We’re going to use what I call “numbers crunching” to check. That is the same as using the nines-remainders. You do know how to get the nines-remainder of a number, don’t you? It’s very simple, but it takes a bit of explaining. [...]]]> To check multiplication of single digits by longer numbers with playing cards:

We’re going to use what I call “numbers crunching” to check. That is the same as using the nines-remainders. You do know how to get the nines-remainder of a number, don’t you? It’s very simple, but it takes a bit of explaining.

It also pays to know why checking with nines-remainders works. Both of those things are beyond the scope of this article, but I’m working on a booklet and a video about how to check your answers for all of the basic operations of math using “number crunching”. There are lots of tips and shortcuts that make this method absolutely simple and effective. Let me know if you’re interested by using the “Contact” box near the upper right hand corner of this page.

(This video will be re-edited and uploaded by the end of Wednesday, April 30)

If you know about crunching, you’ll be interested to know that practicing with cards like this is perfect for checking with crunching. It turns out that if you crunch all the digits from zero to nine, you get a crunch number of 0.

Since we’ll always use sets of cards to represent the digits from zero to ten, we’ll always get a crunch number of 0.

So take whatever digit you were multiplying by, you’d have to multiply it by 0 to get your final check number. As you know, anything times zero is zero, so whenever you practice with cards like this, your check number will always be zero!

So if the crunch number of your answer is anything but zero, you have made a mistake somewhere.

Starting with ten cards is pretty easy. It turns out that as long as you use complete sets of all ten digits from 0 to 10, you will always have a check number of 0, no matter how many sets you use. So you can use ace to ten of as many suits as you like (as long as you remember that the tens count as zeros, and aces as ones).

That makes sense, doesn’t it? Because if a single set of zero to nine crunches to 0, then two sets must also crunch to 0, because 0 + 0 still equals 0.

In a very few days you should be able to work yourself up to multiplying any single digit number by a full set of forty cards (four sets of ace to ten, with all suits) within a few minutes. And then another minute or so to check them.

One of the benefits of doing it like this is that you are going to have to do all the additions and subtractions to get the nines-remainder of a forty or forty-one digit number in order to check it. That’s great practice in those two operations.

This is a fantastic morning exercise for children or adults. When you do something like this before breakfast, your mind becomes much more awake than it would have been.

Watch the video, then try it.

By the way, you may have noticed that at some point in video, I say, “The number has to crunch to nine,” where you may have thought I meant to say, “zero.” But remember, in Mod 9 (nines-remaindering, or “crunching”) zero is nine.

Did you know that technically, you can use any digit-remainder to crunch with, not just the nines-remainder? Most of the shortcuts (ask me about them) don’t work with nines-remainders other than nines, though, so that’s why we use nines, mostly.

Elevens-remainders are good to use as well. They have some shortcuts, just not as many as the nines, though. If you need to be absolutely sure of your answer, it’s best to check with the nines, and the elevens. I’ll have more posts up soon about each of them, and they’ll be thoroughly covered in the booklet that will be out soon.

Remember, a little bit of knowledge can be dangerous; so when you use numbers-crunching, be aware that it is “just a trick” until you understand it more deeply.

Original Photo by Norsehorse Edited by Brian

Ah, I love it when readers beat me to the punch!

The comments to the original post pretty much sum up the paradox and it’s solution very well.

Khaled’s and Mark’s comments illustrate perfectly one of the things I wanted to point out about this [...]]]>

Original Photo by Norsehorse Edited by Brian

Ah, I love it when readers beat me to the punch!

The comments to the original post pretty much sum up the paradox and it’s solution very well.

Khaled’s and Mark’s comments illustrate perfectly one of the things I wanted to point out about this puzzle. That point is:

Just because something is phrased a certain way is not reason to assume that that phrasing is the best way to represent the problem. And one way to critically examine the situation is to reframe it in a mathematical equation.

Khaled said, “Interesting how, once you assume that you can implicitly trust a given source, you can be led through any logic, or illogic, and have a lot of trouble pulling yourself back to a critical mindset.”

How true. Then Mark gave a good method to understand how to see where the paradox lies when he said, “I started to write an equation, because properly written equations can solve all counting problems, but then realized that this was pointless, because adding 2 dollars to the 27 dollars the guests paid did not reflect what happened.”

Exactly! The question was phrased to lead you to believe that because the facts were a certain way (which it accurately represented) you had to see it in a certain way (which was anything but accurate).

This kind of paradox is harder to identify than simply by “fact checking.” If you do a diligent fact-check of the problem, you’ll find that no facts are misstated. In fact, everything in the entire problem is on the up-and-up, except for the last sentence – “Adding the two dollars that the bellboy kept would make a total of $29 dollars.”

There’s the rub. Why would you add the two dollars that the bellboy kept?

