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06 Dec 2005 - 23:11
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Looking at the Saxon Maths example, I understand why so many of my friends didn't get why they minus the +5 first - they were just told to undo the operation. As I had had BODMAS drilled into me from the start it just seemed logical to get rid of the +5 etc first (because you are undoing an operation), but they probably had never had the BODMAS drills, and without being explictly told, they would do it from left to right and get the wrong answer. -- SamanthaRawson - 06 Dec 2005
wow What does 'BODMAS' stand for? Yes, I was intrigued by this, too. I don't remember being told to do the addition first. I think I just kind of picked it up. I'm thinking I was one of those kids who could do huge amounts of incidental learning. I'm wondering if I had reasonably good 'pattern discrimination' or some such.....so that I'd notice that certain things recurred, like getting rid of the +5 first, and I'd just start doing them. I've mentioned that I was taught almost no punctuation or grammar (that I know for sure). Over the years I've noticed how people do it. -- CatherineJohnson - 06 Dec 2005
This is one of the things I keep seeing with Christopher. He doesn't just 'notice' stuff. You need to point things out to him, and teach it. He remembers things very quickly; he has the kind of memory I had as a child. (He remembers everything, but remembers them inflexibly, which was & is true of me, too.) But he needs direct instruction, broadly defined. -- CatherineJohnson - 06 Dec 2005
to get rid of the +5 etc first (because you are undoing an operation wait—I don't follow I think you've just said 'undo an operation' twice, haven't you? -- CatherineJohnson - 06 Dec 2005
My skepticism meter is beeping. The item on the right (with the standard deviation) seems like a traditional assignment with less structure followed up by a tricky word problem.
BODMAS stands for Brackets Of Division Multiplication Addition Subtraction, and it's a mnemetic trick to help people remember which part of a sum to do first, brackets then multiplication/division, then addition/subtraction. I assumed, since we were undoing the problem, you would automatically do the addition/subtraction first, then any multiplication/division. I didn't even have to think about it, I just did it. Most other people in my class struggled. -- SamanthaRawson - 07 Dec 2005
I would need to meddle as they were working on it to make sure they were not going off on a wild goose chase I love it! I'll bet I've lost the reference, but I read a terrific piece of research on college math courses. They looked at the Super Math Brain students, and compared them to the Not a Math Brain students. The big difference between the two groups was that the Not a Math Brain students 'went off on wild goose chases.' That was the exact expression they used. I've had some hilarious moments ever since reading that. I'll be working some math problem I don't understand, sitting there beavering away (mixed metaphor alert!), going, going, going, and I've left the vicinity of the correct answer many eons ago. Wild goose chase, for sure. -- CatherineJohnson - 07 Dec 2005
wait! Is BODMAS the same as Please Excuse My Dear Aunt Sally??? Not quite, I guess. This is for solving equations; that's for simplifying expresions, right? -- CatherineJohnson - 07 Dec 2005
"Is BODMAS the same as Please Excuse My Dear Aunt Sally???" The mnemonic device I prefer for the order of operations is the nonsensical acronym PEMDAS. -- CharlesH - 07 Dec 2005
The Interactive Mathematics activity seems like a reasonable one, as long as the students have the mathematical background for it, and, as long as what's on the paper we see there is not all there is. There has to be some kind of follow-up, IMO, with the mathematical reason for what's going on explained to the students. I think this might be part of why "reform" math has made itself such a bad name. If the teachers aren't explaining what's going on, OF COURSE the students will be lost. Students get lost even when we DO explain what's going on. Maybe the "best" way to teach math combines principles of direct instruction with some of these "reform" type activities. I can see this being a good activity for a high school class, or a non-calculus-based statistics class at the college level. (In a calc-stats class, I don't think I'd devote time to it in class, but I might assign as a homework problem to investigate what happens in a suitably general case, and provide a proof.) -- PaulMiller - 07 Dec 2005
But you can divide by 4 first, you just have to divide both sides of the equation by 4, so you get (4x+5)/4 = 17/4 which simplifies to x + 5/4 = 17/4. Then you subtract 5/4 from both sides to get x = 17/4 - 5/4, or x = 12/4, or x = 3. I actually don't like them writing "we always use the addition rule first" because you can do anything you want to that equation as long as you do the same thing to both sides. You use the addition first because it's much easier that way - no fractions or division to do in the first step, and only one division to do in the second step. There are some things we "always do" because they are written into the syntax of math and give it meaning - parentheses first, then exponents, etc. But that is not the reason we always undo the addition first. As long as a child understands what an equation is, and how it remains unchanged when you do the same thing to both sides, they don't need to memorize "undo the addition first." -- StephanieO - 07 Dec 2005
First: "Making Friends with Standard Deviations" is a really sick-making title. Shouldn't the students have left this sort of thing behind in kindergarten? Second: I think this would make more sense if the students understand the formal definitions of "mean" and "standard deviation" first. 1) This is the definition of mean; this is the definition of standard deviation. 2) If you were to add a constant to each member of the set, what would you expect? 3) Here is a set; find the mean and standard deviation. 4) Here is a second set with each member equal to the first set plus a constant; find the mean and standard deviation. 5) Did the result match your expectations? 6) This is what you should have seen, and why you should have seen it. 7) If you want confirmation, try it again. It's hard to see patterns in the noise. It's far easier if you know what to look for. -- DougSundseth - 07 Dec 2005
When I wrote that the activity was reasonable, I said so assuming the students have the "mathematical background" for it. Knowing the definitions of mean and standard deviation pretty much constitute the "mathematical background" for this activity. I think the questions guiding this activity are quite adequate. The pattern that'll show up is quite simple. All it takes is understanding of the definitions to predict what will happen (and, to prove it, too.) -- PaulMiller - 07 Dec 2005
If the students already understand mean and standard deviation, the activity is trivial. The only time it could possibly be of use is as a method to teach. If you are teaching, make the teaching explicit. "Keep repeating this process until you see a pattern." (Note that you've found a minimum of three means and three standard deviations at the point that this instruction is given.) If you intend to treat math instruction as a scientific experiment, do it right: Hypothesize, predict, test the prediction (and if necessary refine, rinse, and repeat). This isn't the same as faff about, look for a pattern, speculate about why the pattern might exist. -- DougSundseth - 07 Dec 2005
about Saxon: ditto what StephanieO? said. It is inflexible to direct kids that "we must" do this and "we always" do that. Depending on the whole numbers, one way or the other might be less likely to lead to computational error. Maybe. Having students do it both ways would be great if there was time. about IMP: so how many kids in a group? who runs the calculator? who writes down the numbers? who checks the numbers? who sits around idly while numbers are slowly and steadily punched into the calculator? This is the thing that drives me crazy about constructivist math. Apparently, kids just won't believe the result unless they generate the several sets of five numbers themselves. Don't ride on wheels you didn't invent. Don't trust anyone over thirty. Own the numbers! We shall overcome! Get excited about generating several sets of five numbers with your peers, and talk about it? These projects don't eliminate everything that was ever felt to be wrong with lectures given by teachers at the front of the room -- they just move it all into the context of small peer groups. -- BeckyC - 07 Dec 2005
There is a huge, HUGE difference between "know the definitions of mean and standard deviation" and "UNDERSTAND mean and standard deviation." Just ask anyone who teaches freshman calculus if their students understand the chain rule or L'hopital's rule. I certainly didn't when I was an undergrad. I still submit that the mathematical background for this activity is only to "know the definitions" of mean and standard deviation. The point is to get an increased understanding of the concepts. Otherwise, as Doug has so rightly pointed out, it is pointless to even do the activity. I mean, it's not like I have to do this activity to tell you that when you add a nonzero number to every member of a data set, the mean changes by exactly that number, and the standard deviation changes not at all, even if you're looking at a continuous probability distribution rather than a discrete data set. But, I took a year of mathematical statistics as an undergrad, and understand mean and standard deviation well enough to prove this statement mathematically, so I'm obviously not the target audience. There's an old saying: "Research is what you're doing when you don't know what you're doing." This is certainly true in mathematics. Sometimes we just don't know enough about the things we're dealing with to make appropriate hypotheses, and we have to cast about in the dark a bit to find our way. This is the price we pay as mathematicians for being able to publish our work as "theorems" rather than "theories." In the appropriate level course, if I can inject a little bit of discovery for those who can take it, I'm glad to do so. Like I said before, you won't catch me doing this activity in a calc-stats course, but, in a non-calc-based stats course, this is a reasonable and decent activity. The point is to get students to the point where they do understand mean and standard deviation. Why, if we all agree that students have to learn to mastery the algorithms for basic operations on numbers in order to develop number sense, do we object to an activity that tries to do the same thing for concepts like "mean" and "standard deviation"? How can a student form hypotheses about these concepts if he doesn't understand them? Now, coming back to the idea of "mathematical followup" I said would be necessary with this activity... after the students play around for 10 minutes or so, then it's definitely time to explain what is going on. By that I mean to provide a mathematical proof of the statement I made above, or, at very least, a mathematical argument that demonstrates why what's going on should be happening. Like I said, in the calc-stats course, I'd expect the students to generate a mathematical proof themselves. By the way, I certainly hope whatever book this activity comes from is teaching them the difference between population standard deviation and sample standard deviation. If not, then they will never get an understanding of sampling. -- PaulMiller - 07 Dec 2005
