The Meaning of Letters in Algebra - More About

About the meaning of letters in algebra: 4.25

Research around the world has shown that students draw on what they know of other symbol systems when they are trying to understand symbols in algebra.

Common errors in interpreting letters can be explained by:

• students drawing analogies with other symbol systems (e.g. letters in algebra are abbreviations or initials for things)
• misleading teaching materials (e.g. fruit salad algebra and variations)
• students trying to write mathematical ideas that cannot easily be written in algebra (e.g. trying to say in symbols that when x increases by 1, y increases by 4).

A study that compared students, progress in algebra in different schools revealed that misinterpretation of algebraic letters as abbreviated words or labels for objects was a persistent difficulty in schools where teaching materials explicitly presented letters as abbreviated words (e.g., c could stand for a cat so 5c could mean 5 cats). The study suggests that ‘the letter as an abbreviated word’ can be a particularly difficult misconception to eliminate, if it has been caused by certain teaching materials or teachers’ explanations.

In contrast, teachers seemed to be able to correct the naÏve misinterpretation that the value of letters was related to an alphabetic code quite easily, by explicitly pointing it out. This misconception is often evident in puzzling numerical answers to algebraic questions (e.g. occasionally we see students who evaluate 2n as 28 because n is the 14th letter of the alphabet). Not all misconceptions are deep.

Students experience further difficulties when numbers are combined with pronumerals. In other symbol systems, such as chemical formulae, Roman numerals or indeed in our own number system, putting together two or more symbols implies addition in some way. For example:

35 = 30 + 5,
XVI = X + V + I,
CO means a molecule comprising 1 atom of carbon and 1 atom of oxygen.

Over generalising Roman numerals, a few students will write h10 instead of h + 10 and write 1y instead of y - 1 (just as 4 = 5 - 1 is written IV)

In algebra, putting symbols together does not mean addition, but multiplication instead. So students need to learn a new set of rules that are not intuitive: 3 + x is not written as 3x. Instead 3x means 3 × x and this in turn means x + x + x, just as 3 × 4 = 4 + 4 + 4.

 

Reference

MacGregor, M. & Stacey, K. (1997). Students’ understanding of algebraic notation. Educational Studies in Mathematics, 33, 1 – 19.