Exactly I saw many students they run away from math problems even I saw that math is the only subject in which student can get full marks for any question as if every step and problem is correct there is no chance to deduct the marks.
My Journey With Mathematics
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Otis
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Hi Peter. Did you run away from harpazo's pretend problem?
For me, it's a question that involves three equal pieces which add to make 72. That is, the smaller of the two numbers asked for must be one-third of 72.
I used the multiplication table (6×12=72) and this property
\(\displaystyle a \cdot b = \frac{a}{2} \cdot \frac{2b}{1}\)
to mentally calculate 72/3.
If anyone interested does not understand my approach, please ask!
I used the multiplication table (6×12=72) and this property
\(\displaystyle a \cdot b = \frac{a}{2} \cdot \frac{2b}{1}\)
to mentally calculate 72/3.
If anyone interested does not understand my approach, please ask!
\(\;\)
I would believe that if the statement were re-worded to say "A great many high-school students feelings about word problems range from dislike through hatred and fear."
But the statement begs the question of why is that so. And it is that question that is important. If you have difficulty doing the highly stylized and unnaturally simplified problems called "word problems," you will never be able to apply mathematics to the actual problems to which mathematics can give answers or help to give answers.
My personal feeling is that the inability to master word problems comes from a defect in teaching. We do not stress, at least not enough, that mathematics is a language the purpose of which is to find an answer from what is already known. Now I do not believe that there is a universal technique to translating a problem described in a natural language into the language of mathematics. Indeed, many problems cannot be translated at all, such as what should I get my wife for her 75th birthday. So perhaps the first step outside a classroom is to ask yourself, "Might I be able to translate this problem into a mathematical problem that I know how to solve?' So, I might suspect that a problem could be solved by measure theory. I have not a clue about measure theory. When I hear that some theory is based an atomless measure space, I know that I must (1) give up, (2) take a course, or (3) find an expert.
Now this of course is not an issue for students in a class or reading a text. Their word problems are specifically designed to let them exercise techniques that they have already been exposed to. So, given that I suspect math can help me, my next question is whether I can translate what I do know into mathematical language. If the answer is yes, then I do that and write it all down.
students should be strongly encouraged to do this. I might go so far to say that it ought to be a requirement on homework for high school students. At least it would stop them from saying "I don't even know where to start."
My next question is whether I can translate what I am looking for into mathematical or quasi-mathematical language. If the answer is yes, then I write that down. In an algebra class, that would usually be something like
[MATH]x = \text {a numeral}[/MATH] or [MATH]x = \text { an algebraic expression.}[/MATH]
Now we no longer have a word problem. We are now concerned only about what mathematical techniques are in our tool box let us go from what we know to what we want.
I have worked with students who can do mechanics til the cows come home, but (to retain my bovine metaphor) are buffaloed by translation. That indicates to me a serious deficiency in how the subject is taught.
Otis
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I think most of us would agree that instruction is a major reason, Jeff. (Too much public education in the USA is clearly problematic.) Yet, my feeling is that students lack also motivation to read and practice sufficiently outside of class. Either they're not interested in the subject (or school in general) or they don't get decent mentoring and encouragement at home, or both.
Poor instruction notwithstanding, for those students who are competent at solving exercises once they've been provided expressions and equations, what else is preventing them from translating word problems into math statements if not a lack of exposure/practice?
Can we agree that, in addition to serious issues with instruction, the situation is borne also by societal failings and basic human nature?
?
My academic training was in history. I am consequently highly allergic to monocausal explanations of social phenomena: 99% of the time such explanations are simply wrong. Moreover, it is frequently difficult to assign qualitative, let alone quantitative, weights to causes. So arguments about what is the most important cause can seldom be resolved and distract attention from what can be done to improve things..
There are obviously cultural and genetic causes that account for differences in the educational performance of different individuals, and perhaps they far outweigh deficiencies in the educational process itself. But we can do nothing whatsoever to change the genes of individuals, and culture is susceptible to only very slow change. Moreover, to the extent that parents suffering from poor education themselves do not know how to facilitate their children's education effectively and do not know what they should demand from the public schools, we are essentially blaming the victims of past poor education for current poor education.
The things that we could do are (1) start improving the curriculum immediately, (2) recognize that permitting ineffective teachers to stay in classrooms is to sanction politically and to reward economically the systematic abuse of children, and (3) alter the ridiculous pay structure that rewards people for administering rather than teaching, discourages young adults with high economic potential from considering public teaching as a career, and rewards seniority rather than objective metrics of performance.
