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25 Jan 2006 - 03:24
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Also, my experience is that understanding multiplication is a slam dunk, using those rectangular grid analogies, at least through 9x9. Memorizing the tables is sometimes unpleasant, though. Doing multidigit multiplication was my personal bete noire, but other people don't seem to have had much trouble with it. From what I understand, long division is hard for practically everybody. -- CarolynJohnston - 25 Jan 2006
It's one thing to understand a topic or to not understand it. With multiplication, though, even if the student understands it, there is a need for achieving automaticity. I know students that can think a second and give me the correct answer when I ask what 6 x 7 equals. However, they wouldn't recognize that the fraction 28/42 can be reduced because both 28 and 42 are multiples of 7. If asked to reduce it, they'd methodically test for factors to reduce the fraction. This takes so much time and effort, and is so prone to error. The problem is not that they don't understand how to reduce fractions; it's that they are not automatic with their multiplication facts. Because of the same root cause, a simple algebra equation like 3x = 36 is not a simple exercise in (instantaneous) mental math. It becomes a place to apply long division, which is again costly in time, effort, and chance for error. -- DanK - 25 Jan 2006
I think this is a list of common sticking points, as opposed to universal sticking points that every kid will hit. I can recall having no problems with fractions, decimals, long division, etc, although sometimes I simply memorised the way you solved the problem, and the explanation the teacher gave, and only years later looking back really understood why they worked. Sometimes this understanding came after I'd got my A+ on first year engineering math (before I ran into the "actually needing to study for long periods of time problem") so I don't think lack of deep understanding is an important sticking point for continuing in maths. I did however make a mistake in the multi-digit multiplication formula at first. I understand now it was a very common mistake - I would carry the digit and then add it to the digit in the next place before doing the next single-digit multiplication, rather than adding it to the next answer. So it makes sense for teachers to be looking out for common sticking points, even if they're not universal. -- TracyW - 25 Jan 2006
Caveat: My son is too young to address this from a great deal of personal knowledge, so I'm working from anecdotes. Weight the following as you see fit. Order of operations must be learned to automaticity. The various meanings of negative numbers must be thoroughly understood. (Negative coefficients; negative exponents; addition, subtraction, multiplication, and division by negative numbers) Distributive property of multiplication must be automatic. (x+1)(x+1) = x2 + 2x + 1, not x2 + 1. Addition of fractions (LCD) must be understood. (Complex dividends and divisors in algebra seem to be especially problematic.) Hmm, you might want to take a look at Moebius Stripper;s Precalculus Bingo post from early last year. -- DougSundseth - 25 Jan 2006
It took me forever to understand: a) multiplication of fractions b) why, in division, you multiply by the reciprocal (I need to check with everyone to make sure my understanding of multiplication of fractions is right...) -- CatherineJohnson - 25 Jan 2006
I think it would be a fantastic service to collect a list of sticking points. This list strikes me as, in essence, a list about Everything. A common sticking point in math is math. I've never seen a child have trouble with skip-counting, though we'll see with Andrew. I did it for the first time with him today. This is one of those places where the constructivists are right: in my experience, the individual skills aren't that hard to learn. What's hard is: a) having some idea why they work and/or are done the way they're done b) making connections between those lone skills and other skills and applications (this is the story problem abyss) c) recognizing 'equalness': recognizing that two different number expressions or conventions are exactly the same thing For instance, 2^-2 is the same thing as 'the reciprocal of 2' is the same thing as 1/2 This is HUGELY difficult for kids (and no doubt adults) just learning math -- CatherineJohnson - 25 Jan 2006
Another thing, AND THIS IS IMPORTANT — plus it's something 'Math Brains' don't notice (I don't think): people who are learning math need to know:
In my experience teaching my math booster class and my own kids, for elementary school: 1. Memorizing addition, subtractions, mulitiplication and division facts. If a student does not do this or does it imcompletely, they can't understand all of the procedures based on these facts. 2. Even though they don't understand necessarily how it works, most kids can get multidigit multiplication and long division unless #1 is true. 3. Adding and subtracting of fractions is absolutely key. Most kids get stuck on two areas: equivalent fractions and lowest common denominator separately and then again when they have to combine them into one procedure. Also, when they have to add and subtract mixed fractions. If they change them to improper fractions, they use the same procedures. If they leave them as mixed fractions, they have a hard time learning to "carry" and "borrow". -- AnneDwyer - 25 Jan 2006
This is one of those places where the constructivists are right: in my experience, the individual skills aren't that hard to learn. What's hard is: a) having some idea why they work and/or are done the way they're done b) making connections between those lone skills and other skills and applications (this is the story problem abyss) c) recognizing 'equalness': recognizing that two different number expressions or conventions are exactly the same thing Which is the essence of moving from the inflexible to flexible knowledge stage. -- KDeRosa - 25 Jan 2006
You are probably going to get a lot of comments on negative and zero exponents. Since we will all address it a bit differently, maybe something will click. Anyway, start with realizing that whole number exponents just tell you the number of times the item appears as a factor. That is a^3 = aaa = a times a times a So a^2 times a^3 = a^2a^3 = aaaaa = a^5 The rules for exponents (exponential notation) are based on this observation. You ADD their exponents when multiplying two factors with the same base. If you have (aaa)/(aa) the answer is a. So you SUBTRACT their exponents when dividing when the bases are the same. And you know that (aaa)/(aaa) = 1 so that's why a^(3-3) = a^0 = 1. -- SusanJ - 25 Jan 2006
The other day, when Christopher had to learn negative exponents, Ed had never seen a negative exponent that he could remember. I certainly had never seen a negative exponent... You're kidding! I am gobsmacked! -- CarolynJohnston - 25 Jan 2006
Is there a point at which negative exponents became 'standard'? Remember, Ed took calculus-for-engineers at Princeton; he has NO memory of ever seeing a negative exponent. I'm wondering if they came into widespread use after he was there. (Not that they didn't exist, but that people weren't using them as much??) Anyway, I'd never seen a negative exponent. Given the way memory works — you can 'recognize' things you can't 'recall' — I'm 99% certain of that. -- CatherineJohnson - 25 Jan 2006
Ken Which is the essence of moving from the inflexible to flexible knowledge stage. yes, absolutely However, I don't take this to mean that we should 'practice, practice, practice' until finally flexible knowledge emerges. Obviously we should do that, but that's not all we should do. Many, many times, working through Saxon, I've suddenly seen a connection because Saxon directly pointed one out to me. Now, I was 'ready' to see it, because I'd been practicing. BUT SAXON DIRECTLY INSTRUCTED ME IN A 'CONNECTION' WHEN I WAS READY TO BE DIRECTLY INSTRUCTED. Ideally, I would do an Engelmann-like direct teaching of connections -- by which I mean a field-tested Direct Instruction In Connections and Concepts. I would test my lessons on real kids (or teens or adults) to discover which direct lessons on 'conceptual connectedness' work, and which don't. Here's an example from my own math re-learning. I had no idea that a fraction was a division problem. NONE. My knowledge of math, all of which had been practiced to fluency and probably to mastery, REMAINED FRAGMENTED. I mastered long division yeras ago. I mastered fraction addition, subtraction, multiplication, & division long ago. I had no idea fractions were division problems and division problems were fractions. NO IDEA I had NO IDEA that the reason you 'move the decimal over' when you're using long division to divide decimals is that you are simply multiplying the numerator and the denominator by the same number!!!! -- CatherineJohnson - 25 Jan 2006
Finding all this stuff out made math a WHOLE LOT more fun, let me tell you. -- CatherineJohnson - 25 Jan 2006
Is there a point at which negative exponents became 'standard'? Remember, Ed took calculus-for-engineers at Princeton; he has NO memory of ever seeing a negative exponent. He must have encountered them there, and almost surely also in high school. You need to use negative exponents in order to take derivatives of functions like 1/x, and other powers in denominators. Perhaps it was too painful? Bring in the recovered-memories experts... -- CarolynJohnston - 25 Jan 2006
I have a copy of an old high school textbook called "Intermediate Algebra" copyright 1945. The authors, Edward Edgerton and Perry Carpenter, were from high schools in New Jersey and New York. The book has about 500 pages. The frontispiece is a picture of a huge crane placing something in a tanker in a shipyward and below the picture caption it says, "War is teaching United States citizens a belated lesson in the supreme intellectual discipline -- mathematics." From page 155: "Hence: any quantity with a negative exponent is equal to the reciprocal of that quantity with the corresponding positive exponent. From the above principle it also follows that Any factor of the numerator of a fraction may be transferred to the denominator, or any factor of the denominator transferred to the numertor if the sign of its exponent is changed." -- SusanJ - 25 Jan 2006
