Changing Perspectives in Math Class: From Memorization to Understanding

[Mhorton91]

So my semester started yesterday, and my first class on: Monday, Tuesday, Thursday, and Friday is pre calculus. I walked in, sat down, and it began.

My professor is the head of the math department at the college, and within 5 minutes of the intro lecture, it's clear that he really loves math (which I feel like is an important aspect to being a good teacher)... He did all the normal stuff professors do on the first day of class, introduction, ask what everyone is majoring in, tells us the main objectives of the class, ect..

Then he starts into what, is honestly the most eye opening moment I've ever experienced in any class, from K - now... He writes out the standard form of a quadratic equation ax²+bx+c=0... I gave it absolutely not a second thought, I've seen it about a million times since high school, he then writes the quadratic formula next to it (I have no idea how to input a formula) but you all know what it is, but again, I've seen this a million times. Here is where this class broke away from all other classes I've had, he didn't just tell us to "plug in these variables" or "solve this quadratic equation", he said "How do you get from this (standard form), to this (quadratic formula)" He had everyone attempt to derive the formula from the standard form, he called my attempt "a good idea" however it was wrong...

So, the moral of this whole post is just that, even though something like deriving the Q.F. from the standard form is something that most of you can do in your sleep, that isn't the big point, This has been the first time an instructor has said "This is why this formula does what it is supposed to" instead of "memorize this formula" and I think it is the coolest thing ever. He did the same thing with a few other basic concepts. One of the ones that was really simple, but, again, my brain hadn't been taught to think about past just remembering the rule was, "why do you flip the inequality sign when you multiply by a negative"...
But yeah, this is a pretty long post, with no question asked. So, thanks for reading it!
Marshall

 

[Mentallic]

My class in year 10 was expected to be able to derive the quadratic formula, but then again, it was the top math class and the students there did have a keen interest in understanding the subject from top to bottom.

I think I might ask a few of my friends whether they know why the quadratic formula is what it is.

 

[HallsofIvy]

The most straightforward way to derive the quadratic formula is to "complete the square". Unfortunately, most students either never learn to "complete the square" or quickly forget it. In any case, in a good college math course, you can expect a fair amount of time devoted to showing how to derive formulas, not just use them. (In other words, you will be expected to think, not just memorize.)

 

[Borek]

Sadly, what you wrote means a failure of the education system that you experienced so far.

Not your fault, and good for you that you are starting to see what it is all about - you are not yet lost completely

 

[pwsnafu]

My class in year 10 was expected to be able to derive the quadratic formula

My year 9 exam had derive the quadratic formula from first principles as one of its questions.

 

[MathJakob]

I had something similar happen to me once... I had been using all sorts of formula and knowing that it worked, but not knowing why it worked. I think a big part of mathematics is not just knowing the answer, it's knowing why the answer is the answer it is and understanding why certain things work and why they don't.

 

[1MileCrash]

This has been the first time an instructor has said "This is why this formula does what it is supposed to" instead of "memorize this formula" and I think it is the coolest thing ever.

It's good that you thought this was a good thing for the teacher to cover, I suspect not everyone felt the same way.

A lot of people (engineers) in my physics classes used to dislike whenever equations were derived in class, because they didn't care about how or why it worked, they just wanted to get to the answer.

This is a good indicator that you would enjoy some of the higher levels of math, where study of objects and proving qualities about them is the focus, and finding solutions to things is sort of less common.

 

[AlephZero]

Not your fault, and good for you that you are starting to see what it is all about - you are not yet lost completely

And you just took the first step towards understanding "classic" math jokes like:

Q: How do you make a cup of instant coffee?
A: Fill kettle with water, boil kettle, etc, etc...
Q. How do you make a cup of instant coffee, if you already have a kettle full of boiling water?
A: Empty the kettle, then use the previous theorem.

 

[pasmith]

If memory serves, my introduction to solving quadratic equations was essentially "you solve quadratic equations by completing the square". And then we completed the square for an arbitrary quadratic and thus obtained the formula. At about that time we were taught how to sketch the graph of a quadratic function by completing the square.

 

[stardust]

Personally, I've always felt that math(at least in grade school) is largely taught backwards. There is insistence on rote memorization, with no clue why this or that needs to be memorized. Questioning is discouraged, and classes are designed to move on a tight schedule regardless of whether a student has mastered the material or not. Finally, at later point you are expected to start abstracting concepts and use mathematical reasoning, when all that time you've been trained just to memorize math like a recipe. It's no wonder so many people hate math, and don't see it's beauty.

