How do you solve simple math problems?

[ScarlettCatherwood]

I'm curious to see if anyone else does mathematics the way I do, in my head. Using this method, although dependent on who I'm up against, I can almost always reach a conclusion much faster than someone else. At least when it comes to my peers. I think an illustration is the best way to explain.

Subtract 77 from 122. Most would borrow from the 2 in order to subtract the 7 from the 2.

I do it differently. Instead of borrowing, I add 5 to 122, making it 127. That makes it mucher simpler. 127 - 77 = 50. Then, since I added 5, I subtract it from the sum, giving me an answer of 45.

I don't add numbers the way teachers taught us in school, either. For example, I don't carry anything when adding 939 and 237. First I add the hundreds place, then the tens, and finally the ones, and put the sums together. 900 + 200 = 1100, 30 + 30 = 60, 9 + 7 = 16.

1100 + 60 + 16 = 1176

I'm still adding of course, but backwards, which makes it much faster and simpler.

Also, if someone were to aske me to add 13 and 9, I would change it to 19 and 3. And instead of adding 3 and 9 and carrying a 1, I would just add 9 and 1, then 10 and 2. That gives me twelve, plus the 10 from the tens' place, which is 22.

This happens automatically. I don't think about it. It's just the direction I go in, because it works more efficiently, in my experience anyways.

Division as well. 642 divided by 3. 600/3 = 200, 42/3 = 14. So, 642/3 = 214. Also, I don't know what 42 divided by 3 is off the top of my head. But 42 divided by 6 is 7, and 6 divided by 2 is 3, so I multiply 7 by 2 to get 14. Although, of course, this can't be done so easily every time. In this example the divisor goes into the dividend perfectly. But still, I did not have to get out a piece of paper and do long division. And that's the whole point to these weird little methods. If I don't have to use pen and paper, why would I?

I've been doing simple math like this for as long as I can remember doing math. I find it to be much easier and quicker. I remember once, when I was five or six, I was in the car with my mom and my aunt. They were adding several numbers trying to get a total number of miles, why or for what I don't remember. But I do remember hearing them talking out loud, carrying numbers and repeating them so that they wouldn't forget. Using my backwards method, within seconds I had the answer. They looked at me a little strangely, so I explained it to them. They didn't understand it, and still don't. To this day when I try to explain it to people they tell me I'm ridiculous. They say I'm making it unecessarily hard, and why can't I just use the simple way? Family members say about it, condescendingly, "If I have to know how her crazy mind works then I'll never figure it out, and don't want to."

So, do you think I'm making it unecessarily difficult? How is that possible when it's so much easier this way?

Is there a term for these sorts of methods? If so, what is it?

Mostly, however, I'm just curious to see if anyone else does simple mathematics like this. I haven't met anyone, that I know of. Although, admittedly, I don't go around asking people to demonstrate their brain processes when solving simple math problems. So I don't really know if this is common or uncommon. Though I'd tend to think there are other people who do it like this.

[Grames]

Solving a different problem than the given problem creates the appearance of introducing complexity. But since the different problem is so much more easily solved you have a net gain. This is perfectly reasonable and is a technique used in many professional fields.

[~Sophia~]

Math was thought like that long time ago.

There is a program which teaches this method. It is called Math U See. My son's school follows this program so he does math in a very similar way. I think Montessori also does something similar: kids understand what it is that the "carry over" represents.

[Marc K.]

I think you would really enjoy the book "Surely You're Joking Mr. Feynman!" by Richard Feynman. There is a chapter in there called "Lucky Numbers" in which he uses similar calculating techniques but on much more difficult problems. He turns it into a sporting proposition too, hilarious.

The book is a joy to read I recommend it to everyone. Feynman was a highly regarded physicist who worked on the Manhattan Project and who famously solved the Spaceshuttle Challenger accident. The book is a series of funny stories told by Feynman about his life. From mindreading to cracking safes to playing the bongos he keeps you in stitches all from the perspective of a supremely rational mind. Thumbs way up.

[agrippa1]

The book is a joy to read I recommend it to everyone.

One of my favorites.

If you're interested in Physics, there are a couple of audio recordings of his lectures that capture his brilliance and humor. I think they're called "Six Easy Pieces" and "Six Not-So-Easy Pieces." He was a unique individual, with a great sense of life.

[brian0918]

I do this so much it has become instinctive. Anytime I find a math problem that looks complex, I find a similar one that is simple, and then figure out the difference between the two. Of course, doing the more complex problem would help build my short-term memory, so it's a trade-off.

