<p>My younger son has an aversion to memorizing things with numbers. I think he still is apt to multiple 6*7 by adding 14 three times. He calls me for his social security number regularly. I keep suggesting he try to memorize it, he does and he’s always got a number reversed or wrong. He had a terrible time remember the shortcuts he learned in precalc and was always figuring things out from first principles. Interestingly, he scored in the 90th %ile on the WISC. His precalc teacher was actually quite impressed with his understanding of the material even though he was greatly slowed down by having to figure it out all over again on tests. But yeah, I can imagine he might not know how to do long division. But he’s actually got very good number sense and can do mental math much better than me. (He’s very good at the 99+43 is the same as 100+42 sort of thinking.) I find his math brain fascinating. So different from his older brother who could look at all sorts of complicated math problems in 3rd grade, know the right answer and like Quantmech’s kid be unable to explain how he got it - it was just blindingly obvious to him.</p>
<p>I tested at a middle school math level in college. My parents knew I was a little behind but they had no idea it was that bad, and I don’t think they could have. </p>
<p>In elementary my mom did flash cards with me every day, several times a day. We practiced arithmetic with everything from pencil and paper to m&ms spread out into math problems on the table so i could count. I completed a small library worth of homeschool math books. My mom sent me to centers for struggling students to get specialized tutoring by highly trained educational experts. I was evaluated, and diagnosed, with a math learning disability-- but the school told her my english scores were too high for me to be disabled and dismissed the diagnosis. </p>
<p>My mom hired my second grade teacher to keep working with me on math concepts over the summer before third grade to try an get me to grade level before school started back up. In third grade, in addition to regular math homework, we had timed multiplication tests every other day–you started with a test of twos, and if you hit 95% or more you moved on to threes, otherwise you repeated until you passed. It was intended that we’d get through 12 by the end of the school year, and special tests were written to 14s for the two top students in the class. I never got further than the sevens, I ran out of time before I could ever get to 12 even despite drilling flash cards at home. </p>
<p>In fourth grade I remember being sick the day we learned to count back change. My teacher sat with me during recess to teach me the lesson and I could NOT understand a single thing she was doing. She accused me of playing dumb and gave up. My sister ended up teaching me. </p>
<p>In high school I started really failing math. We had four math tracks, one regular, one slow, one advanced, and one special ed. I was on the regular track. When I failed, my teachers wouldn’t let me move to the slow track and actually wanted to put me on the advanced track-- I was so “smart,” I’d “catch up” when I felt like “really trying.” This is what they told my parents, and my parents-- thinking the teachers were the experts-- believed them. When I failed tests I got grounded for not trying. Knowing how hard I was really trying, I began to believe I was just stupid. I worked so hard for Cs and Ds just to be told I must not have been trying. I went to the math tutoring lab offered by my school and was told, “You don’t even know THAT? I’m sorry , but you’re wasting our time and it isn’t fair to the other students. You can’t come to math tutoring anymore.”</p>
<p>It wasn’t until I was already in college that we started to truly understand the depth of the problem. I now understand it will usually take me ten times longer than anybody else to grasp a math concept, and drilling only served to make me feel stupid and defeated-- we were either drilling techniques I could never really learn or moving on too quickly for me to ever catch up. As I know now, for most things it’s simply a matter of drilling a different technique that works better with the way my brain works, or giving me enough time to really get it before I have to move on. We had no way of knowing that when teachers kept telling my parents their methods worked on every kid and it was just a matter of time and my effort for me to catch up. My parents were deceived. </p>
<p>I think teachers have too many students and there aren’t enough funds or resources to identify kids who are struggling and figure out why and how to help them. It is too much work and the teachers do not have time. Parents can only do so much to supplement what their kids are learning in school, and when they are told by the so called experts that everything is fine and it just takes time-- or that the problem is really their child refusing to learn-- I’m not sure what we can expect them to do. My parents thought they did all the right things and went above and beyond what most parents would do and we still got nowhere.</p>
<p>I haven’t read through the whole thread, so maybe this has already been addressed. But what is the benefit of dividing a three digit number by a two digit number by hand, if we can find the answer using a calculator?</p>
<p>That depends on how much math you’re going to do, of course, but in Algebra II and Precalculus, we expect students to divide one polynomial by another. The process for doing that is essentially long division. (Even synthetic division relies on an understanding of the process of division if you’re going to understand the meaning of the remainder.)</p>
<p>And we teach that skill in algebra and precalc classes so that students will be able to integrate rational functions by decomposing them into partial fractions in Calculus II.</p>
<p>Aside from that, it is useful to know when the answer your calculator has given you just can’t be right. (We all key in the wrong thing sometimes.) If you don’t understand the process of division, you probably won’t recognize when your answer is ridiculously wrong.</p>
