Learning Mathematics
Why Learn Mathematics?
Technology is everywhere around us, and we need mathematics to master it!
In fact most toppaying jobs need good math skills:

And Mathematics is not just numbers, it is about patterns, too!
So jobs like fashion and interior design can benefit from math skills.
Mathematics is also useful in everyday life:
 Investing money (interest rates, profits, etc)
 Estimating costs
 Shopping (is it really a bargain?)
 Understanding Computers
 Designing rooms and gardens
 Planning trips
Mathematics also improves your mental ability as it teaches you logical ways of thinking.
And it is just plain fun: what other subject is about solving puzzles?
How to Learn
This advice is based on research, read Bibliography for more.
There are two major steps:
Get the information ... read, listen to a teacher, watch a video.
Use the information ... sketch it, explain it to someone, answer questions.
Using it is so important! It ...
 strengthens your memory: the connections between the neurons in your brain become stronger, making it easier for you to remember later on.
 helps to identify gaps in your understanding, you can then reread with purpose.
 involves actively searching for, organizing, and connecting information in your mind, which can lead to deeper understanding and better memory.
Just reading or watching videos can give you a false sense of confidence in your knowledge, which may not really be that good.
Do plenty of practice each day. It is OK to go back and reread. You will feel the knowledge clicking into place!
Study Schedule
Create a schedule where you study different topics for short periods of time each, so that:
 Any one topic is covered over several sessions spaced out over time
 A study session covers several different topics
This is based on the ideas of Spaced Learning and Interleaving:
 Spaced Learning: study a topic for short, spacedout, intervals. This helps the brain to process and retain complex concepts better, improving longterm memory (while cramming all at once relies on shortterm memory).
 Interleaving: Mix up related topics in a session. Studies have shown that practicing math questions in this mixedup way helps you remember and use what you've learned better. It also helps you relate different concepts.
How to Read Mathematics
Mathematics says a lot in a short space.
Example: In English we say "We don't know what staplers or trays cost, but we do know that the office manager bought 15 staplers and 11 trays for a total cost of $73"
But in Mathematics: 15s + 11t = 73
So it is good to reread, go back and forth and play with the ideas.
Reading Mathematics is different to reading English 
Read it, think about it, read again, write it down or sketch it out, and then use it. This all helps to get the ideas into your mind.
Example: Converting Celsius to Fahrenheit
°F = (°C × ^{9}/_{5}) + 32
 Read it once first to see there is °F (which means Fahrenheit) on one side, and °C (Celsius) on the other side, with some calculations.
 Now go over it again and see that °C is multiplied by 9/5 and think "I wonder why that is done? Why 9/5?"
 Then notice that 32 is added ... why is that?
 Maybe you could make a sketch as shown below.
 Use it yourself, do a few conversions and see how it works.
 Explain it to someone else, even your dog.
Make Sketches
It really helps your understanding when you sketch what you are learning.
Make large and bold sketches with plenty of labels and notes.
Like this sketch about Celsius and Fahrenheit:
Sketches are also very helpful when doing questions. So become a good sketcher.
Work Neatly
Working neatly helps you think more clearly and also gives you good mental habits. 
Have pride in your work, even if nobody else will see it.
Take Your Time!
Math is not about reading pages ... it is about building concepts in your mind.
So don't think "I read 2 pages today", instead think "I understand graphs better now".
Important: If you skip past a section, the rest may not make sense. You will get confused, frustrated, and you will begin to hate the subject.
The cure?
 Go back to where it made sense,
 then move gently forward again,
 do plenty of practical things like solving questions and doing sketches
And you will soon be "back on track"
Practice, Practice, Practice
Do lots of questions.
That is why we have the Mathematics Question Database.
If you need to pass an exam, find past exams and do them.
Aim For 85% Success
Research* has shown that we learn fastest when we get things right about 85% of the time. So aim for a level of difficulty that slightly pushes you out of your comfort zone but doesn't result in too many failures.
Read a Lot
Get some books, and read them. Spend time on math websites (like this one!), and join a forum (like the Math is Fun Forum).
Come Up With Your Own Ways
You have your own learning style.
Don't just follow the steps you are shown, try your own ideas!
Play with the ideas you are learning.
And try reading about the same subject from different places, you may find some that make a lot more sense to you.
Your mind is an amazing and unique tool, and you want to use it the best way you know how.
And studying mathematics is a good way to improve it!
All About Ideas
It is more important to know the ideas than to remember the formulas.
If you know how things work, you can always recreate the formulas when you need them. And you also gain the ability to do more clever things with your ideas.
*Bibliography:
For further research look up Retrieval Practice, Spaced Learning, Interleaving, Elaboration, Concrete Examples and Dual Coding
 Retrieval Practice Produces More Learning than Elaborative Studying with Concept Mapping (Jeffrey D. Karpicke and Janell R. Blunt) Science 20 January 2011: 1199327
 Testing Improves Memory https://www.psychologicalscience.org/index.php/news/releases/testingimprovesmemory.html
 Learning is Optimized When We Fail 15% of the Time https://news.arizona.edu/story/learningoptimizedwhenwefail15time
 Practice testing protects memory against stress
https://now.tufts.edu/newsreleases/practicetestingprotectsmemoryagainststress  Retrieval Practice: Beneficial for All Students or Moderated by Individual Differences? https://journals.sagepub.com/doi/full/10.1177/1475725720973494
 The right time to learn: mechanisms and optimization of spaced learning https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5126970/
 Interleaved practice enhances memory and problemsolving ability in undergraduate physics https://www.nature.com/articles/s4153902100110x
 How Much Math, Science Homework is Too Much?
https://www.apa.org/news/press/releases/2015/03/mathsciencehomework.aspx  The Worst and The Best Study Tips & Habits based on Psychological Research
https://cognitiontoday.com/2019/04/theworstandthebeststudytipshabitsfrompsychologicalresearchhowto/  Exploring Individual Differences in Children's Mathematical Skills: A Correlational and Dimensional Approach https://www.eurekalert.org/pub_releases/2013−12/nuosnmg121313.php
 Teaching the science of learning https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5780548/
 Drawing 'integral' to science learning https://www.nottingham.ac.uk/news/pressreleases/2011/august/drawingintegraltosciencelearning.aspx