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  • #16
    Meanwhile, if I sit down and concentrate, I can calculate square roots in my head. To several decimal places if necessary. Takes a while, mind.

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    • #17
      Quoth Chromatix View Post
      Meanwhile, if I sit down and concentrate, I can calculate square roots in my head. To several decimal places if necessary. Takes a while, mind.
      I'm not sure if I hate you or I'm jealous of you.

      My kids think it's hilarious that I got As in both college statistics classes I took but have trouble with multiplying or dividing fractions.
      Sorry, my cow died so I don't need your bull

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      • #18
        Quoth ADeMartino View Post
        In school, I was a whiz with regular arithmetic, but bombed badly when it came to anything beyond basic algebra. For this reason, I thought I was a complete idiot.

        Not so much, as it turned out once I got to the real world and realized how many people have absolutely no clue how to do simple grade-school arithmetic.

        As an example, I was at a convenience store buying a soda pop and a bag of chips (breakfast, mmm-MMMM!). The cashier had given me my total, I'd handed her the money, and just as the drawer popped open, the lights went out.

        I was astounded when she couldn't figure out my change from the information she already had - my total ($2.28) and the amount of money I'd just handed her - $5.00. I had to give her the answer - $2.72 - but honestly, that wasn't what disturbed me. LOTS of people can't do the arithmetic in their heads. But she couldn't even do it on PAPER.

        I was even more astonished when I realized that her name tag had the word MANAGER on it.
        I astound the kids I work with all the time, by being able to "count change back" (if you know what I mean by this phrase, you're old like me ) In your example, you gave her a $5.00, so I reach into the till and starting with $2.28 - your total - I grab: two pennies, two dimes, two quarters, and two singles (2.28 + 0.02 = 2.30 + 0.20 = 2.50 + 0.50 = 3.00 + 2.00 = 5.00). Even my managers are amazed by this, since they're both under 35

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        • #19
          Honestly, fractions are not too difficult as long as you can memorise the procedures for them. In some ways it's similar to basic algebra. The most difficult part, I find, is simplifying fractions, because that means factorising the components to see which common factors you can remove - and if none, which smaller denominator is likely to result in the least error. It may help to know that factorising integers is considered difficult even for computers - although most integers you will find in a fraction are small enough to not be a problem at that level.

          But I do know that the quality of teaching matters a huge amount. For reasons I still can't fathom, I was sent not to the local school but one miles away, closer to where my dad worked - even though it was almost never him that took me there. And it happened to be in Toxteth. As you can imagine, it had a lot of economically deprived pupils from the lower strata of society - and the style of teaching reflected that.

          So there I was, son of an engineer, being taught arithmetic by the extremely tedious method of filling in pages upon pages of two-plus-two-digit sums. I think I tended to get them right, but progression towards anything more challenging or interesting remained glacially slow. Nobody seemed to have any interest in teaching faster and less boring methods of doing them, or even hinting that such methods existed.

          Similarly, I was soon able to read considerably faster than I could speak the words out loud. But the teachers didn't know any other method of verifying that I could in fact read the books, so they got rather cross when I refused to read out loud because it was so much more efficient to just read.

          Unsurprisingly, I was soon bored out of my skull, and getting in trouble for expressing that fact.

          Which is why it was the Headmaster who ended up being the first to explain a simple optimisation that I could use - by that time I was studying in his office, presumably to keep me out of trouble. I had just been introduced to long multiplication at the age of seven, and so I hadn't memorised the times tables yet, and was instead working them out by repeatedly adding. The optimisation was for the tens, which was simply to shift everything to the left by a digit and insert a zero. This was such a revelation that I first checked to make sure it was actually correct.

          Not long after that, I was removed from that school and introduced to correspondence courses, which first measured what I could already deal with reliably and then sent me materials for everything else. Unfortunately the materials were published in the US - so the history materials were based on American history, and my parents had to go through the English materials to correct their spelling to British format before giving them to me. But at least that meant I was aware of the differences between the two.

          It turned out that, despite the limited material at school, I could already deal with fully half of what the (primary level) correspondence course could possibly teach me, without having first been taught it explicitly. The rest I blasted through in about three months flat, partly because my parents were willing to take a liberal view of my demonstrating that I had learned the technique - after practicing on a few of the easier problems, I could tackle the last row of the harder ones, and if all those came out right, that was done.

