I heard something the other day about a suggestion made to the effect that people under a certain age who drop out of school should be denied getting a driver's license or something like that. Whatever was really said or whatever was meant, I saw an irony in this item because of something that happened a long time ago as a result of which I made a comparison to help the teachers I was responsible for in the schools where I worked. It is a story I cannot tell without giving a lesson in mathematics and so, I hope the reader is in a good mood which I count on -- this being the season to be cheerful. As to the math teachers among you, I know you will like the story and maybe find the lesson useful too.
Before I started my own school I was teaching at a private school in Montreal. I was head of the microprocessor department and was also in charge of the remedial math course, a preoccupation of the school that started small but grew in importance as it became obvious that the students who took remedial math with us – having been classified at the start of the year as lagging -- ended up doing better than the students who were classified as advanced and were deemed not to need remedial math.
Eventually, things got so busy that it became necessary to appoint another teacher in remedial math to lighten the load on me. I knew he was good at teaching digital circuitry and the math that goes with it. And so, I was surprised when the students from remedial math told me early on that he was not doing well in that department. I called the teacher in my office, told him what happened and asked him to teach me the course exactly the way he was teaching it to the students. The lesson was on logarithm, a subject in which most teacher do badly because they teach it the way it was taught to them which is really bad; and this teacher was no exception. In fact, the lesson on logarithm is the point at which most students begin to hate mathematics, even quit school because of it. And to be honest, I felt like leaving the office when the teacher started the lesson on logarithm by trying to define the characteristic and the mantissa of a log without preparing me with a general view of what the subject of logarithm was about and why I should want to learn it.
When he was done, we reversed roles and I started teaching him as if he were the student and I the teacher. I went about it this way: Every problem we tackle in mathematics can be stated in a straightforward manner or stated in the reverse manner. For this reason, every mathematical operation has its own reverse operation. For example, the reverse of addition is the subtraction; the reverse of multiplication is the division. When someone says I had 5 dollars and was given 3 more dollars, you find how much money this person now has by doing 5 plus 3 equal 8 dollars. Stated in reverse, the problem sounds like this. I have 5 dollars, how many more dollars should I be given to have 8 dollars in total? To find the answer you do 8 minus 5 equal 3 dollars. You conclude that you can find the answer to a problem that is stated in a straightforward manner by doing addition; and find the answer to a problem that is stated in a reverse manner by doing subtraction.
In another set of problems you may be told that someone has 4 boxes, each of which contains 3 dollars. You find how much money this person has by doing 4 multiplied by 3 equal 12 dollars. Stated in reverse, the problem sounds like this. I have 12 dollars distributed equally inside 4 boxes; how many dollars are there in each box? To find the answer you do 12 divided by 4 equal 3 dollars in each box. Again, you can see that because division is the reverse of multiplication, you were able to solve the problem whether stated in a straightforward manner or the reverse manner.
To solve another set of problems we encounter in the natural world as well as the artificial world we created around us requires a mathematical tool we call the exponential. And this exponential has a reverse operation we call the logarithm. One tool is used to solve problems that are stated in a straightforward manner and the other is used to solve problems that are stated the reverse manner. For example, if you are a stock market genius (or a crooked insider), if after each trade you manage to grow your money to 3 times the size you started with and if you do 4 such trades a year, how many dollars will you have at the end of the year for every dollar you invest at the beginning of the year? You find the answer by doing 3 raised to the power of 4 which means 3 multiplied by itself 4 times in a row, and this comes to:
3 x 3 x 3 x 3 = 81 dollars.
Thus, if you begin the year by investing 1,000 dollars, you will have 81,000 dollars at the end of the year. As you can see even Warren Buffet could not do as well. In fact, this was an exaggerated example of compounded growth. I deliberately exaggerated the growth to make my point clear but, in real life, growth never happens by tripling with every iteration; it happens by growing a small percentage each time. This makes the math just a little more complicated but not by too much. So then, let us take an example. You inherit 1,000 dollars which you know you will not need for the next 5 years. You go to your bank manager and tell her you want to invest the money in a profitable way. She says you are lucky because this is an inflationary period and she can give you 10% interest that will be compounded over the next 5 years. She writes you a certificate, you go home and sit down to calculate how much money you will have at the end of the period. You reason that a dollar growing by 10% will be worth 1.1 dollars at the end of the first year. At the end of the second year it will have grown to:
1.1 x 1.1 = 1.21 dollars
At the end of the third year it will have grown to:
1.1 x 1.1 x 1.1 = 1.331 dollars
You now see a pattern and realize that there is a shortcut to this operation. Instead of multiplying 1.1 by itself 5 times in a row to find by how much your dollar will have grown, you raise 1.1 to the power of 5 and get the correct answer which is 1.61051 dollars. Thus, the 1,000 dollar certificate that the bank manager gave you will be worth 1,610 dollars and 51 cents after 5 years of growth by compounded interest. This was a problem stated in the straightforward manner and solved with the use of the exponential.
