Stuff that you struggled with in school

[TheRockARooster 50,768]

7 minutes ago, SasQ said:

Whoa, one night and so many replies already!
Some of them very useful, some of them less useful, some of them begging for more clarification.
I'll try to address them all, but instead of doing it in sequence, I'll group them by subjects. I also emphasise in bold red the things that drew my attention.

Let's start with maths, because this seems to be the subject most people complained about (and this doesn't surprise me the slightest, of course ).

Memorization is a very bad way of learning anyway, because keeping stuff in your memory requires constantly fuelling the neurons that keep that memory alive (that's why it feels so exhausting), and it is very short-term, because once you stop powering those neurons, the information is lost forever :q  People forget stuff, that's unavoidable. (Well, unless you drilled it long enough to basically have it burned into your wires for good :q But this is also bad, because it makes it even harder to unlearn it if it turns out that you've learnt it the wrong way ). A better way of learning is by actual understanding, because once you understand something, there's no way you can forget it. If you do, you can easily bring it back by quickly deriving it from scratch once again, because now you know how it works and where it came from.

Memorization strikes again As for the Unit Circle though: you mean the values of trigonometric functions for those "special" angles? There's a nice trick to remember them, as simple as 0,1,2,3,4
    sin(0°)    →   √[0/4]  = 0
    sin(30°)   →  √[1/4]  = 1/2
    sin(45°)   →  √[2/4]  = √[1/2] = 1 / √2
    sin(60°)   →  √[3/4]  = √3 / 2
    sin(90°)   →  √[4/4]  = √1 = 1
and the cosine works the same, just backwards (you can just take the fraction for sine, and complement it with whatever needs to be added to get 1, and take a square root of that instead to get the cosine; e.g. if you used 1/4 for sine, you use 3/4 for cosine, because 1/4 + 3/4 = 1).

BTW it's funny how they call them "special" instead of "simplest", because that's the simplest values of trig functions one could come up with, and usually the only ones dealt with in schools It's a lot more fun to come up with values for angles in a pentagram, or a 12-gon, or some other crazy polygons Schools never show you that, because that's where the real fun is :q

Also, in schools they always teach you that trigonometry is all about triangles (that's what the name means after all, right? :q ), but the truth is, originally it was all about rotations and going in circles, because ancient mathematicians used it for astronomy (calculating positions of stars & planets in the night sky). That's where the unit circle comes from as well. And it is much easier to understand it this way, as coordinates of a point moving around a circle, instead of with graphs of sine waves and solving triangles (which is just one of the applications of trigonometry).

Yeah, that's one of the problems with how maths and physics are being taught these days. They just give you formulas but they don't explain where do these formulas came from and how they work – because they usually don't know that either And if you try asking them questions about it, they get irritated, because they don't want you to notice that they're no smarter than you in this regard. The only way they can keep looking smart is by making you feel dumb. And so they do.

Fortunately, this is something that can be easily fixed by people like me who always dig deeper and try to understand how stuff works and why I've already found a lot of ways of explaining those formulas, in a visual way, with geometry. I also focus a lot on showing my students how they can derive those formulas themselves if they had to figure them out from scratch, which is basically what you have to do anyway when you forget them – because people forget things :q that's why memorizing stuff is such a waste of time and energy. But when you know how stuff works and where those formulas came from, you can always derive them again when needed, and that way you never have to memorize them ever again

That's another problem with how maths are being taught in schools: They totally ignore this important aspect of WHY. Why do we learn about this stuff in the first place? Why do we need this? Why did people come up with it? What they needed it for? What problems they are trying to solve? How did they come up with this particular solution?

Math is often being taught as something "set in stone" – discovered long time ago by some ancient sages / geniuses and all we need to do now is learn those laws and memorize them. This makes an impression that math is finished; that there's nothing more in it to ponder about or experiment with, nothing more to discover, especially for ordinary people like you and me who clearly aren't like those ancient geniuses. But this is wrong! Very wrong! The real fun of math lays in playing with it and discovering all those patterns on your own, because then it's much more exciting and it makes you feel that you can be a discoverer too! Those ancient geniuses weren't really geniuses – they were ordinary people like you and me, who just happened to be playing with those concepts long enough to figure out the patterns, and you can do that too if you enjoy it. There's still a lot of unanswered questions in mathematics, and one day you might be one of those who will answer them. Sometimes the answer might even be something obvious – well, obvious once you figure it out, but not so obvious when you don't know it yet – and that's the only reason why other people didn't find it.

Mathematics originated from real-life problems and people trying to solve them. Things like measuring areas of fields, finding distances between cities, figuring out how planets move in the night sky, calculating dates, finding ways to trade stuff without being duped, balancing accounts, making convincing arguments, aiming missiles at their targets or navigating space ships in the cosmic space, calculating forces on bridges and buildings so that they didn't fall apart, finding patterns in numbers to predict the stock market, etc. So there's plenty of applications, but they usually don't tell about them to you, which of course makes people feel that all this stuff is totally unnecessary and waste of their time, or even makes them hate maths :q

That's why I always do my best to explain stuff by showing how it originated from some real-world problems, ask people for trying to figure out their own solutions before showing them how other people did it (so that they understood better what was the problem really about and what was the difficulty in solving it, because then they have more appreciation when they learn the solution; and even more if they come up with the solution themselves ).

