Consider the following piece of reasoning:

$$x+3 = 5$$ $$x=5-3$$

Based on my own experience back in school, and helping people with their math homework, as well as in talking to other people about math education, it seems that in the minds of most students, when they do the above, they are "moving the three to the other side of the equation". I've seen students act genuinely surprised when you point out that actually, you're not "moving" anything, you're simply saying that it is a true fact that if $x$ is a number such that $x+3=5$, then it must also be the case that $x=5-3$.

It seems to me that students perceive a lot of math as just being "rules", like the rules of chess, rather than "facts", things that are actually true. When they solve a problem, they think of themselves as playing a game using certain allowed moves, rather than proving that something is true based on what they already know.

Do students know - do they feel in their gut - that the Pythagorean theorem is true in the sense that it would check out if you measured the actual side lengths of an actual right angle triangle? Or do they just think of it as a rule for finding $c$ in an exercise?

What I'm looking for here are scientific studies, or any research at all into this distinction between "facts" and "rules" - not anecdotal evidence. I want to know if this distinction has been discussed before in the literature, and what research has been done into finding out if my suspicions are correct.

  • $\begingroup$ I may be missing something, but when I do proofs I'm playing a game, and that is how I teach it. The game has rules, depending on your axioms and chosen logical system. I do not regard any of it as facts while I'm playing. Is it a fact that for every point and line there is a single line through that point parallel to the first line? Is something a fact only if you can construct an example of a theorem, or are negative proofs allowed? $\endgroup$ – Richard Oct 26 '15 at 14:31

I think what you are thinking of is found in maths research described as 'processes' and 'objects'. See, for example, Chapter 2 of Alcock--Simpson.

The big idea is that many (most?) students will start from think of $5+3$ as the process of starting with $5$ and then adding $3$ to it. That is different to seeing $5+3$ as a number (that we more conveniently call $8$). The change in thinking from one to the other (or better, to being able to switch between the two ideas at will) comes with familiarity, but can take a very long time if nothing happens to show the limitations of the first approach.


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