It really depends which direction in mathematics you want to go. Even as a math major, I didn't really understand how vast it was until I got into abstract math.
My favorite way to learn is browse Amazon for "Dover Books on Mathematics." They are generally had for a penny + shipping if you don't mind buying used.
A good intro into modern mathematics: https://www.amazon.com/Concepts-Modern-Mathematics-Dover-Books/dp/0486284247
A mind for numbers, as mentioned below is very good. There is also a good book for the ‘quicker calculation’ called ‘how to calculate quickly’ that is super easy to dive into and get good value from.
https://www.amazon.com/How-Calculate-Quickly-Arithmetic-Mathematics/dp/048620295X
I've seen Larry Wasserman's All of Statistics recommended as an intro to statistics for mathematically competent readers (e.g. it's the textbook for the Berkeley masters'-level intro statistics class). I haven't read it personally though so can't speak directly to quality or the level of rigor.
This book will introduce various fields of math in an entertaining and accessible manner, aimed at someone around your level:
https://www.amazon.com/Concepts-Modern-Mathematics-Ian-Stewart/dp/0486284247/
Concepts of Modern Mathematics by Ian Stewart.
“According to Professor Stewart, an understanding of these concepts offers the best route to grasping the true nature of mathematics, in particular the power, beauty, and utility of pure mathematics. No advanced mathematical background is needed (a smattering of algebra, geometry, and trigonometry is helpful) to follow the author's lucid and thought-provoking discussions of such topics as functions, symmetry, axiomatics, counting, topology, hyperspace, linear algebra, real analysis, probability, computers, applications of modern mathematics, and much more.”
In the past I'd bought a small Dover book that was trying to teach anyone interested how to do fast mental math, with providing the needed methods/tricks and many many many drills to get sharp. I did try for sometime, but then I left it on the side.
Anyhow, this is the book. It might interest someone.
I would highly recommend reading Who's Number 1, it covers many different rating systems of how to rank/rate teams, build predictive models, and explains the math behind the different systems. It wasn't until I read this book that things like Elo & SRS really made sense.
How about selected chapters from Stewart's Concepts of Modern Mathematics? It has a pretty wide range of jumping off points and is a relatively affordable Dover book. You could go into more or lesser detail on these topics based on the students' backgrounds.
Another idea would be to focus on foundations like set theory, logic, construction/progression of number systems from ℕ -> ℤ -> ℚ -> ℝ -> ℂ , and then maybe move into some philosophy of math. There could be some fun and accessible class discussion, such as having them argue for or against Platonism. [Edit: You could throw in some Smullyan puzzle book stuff for the logic portion of this for further entertainment value.]
I would have taken that problem and done (60+40) + 7 + 4. going from left to right is kind of an old trick when it comes to adding, subtracting and multiplying. Given, this isn't exactly what seems to be taught in classrooms today, but it's very close.
This is a good book that shows you much more than the multiplication/division shown in this video:
https://www.amazon.com/Count-Like-Egyptian-Hands-Introduction/dp/0691160120
It's actually pretty fun to go through the exercises.
Since he's 8, I'm assuming he's currently doing multiplication and division and stuff. Everyone in this thread is talking about a bunch of advanced stuff which is great, but I would recommend as a first step How to Count Like an Egyptian. It's a book about how the ancient Egyptians added, subtracted, and their unique numeral system. Anyway, I think the key here is not to overwhelm your kid with math; every kid has a fancy or two. You don't want to go too deep, just keep him interested! And this book is an example of something that's made for kids, but is really cool and also teaches history and culture!
Also, I feel like this would be a nice way to feel him out! If he's still interested, you can go deeper into more serious math later. There's also a bunch of random math puzzles that are fun. I'll edit this after looking for them.
And his class has a book, which he wrote, and might go more in depth.
You are most welcome.
You may want to move on to something more advanced afterwards. All of Statistics was written for learners who want to learn state of the art probability and statistics quickly. It assumes knowledge of calculus and linear algebra though. Unfortunately, I don't know of any video series that covers this content concisely.
I enjoyed Concepts of Modern Mathematics when I was in high school. It might be a little basic and it's a bit uneven in places. But it's a really good lay account of the basic notions in "modern" mathematics. It doesn't really mention so much the various fields. For that, surfing Wikipedia is hard to beat.
