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I've never seen any really good expositions of elementary mathematics (middle school or earlier). A good college-level textbook, written for people with an interest in mathematics, reads like a novel or an impassioned essay. The most famous example I can think off the top of my head of is Spivak's Calculus - the author has a vision for the subject, and wants to communicate their passion to the student.

As an example, the rule for multiplying fractions together takes some quite clever and pretty arguments to prove, but in math class and in classroom textbooks, this is just presented as:

Fraction multiplication rule: Multiply the numerators and the denominators together separately.

Looking at the difference in presentation, you'd almost think elementary arithmetic and geometry aren't part of mathematics. Where are the proofs? Is there even a single book in existence that proves the above fact at all?

Are there any books about elementary math written like actual math books? Note that I am not looking for things like "The Number Devil", which (to be fair I haven't read it) you probably wouldn't mistake for an actual textbook. I'm looking for systematic expositions of basic math, covering (and proving!) all the theorems, rules of computation, and definitions covered in elementary to high school (not necessarily all in one book - that's a lot of material).

I don't necessarily want it to be readable by 8 year olds, that might be asking a lot, but it would be nice to have something you could recommend to a bright 12 year old to give them a more solid foundation and inspire more of an interest in math. Something well written, that wants to make the subject interesting and beautiful, is preferred to a dry going-through-the-motions proof compendium, but of course that's basically like saying "make sure the book is actually good".

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    $\begingroup$ @StevenGubkin I hadn't thought of looking at books for teachers. Are you thinking of one of hers in particular? Could you provide any kind of "book review" in an answer? Specifically, does she provide proofs for everything? $\endgroup$
    – Jack M
    Commented Apr 21, 2014 at 19:39
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    $\begingroup$ @StevenGubkin That's exactly the kind of attitude I'm not looking for! Part of my concern is indeed that many people seem to take these for definitions - but they aren't! Part of the point of my question is to try and find out if elementary school educators are even aware that this stuff can be proven. $\endgroup$
    – Jack M
    Commented Apr 21, 2014 at 19:48
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    $\begingroup$ @paulgarrett From a strictly logical perspective, the rationals are very often defined as equivalence classes of $\mathbb{Z} \times \mathbb{Z}/\{0\}$, with $(a_1,b_1) \cong (a_2,b_2)$ if $a_1b_2=a_2b_1$. Then multiplication of fractions is defined by $(a,b)(c,d) \cong (ac,bd)$. What needs to be proven is that this definition is even well defined. $\endgroup$
    – Steven Gubkin
    Commented Apr 22, 2014 at 13:12
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    $\begingroup$ @StevenGubkin, I am well aware than "X is often defined as Y", but that is a separate question from the actual mathematical function of either. Also, the possibility of writing things in first-order logic or other formalities does not lend anything greater gravitas or correctness. The function of the rationals is as field of fractions of the commutative ring $\mathbb Z$, and that requires/entails certain properties (apart from any notation). The construction as equivalence classes of somehow-denoted pairs of integers, and notations about that, are secondary to the function. $\endgroup$
    – paul garrett
    Commented Apr 22, 2014 at 13:28
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    $\begingroup$ @StevenGubkin The StackExchange button at top left - it's in the drop down. You can ping people using @ once you're in there. I'll edit a proof into the post if you do think it's important, though. $\endgroup$
    – Jack M
    Commented Apr 22, 2014 at 13:39

4 Answers 4

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I think you'll find some of what you want on Berkley mathematician H.H. Wu's homepage.

More precisely, see: Pre-Algebra (pdf) and Introduction to School Algebra (pdf).

Note: I mentioned the same homepage (and the two pdf textbooks) in an earlier MESE post here; I would have just re-posted this as a comment, but I believe it is the actual answer to your question (!). I also ought not mark your question as a duplicate of the one I answered earlier, since they are quite different.

