## Teaching History

There was a part of me since my youth wanting to be a historian. I loved history for there was nothing boring about it. At age 18 I was redirected to engineering and math. My love of history has never abated. I was a voracious before having children. There were historical happenings being researched by me. Along come children and history takes the backseat. My research is on hold for my time is dedicated to my family.

This summer we have started a different study of history. We are listening to some history lessons each night. Doing this during the school year is not feasible. We simply do not have the time with all of the activities going on. An hour or two a night listening to a type of history lecture isn’t a bad way to spend an evening. We do not even have to do this each night. Upon reaching the end of the time, we started a discussion about what we listened to. The two youths or should I now say young adult and youth were not prepared to have this discussion. They were forced to provide information about the topic we listened to and their opinions.

If some university were willing to hire me as a professor without a masters degree let alone a doctorate, there is one part of my class lecture I would include. You see, four to six books would be required for my class. In time I would write a hundred or so page booklet about the class for the student to use. There would be two exams and a final. All of this is standard for a history course. What would make my class possibly unique is what is done in class. My lecture would not take the full 50 or 55 minutes. The last fifteen minutes or more would be class participation worth at least 25% of the student’s grade. The student can’t miss a class and they MUST engage during the class. A discussion about the lecture and what was read is the last part of my class. There are no right or wrong thoughts. This is where a student can grow and begin to think and understand what they are hearing and reading. Once you get past the facts of history, you now have opinions.

Last night our discussion was do you think Alexander, Caesar or even Napoleon are great? Why do you think they are or not. Alexander the Great is great for some reason and do you agree? Does killing hundreds of thousands of people justify you as being great? Are you looking at their military prowess only? Would you consider Ghengis Kahn great? Yes, he brought peace and security to the people of the steppe but at a cost of 20 to 50 million lives? Is he still great and does his military prowess continue to make him great? If so, then should we not consider Adolf Hitler great, too? Stalin should be considered great even though he doomed millions. Is Archbishop Damaskinos Papandreou not great for he didn’t conquer countries and kill millions? (Yes, I want you to look up the archbishop.)

If I have the chance to teach history there would be reading, lecturing and discussions. Homeschooling can provide a similar opportunity. We as parent-educators need to read the books, listen to any lectures and be willing to engage our children in discussion. There can be a disagreement but no one is wrong. Sometimes there is no answer.

## Logarithm

My experience in transcribing my thoughts and knowledge is limited. Many times I do so in a hurried manner within a short period of time. Today I spent a little more time with this. I know if I take a few weeks this will come out more clearly, so I hope this give a general understanding of logarithms.

The definition of a logarithm is the “mathematical raising of a base number” to produce a given number. What does this mean?! We begin with looking at two graphs. The two graphs below are related. Can you see the relationship between the two?

The first graph represents 2^x where a = 2. The second graph is the log2(n) where n is the exponent of 2. If you rotate the second graph 90° to the left and then flip it left, you will duplicate the first graph. The logarithm function is the inverse of the exponent function.

Take 2^3, 2 raised to the power of 3. 2 is the “base” number and 3 is the exponent. The resultant of this is the value 8. The exponent is also called a logarithm. In our 2^3 = 8 example, 3 is the logarithm of 8 with a base of 2, Log2(8) = 3. Unfortunately, I am unable to properly write a log for the base should be a subscript. Keep in mind you write a logarithm as log#(#) where # represents a number.

The definition is:

**loga(n) = x means a^x = n**

Here are a few examples:

10^2 = 100 and log10(100) = 2

4^(-2) = 1/16 and log4(1/16) = -2

3^1 = 3 and log3(3) = 1

Take a look at this example: 7^0 = 1. Can you write the log for this? The answer is log7(1) = 0. Any log of one will be zero. You now have another defintion:

**loga(1) = 0**

What about this example: 5^1 = 5? Well, the log5(5) = 1, so you now know that any logarithm of the base itself is 1:

**loga(a) = 1**

Here is a tricky example to try. What is log2(2^3) ? The answer is simple when you realize a few things. You could take 2^3 and get 8, and now you have log2(8) which is equal to 3. Did you notice anything? Look at log3(3^5). What is the answer? It is 5. Hence we have another definition:

**loga(a^x) = x**

This is one of the areas I see students having difficulty. They do not understand the concept and go through school memorizing all of this without understanding. Math teachers or any mathematician likely does not recognize this deficiency. I struggled with this entering college. There are plenty of people that graduate with engineering degrees that do not understand this.

