How much HS Math is necessary for a liberal arts college?

"Mathematics

No matter what your field of interest, mathematics will be essential for your higher education. It is the language, as Galileo put it, in which the book of nature is written. Today it is the common language describing new discoveries at the frontiers of science, of economic prediction, and of models of climate change.

To acquire the mathematical background you need at Harvard, you should study mathematics every year in secondary school. But simply taking mathematics is not enough. You should acquire the habit of puzzling over mathematical relationships. When you are given a formula, ask yourself why it is true and if you know how to use it. When you learn a definition, ask yourself why the definition was made that way. It is the habit of questioning that will lead you to understand mathematics rather than merely to remember it, and it is this understanding that your college courses require. In particular, you should select mathematics courses that ask you to solve hard problems and that contain applications ("word problems"). The ability to wrestle with difficult problems is far more important than the knowledge of many formulae or relationships.

By the time you get to college, the concept of a function, and its representation by a formula, a graph, or a table, should be second nature to you. For example, you should be able to sketch a graph of the time required to drive from Boston to New York as a function of average speed; or of the number of bacteria in a colony as a function of time given that each one divides in two every twenty minutes. A qualitative understanding of graphs – the ability to sketch and interpret graphs without plotting or reading specific points – is as important as the ability to draw graphs point by point. For example, does a given graph indicate that the concentration of a pollutant in a lake is leveling off, or increasing steadily?

In particular, you should be thoroughly familiar with the graphs and behavior of exponential and logarithmic functions, including doubling times and percentage growth rates. The trigonometric functions, and the ideas of amplitude, period, and phase, are important. Scientific notation and the ability to estimate orders of magnitude are frequently used. An increasing number of fields use the basic ideas of probability and statistics, such as mean, median, mode, and standard deviation.

If you are well-versed in algebra, functions, and graphing, secondary school calculus will enable you to take more advanced introductory courses in mathematics, physics, and chemistry in college. But do not rush into calculus. It may surprise you to know that success in first-year quantitative courses at college is determined more by the strength of your proficiency in algebra, functions, and graphing than by whether or not you have studied calculus in secondary school. Courses in the natural and social sciences often depend more on a real understanding of the behavior of different kinds of functions than on the ability to use calculus.

In the last analysis, however, it is not what courses you have taken, but how much you have thought about mathematics, that counts. More important than the knowledge of a specific mathematical topic, is the willingness to tackle new problems."

-From their website.

Muezzin-Bar'chu wrote:...From their website.

harvard university's website?

http://www.admissions.college.harvard.edu/apply/preparing/index.html

Muezzin-Bar'chu wrote:http://www.admissions.college.harvard.edu/apply/preparing/index.html

thank you.

good advice, i agree with most of it, but i vehemently disagree with one thing -- the notion of learning physics without calculus. i think this is most unproductive and confusing. basic kinematics is so intimately tied with the idea of a derivative that it is pointless to attempt teaching elementary physics without calculus. i know universities have physics classes for liberal arts majors which have titles like "physics for future presidents", but i aver and am willing to defend the notion that physics at the college level should not be taught without introducing at least the basic ideas of calculus.

i completely agree that it is better to spend the first year and half of high school completely ignoring calculus and developing strong fundamentals in algebra, coordinate geometry, probability theory, and trigonometry and get very proficient and comfortable in plotting and examining functions. one thing the website doesn't mention which i would like to emphasize is to also develop adroitness in mathematical manipulative skills. with the advent of symbolic math software this is becoming more and more rare, but i think it still has relevance because it makes different types of neuronal connections which are very valuable. i am glad my kid's high school teacher agrees with me on many of these issues. i sent her a copy of hall & knight which she has been using to make up homework problems.

all this is consistent with my earlier recommendation that if a student wants to learn calculus in high school, but is not sure that s/he will major in mathematics, physics or engineering, it is best to take ap calculus ab, and not ap calculus bc.

Jeremiah Mburuburu wrote:all this is consistent with my earlier recommendation that if a student wants to learn calculus in high school, but is not sure that s/he will major in mathematics, physics or engineering, it is best to take ap calculus ab, and not ap calculus bc.

based on what i have seen at the college board website, i agree with you.

Are you suggesting that the basic one-year of Introductory Physics recommended at HS level requires Calculus? Or are you talking about taking Physics in a liberal arts college without also concurrently taking a Calculus course?

Does it matter that the student is interested in majoring in Philosophy or Art History at the liberal arts college?

Muezzin-Bar'chu wrote:Are you suggesting that the basic one-year of Introductory Physics recommended at HS level requires Calculus? Or are you talking about taking Physics in a liberal arts college without also concurrently taking a Calculus course?

