It seems there is an overall agreement that Geometry is the right place to introduce proof in Basic School. However, number theory (arithmetic) looks like to be a more simple environment (consider, for instance, a sentence like this one: "if n is integer and multiple of 2, then n is multiple of 4").

Are there any references (books, scientific articles, thesis) that had studied this question, I mean, the use of number theory instead geometry to introduce proof in Basic School?

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    $\begingroup$ The statement "if $n$ is an integer and a multiple of $2$, then $n$ is a multiple of $4$" is false. Perhaps you should change it to a true statement (assuming you intend students to prove that certain statements are true). $\endgroup$ – Joel Reyes Noche Oct 8 '17 at 14:43
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    $\begingroup$ The Ross Mathematics Program uses elementary number theory, but that program is for advanced students in secondary school. It was fantastic preparation for me and others in mathematics but is much too advanced for a Basic School or for average students. $\endgroup$ – Rory Daulton Oct 9 '17 at 0:08
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    $\begingroup$ @JoelReyesNoche, false propositions are part of the package! :) $\endgroup$ – Humberto José Bortolossi Oct 15 '17 at 19:20
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    $\begingroup$ What is Basic School? How old are these "school children?" Might they be too young to handle proofs? I understand they are too young to understand the basic rules of logic. If so, they are certainly too young to handle number theory, one of the most difficult courses in university as I recall. One's spatial sense is absolutely useless in number theory. $\endgroup$ – Dan Christensen Oct 17 '17 at 3:18
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    $\begingroup$ @DanielR.Collins, yes, Geometry has its fundamental role and it shouldn't be removed! My question is: to introduce mathematical language, rules and proofs, is it better to start with geometry or arithmetic? $\endgroup$ – Humberto José Bortolossi Oct 18 '17 at 13:25

Extracurricular math, or math enrichment programs often do introduce proofs in the realm of number theory. At the earliest level, it can include direct proofs such as "even + even is always even", "odd + odd is always even", etc.

The geometry proofs are usually taught as very structured -- a build-up from definitions and earlier theorems to the result, justifying each step, usually formatting the steps on the page in a particular way. There are many examples of interesting statements to prove, at a similar level of difficulty. Plus there is a benefit to visualizing things.

I think the answer to your question is that beyond very simple statements, number theory is not as simple an environment as it might seem:

  1. Some statements in number theory are reasonably easy to prove (e.g. that $3$ divides $m^3 - m$), but it's not so easy to structure these arguments as a sequence of simple steps.

  2. For many statements, to prove them rigorously, you need a proof by induction. But that's certainly a higher level of difficulty.

I think that's why number theory proofs are more commonly seen in advanced math programs, rather than in standard curriculum.


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