coming back to the idea of "mathematical followup" I said would be necessary with this activity Another name for "mathemtical followup" is "mathematical closure", and it may be lacking, especially in elementary schools. Teachers can fall into the trap of believing that 99% of their work is done when the "experiment" has been set up and the students have worked in small groups. Due to the primacy of both Discovery and Peer Learning. -- BeckyC - 07 Dec 2005
But you can divide by 4 first, you just have to divide both sides of the equation by 4, so you get (4x+5)/4 = 17/4 which simplifies to x + 5/4 = 17/4. Then you subtract 5/4 from both sides to get x = 17/4 - 5/4, or x = 12/4, or x = 3. I haven't gotten far enough in my re-education to think analytically about the IMP problem. With the Saxon passage, I've had the exact experience Becky is talking about. This will sound awfully low-level (and I've told the story before), but one day, thinking about teaching math concepts as well as procedures, I said to my neighbor, 'If you're subtracting 19 from 30, why can't you subtract the 9 from the 0?' It actually took her a moment to realize that, obviously, you can. (I would have figured it out myself! But she's a statistician. She got there first!) That was a fabulous moment for me; I really saw, in a way I hadn't seen before, what a marvelous invention the 'simple' addition algorithm is. (I still don't see—not really—the beauty and genius of the base-10 system. I 'know' it, I use it, I can explain it.....but my understanding is quite superficial still.) I've mentioned before that deSaussure said that meaning comes from difference. I believe that. For me, the Saxon lesson would be better if you pointed out that, yes, you could divide first, but that would be inefficient and inelegant. -- CatherineJohnson - 07 Dec 2005
There are some things we "always do" because they are written into the syntax of math and give it meaning - parentheses first, then exponents, etc. But that is not the reason we always undo the addition first. I think this is VERY important. As a person re-learning math, I've been constantly frustrated by the need to know whether a 'rule' is a rule because it's more efficient and elegant to do things a certain way, or because it is written into the syntax of math (I love that way of putting it). Certainly for me, this is much more important than any text seems to grasp. In my case, it's important because I have sometimes spent hours and hours trying to understand conceptually why an 'elegance' rule is a syntax rule. Talk about a wild goose chase. (I wish I could remember examples. I'll start keeping notes when this happens in the future.) Actually, the addition example probably falls into that category. I was thinking the addition algorithm, in which you 'must' borrow in order to subtract the '9' from the '0,' was somehow 'syntactical.' Obviously, that betrays a very superficial grasp of the algorithm, BUT—and you'll just have to take my word on this—it doesn't betray a complete lack of understanding of the algorithm. I think I have it! This just came to me. We all know that students (all 'learners,' learning any subject) go through a stage of inflexible knowledge. I believe that when you are in the stage of inflexible knowledge you can especially benefit from being told that, yes, you could divide first, but mathematicians don't divide first because it's inefficient. I suspect that the inflexible stage is also the point at which it's helpful to show learners that there's 'more than one way to do it.' By that I don't mean a zillion and one ludicrously complicated Russian peasant tally mark multiplication schemes. I mean, almost certainly, TWO WORKABLE WAYS TO DO AND/OR EXPRESS THE SAME THING if possible. This is another Lost In Translation moment. Constructivists have accurately noticed that there are many times learners (I loathe that word) do benefit from being shown that it's possible to solve this problem another way. But they've turned this insight into a fetish—a fetish and, even worse, a lesson in multiculturalism. -- CatherineJohnson - 07 Dec 2005
Here's another inflexible knowledge moment. Christopher's teacher is doing almost entirely procedural teaching. She has to; she's racing through material at a clip of practically one new topic a day. (I'll have to post the TOC one of these days.) So the other day she taught them the cross-multiplication procedure for figuring out what a missing numerator or denominator is: 2/4 = 4/ I was confused when I started working on this with him; I guess I didn't quite realize what she'd taught in class. So I was setting this up as an equation: 2 x = 4 x 4. Naturally, Christopher was outraged at this intrusion into his COMPLETE-UNTO-ITSELF CROSS-MULTIPLICATION PROCEDURE, so we had eye-rollling and major sighing until at some point I said, 'Look at this. This equation I've written _is exactly what you're doing_'—and he saw it! His eyes widened, and he said, "Oh!" in a surprised, happy voice. Since we have the goal of teaching math conceptually as well as procedurally, I think these moments are important. But you need a superb curriculum, which we don't have. Ms. Kahl has to race through huge numbers of topics, and there's ZERO time for getting the point. Plus she's brand-new to teaching, and the department has not (to my knowledge) given any serious thought to formative assessment—or to the question Paul raised of how you incorporate formative assessment into a course that has to tear through as much material as this course does. Instead, the department philosophy is: these are the gifted kids, they need to be challenged. That's it. I have that in writing from the Assistant Superintendent for Curriculum. -- CatherineJohnson - 07 Dec 2005
If this were a DI school, Ms. Kahl would have had a nice speaking-to 2 months ago. The situation would have never been permitted to get to this stage where the first semester is practically over, and most of teh class has not learned anything. -- KDeRosa - 07 Dec 2005
Now, coming back to the idea of "mathematical followup" I said would be necessary with this activity... after the students play around for 10 minutes or so, then it's definitely time to explain what is going on. Yes, absolutely. It's not clear to me that anyone has the 'final word' on inductive versus deductive learning. James Milgram's criticism seems to be almost entirely that the main problem with discovery courses is that it takes a phenomenally talented and knowledgeable teacher to teach a good discovery course. Realistically, we're never going to have a vast elementary teacher corps able to pul this off. (Though now that I've read Engelmann I can certainly imagine someone writing a scripted 'discovery' curriculum that would be both excellent and teachable by all teachers.) Wickelgren's complaint was that discovery courses simply take too much time to cover the material they need to cover. Inductive learning is natural to human beings; it doesn't seem right to me that an excellent curriculum would do away with it. But Paul is absolutely right. You can't just send students out to discover something and leave it at that. -- CatherineJohnson - 07 Dec 2005
I ranted at my students once this semester for using this made up operation "cross multiplication." This is just yet another procedure made up by k-12 teachers that students can apply while mathematically flailing about looking for a solution... nothing more. -- PaulMiller - 07 Dec 2005
"So the other day she taught them the cross-multiplication procedure for figuring out what a missing numerator or denominator is:" 2/4 = 4/ I think that the idea of cross-multiplication is more damaging than helpful. What if you have something like: 3/4 = X/4 - 5 I have had students try to use cross-multiplication on this. Cross-multiplication is a special case of a more general rule. The general rule is very simple and teachers should use it all the time to reinforce how basic rules can be applied to all sorts of situations, from simple to complex. -- SteveH - 07 Dec 2005
Here's a good article on why constructivism is so dopey. -- KDeRosa - 07 Dec 2005
"There's an old saying: "Research is what you're doing when you don't know what you're doing." This is certainly true in mathematics. Sometimes we just don't know enough about the things we're dealing with to make appropriate hypotheses, and we have to cast about in the dark a bit to find our way. This is the price we pay as mathematicians for being able to publish our work as 'theorems' rather than 'theories.'" No argument. But this is what you do when you have to. When there is nothing else you can think of. It's extraordinarily inefficient, even when it's more efficient than any other course. Please note Catherine's comments about the amount of material that is being covered in her son's course; I expect the situation is similar in the course for which this problem was written. Why would you use blind-squirrel exploration when you don't have to? "The point is to get students to the point where they do understand mean and standard deviation. Why, if we all agree that students have to learn to mastery the algorithms for basic operations on numbers in order to develop number sense, do we object to an activity that tries to do the same thing for concepts like 'mean' and 'standard deviation'?" It's not the goal I have a problem with; it's the inefficiency of the process they have chosen for reaching that goal. "But they mean well", is insufficient for me to give them a pass. "How can a student form hypotheses about these concepts if he doesn't understand them?" By having the students form even incorrect hypotheses, you get them more deeply involved in the process. If they reasoned correctly, their hypothesis (and understanding of the underlying material) is confirmed; this is the kind of thing that makes students feel really good. If they reasoned incorrectly, there is more incentive to identify the flaws in the reasoning. To reiterate: I don't think the problem is without merit; I think it's clumsy. And the name is execrable. -- DougSundseth - 07 Dec 2005
"How can a student form hypotheses about these concepts if he doesn't understand them?" This encapsulates the entire flaw in the constructivist theory. When the original mathematicians discovered these principles, they didn't do it the same way the kids are being asked to do it. they were experts with extensive domain knowledge, not neopytes with no knowledge. Plus, as Doug points out, it's brutally inefficient to re-discover all these things anyway for what amounts to a dubious benefit in any event. -- KDeRosa - 07 Dec 2005
Steve Sorry to be so far behind....but I don't quite follow. Cross-multiplication is a special case of a more general rule. The general rule is very simple and teachers should use it all the time to reinforce how basic rules can be applied to all sorts of situations, from simple to complex. How do you frame the general rule? Would you frame it as: when a/b = c/d, then ad = bc also, do you bring up the example you do to indicate that your students aren't perceiving that the 'x' here is being divided by -negative one, and so is simply the opposite of itself? If so, I'm sure I would have had inflexible knowledge like that at some stage of the game (though not this go-round; back when I was a child). I should try that on Christopher & see what he does. -- CatherineJohnson - 07 Dec 2005
