Math 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.

Humor

Things you learn in math education forums

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You always get good ideas from forums (or fora), whatever form they are. If you want great insights about math and science education, try attending a PhD forum or seminar. I’ve just been to one. Following are some of the things I learned from the spirited discussion during the question and answer portion from these serious educators.

We complain that our learners are not doing well in their Maths especially in secondary schools. These students are now engineers, doctors, lawyers, and politicians. We trust them anyway (except the politicians).
It is only in math that 1+1 = 2. In real-life, it doesn’t work that way. For example, when two churches combine, you get 3 – the new one and the two old ones. This also applies to political parties.
number theory
On the question of the relevance of your PhD to science education. Short answer by the speaker: I am now relevant to the science education. They now have one learned participant in the science education discourse.
Why do we always expect the teachers to know all their Maths? Answer: It is probably because of our experience of our teachers in first grade as all-knowing. We believe everything teacher say and it was important for us then to have believe them. I think we need to grow up.
Tell me, “Do you know of a mathematician who know all their mathematics?” Why should a math teacher know all their math? This is not fair to teachers. Do you complain in the media when a doctor misdiagnose your illness? $math teachers$
“My conclusion in my review of literature why, despite the extent of research about teaching and learning algebra we still have not solved the difficulty of learning it, is that because algebra is a moving target.”
“I initially thought to explore the reasons of students absenteeism in lectures. But then I thought, why should they when they can find great lectures in the net. Now I do not know how to proceed from here. Will anybody suggest a research question that’s not in the net?”
“In my interview with teachers, most of them said that they don’t really know why students are not getting the test. When they teach them, they seem to understand everything they are discussing and solving. My interview with students confirms this. The students said that they understand everything during the lectures but they couldn’t answer the same questions and problems in the test.”

Humor

The Learning Pyramid

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I attended a lecture today on how to help Year 12’s pass their examinations. One of the slides that captured my attention was the Learning Pyramid. It says that the information retained by our learners is a function of the kind of learning experiences we provide. The percentage shows what is left in the brain after 2-3 weeks. It is very important that teachers take these to heart especially when designing instruction. As you can see in the pyramid, lectures or teacher talk has the least retention rate. I don’t know why most teachers still prefer it, really.

I searched the net for source of this Learning Pyramid. Everyone seemed to be sourcing it to the National Training Laboratories, Bethel, Maine. However, I did make my own original contribution to the learning pyramid – a learning task that has 100% retention rate. Mine is not based on empirical research but from my own experience. This is the reason I blog. And I highly recommend this as a method of teaching and delaying the onset of dementia.

Why Blog — Learning experience vs retention rate

You may also want to know another pyramid – Bloom’s Taxonomy for iPads.

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.

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