## Have Mathematics Exams Become Easier? 853

Coryoth writes

*"The BBC is reporting on a recent study in the UK that found that the difficulty of high school level math exams has declined. The study looked at mathematics from 1951 through to the present and found that, after remaining roughly constant through the 1970s and 1980s, the difficulty of high school math exams dropped precipitously starting in the early 1990s. A comparison of exams is provided in the appendix of the study. Are other countries, such as the US, noticing a similar decline in mathematics standards?"*Readers with kids in school right now may have the best perspective on changes in both teaching and testing methods -- what have you noticed?
## Pay teachers more (Score:5, Insightful)

We've noticed this 'dumbing down' (thanks Idiocracy) for a while now at Uni. The newer mathematics students enrolling in first year are lacking some of the basic skills. Example: a couple of years ago, trigonometric functions and identities were completely removed from the high school syllabus. It goes back all the way to year one at school.

I don't think teachers are being paid enough and they are certainly not valued enough by the community. Once upon a time, the best and brightest minds went into the teaching profession; it had respect and was highly valued. Now, it's whoever wants to become one, winner by default. The best and brightest need to be attracted back. Why would somebody who has the ability to earn more than four times the national average wage go into a job that earns less than the average wage?

## Re:Pay teachers more (Score:5, Funny)

## Re:Pay teachers more (Score:5, Interesting)

## Re:Pay teachers more (Score:5, Interesting)

Teaching the masses in free public schools has never historically been a profession ones chooses if they want to do well financially.The caveat is that you frequently have to go to grad school to be qualified to teach, and grad school prices are rising much faster than public school salaries. Of course housing prices and food prices are also rising faster than salaries. Every career that used to be "just enough to get by" is in danger of falling out of the bottom of the middle class. When you have something like modern public school teaching, where most of the potential creativity and chance to influence young intellects has been replaced with neck deep bureaucracy and a focus on preparing for the next evaluation test, there isn't even a "contribute to the community" sliver lining any more. Public schools in America are broken.

## Re:Pay teachers more (Score:5, Insightful)

## Re:Pay teachers more (Score:4, Insightful)

My favorite idea for "fixing" schools comes from Milton Friedman's book "Capitalism and Freedom". The basic idea is that the government would subsidize education and set some minimum requirements, while the actual schooling would be done by competeing private companies. Parents (or students) could choose which school the kids went to and, if they wanted, could add money on top of the subsidy.

It would solve the quality problem because schools would be competing with other schools. Nobody wants to send their kids to a bad school, so the schools would get better or they'd go out of business. It would also fix the teacher salary problem because better teachers would go to the better schools where they could make more money (hint: that would make them all try harder to be better teachers).

Before anybody yells about poor people getting screwed, look at the current system. Right now poor neighborhoods tend to have worse schools, and the parents in those neighborhoods have no choice but to send their kids to those schools. Under this plan there would always be the option of sending the kids to a better school across town if the nearby school got too bad.

## Privatizing *really* not the answer (long post) (Score:5, Insightful)

Let me start off by owning up to my bias -- actually, twofold. First, my wife is a middle school teacher, and I have volunteered in many different ways at her school as both elective teacher and simple extra pair of hands. Second, I have found very little in Milton Friedman's writings that I can wholeheartedly agree with. The man seemed to think that private enterprise was a panacea for all of mankind's various ills. He somehow seemed to miss the problem that the underlying profit motive is often at cross-purposes with many of the not-really-business areas he advocated for privatization.

To extend this and dig into the meat of your post, let's look at your postulation. Schools are, ostensibly, there to provide a public service. There is some real debate at certain levels in education circles about how much that public service really has to do with teaching, and how much has to do with daycare. No, I'm not just being cynical -- a large part of why schooling in the US plays out the way it does is because, historically, mandatory schooling for certain age groups was instrumental in allowing for the 9-5 working day for both men and women, which became very important during WWII.

So let's say we assume that schools are there to provide the public service of actually teaching kids, with daycare as a nice side-effect. Fine.

Now let's look at the theoretical private company under Friedman's model that would step in to fill this sudden demand for private education. It would ostensibly be a for-profit corporation, given Friedman's leanings, which means a number of things. For starters, the corporation's management is under a legal obligation to ensure that the company makes as much profit as possible -- by deliberately taking in more money than it costs to do business. This is diametrically opposed to how not-for-profit corporations (i.e. most private schools that I'm aware of) operate -- by deliberately spending all funds alloted in the budget for that year in order to ensure that the services provided are the best possible.

With those *very* different directives, a few moments' thought should be enough to show that any for-profit entity operating in the field of public services is going to provide the least possible service at the highest possible rates. We've seen that time and again, in country after country, in sector after sector. Medical services in the US? Check. Water utilities in the UK? Check. Power companies in the US? Check. Major ISPs in Australia, Canada, the US? Mobile communications services just about anywhere? Check.

Fobbing such services off onto the private sector produces other problems as well, as corporations are by their very definition protected by legal limits on their liability. Given the intimate roles that teachers play as

in loco parentis, it is important on many different levels that parents have a serious say in what happens at schools -- which is where PTAs come in. I could well be wrong, but I strongly suspect that no for-profit company would really allow a PTA to have much authority over what goes on.Part of the problem in the Friedman model is the simple issue of motivation. Why would companies suddenly spring up to take over the role of schools? Private schools that exist at present are there in large part because of an organic need in the community, combined with the presence of people with the motivation to be teachers. The Friedman pipe dream instead seems to be based on the profit motive, which is, as noted above, largely incompatible with public services. His model is also flawed in ignoring the very real geographical constraints of schools -- even assuming real market-style competition

## Re:Privatizing *really* not the answer (long post) (Score:4, Insightful)

I've tired to answer some key points, but it's nowhere as elaborate as your post..

