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I remember once hearing a (probably apocryphal) story about a university math professor that tried to teach a gradeschool class about algebra by telling them a few simple axioms and definitions and then making deductions. As the story goes, he thought he would be making things as easy as possible by minimizing the number of things the students would have to learn, but that turned out to be a bad idea because the way you get kids to do good on the SAT is by minimizing how much they have to think.

I am curious about what that lecture series would actually look like. I would like to find a presentation of elementary algebra that treats it from an abstract standpoint, but that requires no prior knowledge. Essentially, I am looking for algebra explained in the "professional" style that the above story depicts.

I have no idea where to find it. Textbooks that present the abstract algebra view of elementary algebra tend to assume you already know elementary algebra. University textbooks about elementary algebra written in the 1700s (when elementary algebra was a dominant research topic) come close, but abstract algebra was not around back then. Does such a book or series of notes exist? Can it exist?

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    $\begingroup$ "...the way you get kids to do good on the SAT is by minimizing how much they have to think." Is that a part of the story, or a claim you are making? If the latter, it seems unsubstantiated to me (at least the way you present it). $\endgroup$ – user138661 May 21 at 19:39
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    $\begingroup$ Maybe you'd like to look for textbooks from the New Math movement. $\endgroup$ – Nate Eldredge May 21 at 19:55
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    $\begingroup$ Not exactly grade school, but in 1998–1999, Luc Illusie was going to teach a course on algebra and Galois theory in the first year at the École Normale Supérieure (I think that would be roughly “advanced undergraduate” or first year graduate in the US). A few of us who had taken his course in algebraic geometry the year before asked him how he planned to explain Galois theory, and he said “oh, very simply, merely as the equivalence of the topos of étale algebras over a field with that of sets under action of the absolute Galois group of that field”… $\endgroup$ – Gro-Tsen May 21 at 20:07
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    $\begingroup$ This question makes me think of the paper pdfs.semanticscholar.org/b44b/… on carrying when making additions and group cohomology. $\endgroup$ – Daniel Robert-Nicoud May 21 at 20:32
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    $\begingroup$ I'm not sure that "New Math" quite answers the question. Here for example is a link to some New Math textbooks. onlinebooks.library.upenn.edu/webbin/book/… If you look at the "Introduction to algebra" book, it is definitely not written in a definition-theorem-proof style (which is what I assume Display Name means by a "professional" style). $\endgroup$ – Timothy Chow May 22 at 14:09
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Hardly "requires no prior knowledge," but:

Klein, Felix, M. Menghini, and G. Schubring. Elementary mathematics from a higher standpoint. Berlin/Heidelberg: Springer, 2016. Springer link.


         


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It is a parody, but "Mathematics Made Difficult" by Carl Linderholm is this, using concepts from category theory in order to explain things like "counting" and "subtraction". The presentation is not strictly correct in terms of the mathematical concepts being wielded, so it's not a great source to actually learn about category theory or elementary arithmetic. People who do know about those things may appreciate the humor.

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You might ask Mark Sapir. I understand he had a series of lessons on words in semigroups with a 4th grade audience in mind. (I don't know which country.) He might know of similar efforts.

Gerhard "For Me , It Was Latin" Paseman, 2019.05.21.

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    $\begingroup$ Several years ago I attended a wonderful talk by Persi Diaconis in the Stanford Statistics Department where he used very sophisticated statistics to analyze properties of elementary-school arithmetic, such as the statistics of how frequently you must perform a carry when adding or multiplying multi-digit numbers. Not what you're seeking, but close. statweb.stanford.edu/~cgates/PERSI/papers/… $\endgroup$ – David G. Stork May 21 at 21:32
  • $\begingroup$ Along similar lines is Dan Isaksen's "Cohomological Viewpoint on Elementary School Arithmetic" jstor.org/stable/3072368 $\endgroup$ – Timothy Chow May 22 at 19:55
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Perhaps you mean the series of school books by Georges Papy titled Mathématique moderne.

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(Image from images.slideplayer.fr)

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    $\begingroup$ These are indeed wonderful, completely unappreciated books by a mathemattician who had a highly original pedagogical talent. He was also a socialist senator in the Belgian Parliament. Once, as he was at the airport waiting for his plane to the USSR, his students cheered him and chanted "Papy russe, Papy russe" (which sounds exactly like "papyrus" in French). He often signed his articles $\not \pi$, because in French "pas pi" (= "not pi") is homophonic to Papy. $\endgroup$ – Georges Elencwajg May 23 at 11:20

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