Weakness in mental arithmetic = weakness in math?

[Juwane]

Halfway through an undergraduate course in engineering, I'm now planning to review math fundamentals from pre-algebra, algebra, geometry to trigonometry and finally calculus because, as you may know, having a solid foundation in math is vital for any engineering course, and I've always been weak in math. I also happen to be very weak in mental arithmetic (adding, multiplying, etc. in head). Even a calculation as simple as 4+7 makes me think for many seconds, and when I can't figure the answer out I use my fingers to count! If this is for addition what can we say about subtraction? For addition involving two negative integers I always use the calculator so as not to make a mistake, even for such numbers as -7-3.

Does having these problems impair one's ability to learn new math concepts? Should I improve on mental math before I start reviewing?

 

[Nabeshin]

As an engineer, this kind of thinking isn't really that important. Seeing relationships between numbers, easier ways to multiply/add/exponentiate/etc. might be important in more of the realm of number theory or something, but not engineering.

That said, if 4+7 takes you several seconds to compute, I imagine going through a numerical problem takes a lot of time. Simple mental computations arise everywhere in physics and mathematics, and the extra time I would imagine to be ridiculous. So for the sake of saving yourself time, I'd say it's a useful skill to practice.

Don't worry too much though.

 

[Juwane]

if 4+7 takes you several seconds to compute, I imagine going through a numerical problem takes a lot of time. Simple mental computations arise everywhere in physics and mathematics, and the extra time I would imagine to be ridiculous. So for the sake of saving yourself time, I'd say it's a useful skill to practice.

But I can use a calculator. Can anyone imagine studying engineering without using a calculator?

 

[xxChrisxx]

Mental arithmatic is not required in the slightest, convenient but not required.

Its all about being able to maniputale formula correctly.

Being good at mental arithmatic only requires practice. The only reason I became good at adding up quickly was becuase I worked behind a bar where you had to add up the prices in your head. Being able to add up 2 bitters and a gin and tonic, whilst pulling drinks and holding a converation, quickly became second nature.

 

[Nabeshin]

But I can use a calculator. Can anyone imagine studying engineering without using a calculator?

Sigh, I guess. I was thinking in terms of something like factoring, differentiating, integrating, where there's quick arithmetic to be done. Or rationalizing fractions, multiplying by conjugates, etc. and I guess you can do all of that on a calculator. I'd argue it's faster (at present day) to do a lot of this in my head that type it into my TI-89. Also I find there's a much greater continuity of thought going through a problem pencil-paper-brain than with a calculator in hand.

Of course this begs the discussion of what the hell is the point of learning it anyways when you can do it all on the calculator, but that's neither here nor there.

The answer to the question of whether or not lacking arithmetic ability impairs your ability to learn math is a definite no in my opinion, but like I said, I find value in it.

-Nabe

 

[benk99nenm312]

But I can use a calculator.

Weakness in mental arithmetic is something you can improve, but only if you want to. Using a calculator is the usual solution for anyone these days, including mathematicians (and math teachers). It is no longer necessary to know what 2+2 is. However, if you ask me, everyone should know, or be able to figure out the answer rather quickly, because it is elemental. It's like knowing the alphabet. I say even if you can do your job as an engineer, it is important to have the basic skill of addition.

There is a book called "Speed Mathematics" by Bill Handley. It assumes some basics, but teaches you new and easier ways to calculate things (addition, subtraction, all of it). It's really cool. Maybe that could help.

Don't loose sleep over this. I can tell you that 97^2 + 15 - 6^2 = 9,388 without using a calculator (seriously, I swear I didn't use one ). Was it necessary... no.. of course not. But I worked hard in order to improve my mental ability, and the fact that I can do that makes me feel a little better about myself. You can too, if you want to.

 

[Juwane]

I can tell you that 97^2 + 15 - 6^2 = 9,388 without using a calculator (seriously, I swear I didn't use one ).

Can you say how many seconds did you take to do that one?

 

[benk99nenm312]

Can you say how many seconds did you take to do that one?

About as many as it took me to type it out... so around 5.

 

[Juwane]

53^2 + 62 - 3^3 = 2898

I took 1 minute 51 seconds to do this one, ending up with a wrong answer: 2571.

Do you have any hope for me?

 

[AUMathTutor]

I think some ability is necessary, but not too much. You should know how to go through the algorithms with pencil and paper... but in your head? I would say only order-of-magnitude calculations need to be done in your head, and those should be easy anyway (if they're not, you're doing something wrong).

