Functions and Graphs (Dover Books on Mathematics)

nikofeyn · 2020-02-26 · Original thread

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...

rramadass · 2019-07-16 · Original thread

I suggest the following approach;

Start with some school textbooks for grades 8-12 i.e. Secondary Education. This is more for a refresher course in the absolute basics.

The above can be supplemented with the following books to develop intuition;

1) Who is Fourier - https://www.amazon.com/Who-Fourier-Mathematical-Adventure-2n...

2) Functions and Graphs - https://www.amazon.com/Functions-Graphs-Dover-Books-Mathemat...

After this is when you enter undergraduate studies and you have to fight the dragon of "Modern Maths" which is more abstract and conceptual. In addition to standard textbooks; i suggest the following;

1) Concepts of Modern Mathematics - https://www.amazon.com/Concepts-Modern-Mathematics-Dover-Boo...

2) Mathematics: Its Content, Methods and Meaning - https://www.amazon.com/Concepts-Modern-Mathematics-Dover-Boo...

3) Mathematical Techniques (i am linking this so you can see the reviews but get the latest edition) - https://www.amazon.com/Mathematical-Techniques-Dominic-Jorda...

Finally, if you would like to learn about all the new-fangled mathematics your best bets are;

a) The Princeton Companion to Mathematics - https://www.amazon.com/Princeton-Companion-Mathematics-Timot...

b) The Princeton Companion to Applied Mathematics - https://www.amazon.com/Princeton-Companion-Applied-Mathemati...

One important piece of advice that i have is to become comfortable with the Symbols, Notation and Formalism used in Mathematics. Most students are intimidated by the Formalism (which is nothing more than a precise form of shorthand to express abstract concepts) and give up on studying Mathematics altogether. This is a shame since it is merely the Form and not the Function of Mathematics.

tokenadult · 2012-09-13 · Original thread

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

http://www.aleks.com

and EPGY

http://epgy.stanford.edu/district/info.html

ALEKS

http://www.aleks.com/

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

http://amc.maa.org/

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.

tokenadult · 2011-08-25 · Original thread

Grothendieck "died" in the sense meant by Paul Erdos (stopped doing new mathematical research) at a surprisingly young age, while Israel Gelfand kept going into his nineties, not "dying" (ceasing to be a mathematician) until he "left" (died in the usual sense of the term).

http://www.math.rutgers.edu/~zeilberg/Opinion62.html

Gelfand was a mathematician who also cared deeply about mathematics pedagogy, and his textbooks are delightful.

http://www.amazon.com/Algebra-Israel-M-Gelfand/dp/0817636773

http://www.amazon.com/Functions-Graphs-Dover-Books-Mathemati...

http://www.amazon.com/Method-Coordinates-I-M-Gelfand/dp/0817...

http://www.amazon.com/Trigonometry-I-M-Gelfand/dp/0817639144...

Gelfand poses delightful problems that give students a workout in arithmetic (and CAN'T be done with a calculator) and that build conceptual understanding and interest in higher mathematics.

tokenadult · 2011-03-22 · Original thread

supplementing the math curriculum we use (Saxon)

For homeschooling, I much prefer Miquon Math

http://www.keypress.com/x6252.xml

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-I-M-Gelfand/dp/0817...

http://www.amazon.com/Functions-Graphs-Dover-Books-Mathemati...

http://www.amazon.com/Trigonometry-I-M-Gelfand/dp/0817639144...

appropriately supplemented by ALEKS

http://www.aleks.com

and EPGY

http://epgy.stanford.edu/district/info.html

tokenadult · 2009-05-07 · Original thread

I think it would be enlightening if you could provide the textbooks you buy.