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Recently, inquiring readers six-year old (!) Julien and his Mom, asked about Triangular Numbers:
Hello, My son is currently working his way through The Number Devil (by Hans Magnus Enzenberger) and is enjoying it thoroughly. He was particularly happy about the triangular numbers in Chapter 5 because I had just coincidentally given him a worksheet which involved calculating how many blocks would be required to complete a series of steps from 1 all the way up to 20. (1+2+3+4+5+6+7…)
He made the connection between his worksheet and the triangular numbers and tried out the (much more efficient) trick the number devil provides (p.101) for solving such a problem:
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21*10=210 However, he immediately realized there will be a problem if you have an odd number of steps.
Can this method be used with an odd number, and if so, how would it work? We have read your page on triangular numbers and used the equation you provide (x^2+x)/2 and that works great. But, he (and I) just wonder if we are missing something about the method discussed in the book (because the number devil doesn’t say it’s ONLY for even numbers).
Thank you for your time!
Professor Homunculus replies:
Hi Julien and Miram,
That’s a great question, and I’m very happy that you are interested in things like this. It is the sign of a mind that knows how to have some fun with patterns. And it is a great inspiration to folks who wonder at what age kids can get interested in math.
I haven’t got a copy of The Numbers Devil anymore, and it’s been a long time since I’ve read it, but it still remains one of my favorite books to introduce real math to people with.
I hope I’m understanding your question correctly. If I am, the following should clear it up for you and your son. There are several ways of looking at this pattern to make it “work” for odds and even numbers. There are probably many more than I will ever know, but here is one of the ways that I like to view it:
Include zero in the mix. It doesn’t change the sum of the number, but it makes the same pattern as above. For example, if you wanted to add the series from 1 to 5: Just keep in mind that adding the series from 1 to 5 is the same as adding the series from 0 to 5.
5 * 3 =15 It works! Let’s try it with 19:
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19* 10 = 190 It works again!
Is that what you were getting at? Did this help solve your son’s dilemma? Please let me know if it did.
All the best, Brian (a.k.a. Professor Homunculus at MathMojo.com )
BTW, readers not familiar with the formula given above ( (x^2+x)/2 ) can find out more about it and triangular numbers in general at the MathMojo.com page at Adding Triangular Numbers
A challenge:
Can any reader see a mental shortcut of how to arrive at the sum of a series from 1 to n? There may be more than one. (Hint: The ones I use are slightly different for even and odd numbers).
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Recently an aquaintance told me that her colleague had a daughter who could use some math tutoring for ninth grade. I don’t do tutoring, per se, because too often they end up being disappointed that you are not just teaching to the test and reinforcing the bad teaching that has already been done.
She said she’d send her friend and her friend’s daughter to my site. Then she called wanted to know if she could give them my number for them to get in touch with me.
They never did.
The next week I spoke my acquaintance again, and she said, “Wait, my colleague is in the next room. I’d like to tell her you’re here.” I told her, “OK, but I don’t really tutor. Tthere are a lot of good resources at my website for a ninth grader, though.”
She came back a few minutes later and said that they couldn’t use Math Mojo because her friend’s daughter’s math teacher said, “She has to do it her teacher’s way.“
What a$%#ing &tard! That teacher should be cited for criminal education neglect. Enforcing the benighted notion that math has to be seen from only one angle is the reason most kids don’t get math in the first place.
I explained the Parallax to my acquaintance, and she completely agreed. I also talked about the reason that we have two eyes and not one is for depth perception and a parallax view. If you only use one eye you are handicapped.
In a nutshell, a parallax is the use of more than one point of view to get an overview of something. That, of course, is not a complete, or entirely accurate explanation, if you want a more complete scoop, check out parallax on Wikipedia.
Only having one method to accomplish anything handicaps you. Having a second method does not degrade the first. It enhances it. It makes each part greater, and it makes the whole greater than the sum of the parts. (No, that is not a logical contradiction. Ever hear of nuclear fusion?)
Unfortunately, the members of the school system who should know this most (math teachers, who should be versed in basic logic) often don’t, and are the greatest enemies to the mathematical reasoning skills of their students.
