Quinnipiac College

When one teaches a general studies mathematics class such as College Algebra or Introductory Calculus, it is typical to begin by defining a few terms. Terms such as rational numbers, real numbers, coeficient, the basic terminology of set theory, what it means for a set to be closed under a certain operation and many more. We, the teacher, take greats pains, to define this words precisely since we use them throughout the course and they are indeed the basis of our lectures. As time goes on we develop a basic vocabulary for the course, whcich includes such terms as function, graph, domain, solution, slope, etc.

At this point you should ask the questions, "What do my students think of all these definitions? Do they see the value in learning the basic vocabulary of the subject?" The answer may surprise you. I have found that most of the students see my attempts to motivate and define basic terms as pure nonsense. For the students the heart of the class is the problems or exercises that they will have to "work out;" they have to learn how to repond "correctly" when they encounter various situations; what they concentrate on is the techniques used to arrive at the right answer; this is what they will be tested on. They see very little relationship between the vocabulary and the techniques: For instance, almost all of your students will be able to solve linear equations and quadratic equations by the end of the semester (They may even know the quadratric formula by heart.) but very few can give you an adequate exlpanation of what it means to solve an equation.

Once you have made this realization, you begin perceive a problem with the way classes are conducted. That is, that the words we are using to explain concepts are not neccessarily having the effect we think they are having. I began to question whether my students really understood the meaning of the words I was using to explain the subject material. This led to a series of experiments which tested how well the students in my College Algebra and Calculus classes understood the definitions of words used in class. The results were shocking. Let me give you a few examples.

I had spent the first two class periods in a College Algebra discussing sets and numbers, eventually doing a quasi-construction of the reals as a complete ordered field. As mathematicians, we are interested in the various properties that say the rational numbers have as opposed to the integers. On the first test I asked the question, "Under which arithmetic operations are the natural numbers closed?"

I did nothing in class to prepare my students for this question other than give a precise definition of what it means to be "closed" in class and use the term carefully and repeatedly in my discussion of number systems. Only three of eighty-three students answered this question correctly. Most of the other students answered that the natural numbers are closed under division and subtraction, thinking that the word "closed" meant the exact opposite of its actually definition: Twenty-nine students gave answers that were so off the wall that it was clear that just about all of the terminolgy I had introduced meant nothing to them. For instance, some of the more bizaar answers I received were: "the real numbers," "absolute value," "true," "false," "p/q," and "prime numbers."

The conclusion I draw from this is that students simply do not have to understand the term "closed" to do arithmetic or to do any of the problems in their algebra book, so they do not bother to try to understand the term. Let me emphasize I am not saying that the students misunderstood the definition or that they applied the definition incorrectly to the situation. The students simply saw no value in learning the term in the first place.

It came to my attention that students did not understand what the words "positive" and "negative" meant. In one College Algebra class in which there was full attendance (a rare occurrence) I announced that I was giving a bonus quiz, the results of which would be added to their previous test scores. The question was this.

Given that w is a real number and w > 0, is w a positive number, a negative number, or neither.

I did nothing to prepare the students for this. All previous quizes had been of the nature: Solve a particular problem. Of twenty-five students, eleven said 'w' was positive, five said it was negative, seven said it was neither, one said it was all three and one simply answered "no". I should add that I was barraged the entire time by students (even some who eventually wrote the correct answer) claiming that the question made no sense. I give no hints except to tell them to answer the question the best they could.

On another occasion in a Calculus class early in the semester I talked about the distance formula one day, and during the next class spent the entire time talking about straight lines, their slopes, and how to find an equation for a line given various information. At the end of the second class I again gave a bonus quiz. This time it consisted of three problems 1) Find the distance between two points on a plane; 2) Find the slope of the line passing through the two points; 3) Find the equation for the line passing through the two points in slope-intercept form.

