How I study and teach

 $$\mathrm{The}~\varepsilon-\delta~\mathrm{blog}$$

How I study and teach

I think it's worth mentioning how I study before my first actual post later today. If you know how I will approach concepts, you'll probably be able to understand why I do things the way I do them.

Anyway, I believe that whenever someone's introduced to a new "field" (like trigonometry, calculus etc.), understanding why exactly learning about and exploring that field is crucial. It rids you of questions like "why on earth am I learning about this?", "what's the point of doing this?", etc. Basically, it gives you an idea of why you're doing what you're doing and keeps up your interest and confidence.

In mathematics, especially, I like to think of it as this :

• We begin from scratch, in the context of that particular field.
• We either feel the necessity to develop that field or accidentally discover some relationship, that "sparks" the field.
• Then, we begin to explore the field. We feel the necessity to develop certain formulae (or discover a relation between two things, by accident), we explore the implications and applications of those formulae and analyze them to unravel interesting details.
• At the end, we understand more than we did before we started learning about this field. We can now apply the knowledge and understanding from this field to other fields and areas. We have intellectually evolved.

I think I should be a little more elaborate about the third point.

When I say "necessity to develop certain formulae", the word necessity doesn't just mean something related to real life applications, it also means the will to understand more of the field, develop some "techniques" to help solve the questions that come up in that field better etc.

I have talked about a particular derivation for expressing $\cos(2\alpha)$ in terms of each trigonometric function at $\alpha$.

I discovered that accidentally. I had two formulae for the length of a chord of a circle with me. In one of them, the chord length was expressed in terms of $\sin\Big(\dfrac\phi2\Big)$ whereas in the other one, it was expressed in terms of $\cos\phi$. As both the formulae gave the same result every time, they can be equated to derive a few multiple and sub multiple formulae in trigonometry.

$\big[\text{Note : I still haven't generalized the formula for all }\phi \text{ such that }0^\circ < \phi \leq 360^\circ$

$\text{I will do so when I will be discussing the multiple and sub multiple formulae later}\big]$

These are some of the formulae that can be derived using the "simple", accidentally discovered relation :

$$\sin(2\alpha) = 2\sin\alpha\cdot\cos\alpha = \dfrac{2\tan\alpha}{1+\tan^2\alpha}$$

$$\cos(2\alpha) = \cos^2\alpha-\sin^2\alpha=2\cos^2\alpha-1=1-2\sin^2\alpha=\dfrac{1-\tan^2\alpha}{1+\tan^2\alpha}$$

$$\tan(2\alpha) = \dfrac{2\tan\alpha}{1-\tan^2\alpha}$$

$$\sin\Big(\dfrac\alpha2\Big) = \pm\sqrt{\dfrac{1-\cos\alpha}{2}}$$

$$\cos\Big(\dfrac\alpha2\Big) = \pm\sqrt{\dfrac{1+\cos\alpha}{2}}$$

$$\tan\Big(\dfrac\alpha2\Big) = \pm\sqrt{\dfrac{1-\cos\alpha}{1+\cos\alpha}} = \dfrac{1-\cos\alpha}{\sin\alpha}$$

By analyzing the formulae, I'm referring to trying to intuitively understand them and what they do and also exploring some details about them.

For example, a thing you might be wondering is why the $\pm$ symbol appears in the latter three of the six formulae written above and not in the former three. It's a good question, I think. I "explored" it a little and here's why it happens :

$\sin(2\alpha) = 2\sin\alpha\cdot\cos\alpha \leftarrow$ Here, we are expressing $\sin(2\alpha)$ in terms of $\sin\alpha$ and $\cos\alpha$. That means that the formula describes a relation and can give the value of $\sin(2\alpha)$ for a given $\sin\alpha$ and $\cos\alpha$.

However, if you want to express the same relation only in terms of $\sin\alpha$ (i.e. $\sin(2\alpha) = \pm2\sin\alpha\cdot\sqrt{1-\sin^2\alpha}$), the $\pm$ symbol will emerge because the value of $\sin(2\alpha)$ for a given $\sin\alpha$ is not unique and the two possible values are the additive inverses of each other (i.e. $\mathrm{value}_1 = -\mathrm{value}_2$).

