<p>atomom and SlackerMomMD: My ninth grader is running into the “explain how you got this” math stuff now. There are questions that he has on his homework that I can’t even answer properly. As long as they get the correct number who cares if they can write it down in complete sentences explaining it the way the book wants? ugh.</p>
<p>OP- As a former Computer Programmer I’m thinking that CS may not be the best choice for her.</p>
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<p>IMO, this “knowing,” meaning memorizing, the procedures is worthless. Sure, you can teach a 3rd grader (or 4th grade?) to do simple operations and place digits in the correct place to end up with a string of digits that is interpreted as the correct answer, but it’s more important to understand why placing digits in that string, in that way, works. </p>
<p>I suspect the reason the procedure is taught rather than the reasoning behind it is because teachers often don’t even know the reasoning. They just know the procedure. I can recall specific events in school where this was the case. </p>
<p>I don’t quite understand the “explain in words how you got the answer.” I don’t think I ever had that. The way we were taught single digit addition, subtraction, and multiplication was simply to memorize. We had these tests where we would just answer a hundred of these as quickly as possible. And in these cases where students are taught to memorize the procedure for multiplying and dividing, I don’t understand what they could say other than “I used the procedure taught in class.”</p>
<p>" My ninth grader is running into the “explain how you got this” math stuff now."</p>
<p>He should just say he got it from the Jolly Green Giant.</p>
<p>Having “Explain how you got this answer” follow up questions are not all bad. This question is really helpful once you get into higher math where the understanding of your process in necessary. Take calculus for example, the students needs to know why the integrated a(t) abd used v(t) to find the original position function. This question flat out tells you whether or not the students knows waft the heck they’re doing. It’s even used in AP classes.</p>
<p>Many years ago, right after leaving college, I worked in the candy department at Filene’s in Boston. I was given the job, by the manager, of explaining to seasonal hires how to use the tare scale and how to calculate discounts and so on. (As an aside, literally the ONLY person who understood how to use the tare the first time I explained it turned out to be a graduate of Pomona who had come to Boston to be a Master’s candidate in Classics at Harvard.) We had a cash register that didn’t do any calculations. Frequently, we had to calculate a 10% or–the horror!-- 15% discount. Many of the young women they hired were incapable of doing this. Some obviously had no idea of the decimal system, for starters. I will never forget one such person. I gave her a rundown, starting by explaining what the decimal point was, the ones column, the tens column, etc. I showed her how to move the decimal point one place to the left and round to get 10%. She seemed to understand what I was talking about. Then a short while later she came to me with an item marked 79 cents (the special character that is a C with a line though it). “There’s no decimal point,” she said.</p>
<p>This is why it is important to understand more that a memorized operation.</p>
<p>Regarding the “explain how” stuff, I would venture to guess that the real problem is that kids who have mastered something are forced to keep repeating it endlessly while waiting for the rest to catch up. One summer I decided to do some arithmentic homeschooling with S, since his first grade class hadn’t done much. I got a couple of workboooks at the homeschooler store. He fully understood how to do two digit addition after one explanation, and completed a page of problems with out a single error. (In K, they had done a lot of great stuff with tens and decimals, and he had on his own produced a chart showing the powers of ten up to some very large number, so he had a good foundation.) He was ready to move on. In school, he would be forced to keep on doing the same thing over and over again, possibly for months, including explanations of something that was blindingly obvious to him, as quantmech suggests.</p>
<p>I loved that story, Consolation! “There’s no decimal point.” Lol</p>
<p>Plenty of kids know how to do math ( follow the instructions) but very few know how to use math. Most schools don’t teach how to use it, just how to do it. </p>
<p>An example: I homeschool my eleven year old. I gave him a word problem that was intended to test mixed units. It went something like " Bob ran a race in 4 minutes and 41 seconds. Tom ran the same race in 3 minutes and 50 seconds. How much faster was Tom?"</p>
<p>When I checked his paper, he had simply written 51, with no calculations. When a asked how he did it, he said that he added to Tom’s time until it was an even with Bob’s. Ten seconds brought him up to 4 minutes then plus 41.</p>
<p>I’d like to think that he was able to solve it quickly, correctly, and mentally because I taught him subtraction as simply the inverse of addition.</p>
<p>That is an example of using math. He’d probably get no credit for the problem if it was st school because he didn’t show any “work.”</p>
<p>I am all in favor of having students understand how the division of large numbers into large numbers works, and I am not a big fan of drill. In practice, I think that some of the methods in vogue currently (which Vladenschlutte didn’t run into), which are supposed to show what’s going on, just don’t succeed. The students tend to approach them as just an algorithm of a different category. I think what it would really take for these methods to work is a one-on-one discussion with each student, asking and answering questions, until the teacher could tell whether the student really “got” it. A close second to that would be to use Montessori methods. Both of these are generally impractical in the public schools, although the Montessori method might have some hope.</p>