"For instance, 2^-2 is the same thing as 'the reciprocal of 2' is the same thing as 1/2" I am not sure I understand correctly. Are you strictly referring to the exponent? (It's a good thing to use different numbers for the base and exponent in examples). An exponent changes signs when moved from the numerator to the denominator or vice-versa. Thus, 1/2^2 or 1/4 in your example. Regarding zero exponents, if the base is a nonzero real number and the exponent is zero, then the result is one as SusanJ? demonstrated. You can demonstrate that with numbers. Take 3^2/3^2 = 3^(2-2) = 3^0. Or 3x3 divided by 3x3 after crossing out is 1x1 divided by 1x1 or one. -- CharlesH - 25 Jan 2006
I first officially encountered negative exponents in high school in 1979. However, I had probably seen them before that in recreational math reading. -- GoogleMaster - 25 Jan 2006
Check out Ask Dr. Math on zero exponents. http://mathforum.org/dr.math/faq/faq.number.to.0power.html You might want to bookmark Dr. Math: http://mathforum.org/dr.math/ There's lots of good stuff. -- SusanJ - 25 Jan 2006
Finding all this stuff out made math a WHOLE LOT more fun, let me tell you. It turns math into something you can trust rather than something that will betray you. Thus, 1/2^2 or 1/4 in your example. Okay, that's what I thought but I wasn't going to say anything. In Saxon 8/7 there is a lot on negative exponents that I have absolutely no memory of being taught. I have no idea how I got through high school or college math. Saxon directly teaches the concept and makes it difficult for you to successfully complete related problems if you don't thoroughly get it. Since some of this feels like new info for me I've had to really focus on some of these chapters because I'll hit the wall if I don't. I had no idea fractions were division problems and division problems were fractions. Same here. Yet, I had some abstract knowledge because I always knew to draw the bar and divide by a number being used as a factor to "get rid of it." I understood it for that kind of problem but I never thought about the "rule" existing anywhere but where the math teacher told me to use it. That probably makes no sense, but it sure has changed how I talk to my one son about it. -- SusanS - 25 Jan 2006
It turns math into something you can trust rather than something that will betray you. I LOVE this observation. This encapsulates exactly what some people love -- and some people hate -- about math. -- CarolynJohnston - 25 Jan 2006
I had no idea fractions were division problems and division problems were fractions. Same here. I find it ironic and quite sad that the some of the parents on this list are saying that they never really understood mathematics back when they were using some other curriculum than the current fuzzy one. The reason I find it both ironic and sad is that apparently this new new curriculum was meant to address this problem .... What this says to me is that it is the teacher's understanding of mathematics and the teacher's ability to appreciate each student's unique misunderstandings that is really important. -- SusanJ - 25 Jan 2006
Check out this web link: http://math.berkeley.edu/~wu/six-topics1.pdf Professor Wu from Berkeley and Professor Milgram from Stanford have written an document outlining an math intervention program for Grades 4-7 for the state of California. -- AnneDwyer - 25 Jan 2006
What this says to me is that it is the teacher's understanding of mathematics and the teacher's ability to appreciate each student's unique misunderstandings that is really important. I agree 100% with this. -- CarolynJohnston - 25 Jan 2006
What this says to me is that it is the teacher's understanding of mathematics and the teacher's ability to appreciate each student's unique misunderstandings that is really important. I think the latter part of this statement is more important than the former. I know that I would make a really horrible math teacher because it would never occur to me in a million years that someone who is perfectly competent at manipulating fractions wouldn't recognize that fractions are just division problems. In order to be able to close the children's gaps, you have to be able to recognize that a gap could exist in a particular spot. And maybe in order to do that, you have to have struggled with it and had your own aha! moment. I have a theory that you don't want the A and A+ students to become the teachers; instead you want the B and B+ students. -- GoogleMaster - 25 Jan 2006
"I have a theory that you don't want the A and A+ students to become the teachers; instead you want the B and B+ students." Unfortunately, K-8 attracts those who probably fared worse. Some actively dislike math and want to do anything that is different than what they had when they were growing up. Speaking of theories, my theory is that K-8 attracts many who want to help special needs kids. Their background and focus is anything but setting and expecting high (explicit content and skills) standards. -- SteveH - 25 Jan 2006
In order to be able to close the children's gaps, you have to be able to recognize that a gap could exist in a particular spot. And maybe in order to do that, you have to have struggled with it and had your own aha! moment. I think this is true. I have turned out to be very good with my LD son because I know exactly what is messing with his head. Even though I never had any LD problems (I always thought I did), the anxiety of not knowing what to do and being too afraid to ask would cause a complete brain shutdown almost daily, particularly if any new material was being introduced. My mathematically gifted son, however, is interesting because he doesn't ever understand what's not to get. Even when he explains to me some shortcut (well, it seems like a shortcut to me) he uses to get to an answer, he can't figure out why I have no idea how he did it. He is a true math head and completely unintimidated. One day I was showing him how Saxon teaches subraction of a mixed number from the number one. I guess that would be subtraction with borrowing but with fractions. Saxon has you convert 1 into a fraction and then proceed. With a whole number higher than 1 you would take a 1, in fraction form, out of the number, add it to your other fraction and then proceed with the problem. He couldn't see why anyone would bother with this. You just subract it. I tried to give him some harder ones, but he could do it in his head with no problem. Eventually, he stared at it and realized that it might be helpful in some way for kids who couldn't do it in their heads. -- SusanS - 25 Jan 2006
Speaking of theories, my theory is that K-8 attracts many who want to help special needs kids. Their background and focus is anything but setting and expecting high (explicit content and skills) standards. But explicit content and skills is necessary for helping special needs kids. (The "But" is directed at the belief, not at SteveH?'s theory). I'm not sure about the high standards bit - my speech just needed bringing up to normal kids' standards. But I wasn't capable of figuring out how to pronounce words correctly on my own, I needed someone to explicitly teach skills. -- TracyW - 26 Jan 2006
I'm thinking we could probably compile an excellent list of sticking points by looking at concepts Saxon covers WAY more than others. I'll be looking out for this as I go along. -- CatherineJohnson - 28 Jan 2006
golly - I hadn't finished reading this thread - lots of good stuff here back shortly - -- CatherineJohnson - 28 Jan 2006
Since I didn't get around to writing the post I'd planned to write, here's something I need in a nutshell: I need for a book and/or teacher to tell me when a concept 'follows naturally' from 'nature' (not one person on the planet will know what that means....) VERSUS when a concept follows logically from the conventions of mathematics = = = = I'll try to explain what I mean. When you start out in math, you're seeing a lot of visual explanations, which, for me, is excellent. "Visual explanations" are probably what I mean by "nature." When you look at a depiction of area, that seems 'natural' — it's something that occurs in physical reality. After I'd seen a number of visual explanations, I developed a 'feeling' that 'understanding' MEANS 'having a visual image.' I felt that pretty strongly. At that stage of the game (a few months ago) I certainly could handle logic; I'd been taught math logic at the h.s. algebra & geometry level years ago. That wasn't the problem. The problem was that 'logic' didn't 'feel' like 'understanding. I have to use the quote marks, because I don't know what I'm saying. All I can do is describe an experience I had, and that I bet lots of students have. SO here's what happened next. I thought I should have a visual image of reciprocals I was extremely bewildered by reciprocals. No problem procedurally; I've been able to divide fractions since I was 11. I also did acquire some workable conceptual knowledge, back then, of what dividing fractions meant. But I had no idea how reciprocals worked, or why you ended up with them, etc. And I kept thinking I was supposed to 'see it' in my mind's eye. = = = = Here's a recent example of a math concept that follows-from-math instead of appearing-in-nature. This happened to Ed & to Kris, both of whom are math-friendly & good at math (for non-math brains). negative exponents I don't think any of us had ever seen a negative exponent. (Maybe Kris had.) So we had to learn them fast. Saxon has a lesson showing that negative exponents, and an exponent of 0, must follow logically from the way the positive exponent system has been set up. Ed, at first, couldn't grasp this at all. He was probably experiencing the same thing I did with reciprocals. Then, when I showed him the Saxon lesson, it started to sink in — 'sink in' meaning: he started to shift over to 'logic.' I did the same thing with Kris, only I didn't have the book. It worked; she got it. Certainly in elementary math, I've needed - and probably a lot of students need - markers for WHEN WE'RE TALKING ABOUT MATH CONCEPTS WE CAN DRAW and WHEN WE'RE TALKING ABOUT MORE ABSTRACT MATH CONCEPTS THAT FOLLOW FROM MATHEMATICAL LOGIC. That may not be the correct way to put it, but it describes a stumbling block I tripped over fairly well. Today, this doesn't seem to be a problem any more. Somehow, math logic now seems 'natural.' -- CatherineJohnson - 28 Jan 2006
OK, now's the time to weigh in -- this weekend I'm going to try to boil down what everyone had to say here. I'm going to try to extract the MAIN IDEAS. Wish me luck. -- CarolynJohnston - 28 Jan 2006
OH BOY, GOOD LUCK where are you putting the page?? do you want to index it on User's Page? This is an important topic. -- CatherineJohnson - 28 Jan 2006