The general rule is very simple and teachers should use it all the time to reinforce how basic rules can be applied to all sorts of situations, from simple to complex. I agree with this, absolutely. It's a little bit of a shortcut around the inflexible knowledge stage....or not around it, but it helps the inflexible stage be not quite so fragmented. -- CatherineJohnson - 07 Dec 2005
Please note Catherine's comments about the amount of material that is being covered in her son's course; I expect the situation is similar in the course for which this problem was written. Yes, and this factor—the time factor—must always be considered. Over and over again, I see educators treating our children's childhoods as if they basically just go on forever. What's the hurry? They can learn long division any time! I speculate that this happens because any adult dealing with just one age group, instead of with the entire sequence of instruction and ages K-12, is dealing with 10 year olds who will be 10 forever.....they're not thinking about the fact that any given 10-year old is going to be 10 for one year only, and then he's going to be 11 for one year only. The Chinese teachers say all teachers should have taught all levels (within elementary school, in their case); you should always have the final goal in mind; you should always be thinking how your teachings gets the child to that goal. Our schools aren't doing that. I find it shocking that no one in our district can give the parents of young mathematically gifted children an answer as to whether their children will or will not be ready for the AP track in high school. They don't have an answer. They say they don't have an answer. -- CatherineJohnson - 07 Dec 2005
Engelmann gives a very detailed reason for exactly why they don't have an answer at the end of his book (which I left at home today). -- KDeRosa - 07 Dec 2005
This is where you see, SO clearly, that this class was made for kids who don't need teaching. My neighbor, who has an M.A. in math, told me that the talented kids 'seem to inhale the book.' -- CatherineJohnson - 07 Dec 2005
Christopher can't inhale a math book. He's very bright, and his memory is fantastic. He'll probably survive this course, and by next fall he'll have the stuff down, because I'll re-teach over the summer and we'll practice like hell. But think about it. The material in this class isn't remotely too hard for him. I'm almost 100% positive the pace isn't too fast, either (though if he were in a 'real' gifted course, with only mathematically gifted kids, he'd be outpaced immediately.) But without a very well put-together course of direct instruction and formative assessment he's failing. -- CatherineJohnson - 07 Dec 2005
Engelmann gives a very detailed reason for exactly why they don't have an answer at the end of his book (which I left at home today). I've got to get busy posting all of his stuff. That book has changed my life. In fact, I've pretty much decided to try to form a committee here in Irvington to press for teaching-to-mastery and formative-assessment. (I'm also going to get the things you've left in Comments pulled up front.) -- CatherineJohnson - 07 Dec 2005
My neighbor, who has an M.A. in math, told me that the talented kids 'seem to inhale the book.' Yes, but are they really getting an appropriate education from it. I think not. Just because they are learning from what is being presented doesn't mean they necessarily learned what they really need or that they couldn't have been accelerated even faster with a better instructional program. Dr. Stat recently ran the numbers and shows that our best and brightest aren't necessarily all that bright. -- KDeRosa - 07 Dec 2005
"How do you frame the general rule? Would you frame it as: when a/b = c/d, then ad = bc " No. I would say that you can multiply every term on both sides of an equation by the same thing. So, for a/b = c/d, you can multiply every term on both sides by 'b' (first) to get a = bc/d Then, you can multiply every term on both sides by 'd' to get ad = bc Actually, at first you would have ab/b = cb/d Then a * b/b = cb/d since b/b is equal to 1 a * 1 = cb/d and a * 1 = a a = cb/d Do the same thing with the 'd' in the other denominator. I remember my algebra teacher being very strict about writing down every tiny little step like I have above and we had to cite the fundamental rule we used. Throughout my math classes in high school the teachers were very strict about not doing too many changes in one step. Equation manipulation was done vertically, with one change per line. For solving linear equations, I was told to collect the 'X' terms together on the left side of the "=" sign and put the constants on the right. Combine the 'X' terms and divide through by the coefficient of the 'X' term. 3X + 5 = 20 - 2X Add 2X to both sides 3X + 2X + 5 = 20 - 2X + 2X Add terms 5X + 5 = 20 Subtract 5 from both sides 5X + 5 - 5 = 20 - 5 Add constant terms 5X = 15 Divide all terms on both sides by 5 5/5*X = 15/5 divide constants to get X = 3 After a while, we were allowed to do multiple steps on one line. Things like cross-multiplication and vague ideas of canceling out really cause problems. Many kids love (!) canceling, but tend to apply it to wrong cases because they don't understand the fundamental rule. "also, do you bring up the example you do to indicate that your students aren't perceiving that the 'x' here is being divided by -negative one, and so is simply the opposite of itself?" No. What they would do is to apply cross-multiplication to the 3/4 = X/4 part and ignore the (-5) term. 3/4 = X/4 - 5 They would get something like 3*4 = 4*x - 5 after cross-multiplication. In my college algebra classes, I would usually tell the students that there is no such thing as cross-multiplication. I would also ask them to explain what canceling means and why and when one is allowed to do it. -- SteveH - 07 Dec 2005
No. I would say that you can multiply every term on both sides of an equation by the same thing. oh, ok! yes, that takes us out of cross-multiplication altogether that's actually what I was shooting for with Christopher (I think!).... He was completely stuck on cross-multiplication. -- CatherineJohnson - 07 Dec 2005
thanks -- CatherineJohnson - 07 Dec 2005
I think what Steve's algebra teacher was doing was a form of "faultless instruction" like Engelmann uses to maek sure that students are learning the one and only correct rule to avoid confusion so that taching could be accelerated. Funny how whenever someone talks about something that worked for them when they were learning it invariably turns out that it is part of DI and is not present in a constructivist curricula. -- KDeRosa - 07 Dec 2005
I remember my algebra teacher being very strict about writing down every tiny little step like I have above and we had to cite the fundamental rule we used. I REALLY believe in this, and I'm having a terrible time getting Christopher to do it. Part of it is, obviously, that it's still quite effortful for him to write period. This is a case where I could definitely use help from the teacher. If she told him to write down each step, he'd do it. -- CatherineJohnson - 07 Dec 2005
yeah, he's getting rules, rules, rules They learned canceling this week—they learn this stuff _in a day_—so last night I tried to show him that the reason you can do it is that the terms you are canceling equal 1, and anything multiplied by 1 is itself. He did see that right away, but he needs lots more practice seeing it. It would be very good for him to do what you did in high school. Write down each step along with the reason why the step works. RUSSIAN MATH does some of that. (Not a huge amount, but there are always a couple of exercises where you do it.) -- CatherineJohnson - 07 Dec 2005
I think what Steve's algebra teacher was doing was a form of "faultless instruction" like Engelmann uses to maek sure that students are learning the one and only correct rule to avoid confusion so that taching could be accelerated. oh, that's interesting Do you have a source I can read on this? I don't think he talks about faultless instruction in War Against the Schools' Academic Child Abuse. -- CatherineJohnson - 07 Dec 2005
I have to read every single word the guy has ever written. -- CatherineJohnson - 07 Dec 2005
Funny how whenever someone talks about something that worked for them when they were learning it invariably turns out that it is part of DI and is not present in a constructivist curricula. I have to say, it's true...... Though I've had some memorable moments discovering things on my own, as I've been re-teaching myself math. That's a quite different situation, however. Since I'm teaching myself, frequently I've had to discover things, because the book didn't tell me what I wanted to know, and there was no one else around who could tell me, either. -- CatherineJohnson - 07 Dec 2005
Re: Faultless instruction. They talk about and give examples of it here. Read all the way through. But beware, there are quizzes. -- KDeRosa - 07 Dec 2005
"Dr. Stat recently ran the numbers and shows that our best and brightest aren't necessarily all that bright." I assume that by "our" you are referring to students in the USA. At the 2005 International Mathematics Olympiad the USA team came in second overall in points and tied for second overall in medals. That's pretty bright. China was first in points with 235, then USA with 213, and Russia at 212. Some other scores: Hong Kong 138, Belgium 74, Netherlands 62, Korea 200, Singapore 145. -- KtmGuest - 07 Dec 2005
The analysis done by "Dr. Stat" compares the percentage of US students at or above the 90th and 95th percentile of the international average. If I understand the Math Olympiad site correctly, each team had six members. The numbers are not especially comparable. Detailed comparison between these is left as an exercise for the student. -- DougSundseth - 08 Dec 2005
Statistically speaking, the math olympiads are errors. We're not going to learn much by analyzing these errors and drawing educational conclusions from them regarding more normal students. -- KDeRosa - 08 Dec 2005
I totally agree that learning "cross multiplication" is a very bad thing. I agree that it is a specialized case of a more general rule. I ran into this exact problem when I was tutoring. The student was in prealgebra, but was failing miserably. It turns out, she didn't have any basic skills past multiplication and division, so we started with fractions. In reteaching her to add fractions, she remembered the cross multiplication method: in the special case where both denominators are primes, the common demoninator is the product of the single denominators. You can then multiply the denominator of the second fraction with the numerator of the first fraction to get the numerator of the first fraction over the common denominator. (For example, in the case of 2/9 + 3/8, the numerator of the first fraction is 8*2.) But she didn't understand that she was actually generating an equivalent fraction. As a matter of fact, she could not generate an equivalent fraction by multiplying. She had absolutely no problem reducing a fraction and getting the equivalent fraction. But she did not know that you could go the other way. When I finally figured out that this was the gap in her knowledge, I was able to demonstrate how to get an equivalent fraction by multiplying a fraction by, for example, 3/3, then having her reduce it to the original fraction. The light bulb went off, and she was able to learn addition and subtraction of fractions. The difficult part of all this is figuring out where the gap is and what the gap is. As the student gets higher up in math, the more gaps there are and the more they rely on "Whenever I see this, I do that". Then they have to unlearn this and learn the correct way. -- AnneDwyer - 08 Dec 2005
