Elizabeth Green NYT Article On Teaching Math

Parents Forum
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<p>No one, including me, objects to kids learning multiplication tables by heart. What I was objecting to before is the fetishisation of that. Every discussion of math teaching I have ever seen includes someone, and usually a lot of people, coming in and saying something like, “How can they claim to be teaching math when the kids don’t even know their multiplication tables?” or, “Why are they spending time on pictures and stories and not memorizing multiplication tables?” Knowing your multiplication tables is a useful thing, but it’s not the same as learning math, and it’s not an indication of quality teaching. And, as I suggested, maybe you lose some kids by stressing it too much, making it a gateway skill rather than something you develop over time. No one in the range of normal intelligence has trouble with multiplication if their daily life requires anything like multiplication, whether or not they have been to school. </p>

<p>Unlike reading, simple multiplication and division are skills that people can and do “discover” naturally. There’s no crying need to spend loads of time teaching it in school, other than everyone expects it to be taught there. If you want kids really to learn math, however, it may make sense not to start by telling them the most important thing in math is memorizing multiplication tables.</p>

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<p>@mathyone:</p>

<p>Do not assume that your school is typical. What type of school is it, BTW? At the magnet HS I went to, teachers taught at most 4 hours a day, but at the regular Jr. high and regular elementary school I went to in rural IL, 6 hours in a classroom per day was par for the course for the teachers (this was several decades ago).</p>

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<p>For the patient, here is a link to a typical SAT forum thread</p>

<p><a href=“Math Question - SAT Preparation - College Confidential Forums”>Math Question - SAT Preparation - College Confidential Forums;

<p>Take the time to click on the link in the last post by Pckeller and see how the formula is described. It is a great example of how memorization differs from the utilization of the formula. </p>

<p>For the interested, I would highly recommend following the threads in which Pckeller, DrSteve, Satquantum, gcf101, and a couple more of the SAT professional participate. I would also recommend following Miter94 to ascertain how a student with a great mathematical aptitude approaches the SAT. The differences between a student who follows the typical HS taught math is easy to spot. </p>

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<p>“A well-prepared student should have NO problems with the time limit”</p>

<p>So you believe all students should be scoring 700+? It’s just a matter of preparing them? Everybody under 700s has trouble with the time limit.</p>

<p>If you have a preparation technique that will get any student who uses it above 700 on all sections, then you are a bazillionaire and posting from your yacht in the Mediterranean.</p>

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<p>The important thing is not to fetishize ANY method of instruction. The idea of teaching kids to understand what they’re doing is really important. But once you do, the “old” methods may still be the fastest and most functional way to solve any number of problems.</p>

<p>Let’s keep it really simple and consider the math problem 14 + 18 =32. In the old way, you’d line up the numbers, add up the right column to get 12, leave the two and carry the one, then add the left column to get three and wind up with 32. But the problem is that if you don’t understand what you’re doing, this is just a series of steps to be memorized. Maybe you master it for the test in June - but when you come back to school in September, you’ve forgotten all about “carrying”, just remember “add the numbers in the right column, then add the numbers in the left column,” and determine that 14+18 = 212, since 8+4 = 12 and 1 +1 =2. Someone who understands the concept would never make that mistake, but someone who has just been taught a formula might.</p>

<p>So, how do we make sure the kid understands? Well, for one thing, we can have him model with blocks or dots or whatever, although that becomes a real time-waster pretty quickly - even the kid who writes 14+18 = 212 would be, in most cases, fully capable of counting out the answer if you gave him fourteen stickers, and then eighteen more. More productively, we can make sure he understands that the “one” in 14 and 18 is really a ten, so that when you have a problem like 14+18, what you are really saying is ten plus four plus ten plus eight - or, alternatively, when you add fourteen to eighteen, you start by adding ten to eighteen, and then add four more. </p>