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    $\begingroup$ Wow, what a set of notes. He really comes out swinging at the start of Pre-Algebra. I really do believe his general approach is the right solution. It can't be about "fun" all the time. Math is serious business. Strangely, I found the idea of these notes very much fun. $\endgroup$
    – James S. Cook
    Commented Jun 12, 2015 at 4:25
  • $\begingroup$ @JamesS.Cook I should just clarify that I am not endorsing these texts for, e.g., students; I can imagine other uses (prospective/current teachers, to deepen content knowledge; mathematicians, to indicate the depth required for a "rigorous" approach to these topics; etc). However, I do agree with your last sentence: I [also] found the idea of these notes very much fun! $\endgroup$
    – Benjamin Dickman
    Commented Jun 12, 2015 at 13:52
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    $\begingroup$ I see, I'm curious, what would you endorse for elementary school or highschool ? My wife homeschools our kids. She's been very happy with the Singapore math curriculum. (we view algebra as a middle school topic) $\endgroup$
    – James S. Cook
    Commented Jun 12, 2015 at 16:24
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    $\begingroup$ @JamesS.Cook The texts I use for elementary school teachers are "Math Matters" and the "Essential Understandings" series. The former text is authored by the PI of EMP from which I draw many resources; the latter is published by NCTM. Rather than a text or curriculum, I would endorse the following three goals for learners: (1) understanding moves from procedural to conceptual; (2) mathematical disposition positive, and math anxiety low; (3) the previous two goals build towards a lifetime love of learning. $\endgroup$
    – Benjamin Dickman
    Commented Jun 12, 2015 at 17:04
  • $\begingroup$ (cont'd) If you all are succeeding in at least the first two goals using Singapore math, then I endorse it! If not, then you might check other curricula for supplementary resources. And, if specific questions emerge in this realm, then I'm sure MESE would be a good place to direct them! $\endgroup$
    – Benjamin Dickman
    Commented Jun 12, 2015 at 17:04
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Below are three books for what I think you want. At one time these books were all very well known, at least in the U.S., probably because virtually every public library (even small town libraries) used to have copies of these books.

Mathematics, Its Magic and Mastery by Aaron Bakst (here is the Table of Contents)

Realm of Numbers by Isaac Asimov

Realm of Algebra by Isaac Asimov

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I have really enjoyed using “Beast Academy” to homeschool my son. I wouldn’t say that it resembles a college textbook at all, but I do think it matches some of your other criteria:

1) “I don't necessarily want it to be readable by 8 year olds, that might be asking a lot.”

It is readable by 8 year olds.

2) “give them a more solid foundation and inspire more of an interest in math. Something well written, that wants to make the subject interesting and beautiful”

I believe it does give them a more solid foundation. I find that it explains things using the same mathematical intuition that I would use, as someone with a master’s in math who loves math and has taught math in college.

The books (for U.S. common core grades 2-5) are written as comics, with characters that are ‘beasts’ attending “Beast Academy”.

I don’t know for sure that they PROVE many assertions, but I’d say that they motivate the material, and make many connections, to help students gain a more thorough understanding. I do recall that they do have some exercises in which there are some informal proofs.

The authors also believe in giving some challenging problems at every level, so that even 2nd graders can start to learn that math isn’t about just having equations memorized, but rather that problems can take some time and mental effort to solve.

After grade 5, they do have books that may be more similar college textbooks, but I haven’t used them yet, so I cannot attest to their quality.

You can find them here: https://artofproblemsolving.com

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    $\begingroup$ artofproblemsolving.com has great materials. Their prealgebra book might meet your criteria: artofproblemsolving.com/store/item/…. But I agree with Joe: Beast Academy helps students learn math with meaning (and joy). $\endgroup$
    – Sue VanHattum ♦
    Commented Aug 16, 2019 at 16:40
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Can I recommend my two-volume book, Elementary School Mathematics for Parents and Teachers?

https://www.worldscientific.com/worldscibooks/10.1142/9696 https://www.worldscientific.com/worldscibooks/10.1142/10014

I wrote it while I was designing a professional development program for primary school teachers and was baffled by the fact that I couldn't find a textbook covering the primary school curriculum in a way that is accessible to teachers (and parents).

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