the books by sanjoy mahajan are also a treat and teach real-world applications of mathematical and scientific thinking.
This is good stuff. Thanks for the pointer.
> 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.
For homeschooling, which for other parents on Hacker News could take the form of "afterschooling," I much prefer Miquon Math
for starting out my children, and then the Singapore Primary Mathematics materials (which now have an edition aligned to United States curriculums standards)
followed up by the Gelfand textbooks
appropriately supplemented by 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)
(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
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
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,"
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
also has some useful FAQ files about studying mathematics at a young age.
Open-ended questions and problems are indeed awesome. Moreover, they are an essential part of a sound education in mathematics, even at the K-12 (primary and secondary schooling) level of learning. But open-ended questions used for teaching purposes should be carefully written for sound teaching points, and teachers using them should have sufficient background in mathematics to guide student approaches to grappling with them. One of my favorite authors on mathematics education reform (Professor Hung-hsi Wu of UC Berkeley) began writing on that issue in 1994 with his article, "The Role of Open-ended Problems in Mathematics Education,"
and he followed up on that article with a wonderful article in the fall 1999 issue of American Educator, "Basic Skills versus Conceptual Understanding: A Bogus Dichotomy in Mathematics Education."
Since then, Professor Wu has written many more useful articles on mathematics education, including guides for parents, teachers, school administrators, and teacher educators on how to apply the new Common Core State Standards in mathematics better to improve mathematics education in the United States.
A good example of a beguiling textbook by a world-famous mathematician with lots of open-ended problems is Algebra, by the late Israel M. Gelfand and Alexander Shen.
Some of the problems in this book are HARD, but they are generally well posed problems of actual research interest to mathematicians, that just happen to be accessible to pupils just beginning to learn algebra.
AFTER EDIT: answering the question kindly posted below, one example I had in mind is that Gelfand asks students to figure out how many different ways there are to group terms in an expression with parentheses as the number of terms increases. This essentially asks the students to discover the Catalan number sequence.
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.
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,"
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
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
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.
Gelfand was a mathematician who also cared deeply about mathematics pedagogy, and his textbooks are delightful.
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.
which includes problems in representing numbers in binary notation and doing arithmetic with binary notation that are very approachable to young learners. (The problems are also very good review for undergraduate math majors
and help adults think more deeply about mathematics, which is why I like teaching with this book as a source of lesson topics.)
Edit after seeing other comment: I also mention to the children in my classes the Babylonian numerals,
in which the implicit base is base sixty. The link shown here mentions speculation from ancient Greece that that base was chosen because it has many different prime factors. That Babylonian system of numerals, whatever its origin, appears to be related to historical relics such as counting sixty minutes in an hour or 360 degrees of arc in a circle.
That's why it isn't trivially easy to be a mathematics teacher. The Russian tradition of mathematics teaching, which goes all the way back to the years when Leonhard Euler researched and taught in St. Petersburg,
does an especially good job of appreciating pure math for its delightful patterns and inherent beauty and elegance while at the same time being well informed by the many applications of math to science and engineering. One of my favorite textbooks for taking a balanced approached to making mathematics teaching interesting, rigorous, engaging, and practical is the late Israel Gelfand's and Alexander Shen's textbook Algebra published by Birkhäuser.
Many of the problems are HARD--the author is not afraid to pose research-level problems to first-time learners of algebra. On the other hand, some of the problems in some sections of the book are very approachable: "How to Explain the Square of the Sum Formula to Your Younger Brother or Sister." One section, "How to Confuse Students on an Exam," is laugh-out-loud funny. I love using this book in math classes that I teach as supplementary weekend classes for third-, fourth-, and fifth-grade-age pupils who like math and who want something more challenging than what is served up by the local school systems. I have clients from seven different counties in my sprawling metropolitan area. Pure math can be fun, and applied math can be fun, and both can be more enjoyable when they are taught hand in hand.
For homeschooling, I much prefer Miquon Math
I can recommend an algebra textbook with fun. And it was written by a mathematician with rather better credentials (theorems that bear his name in higher mathematics) than most algebra textbook authors. The book is Algebra by Israel Gelfand and Alexander Shen
I discovered the Gelfand-Shen textbook through Professor Richard Askey's review of that book.
Askey's review is actually a review of Ma Liping's book Knowing and Teaching Elementary Mathematics that quotes a passage of Gelfand's book in a sidebar. I later saw a glowing description of Gelfand's books in the same series in a bibliography by a U of Chicago mathematics student.
I use the Gelfand textbook to teach supplemental math lessons for gifted elementary-age students. They LOVE the Gelfand problems. They haven't even gotten to the really funny parts of the book yet, which such sections as "How to Confuse Students on an Exam." Gelfand's book(s) exemplify what you are looking for in math textbooks. Of course they are not used in very many public schools in the United States, nor in very many remedial college classes.
I agree with you that mathematics textbooks from the Soviet Union are often excellent. I use two of those in my own supplemental math classes,
Mathematics 6 by Enn R. Nurk and Aksel E. Telgmaa
(currently out of print, as explained on the webpage, as the second English edition is being prepared)
Algebra by Israel Gelfand and Alexander Shen.
Both are full of interesting problems and expect learners to be smart. I particularly like Gelfand's views on mathematics education:
"I would like to make one comment here. Some of my American colleagues have explained to me that American students are not really accustomed to thinking and working hard, and for this reason we must make the material as attractive as possible. Permit me to not completely agree with this opinion. From my long experience with young students all over the world, I know that they are curious and inquisitive and I believe that if they have some clear material presented in a simple form, they will prefer this to all artificial means of attracting their attention--much as one buys books for their content and not for their dazzling jacket designs that engage only for the moment. The most important thing a student can get from the study of mathematics is the attainment of a higher intellectual level."
The late professor Gelfand was optimistic--based on long experience--that many learners can attain a higher intellectual level in mathematics, perhaps because of his flexibility as a teacher: "Students have no shortcomings, they have only peculiarities. The job of a teacher is to turn these peculiarities into advantages."
is an inspirational example of how to make a textbook both humorous and challenging, and is the start of a great series of textbooks on secondary school mathematics from the viewpoint of advanced mathematics. I use that textbook in the supplementary math classes I teach, and the students LOVE the problems. His MacArthur Foundation Fellowship for his work in mathematics education (which I just learned about from the Wikipedia article you kindly shared) was very well deserved.
Interpreting that as a request to name the textbooks I find useful, I'll do that here.
The Gelfand Correspondence Program series
Basic Mathematics by Serge Lang
The Art of Problem Solving expanded series
When a student has those materials well in hand, it is time to work on AMC and Olympiad style problem solving,
and also the best calculus textbooks, such as those by Spivak or Apostol.
By far the best initial reading text is
Let's Read: A Linguistic Approach
but there are many other good reading series, including
Teach Your Child to Read in Ten Minutes a Day
(I devote more time than that to reading instruction, typically, because I use multiple materials)
and quite a few others. There is more junk than good stuff among elementary reading materials, alas.
Michael Artin's book is also a decent text aimed mostly at undergraduates:
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