Of course, the problem can also be stated in the reverse manner whereby it will sound something like this: How long will it take a 1,000 dollar certificate to grow to 1,610 dollars and 51 cents if the going rate of compounded interest is 10% a year? And this is where you will need logarithm to solve the problem because logarithm is the reverse of the exponential. To find the solution we use the formula: The number of years is equal to the log of 1.61051 divided by the log of 1.1. That is:
0.207/0.0414 = 5 years
When I was finished with this demonstration, the teacher was happy and he thought he will have an easy time teaching logarithm from now on because he will begin the lesson the way I did and only then plunge into the definitions of the characteristic and the mantissa. But I told him that such approach will still turn off some students because he should do one more thing before getting into the definitions. This is where my philosophy of teaching comes into play.
As I see it, the problem with the way that logarithm is taught and the way that most subjects are taught is the fact that the teachers start with the generalization then go into the specifics. In other words, they go from the abstract to the concrete when they should be doing it the other way around. The trouble with starting with the abstract is that the students will turn off before the teacher has finished talking abstract. I found that if the teacher does not tell the students early on how the subject relates to what they did previously, how it relates to what they will do later on and why the subject is important for them to understand, he or she will lose the attention of the students in no time at all. To be effective, the teacher must do all this and must give examples to maintain the attention of the students.
In fact, I had many such discussions with teachers, some of whom still insisted that the best way to teach was to go from the abstract to the concrete. This prompted me to make the following comparison to illustrate my point. I would ask the teacher: Suppose you want to teach someone how to drive a car. How would you go about it? He would say, I go to a place where there is little or no traffic, I show the student what to do and I let them do it. And so I ask: Would you begin with a long abstract introduction about the philosophy of driving? And this is where they get my point.
When it comes to teaching logarithm, we must be conscious of the fact that even if we can have a logarithm of any base, we prefer to use the base 10 common logarithm. This is no different from the preference we have for counting with the decimal system which is also based on the number 10. In fact, we see that the common logarithm is used in subjects like the earthquakes and the decibels. Another system we encounter but do so less frequently is the natural logarithm which is used almost exclusively in the calculus. Still, we do not begin to teach the children how to count by telling them there is the binary system, the decimal system, the hexadecimal system and many other systems of counting. On the contrary, we begin by teaching them the decimal system because they have 10 fingers they can see and count on. And when they have learned this system well and they want to go into something like computer science, they learn the other systems of counting, even learn to convert from one system to another. And the same should apply to the teaching of logarithm. And there is a big advantage to getting directly into the common logarithm because it allows us to temporarily skip the definitions and all the confusion that comes with them.
Thus, I told the teacher that following the introduction concerning the straightforward mathematical operations having reverse operations, I found it easy to teach logarithm by doing things this way: I tell the students that the log of a number is how many zeros it contains. For example, the log of 100 is 2, the log of 1000 is 3 and so on. And then I ask them: What is the log of a million, and they say 6 which is correct because a million contains 6 zeros. Now, I tell them this is not the full picture because I neglected to say that if the zeros follow a number other than 1, we have a new ball game. By way of example, I ask: if 2 is the log of 100 and 3 the log of 1000, what might be the log of 500 which falls between 100 and 1000? Most students will not respond but there will always be someone who will say 2.5. And this is the wrong answer; in fact, the log of 500 is 2.699. Now I ask what might be the log of 5000 and only one student in all the years that I taught logarithm guessed the correct answer which is 3.699. Why? I asked and he responded that 5000 has one zero more than the 500, therefore the 2 becomes a 3 while the 699 that follows the decimal point remains the same. In fact, the .699 is called the mantissa while the 2 is called the characteristic.
When a teacher has brought the students to this level of comprehension, he or she can tell them that a number must not necessarily end with zeros to have a log. To find the characteristic in these cases, you count the number of digits and subtract 1. Thus, the characteristic of 5739 is 3; so is the characteristic of 7295 and 8491 and so on. Now the teacher can give a full blown lesson on the logarithm, how to find the log (characteristic and mantissa) of numbers like 532 or 3729 or any number by looking up the tables or using the calculator.
And there is a lesson here for all those who are in charge of education everywhere on this planet. It is this: Every teacher teaches at least one subject in a way that is exceptionally good. If the principal of the school can determine by any method they wish to use what that subject is in every teacher under their employ, they should get the other teachers into a class with the good teacher -- not to learn a subject they already know -- but to learn how to teach it well. This is called identifying the best practices; and if every best practice in every school is picked up by all the teachers and fully utilized, the schools will improve a great deal.
It is time for me to take a rest and recharge. Merry Christmas to everyone of you and a happy New Year. I'll see you here next year.