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rgold
Trad climber
Poughkeepsie, NY
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Jul 27, 2014 - 11:51pm PT
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John, I think one of the problems with upper-level math courses for Math Ed. majors is that the courses were, by and large, originally designed to prepare the students for graduate study leading to research careers in mathematics, and they haven't evolved to reflect the fact that the audience is not going on and will be teaching things far less complex.
Personally, I would be and have been against "diluting" the content, but I do think one ought to ask why secondary teachers should know these things and then make sure the course illuminates the importance and value of the content for that audience. That is a different argument than should they or shouldn't they take this course I think.
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JEleazarian
Trad climber
Fresno CA
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Jul 27, 2014 - 11:53pm PT
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I had a reputation, in my household, of trying to lecture on the subject of problems rather than "just doing it" from both Debbie and our daughter.
Thank you for that comment, Ed. I had the same reputation with my two daughters. Now one is a math teacher married to a math teacher, and the other heading toward a doctorate in music (as she puts it, so Adele and I can have a purpose for the rest of our lives, i.e. supporting her).
And Rich, thank you for your insightful comment about difficulty. I guess it should not surprise us that climbers -- who tend to push things to or beyond the point where they challenge us -- would do the same with academics.
John
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rgold
Trad climber
Poughkeepsie, NY
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Jul 28, 2014 - 12:11am PT
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I guess it should not surprise us that climbers -- who tend to push things to or beyond the point where they challenge us -- would do the same with academics.
Yes, but my point is that the interest in and pursuit of difficulty seems to be a fundamental human trait, not at all restricted to climbers but observable in a vast range of human activities.
The flip side is that if we are searching for something hard to do, that search is conditioned by some inner self-confidence that we will, some of the time, succeed. And so we gravitate to this or that activity, not because it is easy, but because it is hard but we have some sense that we can manage it anyway. The mystery of human nature is that we get pleasure from succeeding at hard things.
If you buy this, then one of the jobs of education is to help to build that sense of possibility in the face of difficulty. We do this through subject matter of course, but how exactly does the subject matter, which is important in its own right, also become a vehicle for conveying that expanding range of strategies analogous to what climbers acquire?
These musings, which in one sense are the idle ramblings of an old man trying to make sense of what he's spent his life on, are still not far from the original topic. There is a lot more going on, and a lot more at stake, than the most efficient rote way to subtract one integer from another.
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jgill
Boulder climber
Colorado
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Jul 28, 2014 - 10:47am PT
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John, I think one of the problems with upper-level math courses for Math Ed. majors is that the courses were, by and large, originally designed to prepare the students for graduate study leading to research careers in mathematics, and they haven't evolved to reflect the fact that the audience is not going on and will be teaching things far less complex
To some extent I would agree, Rich. However, I have always felt that one should be knowledgable of material a minimum of one step above one's teaching level, and in this case I would argue that a HS math teacher who might be expected to teach an elementary calculus course should be aware of the foundations of real analysis and the theory behind elementary calculus (e.g., theory supporting the Riemann integral and the Fundamental Theorem) which is the direction I steered AC, rather than simply more "regular" calculus ( we had a junior-level course for that). Although AC is a traditional stepping stone for grad work, it also provides a foundation for secondary school instruction.
However, topology and complex variables are a bit of a stretch in this regard.
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JEleazarian
Trad climber
Fresno CA
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Jul 28, 2014 - 01:16pm PT
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Yes, but my point is that the interest in and pursuit of difficulty seems to be a fundamental human trait, not at all restricted to climbers but observable in a vast range of human activities. . . . If you buy this, then one of the jobs of education is to help to build that sense of possibility in the face of difficulty.
Rich, you make an excellent point. I've known too many students who didn't make the connection between their ability to overcome initial difficulty in one area of their life, and doing the same thing in other areas of their lives. Your post made me realize that I focused a lot of my effort particularly one-on-one with individual students, helping them to make that same connection. One of my greatest joys is when former students report back on the difference that made.
I made my post about climbers because most I know already made that connection, and recognize their attraction to difficulty.
