Everyone solves problems differently. Some like to work directly with the mathematics head on, while others prefer to have a more intuitive approach. This includes studying simpler cases of a problem, or looking at examples in order to really understand what’s happening. These are all valid approaches, but the point I want to highlight is that these are all strategies. There’s a certain method to tackling a problem. It’s not that you can’t solve a problem through trial and error, but if you want to solve more problems more quickly, your best bet is to figure out a strategy.
If there’s a “most difficult” part in solving a problem, it’s usually the first step. When you first encounter a problem, there are countless directions that you can go in. This can lead to paralysis, but it can also lead to trying things randomly and hopefully get something to work. In particular, I’ve long observed that secondary students come up with a strategy I like to call “combine the numbers”. The basics of this strategy are simple. When given a question (often applicable during a word problem), look at the numbers given in the text, and try and figure out how they fit together to provide an answer. Students know that the answer is usually some combination of simple arithmetic with those numbers, so there’s a good chance that they can solve the problem without completely understanding it.
I’m not just making this up: I’ve seen it happen many times with students. This is a problem, because they are learning a completely different skill than what we want. Instead of learning how to problem solve, students are learning how to find an answer given some numbers. Since that’s not what we want, we need to find a way to direct a student’s thinking to the actual problem.
The way I do this is through asking big-picture questions. When a student is working through a word problem for example, it’s extremely common for them to give it a single read, and then say that they don’t understand. I then ask them what the goal of the problem is. What do they need to find? Then, when the student has a good idea of that, I ask them how this can be done. What do they need in order to achieve the goal? Usually, this is where the student will start giving me a sequence of operations with numbers, but I’ll immediately direct them away from that. Instead, I ask them to tell me what they need to find in words. If the student says they need to find the cost by multiplying five and four together, I’ll insist that they say something along the lines of, “To find the total cost, we need to multiply the cost of one loaf of bread (four dollars) by the number of loaves we want (five).” I’m always trying to separate the particular problem from the more general method.
It’s not that I don’t think it’s important to solve that one problem, but in the grand scheme of things, it’s a lot less important to be able to quickly solve one particular problem versus solving a general class of problems. I want to help students develop tools that will let them solve a variety of problems, instead of only a certain one.
By getting students to be proficient in stating what they need to do in order to solve a problem, they will usually have an easier time starting the problem. I’ve personally found that clearly stating a problem and what I need to do to answer the question often gets me going and working towards a solution. Having a clear goal lets you break it down into subgoals. If I’m looking to prove a result, I ask myself, “What do I need to show in order to come to my conclusion?” This becomes the basis of the proof. It can also be useful to simply get accustomed to translating between the mathematics and describing the problem in words. This seems to get one to think with a slightly different mindset, which could help as well.
When I work with students, I only have a limited amount of time to have an impact on them. I don’t want to use that time to solve one problem with no transfer of knowledge to a new one. I want to get them comfortable with a whole range of problems, so they can work through whatever gets thrown at them. By having them clearly state the goal of a problem, my hope is that students will start to see the underlying similarity between lots of problems they work on. After all, there are only so many ways to ask an “applied” question about area in secondary school. I don’t them to understand the one particular problem we worked on together, and then be unable to do the one during a test because it wasn’t the exact same. Clearly stating what one needs to do solve a problem is a great way to combat that tendency.