More on real-world applications, but there's another thing worth pointing out: "too abstract". People often say this about maths: that it is "too abstract" for them, meaning that they don't see any connection of it to the real world. So what they really complain about is not abstraction, but being too "unearthly", so to speak. Abstraction is really about something else: about noticing a common pattern in different things and "abstracting it out" (taking out just that pattern without all the unnecessary details) and being able to apply it to all sorts of new situations. It is a tool for our brains to actually help us comprehend difficult subjects by grasping a complex idea all at once and naming it, so that we could keep it in mind as a single object instead of a bunch of complexity. It allows us to stop focusing on all those gory details for now, and focus only on the gist of it.

People who play with maths, quickly start noticing repeating patterns in numbers, shapes, formulas etc. And because they're lazy and they don't want to repeat the same arduous tasks over and over again, they abstract those patterns as formulas and laws that they can then apply in other problems more quickly.
Often they start recognizing the same patterns in different areas of maths. For example, they notice that dividing polynomials work pretty much the same way as dividing numbers. Or that all the ways you can rotate a rectangle works in a similar way to all the ways you can multiply positive and negative numbers. Sometimes those patterns are even more surprising: e.g. that there is a connection between arranging playing cards in a square without repetitions, and lines crossing each other in projective geometry. Or those connections can even span multiple domains of science! For example, that the way things fall when you drop them is somehow related to the difference of squares in geometry. Or that locust's mating cycles occur according to prime numbers (it has to do with primes having no common divisors and locusts trying to avoid predators).
Noticing such patterns is the first step for abstraction, and then it allows for applying the same observations somewhere else.

People often say that their minds are unable to comprehend abstract ideas when it comes to maths, but the same people unknowingly use abstract thinking in everyday life, totally unaware of it when, for example, they say stuff like "dogs are man's best friends" – they don't talk about any particular dog or any particular man, but dogs and men in general. They ignore all those unnecessary details and focus just on those features of dogs that are relevant to the case: their friendliness. Same goes when they talk about colours: when you talk about orange (as a colour), you're using abstraction :> You ignore all those unnecessary features oranges have, like their taste, shape etc., and focus only on one particular feature: their colour. Then you abstract that feature and apply it to other things that have the same colour. When someone says "This book is orange", you certainly don't have any problems understanding that they're talking about colour, not that the book is literally a fruit

So why they struggle when it comes to maths? What is it about maths that makes the difference? :q

Well, the problem is that people who work with maths professionally, they worked on it for such a long time that they used to it and they forget that the things that are now bread and butter to them, might be totally incomprehensible for others who didn't walk through all that route. It's those people who start their book with "Let 𝔊 be a simply-connected differential manifold in a normalized Hilbert space ℌ..." and you start wondering where's the Book I where they explain what the heck are those things?! :q  The problem here is not that math is too abstract, but that those authors suck at getting their ideas across :q  And it's quite unfortunate that this became a standard for math and science in general, where you are supposed to know stuff but you are not being told how are you supposed to learn them in the first place.
Same goes with those who just teach maths: they often are in no better position than their students, because they don't understand those subjects well enough themselves either. There's nothing worse than when the blind try to lead the blind. It's a recipe for disaster. There's no way someone can teach something to someone else if one doesn't understand it well ehough either. Unfortunately that's how public education works these days :/

Now for the replies that require more clarification:

What steps are you talking about?

Can you expand on why was that? What were the exact difficulties you encountered? Your statement expresses your feelings, but it doesn't tell much about what caused them. And I'm rather interested in the latter, because that's something I can work with (I can't work with your feelings though, because I'm not the one in control over them).

You meant an inadequate teacher?
Yeah, the fact that with proper teaching/tutoring you were able to ace it, clearly shows how it often depends more on the teacher than on the student. There was nothing wrong with you, it seems, that stopped you from understanding it – it was rather the teachers' fault that they failed at getting their ideas across.

Of course, if a student doesn't want to learn, then even the best teacher can't help :q  And if the student is determined enough, then even the worst teacher won't stop him from learning. But not every student is determined enough, and unfortunately most teachers suck at teaching It's just another paid job for them, often the only job they could get with their qualifications ;q  They know zilch about the stuff they teach, but they're good at pretending that they're smart enough for that job or have sufficient credentials (which doesn't always mean that they understand stuff – they once were students too, and they might just have managed to pass the tests without real understanding, as you do).

Knowing stuff is not enough to be a good teacher. One also has to know how to teach. How to exploit what people know already and use it as scaffolding to let them understand new stuff. Explain stuff clearly enough so that they could understand it.

What's critical thinking?  I've never had such subject when I was in school :q
So I don't really understand what was your difficulty with it. Can you show some examples?

Can you elaborate on WHY a little bit?

---

And there's also a couple of other things I'd like to talk about

That's great to hear! Can you tell us more what is it about it that you like? That might help people struggling with algebra to see what's the real fun in it and maybe let them find some fun in it as well?

That's interesting, considering that algebra is really just a "programming language of geometry" – as I use to call it Why? Well, because when you write computer code, you're basically using symbols and names to manipulate stuff in the computer's memory or on disk – something you couldn't just do directly. Same goes with algebra: those symbols and formulas just represent some geometric objects that you would normally draw on paper or picture in your head, but after a while it would become cumbersome to use, so you replace them with symbols, because they are shorter and easier to manipulate, and you can put them into formulas and equations that can then be easily solved. But under those symbols and formulas, there's always some geometry hiding. Even behind such simple things as (a+b)·(c+d) = a·c + a·d + b·c + b·d. And it is very insightful to try finding such connections between formulas and geometry, because it makes math visible and allows you to picture it in your mind instead of just manipulating abstract symbols It also helps you understand where do these formulas come from.

