[hrothgar]

Interesting article in this morning's Globe

http://www.boston.co...g_from_finland/

[kenberg]

Thanks greatly. I had heard that Finland is a strong success story and this brief article gives some concrete indications of their approach.

Bipartisan moan: If I recall correctly, the first George Bush styled himself as the education president, and Bill Clinton promised that by the year 2000 our children would be the first in the world in math and science. Not quite the way it worked out.

Climbing out of this hole will require some painful honesty. For example, the story notes that in Finland teachers are regarded as cooperating professionals and work with a great deal of autonomy. Why not here, one may ask. It's not so simple. My granddaughter graduated from high school last spring. A first rate school and I would support great autonomy for her teachers. But there are other schools. There is no gentle way around the hard fact that some schools have people teaching math courses who know little or no mathematics. The well prepared teachers are themselves very aware of this problem.

The students in some areas, by the time they reach high school, are badly behind. This is often dealt with by pretending that they are doing something that they are not.

Everyone has their horror stories and I will restrain myself by telling only one: My wife was asked to fill in one day as a tutor for a kid who was taking geometry. The kid brought his homework with him. All of the problems had been checked off by the teacher as acceptable. They were studying the sine law. This is really trigonometry, not geometry. I learned the sine law when I was sixteen, this kid was fourteen or fifteen. Great, an advanced student!! We could wish! None of the work was correct. The kid knew that three of something in a triangle added to 180 but he wasn't sure about three of what. Anyway, if you gave him two angles and told him that the three angles (in degrees) summed to 180 then adding the two numbers together and subtracting the total from 180 was right at the edge of his ability. Sine law problems often lead to "cross-multiplication", which means re-writing an equation a/b=c/d as ad=bc. He was taught that to do this he should circle the numerator on one side and the denominator on the other side, then box the remaining numerator and denominator, and then do something although he wasn't sure what.

The difference between schools like this and the one my granddaughter went to is almost beyond expression. I went to a not particularly good high school in St. Paul in the early 1950s. My close friends went to what was then the best high school in the city. (Best public hs and arguably the best hs.) The difference between the schools existed but was not huge. Of all of the inequalities in the modern USA I think this inequality of educational opportunity tops my list of inequalities that must be corrected.

[hrothgar]

FWIW, I went to a stellar public school in Poughkeepsie, New York.
(Arlington, HS)

IBM had a serious presence in Dutchess County.
Between the IMBers, Vassar College, and the like there was a lot of money available for academics and we had a great cadre of teachers.

By and large, I think that the public schools provided me with a really first rate education.
However, the one area where I think that they really failed me was math.

I hated math all through junior high and high school.
(I really didn't start to appreciate it all all until I reached grad school)

From my perspective: The teachers failed completely to motivate math. They never managed to move the course beyond memorization and make me actually care about the techniques that we were applying.

I understand that math requires a fair amount of rote memorization (though with advanced in software, this may be changing). Where I think that things really need to change is the balance between theory and application. If we don't motivate math with practical, real world problems we're never going to convince kids to care. With me, I didn't start caring until I found that I couldn't solve real economics and (shudder) gambling problems without some serious math.

Another area where we're really missing the boat is linear algebra and differential equations. Linear algebra and diffeq are the fundamental building blocks for most "real" math and 99% of high school graduates in the US never see them in action.

[mike777]

kenberg, on 2010-December-27, 08:38, said:

Thanks greatly. I had heard that Finland is a strong success story and this brief article gives some concrete indications of their approach.

Bipartisan moan: If I recall correctly, the first George Bush styled himself as the education president, and Bill Clinton promised that by the year 2000 our children would be the first in the world in math and science. Not quite the way it worked out.

Climbing out of this hole will require some painful honesty. For example, the story notes that in Finland teachers are regarded as cooperating professionals and work with a great deal of autonomy. Why not here, one may ask. It's not so simple. My granddaughter graduated from high school last spring. A first rate school and I would support great autonomy for her teachers. But there are other schools. There is no gentle way around the hard fact that some schools have people teaching math courses who know little or no mathematics. The well prepared teachers are themselves very aware of this problem.