John
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yanqui
climber
Balcarce, Argentina
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Jul 28, 2014 - 07:24pm PT
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The question posed in the OP is interesting. In recent global rankings the US does poorly compared to other countries in terms of the math skills of high school students (something like 30th in the most recent PISA rankings I'm aware of). The question becomes an exercise in cultural comparison: what differences are there between the places in this list http://www.businessinsider.com/pisa-rankings-2013-12 with scores over 500 and the US, that leads to high school students who have significantly better math skills. Since the US already spends more money than most countries in education, just throwing more money at the problem doesn't seem to be the simple solution. One possible factor is aleady considered in PISA study: relatively speaking US students don't think math is very important. I imagine this is a general cultural phenomenon and (if so) won't be easily changed just by modifying the way math is taught in schools: New Math, etc., all by itself, won't fix the problem.
When it comes to the differences in the way math is taught in (at least some of) the countries which perform better, I thought the book "Count Down: The Race for Beautiful Solutions at the International Mathematical Olympiad" gave an interesting possible comparison. The author suggests that instead of teaching math as a series of formal step-by-step procedures that must be mastered in succesive increments (the US method of education) math can be taught in terms of a problem solving approach: directing students to attack an assortment of challenging problems which can be solved by a variety of methods. I don't know for sure if this characterization is really spot-on, at least in terms of getting to the root of an important cultural difference, but it does jive with the following article from NPR:
http://www.npr.org/blogs/health/2012/11/12/164793058/struggle-for-smarts-how-eastern-and-western-cultures-tackle-learning
Math is certainly difficult, challenging and frustrating. I think the most important ingredient that is missing with respect to US (and Argentine) education is the push (the drive) to learn the subject that comes from the student. I'm not very convinced that a bureaucratic top-down curriculum change can provide the solution. But you never know!
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jgill
Boulder climber
Colorado
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Jul 28, 2014 - 07:40pm PT
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relatively speaking US students don't think math is very important
It's fashionable to joke, "I never could understand math!"
A laughing matter, even among sophisticated adults.
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wbw
Trad climber
'cross the great divide
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Topic Author's Reply - Aug 4, 2014 - 09:49am PT
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I went and had a look at the common core standard for math, and I’m truly puzzled by the responses to them, or at least to the sample I looked at. I can’t imagine any informed scientist arguing against them. They seem to me to propose basic topics for understanding which are really beyond question.
So while you folks were thinking about why we suck, I've just returned from a a very rainy week at your very lovely Iceberg Lake. Interesting responses; I thought it would just be a cursory and predictable "Americans suck" comment from Werner, and that would be the end of this thread.
To return back to the article I originally read, (Ed, my original post was in response to the NYT article you cite), I'll tell you what I have thus far experienced with the Common Core as a teacher.
*The other countries, in particular in Asia and northern Europe, do better at math because their curriculums are not the traditional inch-deep-and-mile-wide of those used in the US. Because these other curriculums give teachers a chance to explore fewer topics more deeply, the students in these countries have a deeper and more profound understanding of mathematics. As a teacher, I am personally all over this idea. I crave having that time with my students so that we can explore the greater depth where the beauty of mathematics truly exists.
Reality of the Common Core: The number of topics to be taught may have been consolidated on paper, but teachers are required to teach at at least as much content as before the Common Core. In Algebra 2, for example, a few minor subjects were removed from the traditional curriculum, and then a whole unit of statistics that includes topics one would encounter in a full stats course were added. Net result: more stuff for students to study at a superficial level. This is personally my single biggest pet peeve about the lie that is the Common Core.
**My school district is on a mission to educate parents about the Common Core. The district math dept. sends out frequent emails attempting to help parents learn how to support their kids with the new curriculum, and warning them that for a period of time they might see their kid's achievement in mathematics decrease. Part of the flaw of the Common Core is that at the elementary level, kids must show many ways to solve a problem.