As I understand his theories, they were actually misunderstood by the politicians implementing them as a panacea to all thier ills.

The trick is to align the profit motive with the actual task at hand. When private companies are paid to run buses in Copenhagen, they are (as I understand it) required to run certain routes at certain frequencies. They are not required to run a service that customers will want to use. Thus, bus service is plentyful, but sucks, and most people will rather bike 15 km in the rain than set foot in a bus.

Similarly, if you subsidise a school according to grades (e.g. you're only paid for >B average students), there's a motivation to neglect the ones that take too much effort to pull above B, or to pressure teachers to over-grade. If you subsidise per student-attendance-day, well, then you create a motive to be a great day-care center.

These are all very high barrier-to-entry industries. A private school can be six kids around a kitchen-table and their parents taking turns as teachers, so while your reservations hold (mostly - most private telecommunications businesses are orders of magnitude more customer-aligned than in their government-past) true for the mentioned businesses, they don't for schools.

Because they can operate the same service at a lower cost, which means money in the pocket.

The more money a business pockets (or sinks in inefficient operations), the more likely it is that a more efficient competitor will appear.

The idea it to create true competition, and true competition means that a loser will lose something real, and the winner will win something real. In a public pseudo-competition, the fight is only for prestige, in private it's for actual money, and people tend to be a bit more rigorous with their money that with their prestige.

("MY school has a superior athletics program" - "Oh yeah, MY school has a better library" - "Oh look, our salaries are exactly the same" - "How about that, let's play golf")

Way to sustains a negative spiral. In a private system, parents, not conjunctures, decides funding.

If you're living in a neighbourhood where the land value declines - if the school is good, you'll keep your kids there, and the school will keep it's funding. If it's struggling, you might even make a donation with the money you saved from property taxes. Now there's a cheap neighbourhood with a good school => more kids => more money.

Competition. Real competition. To win, you must continuously improve yourself. Significant innovation and progress is risky, and is generally awarded.

Responsibility and accountability comes when irresponsibility means losing your job tomorrow, not in four years, and then only if someone will run against you.

## Re:Pay teachers more (Score:4, Insightful)

1. Transport. You just know that "poor" schools will be in poor neighborhoods. Now, poor people don't tend to have the money to drive their kids to school (they most likely have to leave their house before their kids even to get to work somehow), so poor kids would have to either go to those schools or be transported somehow to the "better" ones. How do you plan to solve this problem?

2. "Money on top" from the parents. What should this money pay for, if there is already a standard set? Additional credit? Better teaching material and/or teachers? No matter what that money pays for, it gives the children whose parents can pay some sort of advantage. How does this not disadvantage the children of poor parents?

3. Is an extension of 2: If there is a standard set, why should anyone have to add something on top of it? There are only two possible systems, either the standard is so low that this is necessary (which basically means again that you have "rich" and "poor" schools, because no poor person could afford topping off the governmental funding, thus having to resort to cheap (and bad) schools), or the standards are adequate which in turn raises the question what the money should pay for.

Basically not a bad idea, but you just know how it will turn out: Good schools will require you to fork over extra money, so they can hire better teachers and get better equipment, which no poor person can afford, and the dregs will be left over for the poor kids. That won't change a thing.

## Re:Pay teachers more (Score:5, Insightful)

I wasn't exactly in the public sector, but it was one of the cheaper and thus more popular private vocational colleges. My already modest expectations were far beyond what this enterprise was offering, which is probably why all the grads wound up either in brainless government jobs (lucky them), or call centers.

The day we rid schools of the financial burden, is the day they will start churning out smarter grads.

## Re:Pay teachers more (Score:4, Interesting)

## Re:Pay teachers more; increase top tax rate (Score:4, Interesting)

That's been a bad trend over the past

It's the same "root cause" as teacher's not being able to afford to live in the communities they teach in. Not enough resources into education -- too many resources invested in high-end of life and the adult stages (including, recently, this war that is causing oil prices to go up (war->deficit spending->'printing' money (how close is US debt to 3 T$ (Tera-$)?)->dollar deflates in value as massive 'unbacked-money' is created, commodities (incl oil) go up) -> US goes bankrupt)). But look at how much the rich spend on luxury goods --- increase in cruise ships, vacation spots -- extremely expensive hobbies/sports...so much wealth concentrated in top 1% people -- but it's the 'masses' that are taught in schools -- and that's where the dollar share has been shrinking the most.

There was an opinion piece in the WSJ that tried to show how increasing the top tax rate didn't increase the government's tax-income as a percentage of GDP -- what it unintentionally showed, actually was GDP going up as

the top tax rate rose, and GDP going down as it fell -- so the % going to government appeared level. GDP going

up or down reflects almost directly goes into a rise or fall of the "standard-of-living" of the nation. That meant that as the top tax rate fell, the average standard of living for the nation as a whole fell -- and vice versa.

GDP has fallen to lowest levels in my lifetime under the top tax rate falling from over 70% to the 20-25% it is now. All that was Reagan-& the Bushes rolling back taxes on the rich while using government deficit to inflate the economy. While Clinton didn't raise taxes -- he did manage to get the deficit from around 2-billion to almost breaking even by the time he left office -- now it's up higher than ever.

Bush needs to be out of office so yesterday. I think my postings are too long and people don't get this far...

*sigh*...just supposed to shut-up while the nation is tanking to hell...

