5 Rules For Studying Maths that Every Student Should Know

Authored by Dr Mahya Mirzaei (aerospace engineer, University of Sydney)

Studying maths can feel overwhelming at the start but if you study it in the right way, and have the right mentality, learning maths can be a lot easier than you think and it will be truly enjoyable (yes, it will be, trust me).

Here are five rules that will help you study maths effectively:

Studying maths can feel overwhelming at the start but if you study it in the right way, and have the right mentality, learning maths can be a lot easier than you think and it will be truly enjoyable (yes, it will be, trust me).

Here are five rules that will help you study maths effectively:

Rule 1: The Get Another Text Book Rule

This rule comes from a real story of my own life: When I was in high school, I always excelled at maths but my sister started to struggle with the subject towards the end years of high school. She had completely lost her self-confidence and believed she wasn't "smart enough" to do maths. Now because I loved my sister very much and wanted to help her, I started looking within myself and tried to find out why I was good at maths. Then I suddenly discovered why I was good at maths and my sister wasn't:

So, I used to study maths from textbooks, and whenever I couldn't understand a textbook, I'd say to myself "How stupid is this person, that has written this book in a way that I can't understand it". So, I'd throw that book away, get another book, and I wouldn't give up until I'd mastered that concept.

My sister on the other hand, would read the same book, wouldn't understand the same concept as me, and would say to herself "I am stupid that's why I'm not getting it". So, she would lose her self-confidence, would lose all hope, and would tear up.

And so, the only distinguishing factor between my sister and I was the fact that I believed I could, and she believed she couldn't. And it is by no coincidence that Henry Ford said:

**"whether you think you can, or you think you can't, you're right"**

And this is why your self-belief is just so important. Because if you think you can, you are going to "get a textbook after textbook", you are going to do whatever it takes; and if you think you can't, it's just too easy to give up.

So, I used to study maths from textbooks, and whenever I couldn't understand a textbook, I'd say to myself "How stupid is this person, that has written this book in a way that I can't understand it". So, I'd throw that book away, get another book, and I wouldn't give up until I'd mastered that concept.

My sister on the other hand, would read the same book, wouldn't understand the same concept as me, and would say to herself "I am stupid that's why I'm not getting it". So, she would lose her self-confidence, would lose all hope, and would tear up.

And so, the only distinguishing factor between my sister and I was the fact that I believed I could, and she believed she couldn't. And it is by no coincidence that Henry Ford said:

And this is why your self-belief is just so important. Because if you think you can, you are going to "get a textbook after textbook", you are going to do whatever it takes; and if you think you can't, it's just too easy to give up.

Learning maths is exactly like learning to swim. Now if I'm in the water swimming, and all you're doing is sitting on the beach watching me, will you ever learn to swim? No, never! So long as you're not in the water yourself, attempting to swim, you will never ever learn to swim.

Now that's like learning to do maths problems. You can watch me do a million questions, if you're not doing questions yourself, you will not learn the logical and critical thinking skills required to do maths.

A lot of times I give my students a maths problem to solve, and the way they go about it is usually:

1. They look at it,

2. Try to do it in their head (i.e. try to see the entire solution in their head and find out where they need to start from) ,

3. Since they can't see the complete solution in their head, they say "I don't know how to do it".

So, to know why this is the wrong approach, you need to know how solving maths questions work: solving maths problems work exactly like a jigsaw puzzle. Now, you have all seen a jigsaw puzzle like the image above, right? If you were given this jigsaw puzzle how would you solve it? Think about it for a second…

To solve the above you never think to yourself "oh let me try to do this in my head and figure out exactly where each piece goes from the very start". What you do instead, is that you pick up a piece randomly you see if it results in anything, if it doesn't you'll set it aside, start with a new random piece and continue with this process until you've finally solved the entire puzzle.

And that is exactly how you should approach maths problems. Don't try to find the solution in your head first, then start writing it out. Instead, understand that a lot of the times no one knows what the first step is. So, you just start from somewhere (anywhere) and you see if that approach works or not. Get into a habit of writing things out and working through a problem logically. If where you've started doesn't result in a solution, then try another approach, and continue until you've found your way around the question. Do not give up.