[brian0918]

One of my favorites.

If you're interested in Physics, there are a couple of audio recordings of his lectures that capture his brilliance and humor. I think they're called "Six Easy Pieces" and "Six Not-So-Easy Pieces." He was a unique individual, with a great sense of life.

I used to dig up everything I could find by him. His explanations are unrivaled. Even after all my courses in chemistry, it was still nowhere near as clear to me as when I later read his brief comments on how to visualize the evaporation of water or the scent of a flower.

Another great read is his The Pleasure of Finding Things Out.

[Jake]

I perform basic arithmetic in a very similar way. I add large numbers the way you described, and I sometimes multiply pairs of 2-digit numbers by multiplying all 4 combinations of single digits and adding the results. For the brief time that I was a Math major, I used to race friends in arithmetic.

There's an aspect of my job in which I have to multiply 2 percentages. The first number is usually 110-120%, and the second is always 91-100%. I'm the only one who does this regularly in my head, and I almost got the call sign "Rainman" because people think it's odd, but as you describe above, it's simple. For example, multiplying 116% by 97% directly would take some time, so I instead multiply 116% by 3% (1 - 97%) and subtract the result 3.48% from 116% to get 112.52%. In practice, I only care about whole percentages, and I round down as a safety precaution, so the calcuation is even easier, because I know 116% x 3% is more than 3% and less than 4%, so I just subtract 4%. Everyone else uses a full-page table to lookup the result or scrambles for a calculator/cell phone to punch it in. (For those interested, the first number is torque available from an engine based on environmental conditions, and the second is the engine's efficiency relative to a standard engine.)

To me, the interesting question to answer is, does one do Math this way because they have a better grasp of numerical relationships, or does using such methods increase one's understanding of numerical relationships? I suspect it's both, the method and the understanding reinforce one another.

I think what a lot of people don't realize is that you can often spend a small bit of your time figuring out how to solve a problem faster and with significantly fewer numbers to juggle. This initial time spent is like an investment: it may cost a few seconds to plan, but it often saves you 10 seconds or more in solving.

You can be a smart, motivated person without doing math this way. Unfortunately, when I try to show my peers how they can do the same, I usually run into the anti-conceptual laziness/fear w.r.t. new methods. "That's too hard for me.", "I'd screw it up.", "I'll just stick to what I know.", "Nerd!", etc.

[ScarlettCatherwood]

Unfortunately, when I try to show my peers how they can do the same, I usually run into the anti-conceptual laziness/fear w.r.t. new methods. "That's too hard for me.", "I'd screw it up.", "I'll just stick to what I know.", "Nerd!", etc.

That about sums up the responses I've gotten too.

There is a program which teaches this method. It is called Math U See. My son's school follows this program so he does math in a very similar way.

I'm curious. Does anyone else know of a school that uses this program, or teaches math this way?

[Dante]

Mostly, however, I'm just curious to see if anyone else does simple mathematics like this. I haven't met anyone, that I know of. Although, admittedly, I don't go around asking people to demonstrate their brain processes when solving simple math problems. So I don't really know if this is common or uncommon. Though I'd tend to think there are other people who do it like this.

Yes, a lot of the mental math that I do I do like this. It just makes things simpler, and it seems natural to me.

The only aspect of this that I know is fairly widespread is calculating a 15% tip by splitting it up into 10% and 5%. 10% of whatever the bill is is always easy, and 5% being just half of that is easy as well. I'm pretty sure lots of people calculate tip this way.

[JeffS]

That's really funny. I've always done math the way you describe, but no one I know does it that way (except my kids, whom I taught to do it). I never really knew why I did math that way, or who would've taught it to me until I bought the School House Rock DVDs and watched the math DVD with my kids. The one that struck me was the multiplying by 2s episode (Noah's Ark). One problem presented was 198 x 2. The problem is made easier by multiplying 200 by 2, then subtracting 2 x 2. A light went on for me, "That's how I do math!"

As my kids progressed through school (I'm their teacher), I realized this type of math is just an application of the distribution property of multiplication [e.g. 9 x 9 is really 9(10-1), etc.], and that's the way I teach it to them. As far as solving problems from left to right - well, that's the way we read, why wouldn't we solve the problem from left to right? Generations of children have been taught the slow way to solve math problems.

About 6 months ago I was introduced to Vedic Math, which is very, very close to the way we (you, others, and I) do math. You might want to look it up because it takes the same concepts and extends them to more complicated math (squares of large numbers, multiplying large numbers, dividing large numbers, polynomials, etc.)

