How can I get faster at doing math?

Question

In general, my level of mathematics is good with respect to the competitions that I need to appear in.

However, no matter what the competition is, whether it is an easier level or a more difficult level of Olympiad, or JEE, or just some internal math test, I am always facing a time issue. I get almost all my answers correct, but I’m simply not able to get score beyond a threshold because of the time crunch. Even simple advices like how to get faster at calculations help.

What is it that you advice I should do to get faster at solving problems? Even simple suggestions on how to get faster at calculations help.

Answer 1

Improving your speed in solving mathematical problems requires a combination of practice, strategy, and efficiency. Here are some suggestions that may help you get faster at solving problems:

Practice regularly: Consistent practice is crucial for developing speed. Solve a variety of problems regularly, including both easier and more challenging ones. This will help you build familiarity with different problem types and increase your overall speed.

Develop mental math skills: Strengthen your mental math abilities by practicing mental calculations, such as addition, subtraction, multiplication, and division. Learn techniques like estimation, rounding, and simplification to quickly approximate and simplify calculations.

Focus on key concepts: Identify the key concepts and strategies relevant to the types of problems you frequently encounter. Mastering these core concepts will enable you to solve problems more efficiently and quickly.

Improve problem-solving techniques: Learn and practice various problem-solving techniques, such as visualization, pattern recognition, and logical reasoning. These techniques can help you approach problems in a more organized and efficient manner, saving you time in the process.

Time yourself: Set time limits for solving practice problems and exams to simulate the pressure of real test conditions. This will help you develop a sense of time management and train yourself to work efficiently within specific time constraints.

Utilize shortcuts and tricks: Learn and utilize shortcuts, tricks, and formulas specific to the types of problems you are solving. Familiarize yourself with commonly used formulas, properties, and identities in areas such as algebra, geometry, and trigonometry. This will help you save time by avoiding unnecessary calculations.

Analyze your mistakes: Review and analyze your mistakes and the solutions to the problems you have solved. Identify the areas where you are consistently making errors or spending too much time. Understanding your weaknesses will allow you to focus on those areas during your practice and improve your efficiency.

Build problem-solving intuition: Work on developing an intuition for problem-solving by solving a wide range of problems. With practice, you will start recognizing common problem-solving patterns and become more adept at choosing the most efficient approach for each problem.

Remember, improving speed takes time and effort. Be patient and persistent in your practice, and gradually you will see improvement.

Answer 2

In my view, paradoxically you have to be slower in order to be faster. To be fast at maths you have to be less focused on getting the job in front of you right now done, and more focused during the learning stage on learning deeper insights into what you're doing. When you attain a high level of MASTERY, problems are faster to solve.

I'm slow at learning maths because I need deep insights in order to remember things, and I'm never satisfied until I know something inside out.

Taking a really simple example, to calculate $5 \times (10+4)$ you might say that's $5\times14=70$ and be done with it and move on.

But if you're focused on learning deeper insights you might think about it in multiple different ways:

Multiplication distributes over addition so I can multiply before I add, or after:

$(5\times10) + (5\times 4)=5\times(10+4)=70$

You might think about how multiplying is equivalent to adding the exponents of the prime factors:

$10+4$ has the prime factors $\{2^1,7^1\}$ and $5$ has the prime factor $\{5^1\}$ so the product is $2^{1+0}\times5^{1+0}\times7^{0+1}=70$

Since we write numbers in base $10$ you might think about multiplying by five as multiplying by $\frac{10}2$ so $5 \times (10+4) = \frac{10}2\times14$ then cancel the twos to get $10\times7$

You might think of $14$ as $20-6$ giving you $100-30=70$

Then you might think about the fact that this last example is

$5\times (20-6)=(5\times20-5\times6)$ and ask yourself whether this means that multiplication distributes over subtraction.

Then you might deduce that this is true because subtraction is simply the addition of a negative number.

Then you might think about how addition is an operation on the monoid of non-negative integers, and how subtraction is the extension of rightward transformations on the real line to leftward transformations on the line which are their inverse transformations. And by introducing subtraction you extend the closure of your algebraic operations to include the negatives, and this makes the integers a group according to the group axioms. Multiplying a negative number scales to the left rather than scaling to the right.

A good exercise, is to pay attention to how you solved a problem. Sometimes you do it automatically, and the process is subconscious. Once you become conscious of your method, ask yourself what other methods you might have used, and solve the same problem by those methods. You will learn shortcuts this way and multiple ways of understanding the same thing.

The more insight you build, the more visualisation tools you have at hand on which to pin memories. Then when you have to apply quickly e.g. in an exam, it's not a case of finding your one way to solve a problem. You can see multiple ways and one may instantly jump out as a quick solution. And you can move on with confidence, knowing you've arrived at your answer by multiple different methods.

This of course, requires an investment of more time in the learning stage. But by building deeper insights and a higher level of mastery, you can progress faster when it comes to applying what you learnt.