I recently had the experience of helping a young algebra student with her homework. She is very bright and is in an accelerated math program that teaches algebra in the 7th grade.

However, it was so disappointing to see the curriculum. I was in the same accelerated program 15 years ago and the curriculum hadn't changed a bit. It was still vague and unhelpful. They sent her home for the summer with a large packet of problems to solve without any explanation how to solve them.

I majored in math and I still had to guess at what the worksheets were asking the student to do. They used vague variables and were essentially just teaching rote memorization. It reminded me of my own confusion in mathematics when I was growing up, how one year a variable would mean one thing and the next year it would mean something else. "Solve and explain," means just as little to me now, using math on a daily basis, as it did to the 11 year old version of me.

After so many years, it would have been nice to see the curriculum focus more on application of mathematics and less on how to memorize and calculate.

I've noticed the same thing in all the school curricula I've seen. The word problems are often vague, and all to often completely ambiguous. They require guessing what the question writer had in mind when she wrote the question, which are usually unstated assumptions that apply only to the way math is done in schools. This is not the way to teach the kind of precise thinking and careful definition of terms that one needs to understand why math is the way it is: the “study of precisely defined objects.”

Teachers, if you want your pupils to understand how to create a mathematical problem from a description of a practical one, you must train them first to state the situation with absolute precision and without ambiguity. The best way to do that is, as in the teaching of all virtues, to fastidiously follow that habit yourself. That the questions in math textbooks fail to do that so often is another sign of the enormous institutionality of our education problems.

  > how one year a variable would mean one thing and the next
  > year it would mean something else

I'm probably misunderstanding what you mean, but isn't that the purpose of a variable? In programming, a variable foo in one method which represents Widget objects is completely independent from a variable foo in another method which represents Gadget objects. In math, solving for X in one problem might be to find the missing angle X, but in another problem it might be to determine the elapsed seconds X.

I'm not disagreeing that variables are meant to be variable.

However, it's important to teach context behind variables before asking an 11 year old to solve them or else it becomes a point of confusion. They are asked to use rote memorization to complete the calculations but they don't seed their memory before asking.

Well stated. For further deep thinking and a wonderful read, I recommend the original, "A Mathematician's Lament" by Paul Lockhart (2009): http://www.maa.org/devlin/lockhartslament.pdf [PDF]

I don't see why the software used is particularly important. The point should be the mathematics, rather than specific tools. I don't know how hard it is to transition from Wolfram Alpha, to, say, Matlab, but I don't think it's difficult.

Sorry, replace Matlab with Octave.