Vision in elementary mathematics

What Hacker News says

1. jgordAug 2026
   I have strong opinions on how Calc should be introduced - visually.

   I think we don't cover basics like the distributive rule in school very well, and that it should be a much more nuts n bolts visual / measuring / counting experience.

   Ive attempted to outline how I think this stuff should be taught, by making a video tour of the concepts - from Counting, to Distributive rule / algebra, to Quadratics then the Derivative, here :

   https://www.youtube.com/playlist?list=PLEInJ-Z4qBKYxbK1Mm13g...

   All of these things are covered in some great books :

     W W Sawyer Vision in Elementary Mathematics
     Algebra by Gelfand
     Calculus by Thomas
   

   We have superb resources now like 3Blue1Brown, KhanAcademy and ArtOfProblemsolving.com / BeastAcademy .. so you _can_ get your kids a superb math education, even as many schools seemingly give up on teaching Algebra and Calculus.
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2. krnAug 2018
   > The OP is talking about teaching community college mathematics. That's not introductory university mathematics; if you are lucky it's Pythagoras' Theorem and quadratic equations.

   W. W. Sawyer's books, such as Mathematician's Delight[1] and Vision in Elementary Mathematics[2], seem to be exactly for this audience.

   [1] https://www.amazon.com/dp/0486462404/

   [2] https://www.amazon.com/dp/048642555X/

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3. tokenadultAug 2012
   The problem is easy to state -- we're teaching mathematics in the wrong order.

   Where would we go to find a real-world example of mathematics being taught in the correct order to elementary-age pupils? There are many different schools (and many different settings for learning outside school) in many different countries. What is a good place to look for examples of demonstrably successful practice?

   AFTER EDIT: Thanks for the reply. The thing I would want to look at in such intervention studies is how the teaching effectiveness is measured.

   AFTER FURTHER EDIT: Oh, I see the video "Teaching Math Without Words, A Visual Approach to learning Math from MIND Research Institute" is one I have watched before, and have discussed among other mathematics teachers. Similarly, the publication Mathematics Teaching in the Middle School is a publication I subscribe to as a member of NCTM. I should start out in my reaction to both of these links you've kindly shared by pointing out that my own bent in thinking about mathematics is to think visually, and indeed that is why I like Sawyer's book Vision in Elementary Mathematics (which leads off my longer reply in this thread) so well.

   But although I like visual approaches to teaching mathematics very well, and use them in the mathematics classes I teach, I also like studies of educational interventions to check results. So far, I can't find a publication by independent researchers (after searching Google Scholar and the new Microsoft Academic Search) that verifies the educational effectiveness claims made in the video for the proprietary intervention described there. The company's own website

   http://www.mindresearch.net/cont/research/re_publications.ph...

   provides only a very sparse set of links to published research on the issue.

   Plainly, a more visual approach along the lines of the typical school textbooks used in Taiwan, Singapore, and Japan would be a better presentation of mathematics content than is found in typical United States textbooks. That should be possible to deliver without any computers at all, as it was delivered to pupils in Taiwan in my wife's generation, when Taiwan was still a poor, Third World country.

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4. tokenadultJul 2012
   I am wondering what books would be good to recommend to parents whose children are deeply in love with mathematics, especially if the parents themselves have only an average background in mathematics. That was my situation a decade ago, and I found some helpful books I'll mention here, but I think a lot of Hacker News participants can suggest some books that are even better. I'll be speaking to a group of parents whose children are primary-school age but already well familiar with secondary school algebra and geometry, taking an advanced mathematics summer course.

   Books that I found helpful for understanding mathematics when my oldest son, a mathematics-lover, was growing up include

   Concepts of Modern Mathematics by Ian Stewart (Dover reprint)

   Numbers and Geometry by John Stillwell (Springer)

   Vision in Elementary Mathematics by W. W. Sawyer (Dover reprint)

   The Art of Problem Solving by Sandor Lehoczky and Richard Rusczyk (Art of Problem Solving)

   A frequently recommended book for this need is Developing Math Talent by Susan Assouline and Ann Lupkowski-Shoplik, but I haven't made much use of that book.

   What am I missing? What do you recommend?

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5. tokenadultMar 2011
   Cal Newport, the author of the submitted blog post, draws comments both here on HN and on his own blog pointing out that deep understanding of a subject doesn't necessarily equate to VISUAL thinking about a subject. There is a big literature on "learning styles" and some attempts by some schoolteachers to categorize children by what their preferred learning styles are. When I have taken learning style questionnaires, and when I have asked my wife (a piano performance major and private music teacher) about this, the answer on learning styles is "all of the above." I personally think, based on my observations of successful learners of a variety of subjects, that learning styles are themselves learnable, and a learner with a deep knowledge of a particular subject will know multiple representations of that subject. My wife has had many piano performance courses, and also music theory and ear training courses, and has learned visual representations of music both in the form of standard musical notation and in the form of "music mapping,"

   http://www.amazon.com/Mapping-Music-Learning-Teachers-Studen...

   which she has found very helpful.

   As for mathematics, the subject I teach now, I have always cherished visual representations of mathematical concepts, for example those found in W. W. Sawyer's book Vision in Elementary Mathematics

   http://www.amazon.com/Vision-Elementary-Mathematics-W-Sawyer...

   http://www.marco-learningsystems.com/pages/sawyer/Vision_in_...

   But other mathematicians who taught higher mathematics, for example Serge Lang, recommended memorizing some patterns of multiplying polynomials by oral recitation, just like reciting a poem.

   http://www.amazon.com/Basic-Mathematics-Serge-Lang/dp/038796...

   The acclaimed books on Calculus by Michael Spivak

   http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098...

   and Tom Apostol

   http://www.amazon.com/Calculus-Vol-One-Variable-Introduction...

   are acclaimed in large part because they use both well-chosen diagrams and meticulously rewritten words to deepen a student's acquaintance with calculus, related elementary calculus concepts to the more advanced concepts of real analysis.

   Chinese-language textbooks about elementary mathematics for advanced learners, of which I have many at home, take care to introduce multiple representations of all mathematical concepts. The brilliant book Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States by Liping Ma

   http://www.amazon.com/Knowing-Teaching-Elementary-Mathematic...

   demonstrates with cogent examples just what a "profound understanding of fundamental mathematics" means, and how few American teachers have that understanding.

   http://www.aft.org/pdfs/americaneducator/fall1999/amed1.pdf

   http://www.ams.org/notices/199908/rev-howe.pdf

   Elementary school teachers having a poor grasp of mathematics and thus not helping their pupils prepare for more advanced study of mathematics continues to be an ongoing problem in the United States.

   http://www.ams.org/notices/200502/fea-kenschaft.pdf

   In light of recent HN threads about Khan Academy,

   http://news.ycombinator.com/item?id=2348476

   http://news.ycombinator.com/item?id=2350430

   I wonder what Khan Academy users who also have read the submitted blog post by Cal Newport think about how well students using Khan Academy as a learning tool can follow Newport's advice to gain insight into a subject. Is Khan Academy enough, or does it need to be supplemented with something else?

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6. tokenadultJun 2009
   Is the putative student "bad at math" or "bad at math the way it's taught in high school, perhaps even by a certain teacher"?

   Here is one of my favorite quotations about mathematics teaching, used as my signature tagline on the first online forum where I used the screen name "tokenadult":

   "The proper thing for a parent to say is, 'I did badly at mathematics, but I had a very bad teacher. I wish I had had a good one.'" W. W. Sawyer, Vision in Elementary Mathematics (1964), page 5.

   Yes, the book from which the quotation comes is very good, and very good for learners with a visual learning style.

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