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http://sg.news.yahoo.com/cna/20091010/tap-464-parents-arms-psle-mathematics-pa-231650b.html
Shall we discuss our solutions for the problem included in the above article:
Jim bought some chocolates and gave half of it to Ken. Ken bought some sweets and gave half of it to Jim. Jim ate 12 sweets and Ken ate 18 chocolates. The ratio of Jim’s sweets to chocolates became 1:7 and the ratio of Ken’s sweets to chocolates became 1:4. How many sweets did Ken buy?
Patsy
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32 comments:
Try the following link, as I couldn't see the end of the previous one:
http://tinyurl.com/PSLE2009
Without having read the article:
Let 2c = the number of chocolates Jim bought, and let 2s = the number of sweets Ken bought. After Jim and Ken each shared half of their respective chocolates and sweets with the other, each had c chocolates and s sweets. After Jim ate 12 sweets, his ratio of sweets to chocolate was (s - 12) / c = 1/7. After Ken ate 12 chocolates, his ratio of sweets to chocolate was s / (c - 18) = 1/4.
Jim:
(s - 12) / c = 1/7
7*(s - 12) = c
Ken:
s / (c - 18) = 1/4
4s = c - 18
c = 4s + 18
7s - 84 = 4s + 18
3s = 102
s = 34
Ken bought 2s = 2*34 = 68 sweets.
(And Jim bought 2c = 2*(4s + 18) = 2*(136 + 18) = 2*(154) = 208 chocolates.)
Can you solve it arithmetically? ;->
Ahhh, ratio change problems. Those were my bane in primary school (I eventually mastered them) but they're not new. In fact I call this a "classic PSLE problem" (5 points).
"Jim bought some chocolates and gave half of it to Ken. Ken bought some sweets and gave half of it to Jim."
The trick here is really reading comprehension. The quantities are symmetrical (and if you didn't read carefully, you'd be tricked into thinking the problem is much harder than it actually is).
Patsy: isn't this arithmetic?
It's not algebra as there are no proofs about general properties, etc. or about *all* sweets or *all* even quantities of chocolates.
I would define a problem that requires the solution of two equations with two unknown as "algebra". (FWIW)
It's not algebra until there are general solutions.
(This is what makes most NP-complete problems not algebra, because general solutions are computationally-difficult.)
For example, consider all polynomial equations above degree 4 (quintic equations, etc.)
You can solve them by manipulating the terms; sometimes solutions exist (depending on the equation), but as Abel and Ruffini demonstrated (and later elucidated by my namesake Evariste Galois), there are no algebraic solutions to quintic equations.
Equation solving (even with more than 2 unknowns) isn't really equivalent to algebra.
Many cryptographic techniques are based on the idea of a lack of an algebraic solution, such that it is easy to verify a specific solution but hard to locate one. (Unless of course, you have the proper key, hash, password, etc.)
Thus it's quite important to distinguish equation solving (complicated as it may be -- try public-key cryptography for size!) and algebra, because it is this division that allows information security to exist and prevents crackers from trying to eavesdrop on your electronic bank data, etc.
"Equation solving (even with more than 2 unknowns) isn't really equivalent to algebra."
I never made a claim of equivalence. I made, and reiterate, a claim of set membership.
To support this claim, I offer:
1) Webster's New Collegiate Dictionary, definition 1a - "a generalization of arithmetic in which letters representing numbers are combined according to the rules of arithmetic"
2) Compact OED, definition 2* - "The department of mathematics which investigates the nature and properties or numbers and properties by means of general symbols; and, in a more abstract sense, a calculus of symbols combining according to certain defined laws."
Further, solution of two equations in two unknowns is a core skill taught in Elementary Algebra, and such problems are understood by nearly everybody to be algebra. Words mean what people regularly and intentionally use them to mean.
I'll not argue that there is no sense in which your claim is correct; I don't have the background to make that claim. I will say that in any non-specialized English dialect this problem is algebra.
* Definition 1 is both interesting and surprising to me: "The surgical treatment of fractures: bonesetting."
le radical galoisien wrote:
>It's not algebra as there are no proofs about general properties, etc. or about *all* sweets or *all* even quantities of chocolates.
But, l-r-g, by your critierion, algebra is not taught in US schools at all!
I empathize: I really think of what the US schools teach as “pre-algebra”: when you get into understanding what your own namesake worked out, now *that’s* algebra.
On the other hand, I think the best way to understand Galois theory is to start doing “arithmetic” with symmetry groups, etc., and get a "feel" for it. Then, you’re ready to start studying the *real* algebra.
