If you have ever put together a piece of furniture or completed a do-it-yourself project at home you have probably pulled out a toolbox and used your tools on an as needed basis. Perhaps you had to hammer a nail into a wall and located your hammer to do so. Every now and then though, you cannot find that hammer and are faced with the challenge of hammering in that nail another way. You look around and find something that might work even if not originally intended to do so.
This is what we do, we solve problems with whatever means available to us. We all do it in some way, shape, or form. Doesn’t matter if you are a doctor, an accountant, a student, a drug dealer, or a mobster. You do what you have to with the resources available to you. The same goes for math students. I introduce every student I work with to the concept of the mathematician’s toolbox. As we move from one topic to the next I make sure they understand that this is another opportunity to build their toolbox and make it more complete. As we approach new topics I always ask them what tools can we use to solve some of these problems before even exploring the new tools.
The reason I do this is because I want them to understand there is no single solution for a math problem. You can always approach it a different way with different tools. What makes a mathematician efficient is selecting the tool that requires the least amount of effort for that problem. For instance, although we can simply use the quadratic formula for every factoring problem in the standard form, sometimes there are other tools that do the job more efficiently for particular cases. A students’ ability to evaluate that problem and find the right tool is what will make him or her a stronger all-around math student. Essentially, this is my goal with every single one of my clients.
I challenge you to help your students build their toolbox and learn how they can use their tools to solve new problems. Like I said at the beginning though, we use what we have at our disposal and that won’t always be the most efficient tool. I always make sure to celebrate when my client finds a tool to solve the problem, no matter how inefficient. We then have an opportunity to explore more efficient tools when we look at the problem again. Help your student learn how to look at that nail in the wall and find the best tool for the job. If they do not have a hammer, then help them find an alternate solution.