but are they really getting an appropriate education from it. I think not I just saw this part of your comment. I find that a very interesting question. I'd love to know more about the kids who do inhale their education.....what are they getting, exactly? I agree it's probably not what they should, or perhaps could, be getting. -- CatherineJohnson - 08 Dec 2005
ok.....I see that I did not manage to post my Much Ballyhooed stat on our Very Best Students...... -- CatherineJohnson - 08 Dec 2005
I did, however, get that much closer to finishing Temple's & my new book proposal. -- CatherineJohnson - 08 Dec 2005
For solving linear equations, I was told to collect the 'X' terms together on the left side of the "=" sign and put the constants on the right. Combine the 'X' terms and divide through by the coefficient of the 'X' term. Hey! This is what I was saying the other day! Then everyone told me I was wrong! -- CatherineJohnson - 08 Dec 2005
What they would do is to apply cross-multiplication to the 3/4 = X/4 part and ignore the (-5) term. oh gosh yes, that's pretty bad now that you describe it, though, I can easily see it happening -- CatherineJohnson - 08 Dec 2005
Ken But beware, there are quizzes. Let me tell you......I had some Moments of Hesitation after reading one of the articles you linked to, which had quizzes. I was missing the answers, and if this guy is such a Genius Direct Instruction Champion, I shouldn't have been missing those answers. When I wrote programmed instruction for pharmaceutical reps, nobody missed answers ever. -- CatherineJohnson - 08 Dec 2005
I've just had half a bottle of life-extending red wine with my friend Kriss. Can anyone tell? -- CatherineJohnson - 08 Dec 2005
This may be the same article and it's not by the DI people, so their instruction may not be as clear. -- KDeRosa - 08 Dec 2005
Hi, ktm guest! I assume that by "our" you are referring to students in the USA. At the 2005 International Mathematics Olympiad the USA team came in second overall in points and tied for second overall in medals. That's pretty bright. Were these U.S. kids immigrants? In Count Down the U.S. winners are all immigrants (IIRC). -- CatherineJohnson - 08 Dec 2005
Hi, Anne! I agree that it is a specialized case of a more general rule. Those are the words I need. Thank you! As a re-learner of math, I frequently need to know if some procedure is simply a specialized case of a more general rule. It would be tremendously helpful. RUSSIAN MATH seems always to demonstrate this. -- CatherineJohnson - 08 Dec 2005
The student was in prealgebra, but was failing miserably. It turns out, she didn't have any basic skills past multiplication and division, so we started with fractions. Here's a great story. Christopher's math class was given a fraction pre-test, which is great. Christopher was sick that day, so Ms. Kahl sent the test home after I asked for it (an example of her efforts to work with parents—her view was that he didn't need to take the test, but she sent it home at once when I asked for it). I gave Christopher the test, and he did much better than he'd done on the simple online test I gave him. So I was feeling a bit cocky and chipper until I sat down and analyzed the test. It was 4 pages long, but the whole thing was 'conceptual.' (I'll have to post questions.) There wasn't a single question about the division of fractions; there weren't any addition or subtraction with borrowing; etc. He looked good on the test, but in fact his fraction knowledge is very shaky. And he can't do addition and subtraction with borrowing at all. -- CatherineJohnson - 08 Dec 2005
in the special case where both denominators are primes, the common demoninator is the product of the single denominators One Word.
-- CatherineJohnson - 08 Dec 2005
But she didn't understand that she was actually generating an equivalent fraction. right, and this is why fuzzy math came into being in the first place.... My friend Kris, by the way, is the one who likes TRAILBLAZERS. I may have to get one of the early books, because her son, who is now in 3rd grade, loves math—and is certainly doing math when he's around me..... -- CatherineJohnson - 08 Dec 2005
He's doing all kinds of things I couldn't do at his age. -- CatherineJohnson - 08 Dec 2005
Anne When I finally figured out that this was the gap in her knowledge, I was able to demonstrate how to get an equivalent fraction by multiplying a fraction by, for example, 3/3, then having her reduce it to the original fraction. The light bulb went off, and she was able to learn addition and subtraction of fractions. The difficult part of all this is figuring out where the gap is and what the gap is. As the student gets higher up in math, the more gaps there are and the more they rely on "Whenever I see this, I do that". Then they have to unlearn this and learn the correct way. I've thought about this from time to time..... Are there people who can 'diagnose' gaps really quickly and accurately? I bet there are. I do remember, quite awhile back, reading a study of tutors saying they had no idea where a student's particular gaps were. I'm sure that's true as general rule. -- CatherineJohnson - 08 Dec 2005
My sister says her husband, an architectural engineer who is probably a savant math type, can see his daughter's gaps & unique understandings instantly. He can shift immediately to a second and even third way of showing something as soon as she verbalizes what she thinks she knows. -- CatherineJohnson - 08 Dec 2005
But he has a very unique brain. -- CatherineJohnson - 08 Dec 2005
There wasn't a single question about the division of fractions; there weren't any addition or subtraction with borrowing; etc. He looked good on the test, but in fact his fraction knowledge is very shaky. And he can't do addition and subtraction with borrowing at all. How can this possibly be? Isn't it standard constructivist doctrine that kids with conceptual knowledge can construct these mundane procedural skills for themselves and should thereby be able to solve such problems more adroitly than a traditionally taught student? Say it ain't so? -- KDeRosa - 08 Dec 2005
How can this possibly be? Isn't it standard constructivist doctrine that kids with conceptual knowledge can construct these mundane procedural skills for themselves and should thereby be able to solve such problems more adroitly than a traditionally taught student? Say it ain't so? shooting fish in a barrel again, Ken? Actually, this has become an interesting question to me.....because I'm seeing Christopher grasp concepts pretty well (I think), and then not have a clue how to do the algorithm that goes with the concept..... That took me by surprise, I must say. -- CatherineJohnson - 08 Dec 2005
I'll have to get those problems posted, some of them I wouldn't say they were 'trivial,' though you guys might.... still and all, for him to do so well on this four-page test, with such shaky fraction knowledge, was pretty shocking -- CatherineJohnson - 08 Dec 2005
"The analysis done by "Dr. Stat" compares the percentage of US students at or above the 90th and 95th percentile of the international average." He also looked at the 99.9th percentile. Dr. Stat was attempting to answer questions about "top students." He explicity mentions "winners of competitions in Math and Science." The IMO data is relevant to his discussion. "In Count Down the U.S. winners are all immigrants (IIRC)." Three out of six of the US delegates from 2001(the year of the IMO depicted in Count Down) were immigrants. I don't know where each of the six students from 2005 was born. I would be surprised if they were all immigrants. -- KtmGuest - 08 Dec 2005
The IMO data is relevant to his discussion But not to the point I was making. The mo's represent about the 0.00002 percentile. Dr. Stat only went down to the 0.1 percentile. Its nice that his results generalized to the 5th and 10th percentiles but that fact is irrelevant to our discussion, as is the performance of the mos. We are not going to make curricular generalizations based on either of these group's performance. These facts are mere trivia. -- KDeRosa - 08 Dec 2005
Three out of six of the US delegates from 2001(the year of the IMO depicted in Count Down) were immigrants. I don't know where each of the six students from 2005 was born. I would be surprised if they were all immigrants. What we need to know is whether the other 3 were the children of immigrants. The data I've seen, which may originate with Steinberg, shows that the children and grandchildren of immigrants do successively worse with each new generation. -- CatherineJohnson - 08 Dec 2005
This is funny: (from Count Down):
High-level problem solving has many proponents in the United States. Since inaugurating a major reform effort in 1989, the National Council of Teachers of Mathematics (NCTM) has been urging U.S. math teachers to emphasize the kinds of problems that require students to think deeply about what they are doing. But the cultural inertia of math instruction is very powerful. The eighty-one classes Stigler and his colleagues videotaped in 1994 exhibited virtually no signs of the reforms advocated by NCTM. -- CatherineJohnson - 08 Dec 2005
yeah......if schools would just get off the stick and IMPLEMENT NCTM REFORMS we'd be cleaning up at Math Olympiads -- CatherineJohnson - 08 Dec 2005
Sorry, had to do this:
Lol! I thought he looked depressed. -- SusanS - 08 Dec 2005
oh my gosh!!!! where's you get that?????? That is HILARIOUS! -- CatherineJohnson - 08 Dec 2005
Saxon versus IMP
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Looking at the Saxon Maths example, I understand why so many of my friends didn't get why they minus the +5 first - they were just told to undo the operation. As I had had BODMAS drilled into me from the start it just seemed logical to get rid of the +5 etc first (because you are undoing an operation), but they probably had never had the BODMAS drills, and without being explictly told, they would do it from left to right and get the wrong answer. -- SamanthaRawson - 06 Dec 2005
wow What does 'BODMAS' stand for? Yes, I was intrigued by this, too. I don't remember being told to do the addition first. I think I just kind of picked it up. I'm thinking I was one of those kids who could do huge amounts of incidental learning. I'm wondering if I had reasonably good 'pattern discrimination' or some such.....so that I'd notice that certain things recurred, like getting rid of the +5 first, and I'd just start doing them. I've mentioned that I was taught almost no punctuation or grammar (that I know for sure). Over the years I've noticed how people do it. -- CatherineJohnson - 06 Dec 2005
This is one of the things I keep seeing with Christopher. He doesn't just 'notice' stuff. You need to point things out to him, and teach it. He remembers things very quickly; he has the kind of memory I had as a child. (He remembers everything, but remembers them inflexibly, which was & is true of me, too.) But he needs direct instruction, broadly defined. -- CatherineJohnson - 06 Dec 2005
to get rid of the +5 etc first (because you are undoing an operation wait—I don't follow I think you've just said 'undo an operation' twice, haven't you? -- CatherineJohnson - 06 Dec 2005
My skepticism meter is beeping. The item on the right (with the standard deviation) seems like a traditional assignment with less structure followed up by a tricky word problem.
