Algebra
This book is about algebra. This is a very old science and its gems have lost their charm for us through everyday use. We have tried in this book to refresh them for you. The main part of the book is made up of problems. The best way to deal with them is: Solve the problem by yourself - compare your solution with the solution in the book (if it exists) - go to the next problem. However, if you have difficulties solving a problem (and some of them are quite difficult), you may read the hint or start to read the solution. If there is no solution in the book for some problem, you may skip it (it is not heavily used in the sequel) and return to it later. The book is divided into sections devoted to different topics. Some of them are very short, others are rather long. Of course, you know arithmetic pretty well. However, we shall go through it once more, starting with easy things. 2 Exchange of terms in addition Let's add 3 and 5: 3+5=8. And now change the order: 5+3=8. We get the same result. Adding three apples to five apples is the same as adding five apples to three - apples do not disappear and we get eight of them in both cases. 3 Exchange of terms in multiplication Multiplication has a similar property. But let us first agree on notation.
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What Hacker News says
- You can find some on Amazon.
Gelfand's Algebra: https://www.amazon.com/Algebra-Israel-M-Gelfand/dp/081763677...
Gelfand's Geometry: https://www.amazon.com/Geometry-Israel-M-Gelfand/dp/10716029...
Gelfand's Trigonometry: https://www.amazon.com/Trigonometry-Gelfand-Mathematical-Sem...
- I like books by Ron Larson, particularly his Trigonometry and Applied Calculus books -- the applied calculus title (intended for social science and business majors) vs his "Calculus" book (intended for math, physics, and engineering majors) is much gentler for people seeing this material for the first time. Although I do highly recommend his Calculus book once you have the other book down.
Gelfand also has some nice texts on Algebra, Trig, and Geometry that are reasonably cheap, especially if used.
I'm older and went back to school later in life to study math, and these are the books I learned that material (for the first time -- I flunked math thru high school) from.
Here are the exact titles and ISBN-10s:
Ron Larson, Calculus: An Applied Approach, ISBN: 0618218696
Ron Larson, Trigonometry, ISBN: 1133954332
Israel Gelfand, Trigonometry, ISBN: 0817639144
Israel Gelfand, Geometry, ISBN:1071602977
Israel Gelfand, Algebra, ISBN: 0817636773
And as others have mentioned, Khan Academy is pretty good, although I tend to prefer patrickJMT's explanations a bit more: http://patrickjmt.com/
- israel gelfand, a top russian mathematician, has books aimed at high school students that served as education materials for distance learning in russia. the series covers algebra, functions, trigonometry, graphs, geometry, and more.
https://www.amazon.com/Functions-Graphs-Dover-Books-Mathemat...
https://www.amazon.com/Sequences-Combinations-Limits-Dover-M...
https://www.amazon.com/Geometry-Israel-M-Gelfand/dp/10716029...
https://www.amazon.com/Algebra-Israel-M-Gelfand/dp/081763677...
https://www.amazon.com/Trigonometry-I-M-Gelfand/dp/081763914...
https://www.amazon.com/Method-Coordinates-Dover-Books-Mathem...
the books by sanjoy mahajan are also a treat and teach real-world applications of mathematical and scientific thinking.
https://www.amazon.com/Street-Fighting-Mathematics-Educated-...
https://www.amazon.com/Art-Insight-Science-Engineering-Compl...
- > Gelfand's wonderful basic math texts instead
This is good stuff. Thanks for the pointer. http://www.amazon.com/Algebra-Israel-M-Gelfand/dp/0817636773 http://www.amazon.com/Trigonometry-Gelfand-Mathematical-Semi...
> I haven't seen the physics sections
You should take a look ;) Check out the "Free tutorial" in particular. It is just 7 pages and you can print.
- I am revising some of my FAQ files about elementary mathematics education just now. Let's see what electrons I can paste in here from my drafts.
For homeschooling, which for other parents on Hacker News could take the form of "afterschooling," I much prefer Miquon Math
http://www.keycurriculum.com/products/supplementals/miquon-m...
for starting out my children, and then the Singapore Primary Mathematics materials (which now have an edition aligned to United States curriculums standards)
http://www.singaporemath.com/Primary_Mathematics_Stds_Ed_s/1...
followed up by the Gelfand textbooks
http://www.amazon.com/Algebra-Israel-M-Gelfand/dp/0817636773
http://www.amazon.com/Method-Coordinates-Dover-Books-Mathema...
http://www.amazon.com/Functions-Graphs-Dover-Books-Mathemati...
http://www.amazon.com/Trigonometry-I-M-Gelfand/dp/0817639144...
appropriately supplemented by ALEKS
and EPGY
http://epgy.stanford.edu/district/info.html
ALEKS
is a commerical online site (in which I have no economic interest) delivering personalized instruction in mathematics through precalculus mathematics. The ALEKS website includes links to research publications on which ALEKS is based.
I also recommend the Art of Problem Solving (AoPS)
http://www.artofproblemsolving.com/
(where I first took on the screenname that I also use here on HN) for more online mathematics instruction resources, and I also share specific links to specialized sites on particular topics with clients and with my children. I should note for onlookers that the articles on mathematics learning on the AoPS website
http://www.artofproblemsolving.com/Resources/articles.php?
are very good indeed, especially "The Calculus Trap."