Well, it's pretty much about showing other people how did you come up with some clever observation by showing them all the steps that lead you there from the first principles (i.e. the stuff that everyone can clearly see for themselves). Of course it's easier to do when you came to that observation yourself and you know all the steps that lead you there, but much harder if someone told you that something is true and your job now is to explain why

Problems with proofs in mathematics is that they're often taught as a "finished product" – something that some smart dude already figured out in the past by his ingenious mind that might seem like black magic to you. But rest assured, they were no geniuses. They just don't show you all their work :q  Mathematicians often have to try a lot of multiple approaches and fail until they find the correct way to prove something. But they never show their scratch pads – they only show you their final answer, which might make a false impression that they figured it out by mere luck or some ingenious insight they had. But the truth is, if they told you about all their failed attempts and ideas they tried first, it would become much more obvious to you how they came up with the right solution. And it's very unfortunate that schools don't spend much time on teaching their students exactly that: experimenting with ideas and coming up with their own proofs, and techniques of coming up with such solutions (and trust me, there are such techniques, and if you know them, proving stuff becomes much easier).

There are ways of making geometric proofs fun, as being shown by people who made Dragon Box Elements. It is a video game in which you basically prove geometric theorems, but they "gamified" it in order to make it a lot of fun   You should definitely try it, or just watch the playthrough I linked. I once wrote a commentary to this playthrough on another forum about geometry, so if you're interested, I can translate it to English and post it here somewhere, so that you could understand better what's really going on in that game.

There's also a nice illustrated adaptation of Euclid's "Elements" by Oliver Byrne that I also recommend. It basically teaches you how to prove stuff in geometry with pictures and colours instead of letters and algebraic symbols (but if you know algebra already, it should be pretty easy to connect one with another ).

Phew, that's a wall of text! :q  I'll address other stuff in another post...

It was really hard for me and not what I was used to in math.

 

My brain wasn’t able to adjust and be really good at it.

[SasQ 1,376]

Please, cut the quotes to just the relevant part, we don't need to read the same stuff 30 times ;J

[Fluttershyfan94 5,742]

I've never had anything I struggled with in school actually I have always been good at my subjects. However my social skills have lacked I suppose and that is what has made school difficult or did as whenever there would be group projects I'd have trouble getting into groups. However in university I learned to better apply many of these social skills to manage a group and as such lead it towards our set goal. Yet this is something I learned with time and I no longer really struggle with school but I'm also finished with my education. I have however been thinking of getting myself a masters degree and maybe then continue towards a phd degree. So that in the future I can work as professor at a university and have a tenure as a regular lecturer there in the future that is.

[ExplosionMare 18,062]

@SasQ

Steps to solve an algebraic equation. Like, ways to group the numbers before the actual problem is solved.

[Megas 27,568]

Math and English definitely, though to just math by High School

 

[SparklingSquirrels 21,332]

Thanks for your very in-depth reply! Interestingly enough, I actually didn't mind my statistics class that I took the next year; I think mostly because we actually learned about those real-world applications, like distributions and regressions and probability models. Unfortunately I had to do maybe 90% of the learning myself as the statistics teacher was the soccer coach and was clearly only there to coach   

Another subject I had difficulty with was chemistry, specifically stoichiometry. I had hoped chemistry would be about blowing stuff up, not just another math class where we did equations all day  

Edited August 18, 2020 by SparklingSwirls

[SasQ 1,376]

1 hour ago, SparklingSwirls said:

Another subject I had difficulty with was chemistry, specifically stoichiometry. I had hoped chemistry would be about blowing stuff up, not just another math class where we did equations all day 

Hahah  This reminds me my own experiences with chemistry in school

When I was a kid, I found my parents' chemistry textbook and I was fascinated with drawings of all that lab glassware that looked like in alchemical workshop or a mad scientist's secret lab I didn't understand much, but I understood enough to get excited.

Later on my parents bought me a chemistry book for kids, with all sorts of simple "experiments" one could do safely at home, like mixing juice with oil to make rainbow strata, or mixing baking soda with water to produce bubbles. It was interesting, but not as interesting as explosions, mixtures changing colours, or some actually useful chemistry, like making drugs (no, not that drugs, I mean medical drugs that cure diseases ).

I also read a book about atoms and how they work (you know, protons, electrons, how they attract and repel, how atoms exchange electrons in chemical reactions, stuff like that). My parents kept telling me that I'm "too young for this stuff", but that never discouraged me. Quite the contrary, actually – I wanted to prove them that I'm not too young and I will understand it one way or another. And, to my surprise (and even greater surprise of my parents) – I did understand quite a lot, and I could explain it to them in simpler words, as kids do.

Then I went to school. I had to wait 7 years until chemistry classes began, but the day has finally come. I was VERY excited! Hoping to see some cool experiments, explosions, colourful smokes and flames, mixtures changing colours, and learn how to do such things myself.

MAN I WAS DISAPPOINTED!