Perhaps most parents are now and always will be ignorant or oblivious, and almost certainly children will always be easily distracted and bored. Those obstacles may be forever unrectifiable, but what is not unrectifiable are the obvious deficiencies in US education. I prefer to focus on what can be done before we raise another generation of ignorant and oblivious parents.
A typical algebra text will have 20 or 30 problems that are pure mechanical drills and 6 or 8 word problems. The drills are admittedly easy for teachers to grade and do not require much paper and ink to present. I would encourage texts to have far more (and more diverse) word problems than is currently the case. Furthermore, I would be asking that what is required for word problems is setting the problem up for mathematical solution rather doing the mechanical work of solving it. The mechanical work is boring and error-prone and so frustrating. We already have more than enough drills in mechanics,
There are obviously cultural and genetic causes that account for differences in the educational performance of different individuals, and perhaps they far outweigh deficiencies in the educational process itself. But we can do nothing whatsoever to change the genes of individuals, and culture is susceptible to only very slow change. Moreover, to the extent that parents suffering from poor education themselves do not know how to facilitate their children's education effectively and do not know what they should demand from the public schools, we are essentially blaming the victims of past poor education for current poor education.
The things that we could do are (1) start improving the curriculum immediately, (2) recognize that permitting ineffective teachers to stay in classrooms is to sanction politically and to reward economically the systematic abuse of children, and (3) alter the ridiculous pay structure that rewards people for administering rather than teaching, discourages young adults with high economic potential from considering public teaching as a career, and rewards seniority rather than objective metrics of performance.
Perhaps most parents are now and always will be ignorant or oblivious, and almost certainly children will always be easily distracted and bored. Those obstacles may be forever unrectifiable, but what is not unrectifiable are the obvious deficiencies in US education. I prefer to focus on what can be done before we raise another generation of ignorant and oblivious parents.
A typical algebra text will have 20 or 30 problems that are pure mechanical drills and 6 or 8 word problems. The drills are admittedly easy for teachers to grade and do not require much paper and ink to present. I would encourage texts to have far more (and more diverse) word problems than is currently the case. Furthermore, I would be asking that what is required for word problems is setting the problem up for mathematical solution rather doing the mechanical work of solving it. The mechanical work is boring and error-prone and so frustrating. We already have more than enough drills in mechanics,
harpazo
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How many here know the actor Henry Winkler who played THE FONZ on the famous sitcom Happy Days? I enjoy watching Henry clips online. Not too long ago, I saw an interview where Henry talked about his school days.
Henry aka THE FONZ talked about his fear of math, especially geometry. According to Henry, he just could not understand geometry. He talked about the Pythagorean Theorem and how this theorem made absolutely zero sense to him in high school. In fact, Henry took geometry about 3 or 4 times in high school before he finally passed the class with a D, which does not stand for dandy.
Looking back at his tv career, it is clear to see that geometry was not needed for him to shine as an actor, producer and I think he even directed one or two films. Henry Winkler and so many others have proven that math beyond the four basic operations including fractions is not needed for success after high school. What do you say?
Henry aka THE FONZ talked about his fear of math, especially geometry. According to Henry, he just could not understand geometry. He talked about the Pythagorean Theorem and how this theorem made absolutely zero sense to him in high school. In fact, Henry took geometry about 3 or 4 times in high school before he finally passed the class with a D, which does not stand for dandy.
Looking back at his tv career, it is clear to see that geometry was not needed for him to shine as an actor, producer and I think he even directed one or two films. Henry Winkler and so many others have proven that math beyond the four basic operations including fractions is not needed for success after high school. What do you say?
topsquark
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I suppose that it depends on what kinds of jobs you can get without basic math skills. On the other hand, if you can't do the major operations then you can't even make change at a store, which is pretty much the least mathematical job. I'd opt to learn the Math... It's easier.
-Dan
-Dan
Otis
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I'd say your proof is tantamount to proving that nobody needs to work at all because "so many others" have won a lottery.
The question is not whether it is possible for an individual to be economically prosperous without an education. All of human experience shows that it is possible. So it is a silly question.
The question is whether an individual with an education is more likely to prosper economically than one without in a technological society.
Notice that I distinguish between educated and credentialed. The last thirty years have seen a constant substitution of credentials for education.
The question is whether an individual with an education is more likely to prosper economically than one without in a technological society.