shortcuts That's the other HUGE sticking point for me, or was. It would have been incredibly useful for me to have my textbooks (or a teacher) tell me: THIS IS A SHORTCUT AND HERE'S HOW WE GOT TO IT. For instance, cross-multiplication. If I had learned why it is you can do cross-multiplication, learning the shortcut would have been great. (I remember when I finally figured out, for myself, why cross-multiplication was true. I had no idea!) RUSSIAN MATH does this all the time. They teach all or most of the shortcuts, but they teach them formally, with justification of each step. Also, they'll ask the student to repeat the sequence and write the property that allows you to do each step. -- CatherineJohnson - 28 Jan 2006
He must have encountered them there, and almost surely also in high school. You need to use negative exponents in order to take derivatives of functions like 1/x, and other powers in denominators Just saw this. I never took derivatives of functions, so there was no need for me to have seen negative exponents. I'm 99.99% certain I never saw them. The fact that Ed doesn't remember them at all seems like he really might not have seen them. I may be overstating it.....it may be more that he couldn't remember seeing them.... -- CatherineJohnson - 28 Jan 2006
If it was a sufficiently proof-centric calculus course, he might never have seen the general formula for differentiating functions 1/xn (which is the same as x-n)). But it's unlikely. I'm a bit incredulous! Does he remember what text he used, or if they even used a text? At Princeton, you never know. -- CarolynJohnston - 28 Jan 2006
(Did you notice I did superscripts that time!??? It' s easy!) -- CarolynJohnston - 28 Jan 2006
"They teach all or most of the shortcuts, but they teach them formally, with justification of each step." Some steps you don't want to know like how to arrive at the quadratic formula. It's like what they say about laws and sausage. I'm curious to know how best to explain why cross-multiplication works. I guess one would start out by explaining that a proportion is a statement that two ratios are equal and then go from there. -- CharlesH - 28 Jan 2006
How did you do the superscripts? Very impressive. -- SusanS - 28 Jan 2006
Susan it's html, like this:
I'm puzzling over something related to the previous discussions. The cool thing about natural language is that you can immediately understand sentences you've never heard before and you can also make up grammatical sentences you never heard before. (There's even some indication that possibly some kind of monkeys can do something similar with picture cards.) So I'm trying to understand why this same innate understanding doesn't work for mathematics for some of you. To me saying "three times the sum of four plus two" is exactly the same as saying "three times four plus three times two." But you have explained quite clearly that for you, understanding the previous sentence involves going back and spending a lot of time understanding the distributive law and also order of operations. 3*(4+2) = 3*4 + 3*2 So do most of us have a "language brain" but only a few of us have a "math brain" or is there some kind of critical period where if your math brain isn't activated it stops working or what? -- SusanJ - 29 Jan 2006
For me, the distributive law was something I memorised, along with the order of operations, and then simply applied when occasion called for it. (Of course, the teacher showed us how it worked, with actual numbers). Later on I realised the deeper meaning behind it. I think what got me sailing through maths was that my own interest in it was limited to getting good results on tests so not actually understanding why something worked never bothered me. Just memorise it and get back to my book. It was only later on at high school when we got onto proofs that the "why" of mathematics became more interesting to me. And by then I had thoroughly learnt that understanding was not a barrier to doing maths. Why do we do matrix-multiplication that way? Who cares? Memorise the method and if the reason only becomes clear well after the final exam, well hey, I'll have my A anyway! -- TracyW - 29 Jan 2006
x2 Cool. A bit long-winded, but cool. -- SusanS - 29 Jan 2006
Carolyn: I'll try to remember to ask him today.... (Ed on negative exponents). It's entirely possible I misunderstood; maybe he said they were 'vaguely familiar,' something like that.... -- CatherineJohnson - 29 Jan 2006
To me saying "three times the sum of four plus two" is exactly the same as saying "three times four plus three times two. I haven't read the thread carefully (I will!) so I may be missing a step....I think I'm making a different point about the distributive property. Christopher found it incredibly easy to do what you just did, i.e. to translate a verbal express ("3 x's") into mathematical language: 3x. The Phase 4 kids did this at the beginning of the year, and it was SO easy for him, and SO natural, that I thought, OK, we can do this course. Christopher's problem with the distributive property is exactly what Carolyn mentioned: he doesn't recognize it in a new context. The exact same thing happened to me when I was first re-teaching math. heck, I've got notes on this somewhere......I spent hours, literally, trying to figure out some wonky problem SRA Math, grade 6 had. It was one of those problems where you're supposed to figure it out more or less by guess & check.....and I wanted to know why it worked the way it did. I SPENT HOURS. Finally I showed it to my brother-in-law, who's probaby a math brain, and he said, 'That's the distributive property.' I was mortified. I knew the distributive property COLD. Seriously. I knew it from childhood, I knew it from Saxon 6/5, I knew it was the basis of the multiplication algorithm, I knew it as the basis of mental multiplication, I knew it as the basis of the many 'distributive property' problems Saxon has in 6/5. I also had, in high school, learned to use the distributive property in that particular genre of problem, although I hadn't done such a problem in 30 years. I DID NOT RECOGNIZE THE DISTRIBUTIVE PROPERTY IN THAT PROBLEM. -- CatherineJohnson - 29 Jan 2006
One more thing: I think the distributive property is far more difficult to recognize once the factor has been distributed (assuming I'm saying that right). In other words, using the distributive property to factor, or to 'reverse engineer' an expression is harder than using the distributive property to produce that expression in the first place At least, that's been my experience. -- CatherineJohnson - 29 Jan 2006
Catherine, I think this is evidence of stress. Anyone with a degree in engineering from a good school who couldn't recover the meaning of negative exponents with a few minutes of thought is likely under way too much stress. It's definitely not something I'd expect to encounter in sixth grade, however. In my experience, exponential notation is fairly difficult for high school juniors. -- SusanJ - 29 Jan 2006
Charles You asked about the best way to show why cross-multiplication works. I definitely can't answer that! But I can tell you how I figured it out.....and I think it would have 'worked' for me if a book or teacher had demonstrated it directly. Number one: I never had the slightest problem 'accepting' that the reciprocal of a number is the number that, when you multiply the two, gives you a product of 1. I'd be willing to bet that most kids can 'deal' with this in exactly this way: IT'S A RULE. IT'S A DEFINITION. So.....that was great! If reciprocals are simply defined, from the get-go, in this way I had no problem. Number two: I had zero problem, and I'd bet the ranch most kids have zero problem, accepting AND UNDERSTANDING that any number outside zero divided by itself is equal to 1. Number three: this is going to sound absolutely nuts to you, but it's important. It took me quite a long time (meaning an hour — something in there) to figure out that when you cross multiply, it's exactly the same thing as multiplying a number by its reciprocal. So: 1/2 = 1/2 1/2 x 2/1 = 1 cross multiplication: 1 x 2 = 2 x 1 Number four: equivalent fractions, yes 1/2 = 3/6 so: 1/2 x 6/3 = 1 1 x 6 MUST EQUAL 2 x 3, BECAUSE when you multiply a number by a reciprocal you always get 1, AND BECAUSE the only time you get 1 as a quotient is when the numerator & the denominator are the same number = = = = = I hope that's semi-clear. I think for math-savvy people it probably is. For someone who hasn't spent a lot of time around math, I suspect that last step is going to lose people. I think I could probably write it out in a way that made sense to people who are learning math..... -- CatherineJohnson - 29 Jan 2006
quadratic equations I'm ALMOST positive that by the time a student reaches quadratic equations - this is definitely true for me - you no longer need to see how shortcuts have been arrived at. I now 'get' that mathematicians have been able to figure a lot of things out so I don't have to figure them out myself. I'm guessing that the 'demonstration of shortcuts' is most useful in....arithmetic and possibly beginner algebra. -- CatherineJohnson - 29 Jan 2006
It's fascinating to me how different people understand things differently. Here's how I look at cross-multiplication. It's just a shortcut to applying the rule that you can always do the same thing to both sides of an equation when you happen to have only a simple fraction on each side. a/b = c/d Multiply both sides by b: a*b/b = b*c/d
a = b*c/d Multiply both sides by d: a*d = b*c*d/d = b*c -- SusanJ - 29 Jan 2006
rates This probably isn't a 'sticking point,' per se, but I'll put it in. I had literally never heard of ratios as 'rates' until this year. I think the first time I heard of it was one day when Charles brought 'rates' up in the context of talking about ratios. I don't understand rates-as-ratios. I'm trying to think whether I can come up with a good illustration (I should probably look at Saxon...) hmm... Can't remember whether I did or didn't understand this problem, but here it is: If 12 books weigh 20 pounds, how much would 30 books weigh Saxon says: Since 12 books weigh 20 pounds we can write two rates. (a) 12 books/20 pounds (b) 20 pounds/12 books [first of all, I have a problem using the word 'rate' to apply to weight. Since I've always used 'rate' to means time or speed & I think price.....the language here is an obstacle, not a help. I think that's part of the problem.] To find the weight of 30 books, we could multiply 30 books by rate (b). 30books x 20 pounds/12 books = 50 pounds
At the moment this isn't seeming like one of the Seven Wonders of the World to me, but I know that when Saxon assigns a rates-as-ratios problem, I instantly feel like: HOW ARE BOOKS A RATE?
HOW CAN YOU MULTIPLY BY BOOK-POUNDS (or whatever).....
However, this isn't a sticking point in the sense that kids have to know this - that you'll be blocked from learning future concepts.
(Is it?)