Of twenty-three students, only one used the distance formula to compute the distance in problem 1; the rest attempted to compute the slope. It was as if the words "slope" and "distance" meant nothing. They were mimicking the last thing they saw in class. Some even asked the question, "When you say distance, are asking us to find the slope?"

Repeated experiments and observation of my and other people's classes led to the following conclusion: Students believe that they do not have to really understand the meaning of words in order to "succeed" in math class. It seems that the problems are viewed as stimuli and the students just need to learn how to come up with the proper response,without necessarily understanding the implications of what they have done. Again, my point is not that students are unsure of the definitions of various words or confused about multiple definitions of some words, but are reluctant to apply any meaning at all to many of the terms used commonly in mathematics. A partial list of terms that I have found that many students have never really felt the need to learn are "factor", "term", "coefficient", "solution to an equation", "slope", "absolute value", "variable", and "function". I have seen students misunderstand and misuse even non-technical terms like "twice", "equal", "sum", and "more" when they are in a situation in which they cannot just mimic what the instructor is doing.

Mathematics instructors nation-wide are doing more to incorporate reading and writing into the mathematics classroom as a tool to improve understanding. The problem I am identifying here is a little more specific. We must address the fact that the subject of mathematics contains many terms with very specific definitions. Understanding this basic vocabulary is of paramount importance to one's understanding of not just how to "do" mathematics, but to an overall comprehension of what the basic nature of mathematics is. If we understand what the words mean, solving the associated problems is not as difficult as they are perceived and the techniques we used to solve prolems make more sense.

Public opinion not withstanding, mathematics is probably the most literate of disciplines; everything flows from a profound understanding of the basic concepts and terminology. This is the point we need to be making to our students and our peers. If you were to ask me what my basic goals are when I teach a College Algebra course, I would make a list of about twenty words and tell you, "I would like the majority of my students to truly understand what these words mean."

How do we accomplish this? We could give vocabulary tests and incorporate more writing in general into the testing process. We can give students group projects and make student presentations a larger part of the class. We can try to generate more discussions in class, thereby encouraging students to learn the terminology and to use it correctly. But, most importantly, we need to model for our students how a grasp of the deeper meaning of the basic vocabulary leads to straight-forward, comprehensible solutions to the problems we encounter in mathematics. I'll give three examples.

The definition of absolute value given in most textbooks is this:

The absolute value of x is x if x >= 0 and -x if x < 0.However, the reason that one introduces the concept of absolute value is to give us a way to visualize and compute distance within the real number line. The definition of absolute value could be given as follows:

x = the distance between the number x and 0 on the number line

We then consider the problem:

(1) Solve: x - 5 < 7This problem is going to reinforce the concept of absolute value and what it means to be a solution to an inequality. Start by asking the question: What can we conclude about a number if its absolute value is less than seven? Well, this means that the quantity in question must be less than seven units away from 0 in either direction.

Draw a number line. Convince your students using any manipulates or reasoning available that if a number can be no more than seven units away from 0, then it must be trapped inside the range - 7 to 7. Thus, statement (1) leads to the following statement concerning the quantity, "x - 5"

(2) -7 < x - 5 < 7We should then be able to manipulate this double-sided inequality to arrive at our conclusion that the solution set is:

(3) {x: -2 < x < 12}We should take several numbers inside the solution set and demonstrate that they are solutions, while showing that numbers outside this set are not solutions. Pay particular attention to the numbers -2 and 12. While doing this we would like to express as much as we can in terms of distance.

At first, you might think that this is very trite, or that you already do this, only in a more elegant fashion. However, the idea is to demonstrate to the student that everything flows from an understanding and a willingness to work with the basic definitions. We are modeling how the meaning of words leads to problem solving techniques.

Here is another example. Students are asked to find an equation for a line passing through two points in the Cartesian plane: (2,3) and (5, 9). We have previously discussed slope and why the graphs of certain equations produce straight lines. The students can compute the slope of the line from a formula they have seen before, but you stop them. You tell them, "Look at it this way. Suppose you are traveling along this line. As you move from the point (2,3) to (5,9), how much has the 'y' value changed ? how much has 'x' changed? What exactly is the slope of a line?" The slope of this line is 2.