Remember that maths PhD students spend three years of their lives solving one question, just one question. So don't be disheartened if it takes you 2 hours to solve a question :)

1. They look at it,

2. Try to do it in their head (i.e. try to see the entire solution in their head and find out where they need to start from) ,

3. Since they can't see the complete solution in their head, they say "I don't know how to do it".

So, to know why this is the wrong approach, you need to know how solving maths questions work: solving maths problems work exactly like a jigsaw puzzle. Now, you have all seen a jigsaw puzzle like the image above, right? If you were given this jigsaw puzzle how would you solve it? Think about it for a second…

To solve the above you never think to yourself "oh let me try to do this in my head and figure out exactly where each piece goes from the very start". What you do instead, is that you pick up a piece randomly you see if it results in anything, if it doesn't you'll set it aside, start with a new random piece and continue with this process until you've finally solved the entire puzzle.

And that is exactly how you should approach maths problems. Don't try to find the solution in your head first, then start writing it out. Instead, understand that a lot of the times no one knows what the first step is. So, you just start from somewhere (anywhere) and you see if that approach works or not. Get into a habit of writing things out and working through a problem logically. If where you've started doesn't result in a solution, then try another approach, and continue until you've found your way around the question. Do not give up.

Remember that maths PhD students spend three years of their lives solving one question, just one question. So don't be disheartened if it takes you 2 hours to solve a question :)

This rule is not about being honest with others, it's about being honest with yourself. Many students lose hope and confidence so easily when it comes to doing maths. The "I'm not mathematically minded" or "I'm not smart enough to do maths" comments are everywhere.

The reality is, most people lose hope long before they've given maths a real shot. Remember:

**"you are not asked to do something beyond your reach, you are asked to do your best, and that is within your reach" **

Could you, in your most honest state, look within yourself and say that you have given mathematics your best and you still failed? Most likely the answer is no…. Now trying to do a question for 10 mins, getting frustrated and stopping your studying all-together is not your best. Spending an hour on a topic and then giving up because you don't understand it, is not your best. Remember the "get another textbook rule"? Always apply it.

The most important thing is to always be honest with yourself and really monitor whether you are doing your best or not. One way to ensure you're studying enough is to follow my guide on studying effectively.

The reality is, most people lose hope long before they've given maths a real shot. Remember:

Could you, in your most honest state, look within yourself and say that you have given mathematics your best and you still failed? Most likely the answer is no…. Now trying to do a question for 10 mins, getting frustrated and stopping your studying all-together is not your best. Spending an hour on a topic and then giving up because you don't understand it, is not your best. Remember the "get another textbook rule"? Always apply it.

The most important thing is to always be honest with yourself and really monitor whether you are doing your best or not. One way to ensure you're studying enough is to follow my guide on studying effectively.

Many of my students say they are reluctant to use "the jigsaw puzzle rule" because if they use it, they must spend 0.5-1 hour on only one question, and even then, they may not get it right. They also say they can't spend that long on one question because they have more questions to do and don't want to waste their time on only one question.

What's wrong with the above analogy is that it assumes that if you do 30 questions, that is better than doing one question. That is completely incorrect. If you do one question by spending a lot of time on it, even if your attempt doesn't result in the final solution, your brain is working towards learning the skills of problem solving and critical thinking. These two skills are crucial when it comes to maths, and you cant master them if "you are not in the water yourself", i.e. trying to do questions.

The harder the questions you attempt to do, the better you'll become at doing questions that need outside the box thinking.

Don't forget: Mathematics teaches you how to think and every time you do maths, you are training your brain to learn to think in a logical way. This will not happen if all you do is look at a question and then read through someone else's solution.

What's wrong with the above analogy is that it assumes that if you do 30 questions, that is better than doing one question. That is completely incorrect. If you do one question by spending a lot of time on it, even if your attempt doesn't result in the final solution, your brain is working towards learning the skills of problem solving and critical thinking. These two skills are crucial when it comes to maths, and you cant master them if "you are not in the water yourself", i.e. trying to do questions.

The harder the questions you attempt to do, the better you'll become at doing questions that need outside the box thinking.

Don't forget: Mathematics teaches you how to think and every time you do maths, you are training your brain to learn to think in a logical way. This will not happen if all you do is look at a question and then read through someone else's solution.

Aim for the sky, your dreams are closer than you think!

Dr Mahya Knox

Bachelor of Aeronautical Space Engineer (Sydney Uni)

PhD in engineering (UTS)

Director of education - LearnEd

Bachelor of Aeronautical Space Engineer (Sydney Uni)

PhD in engineering (UTS)

Director of education - LearnEd