Instead of cool reactions and glassware, the teacher just started writing equations on the blackboard Then more equations, and more...
When I asked when will we do some cool experiments, she said that we won't, because that's "too dangerous for students". So I asked if she will do that instead (being adult & all), and she said no, because... the school is low on budget and they don't have money for the ingredients and glassware ;q  She said that she can draw us some reactions on the blackboard if we wish but it won't be of any use if we won't learn how to do chemical equations first

So much for cool reactions, I guess...

My fellow students weren't as much excited as I, so they were bored and uninterested all the time, and of course they didn't get a single thing from what the teacher was writing on the blackboard. They had no idea how to do these equations. And I knew all well why: because the teacher "forgot" to tell them the most important thing in chemistry: that it's all about atoms and how they work. How they exchange electrons, how does it influence the number of other atoms they can bond with, in what conditions they can do that, etc. Without that, those equations were just abstract math to them. Just another random gibberish on the blackboard. They didn't know that behind these formulas, there are atoms doing their dance and exchanging electrons like money, switching partners they dance with, attracting and repelling each other with electric forces (yup, chemistry is basically all about electric forces), and so on... Since they couldn't imagine what's going on behind those formulas, they couldn't understand the formulas either.

I'm glad that I read that book about atoms as a kid – without the knowledge from that book, I wouldn't get anything from what that chemistry teacher said ;q  Knowing how atoms pair up with each other really helps a lot with doing those stoichiometry equations.

Another thing that helped me a lot in studying chemistry was that I avoided chemical formulas that only tell you how many atoms of each element are there in a molecule. This might be enough for doing stoichiometry, but it's definitely not enough if one wants to understand reactions, predict which reactions are possible and which aren't, or predict the properties of molecules (especially those biological ones, like drugs, hormones, enzymes and proteins), because there's multiple ways one can connect the same number of atoms of the same type, but depending on the way you connect them, or the shape of the molecule, it might have very different properties! Heck, it could even make a difference if you connect them all in the same way, just mirror-imaged! For example, our bodies can process only left-handed proteins and right-handed sugars (with the only exception of lactose, if I remember correctly). Depending on the handedness of sugar molecules, they also bend light in different ways – the very phenomenon responsible for how the LCD screen you're reading this text right now works.

Therefore, instead of molecular formulas, I always tend to draw structural formulas – the ones that not only show you how many atoms of each element are there, but also how those atoms are connected with each other, and what shape does the molecule have. This changes a lot! Because it allows you to predict the properties of a molecule, and what reactions are possible for it. Which parts of the molecule will be reactive, which ones will attract other molecules, etc. Sometimes even a flat drawing is not enough – it's better to make a 3D "sticks & balls" model to better visualise its shape. This becomes even more important with those "life molecules", like proteins etc., that can change their shapes to work like microscopic machines and perform some useful work.

Of course those molecules can become very complex sometimes (especially the biological ones). Memorizing giant complex molecules would be an overkill :q  But once you know the basic building blocks of molecules, such as hydrocarbon chains, aromatic rings, functional groups (hydroxyl, carboxyl, carbonyl, amino, phosphate, etc.) and basic structures (aldehydes, ketones, carbohydrates, nucleotides etc.), you can then just build them up like LEGO bricks into bigger and bigger structures Then you don't need to remember how is each atom connected with another – you just remember what building blocks are there and how are they connected.

For example, when you look at this complicated molecule:

it might seem intimidating at first: how am I gonna remember all those atoms and their connections!
But if you break it apart into components, you can notice that there's three phosphate groups on the left, connected to a sugar, ribose, and the other "hand" of that sugar holds a nitrogen base called adenine (which in turn is made of smaller pieces: one hexagonal ring, one pentagonal ring, and the hexagonal one has an "antenna" of amino group attached to it). Adenine connected to ribose is usually caled adenosine. Hence its name of the entire molecule: Adenosine Tri-Phosphate, or ATP in short

This way you can break apart even more complicated molecules, like acetyl-coenzyme A (AceCo-A) that your body uses to break fatty acids in the Krebs cycle:

I'm pretty sure that you already spotted the similarity between this molecule and the previous one (and if you didn't, just take a closer look on its right side). There's our old friend adenosine there!  There are also two phosphates attached, like before, but the third one is attached to the "leg" of ribose instead of its left hand The only new part is that weird chain of atoms attached to the phosphates, but you can break this one apart too! And you'll get pantothenic acid with two amino-acids in a row attached to it (one of them being alanine –  a very popular amino-acid in living organisms, and one of the simplest ones). But all of that is just there for its shape and electric interactions with the enzyme that "holds" this molecule and uses it as "tweezers" to cut and transport pieces of fatty acids attached to its end, two carbon atoms at a time (in a form of acetyl group). The HS- (thiol) group at the end is what really does the job, because it is that single atom of sulphur that grabs the acetyl from a fatty acid and takes it along into the Krebs cycle The more you know about what all those building blocks do and how they work, the easier you can understand how the entire molecule works.