<p>Now, that’s fantastic, because a kid who understands that really isn’t going to make an absurd mistake like saying that 14+18 =212. But here’s the new problem: Saying 10+4+10+8=32 is practical when you’re adding two two-digit numbers, but what if you’re adding nine of them? Or what if the problem isn’t 14 + 18, it is 147,392 + 183, 748? At that point, the best way to solve the problem probably is lining up the numbers, adding the numbers in the ones place, and then the numbers in the tens place, and so on. </p>

<p>The point is, it isn’t that the method is bad, it is that doing it by rote is bad. Once you understand that “carrying” the one isn’t just a step in the process, it is moving a “ten” into the tens column where it belongs, you won’t ever forget it again, since “forgetting” simply doesn’t make sense any more. We can’t throw the baby out with the bathwater.</p>

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<p>So you believe all students should be scoring 700+? It’s just a matter of preparing them? Everybody under 700s has trouble with the time limit.</p>

<p>If you have a preparation technique that will get any student who uses it above 700 on all sections, then you are a bazillionaire and posting from your yacht in the Mediterranean.<<<</p>

<p>Why do I sense a healthy dose of condescension or attempt to ridicule? </p>

<p>Having no problems with the time limit does not mean everyone should ace the test or score above 700. But everyone who is well-prepared should score at her or his aptitude level. The key is knowing how the time limits work. </p>

<p>For instance, math tutors, at least the ones worth their salt, will tell you every problem on the SAT math could be solved in less than the average allotted time, be it 30 seconds or one minute. That is true but it requires that “everyone” to know the best manner to solve a problem. Recognizing known time sinks and knowing what to expect in terms of scores is 9/10 of dealing with time issues. </p>

<p>Students who struggle with the time limits have simply not recognized their own issues, including physical handicaps, or not been instructed properly. </p>

<p>And, as far as a yatch in the Mediterranean, that assumes there is a secret sauce to monetize or … stolen goods to sell as they do in Asia. You do not earn a fortune by telling students to … practice hard and make the necessary efforts. </p>

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<p>@purpleTitan, these are public schools and the hours are for both middle and high school teachers. I don’t have info on the elementary teachers. My own public elementary school had a 5.5 hour instructional day (8:30-3 with a full hour for lunch). The kids were required to walk home for lunch so the teachers were not supervising that. Of course the kids would be taken to art, gym and music where they were handed off to other teachers so that is about 4.5 hours of classroom instruction my own elementary teachers were doing. That 4.5 hours would include recess, which was supervised by classroom teachers I believe, but hardly requires prep time.</p>

<p>As far as early math instruction goes, it clearly makes sense to start kids with manipulative objects so they can understand what the operations really mean, and once they get that, move them to the usual notation which is a lot faster and more efficient. There’s no reason for kids to be going through motions they don’t understand. My kids learned concepts of carrying and multiplication in Montessori preschool, and I am sure most kids could master this if they would use the same materials with a little enthusiasm. My kid already knew the squares and cubes of numbers through 10 when she started kindergarten and she understood what those concepts were. It’s a normal part of the Montessori prek-K curriculum.</p>

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One sees the same type if thing even in Calculus. The first derivatives are done by writing out the definition and reducing it. Once the students have seen the trend, various shortcuts are introduced. It’s still considered important to see where the shortcuts came from to begin with.</p>

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<p>“Students who struggle with the time limits have simply not recognized their own issues, including physical handicaps, or not been instructed properly.”</p>

<p>Indeed, I think that’s ridiculous. Recognizing that you have a slow processing speed doesn’t fix the problem. Nor is there any “proper” instruction that necessarily teaches that kid to make those quick triage decisions in real time. Some endlessly struggle with that.</p>

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<p>I do not think you understood what was meant by “their own issues” in my sentence. It is not only about handicaps such as slow processing or other physical limitations. </p>

<p>And, for that matter, I am not sure you understood what well-prepared meant. Being well-prepared includes.the correct preparation to reach an attainable objective under the existing constraints. That objective is not the same for everyone. The time constraints are manageable with adequate preparation. Other deficiencies are harder to overcome. </p>