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<p>Running into a form of this in 9 or 10th grade is to be expected to some extent. My classmates and I on the slowest academic track were learning proofs and how to apply them towards math problems. </p>
<p>Mandating them for early elementary school kids without sufficient solidification of basic arithmetic/math skills in the way many Ed school theories are put into practice in public schools means a lot of confused and/or frustrated kids.</p>
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“Recently discovered…12th grade”?? How could your friend not have known this much earlier than this (and had more of a chance to do something about it)? This may sound harsh but, doesn’t she pay attention to what her D learns (or not) in school?</p>
<p>GGD–it’s easier to miss gaps in your kid’s education than you may think.</p>
<p>^^ If students are taught using calculators, and there is an expectation that long division isn’t important because, well, we can always use the calculator, this gap can easily be covered.</p>
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Sure, if no one’s paying adequate attention to what the kids are actually doing. </p>
<p>We’re not talking about an advanced concept here in Calculus or something where the parent might have no clue about what’s really being done, but rather something very basic that it seems a parent would typically know whether the kid can do division or not - just like knowing if they can do multiplication, or addition, or subtraction, or spelling. This is a pretty basic thing.</p>
<p>IMO, the 1989 NCTM report set up a generation of people (namely, those younger than about 20 now, unless things change) to be mostly innumerate. Partly due to this report, the SAT allowed calculators starting in 1994; SAT math no longer tested the same thing it did before. Here in NJ, the math curriculum adopted in 1996 incorporated calculators for use starting in grade 2, just when children should be learning place value, multiplication, division, etc. A lack of fluency with multiplication and division leads to problems with fractions, which leads to difficulties with algebra and everything thereafter. Personally, I don’t think students should use calculators before algebra.</p>
<p>The argument that the calculator makes learning long division (or the standard multiplication algorithm) not particularly useful only focuses on the end and ignores the benefits of the means. Should we teach Shakespeare only to those high school students who have decided to become English professors?</p>
<p>As a high school math tutor, my jaw stopped dropping long ago. Now, I’ve begun to expect that my student will not know what 1/2 is as a decimal, or even what 7 times 9 is.</p>
<p>Fignewton – OP here; I agree with you. The friend’s kid in question is in my family’s same school district. The math curriculum spiraled, which meant that every year through elementary school, you saw the same topics again, but there was not always enough time for mastery before the curriculum moved on. The idea was that you would always hit it again. This kid had problems with division in fifth grade, the family worked hard (tutors, etc.) but the curriculum morphed toward algebra after that.</p>
<p>I agree completely with the notion that Shakespeare should not be reserved for those on the English professor track, foreign language study should not be reserved for those who will travel, and that the mental gymnastics involved in learning to do long division are helpful in developing number sense and mathematical reasoning. Besides, we can’t predict who will need what.</p>
<p>I see college-age cashiers (probably students) who are flummoxed when the total is $16, you hand them $20, they punch it into the register and then don’t know what to do when they’re counting out four $1s and you try to hand them an extra $1 so that they can give back a five. I had a cashier try to charge me $3 tax on a $10 item, because the register said so – not thinking that something had to be wrong with the register. I am worried about the critical thinking skills of many HS and college-age people. I don’t know if I am part of a centuries-long line of middle-aged people who have clucked over the younger generation, or if this generation’s thinking skills have truly been subverted in their mental development by reliance on gadgets. If you truly rely on the gadget, the gadget designer is doing your thinking for you.</p>
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<p>I don’t think the above is correct considering I took the SATs in 1994 and 95 and we weren’t allowed to use calculators on the math section at all.</p>
<p>All the math problems on the SAT can easily be done without a calculator. (I teach SAT prep–I have just done all 540+ problems in the blue book without a calculator). There really are very few problems on which a calculator might save you a little time. I wonder if trying to decide if/when/how to use the calculator on a lot of problems doesn’t actually slow down some test-takers. No calculators allowed in my Algebra I class.</p>
<p>I didn’t intend to start a drill vs. understanding debate. I’m not against students understanding what they are doing and being able to explain it. But the newest (changes incorporated this year) elementary curriculum here includes a lot of, imo, unnecessary writing. (I’m talking about third grade math, simple addition or rounding off to tens and hundreds with two and three digit numbers.) If the kids can consistently get the right answers at this level, to me that is enough evidence that they understand it.
