## Information on some of the research that the experimenteal step-by-step solver is based upon.

### 1) Research on how to program computers to perform mathematics using the same techniques humans do.

One of the main researchers in this area is Alan Bundy, and most of his research publications are available here:

An archive of Alan Bundy's research publications

A good introduction to this research is contained in Alan Bundy's book The Computer Modelling of Mathematical Reasoning

Here are some passages from this book:

"Meta Level Reasoning: What mathematical theory do the axioms 1, 2 and 3 above belong to? It is not algebra, because the predicates 'solve', 'collect' etc. do not express relationships between numbers as all the algebraic predicates do, e.g. =, >, etc. 'solve' and 'collect' express relationships between algebraic formula, i.e. they discuss the representation of algebra. Whenever we reason about the representation of a mathematical theory we are said to be operating at the meta-level and the mathematical theory itself is said to be the object-level. Axioms 1, 2 and 3 above are axioms of the Meta-Theory of Algebra." p.173.

The Meta-Theory of Algebra is where a number of the "hidden" techniques of mathematics are contained. Chapter 12 describes some of the hidden techniques and how they work. Chapters 1-11 contain the information that is needed to fully understand chapter 12. However, enough can be gleaned by reading just chapter 12 to get a feel for how the techniques work.

### 2) Research on using expression trees to help students visualize the structure of algebraic expressions.

The step-by-step solver has the ability to show expression tree versions of the equations because the following research paper by Pat Thompson indicates students often have difficulty visualizing the structure of expressions and equations:

http://pat-thompson.net/PDFversions/1987StrucInAlg.pdf

Here are a couple of paragraphs from this paper:

"Typical errors found in previous studies of students' errors in algebra suggest that students studying algebra frequently fail to realize that formulas in mathematical symbol systems have an intrinsic structure. In algebra, expressions are structured explicitly by the use of parentheses, and implicitly by assuming conventions for the order in which we perform arithmetic operations. It is hypothesized that many of students' errors in manipulating an algebraic expression are due to their inattention to the expression's structure."

"Finally, it should be noted that in eight days of instruction these leaving-seventh grade students went from essentially no working knowledge of order of operations to deriving algebraic identities, and did so with some depth of understanding. Even with the limitations stated earlier in this discussion, the fact that such coverage is possible makes us question assumptions that are built into traditional junior high school pre-algebra and algebra curricula about what one can expect of junior high school students in the United States."