(Part 3) Top products from r/matheducation

The Grapes of Math and the other books in that series are pretty good. Kind of a mix of numeracy and recreational math, but for kids.

There's a good book called Innumeracy, but that may be over the head of a child that young.

Once he's had a bit of algebra, A Long Way From Euclid is an excellent book. Even without the algebra, he might benefit by reading it together with someone.

You may also want to look at the library for introductory material on Euclidean geometric constructions, orgami, tessellations, fractals, topology (Flatland?), stereograms, Logo ("turtle graphics"), combinatorics, celestial mechanics.

Or for more hand-on stuff: play with Newtonian physics - build stuff with catapults, marbles, etc. The Art of the Catapult is a good book with plans for various machines.

Also, building kites, or paper airplanes:

The Paper Airplane Book

Paper Airplanes and Other Super Flyers

The Great Kite Book

25 Kites That Fly

Kites For Everyone

The Parrot's Theorem is a light, quick read. Follows the history of mathematics in a murder mystery sort of way. Not Earth shattering, but fun.
Also The Mathematician's Lament by Lockhart. Really does a great job of exploring what mathematics is at heart, and what the focus of math education should be (i.e. learning to appreciate the beauty in number and pattern, etc.)

First, please make sure everyone understands they are capable of teaching the entire subject without a textbook. "What am I to teach?" is answered by the Common Core standards. I think it's best to free teachers from the tyranny of textbooks and the entire educational system from the tyranny of textbook publishers. If teachers never address this, it'll likely never change.

Here are a few I think are capable to being used but are not part of a larger series to adopt beyond one course:
Most any book by Serge Lang, books written by mathematicians and without a host of co-writers and editors are more interesting, cover the same topics, more in depth, less bells, whistles, fluff, and unneeded pictures and other distracting things, and most of all, tell a coherent story and argument:

Geometry and solutions

Basic Mathematics is a precalculus book, but might work with some supplementary work for other classes.

A First Course in Calculus

For advanced students, and possibly just a good teacher with all students, the Art of Problem Solving series are very good books:
Middle & high school:
and elementary linked from their main page. I have seen the latter myself.

Some more very good books that should be used more, by Gelfand:

The Method of Coordinates

Functions and Graphs

Algebra

Trigonometry

Lines and Curves: A Practical Geometry Handbook

The first place I'd look would probably be a physical book - something like this. Encyclopedia-style books with one concept per two-page spread would also be among my go-tos, but those might not be long enough for your purposes. Additionally, do you know anyone who did a history of mathematics class in university? Some of those classes are open to history majors as well as math majors, so they could potentially have more history than math.


Oh, and just for fun: here's a Christmas-themed piece about Pascal's Triangle.


I wish I had something more concrete to offer you. Best of luck!

> popular if controversial amongst math educators

I see some great suggestions being put forward on this thread. I am partial to CGI, which has a very strong research base, but this is technically a program for teachers learning to teach elementary math, not a curriculum for children. However, it is an easy read and gives great insights into how children learn arithmetic and how teachers can guide such learning.

Just a side note: teaching for conceptual understanding (which you seem to understand the importance of) is well accepted and not controversial at all among math educators, only among the general public and a few mathematicians, who sometimes do not understand the importance of a conceptual base in elementary education.

Hmmm. There is an entire academic field dedicated to this question, and most serious stuff will be found in journal articles rather than books. The Singapore Math curriculum is essentially designed for homeschoolers, and they offer some good guides on how to best use their particular product. If you're interested in a more inquiry-based approach, you might check out

http://www.theatlantic.com/education/archive/2014/03/5-year-olds-can-learn-calculus/284124/

And

http://www.amazon.com/Out-Labyrinth-Setting-Mathematics-Free/dp/0195147448/sr=8-1/qid=1167859040/ref=sr_1_1/002-8958891-7740062?ie=UTF8&s=books

As for the journal, the biggest one is the Journal for Research in Mathematics Education, but there are a ton of them. I don't know prices, but it might be worth looking into a membership to the National Council of Teachers of Mathematics or the Mathematical Association of America.

Good luck, it's a rabbit hole!

One of the best books I've read that places mathematical discoveries in their historical contexts: Journey Through Genius. Dunham tells the story of math through different great theorems - why they were historically important, why they are important today - and then walks you through the proof. My copy is at school, so I can't say anything more tonight, but give it a shot.

Good luck!

Depending on the curriculum indicators you need to hit, it might be beneficial to talk with your cohorts in your department. This might not be helpful since the new common core is rolling out but here are my books I'd recommend:

Geometry - Ray C Jurgensen http://www.amazon.com/Geometry-McDougal-Littell-Jurgensen/dp/0395977274

It maybe 14 years old but it does an amazing job of starting easy and cranking up the difficulty. There is no need to have any prior geometry knowledge because it starts you with the very basics to complexities of Geometry. There are certain things I would change in the book, but you can't go wrong with having it as a resource.

And everything else Algebra/Calculus Related, just look for Ron Larson and it's gold. GOLD I SAY!

I also recommend using math folk tales like Stories to Solve to make the problems more interesting. I have four books with these stories from other cultures and times and the kids love them.

Jo Boaler has also done some interesting (contemporary) work on math education. I find her books (like What's Math Got to Do With It?) really reader friendly, but they also give a good overview of some of the issues prevalent in the field and her take on them based on her research.

This was my textbook for an undergrad class I took in math history. I believe it covers everything in your list, and it's all super interesting!

https://www.amazon.com/gp/product/0030295580/ref=oh_aui_search_detailpage?ie=UTF8&psc=1

I think Basic Mathematics is basically a precalculus text. I can't stand normal textbooks, everything is disconnected and done for you. This is written by one of the best mathematicians and will provoke thought and understanding. He knows his audience too, he's good with kids, check out his book Math! Encounters with High School Students. He's also written a 2-volume calculus text that I know has been used well in high school settings.

If you wan't to get ahead of the curve, buy these books. The questions are pretty good (although the explanations are not brilliant).

Sample

1. Edexcel C1
   
   2)Exexcel S1
   
   3)Edexcel C2

The I Hate Mathematics by Marilyn Burns is a classic and fantastic for extending mathematical thinking. She has a whole line of fun books.

So I might actually suggest a 'math for elementary teachers' book for yourself (alternate here), so that you can get an idea of how to approach these kinds of problems with visual aids, such as what kinds of ways you can draw pictures of long division, etc. In other words, most people don't use textbooks with 4th/5th graders, but a lot of worksheets and packets, with activities and explorations drawn from curricula like this.

Have you read Edward MacNeal's book "Mathsemantics?" You might find some inspiration there. MacNeal's thesis is that the difficulties some people have with word problems are actually semantic issues, not mathematical issues. Essentially, they need to learn to establish the logical relationships between things as they read the question, and then the math part comes relatively easily.

I second the Kaplan books and add The Man Who Counted.

Check out Thinking Mathematically
and Betterexplained.

I agree that Khan Academy is a good resource--another one I always recommend to my students who struggle with math basics is this book. A lot of people are embarrassed by their poor math skills. Unfortunately, there are plenty of bad math teachers and since it builds on itself, you can miss one concept and have trouble with so many other things further down the line. Don't get too down on yourself about it! It's not who you are, it's just one aspect of what you know, and with the right teacher/book/explanation, you can understand it.