Similarly, for a given value of just $\cos\alpha$, the value of $\tan\Big(\dfrac\alpha2\Big)$ is not unique but for a given value of $\tan\alpha = \dfrac{\sin\alpha}{\cos\alpha}$, the value of $\tan\Big(\dfrac\alpha2\Big)$ is unique, hence no $\pm$ symbol.

$$\big[\text{More on this later}\big]$$

So, to conclude, for every new formula or definition I come across, I keep the following in mind :

• Why exactly is this necessary $\color{green}{\mathrm{or}}$ what "accidentally discovered relation" was used to derive it?
• What are the various ways to derive this particular formula?
• What are the different versions of this formula? What can be deduced from these different versions?
• Does there exist a different approach towards the formula's function?
• Does this particular definition add something to our understanding of the topic? i.e. is it something new, an extension of a previous definition (like the definition of trigonometric functions for all angles is an extension of the right triangle definition), or just a rigorous version of a previous idea.
• Can the stuff that can be done with the help of this formula be somehow done without it too?
• What does the definition/formula imply and what are it's applications?
• If we're dealing with the definition of something which has already been defined some other way, we need to explore the consistency of both definitions.
• If we already have a rough idea of what the definition is defining, we need to compare both of the interpretations to look for possible misunderstandings.

I would like to point out that in the example of trigonometric functions I've given in point 2, in definitions like these which are extensions, I think it's worth exploring certain concepts with both definitions in mind. I will be doing this in my posts.

I also think that one can never completely explore the implications and applications of a certain concept because if that was somehow possible, we would know almost everything, won't we?

So, one should think of the process of exploring these as gradual and not as something you can accomplish in a day or a couple of days.

An example of different "versions" of the same formula will be the chord length formula which I've discussed about earlier too. For a circle with a radius $R$, let $L_c$ denote the length of the chord that is obtained by joining two points on the circle such that the two radii joining the centre of the circle to the two points make an angle of $\phi$ with each other. Then, $L_c = 2\sin\Big(\dfrac\phi2\Big) = R\sqrt{2-2\cos\phi}$.

These are two "versions" of the same formula and and what we can deduce from these are the multiple and sub multiple identities in trigonometry.

By "a different approach", what I'm saying is "Is there a possible way to approach what the formula is supposed to do?". The term "function" isn't used for the mathematical term "function" here. An example of this would be better. Let's say that we are talking about vectors which are usually treated as arrows in space. A "different approach" would be to attribute a magnitude and direction to the arrow and then exploring the operations on it. I haven't thought through this too well but I will be discussing about it in detail when I will talk about vectors.

Regarding the second last point, a good example would be the definition of refractive index. Let's say that $_yn_x$ denotes the refractive index of material $x$ with respect to $y$, there are two commonly used definitions of $_yn_x$, which are :

• $_yn_x = \dfrac{c_y}{c_x}$, where $c_y$ denotes the speed of light in $y$ and $c_x$ denotes the speed of light in $x$.
• $_yn_x = \dfrac{\sin\angle\mathrm i}{\sin\angle\mathrm r}$, where $\angle \mathrm i$ is the angle of incidence of a ray of light coming from $y$ to $x$ and $\angle \mathrm r$ is the angle of refraction of the same ray once it passes through the interface of $x$ and $y$.

If you calculate the refractive index of any $x$ and $y$ using both definitions, the result will be the same. Exploring the "consistency" of both of these definitions in most likely worth it but I haven't done anything regarding it yet.

Regarding the last point, force will be a good definition.

The mathematical definition of force is $\overrightarrow F = m\overrightarrow a$, where $m$ is the mass of the body and $a$ is it's acceleration. But, what most people intuitively think of it before learning this definition is as momentum ($\overrightarrow p = m\overrightarrow v$).

I will talk in detail about all these examples later in my posts.

So, these are the things I take into consideration while trying to understand a new concept, mostly a new definition or a formula. I will also take these things into consideration in my posts here, i.e. for every formula I discuss, I will explain things like why it is necessary, what are some possible implications and applications of it etc.

If you have any doubts, feel free to ask in the comments.

PS : I'm obsessed with lowercase Greek alphabets. So, I'll leave this little signature in the end. It's just the Greek equivalent of my name.

$\text{~}\rho\alpha\xi\delta\epsilon\epsilon\pi~\sigma\iota\nu\delta\eta\theta$