-- CatherineJohnson - 29 Jan 2006
I DID NOT RECOGNIZE THE DISTRIBUTIVE PROPERTY IN THAT PROBLEM. That's exactly my puzzlement. If I say something like, "The ferret just ate my prune," which is a sentence you've probably never heard, you recognize the meaning immediatetly. Why is math different? I have absolutely no idea but I think it is a fascinating question. BTW, I've never really thought about factoring as a reverse application of the distributive law. There's actually lots of different types of factoring. -- SusanJ - 29 Jan 2006
Susan J yes, definitely: it's a shortcut to applying the rule that you can do the same thing to both sides.....BUT — and here's where the hyperspecificity of the novice comes in — to 'see' that I need a problem with a variable! 2/x = 4/8 If I had been trying to explain to myself why cross multiplication works with this problem, I might have figured out that it was 'do the same thing to both sides' at once! Too bad I didn't try that experiment. But since I was trying to see why cross multiplication is always true in every case, I didn't go at it that way. hyperspecificity is a major, major feature of beginner knowledge. Hyperspecificity and fragmentation (which may be the same thing). -- CatherineJohnson - 29 Jan 2006
I've never really thought about factoring as a reverse application of the distributive law Isn't that how it's taught (that particular form of factoring, of course). I've always been taught that when you factor an expression like: 3x + 12y you're using the distributive property 3x + 12y doesn't look like the distributive property because students are always taught that the distributive property is this: 3 (x + 4y) "Inversing" the distributive property is a HUGE sticking point. Prentice-Hall — and this is something I think the book does right — has the kids 'inversing' the distributive property from the get-go. Even that didn't work. What's more, the distributive property in 'pre-algebra' (beginner algebra) is so mystifying to kids THEY KNOW IT'S MYSTIFYING. I was driving Christopher & his friend Joe around one day, and the subject of the next math test came up, and Joe said, matter-of-factly, "I don't understand the distributive property." -- CatherineJohnson - 29 Jan 2006
Somehow, I don't think that 'inversing' the distributive property is the same thing as shifting from active to passive voice, or whatever the equivalent would be.... It CERTAINLY doesn't come naturally. -- CatherineJohnson - 29 Jan 2006
Factor 3x + 12y This is a two-step problem. Somewhere else we had discussed the difficulty of two-step problems. Could that be the difficulty here? The first step requires factoring each term individually. 3x = 3*x 12y = 3*2*2*y The second step requires checking each term to see whether they have any common factors which in this case they do. So now you pull out the common factor, 3, and put the expression back together. 3*(x + 2*2*y) = 3*(x + 4y) (Or maybe it's three steps since typically you re-multiply any factors that weren't pulled out.) -- SusanJ - 29 Jan 2006
Let me state this more forcefully. Factoring is process. This process is not necessarily the inversing of the distributive law, which is a definition. The distributive law simply says that a multiplier (or divisor) acts upon each term. So 3*(2x + 4y) MEANS 3*2x + 3*4y which can also be written as 6x + 12y. The instruction to "factor" almost always means to pull out the largest common factor. So when you factor according to this rule 6x+12y = 6*(x+2y) which is not simply the inversing of the previous example of the distributive law! -- SusanJ - 29 Jan 2006
hmmm....this is getting a little over my head..... I guess I'm using the word 'inversing' colloquially, to mean 'doing the reverse of what you did to get to the expression you currently have'....is that wrong? This reminds me of Carolyn's post; I'll go find it. -- CatherineJohnson - 30 Jan 2006
here it is! understanding basic algebra moves What I want to show him is the most general way to do it -- that is, to recognize that what he must do is undo the division of w by 2.1, by multiplying both sides of the equation by 2.1. It's the basic trick of algebra; you solve for something by undoing what's been done to it, remembering that anything you can do to an expression in an equation is okay as long as you do it to both sides of the equation. Christopher also wouldn't recognize how to factor: 3x +3y He wouldn't perceive that the distributive property is involved here. He will, at this point, I think, perceive that he should think 'distributive property' when he sees this expression: 3(x + y) He won't (maybe he does now, but he certainly didn't a couple of months ago) perceive that he should think 'distributive property' when he sees 3x + 3y That's what happened to me, struggling with the SRA problem. I simply failed to recognize that the distributive property was involved in any way, shape, or form. The INSTANT my brother-in-law pointed it out, I felt like an idiot, because in fact I had learned this long ago, and I still 'knew' it — and even understood it......to some degree I didn't perceive it -- CatherineJohnson - 30 Jan 2006
One more item for the list. The Doug Carnine book on Designing DI Math Curricula says that decimal division is the most difficult of the decimal operations. No idea how much of a sticking point this is, but it's worth knowing. -- CatherineJohnson - 30 Jan 2006
Catherine, I understood what you meant by "inversing." However, factoring, which is the proper term, is typically more complex than the example you gave. Understanding the distributive law is only part of what you need to know. Factoring is a complex, step-wise process. I'm sorry I didn't note this "gap" when you mentioned this earlier. Here are two examples of factoring to illustrate the process. Problem 1. Factor 14x + 21y Step 1. Fully factor each term 2*7*x + 7*3*y Step 2. Pull out any common factors 7*(2x + 3y) Problem 2. Factor 14x + 21x^2 Step 1. Fully factor each term 2*7*x + 7*3*x*x Step 2. Pull out any common factors 7x(2 + 3x) You will note that in neither problem does the multiplicative factor in front the parentheses in the answer appear directly in the original expression. -- SusanJ - 30 Jan 2006
The Doug Carnine book on Designing DI Math Curricula says that decimal division is the most difficult of the decimal operations. Decimal long division? I don't know about that. Once you know how to do long division, it's easy -- just a matter of rearranging the decimals' positions. Whereas in decimal multiplication, I'm finding that Ben has a huge tendency to put the decimal in the position it would be in if he were adding -- i.e. to line up the decimals and then put the decimal in the same place in the result. This is happening mostly because he's trying not to think at all. Ben would like to learn the whole of elementary math to pure automaticity. My little computer. -- CarolynJohnston - 30 Jan 2006
I don't remember finding it hard.....and I don't remember Christopher finding it hard....(but I could be just not remembering). If Christopher had had a HUGE struggle with it, I'd remember. -- CatherineJohnson - 30 Jan 2006
Actually, I would have said exactly the same thing (and I still do): decimal multiplication is always harder for me. Probably for Christopher, too. I sometimes get confused about how many 'zero's' I actually have, especially with a problem like.....oh gosh, I don't know if I can think of one that actually would illustrate the confusion. The next time I do one, I'll remember to write it down. -- CatherineJohnson - 30 Jan 2006
Understanding the distributive law is only part of what you need to know right, I think I understand that all I'm trying to say is that perceiving that the distributive property is involved in a factoring 'situation' is a sticking point..... a big one -- CatherineJohnson - 30 Jan 2006
Catherine, I think it is a "sticking point" mostly because that is not a natural way of looking at it. Understanding the distributive law is of little help in understanding factoring. In my opinion the "sticking point" is more likely that kids aren't taught the process for factoring individual items to mastery. Note that factoring individual items has nothing to do with the distributive law. They should start with factoring integers. Let me quote something from an old algebra book. "The factors of a product are those quantities which when multiplied together produce that product. A prime quantity is one which has no factors except itself and 1. In all work in factoring prime factors are required. For instance, the factors of 24 are 4 and 6 or 2 and 12 or 3 and 8, but these sets of factors are not prime, since 6, 12, and 8 can all be factored further. The prime factors of 24 are 2, 2, 2, and 3." Factoring is another example of something that is difficult if you don't know your multiplication facts to automaticity. -- SusanJ - 30 Jan 2006
Christopher finds factoring of individual numbers easy, interestingly enough, and I think Carolyn said Ben's the same way. He's also got his multiplication facts down cold. The Prentice-Hall Book opens with a chapter on the properties, which included problems like the one I posted above (I think — I'll check). The kids just couldn't do them (and I think it was lots of the kids, not just Christopher). These were problems on the distributive property specifically. I personally can factor up a storm; RUSSIAN MATH has a fabulous opening chapter on factoring. But I couldn't for the life of me see that the distributive property was involved in the SRA Math problem. -- CatherineJohnson - 30 Jan 2006
But I couldn't for the life of me see that the distributive property was involved in the SRA Math problem. This is good, not bad. Your intuition was correct! The distributive property is a law. Factoring is a process. Trying to make kids think that understanding the distributive property will help them with factoring is wrong from a mathematical standpoint. Can Christopher factor items with exponents such as x^2 or y^3*z^4? x^2 = x*x y^3*z^4 = y*y*y*z*z*z*z What about items with both numbers and letters such as 12x^2? 12x^2 = 2*2*3*x*x If Christopher is given a list of factored items can he find the factors that are in common to all the items? What are the common factors in x*y*3*7 and x*z*2*3? Answer, the common factors are x and 3. You might want to circle the common factors. -- SusanJ - 30 Jan 2006
The distributive property is a law. Factoring is a process. Trying to make kids think that understanding the distributive property will help them with factoring is wrong from a mathematical standpoint. Factoring algebraic expressions is a process in which an understanding of the distributive law is fundamental. Knowing when to apply the law is necessary, but not sufficient, for factoring. It's the second step in factoring; the first step is being able to find the common factors, as you are saying. -- CarolynJohnston - 30 Jan 2006
Carolyn, I (of course) agree that using the distributive property properly is the second step in writing a factored expression when you are simply factoring out a common factor from a sum of two or more terms. What I object to is the idea that knowing the distributive law means that you know all you need to know to be able to factor an expression like 6a^3*x^4 - 15a^2*x^2*y. My intuition is that understanding factoring is much, much harder than understanding the distributive law. It's easy to go in either direction when you write a*(b+c) = a*b + a*c but most factoring problems aren't in that form. I'm also concerned about laying the proper groundwork for factoring expressions like x^2 - xy - 2y^2. -- SusanJ - 30 Jan 2006