[~Sophia~]

I'm curious. Does anyone else know of a school that uses this program, or teaches math this way?

Math-U-See was written by a homeschool dad for the homeschool community.

[SapereAude]

I do math this way too.

Seeing so many here doing the same is interesting for me as in the past most people aorund me found my method odd.

I wonder if it has something to do with the people I'm usually surrounded by (people in low level blue collar jobs) or if there is something about the way people who are drawn to Objectivism tend to problem solve.

Does anyone have a theory about that?

[brian0918]

Someone on Facebook pointed out this video on YouTube, where a meteorologist argues against "TERC" math books that attempt to teach children how to break down a problem into something that can be solved more simply, and then relate the simplified problem back to the actual problem in order to find a solution (the same thing we have discussed here). She instead argues that children should be taught the "standard algorithm" of writing the problem out, handling the digits one set at a time, "carry the 1", "borrow a 1", etc.

The TERC method seems like the best, actually. While her preferred method works with pen and paper, TERC's method allows you to actually solve problems in your head with minimal effort. With her method, there are just too many numbers to remember to do it in your head. TERC also allows you to pick up on common relations between certain numbers, which isn't likely to occur through her simple number crunching.

She claims TERC has no standard algorithm - the algorithm is actually quite simple: "rewrite the problem in an easy to solve way, and then figure out how that differs from the actual problem." What's great is that this is the standard procedure in more advanced mathematics and science (e.g. calculus, physics). So TERC better prepares a student for more advanced methods of problem solving. Richard Feynman, for example, was definitely a TERC-style problem solver.

Ask a kid who is only shown the standard algorithm *why* they need to "carry the 1", "borrow a 1", "move the decimal", "add zeroes to the end", etc, and they probably have no clue, because they don't understand it conceptually. They were taught the concept once, and forgot it. Now they are just performing memorized operations, and might as well be using a calculator.

There is no conceptualization involved in the rote operations of the standard algorithm. It is only memorization. Her method can churn out number-crunchers, but they may not achieve a deeper insight.

Edited September 22, 2010 by brian0918

[ObjectivistMathematician]

I also perform mental arithmetic like this. For the sake of more examples on how I do math, I often factorize numbers when multiplying (not necessarily prime factorization). I often also treat numbers as binomials when squaring them, i.e., in order to square 26, I'll do (20 + 6)^2, and if you remember your binomial theorem, (a + ^2 = a^2 + 2ab + b^2. This means that 26^2 = (20 + 6)^2 = 400 + 240 + 36 = 400 + 276 = 676.

[brian0918]

Suddenly the Montessori method doesn't seem so great. Be sure to watch to the end:

I know this isn't strictly Montessori, since it uses drawings instead of beads, but you can't deny that both methods make large numbers seem difficult to handle.

On Facebook someone made this analogy:

Notice how if the kid had written an M instead of a cube, a C instead of a sheet, an X instead of a line and a short line instead of a dot then she would have been doing mathematics in Roman numerals. There's a reason we don't use Roman numerals anymore: it's because they're rubbish next to the Arabic numerals. Like most 'progressive' suggestions this is actually *regressive* - it is taking children away from using concepts (denoted by arabic numbers) back to using percepts (pictures).

Edited February 27, 2011 by brian0918

[dream_weaver]

That was painful to watch.

[brian0918]

It can certainly be useful to periodically reduce the conceptual back to the perceptual level using the Montessori method, but to restrict children to add/subtract *only* at the perceptual level seems like a huge impediment at an age when learning is so important.

Then again, maybe Montessori only uses the beads much earlier in a child's education than 3rd grade.

Edited February 27, 2011 by brian0918

[aequalsa]

but you can't deny that both methods make large numbers seem difficult to handle.

Sure I can! Those materials, the golden beads, are designed to assist in teaching 1-1 correlation between numerals and quantity and to give a concrete representation of dimension to 4-5 year olds. That girl looks to be 8 or 9, so unless she is developmentally behind or lacking a conceptual understanding of those things I see no reason why they would still be teaching her using them. I have 4 year olds in my class that are adding large numbers using numerals alone.

If that is Montessori, then it is likely an (mis)application added by later Montessorians for older children.

[brian0918]

She's a 3rd grader according to the video, and certainly doesn't show any evidence of being "developmentally behind" *despite* her school. I agree, there is no reason they should still be teaching her this way, and yet they are, according to the video.

Edited February 27, 2011 by brian0918