Reality of the common core: Because of the requirement that a problem or operation be solved (or carried out) in many different ways, which is certainly desirable in a theoretical sense, methods that are absolutely insane (inefficient and incomprehensible, even to mathematically confident parents) are employed that confuse everyone. I was told by my daughter's 4th grade math teacher, who was not confident in her own mathematics, that traditional methods for addition and division did not yield true understanding of those operations. The Common Core methods might strive for understanding, but the true result is that some of the methods used in elementary school confuse everybody, including the teacher! As a high school math teacher, I am often hindered in my attempts to teach concept, by students' inability to accurately calculate. Math operations are tools for engaging in math. Turning them into concepts that have to be looked at in multiple ways by all students in the class simply has the effect of more mistakes being made in the class. Ironically when that would happen, my daughter's teacher would simply take the class to a computer lab for a session of mind-numbing drill-and-kill.
A big part of implementing the Common Core in mathematics is the use of technology. I was on the committee for adopting new textbooks in my district a couple of years ago. I reviewed a lot of textbooks and supporting technology by different publishers of textbooks. We spent
$1.4 million on new textbooks for the district, a substantial portion of which was spent on technology support that came with textbooks. Most of the technology that I saw was really targeted to make teachers' lives easier (for example providing a pre-generated multiple choice test that the teacher does not have to grade).
Reality of the Common Core: The new wave of textbooks written to support the Common Core look very similar to the old textbooks in many cases, and the technology that is provided to support greater understanding on the part of students, is as far as I can tell an attempt to sell teachers on the Common Core by reducing the time they spend on assessing student understanding.
My point of the original post was not to get into the discussion of pay for teachers, or tenure for teachers. My point is that the Common Core is not the answer to greater and deeper mathematical understanding on the part of students. The Common Core is a lie, conceived by politicians to be one thing, and then developed by the educational establishment to be something completely different.
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rgold
Trad climber
Poughkeepsie, NY
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Because of the requirement that a problem or operation be solved (or carried out) in many different ways, which is certainly desirable in a theoretical sense, methods that are absolutely insane (inefficient and incomprehensible, even to mathematically confident parents) are employed that confuse everyone. I was told by my daughter's 4th grade math teacher, who was not confident in her own mathematics, that traditional methods for addition and division did not yield true understanding of those operations. The Common Core methods might strive for understanding, but the true result is that some of the methods used in elementary school confuse everybody, including the teacher! As a high school math teacher, I am often hindered in my attempts to teach concept, by students' inability to accurately calculate. Math operations are tools for engaging in math. Turning them into concepts that have to be looked at in multiple ways by all students in the class simply has the effect of more mistakes being made in the class. Ironically when that would happen, my daughter's teacher would simply take the class to a computer lab for a session of mind-numbing drill-and-kill.
You begin your response with a quote from me which you never actually argue against, so I assume this means you agree with it. You appear to agree that fundamental intellectual strategies such as exploring multiple approaches to a problem are desirable "in principle." So it does seem that, as usual in such things, the devil is in the implementation.
Other pieces I've read have argued forcefully that the implementation phases of at least some Common Core programs were carried out without appropriate collaboration with experienced and knowledgeable classroom teachers.
You also make what I think is a critical observation, which is that progress at "higher" levels requires facility with the routines used at "lower" levels, otherwise the learner's intellectual edifice never rises above ground level because they are busy reinventing the wheel every time they need to take a journey. At some point you have to be able to jump in the car and drive away without mentally recapitulating the entire theory of the internal combustion engine.
So on the one hand we do want our children to be facile with the basic operations of arithmetic and algebra, but on the other hand we don't want them to have learned them in a manner so mindless that they don't understand which operation to choose and what the scope and limits of the operations are, something that is for a very large contingent of students a sad reality. And make no mistake---the consequences of this sad reality is that arithmetic and algebra remain forever in a quarantined compartment of the mind, unavailable for any application that wasn't on the exam, which is to say almost everything.
We as a society and we as individual parents are fully justified in hoping for much more than this sad reality, which as I described it is no more than basic "clerical" competence in the realm of mathematics and its associated domains. We surely ought not to be satisfied with less, and that includes the status quo.
