tag:blogger.com,1999:blog-7718462793516968883.post5335337417977899721..comments2023-03-02T12:12:05.847-05:00Comments on MadMath: The War on StructureDeltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-7718462793516968883.post-9551974338077305792013-07-02T18:03:10.778-04:002013-07-02T18:03:10.778-04:00That's fine -- and for example, I still distin...That's fine -- and for example, I still distinctly remember the similar grade-school exercise that crystallized for me that all numbers divisible by 9 are also divisible by 3, etc. <br /><br />But my usual reply is "you need both" this conceptual intuition, plus the written algorithms for larger problems. Having only one or the other doesn't cut it. Students that never memorized the multiplication table wind up crippled for long division, adding fractions, conversion to decimals, factoring equations, etc. <br /><br />In fact, understanding that our number system was designed to support certain written addition and multiplication algorithms may be the most important intuition of all.<br />Deltahttps://www.blogger.com/profile/00705402326320853684noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-2035411836805643682013-06-25T14:23:09.542-04:002013-06-25T14:23:09.542-04:00There are different ways to get to the structure.
...There are different ways to get to the structure.<br />Interestingly, when I introduced my son to arithmetic, I did not really focus on the traditional, "algorithmic" way of calculating sums and products. My focus was almost entirely on developing intuition with numbers. <br /><br />So addition is simply counting, multiplication is counting by multiples, and division is grouping. <br /><br />So a problem like 3x6 he may solve by counting 3,6,9,12,15,18. This sounds tedious but has a couple of advantages:<br />It is easier to extend this process to problems that don't fit in the multiplication table. For example, 20x19 he may solve as 20x20 - 20, it is simply recognizing that you are counting by twenty and can count backwards as well as forwards.<br /><br />With intuition, it is less likely to make certain types of mistakes with numbers. Although he cannot produce the multiplication table "from memory", he can create it, and through this process will be introduced to certain patterns and properties of numbers which are generally applicable. For example, multiples of 5 always end in either zero or 5. This lets you instantly recognize certain kinds of mistakes.<br /><br />I believe it would be much harder for him to develop this type of fluency when he is older. At that point he would have no choice but to memorize algorithms. He would not be able to really understand numbers though. Many kids don't have that luxury.<br /><br />I have met some adults that believe a billion is equal to a hundred million. I am talking about college educated or even graduate level adults. That is not a problem in calculation -- it is a lack of intuition, of basic understanding of numbers. They simply haven't devoted enough time to it. Gregory Matoushttps://www.blogger.com/profile/02695669454780832427noreply@blogger.com