I’m currently in the process of writing a mathematics book aimed at the high school level. The idea is to introduce high school students (or anyone with a basic background in mathematics, really) to the wonder of mathematics through hands-on problem-solving in topics not usually encountered in high school. Feel free to read any of the drafts linked below and (if you feel so inclined) to provide me with feedback. (If you intend to provide feedback, you probably want to first read this short note to test readers.)
The following chapters are available in draft form:
- Preface/Chapter 0: Boolean Algebra, draft #2 (June 24, 2007)
- Chapter 1: Problem-Solving and Proof, draft #1 (June 24, 2007)
- Chapter 2: Numbers, draft #1 (June 24, 2007)
Planned (possible) topics for future chapters include:
- Functions and Sequences
- Fibonacci Numbers and the Golden Ratio
- Number Theory
- Combinatorics
- Complex Numbers and Beyond
- Sets and Infinity
- Group Theory
- Graph Theory
- Computation
I think it would be nice (especially for some more computationally intensive sections like group theory) to have sample code that could be used in Sage to solve some problems that would be (nearly) impossible to do by hand. e.g. RSA enctyption with large keys. For that reason I would like to point you to Sage: http://sagemath.org which, in case you don’t know, is a Free/OSS CAS that ties many different systems together in one easy to use package.
Sage allows for sharing “notebooks” and uses jsmath to get decent typesetting of formulas. You could create a companion notebook for a chapter, or even publish the entire book as Sage notebooks. I know they are looking for a bunch of examples and introduction stuff that I think this book would be perfect for. There are a number of people on the Sage project who are interested in teaching Math to High School/early undergraduate students, and would appreciate your help. Lots of others are interested in research of course…
Anyway, I just thought you might be interested since I would have loved both Sage and a book like this when I was in High School. I now have my Master’s
-Ivan
“It can seem somewhat non-intuitive that A =) B is true when A is false.
The idea is that A =) B only says what happens when A is true; when A
is false, you certainly can’t say that A =) B is false (even though it may
not be true in any really meaningful sense). As an example, suppose your
friend says, \If it rains tomorrow, I will stay inside.” If your friend goes out
to splash in the puddles during a downpour the next day, then they lied,
no question about it. But suppose the next day dawns bright and sunny,
and your friend goes outside. Did they lie? Well . . . no. Your friend didn’t
say what they would do if it didn’t rain, so their statement was vacuously
true.”
It may be just me but I do not think it is obvious that you are talking about both F T -> T and F F -> F.
I think it would help to say actually state you are talking about F T -> T and F F ->F and then say if its sunny and if your friend stays in or goes out will not be a lie. Your rain example is good but needs to be tied to both statements.
I think for many its not just having having A as False but the having F F turn up T that causes confusion.
It’s Discrete Math. A good book is Suzanne Epp’s Discrete Math with Applications. I dunno if it’s taught in some high-schools. Most high-school students are stuck with calculus. There are questions whether Discrete Math should be taught separately from its applications.
While skimming chapter0-pre2.pdf, I came to the footnote which says
“Unless Australia has just been made up as a part of a giant conspiracy. I mean have you ever been to Australia? . . . Are you sure it was really Australia?”
This is absurd, I live in Australia.
That’s absurd – Smirnoff Green Apple Vodka isn’t alive anyway, and even if it was, an assertion via a website hardly constitutes proof. Stop trying to force your Australia beliefs on the world. You’ll be asking me to believe in “atoms” or “God” next, and I’ve seen neither! It’s all hearsay – just because grownups said it when you were young and impressionable, doesn’t make it true. ;)
This reminds me of the sentiments expressed at the following places:
http://steve-yegge.blogspot.com/2006/03/math-for-programmers.html
http://stackoverflow.com/questions/52176/what-are-the-core-mathematical-concepts-a-good-developer-should-know
+1 for writing about the math that got left behind
I enjoyed reading your drafts very much! I can’t wait for the final version to come out.
