sykick

It seems to me that Russia's desire for the Crimea is rational and understandable. It was an accident of history that the Ukraine ended up with Crimea after the split of the Soviet Union. I understand that my belief on this might not be rooted in reality. What is your view of Russia's Crimean desires?

Collectively Europe outspends Russia in military investments. However, there is quite a bit of duplication in the individual European armies that make a straight comparison misleading. I suggest that military spending is not an accurate enough indicator of relative strength.

Defense spending alone is meaningless if there is not an accompanying ability/willingness to project power. Russia is certainly not shy about projecting it's power and European countries have been when it comes to Russia. Poland, Latvia, Lithuania, and Estonia certainly don't feel their security would be guaranteed in the absence of the United States being in NATO. Thanks to a successful Russian campaign to undermine confidence in the United States increasingly U.S. security guarantees are becoming less reliable.

The shift to a multi-polar world has begun. Will European governments rise to the challenge?

It is neither a big win nor a great improvement. At the time of need of essential goods/services price transparency is not as helpful as you make it seem. The American health system is so complex, has so many competing interests, and so many negative externalities that I doubt any one new regulation can be anything more than superficially beneficial to patients.

I suggest the obvious reason this new regulation doesn’t get as much press attention as you’d like has less to do with Trump and more to do with its futility.

Yes, the inmates are at least partly responsible for the rules of the asylum. There's an answer to a question about the loneliness in pursuing mathematics on mathoverflow.net that says, paraphrasing, that one should only do mathematics if they can't do anything else [1]. That is, if it is such a burning desire that you can't stop. As long as salaries allow one to live comfortably there will be a supply of people wanting academic jobs. The lifestyle is compelling for a lot of people.

I've been led to believe that even after tenure one still needs to be active and that means to publish. There is a lot of variation on salaries within a given department even at the full professor level. The more active/prestigious you are the more salary you make. Those who are active in research and with graduate students don't have to teach as many low level courses or be on as many committees. At least this is true of mathematics departments at large research institutions that I'm familiar with. I'm not a member of a large research institution but this is what I've been told by members of such institutions. My sample size is too small for me to generalize but it sufficiently large for me to hypothesize.

[1] https://mathoverflow.net/a/110063

You wrote: The point is simply that our debt levels aren’t that unmanageable,...

The cost of living is important as this and income level determines the affordability of debt. Just comparing income level and debt level between the two countries is not a deep enough analysis in order to determine if your proposed fix would work.

You are also ignoring the fact that a large number of people who go to college never graduate. These people are completely absent from your analysis of the problem. You are focusing only on the set of college graduates of both countries. I suspect that the rate of college completion for those who enroll at least one semester is much higher in Sweden than in the U.S.

According to your post 1/3 of U.S. college graduates don't have employer subsidized healthcare. This is a huge percentage and you are ignoring them in your analysis of relative debt burden.

Taxes are higher as a percent in Sweden than in the U.S. but in the U.S. at a moment's notice you can get screwed financially really bad in ways that you can't in Sweden. An ambulance ride in the U.S. costs much more than in Sweden and yearly out of pocket expenses for healthcare is much higher. The necessity of a car causes most people in the U.S. to have auto loans and unplanned maintenance costs can send people over the financial precipice.

I suggest that merely looking at the numbers you have does not give a clear picture of the relative situation in both countries.

Public transportation is very bad in the U.S. Even in major cities. For the vast majority of people a car is a necessity in the U.S. Most Americans end up in debt to buy a car. Then there is insurance and the opportunity cost of the government having to spend a great deal of money on road maintenance that could otherwise be spent on things like education.

When making comparisons with Sweden or other countries the cost of healthcare, housing, food, and transportation should be taken into account. Healthcare and transportation costs are much higher in the U.S. I think too we also have a higher per capita income and I don't know how things balance out. One thing for sure though. I'd rather be poor in Sweden than in the U.S.

There is also the fact that many students who start a four year degree end up never graduating. At the college I teach at it is clear that we are marketing to students who should not be going to college and shouldn't realistically expect to get a degree. But we need the money due to lack of state support and as a result standards for passing have gone down. What we have now is a system that has the incentives in the wrong place and this will have bad societal consequences years from now.

Like almost everyone else who comments on this topic I have ideas on how to make things better and like almost everyone else those ideas are largely naive.

Thank you. That’s very nice of you to say.

I deduce from your post that you have very little experience in teaching people at the level of beginning algebra. And while one might know geometrically what a parabola is there is a lot one must know before dealing with parabolas algebraically. I suggest that these things appear easy and obvious because you already know them and that you no longer remember what is hard for people learning this stuff for the first time.