 

[Mhorton91]

The most straightforward way to derive the quadratic formula is to "complete the square". Unfortunately, most students either never learn to "complete the square" or quickly forget it. In any case, in a good college math course, you can expect a fair amount of time devoted to showing how to derive formulas, not just use them. (In other words, you will be expected to think, not just memorize.)

That is how we ended up doing it. Being the first day back from break, I needed some guidance to get my mind back to working properly (during break the only thing I thought about what what toppings I wanted on my pizza)... After I knew that was the route to go, I was able to derive the formula fairly easily!

Sadly, what you wrote means a failure of the education system that you experienced so far.

Not your fault, and good for you that you are starting to see what it is all about - you are not yet lost completely

I hope to get turned around, academically speaking! This instructor seems like a good place to start.

My year 9 exam had derive the quadratic formula from first principles as one of its questions.

When I was in 9th grade, I was learning slope intercept form, the term quadratic had never came in my ears lol... I had no interest in math/science/college until after I was graduated... so I took the easiest classes my school offered (regretting that now)

I had something similar happen to me once... I had been using all sorts of formula and knowing that it worked, but not knowing why it worked. I think a big part of mathematics is not just knowing the answer, it's knowing why the answer is the answer it is and understanding why certain things work and why they don't.

I agree completely, with just 2 days experience I feel like there's something totally different about actually being able to understand why an answer is the way it is.

It's good that you thought this was a good thing for the teacher to cover, I suspect not everyone felt the same way.

A lot of people (engineers) in my physics classes used to dislike whenever equations were derived in class, because they didn't care about how or why it worked, they just wanted to get to the answer.

This is a good indicator that you would enjoy some of the higher levels of math, where study of objects and proving qualities about them is the focus, and finding solutions to things is sort of less common.

I could see what you're saying about the engineers not liking, or caring, about equations getting derived, I read somewhere that for them math and physics are used as tools, and they don't generally care why the tool works, just as long as it does.

And you just took the first step towards understanding "classic" math jokes like:

Q: How do you make a cup of instant coffee?
A: Fill kettle with water, boil kettle, etc, etc...
Q. How do you make a cup of instant coffee, if you already have a kettle full of boiling water?
A: Empty the kettle, then use the previous theorem.

Maybe a few more weeks of actually thinking and I'll fully appreciate that lol. My instructor busted out the oldest math joke I know on the first day of class..

Q: What is the formula for the area of a circle
A: "Pi r squared"

"No, cake are square, pie are circle"

If memory serves, my introduction to solving quadratic equations was essentially "you solve quadratic equations by completing the square". And then we completed the square for an arbitrary quadratic and thus obtained the formula. At about that time we were taught how to sketch the graph of a quadratic function by completing the square.

Last semester in my intermediate algebra class (One class below college algebra), we did factoring for a long time, then completing the square for like a week.

Personally, I've always felt that math(at least in grade school) is largely taught backwards. There is insistence on rote memorization, with no clue why this or that needs to be memorized. Questioning is discouraged, and classes are designed to move on a tight schedule regardless of whether a student has mastered the material or not. Finally, at later point you are expected to start abstracting concepts and use mathematical reasoning, when all that time you've been trained just to memorize math like a recipe. It's no wonder so many people hate math, and don't see it's beauty.

I agree with what you're saying, but at the same time when you said they will follow the schedule even if a student hasn't mastered the material... based on just my own experience that I'm now seeing... as long as we continue how it is being taught, by memorization, and without taking the time to ask and answer questions, could anyone ever really "master" said material?

 

[BobG]

I could see what you're saying about the engineers not liking, or caring, about equations getting derived, I read somewhere that for them math and physics are used as tools, and they don't generally care why the tool works, just as long as it does.

Probably spoken by a mathematician that thinks it's sacrilege to actually use mathematics instead of just ponder it. Of course they care about how the equation was derived and how it works.

Maybe a few more weeks of actually thinking and I'll fully appreciate that lol. My instructor busted out the oldest math joke I know on the first day of class..

Q: What is the formula for the area of a circle
A: "Pi r squared"

"No, cake are square, pie are circle"

Once again - wrong!

pie are 8.54 and always rounded

 

[stardust]

I agree with what you're saying, but at the same time when you said they will follow the schedule even if a student hasn't mastered the material... based on just my own experience that I'm now seeing... as long as we continue how it is being taught, by memorization, and without taking the time to ask and answer questions, could anyone ever really "master" said material?