          And yes, that included fractions.

          The practical upshot was that when I *very* briefly returned to a conventional primary school, in a somewhat less deprived area of the city, I already knew all of the material that was being presented in each class. Instead of being bored to the point of stagnation, I was the annoying kid with his hand permanently in the air. This school, to their credit, rapidly concluded that there was little point in my continued attendance.

          At secondary school I had the distinct benefit of a maths teacher who knew this same method of keeping up with my ability to learn. So by the time I left that school at 16, I had qualifications that I would normally have taken at 18 - and I had spent the last year of that entirely on subjects less fundamental than Maths and Science. I was then able to spend my "sixth form" study time on subjects I was directly interested in, rather than their prerequisites.

          Not all of the teachers at that school were so enlightened, unfortunately. That is another story.

          But yes, I am equally horrified to hear about the typical quality of university entrants these days. Considering that a 18-year-old today would almost have been ready for school when I graduated from university, I am surprised that the average quality of teaching could go down so quickly. Yes, there were a few "special" cases even in my time, even allowing for my own generally higher than average ability, but they were the exception rather than the rule, and more or less to be expected.

          The only explanation that makes sense to me is that teachers in nearly all schools have been reduced to the style I experienced in Toxteth, where the entire class ends up learning at the pace of the slowest student. Allowing ubiquitous calculator use might accelerate that somewhat, but it also avoids teaching people how to cope without one - and also some fundamentals about numbers that are useful for general problem solving. And that hurts the most able students the worst - precisely the ones most likely to enter university.

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          • #20
            Quoth Teefies2 View Post
            I astound the kids I work with all the time, by being able to "count change back" (if you know what I mean by this phrase, you're old like me ) In your example, you gave her a $5.00, so I reach into the till and starting with $2.28 - your total - I grab: two pennies, two dimes, two quarters, and two singles (2.28 + 0.02 = 2.30 + 0.20 = 2.50 + 0.50 = 3.00 + 2.00 = 5.00). Even my managers are amazed by this, since they're both under 35
            At my first job, a little mom-and-pop donut shop, the Pop part of the ownership insisted that all his cashiers know how to count back change correctly. The only problem was that some customers (the ones who weren't regulars) would get annoyed by having their change counted back to them. They just wanted me to tell them the total and dump it in their hands. *sigh* It's a dying art.
            Sorry, my cow died so I don't need your bull

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            • #21
              Quoth Chromatix View Post
              Meanwhile, if I sit down and concentrate, I can calculate square roots in my head. To several decimal places if necessary. Takes a while, mind.
              I was never good at square roots, but I excelled at squares . Had up to 25x25 memorized, and figured out a simple way of figuring just about any square.

              Add 1, 3, 5, 7, 9 (etc) depending on the last square before it. So for 25x25 it would be '25 steps' (1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49 ... so 49 + (24x24)...hehe. Works every time. (for instance 10x10 = 100 + 21 (its 10 steps) = 121 or 11x11) Of course the major flaw in this is you have to know the previous square's total.
              Last edited by Mytical; 03-13-2013, 08:44 PM.
              Engaged to the amazing Marmalady. She is my Silver Dragon, shining as bright as the sun. I her Black Dragon (though good honestly), dark as night..fierce and strong.

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              • #22
                Indeed, I use a similar technique for square roots. In fact, the easiest way to do square roots is to do squares, and adjust the number you are squaring until it matches your target. You would start by estimating and then refine it.

                For example, the square root of 110 would be between 10 (100) and 11 (121).

                If we try 10.5, we can construct the square of that from 100 by adding 0.5x10 + 0.5x10.5, giving 110.25. So 10.5 is pretty close, and a touch high - but 10.4 would definitely be too low.

                The more precise square root of 110 is 10.48809, according to my calculator.

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                • #23
                  Quoth Primer View Post
                  College students cannot add or subtract, much less multiply or divide, and heaven help us if an exponent or a negative number is involved!
                  Unfortunately, college students with no hint of competence at math CAN multiply, even after making a trip to student health services. The gene pool needs a littlelot of chlorine.
                  Any fool can piss on the floor. It takes a talented SC to shit on the ceiling.

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