I don’t show conic sections in elementary algebra. One typically really mentions the phrase “conic section” is pre-calculus which is 3 courses after elementary algebra. Over the past few centuries the order in which concepts are introduced has been developed. It’s not perfect but one should not be so quick to discount the way things are done without knowledge/experience in presenting these ideas to beginning students.

In elementary algebra students can do this for the most part:

Evaluate x^2 + 2x + 4 for x = 3.

Quite a few will struggle with:

Evaluate x^2 + 2x + 4 for x = –3

Almost all will struggle with:

Evaluate -x^2 - 2x + 4 for x = -3

I don't think they'd handle replacing x with z+2.

In elementary algebra they hate fractions. Many struggle with 4 – (–5). I like the approach in the article because of how it relates to factoring and difference of squares. This reinforces those concepts and shouldn't be too great of a leap at this stage. Also, there is a nice geometry behind the (b/2 + z)(b/2 - z) idea. The approach is nice precisely because it isn't a memorization approach. It's an approach that says, "Hey, let's analyze what factoring trinomials is all about and what the relationship between b, c, r, and s are.". It says, we know that r and s have to have the property that r+s is b and rs is c. We know this because of our analysis of multiplying binomials and our experience with factoring trinomials. We are using patterns and pattern recognition and from this we are constructing solutions to an equation that we can't solve by isolating the x like we did with linear equations. This to me, is true mathematics and it's nice to show students this exploration. To show how mathematicians think and approach problems. It's not a black box. It builds upon previous ideas and uses them to solve problems we weren't able to before.

Note that I'll be using this method in elementary algebra from now on but will not be using it to prove the quadratic formula. Indeed I will not even tell them what the quadratic formula is. I will save that for the next class.

Well for one thing the simple substitution you mention is a hard concept. They certainly will have a hard time with f(x-1) and introducing a new variable is a mental block at this stage. At my college we don’t introduce functions until the next course. I suggest your viewpoint is clouded by the fact that you know this stuff so well that you no longer remember what the pain points are for students learning it for the first time.

After learning to solve simple quadratic equations and then the quadratic formula we typically introduce variable substitutions to solve things like x^4 + 5x^2 - 6 = 0.

I doubt I can convince you. I’m just going by my experience teaching the topic. At the time students first learn solving such equations they have just been taught factoring and what it means to factor a trinomial. They know the product of the constant terms in the binomials must be c. It’s also easy to explain that the average of two numbers is the midpoint. And thus if I start with the midpoint then to get to the numbers I took the average of I add and then subtract some number from the midpoint. The geometry makes this easier to explain over using completing the square.

I’ve seen a shocking number of calculus students struggle with completing the square. The merits of the approach in the article are entirely obvious to me but like everyone else I’ve had my share of obvious beliefs turn out to be false.

I’ve taught elementary algebra for many years at a community college. The method described in the article is truly, in my opinion, very nice. I will be using this method from now on in elementary algebra. Students in elementary algebra don’t know what parabolas are. They haven’t been taught to complete the square and don’t know how to work with radicals. They have just been taught what square root is and basic binomial and trinomial factoring.

To me it is obvious that the method in the article is far superior than teaching completing the square. I’m teaching a pre-calculus course this semester and many of my students still can’t complete the square. Pre-calculus is 3 math courses beyond elementary algebra.

All of this is just my opinion of course and I have no data or studies to back up my opinion. I will be using the method described in the article from now on in my elementary algebra courses.

Having taught elementary algebra for many years at a community college I think your perception of what is complicated and easy are not correct. The method in the article is far easier for elementary algebra students than what you describe.

You’ve proved the quadratic formula. This is a formula students in elementary algebra will struggle with. It’s a formula whose proof will be lost on them. What is shown in the article is an easy to apply mechanism for finding the roots. The method in the article is one that I can use in the classroom. There’s no way I’d attempt your explanation in an elementary algebra class.

I teach mathematics at a community college. This includes teaching a lot of elementary algebra courses. These are pre-college level math courses. In elementary algebra we introduce solving quadratic equations by factoring. In the next course intermediate algebra we teach the quadratic formula.

Students in elementary algebra are not equipped to understand change of coordinates. This is too hard of a concept at that stage in my opinion. The method described in the article is a very nice one and is appropriate for elementary algebra. It ties in nicely with factoring and makes solving quadratic equations quite easy. I will be using this method in the future.