If you do a search on "from arithmetic to algebra" as a verbatim phrase, you'll get about 600 hits, with the ones from Google Books reaching back into the nineteenth century. About three out of every four will be about helping students make the transition from arithmetic to algebra -- it has been known for a very long time that that's where we lose many people who are never able to advance much further in math.
As I noted before, 25 years ago RAND surveyed the then-nascent field of educational software and found many effective arithmetic teaching programs and practically nothing that taught any of the important aspects of algebra (abstract relations, strategy, fundamental concepts and so on). Even back in 1988, it was clearly understood that arithmetic training programs should not be the model for developing algebra educational software because arithmetic is taught as procedural training. When you teach the quadratic formula, polynomial factoring or Cramer's Rule as if they were mere complex recipes like long division, you miss the fundamental concepts that are the whole point of studying algebra.
Depressingly, my survey of algebra-teaching software revealed that 25 years later the situation remains the same. Plenty of programs will drill a student on algebraic procedures, but most do not even attempt to teach any sort of insight, strategy or deeper understanding. Even the best merely offer supporting text or "guess the next step," the same wrong-headed approach that the pioneering math educator Mary Everest Boole identified about the arithmetic-to-algebra transition in 1909.
Proceduralism is about performing tasks (now write this number here and do this ...), but conceptualism is the heart of mathematics (how are these numbers connected or related?). In the leap from "what do I do?" to "what is it?" we're losing many students who might otherwise have gained not only the higher incomes, but the much better understanding of the world, to which mathematics is the gateway.
Now, there may be an educational software design company out there right now about to fix this problem, but probably there's not. And parents and teachers who need to get a seventh-to-ninth grader across the gap right now can't very well wait until he or she is halfway through college, or longer, for algebra teaching software that actually teaches algebra.
And yet there is a piece of instructional software right on your computer that can be used to teach all levels of algebra to all levels of student, in a fully conceptual way. It's the spreadsheet.
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You can find plenty of discussion and advice about how to do this from excellent teachers like Tom Button, Sue Johnston-Wilder and David Pimm, Teresa Rojano and Ros Sutherland, but let's just quickly hit the highlights of how exploring spreadsheets, and then exploring with spreadsheets, can provide a conceptual doorway into algebra. If you're interested, you'll find all but limitless resources for this.
Consider, to begin with, that variables and parameters in spreadsheets are very similar to what they are in ordinary algebra. For that matter, Microsoft Excel notation (and most of the Open Office software notation) is either algebra notation outright, or so close that only simple explanations need to be given ("in algebra the multiplication asterisk is understood, in Excel you have to put it in," "what we call a function in algebra is what a formula is in Excel," etc.).
A basic insight of algebra is that a function can be thought of as a rule OR a table OR a graph. (I'm capitalizing because it's the Boolean logical OR rather than ordinary English "or.") In fact, they are three different ways of looking at the same thing. Similarly, a spreadsheet formula can be used to generate a table of data, and the spreadsheet's graphing features can be used to turn it into a graph.