## General request! (Score:5, Insightful)

## Re:General request! (Score:5, Funny)

## Re:General request! (Score:5, Funny)

## Re:Pay teachers more (Score:5, Insightful)

## Re:Pay teachers more (Score:4, Insightful)

Rather than algegra, geometry, etc., as discrete courses, they get jumbled together and reintroduced each year through 3 years. The problem is that there's never enough of it at any given time to actually stick, so you get a lot of students who are going through the motions.

On top of that you get a lot of group work, which basically ends with the one or two students that actually get the course material providing answers to the entire group.

In an atmosphere like that, where the basics aren't really ever taught, I'm not really sure that most students could cope with anything particularly challenging. And that's not even bothering with the switch from more more theory and analysis to more focus on useless proofs.

Proofs can be valuable, but only when the students are being taught to understand the reason why certain corollaries, postulates and theorems have been put forth. Mindlessly regurgitating them without an understanding of the implications isn't particularly worthwhile.

## Re:Pay teachers more (Score:5, Insightful)

Not only do teachers not get paid enough to attract and retain the good ones, but teachers unions and the fear of lawsuits make firing the awful ones nearly impossible.

## Re:Pay teachers more (Score:5, Informative)

## Re:Pay teachers more (Score:5, Informative)

Other than the 10 weeks or so off in the summer, teachers don't really work that little. Most teachers I've known (including mine) put in around 10-12 hours per day and a good chunk on the weekends. Okay, any good teacher. Plus you have to add in all sorts of meetings and weekly side deals all over the place. Once you start doing any extra-curricular activities for your students you're pushing 60-70 hrs a week. No thanks.

## Re:Pay teachers more (Score:5, Informative)

God this sentiment irritates me.

You are evaluating the work teachers do based on your experiences as a student.

When class is in session, I work far more hours a week than my software development friends. Basically, I can't play when school is in session. There literally is no end to what needs to be done.

When class is not in session, I can scale back to about 40 hours a week.

I'm so sure you think that just because students aren't at the school, teachers have nothing to do (eyeroll).

Part of the reason it's hard to get good people to teach is that it's an abusive amount of work for very little pay.

I only taught at the K-12 level for 2 years before I said "screw this." I'm at the uni level now, and while the hours don't go down much, the pay goes up a lot. Also, you don't have to deal with parents!

## Re:Pay teachers more (Score:4, Insightful)

And she would have gotten a lot less stress too. I can't count how often she had to stay late to help a student (sometimes only to have that student not show up) or how many times she had to deal with an irate parent. ("What do you mean my kid didn't get an A? I want my kid to get an A! It's your fault my kid didn't get an A.") She was in a private school and many of the parents seemed to think that, because they paid for school admission, they owned her and were entitled to have their kids on the honor roll. Yes, being on the honor roll was thought of as automatic by parents, not something students earned through hard work and good grades.

She got out just in time too. Apparently, a couple of teachers (good ones, mind you) have been let go because that same group of parents decided to organize to "get rid of" teachers they had a beef with. My wife, on a visit back to the school, overheard some parents discussing which teacher to go after next. When teachers face working conditions like you described, lousy pay, students who don't want to learn, and parents who could care less so long as the teacher gives their kids A's, of course the good teachers will wind up leaving. I'm really fearful about the kind of education that my kids will get. I can only hope that they either wind up with new teachers (who have not yet been beaten down by the system) or are lucky enough to get those rare "diamond" teachers who seem to stay great no matter what pressure the system heaps on them.

## Re:Bludge? (Score:5, Informative)

## Re:Pay teachers more (Score:5, Funny)

## Amateurs (Score:4, Funny)

## Re: (Score:3, Insightful)

## Pay math and science teachers! Supply and Demand. (Score:3, Insightful)

## Back on subject... (Score:5, Funny)

## Stereotype vs reality (Score:4, Interesting)

Interestingly the High School that she is going to is aware of that teachers failings and identified all of his students as likely needing extra help in ninth grade.

This was supposed to be the GATE class too. Now most of these advanced math students have lost the edge they had and are behind other GATE students in the district.

Glad they got rid of him.

## Re:Pay teachers more (Score:5, Insightful)

currentcrop of teachers, the ones the pay increases are supposed to filter out by attracting more capable teachers.## Re:Pay teachers more (Score:5, Insightful)

...can keep up with theshearnumbers of parent classified geniuses.Clearly, your English teacher wasn't paid enough.

But other than that, the problem I see in this country is that the consumers of education have no choice. And like in any other monopoly, the provider gets away with poor quality.

## Re:Pay teachers more (Score:5, Informative)

## Re: (Score:3, Insightful)

## Re:Pay teachers more (Score:4, Insightful)

the majority of people teaching mathematics aren't experts.Um, So? Why should a teacher master differential equations to teach algebra? I'd rather have a good teacher that knows enough math, than a great mathmetician that can't teach. When you require degrees, you restrict more than you enable. My high school physics teacher had a biology degree (and wasn't the biology teacher, but did teach chemistry) the French teacher had an economics degree. The economics teacher had a masters in political science, but no high school diploma or bachelors from college. Oh, and the political science teacher had a degree in education, not political science. Amd they were all good at what they taught.

Throw in the fact that mathematics is one of those subjects where a student can be permanently set back by just one bad teacher and you have a decent part of the problem.Which is why you need a person at the front of the class that connects, and their knowledge of the material is secondary. I know from personal experience tutoring, that I've actually tutored someone successfully in a subject I had no knowledge of. I talked them through, asked them questions, and they were able to learn what they needed with direction, but not someone just giving them answers. Math teachers eed to be teachers first, and mathmaticians low on the list, at least until up until the last coule years of high school and beyond, where the math gets more complicated.