Repeat Question 1b, using a different nonzero number. Add this number to each member of your original set of data, and find the mean and standard deviation of the new set. Keep repeating this process until you see a pattern, and then describe that pattern.Translation: calculate the mean and standard deviation a bazillion times, but you get to pick the numbers. Before I get too catty, I will point out that I do sometimes give my students unstructured problems like this; however, in my class these are the challenge problems on the BACK of the page (to be done AFTER one can do the regular problems if one is bored and is looking for more math to do). If I were teaching this standard deviation activity to college freshmen, I would need to meddle as they were working on it to make sure they were not going off on a wild goose chase. Probably we would work in our groups for a while on tasks 1a and 1b (although I'd probably provide a data set for them to get started on so that I could quickly check to see if they were doing things right). After students had found some numerical answers* we'd come back as a large group to report our findings, summarize what we've found, and make some initial conjectures. I'd send them back into their groups to refine their conjectures. Once that stopped being productive, I'd have the class come back into rows, and I'd collect the ideas of the class; only good ideas would get written on the board. (Bad ideas can sometimes be morphed by, "I think you're trying to say that...") Then I'd use the lecture mode to synthesize their findings. But if I just assigned that activity without managing it? Chaos. *If they couldn't get started on the calculation, we'd be back in rows to reteach that. -- RudbeckiaHirta - 07 Dec 2005
BODMAS stands for Brackets Of Division Multiplication Addition Subtraction, and it's a mnemetic trick to help people remember which part of a sum to do first, brackets then multiplication/division, then addition/subtraction. I assumed, since we were undoing the problem, you would automatically do the addition/subtraction first, then any multiplication/division. I didn't even have to think about it, I just did it. Most other people in my class struggled. -- SamanthaRawson - 07 Dec 2005
I would need to meddle as they were working on it to make sure they were not going off on a wild goose chase I love it! I'll bet I've lost the reference, but I read a terrific piece of research on college math courses. They looked at the Super Math Brain students, and compared them to the Not a Math Brain students. The big difference between the two groups was that the Not a Math Brain students 'went off on wild goose chases.' That was the exact expression they used. I've had some hilarious moments ever since reading that. I'll be working some math problem I don't understand, sitting there beavering away (mixed metaphor alert!), going, going, going, and I've left the vicinity of the correct answer many eons ago. Wild goose chase, for sure. -- CatherineJohnson - 07 Dec 2005
wait! Is BODMAS the same as Please Excuse My Dear Aunt Sally??? Not quite, I guess. This is for solving equations; that's for simplifying expresions, right? -- CatherineJohnson - 07 Dec 2005
"Is BODMAS the same as Please Excuse My Dear Aunt Sally???" The mnemonic device I prefer for the order of operations is the nonsensical acronym PEMDAS. -- CharlesH - 07 Dec 2005
The Interactive Mathematics activity seems like a reasonable one, as long as the students have the mathematical background for it, and, as long as what's on the paper we see there is not all there is. There has to be some kind of follow-up, IMO, with the mathematical reason for what's going on explained to the students. I think this might be part of why "reform" math has made itself such a bad name. If the teachers aren't explaining what's going on, OF COURSE the students will be lost. Students get lost even when we DO explain what's going on. Maybe the "best" way to teach math combines principles of direct instruction with some of these "reform" type activities. I can see this being a good activity for a high school class, or a non-calculus-based statistics class at the college level. (In a calc-stats class, I don't think I'd devote time to it in class, but I might assign as a homework problem to investigate what happens in a suitably general case, and provide a proof.) -- PaulMiller - 07 Dec 2005
But you can divide by 4 first, you just have to divide both sides of the equation by 4, so you get (4x+5)/4 = 17/4 which simplifies to x + 5/4 = 17/4. Then you subtract 5/4 from both sides to get x = 17/4 - 5/4, or x = 12/4, or x = 3. I actually don't like them writing "we always use the addition rule first" because you can do anything you want to that equation as long as you do the same thing to both sides. You use the addition first because it's much easier that way - no fractions or division to do in the first step, and only one division to do in the second step. There are some things we "always do" because they are written into the syntax of math and give it meaning - parentheses first, then exponents, etc. But that is not the reason we always undo the addition first. As long as a child understands what an equation is, and how it remains unchanged when you do the same thing to both sides, they don't need to memorize "undo the addition first." -- StephanieO - 07 Dec 2005
First: "Making Friends with Standard Deviations" is a really sick-making title. Shouldn't the students have left this sort of thing behind in kindergarten? Second: I think this would make more sense if the students understand the formal definitions of "mean" and "standard deviation" first. 1) This is the definition of mean; this is the definition of standard deviation. 2) If you were to add a constant to each member of the set, what would you expect? 3) Here is a set; find the mean and standard deviation. 4) Here is a second set with each member equal to the first set plus a constant; find the mean and standard deviation. 5) Did the result match your expectations? 6) This is what you should have seen, and why you should have seen it. 7) If you want confirmation, try it again. It's hard to see patterns in the noise. It's far easier if you know what to look for. -- DougSundseth - 07 Dec 2005
When I wrote that the activity was reasonable, I said so assuming the students have the "mathematical background" for it. Knowing the definitions of mean and standard deviation pretty much constitute the "mathematical background" for this activity. I think the questions guiding this activity are quite adequate. The pattern that'll show up is quite simple. All it takes is understanding of the definitions to predict what will happen (and, to prove it, too.) -- PaulMiller - 07 Dec 2005
If the students already understand mean and standard deviation, the activity is trivial. The only time it could possibly be of use is as a method to teach. If you are teaching, make the teaching explicit. "Keep repeating this process until you see a pattern." (Note that you've found a minimum of three means and three standard deviations at the point that this instruction is given.) If you intend to treat math instruction as a scientific experiment, do it right: Hypothesize, predict, test the prediction (and if necessary refine, rinse, and repeat). This isn't the same as faff about, look for a pattern, speculate about why the pattern might exist. -- DougSundseth - 07 Dec 2005
about Saxon: ditto what StephanieO? said. It is inflexible to direct kids that "we must" do this and "we always" do that. Depending on the whole numbers, one way or the other might be less likely to lead to computational error. Maybe. Having students do it both ways would be great if there was time. about IMP: so how many kids in a group? who runs the calculator? who writes down the numbers? who checks the numbers? who sits around idly while numbers are slowly and steadily punched into the calculator? This is the thing that drives me crazy about constructivist math. Apparently, kids just won't believe the result unless they generate the several sets of five numbers themselves. Don't ride on wheels you didn't invent. Don't trust anyone over thirty. Own the numbers! We shall overcome! Get excited about generating several sets of five numbers with your peers, and talk about it? These projects don't eliminate everything that was ever felt to be wrong with lectures given by teachers at the front of the room -- they just move it all into the context of small peer groups. -- BeckyC - 07 Dec 2005
There is a huge, HUGE difference between "know the definitions of mean and standard deviation" and "UNDERSTAND mean and standard deviation." Just ask anyone who teaches freshman calculus if their students understand the chain rule or L'hopital's rule. I certainly didn't when I was an undergrad. I still submit that the mathematical background for this activity is only to "know the definitions" of mean and standard deviation. The point is to get an increased understanding of the concepts. Otherwise, as Doug has so rightly pointed out, it is pointless to even do the activity. I mean, it's not like I have to do this activity to tell you that when you add a nonzero number to every member of a data set, the mean changes by exactly that number, and the standard deviation changes not at all, even if you're looking at a continuous probability distribution rather than a discrete data set. But, I took a year of mathematical statistics as an undergrad, and understand mean and standard deviation well enough to prove this statement mathematically, so I'm obviously not the target audience. There's an old saying: "Research is what you're doing when you don't know what you're doing." This is certainly true in mathematics. Sometimes we just don't know enough about the things we're dealing with to make appropriate hypotheses, and we have to cast about in the dark a bit to find our way. This is the price we pay as mathematicians for being able to publish our work as "theorems" rather than "theories." In the appropriate level course, if I can inject a little bit of discovery for those who can take it, I'm glad to do so. Like I said before, you won't catch me doing this activity in a calc-stats course, but, in a non-calc-based stats course, this is a reasonable and decent activity. The point is to get students to the point where they do understand mean and standard deviation. Why, if we all agree that students have to learn to mastery the algorithms for basic operations on numbers in order to develop number sense, do we object to an activity that tries to do the same thing for concepts like "mean" and "standard deviation"? How can a student form hypotheses about these concepts if he doesn't understand them? Now, coming back to the idea of "mathematical followup" I said would be necessary with this activity... after the students play around for 10 minutes or so, then it's definitely time to explain what is going on. By that I mean to provide a mathematical proof of the statement I made above, or, at very least, a mathematical argument that demonstrates why what's going on should be happening. Like I said, in the calc-stats course, I'd expect the students to generate a mathematical proof themselves. By the way, I certainly hope whatever book this activity comes from is teaching them the difference between population standard deviation and sample standard deviation. If not, then they will never get an understanding of sampling. -- PaulMiller - 07 Dec 2005