Why do people who should know better insist that everything must be a zero-sum game, and that their way must be defended at all costs, even though everyone suffers in the long run when they only use that one way?
And of all the logical farts in that teacher’s argument – if her way was so damned good, why does the kid need tutoring? If that way didn’t work for her before, why does she assume more of it, and nothing else, is going to be what helps her most?
Look, I know I should be producing more “nuts and bolts” lessons for people to use. To tell you the truth, the more experience I have with people, the more I start thinking, “What’s the use?” I know that is wrong, and I’m trying to fight it. OK, it’s not wrong – it’s absolutely right. But it’s not helpful. I’m trying to reconcile the two. Anyone got any suggestions before I give the whole “trust your brain” thing up, and become a televangelist or a politician?
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This is the second post in a series about counter-intuition.
(View the first post about counter-intuition here.)
When I ask many elementary school teachers if they teach that you can’t divide by zero, they say, “Of course.”
When I ask them why, things get a little murkier. Usually they will grasp for an answer, like, “Because you can’t divide by nothing. That just makes sense.” Which is meaningless and irrelevant, and is certainly not the answer.
It’s sort of scary that people who are trained to teach our children don’t know the first rule of learning – if you don’t know something that you need to, face up to it, and learn it.
There is a sad tendency for us to want to say we know something more than we want to know it. That is a learned trait. We learn it from ignorant parents who make fun of us when we are young, from a school system that uses grades more to judge us than to help us, and from our peers who in their infinite insecurity need to make fun of anyone who does something wrong, just so we won’t notice their faults.
Our schools train us to want to have “the right answer” more than to want to understand anything. Face it, most of the school experience is about tests. There may be some exceptional teachers that go beyond that, but “No Child Left Behind” has had that become even rarer than it was previously.
This degrades the quest for true knowledge. If you constantly have to pretend you know something that you don’t, you’ll never get to the point of maturity where you finally have to go and learn it. Then, to keep from being found out as an ignoramus, you’ll have to make fun of people who understand more than you do.
Look at your average political talk show host, for example. Raving at things they don’t understand, and decrying “elite intellectuals.” It’s basically a cry for help from them from their own ignorance, but disguised as “political analysis.”
They are hiding in their “anti-intellectual” comfort zone. They are threatened by what they don’t understand. This is the main problem. Admitting you don’t understand something is the first step towards freedom. Freedom from rigidity of thought, freedom from superstition, freedom from ignorance. Once you can imagine that you don’t understand something, you are free to go about finding out an answer, of if one can even be had.
Posturing that you have an answer to what can’t really be known is the depths of slavery of the mind to the will of other, equally ignorant minds.
Imagine if we still believed things like throwing a virgin into the volcano will appease the gods? That sounds ridiculous, but there are people who spout equally silly things today. And a large portion of the population believes them, because, “anyone can see that it’s true.”
It is appalling that in the U.S., anti-intellectualism is actually praised by part of the population. What is the deal – “Don’t be smart like them – be dumb like us?”
An intellectual approach (which does not mean some effete, elite attitude – after all, intellect is not based on emotion) would be, “Let’s look at the facts regardless if they are comfortable or not, and form some opinion from what we can prove, and change or reject the opinion as we find out more facts. And let’s try to use logic to convince those who don’t agree with us. But if their facts and opinions make more sense, we can change our opinions.”
The anti-intellectual approach (which generally does mean some attitude, because this approach is based on ‘gut’ emotions) would be, “Let’s form an opinion based on making us feel comfortable with what we don’t understand, and then we’ll try to find some facts to fit them, and change or reject the facts if they don’t fit our opinions. Oh, yeah, and let’s get real mad at anyone who doesn’t accept our opinions. And if their facts make more sense than our opinions, let’s just deny their facts.”
That is no way to run your mind, your country, your society, or anything else. It is just the same as denying that the Earth revolves around the Sun.
Knowledge isn’t something to be afraid of, nor is it something to be proud of. It is just something to be curious about. Curiosity ends when you think you know the answer, but you don’t care why it is true – you just know “…because…”
Don’t let curiosity end for you. Ever. As the Chinese say, “Curiosity is the best teacher.”
Want to know the real scoop on division by zero (in arithmetic)?