## Re:Pay teachers more (Score:4, Interesting)

Knowing more advanced math is not

requiredto be able to teach high school mathematics; it does, however, make a teacher better able to teach high school math by giving them a better and richer understanding of the material they have to teach. Sure, a terrible teacher who has that extra appreciation of the material isn't a great substitute for a fabulous teacher who doesn't, but in general I think we can expect the distribution of quality teachers to be roughly the same between those who seek a degree in math and those who don't... given that, on average, those with a degree in the subject will be that much better.## Re: (Score:3, Informative)

Ok time to take a Karma hit for telling the truth. Minorities have been screaming for years that the SAT Math section somehow discriminates against them because their scores are so low. So they had no choice but to dumb them down. The more you lower the maximum score, the more equal everyone's score is.

Welcome to the wonderful world of multiculturalism and affirmative action.

## Re:Pay teachers more (Score:4, Insightful)

So "they" had a choice, and blaming minorities is just a way of deflecting from lack of investment in combating the real problems: Poverty - because the common theme when it comes to who underperfom is social situations, not race -, and too little investment in education.

## Re:Pay teachers more (Score:5, Interesting)

Part of the problem, as seen from my view, is the complete and utter dependance on calculators, especially those fancy, programmable Texas Instruments ones, that can practically do the work for you. I have one (it was considered 'required course materials') that I have used maybe a handful of times, preferring my old two-line Casio scientific calculator, particularly now that I know what the little cursive 'i' does.

## Re:Pay teachers more (Score:4, Interesting)

Sorry but I don't believe this is the case at all, the culture of "pay me more" is bullshit. Many teachers and experts can't teach, but there are those in both groups who can. Paying teachers more is not the issue in many places, in Canada highschool teachers after a good decade or so can pull in 60,000-100K per year and student disengagement is at recrod levels. The idea that the private sector will 'solve everything' is also bullshit, it's cultural and it's complicated, people have made the same argument your making throughout history, yet the same problems occur you're not a unique snowflake here.

The problem is really about the culture itself, it goes deeper then that though it's north american insitutional and business culture that is the problem. See here:

See: http://www.youtube.com/watch?v=gG3HPX0D2mU [youtube.com]

Listen to the comments of "calcification" of kids in the school system and adults in the workplace. It makes a lot of good points about self management and responsibility.

I don't agree that all kids are just "lazy", they are disengaged because most of the time we don't allow their curiousity to blossom by killing it early through 'school'. The other problem is that we don't have a place for certain kinds of people in the job market that will pay decent wages. That is the REAL problem, technological displacement, and trying to achieve the impossible (i.e. raising the bar and expectations to unreasonable levels and then being disappointed when kids don't meet them)

Modern schools are often harmful and disengaging enviornments, for many it's positively toxic to someones development. No amount of paying teachers more, or accountability will deal with forced schedules and irrelevant curriculum, the lack of alignment of student curiousity and interest with what they want to learn vs the boring pablum clueless teachers, businesses and government elites, pushing their pablum as 'education'. Many slashdotters can no doubt attest to the low quality of the curriculum and their teachers and school simply not being relevant to what they are interested in, so they 'carve their own path'.

I think something is to be said by not killing childrens motivation and curiousity, which we do very young.

## Re:Pay teachers more (Score:5, Insightful)

I would really like to know what others think. Btw, I am in the US (as one poster asked people to add)

## Re:Pay teachers more (Score:4, Insightful)

But why don't they work? I don't know where you're from but around here one of the major reasons private schools get better results than the neighbouring public school is that most of the private schools have the ability to select the students they take. They take the bright kids and these kids do well. The little shits who don't want to learn / ate lead paint for the first six years of their lives end up concentrated in the public schools which can't refuse them.

All your solution does is increase the education gap between the high-achieving kids and low-achieving kids. I think that goes against the whole point of compulsory education, which is that a rising tide lifts all boats.

## we don't want to upset them (Score:5, Insightful)

## Re: (Score:3)

## First post! (Score:5, Funny)

## Re:First post! (Score:5, Funny)

## Re:First post! (Score:5, Funny)

Don't bruise his self esteem you brute. /. in the 21st century, every post is first post.

Here on

## Re:First post! (Score:5, Insightful)

## Re:First post! (Score:5, Informative)

Studies have found that children are more likely to do better in school if they believe that they're...well, better.Er, no, that was a popular idea in the 60's but recent science [google.com.au] has shown that as students grow up with false praise (to make them think they are better) eventually (early/pre-teens) they realize they're being lied to with counterproductive results including low self esteem and social problems. In the long term it's best to be perfectly honest; by all means praise specific accomplishments at but don't pretend they're doing well when they're not.

---

Stop using tab characters in your code!## Re: (Score:3, Funny)

## Finally (Score:5, Insightful)

Yet every year the exam results get better and the government congratulates itself on improving standards while denying the exams are getting easier.

## Re:Finally (Score:5, Funny)

## Re: (Score:3, Insightful)

## education policymakers need to look good (Score:5, Insightful)

## Re:education policymakers need to look good (Score:5, Interesting)

But maybe we're just being weird here.

## tools (Score:5, Interesting)

doing the math is going to be easier, even if they didn't ask harder questions. However, the amount of automation these days means that most people aren't ever going to have to do the harder math in their daily lives.

Slashdotters are an anomaly because our careers and interests require us to do maths all the time. If the future historians are allowed to slack off on their trig tests, so what? They weren't going to be engineers anyway.

They probably should track out classes more than just "regular" and "honors/AP" though. That way the future nobel prize winning poet who is an over acheiver and the future NASA scientist don't have to compete for the teacher's attention to detail in Calculus.

Just a suggestion.

## Re: (Score:3, Insightful)

Brand X: "Buy one, get one free!"