coming back to the idea of "mathematical followup" I said would be necessary with this activity Another name for "mathemtical followup" is "mathematical closure", and it may be lacking, especially in elementary schools. Teachers can fall into the trap of believing that 99% of their work is done when the "experiment" has been set up and the students have worked in small groups. Due to the primacy of both Discovery and Peer Learning. -- BeckyC - 07 Dec 2005
But you can divide by 4 first, you just have to divide both sides of the equation by 4, so you get (4x+5)/4 = 17/4 which simplifies to x + 5/4 = 17/4. Then you subtract 5/4 from both sides to get x = 17/4 - 5/4, or x = 12/4, or x = 3. I haven't gotten far enough in my re-education to think analytically about the IMP problem. With the Saxon passage, I've had the exact experience Becky is talking about. This will sound awfully low-level (and I've told the story before), but one day, thinking about teaching math concepts as well as procedures, I said to my neighbor, 'If you're subtracting 19 from 30, why can't you subtract the 9 from the 0?' It actually took her a moment to realize that, obviously, you can. (I would have figured it out myself! But she's a statistician. She got there first!) That was a fabulous moment for me; I really saw, in a way I hadn't seen before, what a marvelous invention the 'simple' addition algorithm is. (I still don't see—not really—the beauty and genius of the base-10 system. I 'know' it, I use it, I can explain it.....but my understanding is quite superficial still.) I've mentioned before that deSaussure said that meaning comes from difference. I believe that. For me, the Saxon lesson would be better if you pointed out that, yes, you could divide first, but that would be inefficient and inelegant. -- CatherineJohnson - 07 Dec 2005
There are some things we "always do" because they are written into the syntax of math and give it meaning - parentheses first, then exponents, etc. But that is not the reason we always undo the addition first. I think this is VERY important. As a person re-learning math, I've been constantly frustrated by the need to know whether a 'rule' is a rule because it's more efficient and elegant to do things a certain way, or because it is written into the syntax of math (I love that way of putting it). Certainly for me, this is much more important than any text seems to grasp. In my case, it's important because I have sometimes spent hours and hours trying to understand conceptually why an 'elegance' rule is a syntax rule. Talk about a wild goose chase. (I wish I could remember examples. I'll start keeping notes when this happens in the future.) Actually, the addition example probably falls into that category. I was thinking the addition algorithm, in which you 'must' borrow in order to subtract the '9' from the '0,' was somehow 'syntactical.' Obviously, that betrays a very superficial grasp of the algorithm, BUT—and you'll just have to take my word on this—it doesn't betray a complete lack of understanding of the algorithm. I think I have it! This just came to me. We all know that students (all 'learners,' learning any subject) go through a stage of inflexible knowledge. I believe that when you are in the stage of inflexible knowledge you can especially benefit from being told that, yes, you could divide first, but mathematicians don't divide first because it's inefficient. I suspect that the inflexible stage is also the point at which it's helpful to show learners that there's 'more than one way to do it.' By that I don't mean a zillion and one ludicrously complicated Russian peasant tally mark multiplication schemes. I mean, almost certainly, TWO WORKABLE WAYS TO DO AND/OR EXPRESS THE SAME THING if possible. This is another Lost In Translation moment. Constructivists have accurately noticed that there are many times learners (I loathe that word) do benefit from being shown that it's possible to solve this problem another way. But they've turned this insight into a fetish—a fetish and, even worse, a lesson in multiculturalism. -- CatherineJohnson - 07 Dec 2005
Here's another inflexible knowledge moment. Christopher's teacher is doing almost entirely procedural teaching. She has to; she's racing through material at a clip of practically one new topic a day. (I'll have to post the TOC one of these days.) So the other day she taught them the cross-multiplication procedure for figuring out what a missing numerator or denominator is: 2/4 = 4/ I was confused when I started working on this with him; I guess I didn't quite realize what she'd taught in class. So I was setting this up as an equation: 2 x = 4 x 4. Naturally, Christopher was outraged at this intrusion into his COMPLETE-UNTO-ITSELF CROSS-MULTIPLICATION PROCEDURE, so we had eye-rollling and major sighing until at some point I said, 'Look at this. This equation I've written _is exactly what you're doing_'—and he saw it! His eyes widened, and he said, "Oh!" in a surprised, happy voice. Since we have the goal of teaching math conceptually as well as procedurally, I think these moments are important. But you need a superb curriculum, which we don't have. Ms. Kahl has to race through huge numbers of topics, and there's ZERO time for getting the point. Plus she's brand-new to teaching, and the department has not (to my knowledge) given any serious thought to formative assessment—or to the question Paul raised of how you incorporate formative assessment into a course that has to tear through as much material as this course does. Instead, the department philosophy is: these are the gifted kids, they need to be challenged. That's it. I have that in writing from the Assistant Superintendent for Curriculum. -- CatherineJohnson - 07 Dec 2005
If this were a DI school, Ms. Kahl would have had a nice speaking-to 2 months ago. The situation would have never been permitted to get to this stage where the first semester is practically over, and most of teh class has not learned anything. -- KDeRosa - 07 Dec 2005
Now, coming back to the idea of "mathematical followup" I said would be necessary with this activity... after the students play around for 10 minutes or so, then it's definitely time to explain what is going on. Yes, absolutely. It's not clear to me that anyone has the 'final word' on inductive versus deductive learning. James Milgram's criticism seems to be almost entirely that the main problem with discovery courses is that it takes a phenomenally talented and knowledgeable teacher to teach a good discovery course. Realistically, we're never going to have a vast elementary teacher corps able to pul this off. (Though now that I've read Engelmann I can certainly imagine someone writing a scripted 'discovery' curriculum that would be both excellent and teachable by all teachers.) Wickelgren's complaint was that discovery courses simply take too much time to cover the material they need to cover. Inductive learning is natural to human beings; it doesn't seem right to me that an excellent curriculum would do away with it. But Paul is absolutely right. You can't just send students out to discover something and leave it at that. -- CatherineJohnson - 07 Dec 2005
I ranted at my students once this semester for using this made up operation "cross multiplication." This is just yet another procedure made up by k-12 teachers that students can apply while mathematically flailing about looking for a solution... nothing more. -- PaulMiller - 07 Dec 2005
"So the other day she taught them the cross-multiplication procedure for figuring out what a missing numerator or denominator is:" 2/4 = 4/ I think that the idea of cross-multiplication is more damaging than helpful. What if you have something like: 3/4 = X/4 - 5 I have had students try to use cross-multiplication on this. Cross-multiplication is a special case of a more general rule. The general rule is very simple and teachers should use it all the time to reinforce how basic rules can be applied to all sorts of situations, from simple to complex. -- SteveH - 07 Dec 2005
Here's a good article on why constructivism is so dopey. -- KDeRosa - 07 Dec 2005
"There's an old saying: "Research is what you're doing when you don't know what you're doing." This is certainly true in mathematics. Sometimes we just don't know enough about the things we're dealing with to make appropriate hypotheses, and we have to cast about in the dark a bit to find our way. This is the price we pay as mathematicians for being able to publish our work as 'theorems' rather than 'theories.'" No argument. But this is what you do when you have to. When there is nothing else you can think of. It's extraordinarily inefficient, even when it's more efficient than any other course. Please note Catherine's comments about the amount of material that is being covered in her son's course; I expect the situation is similar in the course for which this problem was written. Why would you use blind-squirrel exploration when you don't have to? "The point is to get students to the point where they do understand mean and standard deviation. Why, if we all agree that students have to learn to mastery the algorithms for basic operations on numbers in order to develop number sense, do we object to an activity that tries to do the same thing for concepts like 'mean' and 'standard deviation'?" It's not the goal I have a problem with; it's the inefficiency of the process they have chosen for reaching that goal. "But they mean well", is insufficient for me to give them a pass. "How can a student form hypotheses about these concepts if he doesn't understand them?" By having the students form even incorrect hypotheses, you get them more deeply involved in the process. If they reasoned correctly, their hypothesis (and understanding of the underlying material) is confirmed; this is the kind of thing that makes students feel really good. If they reasoned incorrectly, there is more incentive to identify the flaws in the reasoning. To reiterate: I don't think the problem is without merit; I think it's clumsy. And the name is execrable. -- DougSundseth - 07 Dec 2005
"How can a student form hypotheses about these concepts if he doesn't understand them?" This encapsulates the entire flaw in the constructivist theory. When the original mathematicians discovered these principles, they didn't do it the same way the kids are being asked to do it. they were experts with extensive domain knowledge, not neopytes with no knowledge. Plus, as Doug points out, it's brutally inefficient to re-discover all these things anyway for what amounts to a dubious benefit in any event. -- KDeRosa - 07 Dec 2005
