# The relation between well-understanding and teaching

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
I believe many users here are teachers now or they were in the past and some might be working as professors at universities. I think if you teach a certain subject you understand it well day after day and you realize that you are learning new things and the concept is becoming fundamental as you teach this subject for a long time which differs than a person who just reads books and watch some videos no matter how many . Is that true , what you think ?

#### Amer

##### Active member
absolutely right, and i think the reason for that when you are teaching a subject some of the students look into it in another way which is new to you
it is interesting

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
I believe when you teach something one of these might happen :

1. Students alert you of things you might have ignored.
3. Teaching encourages you to read more and develop understanding.
4. It teaches you how to think of examples and simplify the concepts.

If you cannot explain it simply , you don't understand it .

#### Fantini

MHB Math Helper
I believe when you teach something one of these might happen :

1. Students alert you of things you might have ignored.
3. Teaching encourages you to read more and develop understanding.
4. It teaches you how to think of examples and simplify the concepts.

If you cannot explain it simply , you don't understand it .
Yes, it is true that explaining it to someone else helps a great deal with your own understanding. However, it assumes you are also interested in delivering the best explanation you can.

I am a TA this semester for multivariable calculus. Teaching is something very important to me therefore I am going to great lengths to ensure I can help everybody during meetings and that they all grasp the concepts in the clearest way possible. I have noticed a few things:

1) Simplifying concepts can be dangerous. I once solved an exercise where it was asked to calculate the volume of a cone, except that the rotating axis was the $y$ axis instead of $z$. The computations remained the same and I told them so, showed how all we needed was to switch the $x$ and $z$ coordinates to polar coordinates and the analysis proceeded the same as usual. However, I noticed they cling too hard to this: instead of a understanding of the situation, what I felt from their reactions was "Oh, so whenever I have this, I do this", and one more rote memorization. This may happen because multivariable calculus is an elementary course and most people taking it are far more interested in magical solutions rather than thought processes.

I've also noticed other departments do not value these as much as we do. I'm taking Electric Circuits Analysis this semester and I did not so as well as I knew on the test. Part of that was because I spent a lot of time explaining what I was doing and the processes behind my computations. I ended up with little time for checking the computations and ended up getting a couple wrong. When I first received the test I had a horrible grade, and was astonished to realize that all the text I had written was as good as trash: they didn't read any of it! The marks were solely for the numbers I had arrived. After I met with the professor he gave me some marks back, but it still leaves a bad taste. How can we teach people to think if we measure them solely by the results? The end does not justify the means.

2) Teaching forces (or should) you to take an active approach: you have to think ahead all the time, not only solve the exercises. This conscious approach helps generate more neurons for long-term memory and comprehension. You can't learn something if you are not paying attention!

These are the couple of points I have right now.

#### soroban

##### Well-known member

I agree completely.

The first few times I taught Calculus 1, I was amazed
how simple the derivations were. .They were crystal-
clear to me, yet I had worked hard to learn these
concepts in college.

Writing down the steps clearly and explaining them
to a class seemed to hard-wire the ideas permanently
in my brain.

To this day, I can still derive the Product Rule and the
Quotient Rule from scratch.

#### Fantini

MHB Math Helper
I find it interesting that you mentioned that. All (most) these concepts are crystal-clear now to us even though they weren't. What changed? Of course, we became familiar with it ourselves. But what processes happened in between? I found this video recently that sheds some light on it:

I believe this resonates with all of us. We have all struggled, we have all battled, we have all fought, we have all been CONFUSED with these concepts long before we achieved understanding. Sometimes we think that if they had just been presented in a more straightforward, in a more concise, in a more clear manner, we would have got them sooner, right? I'm beginning to think otherwise.

As Ackbach, I believe the socratic dialogue method is great. However, I feel it must at times be adapted to specific situations. In order to understand something, this new knowledge has to pass many barriers, such as our natural biases, previous experiences and taste. Moreover, ideally it has to fit with everything else we've learned and make sense as a whole. How can we achieve this goal?

As a reference, I found the video and a lot more interesting information at this question in Quora.

