<p>Poland is an interesting story in academic reforms with many facets to its “revolution.” Delayed tracking until 15, earlier start in primary, introduction of standardized tests, bonus accruing to teachers for performance, and a reversal to stronger standards make it an interesting --yet surprising-- choice for the US to want to emulate. The reality, however, if one believes in the validity of the PISA tests, is that the Poland leapfrogged the United States within a decade of overhauling its educational system. While still just outside the first tiers of countries such as several in Asia and others such as Finland and Belgium in Europe, Poland has made great strides. </p>
<p>As far as the introduction of algebra, there is nothing wrong with the introduction of an "early algebraic thinking” when the algebraic symbolism remains hidden from the naked eye. Students from grades 1 to 5 are able to instinctively solve problems without having seen the symbols per se. The clear lines between arithmetic and algebra are not marked in concrete. </p>
<p>The introduction of standard algebra in 6th grade in Poland is working because of the gradual introduction of related concepts in prior years, and a continuing coherent development towards a mastery in the subsequent grades corresponding to our first year of high school. </p>
<p>I admit that when my kids were in elementary school, I was a big fan of multiplication tables and other “math facts,” too. That’s the way I had been taught math in elementary school – notwithstanding being right in the middle of the New Math – and that’s what I associated with rigor and good training.</p>
<p>It’s only in retrospect that I see how fundamentally dumb it was. One of my kids was great at devising methods to solve problems, but she hated memorizing anything, including times tables. Her math teachers loved her, gave her tons of encouragement . . . but always marked her down because she would make calculation errors. The message she got from that was that she was not good at math, and it became a self-fulfilling prophesy. When in fact she was really good at math, and bored with rote memorization. (It wasn’t even a problem with memorization itself. In 7th grade, she could recite Wallace Stevens’ The Idea Of Order At Key West by heart, because she liked how it sounded and was trying to figure out what it meant. She was great at learning parts in Shakespeare or Sophocles. Times tables were just too trivial to care about.)</p>
<p>As an adult, she has no trouble with math facts. She knows them despite having learned them imperfectly in school. Calculation errors are not a concern, because Excel or some statistical program does the calculations most of the time, and last-digit precision is unnecessary in calculating tips and splitting checks. She is recovering her natural ability at math, and her interest in math, as she sees how real math is used in the real world, or at least that part of it that interests her. But her math education was a disaster.</p>
<p>JHS, please note that I do not disagree with you. </p>
<p>Regarding the books, last Christmas I (finally) had to get rid of the schoolbooks that were cluttering shelves around my parents’ house. I loaded them in my car and drove to a donation center. Without exception, every one of them was a behemot in its own right – perhaps not 800 pages but enough to justify its 100 dollars+ price.</p>
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<p>I do think that there are people in the education establishment who are trying very hard to bring about changes in those specific areas. The problem, of course, is to bridge the theories and the practice and have the courage to fight the forces that are perennially opposed to any changes that … might endanger the deeply encrusted older generations. </p>
<p>The more things change, the more they remain the same.</p>
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<p>We were out to dinner with some parents of younger children the other evening and one of them remarked with some disdain and amazement that they had been in someone’s house where the people had a multiplication table posted on the refrigerator or some other public place. The sentiment spoke volumes. There was no appropriate reponse other than to make a note never to vote for them for the school board.</p>
<p>I think one of the problems with getting any balance in this is that the teachers no longer believe that they should have to do anything that they find to be drudgery. Conscientious parents can get it into the curriculum or textbooks “on paper”, but it never receives any attention whatsoever. And it shows. </p>
<p>Haven’t had a chance to read the articles yet, and I am sure they are interesting but I can’t get past this:
“in Finland, teachers spend only 600 hours/year on classroom instruction, vs. 1,000+ hours here,”</p>
<p>On a block system, there are four 1.5 hour blocks per day, give or take a few minutes. Our middle and high school teachers teach 3 blocks per day. That is 4.5 hours of classroom instruction. We are on a 180 day school year. All quite typical I think. That works out to 810 classroom instructional hours. More than the 600 in Poland, but I think someone’s basic math skills are weak. If you subtract out all days missed for snow, the half days they count as full days, and not to mention some of the wild excuses they make here for canceling school, I suspect it’s barely 750 hours in practice.</p>
<p>My Algebra II teacher was great–he emphasized learning how a theorem worked in the first place, and would always show us the proof–even if no one was listening. It gave us a greater understanding of how we got these random rules in the first place.</p>
<p>Honestly, I did feel like elementary school math curriculum was too easy. I learned multiplication at the beginning of the first grade, while we only went over it in third grade. In consequence, I never tried in math class; it was always too easy for me. Even though I’ve skipped a grade of math, I find my math class incredibly easy and practically sleep through it (although my teacher is very good).</p>
<p>I think that they go over the rudimentary concepts for too long in elementary school; and the fact that they don’t give those who need extra help extra help and that they keep the ones who already get the concepts bored annoys me.</p>
<p>mathyone, maybe they are counting the time for prep work. I know my friend who teaches elementary school puts hours in after class not just correcting homework, but filling out paperwork for IEPs and 504 plans and whatever else the state requires.</p>
<p>I’m not seeing how memorizing the multiplication tables is really that onerous. It comes in fairly handy, actually. It seems like people want an all-or-nothing approach. Why not a sensible combination of “understanding” and “rote practicing”? You can’t learn to play the piano just by taking a music theory course. You also can’t learn to read and interpret music just by sitting around banging out the same piece day after day.</p>
