History UsefulNotes / Mathematics

If you get to higher math, like calculus, you'll even find yourself butting up against the intellectual capacity of your calculator. They only take so many digits at a time, use only so many decimal places before saying "that's enough". In higher math calculations, these flaws can add up, ending in the wrong answer -- and the RealLife problems calculus handles can be expensive, destructive and even deadly when the math is done incorrectly.

Just having the right answer isn't the point. It's [[ItsTheJourneyThatCounts you understanding how you get there]], which is what your teacher wants to see.

You'll see this idea here and there. Is it true? Kinda yes and kinda no.

Yes: math -- as a pattern -- helps us understand our universe through science. Math helps us rationally understand and prove science.

No: Math is a pattern that closely follows reality, but we don't know ''for sure'' if reality follows math or math follows reality. Oddly enough, math has models (asymptotes) that show that some knowledge can be increased for eternity yet never get to full understanding. That's deep into philosophy, that discussion.

[[EverybodyHatesMathematics It is also hated by roughly 99.9% of all people]]. This is the major reason that there aren't more scientists, doctors, engineers, or Wall Street workers. This is probably due to the way it is taught: 6 to 8 years of number crunching, most of which a calculator can do faster and with less chance of error, followed by massive, abstract generalizations (algebra and calculus). To make things worse, students are rarely told ''why'' they need to learn the current topic: science teachers tend to wait until after you've learned the math to show you the uses for it, and a lot of math teachers focus on the techniques rather than the applications. (Or they try to demonstrate the applications by assigning word problems--without realizing that "translate this word problem into math" is ''also'' a skill that needs to be taught, and that assigning said problems to students who don't have that skill won't help.) Worst of all, many elementary school teachers are poorly trained in mathematics, so they don't know good problem-solving techniques, they don't know the particular real-world applications of any given topic, and they don't ''like'' it enough to teach math for its own sake.

So... when you budget your money before going to the grocery store, or design an airplane or a building, or figure out how to launch a spacecraft to Mars you're simply using math to model a possibility to see if everything works without actually doing it yet.

Mind you there are thousands of different kinds of real-world tools for thousands of different purposes - math is no different. Math is one big, messy toolbox of both simple and complex tools.

* Some tools have no immediately obvious use at first glance, like a home computer to TheEveryman pre-internet.

** In precalculus you learn a lot of stuff about functions so you can understand what Calculus simplifies, but doesn't replace.

What makes a math class so dull and "when am I ''ever'' going to use this???" is that they are teaching every tool and every use, not just the ones specific or obvious or even helpful to you - because no one knows what you'll need in life. For that, you have to first figure out where you want to go in life and then [[ItWasWithYouAllAlong figure out how you will use math]] to achieve that purpose. Everyone does use some level of math down every path, from housewives to scientists.

The patterns build on other patterns, making a very simple or a very complicated tapestry - and it works.

[[UsefulNotes/MathCurriculum On this page there's a step-by-step list of all the things you learn in school which lead to complex math]], starting from the 1-2-3's and up to calculus.

You will cover everything on the math path even though when you eventually get a job you'll find there will be huge areas that you'll never actually use in that job. So why did you learn it? Math classes are for general consumption. You may never need long division or trigonometry in that job... BUT... suppose ''you're'' the one who figures out how to apply those areas to the job opening up new horizons to everyone and you end up well known and awarded for your EurekaMoment? You may not have figured it out if you hadn't first dipped your toes in the "unrelated" math!

Mathematics has '''concepts''', just like every other subject, but it also has a DifficultySpike in '''calculation'''. The problem with math classes is you can grasp all the concepts but still stumble on the calculations - and your math teacher will mostly be looking for calculations on exams, ''because it shows you understood and applied the concepts''.

On a non-math, interpersonal level the concept of kindness towards others is easy to grasp. Actually doing it when a person you completely hate needs an act of kindness? ''That's the hard part.''

Math and UsefulNotes/{{Science}} both have the same problem. The "cool stuff" used to generate interest in the subjects has little to do with the nuts and bolts of actually working the subject, and that "nuts and bolts" is boring old calculation. Math has playing with mobius strips and fun logical-thinking puzzles, and science has Mentos geysers and other flashy demonstrations. The problem is when you actually get into the math or science class what you will be doing in class is crunching numbers and learning how different equations were nailed together and simplified to create new equations. Nowhere near as fun, but a thousand times more helpful to math and science for opening up new horizons and thus leading to new flashy demonstrations.