A good way to look at the puzzle is to “follow the money,” or mentally picture the flow of what went where, instead of just listening to the arguer’s “logic” and being lead down the garden path.

So let’s follow the money:
$30 went from the men to the manager.
$5 went to the bellboy.
Of that $5, $3 went to the men, and $2 was kept by the bellboy. There is no reason to add two dollars to anything.

The last sentence of the puzzle is added just to throw you off the actual path of the money. Magician’s do this all the time. It’s called “misdirection.” but when magicians do it, they (hopefully) are doing it for entertainment purposes, only.

Magicians thrive on what we call, “the willing suspension of disbelief.” We assume you came to the show to relax your mind and just have fun in the fantasy world of “that which can’t and does.”

On the other hand, politicians, fanatical religious lunatics, some salesmen, and an awful lot of educational policy-makers thrive on the “the unwitting, or coerced suspension of disbelief.” And that makes all the difference.

If you willingly part with something that is not inalienable – (your temporary suspension of belief, your money, etc.) – well, that is your decision. On the other hand, if someone coerces you or tricks you into actually accepting something as real, or takes your money without your agreement (as in a sale, loan, etc.) they are committing a crime.

And stealing your mind is a lot worse than stealing your money, in the long run.

That is one of the reasons I started Math Mojo to begin with.
Quick story:

I used to work in a Job Corps facility. I won’t go into detail, but in a nutshell, Job Corps is a government boondoggle set up to have corporations get money for running educational and vocational programs for deserving sixteen through eighteen year-olds who have been shafted by the traditional system, or their neighborhood, parents, etc. In reality, Job Corps shafts these kids pretty badly, as well.

At the Job Corps, I was a math teacher. Basically, they wanted me to administer cheap computer-generated quizzes covering basic “math facts.” The system was so dismal I cried many nights working past midnight at my desk to try to fix it even minimally.

OK – to the point – there was one female student who was very intelligent and mature for her age, but who’d been hopelessly victimized by her upbringing. She still had some very bad vestiges of the “we’re just victims” syndrome.

One day she came to me and said, conspiratorially, “You know, Mr. Foley, we all have electronic chips planted in our hands, so the government can track us. You know about that, don’t you?”

Oh, man! My heart fell. The best I could come up with was, “Look, you’re a poor kid from the hood, and I’m just a poor schmuck working in a rural government facility for $12 an hour. Why in the world would anyone want to track you or me? Or anyone else in this godforsaken place?”

I had thought I was doing so much good trying to teach these kids some logic, and rational ways to deal with their world through math, but still the walls had been been built so high and wide by their social backgrounds.

That girl was a nice, smart, valuable person. It DRIVES ME NUTS that our society accepts deception, abuse, and coercion of thought by politicians and educational policy-makers. ESPECIALLY by educational policy-makers, who after all, should be the front line against mind-abuse and enforced stupidity. People like that young woman should be nurtured and encouraged, not “kept down” and “inculcated.”

Back to our puzzle. They type of misdirection used in The Case of the Missing Dollar (?) has several names and versions. One of them is “Red Herring.” That will be the focus of the next post here at The Math Mojo Chronicles.

As Mark pointed out in his comment, one great way is to make an equation, which is a “schematic” of the problem, using numbers.

Another was is the method that is used by the subject of Alexander Luria’s (the great neuropsychologist) book, “Mind of a Mnemonist.” A mnemonist is a person who has a phenomenal memory. I don’t mean like your friend who knows baseball statistics. I mean like a person who memorizes every step he takes and can tell you what he ate on September 3, 1966.

It turns out that the subject, “S,” who was the mnemonist, had very interesting ways to look at math problems, too. He didn’t use symbols for numbers. He used graphic images of the situation. Here’s a taste, from page 105:

When faced with this problem:

The price of a notebook is 4 times that of a pencil. The pencil is 30 kopeks cheaper than the notebook. How much is each?

(You may want to ponder how you would solve this yourself before you read on.)

He pictured the notebook and four pencils next to it, with an equal sign between them.

Then he pictured the pencil = 30 kopeks cheaper than the notebook like this: A notebook, then an equal sign, then a pencil and a plus sign and 30 kopeks next to it.

That immediately lead him to see three pencils, an equals sign, and thirty kopeks.

From there it is easy to see that a pencil is 10 kopeks, which makes it easy to see that the answer to the problem is that each pencil costs ten kopeks, and a notebook costs forty kopeks.

To wrap this up:
You are in control of your mind. The more tools you have to solve problems with it, the less you are at the mercy of people who would like to steal and mislead your attention. And the more you are free to explore the world in ways that are beneficial to you.

You can find a very thorough and interesting (and totally understandable) discussion of the motel puzzle at this really interesting blog. Make sure you look at the clever post by walldog in the comments

Another good resource to understand the motel puzzle can be found at Wikipedia.

