I’ve been wanting to do this for so long, and I’m sure many readers have wanted it too – I’m making videos that will help anyone learn to be fast and confident (not to mention amazing) at the basic operations – addition, subtraction, multiplication and division.

The first series of lessons will be free, as I learn the software and make them more professional. After that, I’ll offer them as a course for a monthly subscription fee that will be astoundingly cheap. It will come with podcasts, webcasts, tele-seminars, tons of downloadable worksheets, progress reports, daily journals, a forum and total e-mail support.

By the end of the course, anybody who puts a bit of effort into it will be able to freak out any teacher, job-interviewer, college entrance examiner, or basically anyone with their calculating abilities.

And I guarantee that you won’t have a second of confusion with any of the methods if you follow them from the beginning. All of the addition and subtraction lessons will be abled to be mastered by just about any second-grader, and many first-graders (or even younger). 

The video you are about to watch is not even the beginning. I’ll be pointing you to even easier lessons for toddlers. But this video is a good place to get a feeling for what things are going to be like. 

Remember, this is just the first draft, as I learn the new software (Wacom Graphics Tablet, iShowU video screencapture, iMovie 8, etc.) Thank God it’s on a Mac, so the learning curve will not be too steep!

Before I clean up the drafts, I’ll make sure I put up a video every few days, so in a short time you’ll be able to long rows of many column in your head, faster than someone else with paper and pencil, and more accurately. 

OK, go have fun with the video. It’s about 12 minutes long. The ones in the future will be shorter. 

And please –  leave a comment. If I don’t hear from anyone, I’ll think nobody wants to learn this stuff, and I’ll have to turn my attention elsewhere. But it would be a shame not to offer this to people, because it’s so easy and fun, and it will give you a huge advantage over anyone who can’t do it.

Georg Cantor

Here is an except from that comment:

“Frankly, I don’t care if an elementary school child can add long columns of numbers in their head – it is an almost worthless skill. I do care if they can think about mathematical concepts.

Better to teach them to come up with simple proofs (not memorized proofs) of basic facts in math.

Better that they should understand what a prime number is, and why we care about prime numbers.

Better that they should learn to enjoy slow, deep thought about puzzles and concepts.

That is where the gold standard in math education is.”

I wanted to revisit this thought, because Penny brought up some great points. I don’t disagree with any of them. But I must say that I, as well as a lot of the readers are coming from a different place. Penny is a brilliant research mathematician. A lot of us, on the other hand, basically have a history of thinking that we sucked at math (at least until we came upon Math Mojo, and learned that almost no one sucks at math, but some sometimes the way math is taught sucks.)

I wanted to address some of the points Penny made, because those points made me think a lot this month. Here’s

Added Value

Penny, I agree that the points you brought up are better than simply being able to add long columns of numbers in your head, but I wouldn’t dismiss learning quick addition.

One reason is the practical value. I catch plenty of mistakes at the checkout counter.

But the real reason is that it is one of the first real mental-math skills that gives you the feeling of what I call “numbers juggling.” This isn’t a trivial thing, either. I’ll go into depth about it in a future post, but I’ll touch on it here, because it’s important.

All you need is balls…

I don’t know if you are aware, but I used to be a busker (street-performer) in Europe. I mostly did magic, but I also juggled in my act. I’m not exactly a brilliant juggler, but I’ve gotten to the point where juggling is practically meditation. There is an amazing circular “feeling” you can get while juggling. It is not just from the patterns that the balls make in the air, but also from how your eyes follow the arcs, and other feelings in your body.

When I practice speed-math I get a similar “rush.” Other people have reported the same thing. I imagine that it’s caused by (among other things) the patterns my eyes make while manipulating imaginary numbers in the air.

This feeling makes the whole phenomenon of reckoning with numbers more “plastic.” (People who like to talk about “modalities” and only have a superficial understanding of what “kinesthetic” means might call it “kinesthetic.”)