As a high school student myself, I can affirm to your comparison of real math to school math. Unfortunately, I’m not sure that many highschoolers will read this book. :( I’m the only guy in my grade that (publicly) reads books like this, and any attempt to explain my obsession or convince them to read a book like yours is often met with failure.
Keep on writing!
Thanks Sam! I haven’t worked on this in a while but I still intend to finish the project at some point. Knowing that people are reading and enjoying what I have so far is encouraging, so thanks for leaving a comment.
Keep on reading!
Just a note – when you get to complex numbers, I suggest you DON’T do the standard, but confusing, thing, i.e. define the imaginary unit ‘i’ as “the square root of -1.” I saw a much better approach in an old (1940s) textbook: treat complex numbers as — as best I can describe it — “uninterpreted” ordered pairs that follow a set of rules for addition, subtraction, multiplication, etc. e.g. (1, 0) + (2, 4) = (3, 4), and (2, 5) x (1, 7) = (-33, 19), and so forth. You can introduce the Cartesian interpretation and segue into Polar representation if you like. Then, once you’ve laid all THAT groundwork, demonstrate that (0, 1) x (0, 1) = (-1, 0), and then tie the whole thing in to “the first number in the ordered pair behaves exactly like a real number, while the second number, taken by itself, is clearly the square root of a negative number.” THEN go into the whole “i” thing.
I dunno, maybe that’s too heavy for high schoolers. But maybe not. Find some high schoolers and ASK, I guess.
C.
Hi Chris, thanks very much for the suggestion! That is an extremely intriguing approach I hadn’t thought of before. I’ll certainly give it very careful consideration when I get round to revising this again!
I really like this approach. On the other hand the question is whether students will appreciate it. Those who have prior experience with complex numbers may think it is a needlessly pedantic exercise and will certainly question the teaching method but I think if the student is being exposed to complex numbers for the first time then they don’t know what to expect and it could turn out to be great learning experience for them.
I think this is the standard way to handle complex numbers (at least my complex analysis class at college started that way, and so did the textbook we followed), but as I see it, doing this just involves creating technical machinery without purpose, or at least postponing it. The question of whether it is meaningful to talk of the square root of -1 seems a more natural way to introduce the topic.
That said, I loved your first 3 chapters, and would like to see how you proceed with this book. Is this project still alive? A few more problems (end-of-chapter exercises and questions to ponder over maybe?) and it could be a really cool maths textbook. Very hard maybe, but really rewarding as well.
I also noticed you didn’t handle any geometry in the table-of-contents, though talking about computation (I assume you’re talking about Turing Machines?) seems an EXCELLENT idea! Kudos!
The project is still “alive” in the sense that I still vaguely hope to complete it or at least do something with it at some point. But I haven’t been actively working on it recently (I don’t really have time).
It’s definitely too heavy for high school kids. I’d go with (1) numbers are stretches of the number line (2) If you stretch by 3 then by 3 again, you stretch by 9 – stretches work like multiplying – the square root of 9 is 3 because you do 3 twice to get 9 (3) -1 is actually a reflection, and -2 is a kind of backwards stretch. (4) I can do the reflection -1 by rotating the number line about 0. (5) If I stop half way, that’s a rotation by 90 degrees, Let’s call that dot 1 above zero i. (6) i squared is -1. Weird, but handy. (7) Argand diagram (8) complex numbers.
“It” referring to Chris Chiesa’s (x,y) ordered pairs exposition. It’s too technical to start with – leave that formalism for later.
HI, I’ve only recently seen this, and am reading the chapters slowly. I like them a lot (though the style could be grating, I suppose, I don’t find it too bad actually).
I want to ask why (in Chapter 0, problems 0.39-0.41) you didn’t point out that the converse and the inverse of an implication are logically equivalent? If this was unintentional I suggest putting it in as an exercise.
Get back to working on it! (Please.)
Thanks for the feedback! The equivalence of converse and inverse is implied by Figure 0.5, but you’re right that it could make a good exercise.
You may also consider this online computer algebra calculator:
http://www.vroomlab.com/nhome
there is no download/install.