This article will deal with the actual misconceptions in mathematical concepts, rather than the common interpretation of the title as "misconceptions about the usefulness or uselessness of mathematics".
Firstly, let's address the question of "so what's the big deal if my basic math concepts are wrong?"
Here comes the big misconception that tops it all. Math is not just about numbers and equations, it is about logic and inductive reasoning. Math questions are phrased such that you see a connection between a set of numbers and another, so that the unknown can be deduced from the known. Not grasping the basic concepts prevents you from drawing correct connections between events to predict an unknown event. The events may not have anything to do with numbers, but the logical process of induction is the same.
The examples below may illustrate what that means. They are misconceptions which lead to the kind of answers people tend to give based on an instinctive interpretation of the questions, which may seem correct at first glance but on closer inspection, are in fact wrong.
1. There are six times as many apples as bananas. If A is the number of apples and B is the number of bananas, write an equation to represent the number of apples and bananas.
The common error is to write "6A = B". The answer, of course, is "A = 6B".
2. A book on sale costs $20 after a 20% discount. What is its original price before sale?
You are wrong if you take 120% of $20, giving $24. The answer is $25.
3. I toss a coin five times, and got a head each time. What is the probability of me tossing a head on the sixth time?
It is not one sixth or anything like that. The answer is still half.
All these may seem like elementary school math to you, but the fact is that many adults do commit these mistakes frequently in their daily lives, allowing themselves to be tricked by the play on words and numbers. Recognise them and live smart!