WHY IS TEACHING MATH SO DIFFICULT?
Some thoughts by Harold Brochmann

Thought 0:

Students are sometimes told that mathematics is "important in your daily life" ... or words to that general effect. I'm not sure I believe that. Here is a challenge to anyone who reads this: Send me e-mail [brochmann(at)saltspring(dot)com] and tell me about the last (one or two or three) time(s) you used mathematics "in you daily life" - however trivial. I'll report on the results here in due time.

Thought 1:

The following is a classic illustration of something. Exactly what it is an illustration of can itself be the subject of an interesting debate.

Ask a reasonably large random selection of people this question:

Using the letters B & G, express the following statement algebraically: 'In this class there are twice as many boys as there are girls'

You will find that among 'the educated', somewhat more than half will come up with the incorrect expression.

Don't comment on what they say. Just write it down. Now ask them to translate the equation they wrote as literally as possible. This not the same thing as repeating the original information. Most likely they will have written either B=2G or G=2B

They will usually say:

Boys equal two girls

or

Boys are twice the girls

or

Boys are two times the girls

Or the reverse of these.

Now what does this tell you? It tells you that the letters B & G are not clearly identified in their minds as numbers. If they were, we would hear

The number of boys is equal to two times the number of girls.

Or something similar.

And herein lies one major problem.

The symbols used in algebraic expressions are not clearly perceived as numbers, but rather as objects.

Thought 2:

A corollary of Thought 1 is that because algebraic symbols are not perceived as numbers, the missing operators [2G is better written 2·G, etc.] aren't missed and the algebraic expression 2G takes on a meaning of "two boys". This is not a numerical concept, but a physical one. The notion that the B in 2B is a representation of a number as opposed to "boys" is lost. There is no operation of multiplication involved. Maybe it would be better to leave the operator in.

Thought 3:

Now, here is an interesting wrinkle. Repeat your experiment with different people, but this time use this question:

Using the letters T & S, express the following statement algebraically:

In this school there are twice as many students as teachers

This time, I'm willing to bet, there will still be a high proportion of wrong answers; but not as high as before.

I suspect the reason is that the possibility of 'checking the answer' now exists. In the first version we had no way of knowing whether there were more boys in the class than there were girls; and so no attempt was made to compare the suggested algebraic expression with reality. Here, however, it would be reasonable to assume that there were a greater number of students than teachers, and so some people - and I say some people, will take the trouble to check the expression against what they know logically must be the case.

[By the way, being a retired 'ordinary citizen', as opposed to being connected with a university which lends legitimacy to surveying activities, I have no practical way of conducting field experiments on this. I would be most interested in hearing from people who have or tried experiments along this line]

This problem is discussed in greater detail and more formally in these articles:
http://www.ed.gov/databases/ERIC_Digests/ed313192.html
http://www-sci.uni-klu.ac.at/~gossimit/pap/misconvar.html

Thought 4:

Suppose we present someone with the inverse of the problems discussed above.

Translate the algebraic statement B=2G into a plain English sentence.

Would the respondent now come up with a lucid sentence in which it is objectively clear that he sees B and G as numbers? I suspect not.

I have a feeling that a lot of people will react to what I have been saying with

So what. I know that B and G and T and S represent numbers rather than things. Just because I don't express this in an unambiguous way doesn't mean that I don't understand it.

I would counter that the careless use of language in normal communication frequently leads to misunderstandings. Mathematics is also a language and unfortunately the careless use of misleading, inconsistent and sometimes incorrect terminology in school mathematics is quite common; and that this is at least part of the reason that there are so many problems with learning mathematics. What, one must ask, is the advantage in being careless with terminology in a field which is supposed to be the most precise of all the school subjects?

Thought 5:

The heading of this article is Why is Teaching Mathematics so Difficult. When I taught at the Junior High level I noticed that quite an assortment of different people were assigned to teach mathematics. Even those who were clearly not prepared for it. There was a common conception particularly, I think, among administrators, that mathematics is probably the easiest subject to teach. Don't forget, we are talking grades 7 - 10 here. Just about anyone could do it. They might not like it. But they can do it. The question that forms the title of this article suggests that this attitude is changing.

On the other hand, from the students' and parents' perspective, I have a feeling that mathematics is seen not so much as a subject which is difficult to teach... but rather, one that is difficult to learn.

I know that when I, in informal conversation with people, mention that I used to teach mathematics, the come-back is invariably "I was never very good at math", or words to that effect.

If one were to do a survey of high school students for the purpose of identifying their attitudes towards different school subjects, it would likely be found that mathematics had a relatively low 'satisfaction quotient'. There is also a perception that a disproportionate number of students who do well enough in other subject areas have difficulties with mathematics. Many students look on mathematics as a cross to bear rather than something which is interesting or useful. At the same time everyone knows that mathematics is a 'very important subject'.