Check out:
Why We Don’t Divide By Zero in Arithmetic
and
Division by zero
By the way, Occam’s Razor is not only a tenet of Math Mojo, it is one of the things that guides mathematics. Math and logic are not the same. Math doesn’t always have to be logical (really? How counter-intuitive), but it always must have logical consistency.
The next post in this series will be a real-life and current example of how people intentionally or ignorantly deny and distort facts by starting with opinions.
One more note – don’t confuse intellectualism with academic elitism. Most populists do that to confuse you. Academic elitism is just another form of ”everybody knows.” “Everybody knows that if you went to Hahvahd, like me ,you ah superior.” Of course the opposite is not true either. Just because someone went to an elite school doesn’t mean he or she is an elitist. Stick with the facts – don’t let where someone went (or didn’t go) to school impress you. Judge their arguments on the merits, and leave the populism and elitism to the slobs and the snobs.
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This is going to be the first post in a series about counter-intuition.
I am still working on some nuts-and-bolts resources for Math Mojo (like addition and multiplication booklets and videos and a teleseminar so people can ask questions as I show an amazing way to teach addition) but sometimes I need to write more about the Mojo than the Math.
Actually, the math is the mojo, but the simple skills as taught in schools are devoid of both. The mojo I’m talking about here is the lessons you can get from understanding math that are beyond simple numbers skills. They are about life-lessons, deeper insights into your mind and how you can think more satisfactorily (for yourself, not for me or your teachers). Which leads me to….
One of the tenets of Math Mojo is Occam’s Razor.
Occam’s Razor is generally seen as “The simplest explanation is generally the best one.” This is true, but the phrasing is dangerous. Another way to put it is, “Make things as simple as possible, but not simpler.” This is often attributed to some hack (um, Einstein LOL?) I think it should be “The simplest explanation that makes sense and does not contradict other known facts is generally the best one.” Of course that is more like what William of Occam meant, but unfortunately it is not how many people think.
The simplest explanation for the Earth’s shape is to say that it is flat. Of course it is. “Everybody knows the world is flat.” That is what “common sense” tells us.
The Sun revolves around the Earth. “Everybody knows that. The Sun comes up and goes down, therefore the Earth stands still and the Sun revolves around it. Just like the moon. Anyone can see that. It’s common sense.”
Of course that’s all bull. But hey, the only reason most people know that that the earth is round(ish) is because they were told that. Not because they understand anything about geometry or astronomy or critical thinking. Many people still didn’t believe it until they saw the first pictures of the Earth from space. Some people still don’t believe it.
We know what “everyone can see,” or what we are told. Well, at least many people do.
When someone uses, “…everybody knows that…” in an argument, it does not necessarily mean that he is wrong, it just means that he is an idiot.
Common sense is generally more “common” than “sense.”
There’s no way to collect or parse all the data, but I’d bet the house (your house, anyway) that 90% of mankind’s problems stem from “what everybody knows.” That’s what happens when people think with their guts. It would be better if they thought with the organ furthest from their butts.
It is sad to see people craving simple solutions for complex questions, without considering that there might be more to something than “what everyone can see.”
This reminds me once again that “intuition” is not what it’s cracked up to be. The flat-earth theory is “intuitive.” The round(ish)-earth fact is counter-intuitive. When you start exploring and developing a feel for counter-intuitiveness, you start getting in touch with your math mojo.
Well, of course you won’t be able to enlighten a person who is dead set on remaining unenlightened. But I’m betting that many readers are more enlightened than I, and that those who aren’t would certainly like to understand more. That’s why I write this blog.
In the next post, we’ll discuss a bit of mathematics that is counter-intuitive, and what we can learn from how people think about it.
Here’s the link for the next post:
Math Mojo vrs. Anti-Intellectuallism
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Unfortunately for me, they are not my financial insights. If I had any financial savvy I wouldn’t be a math blogger. But recently I was googling “math mojo” and I came upon the Reading the Markets blog. In this post, the author says, “I think we profit enormously from looking at alternative approaches to a problem.” She mentions this MathMojo blog as an example thereof.