That's my point, the current education system is flawed. The model of education we have now is based upon the Prussian model from the 1800s. This is for almost all countries. We are still teaching people based on a model that's over 150 years old! Not only that, but this model was designed primarily to teach people how to use machines during the industrial revolution. It produces human equivalent of drones. That's why math is taught in that fashion, along with every other subject. For instance, do you often hear of philosophy classes or their ilk being taught in grade school? I'm sure they exist, but they are the exception not the rule. That's because philosophy is not a "useful" subject, not from the perspective of training someone to simply be able to run machines. That's also why children are allowed to pass classes with Cs. Obviously if at the end of the class, if you are still getting Cs on assignments, there is a lack of mastery. But the goal isn't mastery, it's to simply mass produce people-shaped calculators.

 

[WannabeNewton]

That's because philosophy is not a "useful" subject, not from the perspective of training someone to simply be able to run machines.

It's a useless subject from the perspective of anyone who values money.

 

[stardust]

It's a useless subject from the perspective of anyone who values money.

Well, I don't value money. I need it to live in the society I was born into, but I've never cared for it. I consider my time to be more valuable than the acquisition of wealth.

 

[Borek]

It's a useless subject from the perspective of anyone who values money.

You are aware of the fact many sciences - starting with math and physics - were philosophy one day?

Churchill is quoted as responding to a plan to cut money for the arts to fund the war effort by saying: "Then what are we fighting for?" This is not about philosophy, and it is not even a true story, but it nicely explains the spirit.

 

[SteamKing]

That's my point, the current education system is flawed. The model of education we have now is based upon the Prussian model from the 1800s. This is for almost all countries. We are still teaching people based on a model that's over 150 years old! Not only that, but this model was designed primarily to teach people how to use machines during the industrial revolution. It produces human equivalent of drones. That's why math is taught in that fashion, along with every other subject. For instance, do you often hear of philosophy classes or their ilk being taught in grade school? I'm sure they exist, but they are the exception not the rule. That's because philosophy is not a "useful" subject, not from the perspective of training someone to simply be able to run machines. That's also why children are allowed to pass classes with Cs. Obviously if at the end of the class, if you are still getting Cs on assignments, there is a lack of mastery. But the goal isn't mastery, it's to simply mass produce people-shaped calculators.

Some philosophy classes can be taught using the Socratic method, which is almost 2500 years old.

Not every method which is newer than next week is necessarily more valuable than an older method.

I'm sure you could try to teach philosophy in grade school, but what would be the point? It seems to me that the study of philosophy entails a certain amount of reflection about the world and out place in it, but these qualities are not necessarily present in someone who has only consciously experienced the world in a few years of life.

 

[stardust]

My main point wasn't the age of the present educational system, it was the lack of personalized education for each varied individual. It is true that a newer method isn't necessarily any better. That being said, we now have the internet which can provide large educational resources to students without requiring quite as much manpower. This is why I mention that the model that's being used is archaic, because that model preceded the internet. Also, it's attitude of drilling subjects into people's heads without trying to get them to think and reason(fallout from the Industrial Revolution). To me, it's rather pointless to have students take subjects if they're not allowed the time to master the material. The lack of mastery combined with rote memorization is based on putting students on sometimes unreasonable deadlines. Just as bad, students that can go at a faster pace, will often grow bored going the same pace as everyone else. It's unfair to students and it's bad for any advancement of society in general.

As for philosophy, I can't speak for the general populace, but I remember a fair amount of reflection/introspection starting as early as 5. Not because of the education system(which failed in that regard). Rather my uncle would pose questions to me and try to make me think of the implications. I sometimes think that adults have a tendency to grossly underestimate what children are capable of.

Basically we have a system that doesn't place an emphasis on teaching children how to reason for themselves, and people wonder why we have problems getting prospective students into STEM fields.

 

[HeLiXe]

Pretty cool post :) Thanks for sharing your experience in class.

 

[Student100]

My year 9 exam had derive the quadratic formula from first principles as one of its questions.

I think you missed the purpose of this post my friend. :uhh:

Good for you OP. Math is still a lot deeper than simply deriving equations from principles but it sounds like you're off to a good start. Hopefully you can take a few more math classes with this professor. We were never shown in class how to derive the QE during my math education, or how to derive anything really. It was all formula + calculation.

As a result I have a lot of holes in my understanding; luckily my physics education has been a bit better—except for E&M2, what a worthless lecture that was.

 

[mpresic]

Sounds like you have the right attitude to be successful in pre-calculus, college and beyond. Best wishes.

Completing the square is a good technique for doing this.

If you think the quadratic is hard. MATLAB symbolic algebra can solve the general quartic (fourth-degree) equation. It has about 15 densely packed pages of symbols raised to various powers and roots with weird coefficients. Fifteenth century Italian mathematicians figured out the general method for the four roots.

If your professor decides to show it to your class: it's cool.

 

[lendav_rott]

Approach the subject with an open mind. Many people in my engineering class were like "seen it already, don't care who, where or what it is, just give me the answer". I think you're doing great.