The students in some areas, by the time they reach high school, are badly behind. This is often dealt with by pretending that they are doing something that they are not.

Everyone has their horror stories and I will restrain myself by telling only one: My wife was asked to fill in one day as a tutor for a kid who was taking geometry. The kid brought his homework with him. All of the problems had been checked off by the teacher as acceptable. They were studying the sine law. This is really trigonometry, not geometry. I learned the sine law when I was sixteen, this kid was fourteen or fifteen. Great, an advanced student!! We could wish! None of the work was correct. The kid knew that three of something in a triangle added to 180 but he wasn't sure about three of what. Anyway, if you gave him two angles and told him that the three angles (in degrees) summed to 180 then adding the two numbers together and subtracting the total from 180 was right at the edge of his ability. Sine law problems often lead to "cross-multiplication", which means re-writing an equation a/b=c/d as ad=bc. He was taught that to do this he should circle the numerator on one side and the denominator on the other side, then box the remaining numerator and denominator, and then do something although he wasn't sure what.

The difference between schools like this and the one my granddaughter went to is almost beyond expression. I went to a not particularly good high school in St. Paul in the early 1950s. My close friends went to what was then the best high school in the city. (Best public hs and arguably the best hs.) The difference between the schools existed but was not huge. Of all of the inequalities in the modern USA I think this inequality of educational opportunity tops my list of inequalities that must be corrected.

I think Math teachers forget how tough math is for the rest of us and how easy it is to forget this stuff if you never use it.


I bet if you stop 100 random people on the street 99% can't give you the definition of sine and most of them could not explain the definition if you gave it to them.




Noun
•S: (n) sine, sin (ratio of the length of the side opposite the given angle to the length of the hypotenuse of a right-angled triangle) WordNet home page

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In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse of a right triangle can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides.

For example, if one of the other sides has a length of 3 meters (when squared, 9 m²) and the other has a length of 4 m (when squared, 16 m²), then their squares add up to 25 m². The length of the hypotenuse is the square root of this, or 5 m.

The word hypotenuse derives from the Greek ὑποτείνουσα (hypoteinousa), a combination of hypo- ("under") and teinein ("to stretch").[1] The word ὑποτείνουσα was used for the hypotenuse of a triangle by Plato in the Timeus 54d and by many other ancient authors.
----------


In trigonometry, the law of sines (also known as the sine law, sine formula, or sine rule) is an equation relating the lengths of the sides of an arbitrary triangle to the sines of its angles. According to the law,


where a, b, and c are the lengths of the sides of a triangle, and A, B, and C are the opposite angles (see the figure to the right). Sometimes the law is stated using the reciprocal of this equation:


The law of sines can be used to compute the remaining sides of a triangle when two angles and a side are known—a technique known as triangulation. It can also be used when two sides and one of the non-enclosed angles are known. In some such cases, the formula gives two possible values for the enclosed angle, leading to an ambiguous case.

The law of sines is one of two trigonometric equations commonly applied

http://en.wikipedia....ki/Law_of_sines

[hrothgar]

mike777, on 2010-December-27, 09:07, said:

I bet if you stop 100 random people on the street 99% can't give you the definition of
sine and most of them could not explain the definition if you gave it to them.

At the same time, I'm guessing that most of them could understand a Fourier series.

The issue isn't that the notion of a sine or a cosine is intrinsically complex, rather that its not motivated properly.

Focusing on a geometric interpretation of sines and cosines is completely divorced from (almost) anything remotely practical. (How many people use triangulation with any frequency?)

If, instead, you start by focusing on the periodic nature of a sine curve - which is what most people actually care about - you'll probably get a lot better retention....

[hrothgar]

mike777, on 2010-December-27, 09:07, said:

In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. http://en.wikipedia....ki/Law_of_sines

Silly question:

I always thought that the hypotenuse was the longest side of a triangle.

Is this word only used when discussing right triangles?