But for the real fun (that is, reactions and explosions), you can check out NileRed's channel on YouTube Don't be fooled by his young age though – he's really smart! And he demonstrates a lot of cool experiments and reactions. Recently he even afforded to create his own lab instead of renting one from his school. Not only he explains the chemistry behind those reactions, but he also shows the entire process of preparing those mixtures, processing them in the glassware, purifying them at the end, safety precautions when handling with dangerous substances (however, he also shows that there's a lot of myths about those, especially about acids, by demonstrating their reaction on his own skin :J and what to do if you accidentally poured some acid on your body in order to avoid damages). He's a biochemistry student, so he often demonstrates how some popular drugs are made. In one video he showed how to convert Aspirin (acetylsalicylic acid) into Tylenol/Paracetamol (acetaminophen). Very interesting stuff

Damn, I wish I could have such a cool lab myself... :/

Edited August 18, 2020 by SasQ

[Vefka 1,499]

4 hours ago, SasQ said:

13 hours ago, Vefka said:

I love algebra (in fact it's favourite subject)

That's great to hear! Can you tell us more what is it about it that you like? That might help people struggling with algebra to see what's the real fun in it and maybe let them find some fun in it as well?

To have fun with math you need to understand it first (which is not hard, school math only covers basics). I see a lot of people just gave up and labeled themselves "I can't do math", which is so wrong! You don't need to calculate well or think fast, you just need to give it another try.

A lot of math problems are just equations and if you understood basic rules of solving them, it would be a huge milestone. I know that in US (and possibly Europe) different method is used for solving equations, so I can't really help here, but I can show how we do it here and maybe you find this method easier

So let's say we have simple linear equation: 

x+2-4=8

Our goal is to separate variable(s) and numbers. We can do it by moving them in different sides (usually variables to the left and numbers to the right). The main rule is that if you move something to the other side, you have to change the sign (-2 to +2 and vice versa). So let's move numbers to the right:

x=8+4-2

And after very hard calculations we get: x=10

Same goes for multiplying/dividing:

8x=16

Again we need pure x on the left, but the left side is 8 times bigger than we need, so we just divide both sides by 8 and get x=2 (but beware, you can't divide a side if there's + or - without brackets). Same logic goes with squares and square roots

Combine these methods and you can solve easy/medium problems! Learn a few more rules for squared equations and boom, you can solve the most of school level math problems!

[SasQ 1,376]

6 hours ago, Vefka said:

To have fun with math you need to understand it first

Putting it that way makes it kind of a chicken & egg problem though Because what if people don't understand it? :q Then they can't enjoy it either, and when they don't enjoy it, they can't learn it :q  That's why I asked what is it that makes it fun for you

Speaking for myself, I enjoy math because it's like a puzzle – it is challenging, and I like challenges and puzzles. It feels really good when you can solve a difficult problem, kinda like untangling a very convoluted knot. That moment when everything just clicks into places and you can see how simple it is now, is priceless It is also rewarding, because once you solve the puzzle, it often contains key to some other puzzle :J Or some piece of knowledge that you can use as a tool to solve other problems. Math is also like a game: it has its own set of simple rules that are always the same no matter where you are in the Universe and what time is it (which is very comforting: something that is constant in ever-changing world ) But once you learn them, you can then mix and match them in many different ways to discover new things and relations, sometimes even surprising ones. I especially love geometry because it allows me to see the math.

6 hours ago, Vefka said:

I see a lot of people just gave up and labeled themselves "I can't do math", which is so wrong!

True. The problem is that there's always something that lead people to that conclusion – usually the mental struggle they felt when being taught maths the wrong way, by someone incompetent, or when they got lost in the middle of a lecture. They conclude that they "can't do math" because that's how they feel at the moment. It's just that this is not true as a general statement, because once they learn how to do it the right way, the can in fact do it, and they usually can have a lot of fun with it too I know because I tutor people pretty often, and I taught many people who thought of themselves that they're a "lost cause" (often because that's what their teachers told them ), but once I showed them how to do it in a fun way, they started enjoying it again and recovered their confidence once they saw that they can in fact do that.

6 hours ago, Vefka said:

You don't need to calculate well or think fast

True Computers can crunch the numbers much faster than us and with no mistakes, and yet they're very poor mathematicians
Too bad that schools concentrate so much on "plug & chug" approach and solving word problems instead of experimenting, discovering, looking for patterns, logical & creative thinking, looking for different ways of solving problems. Because that's what mathematicians really do. (Sometimes they even solve problems that people didn't have yet )

6 hours ago, Vefka said:

I know that in US (and possibly Europe) different method is used for solving equations

Well, there are different methods of teaching, but math is universal – it works the same way in the entire Universe. (Which is a good thing, because you can talk to aliens with math ;> )

6 hours ago, Vefka said:

but I can show how we do it here

Where is "here", if I may ask? :>  (I'm in Europe, BTW. Poland, to be exact.)
The only differences I noticed is that we write "ln" for the natural logarithm instead of "log", and we used to write "tg" and "ctg" for tangent and cotangent, but it slowly starts to change towards "tan" and "cot" as in the rest of the world. (However, I will never let them take away my ln's ;J )

6 hours ago, Vefka said:

Our goal is to separate variable(s) and numbers.

Calling them "variables" and "numbers" might lead to some confusions later on, here's why:
Those "letters" stand for some numbers as well, it's just that we don't know as of yet what those numbers are. That's why I rather call them "unknowns". Calling them "variables" don't quite work either, because you can have letters in your formula that actually represent constants – numbers that can't change. Or sometimes you rewrite a formula to express one unknown in terms of other unknowns. Which is perfectly fine! You don't have to know all of them at once, and sometimes you even can't: for example, when solving a system of simultaneous equations, with multiple unknowns. You solve it for one of the unknowns, but you still have all the remaining unknowns on the other side of the equation, and you have to use some other tricks to discover their values later on.