Anyway, it is largely semantics. What is not semantics is the question of whether an educated person should actually learn what used to be called “the theory of equations”: symmetric functions and Newton’s identities, Galois theory, resultants, etc.
I think so, but very few Americans are going to agree. I mean, what use is it? (It is useful, of course, in practical engineering applications, but few people know that either.)
Dave
If you mean Group Theory, then say Group Theory.
If you mean the Algebra that is Group Theory, Number theory, Algebraic Number Theory, and the like, then we're not talking about K-12 anyway. So don't use the word interchangeably on this blog, as it doesn't serve to move the conversation forward.
lrg, your claims about computer science, though, are odder. When I have more time, I will address this odd notion of NP completeness that you seem to have.
Number theory is used at the junior and senior international math olympiad (ages 13-16?) I do believe...
There it's hard to often come up with an algebraic solution for a given problem (at least in my experience it was all very case-dependent and highly arithmetic and specific).
Should I say, this was the Singapore Math Olympiad (yeah I never made it to internationals =( )
lrg: "Number theory is used at the junior and senior international math olympiad (ages 13-16?) I do believe..."
Sure, but that's something that in the US only a few "math geeks" would ever mess with. And you admit there is another term for this ("number theory") than algebra. My college's math department would call it "real analysis."
If you ask 99% of folks in the US to define algebra, their definition will involve solving for "x", and certainly the mathematics that is used throughout the physical sciences, which is usually described as algebra, is solving linear and non-linear equations. And until my students come to college with a better understanding of those basics, I'm not going to worry too much about their ability to do number theory!
In the US, the only kids who do number theory as teens go to math camp.
And there's no reason not to use the words "number theory" and "group theory" to describe it then.
I like it when the conversations here are englightening; us all getting in the weeds about what constitutes algebra is not helpful for parents trying to figure out what their children need to know; the parents who already know what abstract algebra is don't need this discussion; the ones who don't will just get confused.
that said, my real analysis was absolutely NOT my algebra course. sigh sigh sigh.
In the US, the only kids who do number theory as teens go to math camp.
And there's no reason not to use the words "number theory" and "group theory" to describe it then.
I like it when the conversations here are englightening; us all getting in the weeds about what constitutes algebra is not helpful for parents trying to figure out what their children need to know; the parents who already know what abstract algebra is don't need this discussion; the ones who don't will just get confused.
that said, my real analysis was absolutely NOT my algebra course. sigh sigh sigh.
yeah, but an example of a junior question I do believe, is stuff like
calculating (1/2)+(2/3)+(3/4) ... (1999/2000)
or
1996^1997 + 1997^1996 + 1998^1995...
The difference from school arithmetic I suppose is that you're allowed to use calculators.
(Calculators don't go past 10^100 or 10^150 so something like 1996^1997 would kill..).
It's sorta like what goes on in US high school math competitions, but just more intense.
actually in high school math competitions in the US you do write out algebraic solutions to things so actually...
(Well those are the "easy" questions that seem straightforward -- later on you're supposed to find some special numbers with certain properties that you imagine would be hard to apply to other numbers.
Allison,
On another thread, I mentioned Chrystal's “Textbook of Algebra: An Elementary Textbook for the Higher Classes of Secondary Schools and for Colleges,” a book originally published in 1889 and updated in 1900 and 1904. Have you ever looked at that?
It seems that a century ago, advanced high-school algebra covered stuff that is simply not learned by most people today at all – not “modern algebra” a la van der Waerden, Jacobson, etc.. but also not current high-school algebra.
So, there is a real question of substance here: should high-school algebra be restricted to the very thin gruel that is now taught, or was Chrystal right over a century ago that high-school kids should and can learn a lot more.
Dave
Using Singapore's bar methods, you can solve this problem without algebra. I just didn't post the bar diagram solution before b/c there was no easy way to draw the bars proportionately on the computer. So let's just pretend my bars are equal =).
The diagram:
|----Chocolates----|Sweet|
|````````````````|`````|
|________________|_____|
|__|_|__|_|__|_|_|_|+12| => B1
|````````````````|`````|
|________________|_____|
|cccccccccccccccc|ssssss| => B2
|````````````````|`````|
|________________|_____|
|+18|__|___|__|__|_____| => B3
|````````````````|`````|
B1: Jim, final (pretend each unit |_| is equal in length)
B2: Initial
B3: Ken, final (pretend each unit |_| is equal in length)
From the diagram:
Jim has a total of 8 equal units, 4 chocolate and 1 sweets.
