Thanks to all of you for the interesting comments on this online book, which came to this thread overnight in my time zone. I am a mathematics teacher, and this

Thanks to all of you for the interesting comments on this online book, which came to this thread overnight in my time zone. I am a mathematics teacher, and this link was recommended to me by a parent just more than a day ago. To answer one frequently asked question in the previous comments, no, I don't think any school anywhere includes teachers who use all these tricks in their teaching, and, yes, I think most of these classroom teacher tricks are specific to the United States. In the United States, the great majority of elementary school teachers are generalists, teaching all school subjects to their pupils, and their higher education does not prepare them well for teaching mathematics.[1] By contrast, elementary teachers in many countries where students learn more mathematics more thoroughly are subject-matter specialists, with mathematics teachers teaching only elementary mathematics, and other teachers teaching elementary pupils other subjects. My teaching is in two contexts: since 2007, I have taught voluntary-participation, extracurricular courses in prealgebra mathematics with additional advanced topics to middle-elementary age (mostly fourth grade) pupils in weekend supplementary classes. Many of those students are quite advanced intellectually for their age. They come to my classes (mostly through word-of-mouth recommendations from their parents' friends) from a ten-county expanse of Minnesota. Just this school year (that is, just since August) I am also on the faculty of an independent school, teaching all sixth grade pupils and "honors" seventh grade students mathematics at about the same level, although it is my intention this school year to move the seventh graders along into topics that can properly be called "beginning algebra" topics (to solving quadratic equations and graphing systems of two linear equations on the Cartesian plane). The school where I teach is reforming its mathematics curriculum with advice from a nonprofit consulting organization.[2] The reform program at my school is informed by international best practice in primary and secondary education and by what mathematics background is necessary to succeed in higher education in universities like MIT. (The founder of the consulting organization is an MIT alumna.) As the submitted ebook says, the hardest thing for a teacher to do is to encourage students to think rather than just rely on a mindless trick. This year I will have to set homework and tests that I write myself to probe for deep understanding of mathematical concepts and relentlessly try to find out how (and even whether) the learners in my care think about mathematics. Most of the rest of today on my weekend schedule is slated for writing a major unit test for my seventh graders, who use an excellent textbook[3] that is part of a textbook series that doesn't provide teachers with ready-made unit tests. The textbook is based on problem-solving and explaining mathematics from first principles (Chapter 1 takes the field properties of the real numbers as an axiom system to explain many principles of arithmetic) and is the best textbook for its topic available in English. [1] http://condor.depaul.edu/sepp/mat660/Askey.pdf http://condor.depaul.edu/sepp/mat660/Askey.pdf http://www.ams.org/notices/199908/rev-howe.pdf http://www.ams.org/notices/199908/rev-howe.pdf http://www.amazon.com/Knowing-Teaching-Elementary-Mathematics-Understanding/dp/0415873843 http://www.amazon.com/Knowing-Teaching-Elementary-Mathematic... https://math.berkeley.edu/~wu/Stony_Brook_2014.pdf https://math.berkeley.edu/~wu/Stony_Brook_2014.pdf https://math.berkeley.edu/~wu/MSRI_2014_1.pdf https://math.berkeley.edu/~wu/MSRI_2014_1.pdf [2] http://www.mathwalk.org/ http://www.mathwalk.org/ (I liked the old website of this organization better than the new website.) [3] http://www.artofproblemsolving.com/store/item/prealgebra http://www.artofproblemsolving.com/store/item/prealgebra The Art of Problem Solving website is a treasure trove of interesting mathematics education resources for learners of all ages.