Notice that I distinguish between educated and credentialed. The last thirty years have seen a constant substitution of credentials for education.
harpazo
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Can you further explain what you mean here?
harpazo
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I myself think that most people in our technically advanced society should at least know Algebra 1 and 2 very well, you know, like drinking water. What do you say?
About what!
harpazo
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You said:
"The question is whether an individual with an education is more likely to prosper economically than one without in a technological society."
Do you think a person with proper education can prosper after the campus life in a highly technological world? Same question in terms of people who are poorly educated.
D
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Depends on how you define "prosper"?
Peter Greene, the legendary guitarist of FleetWood Mac - quit the group at the top and became a night-janitor at one of the Seattle office buildings (at least thus the legend goes!). And then the FleetWood Mac became fashionable again in the 80 - 90 and the recording company cut a huge "royalty" cheque. The legend goes - Peter Green refused the cheque and said "I don't need that - I got all I need".
Did Peter Greene prosper? Was he happy? - he did not know geometry but surely could bend those chords!!
The point is - education gives you options! If you know geometry - you could be a high-school math-teacher or night janitor at the high school.
Without geometry - your only choice is to be night janitor at the high school.
Harpazo - Did not you ask the same questions (not very distant past) as MathDad in this forum?
Why are you trying to bring back the same topic?
If you need to "troll" - please bring in new topics!!
You have not paid attention to what I said.
First, I carefully limited myself to economic prosperity rather than to a more general evocation of a "good life." (See SK's response above.)
Second, I was discussing probabilities rather than certainties. People with virtually no education and little intelligence may have great financial success, particularly in sports or entertainment. However, if we take a random sample of 1000 people who never graduated from high school but have IQs over 140 and another random sample of 1000 people who graduated from Harvard Law and, on average, have the same IQ as the average for the first group, which group do you think would have, on average, higher income and higher net worth?
Third, I carefully distinguished between educated and credentialed. Having a degree in art history may very well be good for the soul. I wish that I had known of Hans Memling decades before I ever saw one of his pictures. But having such a degree does little to prepare you for succeeding economically in a technological world.
I went to Columbia back when it was still serious about a well rounded education. I am still convinced that such an education provides the best opportunity for a life that is fulfilling in many dimensions. I was forced to study classical music, European art, western ethical and political philosophy, foreign languages, and the canon of western literature in addition to pure mathematics and physical science. Not only do I view such an education as broadening the mind generally, but its breadth is an economic advantage that admittedly not everyone chooses to exploit because you have been trained to learn new things. I view with disgust universities changing themselves into trade schools and enclaves of dubious "social science." (I am not prejudiced against the social sciences when they are pursued rigorously and without ideological intent. I took several classes in sociology at Columbia when Daniel Bell, Amitai Etzioni, and C. Wright Mills were on the faculty. And I got my degree in history, which I consider a social science, and studied economics in graduate school.)
Do I think an education is necessary to be economically successful? No.
Do I think a serious, well-rounded education is more apt to lead to a fulfilling life? Yes.
Do I think a serious, well-rounded education is an economic advantage? Yes, but it may be a relatively small advantage.
Do I think being mathematically knowledgeable is a sizable economic advantage in a technological society? Yes indeed, but it is no guarantee.
Do I think degrees have any meaning? No, universities sell degrees like papal indulgences. When I was hiring people, I paid virtually no attention to what degrees they had been awarded. A marketing degree from West Virginia University says nothing about anyone's diligence, ethics, competence, collegiality, or intelligence. I hired a guy with a physics degree from Harvard. He was intelligent, abrasive, and unscrupulous. He soon left my employ.
harpazo
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1. Please ignore my posts from now.
2. This problem goes WAY BEYOND NOT KNOWING HIGH SCHOOL GEOMETRY.
3. Go back and read what I said again without your reply.
4. You enjoy bringing up the past. I suppose you are perfect, KHAN.
5. MOVING ON....
harpazo
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As soon as I read the opening line about me not paying attention, I stopped right there. Thank you for agreeing with Mr. KHAN about me being a troll, whatever the heck that is. Moving on. People here take something so simple and turn it into a psychiatric session. Lord have mercy!!! Moving on....
I have had it with you. I spent considerable time and effort writing seriously intended responses to your posts. But you get in a huff and admit you do not even read them. I did not call you a troll. But I suggest you do move on to some place where people will indulge your whining.
harpazo
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I went back to read your post. Interesting information.