## Proof of log product rule

Youth 2 doesn’t know this, yet. He will be doing proofs for pre-calculus with me. They will be graded quizzes. He is going to hate it, but this will help him with understanding math. I believe we as teachers need to have our students understand how things are done whether it is in math or not.

One teacher I had in school, gave us the same quiz each day until everyone passed. If you passed the quiz, getting everything correct, you didn’t have to take it. I was one of the first. It would take many weeks before everyone passed with a 100%

Below is the proof for the log product rule. It is done in a pictorial format.

The product rule is: loga (xy) = loga (x) + loga (y)

You need to write the two logs in exponent form.

The next step is to multiply x and y together.

You next take the log of both sides. You are able to bring out the (m+n) as shown. Loga(a) = 1. If you substitute back m and n from the top, you get loga (xy) = loga (x) + loga (y).

Hopfully this rough proof helps you understand where the product rule comes from.

## Math Advice – Logarithms

One area of mathematics not covered by many students are logs. I tend to blame teachers for not teaching this. I believe many school teachers are not true mathematicians but general educators. It could be they do not know how to translate this topic for students. This was one area where Youth 1 wanted to do it his way; the harder way.

Instead of going through proving, I want to list what a student should know when entering higher mathematics. You really see this in pre-calculus at the start.

**Laws of logarithms**

This covers only log and not the natural log. Knowing the log relationship helps when working algebraic problems.

## Failure Can Be An Option

One of the hardest acts to do as a parent is to let your child fail. We want to serve as a safety net for our children. This is necessarily not a bad thing to be for our children will always face difficult times and need our help. There are those time where being the safety net is the wrong decision for parents.

As I was instructing my oldest in math, he would balk on my insistence on learning formulas or how formulas are derived. This became a battle between us. He didn’t understand I was not trying to punish him but help him for the future. He did not like my method of teaching by having him do the bulk of the work figuring how things work. He didn’t want to keep in his memory formula or my personal favorite, the unit circle. Instead of forcing the issue, I allowed him to have his way. I knew the future outcome but was prepared to be a safety net once he failed.

Taking math, chemistry and physics classes at the local university, Youth 1 discovered life was not fair. You were expected to figure things out on your own, and you better know your formulas coming into the course. Not being prepared cost Youth 1 but the lesson learned was valuable.

Yes, my son does not have a 4.0, but he hasn’t gotten anything below a C. He now smiles when I remind him of what he should have done. He has been able to achieve Dean’s List with the extra effort he has had to put in. He understands the mistake he made with me and hopefully to listen to what I say. He now knows my goal was not to punish but to prepare.

It was difficult for me to let go and allow him to fail. Fortunately, his failure was not severe. He will continue to work harder until he finally catches up; I hope and pray.

I believe I was a good parent with this method. It was difficult to watch him struggle, but I was there when he needed me.

## Math Advice #1

When Youth 1 was taking pre-calculus, I told him what to learn and remember. As a teenager, he is designed to ignore me. Now he understands as he smiles when I remind him of what I told him to do.

A unit circle has a radius of 1 and sits at the origin of (0,0). Knowing the unit circle and the trigonometric functions associated with it is a must. You need to know the relationship of the unit circle with radians.

If a student knows the unit circle inside and out, it will go a long way in helping them through calculus, physics and other science related classes.

## One Down With A New Adventure To Go

Youth 1 has finish his last year of school. Ten or eleven years of homeschooling completed with my oldest child. We have prepared our son for the future as best we could. Now is the time for my son to take what should have been learned and proceed upon his path.

This brings me to a potential class for college; it’s a history class. If this is class he selects there will be four books to be read. He doesn’t understand why I’m excited about this. I want to read them, too!

Apart from this class there are also physics and math for my son to tackle. This should not bother him for he’s already tackled these two topics at the university. These two classes are just the beginning of the work needed for science or even engineering.

There are about 48 to 50 classes needed for a degree. A minor is not out of the question if he should decide to put forth the effort for one. It may seem a lot, but it’s not. As each semester passes by, he will gain the knowledge and confidence to continue on with his education.

There is some thought of graduate school after graduation. Right now this is too far away to look at but not far enough away to plan for. Now is the time for Youth 1 to plan his classes and make himself known.

What can go wrong?

Well, he could decide to not finish college. There is nothing wrong with this as long as detrimental events were not the cause. Yes, women can be on the detrimental side of things. This is one adventure I do not want to miss.