Ken has a total of 5 equal units, 7 chocolate and 1 sweets.
One of Ken's units equals one of Jim's plus 12 => K = J + 12.
Seven of Jim's units equals four of Ken's units plus 18 => 7J = 4K + 18.
From my Primary Mathematics Challenging Word Problems 6 (CWP6), it seems that in Singapore by the time of the PSLE they are taught to write simple equations in terms of an unknown, and they can substitute a number into that equation and evaluate the result. So, I think it is reasonable, for a challenging problem, to assume that some students would be able to combine those two skills, and do a simple substitution.
So, 7J = 4(J + 12) + 18 =>
7J = 4J + 48 + 18 =>
3J = 66 =>
J = 22
So, K = J + 12 =>
K = 34
So, one of Ken's units is equal to 34, and, since Ken had one unit's worth of sweets to begin with, he originally bought two units of sweets total, which is 68 sweets.
It requires a bit of an extension of what they are taught, but nothing too unreasonable, imo. There are problems in the CWP6 book that are worse than this one.
Those bar models seem to outlive their usefulness at about book 5, in my opinion. I have kids working through Singapore 5 and 6 and I am steering them towards algebra/arithmetic for solving those types of problems, setting up equations rather than bar models. It is tricky to set up bar models for changing ratios and I don't think it is really worth the effort especially considering the fact that the time perfecting the bar model approach could be time spent learning algebra. (When I say "algebra" I mean setting up equations with variables and solving them, not group theory.)
Those bar models seem to outlive their usefulness at about book 5, in my opinion. I have kids working through Singapore 5 and 6 and I am steering them towards algebra/arithmetic for solving those types of problems, setting up equations rather than bar models.
I think the bar models at this level can be good as an aid to setting up equations, if you are having trouble visualizing the relationships, but less useful in actually solving the problem. My kids aren't that far yet - my oldest is 3yo =) - but I plan to teach them to set up and solve equations for those sorts of problems, too.
Dave,
Just found your comment about Chrystal's Algebra.
My local university library (U of MN) doesn't have it (really, they have a lousy library system.) But Google Books does.
It looks great. To my cursory ten minute reading, it looks like what's standardly called precalculus plus a great deal of what's included in complex variables plus all of the numeric methods that computers eradicated from our lesson plans, and all of those tricks that you learn in physics class from your teacher but never found in any book.
To the bigger issue, I am not arguing that we keep our definitions clear so that we can keep a limited notion of algebra in high school. I'd love to see some heft there. As k-8 is currently constituted, that's of course impossible. Doing what's possible to make that no longer impossible is my goal.
But I wonder if there isn't a bigger institutional reason for the thinness of modern high school curricula: that people used to only have 8 years of schooling, so you had to teach everything they needed to run their lives and their businesses in those 8 years.
Now that we have 13 years, why, like an ideal gas, we've expanded to fill the available space and the density has dropped accordingly.
High school now just delays entry into the workforce. Soon college will do only that as well.
You said somewhere else that we "forget" to discuss how bad college is: what thin gruel it is, too.
I don't forget. I don't bring it up because a) it seems cruel to the parents trying to educate their children so they can succeed at college to dash their hopes about the value of tha college anyway, and b) it just seems off topic, and a big enough off topic to be its own blog.
But here is my short version: 90% of colleges and their majors are a complete waste of time, unless your goal is drinking, drugging and whoring on the taxpayer or daddy and mommy's dime. The therapeutic model of liberal arts, where the disciplines are all "studies" courses rather than the old 7 majors that believed in truth, has undone the academe. What's left is political correctness and victimology. No one is taught to read, write, or reason.
Yet everyone plays the game and wants their kid there anyway. Why? because parents want their children to find appropriate mates, the children want 4 years of free booze and sxe, professors want tenure and need the childrens' approval to get it, and administrators want the cash. The incentives are perverse with no end in sight.
Engineering and science schools are only moderately better; they produce some limited value. But as all non-eng/sci courses are just as postmodern drivel as at the lib arts schools, the student is not being taught to read, write, or reason about ideas. It will take a good engineering student a long time to find out that there were truths in the world of philosophy or history that were worth his while. For many, his own hubris won't be tempered, so he too will go on for quite a while believing that all problems can be solved if the right person were in charge of them. It will take a very long time to figure out that the State is not that person.
I think the project of English speaking peoples can't recover from this decline, so that's the other reason not to discuss this aspect of education. Others may have more optimism.