This perceived plasticity makes math, and many other abstract concepts more understandable to me than they otherwise would be.

I imagine that many people could use this to their benefit, and that’s one of the reasons I started Math Mojo.

I’m no Einstein, but…

Penny went on to mention this about her daughter:

“She is now a molecular biologist. She is still no whiz at adding numbers in her head!”

Of course, not everybody has to learn every math skill, but I don’t think there’s a reason not to learn something. Einstein had a great memory and orientation for concepts, but he still occasionally had to phone his wife to find his way home. Is that good?

Some lightning calculators were idiot-savants, it is true. But that does not mean that lightning-calculating is useless (it’s the skill that gave them the “savant” part of that label, after all).

Other lightning calculators include the astronomer Trueman Henry Stafford, the physicist André Marie Ampère, the mathematicians Srinivasa Ramanujan, Wim Klein, John Wallis, Sir William Rowan Hamilton, Norbert Wiener, A.C. Aitken, and no less than Johann Friedrich Gauss. And of course let us not forget that Brittney Spears is noted as an accomplished calculating prodigy. (Ok, I only threw that in there to see if you were paying attention.)

These people generally exhibited their prodigious calculating abilities when they were young, and their skills waned as they got older.

They’re only going through a phase…

It’s interesting to consider if it could be that as they matured mathematically, their fascination with pure calculations wore off as they investigated more fascinatingly complicated things. And could it be that they would never have developed that maturity if they had not gone through the lightning-calculating stage of their development?

Could it be that their fascination with math was developed so intensely by that phase, where it might not have been otherwise? Maybe some types of personalities are a perfect match for ready-reckoning, and we’d do them a disservice by dismissing it as trivial.

All I know is that we are letting a lot of children through the cracks, and it would be a shame not to give them the opportunity to grab on to something that might turn them onto a path that might lead them to higher knowledge on their own terms, not just what we think should be valuable.

When they have to re-write simple arithmetic, we are teaching them that using a crutch is good. It’s the right way, it’s the only way. That sucks!

Understanding and practicing mental math is good for visualization, mental flexibility, understanding that new pathways can be better.

For most people, not doing arithmetic mentally is like not learning to run. “I can walk already. I’ll just learn how to build wheelchairs and cars. That’s progress.” And never learn to run? But what “value” does running have? Oh, come on, that would be just a ridiculous question.

I’d like to thank all the readers who know a lot more about math than I do, and come and comment on this site. It gives me so much opportunity to think and communicate. I hope it does the same for Math Mojo readers.

Mission Position

Part of the mission of Math Mojo is to be a liason between the people struggling with math, and the people who love, enjoy and understand math. Normally they never meet.

What is an “enhanced podcast?” It’s one with visuals, like a PowerPoint presentation. Not every browser can see it, though, although most can. You may need a fast connection to hear and view it – at least a bit faster than dial-up, although it will work with dial-up if you have a lot of patience. You don’t need iTunes or an iPod to listen to or watch it, although if you want to subscribe to it, iTunes is the way to go. iTunes is free, and if you don’t have it, you can find out all about it and get it at apple.com.

What is “subscribing” and why should you do it? When you go to the above site, to view the podcast, there will be a button on it, from which you can “subscribe.” That means that every time a new episode is published, it will automatically be sent to your computer the next time you open your iTunes program. That way, you will always be up-to-date with new podcasts from MathMojo, without having to do anything further.

A word about the podcasts. They were made on a mac. I love my mac. I never was a geek, but this thing is user-friendly. It’s user-promiscuous! Using Garageband, iWeb and a dotMac account, it is pretty simple to do podcasts. I hope to get more heavily into this technology, because it is a great way to communicate with the world.

I’ll also be putting up some videos on this blog, and on the main Mathmojo.com site, and on YouTube in the near future, so stay tuned.   By the way, if you are at all interested in the kind of magic I do, you can check out a very old video I made (in about ’91 or so, when I was living in Germany), below. 