If we could ever get down to the essence of the matter, I would say that mathematics and science have to be taught as subjects that are accessible to human reason, because they are the product of human reason. If this sounds obvious, it has not, in general, worked out that way.
The danger of reactions to the Common Core is that they end up throwing out the baby with the bath, and push us back to the bad old days when the "mind-numbing drill-and-kill" referred to above was in fact the entire content of elementary mathematics schooling. In today's politicized atmosphere, there are plenty of voices advocating for just such a regression. They may even win, but it won't be any kind of victory for our children.
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jgill
Boulder climber
Colorado
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Much of what is said here assumes all students are capable of developing or enhancing skills in critical thinking. As a retired college teacher my impression was students seemed either to have these skills or not when they reach college age. Those of you posting here might address this issue regarding elementary/secondary-age students. Nature or nurture? Both probably but in varying degrees. Like mathematical or musical ability, is there a latent talent in critical thinking?
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JLP
Social climber
The internet
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I've skimmed the thread and believe most of you are missing the real problem entirely.
Much of what is said here assumes all students are capable of developing or enhancing skills in critical thinking.
Yes - what kind of students are showing up to class these days, where did they come from, what are their examples at home?
My take is simple - more and more we are breeding a lazy, dull, consumption driven youth. The kids would rather Facebook and eat Twinkies than actually have to think. Our systems promote and reward a deep need for instant gratification and consumption. The strain of learning math isn't compatible with these systems.
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wbw
Trad climber
'cross the great divide
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Topic Author's Reply - Aug 4, 2014 - 03:37pm PT
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Rgold, I'm not against a national curriculum or national standards. I guess if pressed, I do agree with you that it is the implementation of the Common Core that is problematic. Just out of curiosity, what was it in the Common Core that impressed you that wasn't previously written into other math curriculums? What part of the Common Core is your baby in the bathwater?
I am however, against a national curriculum that is supposed to accomplish reform by drastically changing the old curriculum, when in fact it is simply more of the same: too much to teach/learn in a meaningful way in the allotted time.
Common Core mathematics, as I understand it was the result of a commitment by politicians to produce a new national curriculum, that would allow teachers the time to actually get kids thinking about mathematics in a more meaningful way. Predictably, "educators" couldn't resist writing so much content into the curriculum, so then it became this thing that justifies all these "new" methods of teaching math, particularly at the elementary level. . . because after all, "traditional methods of teaching math simply do not work" (the premise of the original NYT article I cited).
But make no mistake, the Common Core is an inch deep, and a mile wide, just like the curriculum it is supposed to replace. The Common Core has become the current trendy flashpoint for math education reform, while at the same time it contains the inherent flaws of the old curriculum.
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Chaz
Trad climber
greater Boss Angeles area
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School teacher is the #1 profession in my family. My dad, step brother and his wife, my cousin, her husband, and her daughter are teachers. I might be forgetting someone.
So when we get together, I hear a lot of shop talk.
From what I gather, teachers mainly want two things ( except for my step-brother's wife, she wants an SEIU shirt in a 6X, because 5X is tight on her ).
They want to be able to eject poor and disruptive students, like I was ejected at the end of my academic career. Having to explain things two, three, four, five times cuts instruction time in half, by two-thirds, three-fourths, etc. for the students who are bright enough to grasp the material the first time around.
At some point, those who want an education need to be separated from those who don't. The sooner the better for everyone involved. Second or third grade isn't too early. I knew in the second grade my mission in life was to do whatever it took to get out of school, but it took the school until I was a junior in high school to finally concur.
And they want their administration to back them up on obvious issues. For example, if a student tells his mom his teacher called him a "poopy head", and mom complains to the principal, the proper thing for the principal to do is to laugh in the parent's face and send her on her way. NOT call the teacher on the carpet to address the issue as if it actually happened.
As far as math goes, my cousin teaches math at the same Junior High I attended. She is head of the math department because she's the only math teacher there who can do long division.
