini-tions, they are not likely to contribute much tostudents’ understanding of the concept. Whilelanguage is crucial in the development of theconcept of variable, the multi-faceted nature ofthe concept makes the use of a formal defini-tion problematic.

Learning with definitions

The definition is an important language formin the register of mathematics. Students needto understand the structure of a definition sothat they can make sense of the definitionsthey encounter and so that they can constructtheir own definitions as part of organisingtheir thoughts about the concepts they haveexplored. However, I am suggesting in thisarticle that there are many mathematicalconcepts for which the use of a single formaldefinition is not helpful in developingstudents’ understanding of the concepts.Rather, more might be gained by havingstudents use language, both mathematicaland everyday, in some of the other languagetasks that have been suggested (for example,Shield & Swinson, 1997).

Reference

Shield, M. J. & Swinson, K. V. (1997). Encouraging learning inmathematics through writing. The Australian MathematicsTeacher, 53 (1), 4–9.

White, P. & Mitchelmore, M. (1993). Aiming for variable under-standing. The Australian Mathematics Teacher, 49 (4), 31–33.

Fractions and decimals are often viewed bystudents as two separate sets of numbersrather than different representations of thesame set of numbers. Often we reinforce thismisconception with instruction that firstfocuses on computations with fractions only,or computations with decimals only. We canhelp students develop a clearer understandingof the concept of ‘representation’ of a numberas a fraction or as a decimal by showing bothnames for a number on the number line;

To accomplish this, students need to beable to find the equivalent fraction for adecimal representation of a number and viceversa. Thus, we become familiar with the frac-tional representation of decimals, writing eachdecimal as a fraction whose denominator is apower of ten. Consider the following examples:

1.

2.

3.

4.

When the decimals are non-terminating,but repeating, there is a procedure sometimestaught in junior high school to find its frac-

0 1 2 3

= 0.5 = 2.512

52

Mal ShieldQueensland University of Technologym.shield@qut.edu.au

DAVID PAGNI

Working from fractions to the decimalrepresentation, we simply divide the numer-ator by the denominator. Consider thefollowing examples:

1.

2.

3.

4.

When a sequence of digits repeats, thissequence is called the repetend and we indi-cate this by placing a bar over those digits. So,a fraction is a number of the form a

b where aand b are whole numbers, and b ≠ 0. A frac-tion has a decimal representation that can befound by dividing a (the numerator) by b (thedenominator).

The final step to showing students thatfractions and decimals are just different waysto represent the same point on the numberline is an investigation whereby studentsperform side-by-side computations that alsoprovide equivalent results. Thus, if we add twofractions and their respective decimal repre-sentations, the sums should agree. Considerthe next examples:

1.

amt 60 (4) 29

tional representation. (See also Beswick in thisedition. Ed.) Consider the following examples:1. 0.11111…

Let x = 0.11111…then 10x = 1.11111…so

2. 0.12121212…Let x = 0.12121212…then 100x = 12.121212…so

3. 0.123123123…Let x = 0.123123123…then 1000x = 123.123123123…so

4. 0.49999…Let x = 0.49999…then 10x = 4.9999…so

amt 60 (4)30

2.

Note that we are able to do this because of theinfinite repetition of the repetend. This makesa nice connection between the two representa-tions that students can explore with morecommonly used fractions like

recalling that

The results would hold true for subtractionof two fractions and their respective decimalrepresentations. Consider the next examples:1.

2.

However, we get into trouble trying tomultiply (or divide) two decimal representa-tions with infinitely recurring repetends.Students can, however, multiply and divideterminating decimals and their equivalentfractions. Consider the next two examples:

1.

2.

Lastly, students can also multiply (ordivide) both representations by wholenumbers and notice that the results agree.Consider the following two examples:

1.

2.

Investigating the relationship between frac-tions and their equivalent decimalrepresentations helps clarify to students thatboth representations stand for a single(rational) number on the number line. Sincewe teach students to perform computationswith fractions and also to compute with deci-mals, performing these computationsside-by-side with consistent results furtherreinforces this, and pulls these two represen-tations into focus. In a way, we remove themystery of these representations, showingthat we can perform computations in any waywe like; the results are equivalent. We havebut to ‘translate’ the result into the represen-tation we want.

David PagniCalifornia State University, Fullerton, USAdpagni@fullerton.edu

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