Click here to view the embedded video.

This post is about meaning and math

First we’ll learn a math concept - Lowest (or Least) Common Denominator (or LCD). Then we’ll talk about how it’s sometimes used in everyday life.

In layman’s (non-mathematician’s, in this case) terms, the LCD is the largest partition of something that will also go into another thing of the same kind.

What the heck does that mean? Well, if you have, say, 1/4 of a apple blueberry crumble pie (you can tell where my mind is this balmy upstate Memorial day), and you have 1/5 of another apple blueberry crumble pie, how much apple blueberry crumble pie do you have altogether?

Changing the Denominators

You probably know that you can’t add fourths and fifths without turning them into something else, because the denominators (lower numbers of the fractions) are different.

What do you change those denominators into? You change them into the lowest common multiple. That means the lowest number that both of those denominators go into.

The lowest number that both five and four go into (without remainders) is 20. So the lowest common multiple of 4 and 5 is 20, which means that the lowest common denominator of 1/4 and 1/5 is 1/20.

Now, how many groups of 1/20 do you need to make up 1/4? Well, when you turned that 4 (the denomenator) into a 20, you had to multiply it by 5, so you have to also multiply the 1 (the numerator) by 5, giving us five twentieths. That means that in 1/4 can be expressed as 5/20.

A good way to look at it, is to imagine that you want to keep to things in proportion (the numerator and the denominator). If you make one bigger, you have to make the other equally bigger. Like when you grow; If your head grows, the rest of your body should grow along with it. Usually.

So far we’ve used the lowest common multiple to turn the fourths in to twentieths. Now we need to turn the fifths in to twentieths, so both numbers will have a common (same) denomenator.

Since twenty is the lowest common multiple, we also want to turn the one fifth into twentieths. To turn 5 (the denominator) into 20, you have to multiply by 4. So you aslo have to multiply the 1 (the numerator) by 4, thus giving you four twentieths. That means that in 1/5 can be expressed as 4/20.

Adding with the new (common) Denominators

So if you want to add one-fourth (which, as we now know is 5/20), to one fifth (which, as we now know is 4/20), instead of saying 1/4 + 1/5, you can state it as 4/20 + 5/20. That is easy. Four of anything plus five of the same thing is always nine of them. So 4/20 + 5/20 = 9/20.

You couldn’t have added two uncommon things without changing them into groups of the smallest unit they had in common.

One more quick example:

2/3 + 6/7 =

• The lowest number that 3 and 7 have in common is 21. So that is our LCD.
• You had to multiply the 3 by 7 to get 21, so you must do that to the 2 in the numerator as well, which gives you 14. So the first fraction would be changed to 14/21.
• You had to multiply the 7 by 3 to get 21, so you must do that to the 6 in the numerator as well, which gives you 18. So the first fraction would be changed to 18/21.
• 14/21 + 18/21 = 32/21, which is the answer.

This answer can be turned into a mixed fraction (a whole number and a fraction) which makes it more convenient to work with. It’s also usually what schools want you to do (although they are seldom sure why they want you to do it). This isn’t a lesson about simplifying and mixed fractions, but here’s the skinny on it:

32/21 is the same thing as 21/21 + 11/21.  21/21 is the same thing as 1, so 21/21 + 11/2 = 1  11/21 (that’s “one and eleven twenty-firsts.”)

You basically divide the numerator by the denominator, and write the answer (the quotient) as a whole number. If there is a remainder, that becomes the new numerator of the fraction.

One more example of that? OK:

53/16 (it even looks like a division problem).

• 16 goes into 53 3 times, giving you 3, remainder 5 (because 16 x 3 = 48)
• The 3 is the whole number, the remainder goes over the 16, giving you 3  5/16 (“three and five sixteenths.”)

What about the meaning? I’ll get to that when I update this post.

I sure hope this helps you understand the basics of using the Lowest Common Denominator to add fractions.

All the best,

Brian (a.k.a. Professor Homunculus)