I'm not quite certain myself why, precisely it is so important. Perhaps it has something to do with logical thinking. Doing mathematics teaches you to think logically. Something like that. Utter hogwash of course; but certainly some people used to think this. I suppose some still do.

It seems there are some difficulties.

I think one of the problems has to do with the fact that school mathematics is not really about understanding. It is about skills. Please understand that I'm talking here about the realities of the classroom rather than what ought to be.

A skill is defined in terms of being able to get the right answer to a particular type of 'problem'. What is involved is some sort of manipulation of algebraic symbols. Simplifying complex fractions. Stuff like that.

But being able to do something does not mean the same thing as understanding what one is doing. Further, there are many different levels of understanding any one thing, often involving some metaphor or other.

Though the term is frequently used in education circles and often by parents and students as well, there is not even a hint of agreement among educators precisely what is meant by understanding. One might, for example, ask if metaphorical understanding is real understanding.

Most times understanding is understood to be metaphorical. Several valied metaphors, depending on circumstances. Several valid ways of understanding. No?

It is generally recognized that teaching for understanding, however it is defined, is much more difficult than teaching skills. Also, few educators know, or even claim to know, how to test for understanding in a practical way in a school setting. This means that those teachers who don't "teach to the exam" but rather "teach for understanding" are very safe because there is no way to determine whether or not they are achieving what they purport to achieve. No accountability in other words.

I'd be delighted to hear from someone who disagrees with me on this point.

Thus in school mathematics classes most activity is directed at skill building through abundant practice exercises - read drills.

The reason all of this matters so much and its connection with the topic under discussion, is that it is only through personal understanding that the insight needed to learn on one's own can achieved.

Thought 6:

I think that there is a distinction between Mathematics and School Mathematics. A discussion of this topic would take a great deal of space and won't be attempted here beyond an initial comment.

In the school auto shop students repair cars. Not very different from what happens in real auto repair shops.

In Physics classes, students experimentally determine the physical properties of objects and the mechanical properties of dynamic systems. Not essentially different from the activities of real physicists.

In English class students discuss the merits of literary works and express themselves in writing. Not essentially different from the activities of critics and authors.

But what happens in the Mathematics classroom bears practically no resemblance to the activities carried out by mathematicians in the real world.

Thought 7:

Another problem with school mathematics is that students are frequently unclear about how what they are doing 'fits into the scheme of things'. There seems to be little opportunity to stand back and look at the overall view of what we are trying to do, what we have done, where we are going and why we are doing it. They cannot see the forest because there are so many trees.

School mathematics seems to consist of an endless series of topics amongst which there seems to be no obvious connection.

It's easy enough for the mathematics teacher to rationalize here by stating that it is necessary to have this or that as a background before you can do, or even 'understand' such and such and so on. It can all be justified. But I really don't believe that the student sees the broader picture - and so the whole thing becomes a series of isolated rote type of activities.

And where, when all is said and done, does it lead? What is the justification offered the student. "So that you can use mathematics to solve problems". Right. What kind of problems? Well we see examples at the end of the chapters. Something to do with the length of farmer's fences. Here is a good one:

If Billy has one more orange than Peter has and Peter and Billy have five oranges between them, how many oranges does Billy have?

Is this what mathematics is about? How many problems in school mathematics are convincingly important enough for anyone to care about? Not many.

And the reason is that the 'word problems' are selected to practice the 'theory' that has just been covered. Another approach would be start with the problems - the justification for doing math - and then developing the math needed to deal with the problem. In other words perhaps we should reverse the sequence of what we are doing. I think it makes sense. Do you?

Thought 8:

I have talked about careless use of language and the problems that can arise from this. Here is some more specifics on this topic.

There doesn't seem to be much emphasis placed on the distinction between the symbol indicating addition, operation of addition, the result of addition and the sign associated with non-negative numbers. Plus, and, add, sum and positive are sometimes used interchangeably, as in 2 plussed with 3 is 5 and 2 summed with 3 is 5 as opposed to 2 added to 3 is equal to 5

"Addition" is the name of an operation. "Plus is the name of the symbol. "Add" is the directive to carry out the operation. "Sum" or "total" is what you get.

"And", despite common usage, has nothing to do with addition whatever! "2 and 3 is 5" simply makes no sense until you come to Boolean operations at which time it is classified as a false statement.

Similarly, whenever I hear students refer to negative numbers as "minus" such and such I cringe. What possible justification can there be for not using the terms "subtract", "difference", "minus" and "negative" in the correct context.