6 hours ago, Vefka said:

We can do it by moving them in different sides

That's another thing that may confuse students, because they may think that they can solve equations by just moving symbols around at random. Some time in their lives (if they survive in math long enough) they may be really surprised to discover that sometimes you can't just move things around – some operations are not commutative. (Yes, you heard it right: there are times when a·b is not the same as b·a ). Also, sometimes moving things around won't do them any good – it may even make things worse :q

That's why it is very important to understand WHY can we move some things around in the equation, and what moves bring us towards the solution (because some moves can take us away from it).

6 hours ago, Vefka said:

The main rule is that if you move something to the other side, you have to change the sign

Ha! But WHY though?!  That's the thing that confused most of the students before they came to me. It sounds very much like an arbitrary rule that just happens to work, without any explanation WHY does it work. And why don't we do the same when dividing, or cancelling powers?

Instead of learning multiple arbitrary rules, you just really need to know ONE rule (to rule them all, and in the darkness bind them, muhahah ):
That whatever you do to the equation, you have to do the same thing to both sides, in order to keep it balanced.

Equation is like a balance scale: there might be different stuff on one scale than on another, and some things might be hidden in boxes (unknowns), but whatever is there on the left scale, must "weigh the same" as the stuff on the right scale. You can't just add some more stuff on one scale without adding the same amount of stuff on the other, because that would break the balance. If you add or take something to one side, you have to add or take the same amount of stuff from the other side.

There's even a reason why (interestingly, I found it in Euclid's "Elements" – a book about geometry, not algebra): In one of the "common notions" he says that when you have two equal things, and to each of them you add (or subtract from) the same amount, then the two results will remain equal. In other words, if the scales were equal to begin with, and you add the same amount of stuff on both sides, they will remain equal afterwards.

Only those operations that keep the equation balanced are allowed, because we want the equation to remain true.

So when you "move" some number from one side of the equation to another, you're not really "moving" it: you subtract it (take it away) from one side, so it disappears there, but you have to subtract it from the other side as well to keep the balance. Therefore it seems that a negative counterpart of the original number has just appeared there out of nowhere. But in reality, you just subtracted the same amount from both sides.

6 hours ago, Vefka said:

Again we need pure x on the left, but the left side is 8 times bigger than we need, so we just divide both sides by 8 and get x=2

Ah, so I see that you know that rule here (for division), but probably didn't noticed that it's just the same rule for addition/subtraction

It's good that you also explained WHY do we need to divide this time, because that's another thing people often have trouble with: they don't quite know what rule should they apply next. I guess that's what @ExplosionMare's problem was when she said that she have difficulties if there's more steps involved – some moves may push you farther from the solution, while others bring you closer to it. And the way to know which one you should use is this:

You look at what operation was used to produce that complex expression, and you apply its "counter-spell" to reverse that operation and make it simpler again

For example, if our unknown is multiplied by 8 in our equation, we divide both sides by 8 to "undo this spell" (division is a counter-spell for multiplication, and vice versa). If there's something added to our unknown, we subtract that thing from both sides (because subtraction undoes addition, and vice versa) to get that unknown alone again. If the expression on one side is squared, we undo this by taking the square root of both sides (because square roots undo squares, and vice versa – they are like "fractional powers", so if someone raised something to the 2nd power, we can undo this by raising it to "1/2-th power", which is the same as taking the square root).

Every mathemagical spell has some counter-spell Subtraction undoes addition, division undoes multiplication, taking n-th roots undoes raising to n-th powers, logarithms undo exponents, integration undoes differentiation, etc. And the key is to use the right counter-spell that will make the equation simpler, not harder After all, our goal is to discover the value of one of the unknowns. And the way we can learn its value is by untangling the equation in such a way that we had that single unknown alone on one side of the equation, while everything else (either known or unknown) on the other side. Because then, we know that this particular unknown is "the same as that other stuff on the other side", so we've basically just discovered what it stands for And of course we don't want it to look like this:
    x = 3x + 2
because then, in order to know what x is, we would have to know how much 3x is, so we would have to know x to begin with

There are some rare occasions when that's actually a good thing, though. For example, if you have something like this:
   x = √[1 + x]
you know that x is equal to whatever there is on the right side, so you can take that entire right side and substitute it for x inside of that right side and get:
  x = √[1 + √[1 + x]]
then you do that again:
  x = √[1 + √[1 + √[1 + x]]]
and again:
  x = √[1 + √[1 + √[1 + √[1 + x]]]]
and... you get the idea ;>
  x = √[1 + √[1 + √[1 + √[1 + ...]]]]
And while at first it may seem that you made things much worse, it's actually a nice "formula" that tells you how to compute a certain number that the ancient Greeks loved so much :> If you take your calculator, start with 0, add 1, then square-root it, then add 1 again and square-root it again, and add 1 and square-root it again... and repeat it enough times, you will soon start noticing that the digits are starting to settle on a particular value: 1.61803398…  also known as the Golden Ratio How cool is that?

So as you can see, you don't always have to actually solve an equation in order to figure out the value of the unknown. Sometimes you can leave it just partially solved, and use it as an algorithm for computation. That's pretty much how Archimedes calculated the value of π in the ancient times.