This particular post was about becoming and expert with math “facts.”

I have one observation that I feel has been terminally left out of this [...]]]> I just read an interesting and valuable post by a concerned parent at “Mindless Math Mutterings” (which are anything but). I like that blogger’s thoughtful posts about education.

This particular post was about becoming and expert with math “facts.”

I have one observation that I feel has been terminally left out of this discussion, though:

Although the basic building blocks must be able to be used immediately when needed, what we generically call “memorization” is a dead-end for most people.

We require children to sit and “memorize” in order to learn, but we don’t teach them how to memorize. Memorization is a skill, just like other thinking skills, that needs learning and tweaking. Rote memory is a terrible myth. I mean, it works for some, but it is not the only, nor is it the best, way for most people.

Before society did things by “rote,” the elites learned the art of memorization (ars memoria). We lost that as paper and books entered the general culture. (With every blessing comes a hidden curse?)

Since we lost our “art,” we are at a loss to do something that was commonplace before. Sort of like cars and horseback riding.

The thing is, most intelligent people know that we still need to be able to “ride the horse” of memory. But telling a kid to “just sit there and memorize it” is like telling him to “get on that horse and ride” without teaching him how to ride. Could be done. Wouldn’t want it to be the default method though, right?

In “The Mind of a Mnemonist” by Alexander Luria (One of the founders of modern neuropsychology), the subject (the mnemonist “S”) was observed solving mathematical problems. His methods were entirely counterintuitive to what most educators today would tell you are the “correct” ways, yet he could solve things in seconds that the “experts” would be stupified by. And he did not do it with mnemonic solutions to formulas. He did it with reasoning and visualization skills that are entirely ignored in public schools, and pretty much anywhere else. You have probably never heard of them. If you want a fascinating, eye-opening read, get a copy of that book.

By the way, mnemonics is the expert use of memory devices. If it isn’t expert, it’s just a bunch of memory tricks. It’s named after the mother of the younger Muses, Mnemosyne.

So math “facts” (a counterproductive word if I ever heard one) need to be ingrained, but we need caring, informed people to give kids the best methods and help them explore in a meaningful way.

No, this is not a call for those idiotic TERC etc. methods. They are about selling a method to people who don’t understand them, and then telling them to teach it to kids. Bad mojo!

Better mojo is to learn to love the things about math that you want to teach a kid. And that is easy. There are soooo many cool ways to multiply that should be explored, thoroughly. But they should not be relied upon to be the default method for multiplication.

Remember Bruce Lee? He created Jeet Kune Do, which was a mixture of many martial arts and straight-up, artless streetfighting techniques.

He explored enough to be able to know what worked for him, and most people in general. In other works he became an expert. And I mean a real expert.

I’m sure you see where I’m going with this.

Despite it being a shameless plug, I am going to once again mention that if you want to get your “multiplication facts” (God, I hate that term!) down, the Jeet June Do of multiplication starts with “Numbers Juggling - Times Without the Tables”.

And as far as memory goes, you could not do much better than to go right to the best memory teacher of the 20th century, Harry Lorayne. Get his book, “How to Develop A Super-Power Memory.” You can find it in amazon.com.

Yes, it’s a hokey name. It was originally written in the 50′s. It’s been re-edited and re-released, and it still blows all other books (his included) out of the water.

Readers of this blog may know that I have been a professional magician for over thirty years. I learned a lot of my material from Harry Lorayne books, and seeing him at meetings and lectures around New York City in the 70′s. They guy is amazing, and he is a great teacher.

‘Nuff said?

- B

As I worked, I used a catch (see last post) to collect my thoughts. The one I used is my favorite – it’s a mnemonic device. I [...]]]> Just had a great evening mulching and edging our apple tree. I ran out of mulch and ground paper, but I can get more tomorrow.

As I worked, I used a catch (see last post) to collect my thoughts. The one I used is my favorite – it’s a mnemonic device. I used the rhyming peg method, (“one is the sun, two is a shoe, three is a tree,” etc.) and only needed to get to “door” for the four Ideas I had been mulling over.

(Note: After I’ve finished posting all the lessons on basic math I want to put up on the MathMojo site, I’ll start working on Memory Mojo, and explain all the memory hacks I have been using, with lots of tips).

If you read the last post, you remember that I’d been pondering some Ideas as I mulched the tree, but had forgotten them by the time I wanted to write them in this blog. So I went back with a catch, and hoped that by resuming mulching, I could get back into the thoughts I was having.

It worked. The first thought I wanted to record was this:

I a had already mulched half of the ground below the tree last year. The entire area is about – well, you figure it out – the radius is about 18 feet. The formula for the area of a circle is pi times the radius squared. Pi can be rounded to about 3 for our purposes here.