We have often seen this sort of thing:
2 - 3 = - 1 [two minus three equals minus one]

It boggles the mind why we allow this kind of sloppiness in our mathematics classrooms.

Anyhow, this is the third time I've come back to this theme. Better leave it alone.

Thought 9:

"One more kick at the can."

All computer languages - except for BASIC which is long gone - make a distinction between the assignment operator "=" and the logic operator "=".

In school mathematics, for whatever reason, this distinction is never made. In x = 3 and 2 + 3 = 5 the "=" sign is presumed to have precisely the same meaning. And this is supposed to be an exact subject?

Thought 10:

If there was such a thing as a Social Contract between society and the public school system specifying what should be done in the mathematics program, ask yourself what would be included in the curriculum.

I'd say: statistics, so that the citizen can understand the economic information on which political decisions must be made.

And also personal accounting and budgeting so that citizen will be spend wisely, save for his retirement and generally be financially responsible.

Are these topics taught in the public school system? In some systems they are, but only to a to a token degree, and usually reserved for non-academic streams. Is that because 'academic students' don't need to be responsible citizens, or is it because these topics are too easy for most students?

On the other hand, factoring is taught. At length. Monomials are factored. Binomials are factored. Trinomials are factored. Complex polynomials are factored. What on Earth for?

To make responsible and competent citizens?

Because you need to be able to do this to solve "equations"?

Let's get 'real'.

Thought 11:

The word "equations" as used at the end of the last 'thought' was placed in quotes. I wanted to draw attention to it.

Here is an example of an equation: 2 + 3 = 5. Here is another x = 7.

This is not an equation: x + y = 7. This is a relation. One difference is that there are very few problems that result in equations - only puzzles. This is using the definition of problems and puzzles that I deal with in another "thought".

In any event, the build-up to relations is very badly handled in most, if not all, school mathematics texts; and causes a great deal of unnecessary difficulty. I really must elaborate on this topic another time.

Thought 12:

The computer is arguably the most significant mathematics related invention in many centuries. It has made possible solution procedures to mathematical problems that are not only effective, but much faster and easier than traditional approaches.

In fact, much of what is covered in traditional school mathematics is rendered obsolete by computer based techniques.

Here we are three decades into the personal computer revolution. Has the school mathematics curriculum acknowledged computers yet?

Thought 13:

School mathematics is taught backwards.

Consider the typical scenario. Introduction of the concept. Practice on the concept. At the end of the chapter some "word problems" the solution process to which involves applying the concept.

From the general to particular.

Yet we all know that learning always takes place the reverse of this.

From the particular to the general.

Concepts cannot be 'taught' on schedule. Concept formation is a generalization process. Generalization from specific examples.

And we wonder why mathematics is difficult to teach?

Thought 14:

Yesterday I had my regular grade 9 student come in for his weekly tutoring session.

At school he has covered solving equations in one variable. He has also covered solving 'formula' for a specified variable. For example he has just done: A = h(a + b)/2... solve for a. He has also substituted values into formulae.

Now he is being asked to write equations in one variable of the type "Two consecutive numbers add up to 45. What are they?" or "The sum of two numbers is 5 and their difference is 1".

He is supposed write: x + x + 1 = 45 for the first one and x - 1 + x = 5 for the other.

He had difficulties with this. I said "Use L to represent the larger number and S to represent the smaller. In each of the two questions you get two equations (I suppressed the desire to call them relations. For the first:
L = S + 1 and
L + S = 45

For the second one you get:
L + S = 5 and
L - S = 1

... well, you know the rest.

Yes they do that in the next grade. But he had no trouble with it this way. I think it's more obvious to use two variables; especially if you use letters that remind you of what the variables are supposed to represent.

I'm not aware of any textbooks that do this. Are you?

Thought 15:

The next chapter in my student's book is called "Congruent Triangles". Just like that. Out of the blue. What's the connection?

I think what really happened was that some members of the Curriculum Committee wanted to scrap Euclidian Geometry; but had to bow to pressure from the conservatives (Euclidian Geometry is also called "Reasoning") to keep some of it. So they kept "Congruent Triangles" as a topic. The application (presumably finding the heights of trees and such like) isn't coming till next year. An esoteric topic in isolation from any connection with reality. SSS, ASA and SAS is tradition.

Q: What do we have to learn this stuff for?
A: So you will pass the course so that you will be able to do more of it next year

Q:What do we have to learn this stuff for?
A: Unfortunately you haven't taken enough math to understand the answer. You will have to wait a couple of years. Then you will understand it.

Q: What do we have to learn this stuff for?
A: You need it in your daily life

You got a better answer? Is it true?