Does it matter that the student is interested in majoring in Philosophy or Art History at the liberal arts college?

i am talking about introductory physics both at the high school level and especially at the college level. i liked one of the suggestions JM made earlier to not label things as physics or calculus and teach it as a continuum. i think rudimentary notions of a derivative and how to use it in concepts like the instantaneous velocity and acceleration are not very difficult to teach. IMO it is more effective to do it this way instead of avoiding calculus in physics altogether to begin with and then doing a repair job later.

MaxEntropy_Man wrote:i completely agree that it is better to spend the first year and half of high school completely ignoring calculus and developing strong fundamentals in algebra, coordinate geometry, probability theory, and trigonometry and get very proficient and comfortable in plotting and examining functions. one thing the website doesn't mention which i would like to emphasize is to also develop adroitness in mathematical manipulative skills. with the advent of symbolic math software this is becoming more and more rare, but i think it still has relevance because it makes different types of neuronal connections which are very valuable. i am glad my kid's high school teacher agrees with me on many of these issues. i sent her a copy of hall & knight which she has been using to make up homework problems.

what are "neuronal connections?"

isn't the wide availability of symboloic math sw a strong reason to pay less attention to manipulative skills, and emphasize fundamental ideas, modeling and application?

Muezzin-Bar'chu wrote:

Does it matter that the student is interested in majoring in Philosophy or Art History at the liberal arts college?

this allergy to calculus is a little hard to understand. physicists, engineers, and chemists are expected to know english and have a fair bit of exposure to the humanities and social sciences, but art history majors can get away without learning what i think is one of the great classical achievements of human thought.

http://www.varsitytutors.com/math-tutoring

The ad for this website keeps popping up now on SuCH. Interesting that there are so many 'certified' tutors.

Jeremiah Mburuburu wrote:
isn't the wide availability of symboloic math sw a strong reason to pay less attention to manipulative skills, and emphasize fundamental ideas, modeling and application?

based on my personal experience, mastery of symbolic math software and effectiveness in using it is enhanced by manipulative skills.

These days the knowledge of basic Math (the high school level) is helpful no matter what type of subject a person wants to study in the college. I have heard people talking about (in terms of) Math even in the language courses.

MaxEntropy_Man wrote:

Muezzin-Bar'chu wrote:

Does it matter that the student is interested in majoring in Philosophy or Art History at the liberal arts college?

this allergy to calculus is a little hard to understand. physicists, engineers, and chemists are expected to know english and have a fair bit of exposure to the humanities and social sciences, but art history majors can get away without learning what i think is one of the great classical achievements of human thought.

good point. i am reminded of this line from john gribbin's review of susskind's book, "the theoretical minimum".

They do so with great success, and in the process they pull the rug out from under the all too common attitude that, while a physicist who doesn't appreciate art is a philistine, an artist who doesn't appreciate physics is only doing what comes naturally.

http://online.wsj.com/article/SB10001424127887323968304578250253842601938.html

bw wrote:

MaxEntropy_Man wrote:

Muezzin-Bar'chu wrote:

Does it matter that the student is interested in majoring in Philosophy or Art History at the liberal arts college?

this allergy to calculus is a little hard to understand. physicists, engineers, and chemists are expected to know english and have a fair bit of exposure to the humanities and social sciences, but art history majors can get away without learning what i think is one of the great classical achievements of human thought.

good point. i am reminded of this line from john gribbin's review of susskind's book, "the theoretical minimum".

They do so with great success, and in the process they pull the rug out from under the all too common attitude that, while a physicist who doesn't appreciate art is a philistine, an artist who doesn't appreciate physics is only doing what comes naturally.

http://online.wsj.com/article/SB10001424127887323968304578250253842601938.html

i don't know if we have talked about this, but here is a gem of a little book born out of an effort to teach art history majors a little probability:
http://www.amazon.com/Reasoning-about-Luck-Probability-Physics/dp/0521442176

MaxEntropy_Man wrote:

bw wrote:

MaxEntropy_Man wrote:

Muezzin-Bar'chu wrote:

Does it matter that the student is interested in majoring in Philosophy or Art History at the liberal arts college?

this allergy to calculus is a little hard to understand. physicists, engineers, and chemists are expected to know english and have a fair bit of exposure to the humanities and social sciences, but art history majors can get away without learning what i think is one of the great classical achievements of human thought.

good point. i am reminded of this line from john gribbin's review of susskind's book, "the theoretical minimum".

They do so with great success, and in the process they pull the rug out from under the all too common attitude that, while a physicist who doesn't appreciate art is a philistine, an artist who doesn't appreciate physics is only doing what comes naturally.

http://online.wsj.com/article/SB10001424127887323968304578250253842601938.html

i don't know if we have talked about this, but here is a gem of a little book born out of an effort to teach art history majors a little probability:
http://www.amazon.com/Reasoning-about-Luck-Probability-Physics/dp/0521442176

yes - you did mention this book a while back. i recall checking it out from the library.

