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.
I found in teaching elementary algebra to disadvantaged adult learners that the progression from table to graph and then to formula/function/equation might have been the most powerful tool for "selling" algebra I ever encountered. Anyone can see intuitively that for many problems, if you just make a table big enough, trying out all the possible values will lead to a solution. From there, it's a short step to, but what if there are millions of values? Then they're ready to graph it and the answers are right there at the intersections. From there it's just the step to, but what if you need an answer more exact than the line you can draw, or more dimensions than two? Well, by then, they're used to the idea of formula/function/equation as description, and if a point satisfies more than one description, it's a solution. And they've crossed over to doing algebra.
The spreadsheet provides a natural bridge from arithmetic procedure to formula/function. At first, students will just input numbers, as if the spreadsheet were a calculator. Then they see that if they input variables, they don't have to type nearly so much, and then that this means having not just this answer this time, but all the answers to all problems of this type, all the time.
It's a wide and easy-to-cross bridge from the specific to the general, and from procedure to concept.
One of the big changes in mathematics in the last 30 years has been the idea of experimental math, i.e., of exploring how numbers work by setting up numerical processes and looking at the results; it's at the heart of chaos research, for example. Just as the computer has become the equivalent of the telescope or microscope for mathematicians, Excel can be used as an amateur "scope" for exploring numbers, in a way very much analogous to the way countless students have gotten a handle on science by finding a planet in the sky or exploring the ecology of pond water. Among other things, I've used Excel to teach how every fraction is a division, and division is equivalent to multiplying by the inverse. It could easily be used for many other projects beginning even from a very early age in arithmetic.
Spreadsheet algebra is such an effective and intuitive idea that it has been re-invented several times in the last 20 years, and some Googling around will turn up immense amounts about it. (Caution: "spreadsheet algebra" is also a term used in advanced mathematics research for a kind of non-linear matrix algebra, so your Googling may very well turn up an article or two that's a bit beyond you. Don't worry, just keep looking!)
Although there still needs to be a human being there to guide the student in exploring and using the spreadsheet, as a teaching device for actual algebra (as opposed to a drilling device for standardized tests) the spreadsheet still beats out thousands of purpose-designed products.
Doesn't that suggest something to you, educational software people?