Cheers.

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
I also think that teaching helps our understanding because when you want to explain a certain concept you think of the easiest way to deliver the idea so students can inhale it easily. This , I believe , gives you the ability to think of alternative solutions of a certain problem.

#### topsquark

##### Well-known member
MHB Math Helper
1) Simplifying concepts can be dangerous. I once solved an exercise where it was asked to calculate the volume of a cone, except that the rotating axis was the $y$ axis instead of $z$. The computations remained the same and I told them so, showed how all we needed was to switch the $x$ and $z$ coordinates to polar coordinates and the analysis proceeded the same as usual. However, I noticed they cling too hard to this: instead of a understanding of the situation, what I felt from their reactions was "Oh, so whenever I have this, I do this", and one more rote memorization. This may happen because multivariable calculus is an elementary course and most people taking it are far more interested in magical solutions rather than thought processes.

I've also noticed other departments do not value these as much as we do. I'm taking Electric Circuits Analysis this semester and I did not so as well as I knew on the test. Part of that was because I spent a lot of time explaining what I was doing and the processes behind my computations. I ended up with little time for checking the computations and ended up getting a couple wrong. When I first received the test I had a horrible grade, and was astonished to realize that all the text I had written was as good as trash: they didn't read any of it! The marks were solely for the numbers I had arrived. After I met with the professor he gave me some marks back, but it still leaves a bad taste. How can we teach people to think if we measure them solely by the results? The end does not justify the means.
I got my first teaching "assignment" as a student at Alfred University doing a help session for Engineering students. It was, frankly, horrible. When a student first encountered a problem the first and only question was "what equation do I use to solve it" and on to the next question. It's simply a part of things. Engineering isn't about solving things...it's about creating them.

I have since gone on to teaching nursing, more engineering, and base "I have to take this to get it out of the way" students. Hey, I give them what they need to do what they need to do.

But I really (heart) the ones who want to know more.

-Dan

#### Fantini

MHB Math Helper
I also think that teaching helps our understanding because when you want to explain a certain concept you think of the easiest way to deliver the idea so students can inhale it easily. This , I believe , gives you the ability to think of alternative solutions of a certain problem.
Zaid, I am not sure if you read my previous post. I just said I think that explaining the concept the easiest way to deliver the idea may not be optimal. However, I will agree that considering alternative explanations to show someone else aids in discovering new solutions to a problem.

I got my first teaching "assignment" as a student at Alfred University doing a help session for Engineering students. It was, frankly, horrible. When a student first encountered a problem the first and only question was "what equation do I use to solve it" and on to the next question. It's simply a part of things. Engineering isn't about solving things...it's about creating them.

I have since gone on to teaching nursing, more engineering, and base "I have to take this to get it out of the way" students. Hey, I give them what they need to do what they need to do.

But I really (heart) the ones who want to know more.

-Dan
Should they be asking "what equation do I use to create it" then? Just kidding . We all know what the right question is. It's "Will this be on the test?" (Joking again XD)

I understand you Dan. I just try to be enthusiastic while teaching others and doing the best I can. Even if they are not thoroughly interested in what you are trying to convey, it does pay off to put all of yourself into it.

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
Zaid, I am not sure if you read my previous post.
Yeah , I did. I think I got lost in the language used. Anyways , I agree that there might be some consequences of trying to explain the concept the easiest way. Sometimes when I try to explain something for a friend I generally have a broad understanding of the concept but I try to explain that concept in such a way that I inform him that he will encounter it in the future and it will be explained in more details. Generally if you try to explain something in an easier manner you have to be careful not to abuse the concept or assume things that you are not allowed. Also when you want to prove a theorem , it is a good practice to know different proofs of the same theorem so that you think what is easier for the students to inhale.

#### MarkFL

Staff member
My least favorite question from friends and family, and it seems to be asked frequently, is when I am explaining a recent problem I found interesting, and they say, "When would you ever use something like that in real life?" I usually point out that even if there were no applications, and usually there are many, it would still be interesting in its own right. I have often wondered why this does not get asked when discussing literature, art or music. No one has ever asked, "When would you use that song or book in real life?" I guess the general population can more readily see enjoying these things for their own sake.

