Tackling Algebra

I had a thought earlier: an interesting way to introduce algebra. It seems to me that many students who struggle have a hard time with the notion that a letter can "hold" the value of a number. I've heard people say they were good at math until we started using letters. This implies, to me, difficulty in understanding symbolism and representation, at least with numbers (and letters). An "a" has the value of the "eh" sound, and a "4" has the value of this many "items": / / / /. How can an "a" suddenly have a value of "/ / / /"?

There are a couple ways I can think of for getting around this sticking point. The first would use roman numerals, which are usually already introduced in elementary (or middle) school. Just explaining that the concept of "/ / / /" can be represented by either the symbol "4" or "IV" starts to introduce students to the idea that multiple symbols can be used to represent a concept. Once students are comfortable with that idea, it could be extrapolated: let's make a "new" number system with whatever symbols you want.

Another way to lead to the same point would be to start with "secret" codes. Kids generally become familiar with breaking codes in some way or another (e.g. the backs of cereal boxes). Have them start by making an alphabetic code since this will be most familiar. Once that's okay, have them add numbers to their code. At this point, you could go two ways: to the same place the previous idea brought you or to "higher" level algebra.

For the first way, either of the previous methods brings students to the understanding that concepts can be assigned any symbol you choose. Therefore, we can "change" the symbols for the numbers. Start by using their code to do basic math (e.g. :)=4, *=5, :)+*=?). At some point, you could bring in letters as symbols and move toward "actual" algebra (e.g. a+4=10, a=?).

The other way would build on this idea of a secret code. Students would write a series of math equations using their symbols and their classmates would have to figure out what the symbols are. For example, :)-4=* , :)+2=%, *+%=:) or similar but using the code without the numbers. This might be difficult to do with a full set of numbers, so just doing a defined range (e.g. 1-3) could work better. I'm not sure how well it would work, but at least it's framed as a game which could keep students interested longer. It also doesn't have to be called "algebra", and sometimes fear can be the biggest inhibitor to learning.

In any case, it would be interesting to sit down with some kids, young or old, and see how they respond to these ideas and how well they help the student to grasp the basic concepts necessary for algebra.