Allison,
It’s not going to surprise you that I am in sympathy with much of your post: I actually married in to a family of Chinese immigrants, and the difference in attitudes does make credible the common perception that the future belongs to Asia.
On the other hand, my great-grandmother, who was born in 1883 (I knew her through most of my college years since she proved to be quite long-lived), had attitudes not that different from my Asian in-laws: I think she was not untypical of her generation of Americas. So, cultures can change. The West, and America in particular, have shown enormous dynamism in the past – and there are conversations going on all around the country similar to those here at ktm. A lot of Americans are starting to figure out that some things have gone very wrong.
The future may yet surprise us.
I do have a very specific, practical question for you relating to education.
You went to MIT, right? My wife and I graduated from Caltech in the mid-‘70s, and while Caltech has many negative aspects – not much value placed on good teaching, poor support structure for students, very strange student life – the one thing it did very well was to not stand in the way of students’ learning.
You could basically quiz out of any course if you could honestly prove that you already knew the stuff. There was a good chance that you could skip a pre-requisite course and jump into an advanced course even if you could not prove you knew the pre-reqs – it was you who were looking at a possible “F” after all – although you were given fair warning if you wanted to try that.
And, there was almost no busy work in classes. The homework problems might be nearly impossible, but I remember no assignments at all that wasted my time on repetitive, pointless work (and I was one of the better students).
Was MIT similar in these respects?
I ask because my kids are going to be advanced way beyond normal high-school grads, even those with lots of AP credits. It’s not that they are geniuses, but they will have covered a fair amount of the under-grad curriculum (not all, of course) when they enter college. Does MIT work together with students who are advanced like that to maximize what they can actually learn, as Caltech does?
You may have written about this and your MIT experience elsewhere; feel free to direct me to those comments if you have.
By the way, your comments about Mike Artin’s course have convinced me that I need to get hold of his book and go through it: I know bits and pieces of abstract algebra (I hold patents on uses of Galois-field theory in engineering applications), but I never went through a systematic course on it.
I also had to smile at your description of the brilliant lecturer who made everything seem so clear in class – and then you realized you really had no idea how to do the homework! That fit the two classes I took from Dick Feynman to a “T.” (Although I do not regret having taken the classes: he was a great guy, and we knew what we were getting into.)
I hope you can give me some insight into your MIT experiences as I asked above.
All the best,
Dave
forty-two & anon: Those bar models seem to outlive their usefulness at about book 5.
My sons went back to visit their elem. teachers after a couple of months into algebra and immediately told the 6th graders: "You never need too draw a bar model again-Algebra is the best!"
Of course, they don't see that it was easy because they had the Singapore Foundation since 1st grade.
my impression is that singapore is wonderful and amazing...until 6th grade. After 6th grade, the spirit seems to be gone and while students may be prepared for algebra, a book like Jacobs' may be best for actually bringing them there.
Dave,
I was at MIT from 1989 to 1995. I wonder how much has changed in the last 20 years--could be great, could be little.
MIT is a different weird place than Caltech (I had friends who went there as ugrads and grads.) But as of 1995, on the books, they still had a rule that said that you could take a test in ANY course at all on the first day of the term and receive that grade for the course itself.
No less than Bibi Netanyahu once said (I was at a talk of his being given at MIT when I heard him say it) that that rule at MIT had made it the best school in the world; he told a cute story of how he and a few other folks were waiting to take a test on the first day, and one of the students brought out his slide rule. The other students yelled and insulted him until they basically shamed him into taking the test without it. I loved that story--NO CALCULATORS ALLOWED! :)
I never really interacted with any engineering students. I knew kids in physics, math, chemistry, mat sci, and computer science (almost no EEs, even.) In those courses/majors, when I was there, this rule was seldom used by anyone. There were few students encouraged to move on in this way.
That said, there wasn't much need. I had never heard of anyone who was forced to take the pre reqs for a course if they didn't want to. They often made kids retake courses that the kids thought they shouldn't have to: like O Chem, or multivariable calc, or linear algebra. But they were right, and if they really weren't, well, then, you could take that test, couldn't you?
I knew a handful of triple majors, and one quadruple major, who at the time was supposedly the first person to ever be a quad major (math, physics, CS and EE...so highly inter related.) They were all straight A students, and their advisors put up no resistance to them taking 5, 6, 7 courses or more; the quad major routinely took 2 courses given at the same time, but as an A student no one cared.
It was really up the individuals in your department. If some big tenured prof didn't approve of what you were doing, and your advisor did approve but was simply an asst prof, you lost.