#### Fantini

MHB Math Helper
Most people apply the functional learning mindset to subjects which they don't enjoy (frequently mathematics). If I am "forced" to learn this, it has to be useful for me, else why would I bother waste energy and brain resources on this thing? This is lame, to say the least, but it requires a major cultural effort to break free.

They possibly do not apply the same thought to literature, art or music because of another misconception that "art should be for the sake of art", whereas mathematics is something belonging to the "industrial" side of things. (I disagree entirely with both.)

#### Petrus

##### Well-known member
My least favorite question from friends and family, and it seems to be asked frequently, is when I am explaining a recent problem I found interesting, and they say, "When would you ever use something like that in real life?"
That is something I ask myself many Times! I find it more intressting when I know how I can use it in real life!

Regards,
$$\displaystyle |\pi\rangle$$

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
My least favorite question from friends and family, and it seems to be asked frequently, is when I am explaining a recent problem I found interesting, and they say, "When would you ever use something like that in real life?"
It is always important when explaining a concept that you at least mentions some applications. That encourages you to learn more about the concept and master it. Of course it is sometimes hard to find applications. I always think of Number theory which was developed nearly in the 17th century by Fermat. It was thought to be a dead subject and there were no applications for .It was just for fun. Then it was the appearance of computer science that proved the importance of this branch of mathematics. This story tells us that we can not judge that a certain concept is not useful even if absurd. There might not be applications for the time being but who knows what happens in the future ?

#### Fantini

MHB Math Helper
If only that would satisfy them, Zaid.

"What applications does this have?" "Well, none that I know right now...but it might have." "When?" "In distant future, in a galaxy far far away, someone or something may have found a practical use for this." "I'm convinced now. Show me the theory."

Jokes aside XD, while applications are valuable they should not be the end goal of learning. Besides, there are numerous other skills people could acquire that are incredibly useful in their daily lives, with applications everywhere, and yet nobody does. I don't think we need to motivate how knowing a little of programming, electronics and chemistry can be a complete lifesaver.

#### Deveno

##### Well-known member
MHB Math Scholar
Personally, I have always taken great pride in the utter uselessness of mathematics. To me (and this a deeply-held personal belief, not necessarily even TRUE), to justify mathematics as "means to an end" (like building better bridges, or spacecraft, or whatever) is evil incarnate. You see, I don't believe that "ends" even exist at all! Only "means" exist, there's never any "all stop" where the final score is tallied...life is a PROCESS.

Furthermore, I think one cheats one's self out of one of the most stunning parts of being alive by doing so: the sheer beauty and mystery of existence, the joy of being.

If you take no joy in the teaching and/or learning of mathematics, by all means stop right now, and do something else! You can thank me later

While I am rather reticent to discuss spiritual matters on a math website, I feel I commit only a minor transgression by suggesting that the same beautiful being that whispered to Pythagoras also whispers in MY heart when I think of roots of unity, or vector spaces. Mathematics may be (perhaps) a great work of fiction, but there is depth and brilliance in it to rival Beethoven, or Shakespeare, or Rembrandt. Some of the truest things in our world of experience and thoughts/feelings about experience are "unreal".

******

On the main topic, teaching someone CAN distill ideas in one's own head. I think this has more to do with communication than teaching itself: externalizing our thoughts makes us more familiar with what they are, and what has heretofore been hidden to us as things we already knew, but didn't KNOW we knew. Sometimes formal teaching is even a hindrance to this process...if you are teaching a subject to those with no feeling for it, all you are doing is being a poor substitute for a book, or google. While there is certainly a time and place for this sort of thing, it is not necessarily very illuminating.

#### topsquark

##### Well-known member
MHB Math Helper
Should they be asking "what equation do I use to create it" then? Just kidding . We all know what the right question is. It's "Will this be on the test?" (Joking again XD)
The other most asked question is "Will there be extra credit?"

-Dan