why algebra is difficult Archives - Mathematics for Teaching

The multiple meanings of letter symbols in algebra- Part 2 of x

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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.

Algebra

What Makes Algebra Difficult is the Equal Sign – Part 1 of x

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Algebra is one of the most researched topics in mathematics education. And most of these studies are about students understanding of algebraic concepts, particularly equations and the 24th letter of the English alphabet. With the volume of studies, one wonders why until now algebra many learners still have difficulty with the subject. I read a remark somewhere comparing the search for effective means of teaching/learning algebra similar to that of the quest for the holy grail.

I’m not about to offer in this post a way of making learning algebra easier. I have not found it myself. But let me offer an explanation why algebra is illusive to many first time learners of the subject. I adhere to the belief that once you know where the problem is, you have solved half of it. Sometimes, it could turn out of course that the solution of the other half of the problem is learning to live with it.

Consider the following familiar symbols we write in our blackboard. I will label each string of symbols, A and B.

What do the math symbols in A and B mean? How does A differ from B? How are they similar?

Let’s start with the ‘visual’ similarity. They both have an equal sign. They both show equality. Are they both equations? The statement 12+4x=4(3+x) is an equivalence. It means that the right hand side is a transformation of the left hand side. This transformation is called factoring, using the division operation. The transformation from right to left is called getting the product and you do this by multiplication.

Would you consider statement B an equivalence? It certainly not. You can test this in two ways. One, try to think of an transformation you can do. Two, you can test a few values of x for both sides of the equality sign to check if it will generate equal values. You will find that only x=-5.5 will yield the same result. This means that statement B is not an equivalence but a conditional equation. They are only true for certain values of x. This is what we commonly call equation.

I have shown that we have used the ‘=’ sign in two ways: to denote an equivalence and an equation. How important are the distinctions between the two? Is it so much of a big deal? Are they really that different? Let’s fast forward the lesson and say you are now dealing with function (some curriculum starts with function). Let f:x?12+4x, g:x?4(3+x), and h:x?2x+1. Their graphs are show below. Note that functions f and g coincide at all points while function h intersect them at one point only.

The graphical representation clearly show how different statements A and B are and that the ‘=’ sign denotes two different things here. Now, if you notice the graphs above, the function notation also use the ‘=’ sign. Is it use the same as in A and B? Try transforming. Try solving. It’s different isn’t it? In function notation such as f(x) = 12 + 4x, ‘=’ is used to denote a label or name for the function that maps x to (12 + 4×0. This meaning should be very clear to students. Studies have shown that learners misinterprets f(x) as f times x and tried to solve for x in the equation.

In 13 – 5 =____, what does ‘=’ equal sign mean? Ask any primary school learner and they would tell you it means ‘take way’ or ‘do the operation’. You may be interested to read What Pupils Think About the Equal Sign and Teaching the Meaning of Equal Sign.

I have presented four meanings of ‘=’ in mathematics: equivalence, equation, to denote a name for a function, and to do the operation. My point is that one of the factors that make algebra difficult is the multiple meaning of symbols used. We also use of the word equation to everything with ‘=’. Students need to be able to discern the meaning of these in the context to which they are used if we want our learners to make sense of and do algebra.

In Part 2, I talk about the multiple meanings of the letter symbols as source of students difficulties in algebra. You may also want to read Making Sense of Equivalent Equations and Expressions and Equations, Equations, Equations. If you want some references for Algebra teaching you can try Fostering Algebraic Thinking.

Algebra, Math videos

Top 10 errors in algebra

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Mathematics is indeed a universal language. Even errors are universal. Here are the top ten errors in algebra which are beyond borders and colors.

#10. Squaring the negative. A minus a squared unless it’s been snared: $-8^2\neq 64$

#09. Logarithms: The log of the sum ain’t the sum of the log: $\log(a+b)\neq\log_a+log_b$

#08. Shifting function: Add to y go high, add to x go west: $y = (x+3)^2$

#07. Inequality: Multiplying the inequality by a negative flips the inequality: $-3(x<5) \neq -3x<-15$

#06. On exponents: When in doubt, write it out: $x^4 = x.x.x.x$

#05. Fractional exponent: Don’t flip over the root. $25^{\frac{1}{2}} \neq \frac{1}{25^2}$

#04. Subtraction: Don’t forget to share the minus and the negativity. $x-(3+x) \neq x-3+x$

#03. Cancellation: Cancel factors, not individual terms. $\frac {x}{x-5} \neq \frac{1}{-5}$

#02. Quadratics: Remember exponents 2, answers 2. $x^2=25, x=5, x=-5$

#01. Squaring: Don’t forget to FOIL. $(x-3)^2 \neq x^2 -9$ .

Here is a funny video of common algebra mistakes listed above: Continue reading “Top 10 errors in algebra” →

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