<p>Exactly. I thought our elementary school did a pretty decent mix. Every year the 5th kids would vote that Math was their favorite subject (though there was always a pretty strong contingent for recess.) :D</p>
<p>@mathmom, I know teachers put in time outside the classroom for prep, meetings, paperwork, etc. Of course that is important but I don’t think that is considered under “classroom instruction”. Maybe I misunderstand that term, but the actual hours spent teaching as opposed to prep time were pretty clearly spelled out in our local teachers contract and it was not easy for our school board to increase them to the numbers I cited above.</p>
<p>The multiplication tables as usually learned consist of 144 facts, some of which are pretty trivial to remember. Is that really asking too much? Should that take multiple years to learn? I think one of the problems of math education is the idea that we can always use a calculator. (And I don’t generally see people whip out a calculator in the grocery store to compare prices.) Kids are using calculators too much and too soon and they fail to develop basic numerical sense. Plenty of high school kids have forgotten what they might have once known about this because of years of calculator use. Not being able to do the simple arithmetic in their head is a major distraction when they are trying to learn more complicated concepts. “And now this equation simplifies to…” Wait, let me get out my calculator. How did they get that 3 from that 21? What did they do!!!</p>
<p>We can always look up every word we don’t know in a dictionary so maybe we shouldn’t teach vocabulary. In fact, we have software to read to us so why bother learning all those letters? It’s 26 facts, 52 if you count the capital letters and that’s a lot for a kid to remember! And why study history at all, it’s all written down somewhere and you can just google it if you need to know something. Foreign languages? Way, way too much to memorize. Science? Too many facts. Just look it up if you need to know something. Why learn anything when you can look everything up?</p>
<p>I also don’t understand reluctance to make kids memorize. Asian kids have to memorize thousands of characters just to learn how to read. (Maybe it’s having to put in so much effort just to read that teaches Asian kids to put more effort into their schoolwork? Just thinking out loud.) The ability to memorize is a skill – mental and physical – like many others and it can come very handy in later. Without it, you will never learn a foreign language, for example. Think of it like calisthenics for the brain. </p>
<p>I don’t see the big deal about having to memorize the times tables, either. Sometimes there are just basic facts you need to get under your belt to go on to master any subject – whether that’s time tables in math, learning the rules of conjugating an -er verb in French, or the scales in music.</p>
<p>Indeed, I’ve been interested in learning music theory (an area that has always interested me, but that I’ve never studied) but I know that I will have to do some memorization of different scales, chords, etc. so that I can get to the point I want to get to - which is to read some articles written from a music theory perspective that are of great interest to me. Right now I’m not interested in taking the time to do that, so I’m going to stall out. But that’s not because I’m “dumb at music” - I am just not putting in the effort it takes. </p>
<p>We got a note from my son’s second grade teacher, asking us not to show our kids the “old” way of subtracting, because it would confuse them. Like the convoluted way they were using wouldn’t?? Then I got the SAME note when my daughter got to 2nd grade! I figured out it was a canned note from the curriculum authors. Lovely.</p>
<p>I agree that the over-reliance on calculators is a problem. My SAT/ACT students are turning to the calculator for problems like 4x15 and 100/5. They’ll get the right answer, but the extra time will kill them on the test. Now, I mostly think it’s silly to have such time-pressured tests anyway, and I’d like to see the time limits expanded or abandoned. But this is the system we have, and the kids are at a big disadvantage on the test if they can’t do those problems quickly in their heads. Using the calculator also interferes with the short-term memory process they’re engaged in on a problem. While they’re getting the calculator and typing in the digits, they’re forgetting that they were asked for the area, not the perimeter (or whatever).</p>
<p>The problems with teaching math begin early on - when my daughter was going to school to become a teacher, she had an observation in a classroom where the teacher started off teaching fractions by saying “this is really hard and kind of scary, but we have to learn it”. My daughter was horrified! </p>
<p>I have always struggled with math, but I did have to pass algebra in college and I was fortunate to have a really wonderful math teacher who walked through each kind of problem, how to identify what the problem was, how to solve it, and then applied what we had learned to word problems where we had to figure out what kind of problem it was from the words, and so on. It was the first time in my life I ever got an A in a math class. </p>
<p>First off, the time limits are not silly but ESSENTIAL to the integrity and nature of the SAT. Without time limits, the test is trivial. Without time limits, you need an entirely different test. A well-prepared student should have NO problems with the time limit, safe and except students who deserve the extra time for medical reasons. </p>
<p>The reliance on calculators is a bad habit learned in school where a graphical calculator is IMPOSED by the teachers. On the regular SAT, it is best to use the calculator for a few problems, or not all. I never used one and always recommended to store it away. On the other hand, the SAT Subject tests in math are mostly rewarding the mastery of a TI-89. </p>
<p>Ultimately, calculators should be prohibited on the regular SAT. </p>
<p>This is no different from learning the alphabet. The calisthenics for the brain is a PERFECT analogy. Kids love puzzles, and the ones who keep up with them will find standardized tests easy and perhaps fun. In fact, it is NOT truly memorizing and regurgitating but finding ways to increase the immediate recall through pragmatic uses. </p>
<p>This said, I believe that some are wired in a way that they “see” math just as some “see” music or art forms. I do, however, not believe it is gender based. Having been beaten for two decades by my younger sister on every board game and puzzle, I can attest that girls do … get math. And English. And sciences. Well, you get the picture! Girl power! </p>