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jgill
Boulder climber
Colorado
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She is head of the math department because she's the only math teacher there who can do long division
Oh, wow. I fear you are not joking.
;>\
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Chaz
Trad climber
greater Boss Angeles area
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She also tells me none of the students at that Junior High are given "word problems" to do in math class.
That baffled me, because life is a series of math word problems.
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rgold
Trad climber
Poughkeepsie, NY
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wbw, it is entirely possible that "the curriculum" per se isn't the problem. Since I only skimmed the common core, I don't think it wise for me to start elevating particular examples as models of right-headedness. My observation was that I didn't see anything I thought a scientist would disagree with.
The real question is, as I said before, are we going to try to teach mathematics as a subject accessible to human reason, or are we going to insist, quite possibly under the pressures of high-stake exams mandated by partially if not wholly clueless administrations, to make it into a set of mystical catechisms, uttered reflexively in response to predetermined stimuli? Of course, I'm being hyperbolic here for rhetorical effect, but at heart this is the issue.
John, I think some of the difficulties with "critical thinking" we see are actually the side-effect of an education system that has in some cases promoted mathematics as a field not accessible to critical thinking. After twelve or thirteen years of that for some kids, it may well be too late to convince them that the subject is very different from what they have come to believe. The school mentioned above that doesn't include word problems in the mathematics curriculum is a stunning example of how math education can jump the tracks and end up not being about anything recognizable as mathematics.
Part of the process of critical thinking in any scientific context and probably in almost all contexts involves multiple attempts, incorrect formulations, and periods of clarity and confusion. Much of our education system, at all levels, hides these processes in favor of communicating only the polished end results. Most students know that it takes years of practice to learn athletic skills, and along the way there will be many failures, many plateau periods were no progress seems to be made, and times of regression when things actually seem to get worse. But when it comes to mathematics, we show them the perfect-10 uneven bar routine but none of what went into producing it, and then we're surprised when a huge majority of students conclude that they can't do it.
One of the things we seem to be terrified of is confusion. Heaven forfend the children---or the teacher, or the parent---should ever be confused. But confusion is an integral component of intellectual progress. You have to learn how to deal with it and ultimately resolve it in order to be any kind of scientist, but you can't learn that if everyone is trying at all moments, not only to banish confusion from the process, but to suggest that the existence of confusion is some kind of failure. Now you don't go about confusing kids on purpose, but somehow there needs to be opportunities in which confusion occurs, is worked through, and resolved. From that comes pleasure and the self-confidence to confront the next difficulty.
One of the things we hear with great frequency is that lessons based on the common core are confusing. I'm not saying all confusion is good, but I am saying that the existence of some confusion is not, by itself, necessarily a fatal flaw. The real question is what happens next. And I might add that when adults go ballistic because they are confused by Johnny's homework, you can be sure that what Johnny really learns is that uncertainty frightens the daylights out of the grown-ups and is certainly to be avoided at all costs. And there goes Johnny's critical-thinking opportunities down the toilet.
I am not knowledgeable about primary and secondary education. I don't know if there is any hope for motion in the direction I mentioned. The NCTM has been trying to promulgate many of these ideas for years, apparently with little more than pockets of success. The wonderful teachers I had in high school were able to do the kinds of things I mentioned; there is nothing new or even remotely radical about my mentioning such things more than sixty years later.
The question of technology has been raised. I use it every day, but I don't think it is likely to have any impact on developing the kinds of intellectual skills I mentioned, so to some extent it is a very expensive distraction. I also think it isn't being used properly. We should be using it not to replace ordinary arithmetic with small numbers, but rather to do calculations that involve very big and very small numbers, calculations that are not practical for hand arithmetic. As I asked earlier in the thread, why aren't kids computing how many blades of grass are on the school lawn?
Of course, when technology enters the picture, the users need more knowledge, not less. The black box is perfectly happy to give you garbage out for the garbage you put in.