Brand Y: A few pence cheaper, and a larger pack too.

Brand Z: "25% off!"

How many people today can't work out which is best?

(UK supermarkets even do most of the work for you, below the price for every product is printed something like "1.50 per kg", so it's

veryeasy to compare prices -- you only need to work stuff out if there's an item on multi-buy promotion, in which case t## Re: (Score:3, Insightful)

And yeah, I do carry my TI-89 with me, but I'm an Aerospace Engineer. Without that, my mechanical pencils and my ID card I'd be naked!

## (Anecdotal) Proof of Impact? (Score:3, Interesting)

When kids start Algebra I with a TI-89 that is drawing tangent lines and running linear regressions (in between games of tetris) for them, they don't learn any of the basic skills.

## Re:(Anecdotal) Proof of Impact? (Score:4, Informative)

Purchases from a modern store are not as simple as "I give you some money, you give me some product", even if it may look that way from the point of view of the customer.

## Re: (Score:3, Informative)

doing the math is going to be easier, even if they didn't ask harder questions.

I agree with you; sporting records are regularly improved upon, but no-one is complaining about sprinting becoming easier.

That said, in the linked PDF a 1951 question is stated as:

Solve the equation:

9 * (1-x^2)/(1+x^2) - 7 * 2x/(1+x^2) = 3

A 1970 question is:

Show that (x â" 3) is a factor of

x^3 - 5x^2 - 18x + 72

then find the three points where y = x^3 - 5x^2 - 18x + 72 meets the x axis

While a 2006 question is:

Find a and b when

x^2 + 8x + 21 = (x + a)^2 + b

Use your answer to find the minimum value of

x^2 + 8x + 21.

I can see why someone might say the 1951 question was harder than the 1970 question which was harder than the 2006 question.

## Re:tools (Score:5, Interesting)

Calculators on tests have made tests easier, but this is a good thing. Can you imagine having to figure out sines and cosines by hand anymore? What calculators do is make it easier to get to more advanced topics. Knowing how to add 1234+2345 in my head is just no longer a necessary skill. I rather students practice the properties of math, and write things out on paper anyway. (#1 problem with algebra and calculus students, they try to do too much math in their head) Calculators are not going away any time soon, and anything that encourages the entire population to do more math is a good thing in my book.

Secondly, while historians may not need to know trig, it is imperative that as a nation we raise our mathematical abilities. Great math and science students generally did not learn everything at school. Having a parent that can help out with some algebra homework (or even better understands the value of math) will make it much MUCH more likely that the child will grow up with an appreciation of mathematics. If we as a human race want to push the maths and sciences as far as we can, then we much raise the math and science ability of the entire population.

Just so everyone knows where my loyalties lie, I am a mathematics major, I am a math tutor, and hopefully eventually you'll see me teaching mathematics at a University near you!

## We used Math Handbooks... (Score:3)

## Re:tools (Score:5, Informative)

Take an equilateral triangle of side length 2. Cut in half, so you have hypotenuse length 2, base length 1, and vertical length sqrt(3). Now you can find the cosines and sines of both 30 and 60 degrees (or pi/6 and pi/3 radians, respectively).

Now take a right angle triangle with base and vertical length 1, and hypotenuse length sqrt(2). Now you can find the sine and cosine of 45 degrees (pi/4).

So with a few simple skills: basic geometry, SOHCAHTOA, Pythagoras's theorem, you can find the sine and cosine of 3 different angles. Now learn your CAST rule (where the different trig functions are positive based on the quadrant) and you can do it for up to 12 different angles. Then learn your double angle formulas and you've got another 4 angles. Then learn the period of trig functions and you can now find it for any of those 16 angles plus the period of the function. Anything other than that, and yes, you'll need a calculator, but knowing those rules (which can be taught progressively throughout high school) and you'll find doing certain things much easier. Now, granted, trig isn't for everyone. However, it's not unreasonable to expect people to do certain calculations sans calculator. Like multiplication, addition, and division.

## Re: (Score:3, Interesting)

I definitely agree with you, when tutoring trig students the first thing I do is have them reproduce the unit circle and the trig values for multiples of 30 and 45 degrees. Trig exams force students to learn this by asking things like, what is the sine of 60 degrees? sqrt(3)/2 is the correct answer, but generally not one that most calculators can spit out.

However, eventually when most calculators are able to spit out an answer like sqrt(3)/2, students may no longer need to know their 30-60-90 and 45-45-90

## Re:tools (Score:5, Insightful)

Sure, not everyone is going to go on in mathematics; some will be poets, some will be historians, and so on. It is also true, however, that most people don't have their future that well written by the age of 16, and having a solid enough background in a variety of subjects, including mathematics, literature, and history, to be able to keep future options open to exploration is important.

## Good Timing (Score:5, Interesting)

## That time of the year, already? (Score:3, Insightful)

pass rates go up - exams are getting easier. education system in decline

pass rates go down - teachers not able to communicate with students. education system in decline

## Blame? Look at the No Child Left Behind Act (Score:5, Interesting)

Kids aren't dumber, they just aren't given the opportunity to fail. If they aren't given the chance to make mistakes, they don't learn from them, and unfortunately, that is where the NCLB is leading us.

## Re: (Score:3, Funny)

We in the U.S. did the same thing to the Presidency eight years ago, and have gotten similar results.

## Re: (Score:3, Interesting)

I'm a licensed teacherSince surely this is at least in part teaching issue, and seems to be commonplace in most of western society (not just nations with the NCLBA or equiv.) doesn't this really suggest a drop in teaching quality/ability? Are we handing out 'licenses' to the wrong people, or too easily (to people with integrity issues who bend to NCLB standards) or some other flawed way? Should we be handing out 'licenses' at all? Can teaching ever be taught, really? Are teaching unions a help or a hindr

## For the non-united-statesians... (Score:5, Informative)

For people not in the U.S., NCLB is the controversial No Child Left Behind [wikipedia.org] act.