Continuing, you now tell them that you know a point on the line (In fact, you know two.) and you know the slope. This is enough information to compute an equation, given that we have assumed we are dealing with a straight line. At this stage many students have been taught to use the point and the slope to solve for the y-intercept.

(4)y = 2x + b; use the point (2,3)

3 = 2 ú 2 + b; b = -1;

y = 2x - 1

Of course this works, but it does not serve the purpose of reinforcing the concept of slope. We want the student to think about the meaning of the word "slope" as he or she is doing the problem. We start again. (2,3) is a point on this line. We would like develop an equation that tells us whether an arbitrary point on the plane is on that line. We point out that the line itself would consists only of pairs (x,y) which are solutions to this equation. What would it take for a point to fall on this line? Assume (x,y) is on this line. Then, as we travel from the point (2,3) to the point (x,y) the change in the y coordinates "y - 3" divided by the change in the x coordinates "x - 2" must be equal to the slope. (Use a diagram within the Cartesian coordinate system to illustrate this.) We obtain the equation

(5)[From editor: equation missing]By doing a little algebra we can obtain the same answer as we did in (4).

We should then discuss with our students which method is preferable: (4) or (5). It has been my experience that most students will say that the first method they learned, probably in high school, is by far the easiest and best method. I usually try to make the point that the real goal of this particular problem is to reinforce the concepts of " slope" and "equations and their graphs" and that my personal preference for (5) is that it comes from thinking about and using the definition of slope. Students usually think that they completely understand what slope is, but most really only have a very fuzzy idea of what the word means. We need to make students understand that the deeper they understand this term, the more clear-cut and obvious the above computations will become.

The final example can use in any class in any discipline. We present the students with the following facts:

The Revolutionary War began in 1775. The Declaration of Independence, which we celebrate as the official birth of our nation, was written in 1776. The battle of Yorktown, essentially ending the Revolutionary War was fought in 1781. The Constitution was signed in 1787 and ratified by all thirteen colonies by the end of 1788. The Bill of Rights was ratified in 1791.

We then ask the question: "What year was George Washington first elected President of the United States?"

You will be surprised how many people answer 1776 or 1777 to the above question. The reason for this is obvious to me. The students are not deciphering the above information, but giving the standard answer they have always given when asked a question about the birth of our nation. When I did this in my class over eighty per cent of the students gave an answer before 1782. This leads to a short discussion about the Constitution. We point out that the office of President of the United States is defined in the Constitution. So how can we have a President before we have a constitution? We make the general point, "You really can't do anything with a concept until we have a reasonable definition of the word." The answer to the above question is 1789. By the way, the Bill of Rights was the first ten amendments to the Constitution, so it was possible to elect a President without the Bill of Rights.

Students have complained that I have no right to ask the above question in a mathematics class, but my better students really seem to appreciate the point I am making. Words matter, whatever discipline you are discussing. We should never attempt to make a conclusion or take an action based on a word or words to which we have attached no meaning.

In conclusion, let me say that though the reader may have seen the first part of this article as being pessimistic; that was not the intent. I believe that we can overcome this problem, once we acknowledge its existence. Instructors are encouraged to try variations of the above experiments, but not to take the results too much to heart. Students come into all classes with a certain amount of intellectual baggage. Unfortunately, I have observed that part of this baggage is the belief that terminology and definitions are not important in the study of mathematics. (Students may very well feel this way about many disciplines.) It is our job to convince them that the opposite is true; that almost everything in mathematics flows from the meaning of specific words. When an instructor is defining his or her goals for a course, she or he should make a list of the basic vocabulary. Having the students develop a deeper understanding of the basic terms should be one of the primary goals. This is true of all mathematics classes from first grade through graduate level courses.

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