They do still teach the arithmetic algorithms in schools nowadays, yes?

 

[djeitnstine]

Does having these problems impair one's ability to learn new math concepts? Should I improve on mental math before I start reviewing?

No but I find it helps one concentrate on one's work better, double check an answer and even follow what the professor does in class!

I'm not demeaning anyone but I do remember an incident where the professor had something like 1 = \frac{4 \mu_0}{2 \pi}x + \frac{1}{3} then the professor said \frac{4 \mu_0}{2 \pi}x = \frac{2}{3}

and this kid asked how in the world did he get 2/3 from. Needless to say many people got a kick out of it.

Learning to do mental arithmetic can save you many trivial steps especially when someone is trying to explain something to you. IMO its better that way because I used to be a tutor and I would pull my hair out when my students asked my to show a trivial step =(

 

[benk99nenm312]

53^2 + 62 - 3^3 = 2898

I took 1 minute 51 seconds to do this one, ending up with a wrong answer: 2571.

Do you have any hope for me?

You betch-ya champ.

I already mentioned it, but I should again. That book "Speed Mathematics" by Bill Handley is one of the best books ever written. I wouldn't be able to do that without reading that book.

I'll show you a cool example on the back cover of the book.

96 x 97 = ?

The usual method of 7 x 6... carry the 4... and so forth is way too slow. There is an easier way. We use what's called a reference number (in this case, 100). It's hard to explain what that is, but you'll see after this.

So, what we're going to do is subtract both 96 and 97 from 100.

100 100
-96 -97

We end up with 4 and 3, respectively.

Now, we subtract diagnally. Either 96-3 or 97-4. Either way, you get 93. (These are the basics you would make sure to know before moving on to this.)

93 x 100 (the reference number) = 9,300. That's simple enough. You're just adding two zeros to 93.

After that, we will simply add the product of the 2 numbers we found earlier (3 and 4) to 9,300.

3 x 4 = 12 <--(basics, just memorize problems like these)

9,300 + 12 = 9,312

See. Easy, isn't it. With practice, anyone can do that mentally.

 

[xxChrisxx]

Thats mega...

 

[benk99nenm312]

Thats mega...

Tell me about it. I nearly wet myself after I read that through the first time. It still gets me excited.

 

[Juwane]

That book "Speed Mathematics" by Bill Handley is one of the best books ever written. I wouldn't be able to do that without reading that book.

I know that book, and I also have "Secrets of Mental Math" by Arthur Benjamin and "Shortcut Math" by Gerard Kelly, and both these books contain some pretty neat tricks to do mental math. However, the problem with these books is that they often show you how to do ridiculously large numbers, numbers for which someone would ask for a calculator 99% of the time anyway. And who would remember the steps for the example you showed? It may be easy using pencil and paper, but then doing it using the traditional method (right to left) is much easier, for me anyway.

 

[benk99nenm312]

I know that book, and I also have "Secrets of Mental Math" by Arthur Benjamin and "Shortcut Math" by Gerard Kelly, and both these books contain some pretty neat tricks to do mental math. However, the problem with these books is that they often show you how to do ridiculously large numbers, numbers for which someone would ask for a calculator 99% of the time anyway. And who would remember the steps for the example you showed? It may be easy using pencil and paper, but then doing it using the traditional method (right to left) is much easier, for me anyway.

That's cool. I guess if you want the basics down, you could do something I used to do. You could write out a page of 50 basic math problems, and then time yourself on how long it takes to complete it. I remember I got about 23 seconds once on a 50 question multiplication paper (you know.. 5 x 7, 8 x 9). That's the most effective way I know of to memorizing and sometimes computing the basics.

 

[Juwane]

But all that apart, should I go ahead with my review without worrying about mental arithmetic, which perhaps I can improve on while doing the review?

 

[benk99nenm312]

But all that apart, should I go ahead with my review without worrying about mental arithmetic, which perhaps I can improve on while doing the review?

I would practice a bit before doing the review. I wouldn't think it would take long. Maybe a week of somewhat hard work and that should suffice.

 

[Juwane]

Maybe a week of somewhat hard work and that should suffice.

I've got only four months for the review, and I have to do a lot of things in it: algebra, geometry, not to mention calculus.

 

[benk99nenm312]

I've got only four months for the review, and I have to do a lot of things in it: algebra, geometry, not to mention calculus.

I don't know what to tell you, other than I think its necessary. You should have enough time to work on it.

 

[owlpride]

And who would remember the steps for the example you showed?