But most advisors didn't push you to take interesting courses out of your major, and they didn't much try to sway people out of dumb ideas either--they weren't really involved. And if they hadn't come from MIT, they didn't really understand, so they didn't always give appropriate advice.
But for the good student, they'd probably get out of the way.
I wasn't a good student, so I don't know what "busy work" meant. I needed more preparation than I ever got in every single course I took. I felt at the time and since that I was allowed to pass freshman physics when I should have been flunked. I wanted to retake it, but was discouraged from doing so by my advisor and the dept.
If you intended to go to grad school, you were expected to finish your major early enough that you were taking grad courses by senior year. But this was not said out loud early or often.
That said, I knew a lot of kids who were bad students, and a lot of kids who were excellent students with minor dust ups, and once you fell off the perfect path, you were in the dean's world heck. I knew one student who had straight As until his 1st term sr year. That year, he had no money and bought no textbooks; he took and failed grad quantum after a fiasco where the ugrads were not told it would be C-centered for them.
The dean's office REFUSED to let him take more than 4 courses the next term, thereby removing him from the grad rolls for his senior year. Nothing--not letters from his tenured faculty advisor, his thesis advisor, etc. would change their minds. Similarly, the dean's office did lots of meddling if strange things were happening in your dorm that year, so to speak. Basically, once you entered their attention (Rather than your academic advisor or dept's) it was a crap shoot.
"Just found your comment about Chrystal's Algebra.
My local university library (U of MN) doesn't have it (really, they have a lousy library system.) But Google Books does."
I think it is also available as PDFs here: http://onlinebooks.library.upenn.edu/webbin/book/lookupid?key=olbp36404
-Mark Roulo
My local university library (U of MN) doesn't have it (really, they have a lousy library system.) But Google Books does."
more or less. There's a page with someone's hand superimposed and (I can't make this up) what appears to be a condom on one of the fingers.
Allison,
Thanks for the info about MIT. While Caltech and MIT are different in many ways – e.g., MIT being much larger and part of the Boston academic scene – it does sound as if they are somewhat the same, both for good and ill, in how they deal with students.
You wrote:
>I wasn't a good student, so I don't know what "busy work" meant. I needed more preparation than I ever got in every single course I took.
Well, yeah, I do know that feeling. I was in the upper fraction of my class GPA-wise, but at Caltech they wanted even the better students to feel that there was a lot more to the subject than what we had yet grasped. And, believe me, we certainly got that message.
I once had an oral final with Dick Feynman – I was terrified he would flunk me, since, by my standards, I had not grasped even 25 % of the class. In fact, he actually gave me an A-; it turned out that he knew we were not fully grasping the material. He was merely trying to give us a flavor for the subject, and wanted to see if we were paying attention and trying to understand. I was indeed paying attention, even if I felt that I was failing more often than succeeding at actually understanding. I’m more proud of that A- than of any A+ I ever got.
One thing I did learn from Caltech was the difference between “passing the class” or even “getting an A+ in the class” vs. *truly* understanding the material.
Maybe that was a big part of what they were trying to teach.
I should make clear, by the way, that I only really quizzed out of one class – freshman calculus (this was less common back then than it is now). I also took the two classes with Feynman a year ahead of when I would have “officially” taken them, but that was fairly common with those two courses.
Similarly, if my kids should go to Caltech or MIT, I’m not sure that they will quiz out of lots of classes: in a sense, simply getting in to either school means you are automatically taking more advanced, demanding classes. But the fact that students *can* quiz out of classes does show the schools’ commitment to letting a student learn to the maximum level, and that is a sort of guarantee even if most of us did not find the need to take advantage of that.
I did have a friend who entered graduate school in math at Caltech at age fifteen (while simultaneously taking undergrad courses in other subjects to fill the breadth requirements for his bachelor’s). Obviously, his quizzing out of the whole undergrad math curriculum put him way beyond my level in math (or where I expect my kids to be), but, again, the fact that Caltech would let him do that did show their willingness to recognize exceptional ability where it was clearly proven.
Even many elite schools are not like this. My kid brother did has bachelor’s at Stanford, where I also went to grad school, and both the undergrad classes there and the intro grad classes struck me as being like high school: “show your work” (even if you had the right answer and clearly understood how to do it); drill on plugging numerical values into simple formulae; etc.
So, I do think the MIT/Caltech approach is superior in that regard, even if Caltech (and, from what you have said, MIT) also has some serious pedagogical shortcomings in other respects.
Anyway, thanks for your information about MIT: one of my kids has been thinking about MIT from an early age, so I thought maybe I should find out what I could about how it actually works.
Dave
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