sticking points in math education
Rick Garlikov, the subject of the other day's post on TeachingBinaryLikeSocrates, has a mentoring service for students and parents in Alabama. You can pay to have your child mentored and tutored, or you can pay to be mentored in teaching your own kid. I love the latter idea, actually. Teach a parent to fish, and he'll eat for a lifetime. Anyway, on his web page about parent mentoring, Rick has a section in which he discusses the areas in math where kids tend to flounder (you have to scroll way down to see it). He also has a section about verbal subjects where kids tend to flounder, but we'll cover that in a different post. Here's Rick's list of sticking points in math education:- Understanding counting by groups, such as groups of two, five, and ten
- Seeing numerical relationships in general and knowing to look for them
- Place-value and adding/subtracting that requires regrouping or what used to be called borrowing and carrying
- Understanding multiplication and division
- Fractions
- Decimals rate/time/distance problems
- What algebra is about; how it works in general
- Geometry proofs and theorems and their point
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Also, my experience is that understanding multiplication is a slam dunk, using those rectangular grid analogies, at least through 9x9. Memorizing the tables is sometimes unpleasant, though. Doing multidigit multiplication was my personal bete noire, but other people don't seem to have had much trouble with it. From what I understand, long division is hard for practically everybody. -- CarolynJohnston - 25 Jan 2006
It's one thing to understand a topic or to not understand it. With multiplication, though, even if the student understands it, there is a need for achieving automaticity. I know students that can think a second and give me the correct answer when I ask what 6 x 7 equals. However, they wouldn't recognize that the fraction 28/42 can be reduced because both 28 and 42 are multiples of 7. If asked to reduce it, they'd methodically test for factors to reduce the fraction. This takes so much time and effort, and is so prone to error. The problem is not that they don't understand how to reduce fractions; it's that they are not automatic with their multiplication facts. Because of the same root cause, a simple algebra equation like 3x = 36 is not a simple exercise in (instantaneous) mental math. It becomes a place to apply long division, which is again costly in time, effort, and chance for error. -- DanK - 25 Jan 2006
I think this is a list of common sticking points, as opposed to universal sticking points that every kid will hit. I can recall having no problems with fractions, decimals, long division, etc, although sometimes I simply memorised the way you solved the problem, and the explanation the teacher gave, and only years later looking back really understood why they worked. Sometimes this understanding came after I'd got my A+ on first year engineering math (before I ran into the "actually needing to study for long periods of time problem") so I don't think lack of deep understanding is an important sticking point for continuing in maths. I did however make a mistake in the multi-digit multiplication formula at first. I understand now it was a very common mistake - I would carry the digit and then add it to the digit in the next place before doing the next single-digit multiplication, rather than adding it to the next answer. So it makes sense for teachers to be looking out for common sticking points, even if they're not universal. -- TracyW - 25 Jan 2006
Caveat: My son is too young to address this from a great deal of personal knowledge, so I'm working from anecdotes. Weight the following as you see fit. Order of operations must be learned to automaticity. The various meanings of negative numbers must be thoroughly understood. (Negative coefficients; negative exponents; addition, subtraction, multiplication, and division by negative numbers) Distributive property of multiplication must be automatic. (x+1)(x+1) = x2 + 2x + 1, not x2 + 1. Addition of fractions (LCD) must be understood. (Complex dividends and divisors in algebra seem to be especially problematic.) Hmm, you might want to take a look at Moebius Stripper;s Precalculus Bingo post from early last year. -- DougSundseth - 25 Jan 2006
It took me forever to understand: a) multiplication of fractions b) why, in division, you multiply by the reciprocal (I need to check with everyone to make sure my understanding of multiplication of fractions is right...) -- CatherineJohnson - 25 Jan 2006
I think it would be a fantastic service to collect a list of sticking points. This list strikes me as, in essence, a list about Everything. A common sticking point in math is math. I've never seen a child have trouble with skip-counting, though we'll see with Andrew. I did it for the first time with him today. This is one of those places where the constructivists are right: in my experience, the individual skills aren't that hard to learn. What's hard is: a) having some idea why they work and/or are done the way they're done b) making connections between those lone skills and other skills and applications (this is the story problem abyss) c) recognizing 'equalness': recognizing that two different number expressions or conventions are exactly the same thing For instance, 2^-2 is the same thing as 'the reciprocal of 2' is the same thing as 1/2 This is HUGELY difficult for kids (and no doubt adults) just learning math -- CatherineJohnson - 25 Jan 2006
Another thing, AND THIS IS IMPORTANT — plus it's something 'Math Brains' don't notice (I don't think): people who are learning math need to know:
- when is a 'rule' a rule because it works mathematically
- when is a 'rule' a rule because it follows logically from other, 'prior' rules
In my experience teaching my math booster class and my own kids, for elementary school: 1. Memorizing addition, subtractions, mulitiplication and division facts. If a student does not do this or does it imcompletely, they can't understand all of the procedures based on these facts. 2. Even though they don't understand necessarily how it works, most kids can get multidigit multiplication and long division unless #1 is true. 3. Adding and subtracting of fractions is absolutely key. Most kids get stuck on two areas: equivalent fractions and lowest common denominator separately and then again when they have to combine them into one procedure. Also, when they have to add and subtract mixed fractions. If they change them to improper fractions, they use the same procedures. If they leave them as mixed fractions, they have a hard time learning to "carry" and "borrow". -- AnneDwyer - 25 Jan 2006
This is one of those places where the constructivists are right: in my experience, the individual skills aren't that hard to learn. What's hard is: a) having some idea why they work and/or are done the way they're done b) making connections between those lone skills and other skills and applications (this is the story problem abyss) c) recognizing 'equalness': recognizing that two different number expressions or conventions are exactly the same thing Which is the essence of moving from the inflexible to flexible knowledge stage. -- KDeRosa - 25 Jan 2006
You are probably going to get a lot of comments on negative and zero exponents. Since we will all address it a bit differently, maybe something will click. Anyway, start with realizing that whole number exponents just tell you the number of times the item appears as a factor. That is a^3 = aaa = a times a times a So a^2 times a^3 = a^2a^3 = aaaaa = a^5 The rules for exponents (exponential notation) are based on this observation. You ADD their exponents when multiplying two factors with the same base. If you have (aaa)/(aa) the answer is a. So you SUBTRACT their exponents when dividing when the bases are the same. And you know that (aaa)/(aaa) = 1 so that's why a^(3-3) = a^0 = 1. -- SusanJ - 25 Jan 2006
The other day, when Christopher had to learn negative exponents, Ed had never seen a negative exponent that he could remember. I certainly had never seen a negative exponent... You're kidding! I am gobsmacked! -- CarolynJohnston - 25 Jan 2006
Is there a point at which negative exponents became 'standard'? Remember, Ed took calculus-for-engineers at Princeton; he has NO memory of ever seeing a negative exponent. I'm wondering if they came into widespread use after he was there. (Not that they didn't exist, but that people weren't using them as much??) Anyway, I'd never seen a negative exponent. Given the way memory works — you can 'recognize' things you can't 'recall' — I'm 99% certain of that. -- CatherineJohnson - 25 Jan 2006
Ken Which is the essence of moving from the inflexible to flexible knowledge stage. yes, absolutely However, I don't take this to mean that we should 'practice, practice, practice' until finally flexible knowledge emerges. Obviously we should do that, but that's not all we should do. Many, many times, working through Saxon, I've suddenly seen a connection because Saxon directly pointed one out to me. Now, I was 'ready' to see it, because I'd been practicing. BUT SAXON DIRECTLY INSTRUCTED ME IN A 'CONNECTION' WHEN I WAS READY TO BE DIRECTLY INSTRUCTED. Ideally, I would do an Engelmann-like direct teaching of connections -- by which I mean a field-tested Direct Instruction In Connections and Concepts. I would test my lessons on real kids (or teens or adults) to discover which direct lessons on 'conceptual connectedness' work, and which don't. Here's an example from my own math re-learning. I had no idea that a fraction was a division problem. NONE. My knowledge of math, all of which had been practiced to fluency and probably to mastery, REMAINED FRAGMENTED. I mastered long division yeras ago. I mastered fraction addition, subtraction, multiplication, & division long ago. I had no idea fractions were division problems and division problems were fractions. NO IDEA I had NO IDEA that the reason you 'move the decimal over' when you're using long division to divide decimals is that you are simply multiplying the numerator and the denominator by the same number!!!! -- CatherineJohnson - 25 Jan 2006
Finding all this stuff out made math a WHOLE LOT more fun, let me tell you. -- CatherineJohnson - 25 Jan 2006
Is there a point at which negative exponents became 'standard'? Remember, Ed took calculus-for-engineers at Princeton; he has NO memory of ever seeing a negative exponent. He must have encountered them there, and almost surely also in high school. You need to use negative exponents in order to take derivatives of functions like 1/x, and other powers in denominators. Perhaps it was too painful? Bring in the recovered-memories experts... -- CarolynJohnston - 25 Jan 2006