6 hours ago, Vefka said:

but beware, you can't divide a side if there's + or - without brackets

Hah, I thought you're going to warn about something much more dangerous (if done carelessly): dividing by something that might happen to be 0 When you divide by zero, weird things may ensue. You can even prove that 1 = 2 (which is obviously false)

As for dividing without brackets – well, actually you can, as long as you divide every term by the same thing. For example:
     6x + 8y = 20
you can divide both sides by 2, which is a common factor here, as long as you divide every term by 2:
    (6/2)·x + (8/2)·y  =  20/2
       3x + 4y = 10
This is really a "shortcut" for first extracting that common factor from each term and putting it outside of a bracket, and then dividing by it, like this:
      6x + 8y = 20
     2·3·x + 2·4·y = 10      (noticing the common factor)
     2·(3x + 4y) = 2·10      (putting it outside of the bracket, which is just the distributive law in reverse)
         3x + 4y  = 10        (dividing out both sides by 2)
But that would be too much of a hassle for such a simple thing, and once you get the feel of how it works, you can shortcut it that way.

6 hours ago, Vefka said:

Same logic goes with squares and square roots

Yeah, as long as they're already the outer operation (a.k.a. a perfect square), like this:
     (x+1)² = 2       (everything is squared on the left, so we can take the square root of both sides to undo the squaring)
      x+1 = √2
However, when you have something like this:
    x² + 3x = 4
things might get a bit tricky And even more tricky when you have this:
    x³ + 6x² = x + 30
or this:
    x⁵ + 15x⁴ + 67x³ + 45x² = 284x + 420
That's where the real fun begins :> (and where algebra meets geometry)

6 hours ago, Vefka said:

Learn a few more rules for squared equations and boom, you can solve the most of school level math problems!

True, because schools rarely come out of the comfort zone of linear and quadratic equations. (They used to when I was a student, but they don't do that anymore these days, even quadratic equations are slowly disappearing from the curriculum, which is very sad: even ancient Babylonians could solve such problems 4000 years ago, and we are no worse than them, are we? :q )

Edited August 19, 2020 by SasQ
Confused left and right, always does %D

[Splashee 28,563]

2 hours ago, SasQ said:

6 hours ago, Vefka said:

...

Putting it that...

That was the longest reply in history, even broke my poor Android phone scrolling finger.

 

 

I would say, reading loud to the class was the worst, always been

[Vefka 1,499]

5 hours ago, SasQ said:

Well, there are different methods of teaching, but math is universal – it works the same way in the entire Universe. (Which is a good thing, because you can talk to aliens with math ;> )

Of course! But there's different... steps' names in methods. You make up different algorithms with using the same fundamental rules. 

I read some articles about how we and US people solve problems and indeed our algorithms are quite different, but if you look deeper, you'll see that they're essentially the same, yes. But for different people different algorithms are understandable, that's what I meant

5 hours ago, SasQ said:

Where is "here", if I may ask? :>  (I'm in Europe, BTW. Poland, to be exact.)
The only differences I noticed is that we write "ln" for the natural logarithm instead of "log", and we used to write "tg" and "ctg" for tangent and cotangent, but it slowly starts to change towards "tan" and "cot" as in the rest of the world.

In Russia we use "tg", "ctg" and "ln" and don't even know that different spelling is possible :>

5 hours ago, SasQ said:

That's another thing that may confuse students, because they may think that they can solve equations by just moving symbols around at random.

But they can

Not at random of course, the goal is to separate desired variable from other stuff. If it wasn't reliable method, it wouldn't be the standard (yep, we do it this way even in universities) for the entire culture.

Until 10th grade I was just moving things from side to side without understanding. But was it fun? Of course! I was solving puzzles and it doesn't matter how I solve it. Was it difficult? Hell no! I passed every algebra exam with excellent mark!

5 hours ago, SasQ said:

Ah, so I see that you know that rule here (for division), but probably didn't noticed that it's just the same rule for addition/subtraction

No, no, of course I understand it

It's just much more convenient for me to move things. I was taught like this from the beginning and it's the same simple true as 2+2=4

Let's put it like this:

x+8=4

We need our precious x alone, but +8 is stopping us. What can we do? Counterattack +8 with -8 of course! But equation called like this for the reason, so you have to apply -8 for the other side as well and we get:

x=-4

I agree that this is better way of explanation, but practically I'd just move 8 to the other side and change the sign to the opposite

5 hours ago, SasQ said:

As for dividing without brackets – well, actually you can, as long as you divide every term by the same thing. For example:
     6x + 8y = 20
you can divide both sides by 2, which is a common factor here, as long as you divide every term by 2:
    (6/2)·x + (8/2)·y  =  20/2
       3x + 4y = 10
This is really a "shortcut" for first extracting that common factor from each term and putting it outside of a bracket, and then dividing by it, like this:
      6x + 8y = 20
     2·3·x + 2·4·y = 10      (noticing the common factor)
     2·(3x + 4y) = 2·10      (putting it outside of the bracket, which is just the distributive law in reverse)
         3x + 4y  = 10        (dividing out both sides by 2)
But that would be too much of a hassle for such a simple thing, and once you get the feel of how it works, you can shortcut it that way.