This is a question your math teacher will roll their eyes at and often say something about how if you find yourself without a calculator you'll know how to find the answer. To be fair to you, we ''do'' live in a world where computers and calculators are everywhere. It is unlikely that in everyday life you'd find yourself without one ''and'' need to do higher math in an emergency.

However: a calculator is just DumbMuscle doing the heavy lifting in math. There's more to math than calculating, which is all a "[[ExactlyWhatItSaysOnTheTin calculator]]" does. For example:

If you get to higher math, like calculus, you'll even find yourself butting up against the intellectual capacity of your calculator. They only take so many digits at a time, use only so many decimal places before saying "that's enough". In higher math calculations these flaws can add up, ending in the wrong answer - and the RealLife problems calculus handles can be expensive, destructive and even deadly when the math is done incorrectly.

Just having the right answer isn't the point. It's [[ItsTheJourneyThatCounts you understanding how you get there]] which is what your teacher wants to see.

Believe it or not, for many people math is a hobby. You don't even need to buy anything or go anywhere.

Yes: math - as a pattern - helps us understand our universe through science. Math helps us rationally understand and prove science.

No: Math is a pattern that closely follows reality, but we don't know ''for sure'' if reality follows math or math follows reality. Oddly enough math has models (asymptotes) that show that some knowledge can be increased for eternity yet never get to full understanding. That's deep into philosophy, that discussion.

** Sadly the best way to learn math is to roll up your sleeves and do it with paper and pencil even when you don't want to. Many have tried to find a better way, but all have failed at that.
** If you get the wrong answer but have ''shown your work'' on how you got there, the teacher may see that you got concepts correct even if the input was wrong, and give you some credit.

** Before the new millennium, teachers did not like calculators in classes - these teachers learned without them and felt that practice made a good math mind. Now they're accepted in higher math as it's unlikely to find yourself without a computer of some sort. Before scientific calculators made trig easy, students had to have tables for the sine, cosine and tangent! But don't worry (or do!) - there are lots of things a teacher can ask you to calculate that a calculator can't do - like solve for x, or giving an exact fractional answer, and the old standby: "show your work". A calculator will help tremendously if you can't do the more complex basic operations in your head, though!
* When you make a mistake on your homework, don't crumple up the paper and start anew - find the problem and note it on the paper to the teacher where and what you did wrong. Then do it correctly and show your work. That's the kind of thinking math is trying to foster, not simply "getting the right answer".

* On a test, look at a word problem and circle the important things. Underline the actual question that needs to be answered - it may be buried in the question!
** If you list the total cabbages that AliceAndBob bought individually at the store and [[FailedASpotCheck not notice]] that the question actually wants to know how many 300-cabbage loads it will take to transport them in Bob's car, ''you've actually stopped a step short of the true answer and you'll to be wrong''.

** If you go to any Ivy-league math class with the very tip-top minds, you'll see even the professors making simple mistakes - missing a negative sign, skipping a step, forgetting to carry a one, using the wrong equation. If they can do that AND get a doctorate, you can surely pass a low-level class such as algebra.
* Don't compare yourself to people who seem to have "math brains". Yes, there are some who simply ''get it''. But that doesn't make them any better or worse than you. Classes ''aren't'' weeding people like you out - people weed ''themselves'' out. '''All of math is logical, work at it and you'll get it.''' Just because you can't rattle off the factors of 256 without thinking about it first doesn't mean you won't make a fine mathematician.

* If you multiply a number by another you're lining up a tile pattern on a number of rows up and down by a number of rows side to side and then adding them all up.

A pretty pattern of floor tiles is the same all the way across. If they're all square, they'll still be square on the other side. '''The tools of math are patterns, just like that.'''

The patterns math looks for help us make sense of the mess of life.

* If you add two unequal numbers the answer will ALWAYS be higher than either of the two numbers.

* If you multiply a number by another you're lining up a tile pattern on a number of rows by a number of rows and then adding them all up.

