Do you think that facility in mental arithmetic should be encouraged in schools?

I think mental arithmetic is only necessary with reasonably small numbers. I see no problem with using calculators for problems like 999×999. but, I do think people should be able to do $70×7, to be able to figure out 30% off while shopping at a store, or 9×2, so they know to leave just under $18 on a $90 restaurant check. And, how to make change with simple subtraction. Practical math that comes up in daily life. However, most of my friends don’t even understand the math I mentioned, so I don’t think they are going to care to do the more advanced math you want people to do in their heads.

I think it would be helpful if there were actually a mental math course taught in colleges and certain high schools. It could really change the way people do “small numbers math”. As it is, we’ve come to rely on computers and machines to do all the work for us which has dulled our skills.

@digitalimpression College? If people can’t do some math in their head by college, I don’t believe they will ever be able to do it. They certainly are not math people, and they probably would not care about doing it.

@JLeslie They still teach basic algebra in college too. Everyone should know that as well.. yet it’s clear that many don’t. Either way, the class wouldn’t be teaching them basic math.. rather a better way of doing it as in @LostInParadise ‘s example.

Your own example proves my point. 999×999 is actually very simple to do in your head.. yet you see no problem using a calculator.

@digitalimpression How often does 999×999 come up? My point is people who are not “math people” freeze up. They aren’t doing much of anything in their head with math. Basic Algebra is taught in college, and yes they should already know. Algebra was a class at my university to weed people out of the business program. A lot of people seemed to really struggle with that class. I never took it, I took Business Calculus in Jr. College and didn’t have to take that weeder class.

For math people the numbers kind of move around in our heads, flip around on either side of the equal sign, we drop off numbers like zero, move decimals left to right with little trouble. Other people can’t do it. My college educated friends, as I said above, can’t understand $70×7 gives you the 30% off the price, something people deal with every time they shop, some sort of percentage off. They go through all the steps in their mind of 70 x .3=21, 70–21=49, seems much longer and harder to me. When I explain it, they either don’t get it or don’t trust it. Same with the tip thing. I cannot for the life of me understand why they cannot figure out 20% or why people cannot figure out the 24 hour clock, that is all simple to do in our heads. I don’t even subtract 12 for the clock, I just subtract 2 in my head. Two. Why is that difficult? 1400 hours, I ignore the 1, and just subtract 2, 2:00. But, I feel confident with numbers. I shortcut things regarding numbers, reduce it to the least amount of steps in my head.

The tricks for doing math in our head would be useful to be taught at a young age I think, but once someone is in college, if they are in a fairly basic math class, then they just are not very good at math. A professor might give a tip or two, but to really focus on it is pointless I think.

@JLeslie I don’t see a problem with people practicing/being taught these skills. Clearly you do. Next question.

Teaching mental arithmetic should start in elementary school. It is not rocket science. Anyone can learn it. If you want to teach 30% off is the same as 70%, teach 12 apples minus ¾ of them. That leaves you with ¼ of the apples. ¼ times 12 is 3. With practice this becomes as natural as breathing.

@LostInParadise To do this in elementary school it would have to be a private school. In public schools, they are way to concerned with writing answers down and rehearsing the answers for standardized tests.

We do mental math. It’s the only way to get my gifted son to do math as writing things down is too tedious for him. I ask the problems. He answers. I write it down.

In a public school setting, they’d need to loosen up with the need for paper. That is not going to happen anytime soon unless there is some type of major transformation.

Like @JLeslie, I see no need for this with large numbers. It’s vital, IMO, with small digits. Beyond that, it’s fine for people that like doing large sums in their head. However, an important thing to remember when trying to teach an entire class is that you need to teach to middle ground, not just to the excelled students. When things regularly go over a student’s head, they tune out. Once tuned out it’s difficult to get them to hear your ideas.

