This is a tricky subject to write about because it is by nature subjective, see what I did there. Double pun for the win! Recent events at my job got me thinking about what is the proper way to teach fundamental mathematics, is there such a way? When I say fundamental, I mean the basic building blocks of mathematics. The use of arithmetic and what numbers mean to us. Is there a proper etiquette in teaching mathematics? If not, should there be one? The main source of my inspiration comes from Polya's How To Solve It: A New Aspect of Mathematical Method.

Before we dive into that rabbit hole, I recently had a conversation with a friend about mathematics being universal. Its a popular accepted notion that mathematics is a language that is universal. We all can understand it in such a way that it makes total sense in our own language. His point to me was that literally the notation is not universal, look at calculus. Calculus was invented by two people, Isaac Newton and Gottfried Leibniz. Just looking at Europe, certain followers under Isaac Newton use dot notation above the variables.

Followers under Gottfried Leibniz, particularly the Bernoulli family, use the elongated "S" as the acceptable notation.

So you can see how mathematical symbols differ, which is what he was talking about. Take the *equal* sign, "The most commonly used, and most commonly misused" as professor Martin J. Mohlenkamp states in his paper. I don't know the professor, but what he wrote is exactly what I was looking for. My point to my friend was that there is a universal way of teaching the basic fundamental properties of mathematics. No matter what language you teach mathematics in, it comes across the same idea in the end.

Me and my friend both knew each others points of view by the end of the conversation, I just thought it was worth mentioning his point of view as well as mine. I often rely on him for contrast, helps keep the logic seem "*logical*".

So now back to the main point, what is mathematical etiquette? Well for me, it was defined back before I graduated high school. I don't remember exactly when I ran into this definition, but the principle of it has stuck with me through out the years.

When your teaching mathematics one of the most valuable tools at your disposal is contrast. This is arguably the greatest tool of all for any teacher. In math classes, as a student, you will often witness other students asking questions because they got lost in the steps. I am sure almost all students who studied math have at one point or another heard the phrase:

I don't understand how you got to <insert idea>

This is a natural response to occur in such a study, and I believe it has to do with the etiquette of mathematics.

The example that was given to me about this specific principle has to do with adding a multiplication symbol where in fact multiplication is meant to be applied. Multiplication is generally denoted as this:

is the same as

or

is the same as

Now a lot of teachers will argue that there is a certain level of understanding that is expected of the student. I have witnessed many teachers claim that the student should be responsible for understanding at least the prerequisites for the current course. I can understand why they say such things, I truly do. To a certain point, the teachers are correct about a student being prepared for a certain class. But this is not the point I am trying to argue either.

I asked my teacher, who I think was a substitute teacher that day, why he would bother to add a multiplication symbol where it was generally understood? His response was this:

Because it is the right thing to do

I followed up with what I thought was a witty reply:

How do you know what is right?

I remember now, this happened when I was still in elementary school. For this is where I picked up this philosophy. He replied back to me with such a profound statement:

Because it is the harder choice to make

Seems kind of funny to learn what I call mathematical etiquette and sense of morality all at the same time. He got me with that one statement, I was speechless after that.

In an ideal setting, like the ways of old, people used to study subjects that they had interest in. They studied one main subject and focused on any other subject that might coincide with the main subject. Kind of like how "genius" is often said to coincide with "madness".

So naturally in an education setting like this, students tend to know the general notation and symbols. I would like to think the a student back then was a lot more prepared then the average student of today. The reason for this is because today's education does not follow just one subject, it follows many.

Students are expected to learn art, chemistry, English, government, history, Latin, physics, physical education, etc... To me education has become a bit of a harlot, in the traditional meaning of the word. No longer do students have free reign over their own paths in education. There is a lot of focus on passing tests because that is what pays the bills, sort of speak.

Let us not digress any further, so in an ideal setting students should have basic level of understanding of the prerequisites for each class that they take. If the student is in 3rd grade, then he/she should remember lessons from 2nd and 1st grade. In a college setting, if a student is studying *advanced* photography, then it would be expected that they know what a camera is. The difference in digital photography vs 35 mm photography, or what a shutter speed is. The reason for this expectation is because the word "advanced" is represented in the class name. It is not basic photography, it is advanced photography.

Teachers, certified or none certified, know full well that education is not an ideal system. The fact remains that education cost a lot of money in todays standards, maybe in some distant future that will change. So while I agree that students should know basic notation and principles in any study, I don't agree that we should assume students do in fact follow the same logic.

We have students that come to a class from all types of different situations, and each student requires a certain type of style. Just because a student does not understand an idea does not mean it is less valuable in learning then your own interest in the subject matter. In math classes I have found that I have a hard time understanding advanced topics, especially none linear topics. I will ask a question, and often a student will respond

How can you not see it? Its right there in front of you!

The sad part is, when the same advanced student asks me this:

How in the world did you understand where the negative sign disappeared too?

I respond not with a certain vengeance, but I simply just point out where the cancellation occurs. I do this because its the right thing to do, its the harder choice to make. So now that I am teaching students basic fundamentals, which college used to call algebra, I make a certain effort to put the multiplication symbol every where it is needed to be placed. I feel this is an etiquette I can follow and gladly show. I make a habit of showing all the steps needed for anyone to follow, regardless of level of skill. It takes me longer to teach, but I feel it is worth it. Maybe with my efforts, I can cultivate a better mathematician than I. The classroom to me seems far from being ideal, but in due time that might change. Or it could be that the only ideal classroom is in my head.

References:

*Wikipedia, the free encyclopedia*. N.p., n.d. Web. 25 Jan. 2014. <http://en.wikipedia.org>. Path: http://en.wikipedia.org/wiki/George_Pólya.- Polya, George.
*How To Solve It*. 2nd ed. Princeton, New Jersey: Princeton university Press, 1973. xvi-xvii. Print.

*Wikipedia, the free encyclopedia*. N.p., n.d. Web. 25 Jan. 2014. <http://en.wikipedia.org>. Path: http://en.wikipedia.org/wiki/Isaac_Newton.

*Wikipedia, the free encyclopedia*. N.p., n.d. Web. 25 Jan. 2014. <http://en.wikipedia.org>. Path: http://en.wikipedia.org/wiki/Gottfried_Leibniz.

*Wikipedia, the free encyclopedia*. N.p., n.d. Web. 25 Jan. 2014. <http://en.wikipedia.org>. Path: http://en.wikipedia.org/wiki/Bernoulli_family.

- Mohlenkamp, Martin J. "Mathematical Symbols."
*Ohio University*. N.p., n.d. Web. 25 Jan. 2014. <http://www.ohio.edu/people/mohlenka/>. Path: http://www.ohio.edu/people/mohlenka./goodproblems/symbols.pdf.

- Mohlenkamp, Martin J.
*Ohio University*. N.p., n.d. Web. 25 Jan. 2014. <http://www.ohio.edu/people/mohlenka/>.