Approaching Mathematics from scratch as a hobbyist, progression and mindset advice

Peas

Hey folks Mathematics have always been a sore point throughout my life but back in my mind I know that I am missing out on something deep which is beyond my intelligence and understanding. That said, I have been watching a lot of science videos recently which really inspires me to learn about the nature of the universe but I pretty much run into the wall of being clueless about math and physics.

I started learning Math on Khan Academy a couple of weeks back and it is really embarassing to admit that I am already starting to fumble with basic Arithmetics from three digits onwards and even really basic concepts took me pretty long to process in my mind.

12+19=__+20

I think I get the concept but just the thought of processing the numbers drains the heck out of me, not kidding lol. One thing I am also not sure about is if I am thinking in the right way, like how do you folks actually do it? Am I suppose to be already visualizing stuff at the back of my mind at this point? What kind of questions should I be asking myself? Also how comfortable should I be with my fundamentals before I move on to more advanced stuff? I don't even know what is considered "basic".

If anyone have anything to share please feel free to talk about it.

Enc

I'm not a math person, though I use it often in my work. There is no shame in having to write out answers to get to them, lots of folks have to do that. Solving things in your head is really a matter of pattern recognition and rote practice. If you have solved your problem ten thousand times over your career you'll probably have an easy time just identifying the correct number rather than writing out the solution. I regularly do notepad calculations for harded multiplication and division situations, much less formulas, and so do most folks I know who work regularly with numbers.

Math progression in terms of MOOC (massive online open courses, aka free college courses) classes tends to go:
Finite mathematics -> PreAlgebra->Algebra->Trigonometry and PreCalculus (as parallels) -> calculus 1 -> college physics 1 with calculus and calculus 2 as parallels -> college physics 2 with calculus and calculus 3 as parallels -> fancy exotic stuff you won't need probably for 2-3 years of study.

You can find a number of MooCs here: https://www.edx.org/course/subject/math

zepherin

Try this visualization trick. Focus on the singular digits not the whole equation in the above instance. 2+9 is 11 so the last digit is going to end up being 1
1+1+1 is 3. On the other side 1-0 is 1 and 3 minus 2 is 1.

Heffling

This is algebra, and there are a few basic things you do to solve the equations. The first we'll look at is simplifying. In simplifying, what you want to do is perform an operation that doesn't change the value of your equation. From your example:

12 + 19 = X + 20 Then we will simplify the left hand side, which is 12 + 19 = 31. So now we have:
31 = X + 20

The other big thing is equalities. That is one, one thing equals another thing. So if you apply a mathematical operation to one, you must apply the same operation to the other. For example:

31 = X + 20

So if we want to know what X is, we can subtract 20 from both sides as follows:

31 - 20 = X + 20 - 20, which then simplifies using 31 - 20 = 11 and 20 - 20 = 0, giving us:
11 = X + 0, then we can drop the 0, leaving us with:
11 = X

Now, if you want it to appear cleaner, you can swap the sides. So you get:
X = 11

It may not be intuitive that you can just swap the sides, so we can walk through the process as follows, starting from:
11 = X Now we want to subtract 11 from both sides, giving us:
11 - 11 = X - 11, simplify
0 = X - 11, now we want to subtract X from both sides:
0 - X = X - 11 - X, simplify
-X = -11. Now multiply both sides by -1:
-1 * -X = - 1 * -11, simplify
X = 11

You've just proven that you can always swap the sides, because the operations we're performing are equal.

From a learning perspective, if you're just learning how to do algebra or some other mathematics, write everything down and do the work step by step. Don't combine steps, and don't skip straight to solutions. It seems much easier to do so, but this will result in mistakes. Once you have practiced doing the work for a while, you can start to combine and think about visualizing answers. This is why I wrote everything down in a step by step manner above, because you can easily follow the work and if a mistake is made it makes troubleshooting much easier.

As far as fundamentals go, do some practice problems and if you feel comfortable move on. If you get stuck, ask here for help. What you shouldn't do is feel obligated to be totally comfortable, though. A lot of times in learning math, my teachers would move ahead before I totally had internalized a process, but things would click because you keep using that process as you go forward.

Powerpuppies

You probably want to do dozens or hundreds of practice problems for each school year's worth of study. You need to be able to do algebra without thinking about it at all in order to start calculus. You can't think about the calculus you're learning if you're still thinking about the rules of exponent's multiplication.

Heffling is right for moving on between topics within a course, but you don't want to embrace a general practice of moving through topics quickly in mathematics until somewhere around abstract algebra.

Heffling

I think the most powerful tool for me to do most everyday math in my head was learning to use the F.O.I.L. method and finding nearest values when multiplying. So FOIL is a way to multiply two terms of the form (X + A) * (Y + that stands for First, Outer, Inner, Last.