Steve Sorry to be so far behind....but I don't quite follow. Cross-multiplication is a special case of a more general rule. The general rule is very simple and teachers should use it all the time to reinforce how basic rules can be applied to all sorts of situations, from simple to complex. How do you frame the general rule? Would you frame it as: when a/b = c/d, then ad = bc also, do you bring up the example you do to indicate that your students aren't perceiving that the 'x' here is being divided by -negative one, and so is simply the opposite of itself? If so, I'm sure I would have had inflexible knowledge like that at some stage of the game (though not this go-round; back when I was a child). I should try that on Christopher & see what he does. -- CatherineJohnson - 07 Dec 2005
The general rule is very simple and teachers should use it all the time to reinforce how basic rules can be applied to all sorts of situations, from simple to complex. I agree with this, absolutely. It's a little bit of a shortcut around the inflexible knowledge stage....or not around it, but it helps the inflexible stage be not quite so fragmented. -- CatherineJohnson - 07 Dec 2005
Please note Catherine's comments about the amount of material that is being covered in her son's course; I expect the situation is similar in the course for which this problem was written. Yes, and this factor—the time factor—must always be considered. Over and over again, I see educators treating our children's childhoods as if they basically just go on forever. What's the hurry? They can learn long division any time! I speculate that this happens because any adult dealing with just one age group, instead of with the entire sequence of instruction and ages K-12, is dealing with 10 year olds who will be 10 forever.....they're not thinking about the fact that any given 10-year old is going to be 10 for one year only, and then he's going to be 11 for one year only. The Chinese teachers say all teachers should have taught all levels (within elementary school, in their case); you should always have the final goal in mind; you should always be thinking how your teachings gets the child to that goal. Our schools aren't doing that. I find it shocking that no one in our district can give the parents of young mathematically gifted children an answer as to whether their children will or will not be ready for the AP track in high school. They don't have an answer. They say they don't have an answer. -- CatherineJohnson - 07 Dec 2005
Engelmann gives a very detailed reason for exactly why they don't have an answer at the end of his book (which I left at home today). -- KDeRosa - 07 Dec 2005
This is where you see, SO clearly, that this class was made for kids who don't need teaching. My neighbor, who has an M.A. in math, told me that the talented kids 'seem to inhale the book.' -- CatherineJohnson - 07 Dec 2005
Christopher can't inhale a math book. He's very bright, and his memory is fantastic. He'll probably survive this course, and by next fall he'll have the stuff down, because I'll re-teach over the summer and we'll practice like hell. But think about it. The material in this class isn't remotely too hard for him. I'm almost 100% positive the pace isn't too fast, either (though if he were in a 'real' gifted course, with only mathematically gifted kids, he'd be outpaced immediately.) But without a very well put-together course of direct instruction and formative assessment he's failing. -- CatherineJohnson - 07 Dec 2005
Engelmann gives a very detailed reason for exactly why they don't have an answer at the end of his book (which I left at home today). I've got to get busy posting all of his stuff. That book has changed my life. In fact, I've pretty much decided to try to form a committee here in Irvington to press for teaching-to-mastery and formative-assessment. (I'm also going to get the things you've left in Comments pulled up front.) -- CatherineJohnson - 07 Dec 2005
My neighbor, who has an M.A. in math, told me that the talented kids 'seem to inhale the book.' Yes, but are they really getting an appropriate education from it. I think not. Just because they are learning from what is being presented doesn't mean they necessarily learned what they really need or that they couldn't have been accelerated even faster with a better instructional program. Dr. Stat recently ran the numbers and shows that our best and brightest aren't necessarily all that bright. -- KDeRosa - 07 Dec 2005
"How do you frame the general rule? Would you frame it as: when a/b = c/d, then ad = bc " No. I would say that you can multiply every term on both sides of an equation by the same thing. So, for a/b = c/d, you can multiply every term on both sides by 'b' (first) to get a = bc/d Then, you can multiply every term on both sides by 'd' to get ad = bc Actually, at first you would have ab/b = cb/d Then a * b/b = cb/d since b/b is equal to 1 a * 1 = cb/d and a * 1 = a a = cb/d Do the same thing with the 'd' in the other denominator. I remember my algebra teacher being very strict about writing down every tiny little step like I have above and we had to cite the fundamental rule we used. Throughout my math classes in high school the teachers were very strict about not doing too many changes in one step. Equation manipulation was done vertically, with one change per line. For solving linear equations, I was told to collect the 'X' terms together on the left side of the "=" sign and put the constants on the right. Combine the 'X' terms and divide through by the coefficient of the 'X' term. 3X + 5 = 20 - 2X Add 2X to both sides 3X + 2X + 5 = 20 - 2X + 2X Add terms 5X + 5 = 20 Subtract 5 from both sides 5X + 5 - 5 = 20 - 5 Add constant terms 5X = 15 Divide all terms on both sides by 5 5/5*X = 15/5 divide constants to get X = 3 After a while, we were allowed to do multiple steps on one line. Things like cross-multiplication and vague ideas of canceling out really cause problems. Many kids love (!) canceling, but tend to apply it to wrong cases because they don't understand the fundamental rule. "also, do you bring up the example you do to indicate that your students aren't perceiving that the 'x' here is being divided by -negative one, and so is simply the opposite of itself?" No. What they would do is to apply cross-multiplication to the 3/4 = X/4 part and ignore the (-5) term. 3/4 = X/4 - 5 They would get something like 3*4 = 4*x - 5 after cross-multiplication. In my college algebra classes, I would usually tell the students that there is no such thing as cross-multiplication. I would also ask them to explain what canceling means and why and when one is allowed to do it. -- SteveH - 07 Dec 2005
No. I would say that you can multiply every term on both sides of an equation by the same thing. oh, ok! yes, that takes us out of cross-multiplication altogether that's actually what I was shooting for with Christopher (I think!).... He was completely stuck on cross-multiplication. -- CatherineJohnson - 07 Dec 2005
thanks -- CatherineJohnson - 07 Dec 2005
I think what Steve's algebra teacher was doing was a form of "faultless instruction" like Engelmann uses to maek sure that students are learning the one and only correct rule to avoid confusion so that taching could be accelerated. Funny how whenever someone talks about something that worked for them when they were learning it invariably turns out that it is part of DI and is not present in a constructivist curricula. -- KDeRosa - 07 Dec 2005
I remember my algebra teacher being very strict about writing down every tiny little step like I have above and we had to cite the fundamental rule we used. I REALLY believe in this, and I'm having a terrible time getting Christopher to do it. Part of it is, obviously, that it's still quite effortful for him to write period. This is a case where I could definitely use help from the teacher. If she told him to write down each step, he'd do it. -- CatherineJohnson - 07 Dec 2005
yeah, he's getting rules, rules, rules They learned canceling this week—they learn this stuff _in a day_—so last night I tried to show him that the reason you can do it is that the terms you are canceling equal 1, and anything multiplied by 1 is itself. He did see that right away, but he needs lots more practice seeing it. It would be very good for him to do what you did in high school. Write down each step along with the reason why the step works. RUSSIAN MATH does some of that. (Not a huge amount, but there are always a couple of exercises where you do it.) -- CatherineJohnson - 07 Dec 2005
I think what Steve's algebra teacher was doing was a form of "faultless instruction" like Engelmann uses to maek sure that students are learning the one and only correct rule to avoid confusion so that taching could be accelerated. oh, that's interesting Do you have a source I can read on this? I don't think he talks about faultless instruction in War Against the Schools' Academic Child Abuse. -- CatherineJohnson - 07 Dec 2005
I have to read every single word the guy has ever written. -- CatherineJohnson - 07 Dec 2005
Funny how whenever someone talks about something that worked for them when they were learning it invariably turns out that it is part of DI and is not present in a constructivist curricula. I have to say, it's true...... Though I've had some memorable moments discovering things on my own, as I've been re-teaching myself math. That's a quite different situation, however. Since I'm teaching myself, frequently I've had to discover things, because the book didn't tell me what I wanted to know, and there was no one else around who could tell me, either. -- CatherineJohnson - 07 Dec 2005
Re: Faultless instruction. They talk about and give examples of it here. Read all the way through. But beware, there are quizzes. -- KDeRosa - 07 Dec 2005
"Dr. Stat recently ran the numbers and shows that our best and brightest aren't necessarily all that bright." I assume that by "our" you are referring to students in the USA. At the 2005 International Mathematics Olympiad the USA team came in second overall in points and tied for second overall in medals. That's pretty bright. China was first in points with 235, then USA with 213, and Russia at 212. Some other scores: Hong Kong 138, Belgium 74, Netherlands 62, Korea 200, Singapore 145. -- KtmGuest - 07 Dec 2005
The analysis done by "Dr. Stat" compares the percentage of US students at or above the 90th and 95th percentile of the international average. If I understand the Math Olympiad site correctly, each team had six members. The numbers are not especially comparable. Detailed comparison between these is left as an exercise for the student. -- DougSundseth - 08 Dec 2005
Statistically speaking, the math olympiads are errors. We're not going to learn much by analyzing these errors and drawing educational conclusions from them regarding more normal students. -- KDeRosa - 08 Dec 2005