97*96 = (100-3)*(100-4) = 100*100 - 4*100 - 3*100 + 3*4 = 93*100 + 3*4

That's all he did.

 

[symbolipoint]

But I can use a calculator. Can anyone imagine studying engineering without using a calculator?

Can anyone imagine studying Arithmetic and Basic Math in elementary school and junior high school without using a calculator? YES.

What about for Engineering? NO. You want the calculator, at least a scientific calculator, for efficiency. You are no longer trying to learn basic mathematics.

 

[symbolipoint]

I think some ability is necessary, but not too much. You should know how to go through the algorithms with pencil and paper... but in your head? I would say only order-of-magnitude calculations need to be done in your head, and those should be easy anyway (if they're not, you're doing something wrong).

They do still teach the arithmetic algorithms in schools nowadays, yes?

As a reminder, xxChrisxx already told us that the skill was great to have when he was a bartender (see a few posts back).

 

[AUMathTutor]

Bartender? Don't you mean "Drink Engineer"?

 

[Monocles]

Weakness in mental arithmetic definitely does not equal weakness in math. I know plenty of people who are horrible at mental arithmetic (including myself) who are awesome at, well, anything else math related.

I actually had never considered the possibility that mental arithmetic is practice related. I used to be very good at mental arithmetic in elementary school, but once calculators were allowed I began a slow descent into crappiness.

 

[Tobias Funke]

I agree that weakness in mental arithmetic is not such a big deal, but when I think of mental arithmetic it's like 1782-338, or 23 x 10.7. Something like 7+4, I don't know, that might be the point where you should start worrying. I'd definitely work on that if I were you- it's worth it and you won't be under the control of a mindless calculator anymore.

 

[thrill3rnit3]

My belief is that whatever you can do in your head, do it in your head and don't rely on a calculator.

I consider myself "above average to good" in terms of mental arithmetic, and I don't see that hurting (or helping, for that matter) doing more advanced math problems.

 

[Choppy]

I might offer what my be a contrary point of view here.

As a student, I likely would have agreed with many of the posters here and supported the ideas that basic arithmetic can be done using a calculator, it's trivial and you don't need to worry about it. But I would be curious to know if professional engineers hold this opinion.

At the end of the day, it's often the numbers that matter. In my profession it directly translates into the amount of radiation I allow into a patient. For an engineer it would translate into everything from how much weight can a bridge support, to how much heat can an electronic device tolerate before breaking down. These are the factors that you have to sign your name to as a professional.

At the end of the day, sure, just about all of us use a calculator to figure the numbers out. But what happens when you hit the buttons in the wrong order? Or if there's a bug in your code? Part of being a professional is being able to look at what comes out of the machine and recognizing instances when the answer is wrong.

 

[Juwane]

Alright. Let us suppose I really do need to improve on my arithmetic, but how should I go about? Should I first memorize the basic facts (like 7+4=11) before I move on and deal with bigger numbers, or should I just practice doing many different arithmetic problems in my head and wait for an improvement? I want to ask the posters here: When you think 7+4, does the answer come in an instant or after some counting?

 

[benk99nenm312]

Alright. Let us suppose I really do need to improve on my arithmetic, but how should I go about? Should I first memorize the basic facts (like 7+4=11) before I move on and deal with bigger numbers, or should I just practice doing many different arithmetic problems in my head and wait for an improvement? I want to ask the posters here: When you think 7+4, does the answer come in an instant or after some counting?

It comes in an instant, because it is like I said, elemenatal. These types of problems should be memorized. As to how to go about doing that, I already recommended a method a few posts back.

Anything like 4x3 or 6x7 or 9x9 or 9-5 or 7+6... should be memorized. The answers should be easily used in harder calculations, like 96 x 97 for instance.

It is necessary to memorize these. Don't rely on technology to know the answer, know it yourself.

Always remember this retorical question:

Who invented the calculator?
Man did.

 

[ronaldor9]

I know just what you need:

Secrets of Mental Math: The Mathemagician's Guide to Lightning Calculation and Amazing Math Tricks
Practice practice practice!

 

[endless06]

Alright. Let us suppose I really do need to improve on my arithmetic, but how should I go about? Should I first memorize the basic facts (like 7+4=11) before I move on and deal with bigger numbers, or should I just practice doing many different arithmetic problems in my head and wait for an improvement? I want to ask the posters here: When you think 7+4, does the answer come in an instant or after some counting?

i'm sure many people just memorize certain outcomes through experience, but i memorize patterns in the numbers. i haven't read any books about speed math or anything, but knowing and recognizing patterns in numbers often allow me to calculate answers from decimals to relatively large numbers.

i think a good place to start is while doing problems, just try to think of the answer before going to the calculator, then verify after.