That's true, but I still would recommend putting common factor outside of the brackets. This way it's more clear, so many people I know fell in this pitfall and thought that you can divide "2+4x=8" by 4 and get "2+x=2"

5 hours ago, SasQ said:

That's where the real fun begins :> (and where algebra meets geometry)

And that's were school never really goes

The maximum we got in mandatory program are double-squared equations which are just square equations inside square equations. Can be solved with simply introducing new variable

5 hours ago, SasQ said:

True, because schools rarely come out of the comfort zone of linear and quadratic equations. (They used to when I was a student, but they don't do that anymore these days, even quadratic equations are slowly disappearing from the curriculum

Oh, so you say squared equations are disappearing from schools? That's interesting to know. In Russia (and likely in CIS countries) such equations are mandatory and included in basic program (1-9 grades after which you can leave school) and if go to the last two optional grades, you have to learn the basics of math analysis (logarithms, trigonometry, advanced actions with functions, etc). And it doesn't matter if you go to literature or chemistry-biology in-depth class, you have to learn it anyway (just not so in-depth). Unfortunately it goes both ways and I have to learn semi-advanced social studies, history, etc 

[Evil Pink One 13,506]

Hmmmmm being normal...

Spoiler

 

[Kyoshi Frost Wolf 41,349]

Almost everything. There were the struggles with the typical subjects. Math, was horrible with damn near anything above pre-algebra, history mostly because it was usually US history and I couldn't care less about the subject anymore. Wasn't great at homework because I wasn't good at doing it at home. I wanted to actually live my life out of school so I usually tried to do any of that work at school. Then there were tests. Wasn't my cup of tea either. Hardly aced tests, was usually middle of the road or lower.

Another thing I was bad at? The entire social aspect. Other students always stressed me out and if I ever tried to be social, it would usually be awkard and people would think I am a moron. I guess that's not incorrect. School overall wasn't a good experience for me. By the time I was in my junior year in High School, I started to become quite depressed and riddled with anxiety.

[RaraLover 10,414]

The biggest thing for me was English. I never did good in it, but always tried my best. I think I'm just not good enough at writing essays. 

[TBD 17,254]

Anxiety. Ever since what happen in that one unforgettable semester in college, I don't think I could ever recover from that anytime soon.

[Mlppotterwhoovian 381]

Math above the 7th grade level,it all started to look like gibberish lol!

Also i tended to be anxious and i also wanted A-A+ grades and cried when i got anything lower,thinking i was a failure 

Oh and some of the crazy science formulas gave me nightmares

 

[Mitternacht 45]

The topic that I struggled with by far the most was any Math related class. My learning style is very much hands on, a visual learner if you will, however I excelled in History and Government so memory wasn't the issue. Numerically, my mind had trouble grasping mathematics specifically in how equations were assorted. Algebra II was when I began to really lag behind because we worked more frequently with letters and the problems became larger and more complex, like long division and such.

Years later I would drive myself to understand such problems in my own time as seeing the answers in a text book, then looking at my work in contrast never truly aided me. There are so many learning styles out there and if a subject doesn't come naturally to someone or they find the subject not interesting, learning can be a challenge, certainly! Plus, I was easily distracted in math classes. The subject never caught my attention. Perhaps it was a perspective issue for me..

The only reason why I passed was because I was taken off of cross country practice until I sustained a passing grade! - and I loved running, especially with my friends. Ultimately, that was my motivation in trying to understand math.

[Vefka 1,499]

My own language. Literature language and everyday language are two fucking different universes and I hate it so much

[The Kaeya Simp 13,988]

1 hour ago, Vefka said:

My own language. Literature language and everyday language are two fucking different universes and I hate it so much

I had the same problem but with english (but then again my accent is different from most accents here cause we like being different lol)

[King of Canterlot 9,603]

I remember when I was in elementary and middle school I used to struggle a lot with my math classes. It's not that I couldn't understand what was being taught, it was more that it took me longer to get the hang of new material that we were learning. I was put into slower paced math classes in both middle and high school to help give me more time to understand and master the new material that was introduced, and it actually did help a lot. By the time I was a Sophomore in high school I felt that I was grasping the concepts a lot easier and even a little faster than I had been in previous years, and that feeling carried over to my Junior year as well   

[Astralshy 26,441]

The topic I had the most problems with it was English.  I got that covered when in middle school I started to watch cartoons (and later movies) in original English language like Disney’s gummy bears for example. That is now 20 years ago and I can say it was very good starting point for improving my passive English. With the other subjects I had little to no issues.

The most problems I had in school were of being the underdog in class because of living in a small town where my mother came from a big city. I got that also covered with time, but this took more time and effort.

[Cash In 22,230]

I wasn't bad at it, but I definitely struggled with Math. Oddly enough, I hated English more and that was my best subject.

[Clawdeen 12,189]

Tbh I struggled in anything that wasn’t English due to zoning out in class all the time.

[BritishBrony2012 796]

Does have to be a subject? most subject I'm okay cos I'm there to learn but my problem is socialzing, follow students, few teachers and dinner ladies. Kids or called students can be judgmentle, had people give me hard time for being me, and even got push once by a guy who hang out with his friends and even got kick when I try to run to class. Don't understand somw people there two guys remeber talk with them and get on well, I did this bump into their shoulder like messing around but they don't like it thought how friends does that thought we friends. Some girls give hard time also and one say to someone that she doesn't like me that why she give me a hard time. Don't get these people. Also there horrible teachers and also useless ones and one dinner lady is a b****, even she did give problem twice. That how much I can tell here, I can write a blog or book on it.

 

As for subject, Physical Education It about being in team/group/pair, most I get blame for lost, pick last, been critics and even, when tackling in football/soccer people think it's foul/bad play. excerise, yeah right how can you when been making fun of.