09/12/2011 | Thomas Clark
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Why is it that students have such difficulty comprehending the directions for sets of problems in mathematics texts? Several possible reasons surface immediately. The technical vocabulary of the mathematics being taught may be too sophisticated or “polysyllabic.” The child may not understand the “examples on the board” or those in the book. The interchangeability of terms may confuse the child. The child may be prejudging on the basis of past experience — or failure — with laboriously involved explanations for sets of problems.
There may be other factors as well, but one element seems to be present throughout. The student doesn’t understand, and probably has never been exposed to, the structure and syntax of the language of mathematics. It can be demonstrated very quickly and easily that there is a “method” in the supposed madness of the symbolism of math.
Personally, I have found that even relatively young students are entirely capable of grasping the language of mathematics at least as efficiently as the written form of their spoken language. In fact, what seems at first to be just a jumble of symbols, turns out to be a much more manageable group of expressions than the myriad of combinations at our disposal in the English language.
At the risk of oversimplifying the issue, let me share with you the general nature of that structure. Of course, you will have to decide just which elements and how much exposure your students can handle at this point in time. At the very least, you, personally, should be able to view the “jargon” of the subject with more confidence, and less apprehension, thereby making you a better instructor of the subject.
The first thing we should notice about mathematics is that there do exist, in fact, MATHEMATICAL PARTS OF SPEECH. In the English language, it has traditionally been accepted that there are eight basic parts of speech. In mathematics, however, there are only five, and several of them correspond nicely to the English parts of speech. One is number symbols — the “things” of mathematics — and they parallel the nouns in our spoken language. Another is operation symbols — the “actions” of mathematics — and these behave like the verbs in English. Of course, EVERY language has its “things” and its “actions.” Mathematics is no different. Then there are the relation symbols which show comparisons, the grouping symbols which do exactly what their name implies, and the placeholder symbols — usually called “variables” — which signify an unspecified thing, much like a pronoun does in English. That’s it. All symbols in mathematics will fall into one of those five categories. Do you realize how comforting that realization can be to a student who perceives math symbolism as chaotic?
The next step in conquering the language of mathematics is to put the various types of symbols together in meaningful combinations. Recalling our experiences in English grammar, we remember that an expression had to include a subject and predicate to be called a sentence. Likewise, in mathematics, there are certain requirements which must be met before the expression is meaningful. Unlike English, however, there are only FOUR TYPES OF EXPRESSIONS based on the presence, or absence, of placeholder symbols and relation symbols.
For example, the expression 3 + 4 has no placeholder — so it is “closed” — and no relation — so it is only a “phrase.” In addition, there is only one thing we can do with a “closed phrase,” and that is to “evaluate” it.
Now consider the expression 3 + 4 = 7. It does have a relation — so it is a “sentence” — but it still has no placeholder — so it is still “closed.” Further, the only thing we can do with this “closed sentence” is to tell whether it is true or false.
And what about the expression n + 5 ? Since it has a placeholder, it is considered “open” for modification or input. Of course, it has no relation symbol, so, as before, it must be just a “phrase.” Now, what can you do with an “open phrase”? Obviously, all we can do is to replace the placeholder with some number — making it a “closed phrase” — and then evaluate it.
Finally, look at the expression n + 5 > 9. It has a placeholder — making it “open” — and it also has a relation symbol — making it a “sentence.” In addition, the only thing we can do with this “open sentence” is to substitute a number for the placeholder — making it a “closed sentence” — and then tell if it is true or false.
Again, that’s it. There are only four types of mathematical expressions, and that is all you can do with them. Check it out in the math program you are currently using. At the elementary level, there are a lot of “closed phrases” and “closed sentences,” and not so many “open” expressions. At the middle and high school level, you will find very few closed expressions and a lot more “open sentences” — often called equations and inequalities. That really is the essence of Algebra anyway, isn’t it? Now, if you examine the “instructions” for the various sets of problems, you will find that, no matter what the wording is, you are being asked to do “the only thing you can do” with that type of mathematical expression.
If we had the time, we could even carry this to a third level of language, that of “translating” back and forth between mathematics and English. The point is, if mathematics is Greek to you, you simply must learn how to “speak” the language, and that means understanding how mathematical expressions are built. Only then will you be able to concentrate on the actual development of a concept without getting bogged down in terminology.