I have a copy of an old high school textbook called "Intermediate Algebra" copyright 1945. The authors, Edward Edgerton and Perry Carpenter, were from high schools in New Jersey and New York. The book has about 500 pages. The frontispiece is a picture of a huge crane placing something in a tanker in a shipyward and below the picture caption it says, "War is teaching United States citizens a belated lesson in the supreme intellectual discipline -- mathematics." From page 155: "Hence: any quantity with a negative exponent is equal to the reciprocal of that quantity with the corresponding positive exponent. From the above principle it also follows that Any factor of the numerator of a fraction may be transferred to the denominator, or any factor of the denominator transferred to the numertor if the sign of its exponent is changed." -- SusanJ - 25 Jan 2006
"For instance, 2^-2 is the same thing as 'the reciprocal of 2' is the same thing as 1/2" I am not sure I understand correctly. Are you strictly referring to the exponent? (It's a good thing to use different numbers for the base and exponent in examples). An exponent changes signs when moved from the numerator to the denominator or vice-versa. Thus, 1/2^2 or 1/4 in your example. Regarding zero exponents, if the base is a nonzero real number and the exponent is zero, then the result is one as SusanJ? demonstrated. You can demonstrate that with numbers. Take 3^2/3^2 = 3^(2-2) = 3^0. Or 3x3 divided by 3x3 after crossing out is 1x1 divided by 1x1 or one. -- CharlesH - 25 Jan 2006
I first officially encountered negative exponents in high school in 1979. However, I had probably seen them before that in recreational math reading. -- GoogleMaster - 25 Jan 2006
Check out Ask Dr. Math on zero exponents. http://mathforum.org/dr.math/faq/faq.number.to.0power.html You might want to bookmark Dr. Math: http://mathforum.org/dr.math/ There's lots of good stuff. -- SusanJ - 25 Jan 2006
Finding all this stuff out made math a WHOLE LOT more fun, let me tell you. It turns math into something you can trust rather than something that will betray you. Thus, 1/2^2 or 1/4 in your example. Okay, that's what I thought but I wasn't going to say anything. In Saxon 8/7 there is a lot on negative exponents that I have absolutely no memory of being taught. I have no idea how I got through high school or college math. Saxon directly teaches the concept and makes it difficult for you to successfully complete related problems if you don't thoroughly get it. Since some of this feels like new info for me I've had to really focus on some of these chapters because I'll hit the wall if I don't. I had no idea fractions were division problems and division problems were fractions. Same here. Yet, I had some abstract knowledge because I always knew to draw the bar and divide by a number being used as a factor to "get rid of it." I understood it for that kind of problem but I never thought about the "rule" existing anywhere but where the math teacher told me to use it. That probably makes no sense, but it sure has changed how I talk to my one son about it. -- SusanS - 25 Jan 2006
It turns math into something you can trust rather than something that will betray you. I LOVE this observation. This encapsulates exactly what some people love -- and some people hate -- about math. -- CarolynJohnston - 25 Jan 2006
I had no idea fractions were division problems and division problems were fractions. Same here. I find it ironic and quite sad that the some of the parents on this list are saying that they never really understood mathematics back when they were using some other curriculum than the current fuzzy one. The reason I find it both ironic and sad is that apparently this new new curriculum was meant to address this problem .... What this says to me is that it is the teacher's understanding of mathematics and the teacher's ability to appreciate each student's unique misunderstandings that is really important. -- SusanJ - 25 Jan 2006
Check out this web link: http://math.berkeley.edu/~wu/six-topics1.pdf Professor Wu from Berkeley and Professor Milgram from Stanford have written an document outlining an math intervention program for Grades 4-7 for the state of California. -- AnneDwyer - 25 Jan 2006
What this says to me is that it is the teacher's understanding of mathematics and the teacher's ability to appreciate each student's unique misunderstandings that is really important. I agree 100% with this. -- CarolynJohnston - 25 Jan 2006
What this says to me is that it is the teacher's understanding of mathematics and the teacher's ability to appreciate each student's unique misunderstandings that is really important. I think the latter part of this statement is more important than the former. I know that I would make a really horrible math teacher because it would never occur to me in a million years that someone who is perfectly competent at manipulating fractions wouldn't recognize that fractions are just division problems. In order to be able to close the children's gaps, you have to be able to recognize that a gap could exist in a particular spot. And maybe in order to do that, you have to have struggled with it and had your own aha! moment. I have a theory that you don't want the A and A+ students to become the teachers; instead you want the B and B+ students. -- GoogleMaster - 25 Jan 2006
"I have a theory that you don't want the A and A+ students to become the teachers; instead you want the B and B+ students." Unfortunately, K-8 attracts those who probably fared worse. Some actively dislike math and want to do anything that is different than what they had when they were growing up. Speaking of theories, my theory is that K-8 attracts many who want to help special needs kids. Their background and focus is anything but setting and expecting high (explicit content and skills) standards. -- SteveH - 25 Jan 2006
In order to be able to close the children's gaps, you have to be able to recognize that a gap could exist in a particular spot. And maybe in order to do that, you have to have struggled with it and had your own aha! moment. I think this is true. I have turned out to be very good with my LD son because I know exactly what is messing with his head. Even though I never had any LD problems (I always thought I did), the anxiety of not knowing what to do and being too afraid to ask would cause a complete brain shutdown almost daily, particularly if any new material was being introduced. My mathematically gifted son, however, is interesting because he doesn't ever understand what's not to get. Even when he explains to me some shortcut (well, it seems like a shortcut to me) he uses to get to an answer, he can't figure out why I have no idea how he did it. He is a true math head and completely unintimidated. One day I was showing him how Saxon teaches subraction of a mixed number from the number one. I guess that would be subtraction with borrowing but with fractions. Saxon has you convert 1 into a fraction and then proceed. With a whole number higher than 1 you would take a 1, in fraction form, out of the number, add it to your other fraction and then proceed with the problem. He couldn't see why anyone would bother with this. You just subract it. I tried to give him some harder ones, but he could do it in his head with no problem. Eventually, he stared at it and realized that it might be helpful in some way for kids who couldn't do it in their heads. -- SusanS - 25 Jan 2006
Speaking of theories, my theory is that K-8 attracts many who want to help special needs kids. Their background and focus is anything but setting and expecting high (explicit content and skills) standards. But explicit content and skills is necessary for helping special needs kids. (The "But" is directed at the belief, not at SteveH?'s theory). I'm not sure about the high standards bit - my speech just needed bringing up to normal kids' standards. But I wasn't capable of figuring out how to pronounce words correctly on my own, I needed someone to explicitly teach skills. -- TracyW - 26 Jan 2006
I'm thinking we could probably compile an excellent list of sticking points by looking at concepts Saxon covers WAY more than others. I'll be looking out for this as I go along. -- CatherineJohnson - 28 Jan 2006
golly - I hadn't finished reading this thread - lots of good stuff here back shortly - -- CatherineJohnson - 28 Jan 2006
- integers - especially adding & subtracting negative & positive numbers - are a sticking point in my experience
Since I didn't get around to writing the post I'd planned to write, here's something I need in a nutshell: I need for a book and/or teacher to tell me when a concept 'follows naturally' from 'nature' (not one person on the planet will know what that means....) VERSUS when a concept follows logically from the conventions of mathematics = = = = I'll try to explain what I mean. When you start out in math, you're seeing a lot of visual explanations, which, for me, is excellent. "Visual explanations" are probably what I mean by "nature." When you look at a depiction of area, that seems 'natural' — it's something that occurs in physical reality. After I'd seen a number of visual explanations, I developed a 'feeling' that 'understanding' MEANS 'having a visual image.' I felt that pretty strongly. At that stage of the game (a few months ago) I certainly could handle logic; I'd been taught math logic at the h.s. algebra & geometry level years ago. That wasn't the problem. The problem was that 'logic' didn't 'feel' like 'understanding. I have to use the quote marks, because I don't know what I'm saying. All I can do is describe an experience I had, and that I bet lots of students have. SO here's what happened next. I thought I should have a visual image of reciprocals I was extremely bewildered by reciprocals. No problem procedurally; I've been able to divide fractions since I was 11. I also did acquire some workable conceptual knowledge, back then, of what dividing fractions meant. But I had no idea how reciprocals worked, or why you ended up with them, etc. And I kept thinking I was supposed to 'see it' in my mind's eye. = = = = Here's a recent example of a math concept that follows-from-math instead of appearing-in-nature. This happened to Ed & to Kris, both of whom are math-friendly & good at math (for non-math brains). negative exponents I don't think any of us had ever seen a negative exponent. (Maybe Kris had.) So we had to learn them fast. Saxon has a lesson showing that negative exponents, and an exponent of 0, must follow logically from the way the positive exponent system has been set up. Ed, at first, couldn't grasp this at all. He was probably experiencing the same thing I did with reciprocals. Then, when I showed him the Saxon lesson, it started to sink in — 'sink in' meaning: he started to shift over to 'logic.' I did the same thing with Kris, only I didn't have the book. It worked; she got it. Certainly in elementary math, I've needed - and probably a lot of students need - markers for WHEN WE'RE TALKING ABOUT MATH CONCEPTS WE CAN DRAW and WHEN WE'RE TALKING ABOUT MORE ABSTRACT MATH CONCEPTS THAT FOLLOW FROM MATHEMATICAL LOGIC. That may not be the correct way to put it, but it describes a stumbling block I tripped over fairly well. Today, this doesn't seem to be a problem any more. Somehow, math logic now seems 'natural.' -- CatherineJohnson - 28 Jan 2006
OK, now's the time to weigh in -- this weekend I'm going to try to boil down what everyone had to say here. I'm going to try to extract the MAIN IDEAS. Wish me luck. -- CarolynJohnston - 28 Jan 2006