Unfortunately, I really didn’t know what I was doing. I had read about several methods, and had used a fairly “organic one.” I weeded under the tree, tilled the soil with a hoe, and covered everything with a layer of newspaper and soaked the paper down with a hose. Then the I covered the entire thing with a 3 inch layer of fresh grass clippings from our lawn.

This spring there were lots of weeds poking out of the entire area. Damn!

What this all lead me to think was how there are lots of ways to do things, but the most thorough is usually the best.

What does that have to do with math? Isn’t it apparent? Schools try to cover “curriculum” within certain time frames. If the material is throroughly learned or not is not important. As long as they learn enough to pass a test.

Last year I passed the mulch test. I did it successfully. I weeded, tilled, and mulched. I could say I did the job.

The weeds didn’t care if I could say I did the job or not, though.

Math and reality don’t care how you did on your test. The test is just for the bureaucrats to be able to say they tested you.

Math requires a deeper appreciation for reality than those superficial school textbooks and standardized tests will give you.

Well, that’s a lot of typing for today, so let’s save some of the further insights of today’s mulching project for the next post.

Today’s suggestion: Get a good method of catching your thoughts. A small memo book and a golf pencil are easy to carry around.

Hotcha!

I’m not a gardener, and have never been very good at things like that, but I can weed, I [...]]]> I was out mulching our apple tree in the front yard, when some thoughts came to me. One of them was that I can see how gardening can be such great contemplative avocation.

I’m not a gardener, and have never been very good at things like that, but I can weed, I can mulch, and I can mow the lawn. All of which are sort of brain-dead activities which can lead you into a trance-like state.

If you had been contemplating something just before you get into that state, or have had something preying on your mind, it is easy to let thoughts about it come to you while you are in that state. They can wash over you in a pleasant way. The trick is to catch them, and get them recorded somehow, so you can recall them later, and work them into something and take advantage of them.

Having a pencil and paper, or small voice recorder is a good Idea. Unfortunately, I didn’t have any of those when I was mulching, so I’ll have to try the trick I sometimes use when I wake up from a dream that I want to remember, but can’t recall the details of – I go back to sleep and hope I can pick up where I left off. I’m not going to go to sleep now, I’m going to go back to mulching and weeding (it never ends anyhow!)

Trying to go back doesn’t often work, but often I at least get other Ideas that are useful, and this time I’ll make sure I have a catch.

According to “Mind Performance Hacks – Tips and Tools for Overclocking your Brain” by Ron Hale-Evans, a catch is ” … an advanced system for collecting every thought, from everywhere in your life, and bringing them together.” (Page 43)

Well, we’re not quite going to go that far with this today – I’m not trying to collect every thought, but part of a catch is simply something to record your random thoughts with.

Normally, I try to have a something to write with and something to write on with me at all times – a pad and a pen (with a little lightbulb in it so I don’t have to turn on a light and wake up my patient wife when I want to write an Idea fresh out of a dream); a little car notebook (for when I can pull over and record a thought); a hipster PDA when I am out,

I will be writing more about the hipster, because I believe that using it is one of the best things you can do while learning something, like math techniques, or concepts.

So I’ll be taking my little digital recorder out with me to mulch with this time.

See you at the next post!

It’s so easy to say, “Just memorize it.” That’s the cry of people who want you to learn your “math facts.”

But do they ever teach a child how [...]]]> Have you ever heard of Hintikka’s Paradox? In a nutshell, it’s about, “Is it immoral to ask someone to do something that can’t be done?”

It’s so easy to say, “Just memorize it.” That’s the cry of people who want you to learn your “math facts.”

But do they ever teach a child how to memorize? How cruel to would it be to grade a kid on his reading ability if you never taught him how to read?

That’s the paradox of the memorization problem. Not every child even knows how to memorize.

There are lots of solutions, but I’ve never seen any good ones on math sites, or ever heard of teachers really using any. Let’s think about our part of helping out before we prescribe solutions we don’t provide.

Personally, I think rote memory is one rotten way to learn. Learning mnemonics is so much more fun, creative and effective. I don’t mean silly mnemonics that are taught in some of the baby-style trendy math methods that are out there. I mean a concerted effort to learn serious memory techniques. My favorite book to learn memory from is “How to Develop a Super Power Memory,” by Harry Lorayne. He’s written more since then, but despite its hokey title, it is the most straightforward of all the books I’ve read.

One solution for learning multiplication without rote memory is the method taught in my booklet “Numbers Juggling – Times Without the Tables.” the link for it is on the top of the right-hand column of this blog. I think this is the best solution for anyone who is dissappointed with the traditional “just shut up and memorize it” method.

Make sense?

Trust your brain – learn mental math and memory methods!

In lesson 3 of “Eating Math For Breakfast” you’ll learn a very basic but effective way to memorize a string of digits in order to do more math mentally.