So, if you want to multiply out (X + A) * (Y + , you could do this long-hand form as follows:
(X + A) * (Y + simplify by multiplying out the X + A term
(X) * (Y + + (A) * (Y + then simplify again as
(X)*(Y) + (X)*(B) + (A)*(Y) + (A)*(B)

Now, if you look at these terms in part, (X)*(Y) is what you get from multiplying the First part of each of the terms, (X)*(B) is the multiple of the Outer parts, (A)*(Y) is the Inner parts, and (A)*(B) is the Last parts.

So how is this useful in everyday life? Well, let's suppose you need to multiply 53 times 97 and there's no calculator handy. You could write it out long form, but you can also do this in your head. If you recognize that 53 is the same thing as 50 + 3, and 97 is 100 - 7, then you can approximate the answer as 50 * 100 = 5000. Or you can use FOIL as follows:

(50 + 3) * (100 - 3) =
(50 * 100) + (50 * -3) + (3 * 100) + (3 * -3) =
5000 - 150 + 300 - 9 =
5141.

So you're turning two odd numbers into a series of easily manageable numbers. This lets you use your basic 1 to 10 multiplication tables to handle most numbers.

Heffling

Also, 97 is 100 - 3, not 100 - 7. This is why you write things out!

Antinumeric

Peas wrote: »

Hey folks Mathematics have always been a sore point throughout my life but back in my mind I know that I am missing out on something deep which is beyond my intelligence and understanding. That said, I have been watching a lot of science videos recently which really inspires me to learn about the nature of the universe but I pretty much run into the wall of being clueless about math and physics.

I started learning Math on Khan Academy a couple of weeks back and it is really embarassing to admit that I am already starting to fumble with basic Arithmetics from three digits onwards and even really basic concepts took me pretty long to process in my mind.

12+19=__+20

I think I get the concept but just the thought of processing the numbers drains the heck out of me, not kidding lol. One thing I am also not sure about is if I am thinking in the right way, like how do you folks actually do it? Am I suppose to be already visualizing stuff at the back of my mind at this point? What kind of questions should I be asking myself? Also how comfortable should I be with my fundamentals before I move on to more advanced stuff? I don't even know what is considered "basic".

If anyone have anything to share please feel free to talk about it.

I personally find mental arithmetic incredibly difficult, I struggle with problems such as the one you laid out. However I consider myself reasonably good at maths, I got an A in A-level maths (calculus, probability/statistics, mechanics) and had success with optional maths modules at university (linear algebra, group theory)

I think a key thing is to realise that a lot of maths has very little to do with arithmetic which may be where you struggle. Don't be afraid to jump to things that seem advanced as you'll find the numbers don't matter as much as the ideas.

Try watching some numberphile or 3blue1brown videos, and see if you can grapple with the ideas they are talking about. You don't have to understand everything, or get it first time. Just try and get an idea of what about maths interests you, and how you can do it.


For your specific problem, I find it much easier to approach by turning the numbers into letters,

X + y = w + z

So I don't get anxious over the numbers, then just use a calculator once you've reorganised it.

X + y - z = w.


I wish you luck and hope this helps.

Aldo

I used to suck at maths due to having a really bad teacher and no intrinsic motivation to actually study anything. When I had to pass a few classes later in life I found that I could only learn something when I could apply it directly to a question I was curious about (statistics) and I could only understand it if it was visualized in a way that made sense to me. The abstraction level of some methods just never clicked for me. My high school book tried to explain everything related to statistics by way of drawing marbles from vases. Everything.

One other thing that helped me is to read about the person who discovered a mathematical principal. Just to realize that some human actually figured something out and managed to prove to their peers that it made sense. When you get right down to it: it's all humans figuring something out and then telling everyone else about it.

I thought this video from Hank Green made a lot of sense in that regard:
https://www.youtube.com/watch?v=_ADi5JlFf1E

Also, if you need additional resources or ways to visualize maths:
Vi Hart did a bunch of "imagine you're me in maths class and you start doodling" and from there on she visualizes mathematical concepts by way of doodles. It's all pretty old so excuse the low res and cheap mic, but that is how we used to enjoy the internet back then.
https://www.youtube.com/watch?v=DK5Z709J2eo

And if you just need someone to go over all the maths (like a maths teacher I wish I had had) then check out Professor Leonard: https://www.youtube.com/channel/UCoHhuummRZaIVX7bD4t2czg

El Mucho

If you have access to Audible or The Great Courses I highly recommend a lecture series called No Calculator? No Problem! Mastering Mental math with Art Benjamin. The entire series is about 6 hours. It starts very basic with simple addition and subtraction and works all the way up to large number multiplication, division and square roots without using a calculator or pen and paper. The last lecture also gives some methods for large number memorization that also makes doing the large multiplications much easier when you listen a second time.

Lots of practice is still required but it provides all the methodology and tricks to solving large number problems in your head.