To answer this question, it helps to think of an analogy. When would you say someone is good at English?
When is someone good at English?
Consider a peculiar scenario. Let’s say a person who’s learning the English language memorizes the meanings of thousands of words. This person also learns most grammar rules, and definitions of terms like gerunds, past participles, prepositions, etc. However, he’s never conversed in the English language. Given this context, would you say he’s good at the language? Most probably, you won’t. You’d probably want to have a conversation with this person before you decide.
Vocabulary and grammar help, but are not sufficient.
Vast vocabulary and command over grammar rules help, but by themselves are not sufficient for a person to become good at the language. On the other hand, someone who converses in a particular language all the time will almost always become good at it. This person will build up a decent vocabulary and pick up the most relevant grammar rules through conversation.
Conversational fluency comes down to the ability of a person to think in the English language. If the person doesn’t think in English, chances are they are translating every sentence from another language. This will adversely affect their conversational fluency.
Want to be good at math? Learn to think mathematically.
Math is similar. One could memorize steps and procedures to solve problems. One could memorize all formulae and substitute numbers to find answers to textbook questions. However, given a new situation, this person will struggle to figure out which formula to use or which procedure to apply.
On the other hand, someone who has built a basic understanding of concepts, and fluency with procedures will be much better at solving unfamiliar problems. That’s because this person has explored mathematical ideas and built an intuition for solving problems. They have practised and internalised many procedures. They can spot patterns and build coherent arguments. They can break down complex problems. They can spot flaws in logic. They have built the ability to reason. They can think mathematically.
Mathematical thinking can, and should, start early
For young children, this starts right from the introduction to numbers. Unfortunately, there are many children who are simply asked to memorise numbers, addition facts, and multiplication tables. Instead, a foundational understanding would require them to look at numbers and explore their properties.
For instance, if I put 3 and 4 together, I get 7. But had I put 2 and 5 together, I would also get 7. So the number 7 is flexible and can be built in many ways. This is a crucial mathematical idea. This then leads to the discovery that if I take away 5 from 7, I’ll be left with 2. The intuition for numbers, addition and subtraction is being built without any abstract symbols, jargon or drills. Additional practice with similar problems internalises the idea in a child’s mind.
This play using blocks or other learning toys is the best way to build number sense – the most foundational step for being good at math. This is learning by reasoning. This builds mathematical thinking.
Mathematical thinking helps connect math concepts
This directly affects all subsequent math topics. After all, math is heavily interconnected, For instance, strong number sense allows children to break down questions like 17 x 8 in many different ways to quickly and mentally calculate the answer, without having memorized that fact.
Some would see 17 x 8 as the sum of (10 x 8) and (7 x 8). Others would see it as the difference between (20 x 8) and (3 x 8). Another way to look at it is the difference between (17 x 10) and (17 x 2). All of these help children break down a seemingly complex problem into smaller easier to solve parts. Classic mathematical thinking in action.
Mathematical thinking leads to better learning outcomes
Every mathematical idea including division, fractions, decimals, integers, geometry and algebra can be learnt this way. Enough studies from across the world have shown that math class are way more engaging if run this way. At the same time, they show that children’s attitude towards math is more positive. And most importantly, it leads to better learning outcomes.
What about being able to calculate very fast?
Being able to calculate fast certainly allows children (or more often parents) to show off their skill. However in today’s world, this is not a skill that matters. Smartphones are readily available to calculate pretty much everything. What is more important is the ability to know which calculation should be done in the first place.
For instance, given some data, you’ll have to understand the context and the requirement to decide whether to calculate the mean, weighted mean, median or some other quantity. Once this decision is made, the final step of crunching numbers can always be delegated to a calculator.
Being able to calculate fast is like knowing some very specific grammar rules. The knowledge helps but by itself, isn’t enough. Quite often it may not even be necessary. Some of the best mathematicians were slow at calculations. What made them good was their ability to think deeply about a concept or a situation.
Taking the first step towards getting good at math
Starting early helps tremendously. However, you can learn to think mathematically at any age. Your aim is to build the ability to spot patterns and represent them quantitatively. Break down complex problems into smaller manageable parts. Or, convert an unknown problem to a known one. Figure out (multiple) solutions to new problems. Analyse arguments and identify logical flaws.
Search for a great teacher, a good math program or a book that echoes these principles. School textbooks will most probably not help. Find learning resources that help you think creatively, yet quantitatively. This search is your first step to start thinking mathematically and being great at math.