[PassedOut]

Like most, I have many sad examples of the failure of primary and secondary education in the US. It's not a matter of students not being able to compete at the highest levels: it's an almost complete lack of the most basic skills of communicating, calculating, and reasoning.

For a long time I've had the opinion that these failures stem from a general lack of respect for learning and knowledge in the US. When television commercials use educators as props, the educators are made to appear ridiculous more often than not. In political discourse, people who actually know a subject matter are labeled as part of the despised "elite." And so, although few things in the world are more important than getting kids off to a good start, the salaries and social positions of teachers in the US do not reflect that fact.

Constance and I took our sons out of the educational system to teach them at home. But in doing that, we took full advantage of a program at a local university that allowed high school students to take two college courses per term. That program was very valuable to us, particularly for lab sciences and for other courses (life drawing, for example) that we weren't prepared to teach.

It goes without saying that what we did is not a solution to the problem. Education in the US will have to move several rungs up the ladder of importance before the situation turns around.

[mike777]

Right Triangle's Hypotenuse The hypotenuse is the largest side in a right triangle and is always opposite the right angle. (Only right triangles have a hypotenuse). The other two sides of the triangle, AC and CB are referred to as the 'legs'.



http://www.mathwareh...t-triangle.html

In the triangle on the left, the hypotenuse is the side AB which is

edit

darn it I cannot copy and paste any of these very helpful pictures regarding this discussion.

------------

[kenberg]

I'll get to this somewhat advanced math in a minute but first I want my point not to be lost. I was not so much arguing about whether the sine law should be common knowledge as insisting that a student who cannot add two numbers and subtract the total from 180 is a student in a lot of trouble whether or not he can recite the sine law.

Now as to applications. Yes and no. Taking physics, I always skipped the stuff labeled applications. They were usually both trivial and boring. A twelve year ole, or a fourteen year old, has the luxury (in most households anyway) of being interested in the things that he finds interesting,putting off the practical aspects until later. It's a little touch and go as to what is an application, in the sense of being useful, and what is just an interesting application. For example: Suppose we know the radius of the Earth and the fact that a satellite near the Earth's surface completes an orbit in about 90 minutes. Then, using Kepler, we can work out the distance to the moon from the 28, or is it 29, days it takes to complete its orbit. This might well interest a 12 year old. Beats me whether to call it theory or application. Certainly it will not be of use to him as he delivers papers. Later, after the diff eq, Kepler's laws can be deduced from the law of gravity. Again interesting (to me anyway), again maybe it's theory, maybe it's an application.

Anyway, at more than a few schools, such a discussion is totally irrelevant because the students cannot add two numbers and subtract the total from 180. Calculators or not, how about just learning this trivial task when you are 9 or so. It won't strain the brain.

[mike777]

As to the main point regarding motivation and teachers the following is a true story.

I went back to school for my mba in the middle of my life at a Calif school.

In one class half the students were bus students the other half were teachers working on some kind of graduate teaching degree.

In Calif and most union states the higher level of education you achieve the more money you are paid; No merit pay.

In this class we were broken up into groups for the duration.

The teachers would just come out and say, lets just do the basic minimum required to get a passing grade and complain how the others would push them to improve our group papers and presentations.

The funny thing is a big part of this course was all about motivating your work force as part of a management or leadership class.

In this case the rewards for these teachers was to get the piece of paper and get paid more, not to learn anything.

[mike777]

kenberg, on 2010-December-27, 09:49, said:

I'll get to this somewhat advanced math in a minute but first I want my point not to be lost. I was not so much arguing about whether the sine law should be common knowledge as insisting that a student who cannot add two numbers and subtract the total from 180 is a student in a lot of trouble whether or not he can recite the sine law.

Now as to applications. Yes and no. Taking physics, I always skipped the stuff labeled applications. They were usually both trivial and boring. A twelve year ole, or a fourteen year old, has the luxury (in most households anyway) of being interested in the things that he finds interesting,putting off the practical aspects until later. It's a little touch and go as to what is an application, in the sense of being useful, and what is just an interesting application. For example: Suppose we know the radius of the Earth and the fact that a satellite near the Earth's surface completes an orbit in about 90 minutes. Then, using Kepler, we can work out the distance to the moon from the 28, or is it 29, days it takes to complete its orbit. This might well interest a 12 year old. Beats me whether to call it theory or application. Certainly it will not be of use to him as he delivers papers. Later, after the diff eq, Kepler's laws can be deduced from the law of gravity. Again interesting (to me anyway), again maybe it's theory, maybe it's an application.