Sorry for the late reply.
The only book on rating or modelling I'd recommend is Who's #1?. I sort of hacked it out myself from there. I learned a lot about backtesting and fitting with some quant finance books that helped a lot.
I don't have as much free time as I did in 2014/2015 when I developed the model and started the site. I'm still doing uni but working part time in a different city, I travel back and forth each week. I'll finish uni at the end of the year and have a lot more free time.
The rating startup is still on it's way. I'm using the API I built to run the DoTA site. I made a web app for it over the past summer, but I wasn't really happy with it. I want to try to release it at the start of next year.
The rating thing may be bigger, hard to know. I'm not giving up on the Dota site though. I'm extending the site to all E-Sports this year, with all the work I've done so far all I need to track another E-Sport is to find a datafeed for past matches.
You can get most of the Dota data from the Steam API (what Dotabuff uses). The only thing you'll also need is historical odds to test against. You can get this from D2L easily, but Pinnacle and others you'll just have to start recording it.
This is not really a python question, you might get a better response from /r/statistics or /r/datascience
Ranking things is actually quite difficult since there's really no "correct" answer. This book is a quite interesting "pop" book on the subject.
I'm an applied mathematician, not a statistician, but I'll take a crack at this. If you had variability info (standard deviation) that would be peachy because you could use confidence intervals. Unfortunately, we can't always get [the data that] we want.
So, we conjure a standard deviation for ourselves. This is not a great solution, but we can estimate s approx range / 4. (note, this is a terrible rule of thumb. But we take what we can get). So s approx 5 / 4 = 1.25.
Now that we have this, we can built a confidence interval. If some of your ratings numbers are small (like, less than n = 30), use a t-distribution confidence interval.
Thus, your "true" rating is the lower bound of the confidence interval:
weighted = avg - t_dof * 1.25 / sqrt(n)
Where dof is n-1, n is the number of ratings you have for a particular item, and t is the t-value for the desired width of the confidence interval and dof on a t-distribution with mean 0 and standard deviation 1.
You can get the t-value using scipy.stats.t.ppf using
from scipy.stats.t import ppf
t_dof = ppf(0.975, dof)
For your typical 95% confidence interval.
Check this book out http://www.amazon.com/Whos-The-Science-Rating-Ranking/dp/0691154228
Goes over different ranking methods people use to rank sports teams. It will go over the initial method Massey used which he has since tinkered with but it will get you started.
I'm a machine learning fan boy.
Actually I read this book about rating and ranking before I even considered making this model. At the time all of my models were from machine learning and this was my first more traditional stats model.
I tried several machine learning approaches to the updating feature of my model, but I ultimately come back to traditional stats for this project (for some reason).
In my other projects I exclusively use machine learning, although now that I'm learning more stats I'm doing more of both.
The system is fast to update by normal means - a complete recalc of the model on all of the professional Dota data that I have would take about 10min. Under a second to update for each match as they come in live though.
It would be slow at enterprise level data throughput, I believe I can make the updating algorithm more efficient and then it would be blazingly fast.
My tournament calculator is Monte Carlo. It took about 18 hours for 100k simulations of The International (to produce the odds on my site). This again could definitely be improved, also 100k simulations was overkill when you only need accuracy to 3 decimal places :P
Looks interesting, my model probably won't be too much help to you though. Send me a PM and we can chat on Skype/Steam about it.
Perhaps a good place to start is to learn more about probability theory. Joe Blitzen at Harvard teaches a great intro course on probability: Harvard 110.
From there, a good crash course on more advanced statistics is All of Statistics: A Concise Course in Statistical Inference by Larry Wasserman.
Maybe your school library has Programming for Mathematicians. I have enjoyed reading it, but it is a little-known book.
Take a freshman proofs class (or any "Fundamentals of Math" course) so you'll get a solid grasp on proof techniques, then pick up this book to get a neat overview of some interesting topics. From there it's entirely up to your desire to learn more.
Pretty much all I read these days are textbooks and nonfiction science books. I might poke at stuff like The Trachtenberg Speed System of Basic Mathematics or Negative Math. I'm sure that makes me a real interesting person to talk to about books /s
Great book. This edition is expensive-- if you get it, get a different edition.