OH BOY, GOOD LUCK where are you putting the page?? do you want to index it on User's Page? This is an important topic. -- CatherineJohnson - 28 Jan 2006
shortcuts That's the other HUGE sticking point for me, or was. It would have been incredibly useful for me to have my textbooks (or a teacher) tell me: THIS IS A SHORTCUT AND HERE'S HOW WE GOT TO IT. For instance, cross-multiplication. If I had learned why it is you can do cross-multiplication, learning the shortcut would have been great. (I remember when I finally figured out, for myself, why cross-multiplication was true. I had no idea!) RUSSIAN MATH does this all the time. They teach all or most of the shortcuts, but they teach them formally, with justification of each step. Also, they'll ask the student to repeat the sequence and write the property that allows you to do each step. -- CatherineJohnson - 28 Jan 2006
He must have encountered them there, and almost surely also in high school. You need to use negative exponents in order to take derivatives of functions like 1/x, and other powers in denominators Just saw this. I never took derivatives of functions, so there was no need for me to have seen negative exponents. I'm 99.99% certain I never saw them. The fact that Ed doesn't remember them at all seems like he really might not have seen them. I may be overstating it.....it may be more that he couldn't remember seeing them.... -- CatherineJohnson - 28 Jan 2006
If it was a sufficiently proof-centric calculus course, he might never have seen the general formula for differentiating functions 1/xn (which is the same as x-n)). But it's unlikely. I'm a bit incredulous! Does he remember what text he used, or if they even used a text? At Princeton, you never know. -- CarolynJohnston - 28 Jan 2006
(Did you notice I did superscripts that time!??? It' s easy!) -- CarolynJohnston - 28 Jan 2006
"They teach all or most of the shortcuts, but they teach them formally, with justification of each step." Some steps you don't want to know like how to arrive at the quadratic formula. It's like what they say about laws and sausage. I'm curious to know how best to explain why cross-multiplication works. I guess one would start out by explaining that a proportion is a statement that two ratios are equal and then go from there. -- CharlesH - 28 Jan 2006
How did you do the superscripts? Very impressive. -- SusanS - 28 Jan 2006
Susan it's html, like this:
x<SUP>-n</SUP>-- CarolynJohnston - 29 Jan 2006
I'm puzzling over something related to the previous discussions. The cool thing about natural language is that you can immediately understand sentences you've never heard before and you can also make up grammatical sentences you never heard before. (There's even some indication that possibly some kind of monkeys can do something similar with picture cards.) So I'm trying to understand why this same innate understanding doesn't work for mathematics for some of you. To me saying "three times the sum of four plus two" is exactly the same as saying "three times four plus three times two." But you have explained quite clearly that for you, understanding the previous sentence involves going back and spending a lot of time understanding the distributive law and also order of operations. 3*(4+2) = 3*4 + 3*2 So do most of us have a "language brain" but only a few of us have a "math brain" or is there some kind of critical period where if your math brain isn't activated it stops working or what? -- SusanJ - 29 Jan 2006
For me, the distributive law was something I memorised, along with the order of operations, and then simply applied when occasion called for it. (Of course, the teacher showed us how it worked, with actual numbers). Later on I realised the deeper meaning behind it. I think what got me sailing through maths was that my own interest in it was limited to getting good results on tests so not actually understanding why something worked never bothered me. Just memorise it and get back to my book. It was only later on at high school when we got onto proofs that the "why" of mathematics became more interesting to me. And by then I had thoroughly learnt that understanding was not a barrier to doing maths. Why do we do matrix-multiplication that way? Who cares? Memorise the method and if the reason only becomes clear well after the final exam, well hey, I'll have my A anyway! -- TracyW - 29 Jan 2006
x2 Cool. A bit long-winded, but cool. -- SusanS - 29 Jan 2006
Carolyn: I'll try to remember to ask him today.... (Ed on negative exponents). It's entirely possible I misunderstood; maybe he said they were 'vaguely familiar,' something like that.... -- CatherineJohnson - 29 Jan 2006
To me saying "three times the sum of four plus two" is exactly the same as saying "three times four plus three times two. I haven't read the thread carefully (I will!) so I may be missing a step....I think I'm making a different point about the distributive property. Christopher found it incredibly easy to do what you just did, i.e. to translate a verbal express ("3 x's") into mathematical language: 3x. The Phase 4 kids did this at the beginning of the year, and it was SO easy for him, and SO natural, that I thought, OK, we can do this course. Christopher's problem with the distributive property is exactly what Carolyn mentioned: he doesn't recognize it in a new context. The exact same thing happened to me when I was first re-teaching math. heck, I've got notes on this somewhere......I spent hours, literally, trying to figure out some wonky problem SRA Math, grade 6 had. It was one of those problems where you're supposed to figure it out more or less by guess & check.....and I wanted to know why it worked the way it did. I SPENT HOURS. Finally I showed it to my brother-in-law, who's probaby a math brain, and he said, 'That's the distributive property.' I was mortified. I knew the distributive property COLD. Seriously. I knew it from childhood, I knew it from Saxon 6/5, I knew it was the basis of the multiplication algorithm, I knew it as the basis of mental multiplication, I knew it as the basis of the many 'distributive property' problems Saxon has in 6/5. I also had, in high school, learned to use the distributive property in that particular genre of problem, although I hadn't done such a problem in 30 years. I DID NOT RECOGNIZE THE DISTRIBUTIVE PROPERTY IN THAT PROBLEM. -- CatherineJohnson - 29 Jan 2006
One more thing: I think the distributive property is far more difficult to recognize once the factor has been distributed (assuming I'm saying that right). In other words, using the distributive property to factor, or to 'reverse engineer' an expression is harder than using the distributive property to produce that expression in the first place At least, that's been my experience. -- CatherineJohnson - 29 Jan 2006
Catherine, I think this is evidence of stress. Anyone with a degree in engineering from a good school who couldn't recover the meaning of negative exponents with a few minutes of thought is likely under way too much stress. It's definitely not something I'd expect to encounter in sixth grade, however. In my experience, exponential notation is fairly difficult for high school juniors. -- SusanJ - 29 Jan 2006
Charles You asked about the best way to show why cross-multiplication works. I definitely can't answer that! But I can tell you how I figured it out.....and I think it would have 'worked' for me if a book or teacher had demonstrated it directly. Number one: I never had the slightest problem 'accepting' that the reciprocal of a number is the number that, when you multiply the two, gives you a product of 1. I'd be willing to bet that most kids can 'deal' with this in exactly this way: IT'S A RULE. IT'S A DEFINITION. So.....that was great! If reciprocals are simply defined, from the get-go, in this way I had no problem. Number two: I had zero problem, and I'd bet the ranch most kids have zero problem, accepting AND UNDERSTANDING that any number outside zero divided by itself is equal to 1. Number three: this is going to sound absolutely nuts to you, but it's important. It took me quite a long time (meaning an hour — something in there) to figure out that when you cross multiply, it's exactly the same thing as multiplying a number by its reciprocal. So: 1/2 = 1/2 1/2 x 2/1 = 1 cross multiplication: 1 x 2 = 2 x 1 Number four: equivalent fractions, yes 1/2 = 3/6 so: 1/2 x 6/3 = 1 1 x 6 MUST EQUAL 2 x 3, BECAUSE when you multiply a number by a reciprocal you always get 1, AND BECAUSE the only time you get 1 as a quotient is when the numerator & the denominator are the same number = = = = = I hope that's semi-clear. I think for math-savvy people it probably is. For someone who hasn't spent a lot of time around math, I suspect that last step is going to lose people. I think I could probably write it out in a way that made sense to people who are learning math..... -- CatherineJohnson - 29 Jan 2006
quadratic equations I'm ALMOST positive that by the time a student reaches quadratic equations - this is definitely true for me - you no longer need to see how shortcuts have been arrived at. I now 'get' that mathematicians have been able to figure a lot of things out so I don't have to figure them out myself. I'm guessing that the 'demonstration of shortcuts' is most useful in....arithmetic and possibly beginner algebra. -- CatherineJohnson - 29 Jan 2006
It's fascinating to me how different people understand things differently. Here's how I look at cross-multiplication. It's just a shortcut to applying the rule that you can always do the same thing to both sides of an equation when you happen to have only a simple fraction on each side. a/b = c/d Multiply both sides by b: a*b/b = b*c/d
a = b*c/d Multiply both sides by d: a*d = b*c*d/d = b*c -- SusanJ - 29 Jan 2006
rates This probably isn't a 'sticking point,' per se, but I'll put it in. I had literally never heard of ratios as 'rates' until this year. I think the first time I heard of it was one day when Charles brought 'rates' up in the context of talking about ratios. I don't understand rates-as-ratios. I'm trying to think whether I can come up with a good illustration (I should probably look at Saxon...) hmm... Can't remember whether I did or didn't understand this problem, but here it is: If 12 books weigh 20 pounds, how much would 30 books weigh Saxon says: Since 12 books weigh 20 pounds we can write two rates. (a) 12 books/20 pounds (b) 20 pounds/12 books [first of all, I have a problem using the word 'rate' to apply to weight. Since I've always used 'rate' to means time or speed & I think price.....the language here is an obstacle, not a help. I think that's part of the problem.] To find the weight of 30 books, we could multiply 30 books by rate (b). 30
I DID NOT RECOGNIZE THE DISTRIBUTIVE PROPERTY IN THAT PROBLEM. That's exactly my puzzlement. If I say something like, "The ferret just ate my prune," which is a sentence you've probably never heard, you recognize the meaning immediatetly. Why is math different? I have absolutely no idea but I think it is a fascinating question. BTW, I've never really thought about factoring as a reverse application of the distributive law. There's actually lots of different types of factoring. -- SusanJ - 29 Jan 2006