MaxEntropy_Man wrote:

Muezzin-Bar'chu wrote:

Does it matter that the student is interested in majoring in Philosophy or Art History at the liberal arts college?

this allergy to calculus is a little hard to understand. physicists, engineers, and chemists are expected to know english and have a fair bit of exposure to the humanities and social sciences, but art history majors can get away without learning what i think is one of the great classical achievements of human thought.

i don't find it hard to understand "this allergy to calculus." artists don't use calculus, even if it is one of mankind's great accomplishments, and the automobiles and cell phones these same artists use every day couldn't have been made without using calculus. indeed, not many others use calculus either.

i don't see why anyone who finds calculus fascinating or useful should insist that others learn calculus, or worse, turn calculus into a battle ground where educators ensure that there's parity in the amount of time and effort the artists and the scientists spend to acquire knowledge in subjects in which they have no interest.

the role of educators should be to make it easy and enjoyable for people to acquire knowledge in the fields in which they are interested.

Jeremiah Mburuburu wrote:
the role of educators should be to make it easy and enjoyable for people to acquire knowledge in the fields in which they are interested.

The only problem is no one can be sure that how many will end up working in their chosen field of study (unlike in India). The idea is to teach them the basics in a variety of fields and let them excel in whatever they want at any given time. That is one reason why kids are forced to take even music and painting in middle schools. most may not indulge in music but once in a while you have a Max or a vakapaka or an IMPY get immersed due to interest. I wish I had some exposure so I get to appreciate the music world (pliss to note my deficiencies - one of only few...

MaxEntropy_Man wrote:

Jeremiah Mburuburu wrote:
isn't the wide availability of symboloic math sw a strong reason to pay less attention to manipulative skills, and emphasize fundamental ideas, modeling and application?

based on my personal experience, mastery of symbolic math software and effectiveness in using it is enhanced by manipulative skills.

relatively inexpensive, hand-held, pocket-sized, hardware-software packages such as the texas instruments graphing calculators are 21st-century substitutes for manipulative skills. if one needs manipulative skills to master the device that seeks to render symbolic manipulation easier, then the device one is using is poorly designed; there are likely to be better substitutes.

i would not discard symbolic manipulation altogether. some people enjoy it; others create efficient algorithms based on their knowledge of symbolic manipulation. today however, it's not necessary to be skilled at symbolic manipulation to be a highly effective physicist, engineer, or social scientist.

the sensible scientist may perform the symbolic manipulation once or twice to understand its basis, but does not spend his time developing manipulative skills; he relies on 21st-century tools to perform the manipulations for him. he uses his time and effort to perform higher-order tasks, for example, creating new products, proposing new and improved models of phenomena, and understanding or predicting the behaviour of systems composed of many components.

educators are likely to stimulate a lot more interest in the sciences and engineering, if only they did not insist on symbolic manipulation skills past, say, eighth grade.

Marathadi-Saamiyaar wrote:

Jeremiah Mburuburu wrote:
the role of educators should be to make it easy and enjoyable for people to acquire knowledge in the fields in which they are interested.

The only problem is no one can be sure that how many will end up working in their chosen field of study (unlike in India). The idea is to teach them the basics in a variety of fields and let them excel in whatever they want at any given time. That is one reason why kids are forced to take even music and painting in middle schools. most may not indulge in music but once in a while you have a Max or a vakapaka or an IMPY get immersed due to interest. I wish I had some exposure so I get to appreciate the music world (pliss to note my deficiencies - one of only few...

Carnatic music is not considered cool especially by the TN non-brahmins. Brahmins who want to be secular also stay away from it. If A.R. Rahman gets into carnatic Music then there will be a lot of people following his lead.

Rishi wrote:

Marathadi-Saamiyaar wrote:

Jeremiah Mburuburu wrote:
the role of educators should be to make it easy and enjoyable for people to acquire knowledge in the fields in which they are interested.

The only problem is no one can be sure that how many will end up working in their chosen field of study (unlike in India). The idea is to teach them the basics in a variety of fields and let them excel in whatever they want at any given time. That is one reason why kids are forced to take even music and painting in middle schools. most may not indulge in music but once in a while you have a Max or a vakapaka or an IMPY get immersed due to interest. I wish I had some exposure so I get to appreciate the music world (pliss to note my deficiencies - one of only few...

Carnatic music is not considered cool especially by the TN non-brahmins. Brahmins who want to be secular also stay away from it. If A.R. Rahman gets into carnatic Music then there will be a lot of people following his lead.

people who deliberately stay away from carnatic music for non-musical reasons, regardless of caste, religion, and language, are missing hours and hours - perhaps a lifetime - of interesting and pleasurable listening. i feel similarly about bharathanatiyam. to that extent, i understand uppili's wish for exposure.