Anyway, at more than a few schools, such a discussion is totally irrelevant because the students cannot add two numbers and subtract the total from 180. Calculators or not, how about just learning this trivial task when you are 9 or so. It won't strain the brain.

My mom was a grade school teacher in Chicago. She spent most of her time, making lesson plans and making sure her kids had hats, winter boots and a hot meal.

---

In my current local town there is a huge fight on how to cut 100million from the budget.More than 50% of the students are minorities. If they close poor performing schools, they are called racist and spend years with law suits.

Our local grammer school seems to have highest scores in town. The problem is if they choose to close that school, bus the kids out, spread the "good" teachers around to the "bad" schools, the students leave the system and the school system loses state aid which is based on the population of students going to classes.

[Gerben42]

When cutting, education should be a no-go area. If you cut on education, you will be generating future welfare receivers. Only one thing is worse than cutting education, and that's cutting math education. The current world requires us to know more math than ever before.

And yet, math is so simple to learn because you have to memorize hardly anything. I remember a helping a student who was completely confused with z-tests (evaluating probabilities using the normal distribution). Turns out the teacher explained it in a way that HE could understand, but the students didn't. After some hours with me, the student passed the test with ease, wheras before she wasn't even close to passing.

Quote

Now as to applications. Yes and no. Taking physics, I always skipped the stuff labeled applications. They were usually both trivial and boring.

Of course you have to think a bit about interesting applications. Why doesn't a bicycle fall over? Why do we see a rainbow and what determines its position? What is the tone difference between an approaching and receding police siren? Why is the sky blue? If an elevator falls, what would those inside experience?

[mike777]

Gerben42, on 2010-December-27, 10:59, said:

When cutting, education should be a no-go area. If you cut on education, you will be generating future welfare receivers. Only one thing is worse than cutting education, and that's cutting math education. The current world requires us to know more math than ever before.

And yet, math is so simple to learn because you have to memorize hardly anything. I remember a helping a student who was completely confused with z-tests (evaluating probabilities using the normal distribution). Turns out the teacher explained it in a way that HE could understand, but the students didn't. After some hours with me, the student passed the test with ease, wheras before she wasn't even close to passing.



Of course you have to think a bit about interesting applications. Why doesn't a bicycle fall over? Why do we see a rainbow and what determines its position? What is the tone difference between an approaching and receding police siren? Why is the sky blue? If an elevator falls, what would those inside experience?

First off thanks for being a tutor. Great note on how different students learn much easier in different ways.

Great story it reminds me of the math tutor who helped me as a 14 year old freshman taking Algebra 1. and the tutor who much later in life helped me with statistics.

In fact I wish if anything there was more of a tutor system in education, a system which encouraged many more students in many more subjects.

[mgoetze]

Quote

All teachers are required to have higher academic degrees that guarantee both high-level pedagogical skills and subject knowledge.

In Germany, teachers are also required to have higher academic degrees. I'm pretty sure that this is no guarantee whatsoever of high-level pedagogical skills. From what I hear from fellow students who are working on a teachers' degree, the contents of these didactics courses is pretty much bull****.

Quote

Educational leadership is also different in Finland. School principals, district education leaders, and superintendents are, without exception, former teachers.

I think this is the same in Germany. I'm pretty sure it doesn't do any good whatsoever.

Of course, the fact that the pupil-to-teacher ratio in Germany keeps growing and growing doesn't help matters at all. Lobbyocracy is not a very future-oriented form of government...