The problems I see with students in class and students I tutor have to do with both a lack of drill and a lack of understanding. Things like getting wrong answers due to simple arithmetic errors or not noticing when answers don’t make sense, etc. But the lack of memorization of basic facts (drill), imo, is what makes them truly unable to do math. Not sure if you need to be able to explain in complete sentences that you must carry to the next place value when your sum in a multi-digit addition column is 10 or more. But if you habitually forget to do this (that is, follow the procedure) and frequently get wrong answers, (and you’re way beyond 3rd grade) you obviously don’t have it down. When they make this kind of mistake, students will often say, “Oh, but I was so CLOSE!” (and then I say, “Close equals wrong.”) Same with times tables. Students should be able to do them upside down in their sleep. The reason they don’t know what 8 X 7 equals isn’t because they can’t explain in nice sentences (or pictures) that there are 8 groups of 7 or 7 groups of 8 all added together. They don’t know it because they haven’t memorized it. It doesn’t matter if they can explain how to get the answer when they’re getting the wrong answer. On a recent thread about math tutoring, someone described Kumon as “Drill and Kill.” Yeah, baby! 
I’m for it.</p>
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<p>That reminds me of a funny story a former supervisor turned close friend recounted from an engineering math class during his college days. </p>
<p>After a math exam, several students were loudly complaining about the Professor’s stingy grading practices, including his policy of awarding no partial credit. The Professor’s retort was “You build a bridge! Bridge collapses due to tiny miscalculation. NO PARTIAL CREDIT!!!”</p>
<p>I don’t think you are allowed to “carry” any more in arithmetic, atomom. Now you “regroup” tens into the tens column. Ditto “borrowing.” I think it has expired.</p>
<p>Once, in response to a question that asked how the student got the answer to a problem, QMP wrote, “By looking at it.” This was homework, so I was able to explain, “I know that is how you solved it, but that is not the answer they are looking for.” It can be nearly impossible for a third grader to figure out what he/she is allowed to just see, and what he/she is supposed to have to figure out by some lengthier route.</p>
<p>I’m not anti-drill. QMP’s fourth grade teacher had an arrangement where after a student had scored 100 3 times in a row on the test of each of the basic operations, the student was exempt from further testing on that operation. This made a great deal of sense to me.</p>
<p>Even if there is a lot of drill, a student who knows the answers can generally finish the drill quite rapidly. I am not really fond of homework that takes longer for a student who understands the math than for a student who doesn’t understand it–at elementary school level, anyway.</p>
<p>My DD got to fourth grade without memorizing multiplication tables. No teacher insisted on it. Then she hit division which was of course way difficult because it was harder to manipulate the numbers. After convincing her that the only way to get through that division homework FAST was to memorize those multiplication tables, she finally did the flash cards with me (it was a battle up to then since no teacher pushed it). Two days later she had them down pat and all of her math was easier after that. Drilling is important–it’s brain exercise that improves number manipulation.</p>