As I understand it (I once dated a teacher,) the history of NCLB is basically:

Congresscritter 1: We should improve education.Congresscritter 2: How about we tie test scores to school funding?Rational Person: Wouldn't that just inspire schools to change the tests in order to improve the scores and maximize funding? That's far easier than improving the quality of education, yet it has the same rewards under NCLB.Congresscritters: Shut up! We've got pork in this bill now!## Re: (Score:3, Interesting)

I

wantstandardized testing to make teachers "teach the test" -- so long as the test covers all the material we want students to understand, that's an ideal outcome. It gives schools and teachers and objective reference to determine if their curriculum is complete and accurate. And the## Re: (Score:3, Interesting)

## The teaching of math and science are doomed. (Score:4, Interesting)

While this is true, very few actually pursue higher learning in these fields because all of the emotion and excitement is gone when math and science are taught in this way. The wonderment that inspired so many young engineers during the space race is gone. Teachers need to address and emphasize the larger concepts to get children excited about math and science.

## The "dumbing down" and muddying of math continues. (Score:4, Interesting)

Because the government education establishment in many places has given up on any attempt to maintain the tried-and-true approach to math education that has been employed in the past - building skills step-by-step in such a way that the student's "toolkit" grows in a logical fashion through the different skills, now they are left with a very fuzzy approach that doesn't really build anything on anything, and mostly is concerned with keeping busy doing something that they can pretend is math and pretend that some sort of progress is being made.

The most tragic part of it is that the kids who would have been the real math enthusiasts under traditional teaching methods never get the chance to see the order and beauty of math, because curricula like this completely hide it.

For more info on this, see the Web site mathematicallycorrect.com .

Because the poor government "education" establishment is failing to really teach math, of course they have to put a happy face on the situation by dumbing down the tests too.

## No one is going to say (Score:4, Insightful)

This is because it would be false. You might get arguments about the extent of the change, but none on the direction.

And nothing in education will ever improve in the US as long as the system is union-controlled.

## My unprofessional anecdotal experience (Score:4, Interesting)

## Students are dumber (Score:5, Interesting)

I agree with everyone else, we need to pay math teachers more. In states like TX a public school teacher makes barely enough to live poorly, and with a math degree, they can make double working in private industry. It is a very hard sell to convince mathematicians to go into education.

The other thing we need to do is not be afraid to actually fail someone. This society has made it so that everyone feels its their "right" to graduate high school and go to college. We need to change this and actually fail people when they can't do the work. If someone doesn't earn a degree, they shouldn't be "awarded" one.

## Mensa and testing... (Score:5, Interesting)

## Re:Mensa and testing... (Score:5, Insightful)

On the other hand, they should be smart enough to know that the SAT was never meant to measure your IQ. In fact, they should be smart enough to know that IQ tests themselves only measure certain abilities, and are not really a good measure of intelligence.

I normally score around 135 in IQ tests (of course it depends on things like time of the day, quality of sleep on the previous night, BAL, etc), and in my opinion IQ tests and Mensa-like organizations are only good to inflate egos, as they have little relevance to real life.

By the way, did you know that "mensa" means "fool", "stupid", or "jerk" in Spanish? [wordreference.com] How fitting...

## Re:Mensa and testing... (Score:4, Funny)

x(n) = [3 * x(n-1)] + x(n-2) , where n>3

So it's 313 next, right? Next question please

## Easier or more straight forward? (Score:5, Interesting)

The newer example questions seemed more rationalized, they test whether you know the theory or formula needed to solve the question without throwing you a curve ball.

Would you rather encourage people to continue studying onto more advanced levels with easier tests, or throw them a GOTCHA question which will totally turn them off to the subject matter?

There is a difference between testing knowledge of the subject matter, and giving the test taker a hard time. A "difficult" question might be great to ponder when you have unlimited time, but in a time pressured test, it is not appropriate.

## Re:Easier or more straight forward? (Score:4, Insightful)

facts about mathematics. The earlier questions require you to actually put together a multi-step process to get to a result rather than hand-holding you through it. They also tend to require you to actually lay out the line of reasoning you had to use. That actually requires some mathematics -- actually using and mentally manipulating abstract objects in a logical fashion; constructing lines of reasoning yourself to solve problems rather than just using fixed recipes. I'm not saying the early exams are perfect, but they do have a very distinct requirement that the later ones do not -- they require you to actually think and reason. The later tests are akin to history tests that are nothing but questions like "In what year did Columbus sail to the Americas"; they only require you to be able to regurgitate facts. Now such history exams exist, but they suck too. A real history exam should test your understand of meaning of events (both contextually at the time, and for us today), not just raw facts about events. Likewise a real maths exam should ask for more than just regurgitation of facts about mathematics.## Maths has changed / evolved... (Score:5, Interesting)

Maths in the 1950s was designed for engineers and scientists in that generation. They learned what they needed.

Maths today is exactly the same. The fact is you can't use 1950s standards to evaluate today's exams any more than you can use today's standards to evaluate 1950s exams.

The only real question is - Are engineers and scientists finding their maths education weak?

The answer in my view is no in most cases. In a limited number of careers the maths they received isn't nearly advanced enough but that would have been the case in the 1950s too.

As I said they're using the wrong measuring stick to measure the difficulty of exams. Nobody needs to know half of the useless junk that kids learned in the 1950s when frankly it is less time consuming and more accurate to use a calculator.