 

[arunma]

Honestly I wouldn't worry about mental math so much if all you're doing is reviewing for a test. I can't do mental arithmetic all that well (though I can do 4+7), and I got an undergrad degree in math. As Choppy said, the numbers are often the thing that matter, so you may want to eventually get better at this. But if you're just studying for some exam, I personally wouldn't worry about it at the moment.

 

[symbolipoint]

Alright. Let us suppose I really do need to improve on my arithmetic, but how should I go about? Should I first memorize the basic facts (like 7+4=11) before I move on and deal with bigger numbers, or should I just practice doing many different arithmetic problems in my head and wait for an improvement? I want to ask the posters here: When you think 7+4, does the answer come in an instant or after some counting?

A person should know the basic facts by about the end of second grade (about the age of 7 years old) for Addition, and about fourth grade for Multiplication.

After that, a person still might have some trouble with Division and Fractions, but these can still be studied and improvements made.

 

[Slidingby]

Who said you need math for engineering? There are plenty of bridges ready to fall down around my city! :D

 

[Tobias Funke]

Alright. Let us suppose I really do need to improve on my arithmetic, but how should I go about? Should I first memorize the basic facts (like 7+4=11) before I move on and deal with bigger numbers, or should I just practice doing many different arithmetic problems in my head and wait for an improvement? I want to ask the posters here: When you think 7+4, does the answer come in an instant or after some counting?

Certain things, using involving 7 for some reason, aren't instantaneous for me. 7+8 for example, I'd either do (7+3)+5 or (8+2)+5, but either way I have the answer in 2 seconds tops. Or 7 x 6, for example, I would do as (7 x 3) x 2 since I have 7 x 3 memorized better than 7 x 6. Again, that takes less than 2 seconds.

How much you need to memorize is up to you though. I'm sure there are some people that would be mortified if I, someone with a math degree and a math teacher, told them I don't have 7 x 6 memorized, but I do just fine.

 

[Troponin]

I was never very good at mental arithmetic either, so I've set a mission to practice until I am good.
After receiving Feynman's lectures on physics from my wife for my birthday, I read through his "tips on physics." He starts right off on tips on differentiation (he does a type of logarithmic differentiation) and then talks about carrying around a little notebook to practice basic math whenever you get the chance.

Since reading that...I've been carrying around a notebook like a little Feynman wannabe and practicing. I was surprised how quickly I was able to learn things like squares and roots. Whenever I'm bored, I run through squares in my head and I can do anything up into the 60's within a second or two...either I have the number memorized by now, or a number is near a number that has been stuck in my brain, or I know it follows a pattern that allows me to skip multiplication steps, etc., etc.

It's become a tremendous help in my studies. Being able to accurately estimate an answer helps to tell if I'm on the right track for an answer. And when learning new material, it is a HUGE help. You can run through problem sets to make sure you understand how to set up the problem MUCH quicker.
Instead of setting up the problem, slowly working through the calculation with a calculator and going to the back of the book to check your answer...you set up the problem, get a quick estimation of your answer and see how it checks in the back of the book.
If you've obviously set it up correctly, you can move to the next one.
It has helped tremendously in my mathematical modeling, if nothing more than allowing me the time to work through many more problems in a given study period.

 

[chiro]

Halfway through an undergraduate course in engineering, I'm now planning to review math fundamentals from pre-algebra, algebra, geometry to trigonometry and finally calculus because, as you may know, having a solid foundation in math is vital for any engineering course, and I've always been weak in math. I also happen to be very weak in mental arithmetic (adding, multiplying, etc. in head). Even a calculation as simple as 4+7 makes me think for many seconds, and when I can't figure the answer out I use my fingers to count! If this is for addition what can we say about subtraction? For addition involving two negative integers I always use the calculator so as not to make a mistake, even for such numbers as -7-3.

Does having these problems impair one's ability to learn new math concepts? Should I improve on mental math before I start reviewing?

To me math is simply the science of representation. I think your ability to abstract, recognize patterns, generalize and synthesize information is more important than having say a quick ability to make some algebraic calculations.

I wouldn't worry too much about it.

 

[Saladsamurai]

I would definitely work on it a little. Don't spend days at a time on it, just work on a few problems here and there while doing the other stuff. You are an engineer for Pete's sake, multitask!

