mental math question

Kumon asks kids to do multi-digit multiplication problems without writing down any of the 'carried' digits:

Students master the multiplication tables by practicing until they can answer immediately. Next, students learn up to 4-digit by 1-digit multiplication with mental carryovers.

What do you think?

Interesting post on the subject of mental maths here.

long division & mental math

Barry Garelick:

Investigations and EM both boast about how their programs emphasize mental math. Though they don't seem receptive to the idea of teaching long division. I wonder why not? Long division is an ideal way to hone mental math skills. A problem like 56,098 divided by 84 requires a lot in the way of number sense and mental math. The first step is looking at the maximum number you can have to multiply by 84 and stay under 560. Rounding, 560 divided by 80 would be 7, but a quick glance will tell you that 7 x 84 will be greater than 560, which quickly leads you to "6" for a choice. And so on and so forth. But reformers don't want to hear about all that. It's drill and kill, has no value, and anyway, calculators have made long division obsolete.

Obi-Wan on calculators

It bugs the hell out of my kids, but I love saying it:

The fact that you're asking for a calculator is the surest sign that you shouldn't have one.

I have seen far too many kids so clueless about basic computations that they can't even recognize an answer which is off by several orders of magnitude. They add two numbers and get a smaller number. No clue.

Obi-Wan

I don't do mental math well at all. I was never taught, and when I taught myself how to do it in order to teach my little afterschool Singapore Math class, I didn't practice it enough to learn it well. (I think Saxon Math has some mental math exercises, right? Or am I confusing the two curricula?)

Now that I know a bit about it, I'm a big fan of mental math.


math dunces

Math in the late 1800's

Don Potter sent me a link to an interesting 1879 math book in Google Books, "Graded Work in Arithmetic" by S. W. Baird. It has an interesting visual method of showing fractions similar to how they're depicted in Math-U-See. (The unexpected results of web searches, when "Graded Work in Arithmetic" comes up under a search for "phonics charts." There are, of course, ads for phonics charts, McGuffey Readers, and some other books and visual aids at the end of the book.)

Looking through "Graded Work in Arithmetic" prompted me to look up Baird's "American Mental Arithmetic," which was even more interesting. His explanation of fractions for this book used a different method than his 2nd grade book and showed a number line explanation along the lines of Wu.

Also, his preface had the following statement (page 3 in the book, page 5 in google book pages)

"Many who have broken the habit, in adding, of saying "6 and 8 are 14 and 6 are 20," are still saying in subtracting, "6 from 10 leaves 4" ; in multiplying, "9 times 8 are 72, and 4 are 76" ; and in dividing, "12 [divided by sign] 5 = 2 and 2 remaining." Special stress is laid upon the improtance, in performing operations, of dropping all unnecessary words, since the mind reaches results much more rapidly without them."

Interestingly, Don Potter has told me in the past that his father could multiply with lightning quickness and would always say "five 3's are 15" instead of today's common "five times 3 equals 15." I've only taught Mary a few multiplication facts, but I try to follow the two 5's are 10 pattern when teaching her in case there is something to this...I figure it can't hurt and could be helpful. I still teach what the equals sign is and what it means, but I always have her say the shorter are instead of equals when reciting addition and multiplication facts out loud. (Are may also be less abstract than equals as well as shorter.)

Bailey also states,

"Percentage is taught without rules or formulae, and without the use of the terms base, amount, and difference, although one page is devoted to the after the subject has been completed. The student comes to see clearly that the various exercises in percentage do not need special rules, but are familiar cases slightly modified since the symbol "%" is used instead of hundredths. Interest is taught by the 6% mehtod and by the modification of this method in general use among bankers."

His concluding sentence of the preface is,

"Let it be remembered, that he who relies upon thousands of special rules is but a pygmy beside the giant who can apply a score of general principles to millions of particulars."

Perhaps we're the pygmies today for throwing all this out with the bathwater without trying to figure out if there were reasons for some of the things that were taught and some of the methods that were used to teach in the past.

Mental Math link

I spent the morning with my new friend Angela McIver, who evaluates the math skills of middle-schoolers and older children. I saw some fascinating videos of children struggling through basic math problems in ways that suggested huge gaps in their elementary school instruction.

Angela is a big proponent of mental math, which Reform Math programs not only marginalize, but, in insisting on explained answers, actually mark children off for. Among other things, mental math forces you towards maximally efficient strategies, with all the mathematical insights this entails.

She recently searched the Internet for mental math activities, and just sent me a good link.

Stu Savory: Learn to Take Cube Roots In Your Head

Stu writes:

Most people can't do mental arithmetic any more. What a pity indeed :-(

Almost everyone relies on calculators these days. So today I'm going to tell you how to do an apparently difficult feat of mental arithmetic. Today you will learn how to take integer Cube Roots in your head! It's really dead easy, believe me!

Go to this blog post for the technique