* If you add two unequal numbers the answer will ALWAYS be higher than either of the two numbers. ''That's a pattern''.
* If you subtract two unequal numbers the answer will ALWAYS be lower than either of the two numbers. ''That's a pattern''.
* If you multiply a number by another you're lining up a tile pattern on a number of rows by a number of rows and then adding them all up. ''That's a pattern.''

Patterns build on patterns, making a very complicated tapestry - but it works.

One concept in math ''really does'' lead to another, though it's hard to see when you're actually on it.

If you're looking for a specific "useful notes" about a part (tool) of math Like "What's multiplication?", that page is where to go.

You cover everything on the path even though when you eventually get a job you'll find there will be huge areas that you'll never actually use in that job. So why did you learn it? Math classes are for general consumption. You may never need long division or trigonometry in that job... BUT... suppose ''you're'' the one who figures out how to apply those areas to the job opening up new horizons to everyone and you end up well known and awarded for your EurekaMoment? You may not have figured it out if you hadn't first dipped your toes in the math!

What makes math classes so dull and "when am I ''ever'' going to use this???" is that they are teaching every tool and every use, not just the ones specific or obvious or even helpful to you - because no one knows what you'll need in life. For that, you have to first figure out where you want to go in life and then [[ItWasWithYouAllAlong figure out how you will use math]] to achieve that purpose. Everyone does use some level of math down every path, from housewives to scientists.

Only in class is there a time limit. Many people find math much more enjoyable when they are doing it on their own, at their own speed, for whatever end they choose.

Believe it or not, for many people math is a hobby. You don't need to buy anything or go anywhere.

The point is we can figure out how many apples Alice will have if that happens ''without ever actually moving any apples in reality''. All of math is doing simply that. When you budget your money for going to the grocery store or, design an airplane or a building, or figure out how to launch a spacecraft to Mars you're simply using math to model a possibility to see if everything works without actually doing it yet.

The point is we can figure out how many apples Alice will have if that happens ''without ever actually moving any apples in reality''. All of math is doing simply that, no matter how complicated the math is.

You learn everything on the path even though when you eventually get a job you'll there will be huge areas you'll never actually use in that job. So why did you learn it? Math classes are for general consumption. You may never need long division or trigonometry in that job... BUT... suppose ''you're'' the one who figures out how to apply those areas to the job opening up new horizons to everyone and you end up well known and awarded for your brilliance? You may not have figured it out if you hadn't first dipped your toes in the math!

What makes math classes so dull and "when am I ''ever'' going to use this???" is that they are teaching every tool and every use, not the ones specific or obvious or even helpful to you - because no one knows what you'll need in life. For that, you have to first figure out where you want to go in life and then [[ItWasWithYouAllAlong figure out how you will use math]] to achieve that purpose. Everyone does use some level of math down every path, from housewives to scientists.

* Some tools are outdated and or simply not often used anymore, like lamp lighting or a telephone switchboard operator, but will still work if needed.

* Some tools are outdated and not often used anymore, but still work.
** A an abacus has been replaced by a slide rule which has in turn been replaced by a calculator, but abacuses and slide rules still work. In precalculus you learn a lot of stuff about functions so you can understand what Calculus simplifies, but doesn't replace.

So what IS mathematics? Mathematics is a ''tool'', just like how [[Administrivia/TropesAreTools a trope is a tool]].

...That's it, full stop. Mind you there are thousands of different kinds of real-world tools for thousands of different purposes - math is no different. Math is one big, messy toolbox of both simple and complex tools.

Math and UsefulNotes/{{Science}} both have the same problem. The "cool stuff" used to generate interest in the subjects has little to do with the nuts and bolts of actually working the subject, and that "nuts and bolts" is boring old calculation. Math has playing with mobius strips and fun logical-thinking puzzles, and science has Mentos geysers and other flashy demonstrations. The problem is when you actually get into the math or science class what you will be doing in class is crunching numbers and learning how different equations were nailed together and simplified to create new equations. Nowhere near as fun, but a thousand times more helpful to math and science for opening up new horizons and thus leading to new flashy demonstrations.

Mathematics is a tool that serves as a model of reality.