I was impressed, not just for the ego stroke, but by the fact that the author “gets it.” It turns out that she is a Yale-trained philosopher, so I imagine that she gets it more than I do. But I was glad that my message is getting through. MathMojo isn’t simply about math and arithmetic. It’s about approaching things differently, and training and trusting your brain.
The “Reading the Markets” blog is about “Insights from Financial Literature’” a subject that is Greek to me (sorry about the pun, Brenda). But if it were a subject of interest for me, I know I’d make a bee-line for that blog. Insights always trump information.
If you learn no math from MathMojo, but learn that the “standard algorithm” (or the standard way to do anything) is only one way to skin a cat, and not necessarily the best way, then you’ve “gotten it.”
Getting out of “but the teacher said we have to do it this way” way of thinking is about the best thing you can do for your mental development. Yeah, maybe you have to do it that way in school, to get a grade, but please realize that grading is a way for schools to keep you obedient, not make you enlightened.
Go ahead and give the teachers what they want, but make sure you pursue anything you like to a much higher degree than those minimums they call “standards.”
Be an uncommon denominator.
Hotcha!
Brian (a.k.a. Professor Homunculus )
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Sky and Telescope.com has an interesting article this month. In “A Rogue Star Going Wild?” (no, it’s not about her) it discusses the Homunculus Nebula.
It’s not exactly a math article, but Professor Homunculus likes it anyway. Check it out.
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I’m almost finished putting up the first week of the new “Quick and Dirty Multiplication” Math Mojo course.
In the meantime, a few people have asked me to get to work on a “Quick and Dirty Addition” Math Mojo course.
I won’t be able to get to that for a few weeks, but if you need immediate help, I’ve put together a few resources that I’ve created over the last year or so.
The goal would be to take anyone who can already say, 3+ 6, and knows the difference between the ones column and the tens column, and be able to add huge columns and rows of whole numbers in a short time, accurately, and easily.
Check out the videos below for an example of what anyone should be able to do in less than a month of practice.
Speed Addition Demonstration Video
The video you are about to watch is simply a demonstration of how fast a huge addition can be done with Math Mojo methods. It does not explain anything. All the things you’ll need to be able to do addition like this will be explained in the resource page (see below).
Speed Addition Demonstration
Speed-Checking your Addition
Same goes for this video. It is simply a demonstration of how fast checking huge additions can be done. This checking method is taught in full detail in The See-Say-Write Method of Addition Mojo. (If you order your copy from the resource page that you can get to by filling in the form below, you will get a free practice booklet of over 100 pages bundled with it.)
Speed Checking Addition Mentally
One of the resources I’ve gotten together for this post are an audio file that you can listen to here. It’s about 12 minutes long, and will explain the basic Idea of speed addition. It talks about the reason we count and add the way we do. It explains it so that a child can understand it.
I’ll also be sending you to another Math Mojo page about Learning Addition from the Ground Up, that has a very comprehensive video about basic speed-addition.
After that, I’ll recommend the only thing that requires any payment (and it is only $9.95). It’s my e-booklet, “The See-Say-Write Method of Addition Mojo.” It teaches only two things, but they are amazingly powerful, and are hardly ever mentioned in schools. The first thing it teaches is how to add two 2-digit numbers (like 76+89) in your head, without having to think about it consciously. You’ll be able to do it easier than most people can add 7+9 in their heads.
The second thing it teaches is how to check your answers. Don’t underestimate this! It is one of the best kept secrets in all of arithmetic. It’s almost never mentioned in schools, and when it is, it is glossed over, and not really taught. It is a crying shame. If schools taught this, they would immediately see gains in their student’s math skills.
An amazing bonus is that this same method of checking can be used for multiplication, subtraction and division. It is about the best weapons you can have in your mathematical arsenal, and it is a huge help on standardized tests.
There is a practice pad that is available for the See-Say-Write method, which insures that you actually learn the method, instead of just reading about it and then forgetting it. (After all, what good is it to you to simple “know about” it, when you can actually know it and be able to use it?) That practice pad is normally $5.95, but you can get it free if you order
In the resource page, I’ll also include a special report in PDF file, of how to use the See-Say-Write Method of Addition Mojo to accomplish gigantic additions like the one in the above demonstration-videos.