Susan J yes, definitely: it's a shortcut to applying the rule that you can do the same thing to both sides.....BUT — and here's where the hyperspecificity of the novice comes in — to 'see' that I need a problem with a variable! 2/x = 4/8 If I had been trying to explain to myself why cross multiplication works with this problem, I might have figured out that it was 'do the same thing to both sides' at once! Too bad I didn't try that experiment. But since I was trying to see why cross multiplication is always true in every case, I didn't go at it that way. hyperspecificity is a major, major feature of beginner knowledge. Hyperspecificity and fragmentation (which may be the same thing). -- CatherineJohnson - 29 Jan 2006
I've never really thought about factoring as a reverse application of the distributive law Isn't that how it's taught (that particular form of factoring, of course). I've always been taught that when you factor an expression like: 3x + 12y you're using the distributive property 3x + 12y doesn't look like the distributive property because students are always taught that the distributive property is this: 3 (x + 4y) "Inversing" the distributive property is a HUGE sticking point. Prentice-Hall — and this is something I think the book does right — has the kids 'inversing' the distributive property from the get-go. Even that didn't work. What's more, the distributive property in 'pre-algebra' (beginner algebra) is so mystifying to kids THEY KNOW IT'S MYSTIFYING. I was driving Christopher & his friend Joe around one day, and the subject of the next math test came up, and Joe said, matter-of-factly, "I don't understand the distributive property." -- CatherineJohnson - 29 Jan 2006
Somehow, I don't think that 'inversing' the distributive property is the same thing as shifting from active to passive voice, or whatever the equivalent would be.... It CERTAINLY doesn't come naturally. -- CatherineJohnson - 29 Jan 2006
Factor 3x + 12y This is a two-step problem. Somewhere else we had discussed the difficulty of two-step problems. Could that be the difficulty here? The first step requires factoring each term individually. 3x = 3*x 12y = 3*2*2*y The second step requires checking each term to see whether they have any common factors which in this case they do. So now you pull out the common factor, 3, and put the expression back together. 3*(x + 2*2*y) = 3*(x + 4y) (Or maybe it's three steps since typically you re-multiply any factors that weren't pulled out.) -- SusanJ - 29 Jan 2006
Let me state this more forcefully. Factoring is process. This process is not necessarily the inversing of the distributive law, which is a definition. The distributive law simply says that a multiplier (or divisor) acts upon each term. So 3*(2x + 4y) MEANS 3*2x + 3*4y which can also be written as 6x + 12y. The instruction to "factor" almost always means to pull out the largest common factor. So when you factor according to this rule 6x+12y = 6*(x+2y) which is not simply the inversing of the previous example of the distributive law! -- SusanJ - 29 Jan 2006
hmmm....this is getting a little over my head..... I guess I'm using the word 'inversing' colloquially, to mean 'doing the reverse of what you did to get to the expression you currently have'....is that wrong? This reminds me of Carolyn's post; I'll go find it. -- CatherineJohnson - 30 Jan 2006
here it is! understanding basic algebra moves What I want to show him is the most general way to do it -- that is, to recognize that what he must do is undo the division of w by 2.1, by multiplying both sides of the equation by 2.1. It's the basic trick of algebra; you solve for something by undoing what's been done to it, remembering that anything you can do to an expression in an equation is okay as long as you do it to both sides of the equation. Christopher also wouldn't recognize how to factor: 3x +3y He wouldn't perceive that the distributive property is involved here. He will, at this point, I think, perceive that he should think 'distributive property' when he sees this expression: 3(x + y) He won't (maybe he does now, but he certainly didn't a couple of months ago) perceive that he should think 'distributive property' when he sees 3x + 3y That's what happened to me, struggling with the SRA problem. I simply failed to recognize that the distributive property was involved in any way, shape, or form. The INSTANT my brother-in-law pointed it out, I felt like an idiot, because in fact I had learned this long ago, and I still 'knew' it — and even understood it......to some degree I didn't perceive it -- CatherineJohnson - 30 Jan 2006
One more item for the list. The Doug Carnine book on Designing DI Math Curricula says that decimal division is the most difficult of the decimal operations. No idea how much of a sticking point this is, but it's worth knowing. -- CatherineJohnson - 30 Jan 2006
Catherine, I understood what you meant by "inversing." However, factoring, which is the proper term, is typically more complex than the example you gave. Understanding the distributive law is only part of what you need to know. Factoring is a complex, step-wise process. I'm sorry I didn't note this "gap" when you mentioned this earlier. Here are two examples of factoring to illustrate the process. Problem 1. Factor 14x + 21y Step 1. Fully factor each term 2*7*x + 7*3*y Step 2. Pull out any common factors 7*(2x + 3y) Problem 2. Factor 14x + 21x^2 Step 1. Fully factor each term 2*7*x + 7*3*x*x Step 2. Pull out any common factors 7x(2 + 3x) You will note that in neither problem does the multiplicative factor in front the parentheses in the answer appear directly in the original expression. -- SusanJ - 30 Jan 2006
The Doug Carnine book on Designing DI Math Curricula says that decimal division is the most difficult of the decimal operations. Decimal long division? I don't know about that. Once you know how to do long division, it's easy -- just a matter of rearranging the decimals' positions. Whereas in decimal multiplication, I'm finding that Ben has a huge tendency to put the decimal in the position it would be in if he were adding -- i.e. to line up the decimals and then put the decimal in the same place in the result. This is happening mostly because he's trying not to think at all. Ben would like to learn the whole of elementary math to pure automaticity. My little computer. -- CarolynJohnston - 30 Jan 2006
I don't remember finding it hard.....and I don't remember Christopher finding it hard....(but I could be just not remembering). If Christopher had had a HUGE struggle with it, I'd remember. -- CatherineJohnson - 30 Jan 2006
Actually, I would have said exactly the same thing (and I still do): decimal multiplication is always harder for me. Probably for Christopher, too. I sometimes get confused about how many 'zero's' I actually have, especially with a problem like.....oh gosh, I don't know if I can think of one that actually would illustrate the confusion. The next time I do one, I'll remember to write it down. -- CatherineJohnson - 30 Jan 2006
Understanding the distributive law is only part of what you need to know right, I think I understand that all I'm trying to say is that perceiving that the distributive property is involved in a factoring 'situation' is a sticking point..... a big one -- CatherineJohnson - 30 Jan 2006
Catherine, I think it is a "sticking point" mostly because that is not a natural way of looking at it. Understanding the distributive law is of little help in understanding factoring. In my opinion the "sticking point" is more likely that kids aren't taught the process for factoring individual items to mastery. Note that factoring individual items has nothing to do with the distributive law. They should start with factoring integers. Let me quote something from an old algebra book. "The factors of a product are those quantities which when multiplied together produce that product. A prime quantity is one which has no factors except itself and 1. In all work in factoring prime factors are required. For instance, the factors of 24 are 4 and 6 or 2 and 12 or 3 and 8, but these sets of factors are not prime, since 6, 12, and 8 can all be factored further. The prime factors of 24 are 2, 2, 2, and 3." Factoring is another example of something that is difficult if you don't know your multiplication facts to automaticity. -- SusanJ - 30 Jan 2006
Christopher finds factoring of individual numbers easy, interestingly enough, and I think Carolyn said Ben's the same way. He's also got his multiplication facts down cold. The Prentice-Hall Book opens with a chapter on the properties, which included problems like the one I posted above (I think — I'll check). The kids just couldn't do them (and I think it was lots of the kids, not just Christopher). These were problems on the distributive property specifically. I personally can factor up a storm; RUSSIAN MATH has a fabulous opening chapter on factoring. But I couldn't for the life of me see that the distributive property was involved in the SRA Math problem. -- CatherineJohnson - 30 Jan 2006
But I couldn't for the life of me see that the distributive property was involved in the SRA Math problem. This is good, not bad. Your intuition was correct! The distributive property is a law. Factoring is a process. Trying to make kids think that understanding the distributive property will help them with factoring is wrong from a mathematical standpoint. Can Christopher factor items with exponents such as x^2 or y^3*z^4? x^2 = x*x y^3*z^4 = y*y*y*z*z*z*z What about items with both numbers and letters such as 12x^2? 12x^2 = 2*2*3*x*x If Christopher is given a list of factored items can he find the factors that are in common to all the items? What are the common factors in x*y*3*7 and x*z*2*3? Answer, the common factors are x and 3. You might want to circle the common factors. -- SusanJ - 30 Jan 2006
The distributive property is a law. Factoring is a process. Trying to make kids think that understanding the distributive property will help them with factoring is wrong from a mathematical standpoint. Factoring algebraic expressions is a process in which an understanding of the distributive law is fundamental. Knowing when to apply the law is necessary, but not sufficient, for factoring. It's the second step in factoring; the first step is being able to find the common factors, as you are saying. -- CarolynJohnston - 30 Jan 2006
Carolyn, I (of course) agree that using the distributive property properly is the second step in writing a factored expression when you are simply factoring out a common factor from a sum of two or more terms. What I object to is the idea that knowing the distributive law means that you know all you need to know to be able to factor an expression like 6a^3*x^4 - 15a^2*x^2*y. My intuition is that understanding factoring is much, much harder than understanding the distributive law. It's easy to go in either direction when you write a*(b+c) = a*b + a*c but most factoring problems aren't in that form. I'm also concerned about laying the proper groundwork for factoring expressions like x^2 - xy - 2y^2. -- SusanJ - 30 Jan 2006
| WebLogForm | |
|---|---|
| Title: | sticking points in math education |
| TopicType: | WebLog |
| SubjectArea: | ParentsTeachingKids, TeachersTeachingKids |
| LogDate: | 200601242222 |