In Part 1 of this series of posts about what makes algebra difficult, I discuss the multiple meanings of equal sign learners has deal with to make sense of the subject. With the changing meaning of equal sign and equations comes the changing meaning of the letter symbols.
Teachers would oftentimes introduce algebra by telling their learners that x stands for an unknown number. It is not incorrect but that’s not all. Some teachers also introduce the word variable by saying that x can take any value that’s why x is called a variable. Again, it is not incorrect but that’s not all. I have heard teachers that say that in an equation, the x is an unknown, but in an algebraic expression, the x is a variable because it can take any value. Is it this simple? Let us consider the following example:
In letter A in the figure above, x can take infinite number of values but it is not an expression. It is an equivalence. Is x a variable then? The use of x is actually as a placeholder. In C, x can take any values so it is a variable. But f(x) is a function so x is called the argument of the function. We also have to be careful when we say that a letter symbol stands for a number (or value) because in the function in C, f does not stands for a value but simply as a name for the function that maps x to 8x +12 as I pointed out in the previous article. Because f(x)=8x+12 represents a function, we further distinguish between the values of x and f(x) as independent and dependent variable.
In letter B, x is known as unknown (pun intended) and students usually learn it so well, they apply it everywhere. I tell you a little story of a Year 7 algebra class I observed. The teacher gave the following problem:
The school library charges 3 pesos if a book is returned a day late. An additional 25 centavos is charged for each succeeding days that a book is not returned. How much will Aldo be charged if he returns a book 2 days late? 3 days late? 4 day later? 5 days late? x days late?
A student has this solution:
When asked how he calculated for x days, the student explained that he only added 0.25 to 4.0. The teacher asked what about x? The students said x is an unknown but since it comes right after 5 so it must be 6.
Related to the multiple meaning of “x” are the algebraic expressions. Students learned during the introduction of algebra that 2x represents an even number and 2x+1 represents and odd number. In Equation B above, we say that 8x+12 = 2x+1. But, 8x+12=4(2x+3) so this means that 4(2x+3)=2x+1. Now, how come than an even number is now equal to an odd number? How would you now explain this to your learners? I will leave this to the readers so not to spoil the fun
Salman Usiskin has written numerous articles trying to articulate the multiple meanings of equations and letter symbols. Here are some of his ‘equations’. What is the meaning of the letter symbols in each of the following?
In 1) A, L, and W stands for the quantities area, length, and width and have a feel of ‘knowns’; in 2), we say x is unknown; in 3), x is an argument; in 4), n stands for an instance of the generalized arithmetic pattern; and, in 5) x is an argument, y is the value of the function and k is a parameter. It is only in 5) that we have a feel of variability hence we say x is a variable. It has a different feel from 3) where you don’t get a sense of variability hence in this case, x is more of a placeholder.
The multiple meanings of letter symbols is a source of learners difficulty in algebra. Note, however, that this is also what makes algebra a powerful language and thinking tool.
In my next post I will discuss about the dual nature of algebraic objects as source of learners difficulty in algebra.