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jgill
Boulder climber
Colorado
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John, I think some of the difficulties with "critical thinking" we see are actually the side-effect of an education system that has in some cases promoted mathematics as a field not accessible to critical thinking. After twelve or thirteen years of that for some kids, it may well be too late to convince them that the subject is very different from what they have come to believe (rgold)
The problem seems, in general, to cross academic lines. I recall a study done several years ago that implied there was very little if any improvement in critical thinking skills in all areas between the freshman college year and graduation. I wonder if studies have been done on freshman to senior levels in high schools. It might be there is a window of opportunity at a very early age for a child to develop this skill . . . or not, so that home environment may be crucial. I haven't a clue. Maybe someone else on this thread does.
That being said, I think there can be critical thinking in one area and not another. I've known people who have excellent skills in the humanities, but are flops in math and science. The opposite seems true as well on occasion, for there are technical types who have limited language skills. (a phenomenon not seen on this site, with Ed and rgold and others superb at both ends of the spectrum)
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Ed Hartouni
Trad climber
Livermore, CA
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ah, technology...
so all you math teachers out there, just how does a computer do division? what's the algorithm?
and how do you think they came up with that?
how many times do you do long division every day?
how many times do you do division by estimating?
and then refining the estimate?
what are all the ways you do division?
and why are we, as a culture, fixated on doing long division precisely and accurately?
what's the point?
as has been stated many times in this thread, we confuse the flawless execution of a specific algorithm for a computational exercise as mathematics.
More interestingly, can you describe what the algorithm is doing? can you implement it in an arbitrary base (binary long division? can you do it? hexidecimal? vigesimal? sexagesimal?... ) can you divide one polynomial by another using long division? why?
Can you do it only by multiplying?
what are all the different ways?
just what is division, anyway? and why is it important that we understand it at all?
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Chaz
Trad climber
greater Boss Angeles area
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Dr Hartouni,
Those are the types of questions I used to ask of my teachers when I was a kid. "Why are we doing this?"
In eighth grade math class, we spent a ton of time learning Roman Numerals. Eighth grade. Should have been first grade.
I asked my math teacher what kind of a job I would qualify for, assuming I mastered Roman Numerals.
She told me I should just quit school right then, if the only thing I wanted out of it was to learn something to enable me to market myself.
I should have taken her advice. I would have had a three year head start.
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rgold
Trad climber
Poughkeepsie, NY
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I think Ed asks a lot of important and appropriate questions.
For someone who is going to go on in mathematics at the undergraduate level or above, there are a host of concepts related to "long division," or more accurately to the nature and possibility of carrying out long division, they will have to master.
But even the algorithm itself is important (well, many people are not going to think what I say next is at all important). There are various types of real numbers, and in particular every real number is either rational or irrational. The human recognition of the existence of irrational numbers is typically ascribed to the Pythagoreans, and so is something like 2300 years old at least. The ancient Greek philosophers were far more sophisticated about the concept of number and its implications than most of our current population, and so the labored to produce a logical system capable of embracing what we now call irrational numbers.
Nowadays, one way of describing the distinction between rational and irrational numbers is in terms of their decimal expansions. (Yeah...infinite decimals...what exactly do they mean and in what sense can they possibly denote "numbers?") The rational numbers are precisely the ones with terminating or infinite repeating decimal expansions, whereas the irrational numbers are precisely the ones with infinite non-repeating decimal expansions. How do we know this? The only argument I know for a rational number having a terminating or infinite repeating decimal expansion comes from an analysis of the algorithm for long division.
I mention this because it is not at all atypical; various topics in elementary mathematics are gateways to more advanced results.
Chaz, I think devoting a large segment of the eigth-grade curriculum to Roman numerals is a waste of time. Maybe a little bit so that the person encountering them has an idea how to decode them, since they do still get used. One reason to at least mention Roman numerals is to illustrate what a miracle place-value notation is. You had to be a genius to multiply two three-digit numbers if your only framework for representing them is Roman numerals, and in principle no finite table of products would suffice for all possible calculations. With decimal notation, you learn 55 multiplication facts and you can handle anything! Division with Roman numerals---fuhgettaboutit.
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