Unjustified Criticism of The Way Math is Taught

I have come across many unjust criticisms of US education system in general and the way math is taught in particular, but this one takes the prize: the author, who himself is a math teacher, criticizes a particular problem statement as being unrealistic, the solution to be artificial and the laments that the problem doesn't teach any problem solving skill.

It is not that I see no need to improve the US education system or the way math is taught. There are many valid criticisms and many more ways to improve the system. Carefully chosen and well worded problem statements are certainly one of them. But I found the acerbic reaction to this particular problem statement to be misguided. Let me explain myself.

Let us first take a look at the problem

A youth group with 26 members is going to the beach. There will also be 5 chaperones that will each drive a van or a car. Each van seats 7 persons, including the driver. Each car seats 5 persons, including the driver. How many vans and cars will be needed?

The problem can easily be solved by representing no. of cars and vans as two variables and writing equations satisfying the constraints as stated in the problem. However, the author feels that this particular problem is not worthy of being introduced to students for following reasons:

1. "One, is the problem realistic? Would a real person need to solve this problem?
2. "Two, is the solution realistic? Would a real person solve the problem using a system of two equations?
3. "Three, in what ways does this problem help our students become better problem solvers?"

The author does elaborate on each of the points and you should read the blog post to understand his point of view. What follows is my reaction to his points:

Is the problem realistic? My answer: yes, it is as realistic as it can get. The domain of the problem statement is certainly familiar to most students and they understand the role of chapreones as drivers, the difference between cars and vans and the need to optimally use the vehicles. A real life scenario may have additional constraints in some aspects such as the need to seat friends together, availability of specific types of vehicles only and less constraints in others such as no need to fill all vehicles to their full capacity. But simplification or idealization is routinely done in scientific or engineering problems and I see no reason why it shouldn't be done in a math problem.

Is the solution realistic? Well, given the simplicity of the numbers involved, a student might be able to solve the problem by simple trial and error and I'll accept that as a valid solution. In fact, that is what I would expect from a 3rd or 4th grader. A system of equations with two variables is certainly a more generalized solution and very realistic for the problem. Historically, the system of linear equations arose in the field of transportation as a way to optimize routes and this particular problem and solution are good substitute for the more generic problem.

In what ways does this problem help our students become better problem solvers? First, students learn how to convert a real life scenario to a system of equations. Based on my own experience working with middle school children I can say that this in itself is no mean feat. Second, they are able to relate the answers back to a real life solution. If the teacher encourages alternative solutions, such as the one using trial and error or the one using a single variable, then the students also get a better understanding of the system of equations. Much better than just solving a long list of equations without relating to any situation, real or unreal.

Yes, we can come up with more real problem scenarios from different fields of Science, Engineering, Operations Research or Accounting but will the students find that more palatable? I doubt. Saying that most people don't face situations requiring solution of a system of equations and hence should not be included in curriculum misses the point about teaching maths.