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It's kind of a serious question even if the title seems silly. As math educators, we all know that students link together different algebraic expressions thinking that they mean the same thing, e.g. \begin{equation} (a+b)^2 = a^2+b^2 \end{equation} or \begin{equation} \frac{x+2}{2} = x \end{equation}

These errors can be grouped statistically in categories, such as "false linearity" as in the $(a+b)^2$ case. This student-induced grouping leads to a kind of student-induced topology on the space of all algebraic formulas, where the "student-distance" between two expressions is small if statistically a lot of students think that they are the same expression.

My question is this: has this "topology", i.e. the organization of student errors in algebra, been studied sistematically? Is there any reference in literature that thoroughly deals with the issue of really categorizing and understanding errors in basic algebraic expression manipulation?

I think it would be extremely beneficial to know how to traverse the formula space in order to optimize learning (Of course this taxonomic approach has a sense only for algebraic manipulation, not for complex problems).

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  • $\begingroup$ I believe some manifestation of this kind of error has happened to everybody at some point. It's just a consequence of our inexact brains with their fuzzy pattern matching; only can be purged with enough practice. And, kind of mirroring human behaviour, GPT-4 also does this: what was the story where it claimed that $e^{e^x+e^y} = e^{e^{x+y}}$ or what was it? So using a LLM for investigating such a possible "topology" might be more tractible than using human experiments. $\endgroup$ Oct 3, 2023 at 6:39

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Has this "topology", i.e. the organization of student errors in algebra, been studied systematically? Is there any reference in literature that thoroughly deals with the issue of really categorizing and understanding errors in basic algebraic expression manipulation?

I did a toy research project a while back (prototyped a software system to automatically detect/classify/explain algebra mistakes in student work) and investigated a lot of this stuff while doing a literature search beforehand.

If you Google something like "most common algebra mistakes", then you'll find plenty of online resources (e.g. this and this) detailing common algebra mistakes that educators experience in practice when working with students.

I also found a handful of papers that more generally and systematically describe errors as invalid edges in "procedural networks" (nodes represent granular steps in a problem-solving procedure, and edges reflect how the steps may be arranged in sequence). Some references:

Brown, J. S., & Burton, R. R. (1978). Diagnostic models for procedural bugs in basic mathematical skills. Cognitive science, 2(2), 155-192.

Brown, J. S., & VanLehn, K. (1980). Repair theory: A generative theory of bugs in procedural skills. Cognitive science, 4(4), 379-426.

However, these papers only considered errors in arithmetic and did not extend into algebra. It turns out that the procedural network for algebra is so complicated that it doesn't actually simplify the problem of diagnosing student errors.

To automatically diagnose student errors in algebra, it's more common to use ad-hoc error diagnosis, where a set of common errors are chosen using domain knowledge, and an error detector is hand-crafted for each individual error. Some references:

Erabadda, B., Ranathunga, S., & Dias, G. (2017, July). Automatic Identification of Errors in Multi-Step Answers to Algebra Questions. In Advanced Learning Technologies (ICALT), 2017 IEEE 17th International Conference on (pp. 215-219). IEEE.

Gonzalez, C. S., Guerra, D., Sanabria, H., Moreno, L., Noda, M. A., & Bruno, A. (2010). Automatic system for the detection and analysis of errors to support the personalized feedback. Expert Systems with Applications, 37(1), 140-148.

Hoppe, H. U. (1994). Deductive error diagnosis and inductive error generalization for intelligent tutoring systems. Journal of Interactive Learning Research, 5(1), 27.

Huang, Z., & Tokuda, N. (1996). A syntactical approach to diagnosing multiple bugs in an intelligent tutoring system. IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans, 26(2), 280-285.

For more details, see Appendix 1 of my project writeup (html, pdf).

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