[kenberg]

As to teachers in the USA, I really think the problem is that there is a huge range of talent, far more than would be tolerated in, say, piloting a plane. I taught a math class for future elementary school teachers and one of my very good students was planning a career in early childhood education, meaning 3 and 4 year old children. She had a sharp mathematical mind. At the other end, I had a future high school teacher in an advanced calculus class and she had far less ability. Further, some of the others get their degrees at God only knows where. The best are very good. There is not nearly enough of them to go around.


One young woman took a couple of upper level courses from me. Pretty good. Not remarkable but pretty good. She worked as a waitress while in school and was trying to decide if she really was going to teach after graduation. Better money in being a waitress (attractive, charming, and I would guess good at keeping orders straight and customers in line).

We have a problem. Looking to Finland could help, maybe help a lot, but we need to solve it.

[PassedOut]

kenberg, on 2010-December-27, 21:41, said:

One young woman took a couple of upper level courses from me. Pretty good. Not remarkable but pretty good. She worked as a waitress while in school and was trying to decide if she really was going to teach after graduation. Better money in being a waitress (attractive, charming, and I would guess good at keeping orders straight and customers in line).

Now that is really sad.

[TimG]

It is my opinion that any post graduate degree is an excess when it comes to teaching high school students. In math, for instance, things like Abstract Algebra, let alone Differential Equations, will be of little or no use when teaching Pre-Calculus. There is a balance between enthusiasm for the subject, enthusiasm for teaching, and knowledge in the subject area that needs to be achieved. A PhD in Mathematics won't make up for a lack of interest in teaching; a Masters in Education won't make up for a lack of interest in math (when it comes to teaching math); and of course, a lack of basic math knowledge can't be overcome by enthusiasm and knowledge of teaching techniques.

[kenberg]

I pretty much agree with the view of TimG above. In some area high schools the courses get very advanced and the teachers in those spots need some of the advanced courses mentioned. For example I believe a very decent course in differential equations is taught in some (very few) places. But these schools take very good care of themselves.

I think that there is a real danger in demanding to much in the way of advanced certification. This may sound strange, but here is why: I have seen a recurring pattern. First, someone asserts that advanced knowledge is really important. Everyone nods his head in enthusiastic agreement. Then plans and standards are put into place. Great. Then reality sets in. Although no one says it exactly, ways are found to provide the appearance that the standards are being met without actually insisting that, in any honest sense, they actually are met. A hs teacher called my wife for assistance with a math course he was taking at the so-called graduate level. In this case I forget the details but my wife was shaking her head in dis-belief. He was required to take this as a condition of employment, a college was happy to provide it for cash, calling a graduate level course was a total joke. There is a substantial industry in this pretense.



Anyway, yes, I agree with Tim that a reasonable balance is needed.

[hrothgar]

TimG, on 2010-December-27, 22:36, said:

It is my opinion that any post graduate degree is an excess when it comes to teaching high school students. In math, for instance, things like Abstract Algebra, let alone Differential Equations, will be of little or no use when teaching Pre-Calculus. There is a balance between enthusiasm for the subject, enthusiasm for teaching, and knowledge in the subject area that needs to be achieved. A PhD in Mathematics won't make up for a lack of interest in teaching; a Masters in Education won't make up for a lack of interest in math (when it comes to teaching math); and of course, a lack of basic math knowledge can't be overcome by enthusiasm and knowledge of teaching techniques.

I agree with most of what Tim is saying, however, I'm not sure whether I agree about diffeq

There are broad classes of problems that you can't really touch unless you're using differential equations
From my perspective,

1. Feedback is an incredibly important concept
2. It's quite intuitive and also very easy to explain/illustrate

I think that it can/should be introduced much sooner than it is.
Moreover, I think that modern software makes it practical to do so.

[TimG]

To be honest, I could not really tell you what differentiated Differential Equations from other advanced calculus courses. I was a math major at the University of Vermont. Not an elite school by any means, but decent nonetheless. I entered college with eight credit hours of calculus through advanced placement, so I took Analytic Geometry & Calculus 3 as my first college math course. It wasn't until the fall of my junior year that I took Ordinary Differential Equations. Based upon this, I would suggest that very few high school math teachers would require any training in Differential Equations.