The interactive examples in this post require installing Wolfram CDF player, which is free and works on most desktop computers using Firefox, Safari and Internet Explorer, but not Chrome. The source code is the Mathematica Notebook Handmade Exp Tree.nb, which is available for free use under a Creative Commons Attribution-ShareAlike 2.5 License. The notebook can be read by CDF Player if you cannot make the embedded versions in this post work.
Algebraic notation contains a hidden abstract structure coded by apparently arbitrary conventions that many college calculus students don't understand completely. This very simple example shows one of the ways in which calc students may be confused:
Students often mean to express formula 2 when they write something like \[x\!\!+\!\!2\,\,\,\,y\] (with a space). This is a perfectly natural way to write it. But it is against the rules, I presume because in handwriting it is not clear when you mean a space and when you don't.
Formula 1 can also be written as $x+(2y)$, and if it were usually written that way students (I predict) would be less confused. Always writing it this way would exacerbate the clutter of parentheses but would allow a simple rule:
Evaluate every expression inside parentheses first, starting with the innermost.
Using trees for algebra
Writing algebraic expressions as a tree (as in computing science)
- makes it obvious what gets evaluated first
- uses no parentheses at all.
An example of using the tree of an expression to do calculations is available in Expressions.nb (requires Mathematica) and Expressions.cdf (requires CDF player only) on my Mathematica website. I could imagine using tree expressions instead of standard notation as the normal way of doing things. That would require working on Ipads or some such and would take a big amount of investment in software making it intuitive and easy to use. No, I am not going to embark on such an adventure, but I think it ought to be attempted. (Brett Victor has many ideas like this.)
Transforming algebraic notation into trees
The two manipulable diagrams below show the algebraic notation being transformed into tree form. I expect that this will make the abstract structure more concrete for many students and I encourage others to show it to their students. Note that the tree form makes everything explicit.>
After I return from a ten-day trip I will explore the possibility of making the expression-to-tree transformer turn the expression into an evaluable tree as in Expressions.nb and Expressions.cdf. In the I hope not to distant future students should have access to many transformers that morph expressions from one form into another. Such transformers are much more politically correct than Optimus Prime.