My children make quite a bit of voluntary use of Khan Academy (both watching videos and working online exercises) and I am gratified that my previous suggestions to the Khan Academy developers here on HN
http://news.ycombinator.com/item?id=2760663
have been followed up as Khan Academy developers have communicated with me by email about new problem formats available in their online exercises, which are becoming increasingly challenging.
Besides that, I fill my house with books about mathematics, and circulate other books about mathematics frequently from various local libraries.
I also recommend that all my students use the American Mathematics Competition
materials and other mathematical contest materials as a reality check on how well they are learning mathematics.
In general, I think mathematics is much too important a subject to be single-sourced from any source. Especially, mathematics is much too important to be left to the United States public school system in its current condition. I was rereading The Teaching Gap: Best Ideas from the World's Teachers for Improving Education in the Classroom (1999) last month. It reminded me of facts I had already learned from other sources, including living overseas for two three-year stays in east Asia.
"Readers who are parents will know that there are differences among American teachers; they might even have fought to move their child from one teacher's class into another teacher's class. Our point is that these differences, which appear so large within our culture, are dwarfed by the gap in general methods of teaching that exist across cultures. We are not talking about gaps in teachers' competence but about a gap in teaching methods." p. x
"When we watched a lesson from another country, we suddenly saw something different. Now we were struck by the similarity among the U.S. lessons and by how different they were from the other country's lesson. When we watched a Japanese lesson, for example, we noticed that the teacher presents a problem to the students without first demonstrating how to solve the problem. We realized that U.S. teachers almost never do this, and now we saw that a feature we hardly noticed before is perhaps one of the most important features of U.S. lessons--that the teacher almost always demonstrates a procedure for solving problems before assigning them to students. This is the value of cross-cultural comparisons. They allow us to detect the underlying commonalities that define particular systems of teaching, commonalities that otherwise hide in the background." p. 77
Plenty of authors, including some who should be better known and mentioned more often by HN participants, have had plenty of thoughtful things to say about ways in which United States mathematical education could improve.
In February 2012, Annie Keeghan wrote a blog post, "Afraid of Your Child's Math Textbook? You Should Be,"
http://open.salon.com/blog/annie_keeghan/2012/02/17/afraid_o....
in which she described the current process publishers follow in the United States to produce new mathematics textbook. Low bids for writing, rushed deadlines, and no one with a strong mathematical background reviewing the books results in school textbooks that are not useful for learning mathematics. Moreover, although all new textbook series in the United States are likely to claim that they "expose" students to the Common Core standards, they are not usually designed carefully to develop mathematical understanding according to any set of standards.
The Epsilon Camp website
http://www.epsiloncamp.org/FAQ.php
also has some useful FAQ files about studying mathematics at a young age.
- Funny that I was just teaching that lesson about fraction comparison last Saturday morning, but I was using fraction pairs that are a bit more challenging (from the excellent textbook Algebra by the late I.M. Gelfand and Alexander Shen).
http://www.amazon.com/Algebra-Israel-M-Gelfand/dp/0817636773
I just put problem 40 from the book, which I taught last week to children of third-grade to fifth-grade age, into Wolfram Alpha's natural language interface.
http://www.wolframalpha.com/input/?i=Is+10001%2F10002+greate...
The Wolfram Alpha input and output is convenient for making the teaching point, and could spark a discussion about problem 41, which is
41. Which is greater, 12345/54321 or 12346/54322?
Of course a sensitive mathematics teacher is supposed to recognize at once that what is really being asked for by the second problem is a way to generalize when a/b is greater than (a+1)/(b+1) and when it is not. I will wrap up that part of the lesson next week.
I have posted recently here on HN that "There will continue to be an important role for in-person teachers,"
http://news.ycombinator.com/item?id=3555728
even after online teaching tools become much more fancy. I recommend some good tools (I don't think Khan Academy is the best available online mathematics teaching tool, but its price point is appealing) in that comment. I also include links in that comment to thoughtful recent articles about improving mathematics education. A skillful teacher will teach learners how to use tools, when tools are suitable for getting the answer, and how to use the unaided human brain and speech when that should be enough to get (and EXPLAIN) the answer. A big part of mathematics learning is learning how to use appropriate tools and methods in different circumstances. I don't decry online mathematics learning tools; I model in the classroom using the good-old human brain, sometimes with some help from pencil-and-paper or whiteboard-and-marker calculations, to puzzle through challenging mathematical problems
http://news.ycombinator.com/item?id=2760663
and get reality checks on whether the procedure used to reach a solution is correct or not.
AFTER EDIT: After posting this comment, I asked my Facebook friends (who include a number of professional and amateur mathematics educators, including homeschooling parents who have brought up International Mathematical Olympiad gold medalists) about the blog post submitted here, and one of those friends suggested that the blog post author look closely at the Art of Problem Solving
http://www.artofproblemsolving.com/
model of online mathematics education. "The medium is not the message, because the medium is only stepping in to do (interactively) what you would do in person if you could, and instead distributing the teaching resource more widely, but basically in the same mode." I agree with that suggestion, and with that comment on whether or not the medium is the message if online mathematics education is well done.