That's just my opinion. I honestly think a lot of this kind of "research" is a result of much older people looking at today's maths and thinking "Why aren't they learning what I did?" While completely ignoring what they're learning that the 1950s students didn't.

## I think you are quite right (Score:4, Insightful)

Well, since education used to be so much heavier in the memorization it is no surprise that the tests are "hard" for people today. I remember getting in to this argument with someone I knew. They'd found a test posted online that was a highschool graduation test circa 1900. They used it as an example of how much "harder" school was then and how I couldn't pass it. Well, turned out I could, but only because I'm a trivia junkie. I know lots of useless facts, and there was a whole lot of the test that was full of it. The geography and history sections were nothing but. Things like matching capital cities to states.

Ok, well that's neat and all, but it is quite thoroughly useless. There is no reason to know that. If you want to, great, but don't pretend like it is useful knowledge or that you are smart because you can do it.

So ya, people today had trouble passing the test, but that doesn't mean the test was hard, it meant the test was different.

## Re:Maths has changed / evolved... (Score:5, Interesting)

Absolutely. An engineer in the 40s/50s would need to have in-depth critical skills of geometrical proofs and relationships, nasty algebraic manipulations, and "bag of tricks" mathematics like series approximations, dummy variable substitutions, etc, because computing resources were rare and resource intensive. If you look at the older tests linked in the OP, you can really see a reflection of that need.

As an engineer today however, I have zero need for knowing trig simplification identities, calculus proofs, and the like beyond a high conceptual level, but I have far more need and usage of logical and discrete math fields, programming concepts, vector operations, statistical methods, and other "math" topics that are still completely absent from any high-school math curricula that I've seen.

My wife (a math degree and former teacher) suggested throwing out the "calculus path" of mathematics entirely and retool math education to a "discrete math path". It sounded heretical to me initially, but I've come to believe that she's correct.

## Re: (Score:3, Interesting)

## Re:Maths has changed / evolved... (Score:4, Interesting)

Here, you've provided me an excellent demonstration of

whywe need game theory in high school, because your post neglects the vitally important concept of opportunity cost [wikipedia.org], something that I'd much rather the general populace had exposure to than something as useless as trig identities. The opportunity cost of teaching trig identities when youcouldbe teaching, say, opportunity costs, is way too high. Trig is not even close to thebestthing we could be spending our time on.Besides, optometrists and surveyors are invited to take specialized courses in trig identities, just as the mere fact that I took a course on the mathematics of evolutionary computation doesn't even begin to imply that everybody in high school should learn about that stuff. Time is finite. Opportunity costs are important. Trig identities are too expensive and displacing a lot of stuff that is

bothandmore useful to mathematicians, who, like I said, don't consider them important.(Actually, the disconnect between real mathematicians and mathematical education is truly staggering once you fully understand it. The educational community, and I say this with full consideration to the people involve, wouldn't know math if it bit them on the ass.)

## Re:Maths has changed / evolved... (Score:5, Insightful)

As an engineer today however, I have zero need for knowing trig simplification identities, calculus proofs, and the like beyond a high conceptual level, but I have far more need and usage of logical and discrete math fields, programming concepts, vector operations, statistical methods, and other "math" topics that are still completely absent from any high-school math curricula that I've seen.And I'm afraid that you are, indeed, a victim. You see, the reason why you learn geometric proofs and calculus proofs is to assist with developing problem-solving skills that require an individual to reason a problem from start to finish, much like real life. It scares me that you claim, as an engineer, that all you need to know are the rote mechanics of math (and yes, that is what you describe: number crunching as opposed to critical problem analysis).

Unfortunately, at least in the US, proofs of any type are becoming rare to non-existent in many curricula. I see the direct result of this every day I'm in school and a student stares at me with a blank look on his/her face when I ask him/her to analyze and determine the best course of action for solving for some quantity X given Y and Z.

You didn't mention what type of engineer you are. Computer/software/hardware, perhaps? Then yes, I'd agree that programming logic, vector operations, and the like are probably a valuable intellectual commodity. But I know many engineers who work day in and day out

designing things, and this takes more than a simplistic knowledge of how to perform statistical computations.## Re: (Score:3, Interesting)

Maths in the 1950s was designed for engineers and scientists in that generation. They learned what they needed.

Maths today is exactly the same. The fact is you can't use 1950s standards to evaluate today's exams any more than you can use today's standards to evaluate 1950s exams.

Well, perhaps the people to ask as to whether things are going well or not are professional mathematicians, physicists, and philosphers of mathematics. Conveniently, a bunch of them are busy discussing this over at the n-Category Cafe [utexas.edu]. And yes, there is an element of some material beign dropped in favour of other newer material. It's worth noting, however, that there is a real concern (particularly by Tim Porter and David Corfield) that core material (that is, the essence of mathematics) is being lost in t

## Its called "How to cheat mandates" (Score:5, Interesting)

What it all boils down to is that no matter what standard the Federal Government tries to set someone tries to cheat it. That is why there is always such an uproar versus standardized tests. Down here in Georgia they failed nearly 40% of all students in tested grades versus a standardized test. They knew it was coming. They even had practice tests. Is it all the schools fault?

No. Students seem have this sense of inevitability. They are still of the belief that they don't have to. After all anything else they complain about in school gets changed. I don't see their attitudes as defeatism, its entitlement that they suffer. They don't have to do this, that, or what not. We don't have the right culture in schools, especially city schools among minority students. Until we change the fabric of society the MTV generations will forever think themselves above "working hard". They are all going to be rap starts, professional sports players, or worse win the lottery!