Nothing will be left out.
In a nutshell, the resource page contains:
- Free audio about basic addition
- Free video about basic addition
- Link to the See-Say-Write Method of Addition Mojo booklet with the special offer for a free practice pad
- Free PDF of how to use the See-Say-Write Method of Addition Mojo to accomplish gigantic additions, quickly and accurately.
Anyone should be able to add large examples quickly and easily with these methods. Students will freak their teachers out with their new abilities.
Fill in the form below to get free access to the resource page for speed-addition.
The reason your e-mail is required, is for me to be able to gauge interest in speed-addition. If there is a lot of interest, I will finish the full course as soon as possible. It’s also so I can notify you when the course is finished, as well as send you free addition tips as I add them to the free resource page.
There is absolutely no obligation for you to enroll in the course, buy the See-Say-Write e-book, or anything else. I won’t share your e-mail address with anyone, either. The form is a double-opt in, which means that when you hit the submit button, you will get a confirmation e-mail. As soon as you reply to it, you will get an e-mail sending you the web-address (URL) and password to the resource page.
If you do not wish to remain “opted-in” there is an easy way to opt out in every message I will ever send you. The double-opt in is just a way to keep both of us free of spam. And I will never, ever share your e-mail with anyone, for any reason.
This is your best way to learn an amazing method. Just fill in the form and submit it now for your free access to the resource page.
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There was a wonderful article in the Wall Street Journal today about mathematics in the former Soviet Union. It is worth reading for anyone interested in finding out a little about the inner beauty of math.
Here’s a short except:
what mathematics really is: “It was a wonderful education… Gelfand amazed me by talking of mathematics as though it were poetry.”
In the mathematical counterculture, math “was almost a hobby,” recalls Sergei Gelfand. “So you could spend your time doing things that would not be useful to anyone for the nearest decade.” Mathematicians called it “math for math’s sake.”
How cool would that be? Can you imagine something like that in an American school? I can’t.
So check out Mathematics in the Soviet Union, at the Wall Street Journal’s website.
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This will not have much to do with math, but a lot to do with mojo.
One of my heros, Soupy Sales, died yesterday.
Soupy was a comedian, who hosted a children’s show in the 50’s and 60’s. He was a master absurdist, with an incredible sense of humor, and a wonderful respect for the minds of his audience. His show was booked as a “children’s show”, but his material was aimed a little higher – over the kid’s shoulders with a nodding wink to their parents.
Soupy is among the other master creators, like Lewis Carroll, L. Frank Baum, James Barrie, and J.R.R. Tolkien, who’s books are sometimes considered “children’s books,” but we get a lot more out of them as we mature. It’s fitting that this week also brought us the movie “Where the Wild Things Are” (from Maurice Sendak’s book), which is another example of this.
He was one of the people who shaped my Idea of how to present my magic show, how to do math-magic presentations for children and adults, and even how to write this blog.
“I’ve never done a pretentious show; it’s always had a live feeling, the kind of thing that comes across when you don’t know what’s going to happen next,” Sales told author Gary Grossman in the 1981 book “Saturday Morning TV.”
I only hope I can live up to that.
His show was aired during an era when live television still was the norm. Other great children’s hosts, like Sandy Becker and Sonny Fox, shaped my Idea of mojo. Mojo isn’t just magic, and it isn’t just about the blues, as a lot of people think. Mojo is just a weirder, more open-minded way to approach anything. And that was Soupy, all over.
I remember watching his show when I was a kid, sitting with my dad and my brother on the couch, the three of us cracking up and mugging for each other when Soupy’d get hit by a pie.
Here are some links where some of you other baby-boomers can reminisce about Soupy, and maybe some younger people can go find some comedy-mojo from a different, less nasty era:
Pachalafaka, Soupy!
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How could I have missed it? Or almost have missed it.
Today is Martin Gardner’s 95th birthday.
Who is Martin Gardner, you ask?
I’m glad you asked that question!
You can read more about it in today’s New York Times Science Section.
I’ll write more about him soon, but for now, do yourself a huge favor and check out that article, then run to a library and check out any math books of his that you can find.
Here’s another good article about Martin Gardner in the New York Times.
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