<p>FlyMeToTheMoon, ditto to what Sikorsky said. I’ve brought up polynomial division earlier. If you don’t know how to divide without a calculator, your going to have a bad time once your x’s have multiple degrees. AND you’re going to be SOOL when you don’t have a calculator on your exam. Everyone should be able to divide a three digit number by a two digit number. It’s simple math.</p>
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<p>Because there are all kinds of situations in life when it is handy to be able to quickly do division–preferably in your head, but at least on paper–when you don’t have a calculator handy. Splitting a check for a large group, calculating how much square yardage you need of something while standing in the store aisle, comparing cost of materials, and so forth. </p>
<p>It is very simple. I really cannot believe that teachers wold skip it. They are probably among the math cripples one encounters on a daily basis who can’t do the simplest calculation and are helpless without a calculator. I had this conversation with my son’s 4th grade teacher. She told me that she taught long division, but that some teachers did not. I was astounded then. I gather that the situation is even worse now.</p>
<p>“I’m not an ambi-turner. It’s a problem I’ve had since I was a baby. I can’t turn left.” - Derek Zoolander</p>
<p>Sorry. This scene plays through my head every time I scroll past this thread. </p>
<p>Carry on.</p>
<p>(Terrible at math, no advice to offer anyone.)</p>
<p>(But great at movie quotes.)</p>
<p>I don’t find the “what if you don’t have a calculator?” argument completely worthless, but I think it has less and less value year by year.</p>
<p>In the early 1900s, new manufacturing technology made paper and pencils cheaper and more widely available than they had ever been before, and consequently, American schools started moving from chalk and slate to paper and pencil. People carried on about how that was going to be the ruination of math education in America, because American students were being taught to do arithmetic with paper and pencil instead of mental arithmetic. “And what will they do,” the traditionalists asked, “when they don’t have access to paper and pencil?” </p>
<p>Obviously, by the middle of the century, that was a ridiculous question. Paper and pencils were ubiquitous, and just about the only place where a person wouldn’t have easy access to paper and a pen or pencil was the shower.</p>
<p>During my lifetime, calculators have achieved just about the same saturation as paper and pencils. As I glance about the study where I’m typing this, I see not one but two TI-30s that my wife has left out on the desk. I could locate three different TI-83s in a couple of minutes. Calculator watches are now pass</p>
<p>My oldest child’s second grade teacher sent home a canned note from the Everyday Math curriculum that said something like, “Please don’t show your child the method of subtraction that you used in school. We use the such-and-such method and don’t want the students to get confused.” CONFUSED, LOL??? If you saw the such-and-such method, you would gag! One problem could literally take fifteen lines of numbers. Totally ridiculous. I was a rebel and ignored the instructions! Oh, we also got a note saying that in this curriculum, calculators were allowed, since paper and pencil calculations weren’t used very often in real life. I actually got up and complained to the school board, and they gave me patronizing smiles and ignored me.</p>
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<p>You can’t imagine the sleep that my husband and I lose worrying about our calculations as structural engineers! Once I misplaced a decimal and cost the client a bundle of money. Structural engineering is not a career for the faint of heart. But it IS fun using math to design buildings that we drive by every day.</p>
<p>Three cheers for structural engineers! :D</p>
<p>Lots of science careers (medicine, nursing, pharmacy, engineering) use math which can kill people with a wrong answer. Close just ain’t gonna cut it. And you HAVE to know when that answer you got on the calculator still doesn’t add up.</p>
<p>The loss of the Mars Climate Orbiter (1999?) due to failure to convert English units to Metric units was a famous and costly arithmetic mistake. Not life or death, but a $125 million loss.</p>
<p>Keeping my fingers crossed/holding my breath/saying a Hail Mary when I cross bridges. . .</p>
<p>I have an S with dyscalculia. This is not normal but may not be the result of bad teachers or poor learning.</p>
<p>Basically, dyscalculia is like dyslexia for math. And yes it was not consistently a problem for my S., he could subtract and divide better then he could add and multiply. He struggled with the numbers being in constant motion and sometimes they moved completely out of his mental vision. Try doing calculations with free floating numbers. And his math SAT was low. He made up for it in the other 2 parts but it weighed like an anchor on his overall score. He attends a school with very little math requirements.</p>
<p>And high school was inept at helping him with his dyscalculia. I learned that the only learning disabilities that get extra help are the ones that come with extra funding from the government. Your child can go suck eggs if there is no extra money in helping him.</p>
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Some years ago, a McKinsey consultant was presenting the results of their (very expensive) study that they had performed for my employer. The PowerPoint slides looked wonderful, with animation and a very consistent look and feel, but the conclusion was obviously wrong, as anyone with an elementary school grasp of arithmetic could tell. Not that the McKinsey folks couldn’t multiply and divide (they are, after all, the best and the brightest), but they had stopped doing “sanity check” math in their heads since Excel was so much more accurate-seeming. Since I was far from the best or the brightest, if you showed me a bunch of numbers that you told me were related in some fashion, I couldn’t help but do a quick sanity check just to make sure you hadn’t made an error in entering a formula. </p>
<p>Especially interesting to me was how long the consultant stuck to his result, even when it was clearly based on a significant arithmetic error.</p>
<p>Sikorsky, even if I though I could pull out my phone and use its calculator, I still do the math in my head, because I don’t want to become brain dead. 
