Prealgebra

What Hacker News says

1. bumbledravenDec 2013
   Any artofproblemsolving.com book will be outstanding for this purpose. I suggest starting with Prealgebra by Richard Rusczyk.
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2. tokenadultAug 2011
   the lack of intuitive explanations for math is a major pet peeve of mine

   I'm a very visual thinker, and that is one reason I enjoy the new Art of Problem Solving textbook Prealgebra by Richard Rusczyk, David Patrick, and Ravi Boppana--

   https://www.artofproblemsolving.com/Store/viewitem.php?item=...

   it is full of interesting visual "explanations" and substitute for proofs in a book intended for a young audience.

   That said, I finally realized that I was limiting my mathematical development by insisting that every mathematical idea must appeal to my visual intuition. Some mathematical ideas are proven even if they don't appeal to visual intuition. In the words attributed to John von Neumann, "in mathematics you don't understand things. You just get used to them."

   http://en.wikiquote.org/wiki/John_von_Neumann

   That point of view makes a lot of sense to many of the best mathematicians.

   One more example of really interesting visual explanations of mathematical concepts is Visual Complex Analysis

   http://usf.usfca.edu/vca/

   by Tristan Needham. The book is delightful, and well reviewed, but it is not the sole path toward getting used to complex analysis.

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3. tokenadultAug 2011
   Is there any evidence that being unable to do mental arithmetic has any correlation with understanding complicated mathematics?

   That's a very good question. This question has not been studied as rigorously as it should have been. Here are some suggestive observations. Professor W. Stephen Wilson surveyed many colleagues (other research mathematicians) once to ask if they thought advanced mathematics could be learned without a basic understanding of arithmetic. The responses he received

   http://www.math.jhu.edu/~wsw/ED/list

   included comments such as "I am shocked that there is any issue here" and "That it is even slightly in doubt is strong evidence of very distorted curriculum decisions" and "One of my favorite attacks is that we are _helping_ the students by insisting that they do things by hand because otherwise they can waste a lot of time when the calculator would fail them." One especially thoughtful comment, by a mathematician who has long thought deeply about teaching mathematics, was "It might be argued that we do not really require students to fiercely add, subtract, multiply and divide in our university courses - which is true. But we do require an automatic understanding of these operations and why they work because WE BUILD FROM THERE."

   The longer story about understanding arithmetic--REALLY understanding it--and how that relates to learning beyond arithmetic can be found in the book Knowing and Teaching Elementary Mathematics by Liping Ma

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

   (well reviewed by two mathematicians who study mathematics education)

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

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

   A classic article on the subject is "Basic skills versus conceptual understanding: A bogus dichotomy in mathematics education"

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

   A recent effort to embody strong conceptual understanding of basic skills into a mathematics textbook is Prealgebra by Richard Rusczyk, David Patrick, and Ravi Boppana,

   https://www.artofproblemsolving.com/Store/viewitem.php?item=...

   which points to what young students should be able to do WITHOUT a calculator if they really understand mathematics well.

   One example I know, related to me by an economics professor, is teaching a lecture on economics in which the professor (the one who told me this story) said, as part of a calculation, "20 percent," and then was interrupted by a student who asked, "You just said '20 percent,' but you wrote '.2' on the blackboard. Why did you do that?"

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4. nwatsonSep 2021
   I'm home-schooling my daughter in math because Zoom and it not agreeing with her. She did great last semester and tests several years ahead and has a very intuitive sense of why things work the way they do, I'd like to keep her interest.

   However, we're now on to pre-algebra and using this book [0], "Prealgebra: The Art of Problem Solving". The first chapter is all about axioms, proofs of some sort, breaking down "obvious" conclusions back to their constituent proof-from-first-principles and it's not agreeing with her at all. Since this is the first week I'm struggling to find a way to have it all make sense, and I've concluded we need to kind of skip much of this first chapter (we'll look at the summary) and get to the later content which is more intuitive and applies "obvious" principles, and come back periodically to revisit the more mechanistic content in the first chapter. I think the parent post's description of exploring a topic matches how my daughter will come to understand the whole "algebra stack." (Wish me luck.)

   [0] https://www.amazon.com/Prealgebra-Richard-Rusczyk/dp/1934124... , Amazon: "Prealgebra: The Art of Problem Solving"

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