Believe it or not people, you can punch numbers into a calculator wrong when you are in a hurry. You should be able to detect whether the answer that it returns is reasonable or not.

As engineers we also do a lot of "back-of-envelope" calculations. You shouldn't need a calculator for those.

Trust me, if you have to grab a calculator for every simple arithmetic problem, you are going to look like a jerk.

Just be aware of your weaknesses and make effort in your down time to strengthen them. You'll be fine.

 

[flamrim]

I'm a big fan of the Trachtenberg system

 

[kirab]

Here's what you do:

If you know programming, write a program that randomly generates a list of 100 subtraction problems where the largest integer is 10. So stuff like 10 - 2, 7 - 8 etc. Give yourself 10 minutes to complete it. After you have finished it and have gotten a PERFECT score, create another set of problems and increase the difficulty if you want (by allowing a higher integer to be the max).

After you finish 10 of these sheets, move on to multiplication and division if you want. This is basically what my grade 4 teacher did for us every day before class began, and it helped a lot in doing mental math.

 

[rajdeepghai]

53^2 + 62 - 3^3 = 2898

I took 1 minute 51 seconds to do this one, ending up with a wrong answer: 2571.

Do you have any hope for me?

Even 2898 is wrong...
53^2 + 62 + 3^3 = 2898
53^2 + 62 - 3^3 = 2844

And of course there is hope. While there are several books of shortcuts and quick arithmetic, not too many teach a more fundamental and basic understanding of numbers and patterns.

I have recently found a program on numerical understanding and appreciation that sounded very interesting and good. Sent them a message yesterday and am waiting more details.

Its expensive but if it is worth it, I might take it up. Will keep informed incase anyone is interested

 

[GregA]

I think mental arithmetic a skill well worth training

Don't take this as condescending, but the same advice as I gave to my cousin last year as I was helping her with maths I give to you...spent some quality time reciting you're multiplication tables because though its a pain in the arse to learn them; with multiplication I'd say you'll perform pretty much all your mental calculations resorting to this look-up table in some way.

As for addition & subtraction I'd say that something like 4+7 comes immediately but to try and imagine myself naively calculating it I suppose I'll notice that 3 is less than ten, and for want of another way of putting it 'borrow' 3 from the 4 leaving just a 1 left to add on to the 10 I've made

 

[Hurkyl]

One of my favorite quotes is:

"I'm a mathematician, darn it, not an arithmetician!"

 

[symbolipoint]

One of my favorite quotes is:

"I'm a mathematician, darn it, not an arithmetician!"

That one is great.

What most people need to understand is that mathematically instensively educated people are not human calculation machines. We may know and are able to use number properties, may understand shape relationships, but we do not need to handle even slightly complicated numeric calculations in our head.

We may examine a set of values and features in a situation, assign numbers to parts of these features, transform or transcribe expressions and equations to relate these features, and perform "Arithmetic-algebraic" operations to find a formula for an unknown value which we are interested in having. We can then make meaningful use of this value. Even with this wonderful composition of skills and understanding, some of us can not handle money-sale transactions with counting back change - cashiering work - at least, not efficiently.

 

[Count Iblis]

You could also http://www.centreforthemind.com/publications/SavantNumerosity.pdf"

 

[GregA]

One of my favorite quotes is:

"I'm a mathematician, darn it, not an arithmetician!"

Heh!...yeah I hear you Hurkyl!
If I tell anyone new back home that I'm studying maths they never fail to hit me with some pointless sum they can't add up for themselves (ok sometimes they fail at doing this but not very often)

Though getting back to the op for all practical purposes you don't need to be ****-hot with mental arithmetic; just good enough that if you type say 123 + 456 into your calculator and get an even number you at least know you've screwed up with your button pressing as opposed to blindly continuing

 

[ImAnEngineer]

I try to do as much arithmetic as possible in my head, even on tests. For me it's easier to focus without a calculator and it makes me feel more satisfied and confident if it turned out i was able to do it without the machine. If I'm not sure if the answer is correct, or I have spare time I use the calculator to check my answers.

 

[LSDwhat?]

what ??

On my University a calculator isn't allowed for calculus classes there few where its allowed.

 

[symbolipoint]

what ??

On my University a calculator isn't allowed for calculus classes there few where its allowed.

That which you find is not universal. Many or even most Calculus classes will allow use of at least a scientific calculator on tests; even if a graphing calculator were allowed, you would at least expect that your full solution and steps must be shown in order to receive any credit.