The Mathematics Education Research Blog has recently noted that some articles have been made freely available until July 31st.
I’d like to plug this article in particular entitled From arithmetical thought to algebraic thought: The role of the “variable”.
Many contemporary mathematics educators still maintain that the conceptions of the unknown and the variable as a thing that varies are the same, mainly because they represent unknown numbers and—on the level of symbols—they can be manipulated in the same way. However, the historical-epistemological analysis makes it possible to show, in a decisive way, that they belong to two different conceptual formations (see Radford 1996), unified under the name “variable”.
The authors do experiments complete with flow charts of student thought patterns:
With abbreviations meaning for example:
AL4.3: He/she adds a datum, but he/she considers that the bets are equal.
AL5: He/she translates the problem into a first degree equation with two unknowns.
AL6: He/she explicitly considers the bounds of the problem.
(See the appendix to the article for a full list.)