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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?

Humberto José Bortolossi
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    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). – JRN Oct 08 '17 at 14:43
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    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. – Rory Daulton Oct 09 '17 at 00:08
  • Students most likely got into contact with geometry quite some times before, drawing things, doing "proofs" or "calculations" by drawing and measuring, etc.
    Number theory, on the other hand, might be a completely new topic.     – Dirk Oct 09 '17 at 08:59
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    Note that basic geometry is a prerequisite for any proper college algebra course. For example, basic geometric facts are required to prove: the Pythagorean theorem, distance formula, midpoint formula, slope being constant on a line, parallel lines having equal slope, perpendicular lines having negative-reciprocal slope, standard equation for a circle, etc. – Daniel R. Collins Oct 09 '17 at 14:03
  • @Daniel R. Collins: Could you please clarify, what is required to prove what? – beroal Oct 11 '17 at 17:37
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    @DanielR.Collins, the question is about how to introduce proofs in Basic School: using Geometry or Number Theory? – Humberto José Bortolossi Oct 15 '17 at 19:20
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    @JoelReyesNoche, false propositions are part of the package! :) – Humberto José Bortolossi Oct 15 '17 at 19:20
  • @beroal: A proof of the Pythagorean theorem may at least require knowledge that a straight line has an angle equal to the interior of a triangle. Proof of the midpoint formula, that fact slope of a line is fixed, and that parallel lines have equal slope, each require knowledge of transversals and similar triangles (AAA). – Daniel R. Collins Oct 15 '17 at 22:37
  • @HumbertoJoséBortolossi: I'm pointing out that if you remove the geometry class in the sequence, then you may undermine the scaffolding that expects those subject-matter topics as prerequisites in the following algebra course. I am aware that this is a comment, not an answer to the question. – Daniel R. Collins Oct 15 '17 at 22:44
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    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. – Dan Christensen Oct 17 '17 at 03:18
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    @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? – Humberto José Bortolossi Oct 18 '17 at 13:25
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    @DanChristensen, consider the situation where children is old enough and, in your opinion, they are already ready to be introduced first time to questions about mathematical language and proofs. Would you start this introduction using Geometry or Arithmetic? It seems there is a common sense towards Geometry and I would like to know if someone already had scientifically tested the subject with Arithmetic instead. – Humberto José Bortolossi Oct 18 '17 at 13:33
  • Which seems easier to you? 1. Prove that sum of the interior angles of a triangle is 180 degrees. 2. Prove that if $m$ is a natural number then 3 divides $m^3 − m$. 3. Prove that if A implies B and B implies C, then A implies C. – Dan Christensen Oct 18 '17 at 21:03
  • I would opt for neither geometry nor number theory (arithmatic), but basic logic to illustrate various methods of proof. Example 3 (above) would be an example of a simple direct proof. – Dan Christensen Oct 18 '17 at 22:09
  • @DanChristensen, do you know the work and the software of Dave Barker-Plummer, Jon Barwise and John Etchemendy? What do you think about their approach? https://ggweb.gradegrinder.net/lpl – Humberto José Bortolossi Oct 23 '17 at 21:00
  • Looks promising, but it may be suffering from feature overload. You may prefer my approach. Visit my homepage at http://www.dcproof.com – Dan Christensen Oct 23 '17 at 23:36

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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.

DS.
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