I totally agree that learning "cross multiplication" is a very bad thing. I agree that it is a specialized case of a more general rule. I ran into this exact problem when I was tutoring. The student was in prealgebra, but was failing miserably. It turns out, she didn't have any basic skills past multiplication and division, so we started with fractions. In reteaching her to add fractions, she remembered the cross multiplication method: in the special case where both denominators are primes, the common demoninator is the product of the single denominators. You can then multiply the denominator of the second fraction with the numerator of the first fraction to get the numerator of the first fraction over the common denominator. (For example, in the case of 2/9 + 3/8, the numerator of the first fraction is 8*2.) But she didn't understand that she was actually generating an equivalent fraction. As a matter of fact, she could not generate an equivalent fraction by multiplying. She had absolutely no problem reducing a fraction and getting the equivalent fraction. But she did not know that you could go the other way. When I finally figured out that this was the gap in her knowledge, I was able to demonstrate how to get an equivalent fraction by multiplying a fraction by, for example, 3/3, then having her reduce it to the original fraction. The light bulb went off, and she was able to learn addition and subtraction of fractions. The difficult part of all this is figuring out where the gap is and what the gap is. As the student gets higher up in math, the more gaps there are and the more they rely on "Whenever I see this, I do that". Then they have to unlearn this and learn the correct way. -- AnneDwyer - 08 Dec 2005
but are they really getting an appropriate education from it. I think not I just saw this part of your comment. I find that a very interesting question. I'd love to know more about the kids who do inhale their education.....what are they getting, exactly? I agree it's probably not what they should, or perhaps could, be getting. -- CatherineJohnson - 08 Dec 2005
ok.....I see that I did not manage to post my Much Ballyhooed stat on our Very Best Students...... -- CatherineJohnson - 08 Dec 2005
I did, however, get that much closer to finishing Temple's & my new book proposal. -- CatherineJohnson - 08 Dec 2005
For solving linear equations, I was told to collect the 'X' terms together on the left side of the "=" sign and put the constants on the right. Combine the 'X' terms and divide through by the coefficient of the 'X' term. Hey! This is what I was saying the other day! Then everyone told me I was wrong! -- CatherineJohnson - 08 Dec 2005
What they would do is to apply cross-multiplication to the 3/4 = X/4 part and ignore the (-5) term. oh gosh yes, that's pretty bad now that you describe it, though, I can easily see it happening -- CatherineJohnson - 08 Dec 2005
Ken But beware, there are quizzes. Let me tell you......I had some Moments of Hesitation after reading one of the articles you linked to, which had quizzes. I was missing the answers, and if this guy is such a Genius Direct Instruction Champion, I shouldn't have been missing those answers. When I wrote programmed instruction for pharmaceutical reps, nobody missed answers ever. -- CatherineJohnson - 08 Dec 2005
I've just had half a bottle of life-extending red wine with my friend Kriss. Can anyone tell? -- CatherineJohnson - 08 Dec 2005
This may be the same article and it's not by the DI people, so their instruction may not be as clear. -- KDeRosa - 08 Dec 2005
Hi, ktm guest! I assume that by "our" you are referring to students in the USA. At the 2005 International Mathematics Olympiad the USA team came in second overall in points and tied for second overall in medals. That's pretty bright. Were these U.S. kids immigrants? In Count Down the U.S. winners are all immigrants (IIRC). -- CatherineJohnson - 08 Dec 2005
Hi, Anne! I agree that it is a specialized case of a more general rule. Those are the words I need. Thank you! As a re-learner of math, I frequently need to know if some procedure is simply a specialized case of a more general rule. It would be tremendously helpful. RUSSIAN MATH seems always to demonstrate this. -- CatherineJohnson - 08 Dec 2005
The student was in prealgebra, but was failing miserably. It turns out, she didn't have any basic skills past multiplication and division, so we started with fractions. Here's a great story. Christopher's math class was given a fraction pre-test, which is great. Christopher was sick that day, so Ms. Kahl sent the test home after I asked for it (an example of her efforts to work with parents—her view was that he didn't need to take the test, but she sent it home at once when I asked for it). I gave Christopher the test, and he did much better than he'd done on the simple online test I gave him. So I was feeling a bit cocky and chipper until I sat down and analyzed the test. It was 4 pages long, but the whole thing was 'conceptual.' (I'll have to post questions.) There wasn't a single question about the division of fractions; there weren't any addition or subtraction with borrowing; etc. He looked good on the test, but in fact his fraction knowledge is very shaky. And he can't do addition and subtraction with borrowing at all. -- CatherineJohnson - 08 Dec 2005
in the special case where both denominators are primes, the common demoninator is the product of the single denominators One Word.
-- CatherineJohnson - 08 Dec 2005
But she didn't understand that she was actually generating an equivalent fraction. right, and this is why fuzzy math came into being in the first place.... My friend Kris, by the way, is the one who likes TRAILBLAZERS. I may have to get one of the early books, because her son, who is now in 3rd grade, loves math—and is certainly doing math when he's around me..... -- CatherineJohnson - 08 Dec 2005
He's doing all kinds of things I couldn't do at his age. -- CatherineJohnson - 08 Dec 2005
Anne When I finally figured out that this was the gap in her knowledge, I was able to demonstrate how to get an equivalent fraction by multiplying a fraction by, for example, 3/3, then having her reduce it to the original fraction. The light bulb went off, and she was able to learn addition and subtraction of fractions. The difficult part of all this is figuring out where the gap is and what the gap is. As the student gets higher up in math, the more gaps there are and the more they rely on "Whenever I see this, I do that". Then they have to unlearn this and learn the correct way. I've thought about this from time to time..... Are there people who can 'diagnose' gaps really quickly and accurately? I bet there are. I do remember, quite awhile back, reading a study of tutors saying they had no idea where a student's particular gaps were. I'm sure that's true as general rule. -- CatherineJohnson - 08 Dec 2005
My sister says her husband, an architectural engineer who is probably a savant math type, can see his daughter's gaps & unique understandings instantly. He can shift immediately to a second and even third way of showing something as soon as she verbalizes what she thinks she knows. -- CatherineJohnson - 08 Dec 2005
But he has a very unique brain. -- CatherineJohnson - 08 Dec 2005
There wasn't a single question about the division of fractions; there weren't any addition or subtraction with borrowing; etc. He looked good on the test, but in fact his fraction knowledge is very shaky. And he can't do addition and subtraction with borrowing at all. How can this possibly be? Isn't it standard constructivist doctrine that kids with conceptual knowledge can construct these mundane procedural skills for themselves and should thereby be able to solve such problems more adroitly than a traditionally taught student? Say it ain't so? -- KDeRosa - 08 Dec 2005
How can this possibly be? Isn't it standard constructivist doctrine that kids with conceptual knowledge can construct these mundane procedural skills for themselves and should thereby be able to solve such problems more adroitly than a traditionally taught student? Say it ain't so? shooting fish in a barrel again, Ken? Actually, this has become an interesting question to me.....because I'm seeing Christopher grasp concepts pretty well (I think), and then not have a clue how to do the algorithm that goes with the concept..... That took me by surprise, I must say. -- CatherineJohnson - 08 Dec 2005
I'll have to get those problems posted, some of them I wouldn't say they were 'trivial,' though you guys might.... still and all, for him to do so well on this four-page test, with such shaky fraction knowledge, was pretty shocking -- CatherineJohnson - 08 Dec 2005
"The analysis done by "Dr. Stat" compares the percentage of US students at or above the 90th and 95th percentile of the international average." He also looked at the 99.9th percentile. Dr. Stat was attempting to answer questions about "top students." He explicity mentions "winners of competitions in Math and Science." The IMO data is relevant to his discussion. "In Count Down the U.S. winners are all immigrants (IIRC)." Three out of six of the US delegates from 2001(the year of the IMO depicted in Count Down) were immigrants. I don't know where each of the six students from 2005 was born. I would be surprised if they were all immigrants. -- KtmGuest - 08 Dec 2005
The IMO data is relevant to his discussion But not to the point I was making. The mo's represent about the 0.00002 percentile. Dr. Stat only went down to the 0.1 percentile. Its nice that his results generalized to the 5th and 10th percentiles but that fact is irrelevant to our discussion, as is the performance of the mos. We are not going to make curricular generalizations based on either of these group's performance. These facts are mere trivia. -- KDeRosa - 08 Dec 2005
Three out of six of the US delegates from 2001(the year of the IMO depicted in Count Down) were immigrants. I don't know where each of the six students from 2005 was born. I would be surprised if they were all immigrants. What we need to know is whether the other 3 were the children of immigrants. The data I've seen, which may originate with Steinberg, shows that the children and grandchildren of immigrants do successively worse with each new generation. -- CatherineJohnson - 08 Dec 2005
This is funny: (from Count Down):
High-level problem solving has many proponents in the United States. Since inaugurating a major reform effort in 1989, the National Council of Teachers of Mathematics (NCTM) has been urging U.S. math teachers to emphasize the kinds of problems that require students to think deeply about what they are doing. But the cultural inertia of math instruction is very powerful. The eighty-one classes Stigler and his colleagues videotaped in 1994 exhibited virtually no signs of the reforms advocated by NCTM. -- CatherineJohnson - 08 Dec 2005
yeah......if schools would just get off the stick and IMPLEMENT NCTM REFORMS we'd be cleaning up at Math Olympiads -- CatherineJohnson - 08 Dec 2005
Sorry, had to do this:
Lol! I thought he looked depressed. -- SusanS - 08 Dec 2005
oh my gosh!!!! where's you get that?????? That is HILARIOUS! -- CatherineJohnson - 08 Dec 2005
| WebLogForm | |
|---|---|
| Title: | Saxon versus IMP |
| TopicType: | WebLog |
| SubjectArea: | ImpCurriculum, IrvingtonMath, SaxonMath |
| LogDate: | 200512061810 |