I can’t tell you how many times I’ve been with other women who flutter around looking for a calculator every time a question involving basic arithmetic comes up. By the time they find it, I’ve already got the answer.</p>
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<p>I don’t see too much of this except among some biglaw partners at one firm I worked at, among a couple of college alum friends who weren’t very strong in basic arithmetic/math, and random first/second-time acquaintances. </p>
<p>With the exception of the partners, everyone else I met who had such issues tend to react with the same type of shame/embarrassment I’ve observed from some older adults who are totally/functionally illiterate.</p>
<p>I am sure you do, Consolation, and I usually do the math in my head, too. I didn’t mean to suggest that using a calculator for straightforward computation is better–merely that it’s very rarely impossible, or even impractical. </p>
<p>I would even venture to say that if we were of our grandparents’ generation, we’d do even more computation in our heads than we do. I have taught in a lot of schools that had a grandparents’ day. Year after year, I used to think I figured in my head a lot better than my students did. Until grandparents’ day. Year in and year out, those grandparents can clean my clock! But on the other 364 days of the year, that was of absolutely no consequence, because even if I need a few more seconds, or paper and pencil, or a calculator to do some of the computation that the grandparents can do faster than I, I have access to those things just about anywhere I am.</p>
<p>Consolation, this happens every time I have dinner with some friends. They think I’m a human calculator because they are amazed I can figure out a 15% tip in my head way faster than they can punch numbers into their phone.</p>
<p>Like I said, people have become way too cavalier about basic arithmetic because “we have calculators on our phones! why do we need to know how to do long division / multiplication?” I see the same disturbing attitude toward spelling (because we have spellcheck). I get that some people have learning disabilities that may it near impossible to do this - I’m not talking about them - I’m talking about capable people who decide NOT to use their skills.</p>
<p>My daughter had terrible spelling. At 2nd grade, she spelled like a 4 year old. I did not say 'Oh, well, there’s spellcheck", which several parents and a teacher said to me. I was appalled by this attitude of not wanting to improve and assuming that a computer can take over. I got her a tutor and she worked with the tutor twice a week for 2 years and then once a week for another 2 years so that her spelling improved to the point people could recognize her groups of letters as words.</p>
<p>It is scary that people are becoming more reliant on machines than themselves. Actually what’s scary is that people are comfortable about becoming more reliant on computers/machines than themselves.</p>
<p>Timely thread. Just this week I was in a meeting where the spreadsheet that needed to be discussed had numbers that didnt make sense in many fields. The person who produced it couldnt tell - a risk of just plugging numbers in. It was a waste of valuable time. While calculators are great tools, the reason to understand the why of mathematical operations is that it really makes us smarter it helps the brain learn to see patterns and relationships. I think such learning generalizes to other areas. I also think that spending time playing with numbers (even drill) helps kids see the patterns. Ill second the power of Montessori methods. I just dont understand why Montessori manipulatives are not used more extensively in math education.</p>
<p>Like some on the list, I always play with numbers in my head both for fun and to get a “ballpark” answer to crosscheck against the formal answer. However, I’m not so much against someone not knowing the mechanics of long division. To me the “why does it work” and “what else can we do using this technique” is more fun to teach a kid than to get the actual answer.</p>
<p>To take it a step further - I suspect many engineering students of today don’t know how to do square roots by hand, long-division style, but by putting commas and taking down 2 digits at a time, etc. And in this day and age, the ability to do so is not important. But it has a really cool use. Whenever I’ve shown the technique to middle/high schoolers when I used to help out in math/chess clubs, there was always one or two that I could really get involved. After the 5 minutes spent on learning this, we would spend 30 minutes discussing why it worked, and talk about how this kind of technique could be used elsewhere and what else can be done.</p>