There is a sense of accomplishment in being able to turn a problem involving large numbers into one involving small ones and it ties in with basic algebraic formulas. Another example. You are given a right triangle with one side equal to 49 and the hypotenuse equal to 51. What is the other side? From the Pythagorean equation, the square of the side is 51^2 – 49^2 = (51 + 49)(51–49) = 200. The third side is square root of 200.

As to standardized testing in schools, don’t get me started. The program is a failure by its own standards.

I have the opposite complaint… I was always able to get the right answer mentally, without thinking about it; but they always wanted me to “show my work” – which, if I tried, I’d trip myself up in some way. Or I’d ACE the exams, but not do the homework, and was failing because of it. If I understand the material and am able to show my proficiency on the exams, why should I waste my time on homework? So I stopped taking math classes. School killed something I showed an aptitude for.

I think showing agility in math all through school is absolutely a good idea. My son is okay with Math, my daughter is better, but neither can figure out 15% in their heads. Neither can convert Fahrenheit to Celsius or back again. And they can’t estimate the square root to find a factor of a larger number, so they have problems with reducing fractions.

@zenvelo What ages are they?

@LostInParadise I would just be happy the student knows the formula, who cares if they use a calculator to square or square root or multiply? Who cares is too extreme, I do care they can multiply, but if they get to the point they can do the agebra probably they know how to multiply.

@hearkat The thing is as math gets more advanced it becomes more important to write out the steps. I understand your frustration completely. I was dinged many tmes for not showing all the steps, but still getting the right answers. The problem is sometimes especially in lower levels of math, a person gets the right answer for the wrong reason. The most simplistic example is a student getting 2×2 right, but then every other multipication problem wrong, because they are adding everything, not multipying. But, again, many kids skip steps and know exactly what they are doing, like my examples above where I drop off numbers and skip steps even in my head, because it is just very obvious to me. A good teacher would see what the problem is with the student if they got everything wrong except that one problem, but I think a lot of teachers are clueless, especially elementary ed. Ugh, I know that sounds awful, but elementary ed people generaly (generalization, not all) are not math people in my experience. My girlfriend called me to ask me what she thinks is going on regarding her daughter and math. She was great at doing the homework, but did poorly on the tests. It was dead obvious what was wrong, every tme the question was asked in the reverse of what was expected she got the question wrong (but the right answer if it had been asked the same as all the problems she had practiced). For instance if it asked to put four numbers in order highest to lowest, she would put them in order lowest to highest. It had nothing to do with math, it was reading the question. I really don’t understand how the teacher or her mom didn’t see it. My girlfriend said that many of the moms were frustrated their children weren’t doing well in math in her class. This was a private school.

I found out through my 3rd grade teacher (who mde us memorize the multiplication tables through 12) that you only need to know those tables off the top of your head and mental math comes easy from then on.

To take your question off on a slight tangent:
As part of her university degree a friend was doing research into how people process mathematical problems in their head, she presented me with 10 different mathematical problems and afterwards asked how I’d processed each one in my head to achieve the answer. She said everyone does it differently, despite ending up with the same answer, I found that interesting, I’d always imagined everyone processed mathematical problems in the same way I did, but no, not at all.

@Adagio I wonder if there are similarities with how people do the problems depending on how far they went with math, whether they consider themseves good a math, the school system they grew up in, or maybe the amount of education they have in general? Maybe it seems like people are all over the map in how they approach a problem, but if you group the people in a specific way, within their group they do it similary? I have no idea, these are all questions.

@JLeslie interesting thought…

Yes, I think we should encourage mental arithmetic skills. How often problems come up when we don’t have a calculator is irrelevant. It’s just a good skill to have in its own right. It’s exercise for the brain at a time when an alarming number of people exercise nothing more than their remote control hand.

I think it’s a useful skill to have, but less important to developing “math brain” than most other concepts. All the important stuff is generalized with variables; a good math student is not dependent at all on numbers. Being able to manipulate numbers is nice too, but trivial.