Math is not supposed to be hard. It is supposed to make it easier for us to solve complex problems. So why is it that we so often find ourselves struggling with the very thing that is supposed to make our problems easier to deal with? Here are my thoughts on three of the main reasons why math is hard.
- We aren’t taught to understand the problem, we are only taught to solve it. I see this so often when teaching people. Students memorize formulas or know a process for solving a problem, but they don’t actually understand what problem they are solving and why the process they’re using works. In most classes and courses, there is little time given to exploring the problem organically. This means that students usually don’t understand why the problem is interesting and why their current techniques for solving problems don’t work. This is further exacerbated by textbooks that are written in the ‘most logical’ order (i.e. give the most concise form of the solution). Whilst these books offer a fast way to learn a solution, we often miss out on the process of solving the problem.
Not understanding questions before we learn a solution makes it very difficult to translate problems into appropriate mathematical language. It also means that if the problem is changed slightly, we don’t have any tools to deal with it. This gap is probably the main reason that students struggle with ‘modelling and problem solving’ type questions.
- Abstraction. This problem is extremely inter-related with the first one. Most of high level mathematics deals with abstract objects and students are usually introduced to these when they have very few concrete examples to work with. Again, this is a problem of being taught a solution before being given a chance to explore the question. Even in high school mathematics, abstraction leads a lot of students to question what the point of mathematics is.
Abstraction is essentially the process of trying to generalize a situation or problem that appears in a number of different areas. For example, numbers are an abstraction of counting concrete objects. If you have two monkeys and three more come along, then in total you have five monkeys. Here, the fact that we are counting monkeys isn’t really important. We will get the same answer whether we are counting monkeys, apples, or followers. Hence, we introduce the abstract concept of numbers and say that .
Once we get used to using numbers, we make a further abstraction to ‘algebra’. We see that in the equation , there is nothing special about using 5. This equation will be true if we replace 5 with any other number. Hence we introduce variables and say that , where now represents a fixed, but unknown, number (we often say is a fixed but arbitrary number). This is abstraction: we are now making general statements about numbers.
As you learn more math, you will hopefully see that there are many other things that ‘work’ the same way numbers ‘work’ and so it is possible to abstract further (one direction of this leads to ‘abstract algebra’). Abstraction is very useful. It allows us to link problems that at first appear to be completely different and use a common method to solve both of them. It allows us to write computer programs based on a users input. However, abstraction is also very confusing if you don’t have concrete examples to work with. Again, this confusion essentially arises because not enough time is spent understanding the different problems before teaching the abstract concept.
- Not doing math. Mathematics is something that should be done, not watched or read about. If you only ever watch someone play football, you are unlikely to ever be a good football player. In the same way, only watching someone else solve problems will not make you a good problem solver. If you want to get good at mathematics and really understand it, you need to practice solving problems. You need to make mistakes and work out why you made the mistake. You need to try different approaches and get stuck, then try to work out why you’re stuck and what you might be able to do about it. Doing this will help you understand the problem much more than watching someone provide a textbook solution. It will also allow you to ask better questions about future problems.
You should also practice communicating your solutions to other people. Just knowing the answer is usually not very useful in mathematics (in a lot of mathematical problems, we know what the answer ‘should’ be but we can’t prove that it actually is the answer). It is important to be able to demonstrate to other people why your solution works or how you got to the solution. Think about what made the problem hard for you and whether you have now made it easier for people to understand this. One of the best pieces of advice I have been given regarding communication is to write for a future version of yourself. If you looked at your solutions in a week/month/year would you be able to understand how you arrived at the answer?
So, what can you do to overcome these three problems? I think the most important thing you can do is to ask questions: ask why the problem is important, what makes it hard, why can’t you use maths that you already know, what happens if you change something. Someone out there will be able to give you a good answer, so keep asking until you understand.