We gave up control of our schools to "feel gooders". Now its all about grief counselors and no winners allowed because no one should be a loser. When we removed the reward of success what did we expect? I have seen articles where every student got to walk the diploma line regardless if they graduated just so they didn't feel ostracized. Well tough shit. Your boss ain't going to worry about making a failure to feel good. If you don't perform your in for a world of hurt. I guess you could go into government work, of all categories in the job market they have added more jobs than anyone and everyone knows the saying about how its near impossible to lose a government job.

Schools and students are simply trying to cheat the system. The problem is the schools encourage it because they don't allow for losers. They don't want to hurt little Bobby's feelings so they set him up to fail in life. If they want control of our kids then they should be responsible for them. They get hell bent if someone raises a finger about the Bible in school or complains about sex education yet they are completely aloof when it comes to holding the kids to a standard of education.

Private school was the only recourse I found. Standards had to be met or we might not be allowed to come back. Students were encouraged to be better. I don't see that outside of a few select public schools; you know I hear it all the time how so and so's public school isn't like those others but sorry it is.

## Poor math skills of 1st year physicists (Score:3, Insightful)

## Yes, yes, yes and partly no. (Score:5, Interesting)

Maths has definitivly become a lot easier here. It takes a lot less work to get good grades now, and there's an alarming lack of focus on basic math skills. There's plenty of A-students who can't do basic math. The norwegian school-system is

reallyfucked up though. There's so much focus on getting the trouble-makers through school, so they're allowed to basically take over classes. I mean, we don't want to send them to special schools, because that would stigmatize them! Never mind the 25 other students in the class, they'll just have to sit there and feel neglected.. Not to mention, without consequences these students never learn. I've had students yell at me straight off at 08:15 in the morning because the last test had some questions which weren'texactlyas the ones in the book. They're so mal-adjusted and unfit for real life it's scary.. (ohh.. and just for kicks.. 90% of the worst students are pakestani.. while they make up about 3-4% of Oslo in total..trying to teach them anything is basically a crash-course in becoming a racist)That said, I work with a couple of really old math teachers, and there's a few subjects like probabilites that are completely new them.. so math has changed. Don't be fooled though, they've replaced all the hard'n'gritty stuff with fluffy feel-nice stuff.

In Norway, we've had two big reforms in the last ten years, and both made the hardest paths easier. Ironically, they also both made the maths for students taking vocational education harder. It's so tragic I want to cry :(.

## Then/Now (Score:5, Funny)

Now: Chloe has 7 apples. How many apples does Chloe have?

Tomorrow: Write the number 5.

## Math education in the US is *officially* flawed (Score:3, Interesting)

In our district, the nonsense stops in high school (which is administratively separate), and and I actually think my ninth-grade daughter is learning more math than I did at the same age. But you have to survive elementary and middle school math to get to the high quality teaching. It's such a waste.

## Musings on school in general (Score:3, Insightful)

The biggest is that the school system is not a great way to learn stuff. I remember (but bear in mind that I'm your average slashdotter, not your average person) at a fairly early age drawing 6x6 grids which taught be that 7 has probability 1/6. I remember my father drawing circles in the sand with dots in the center, explaining the basics of chemistry (and he's not a chemist), and me completely getting it.

I remember at age 14 (laughably late by slashdot standards) that a person I knew had written a program that played chess. Being a moderately skilled chess player at the time (1390), I thought that was awesomely cool and wanted to do that myself. That got me started writing C (I had dabbled in

Where am I? Studying CS & Math. Doing the things I chose to study in my own time, not the things I discovered in school.

Contrast this with school. You're forced into confinement (it wasn't until grade 6 or 7 we were allowed to leave school grounds unsupervised) with a bunch of people that mistreat you horribly and wish you the worst, and another bunch of people who really don't give a rats ass. You're bored out of your mind in the classes that interest you because the material is easy and progress through it is slower than your pace. You're bored in the rest as well, because they don't interest you; the disinterest may arise merely from the fact that they are being forced upon you.

And I went to a private school... with the things my mother has said about public schools (and she's worked at one), I think I should be glad to not have attended one. On top of that, I hear the danish school system is better than the one in USA.

More edibles for cognition: John Taylor Gatto (English teacher [johntaylorgatto.com]) says that we he finds companies that don't mind having the kid do some work, the kids do more and better work than the paid staff. My ex-girlfriend (okay, so not completely an average slashdotter

Not wanting to be completely off topic, the article says that work needs to be done on making math chic. The question is: who has the credibility and influence with kids to make math cool? For young kids, the parents have some influence, although not much in the "cool" department. For teens, it's mostly the peers (not the kinds who reset the connection). That's a network effects problem you have to solve. Who else? Rock stars? Quaterbacks? Miss teen south carolina (everywhere such as maps)? I mean, having math be the Hot Stuff wouldn't be bad, but it would imply (not just suggest, as the decline in maritime piracy has) the existence of the flying spaghetti monster.

(for those not picking up logician's humor, everything follows from a contradiction).

## USA Math 1950 - 2008 (Score:5, Funny)

1. Teaching Math In 1950:A logger sells a truckload of lumber for $100. His cost of production

is 4/5 of the price. What is his profit?

2. Teaching Math In 1960:A logger sells a truckload of lumber for $100. His cost of production

is 4/5 of the price, or $80. What is his profit?

3. Teaching Math In 1970:A logger sells a truckload of lumber for $100. His cost of production

is $80. Did he make a profit?

4. Teaching Math In 1980:A logger sells a truckload of lumber for $100. His cost of production is

$80 and his profit is $20. Your assignment: Underline the number 20.

5. Teaching Math In 1990:A logger cuts down a beautiful forest because he is selfish and

inconsiderate and cares nothing for the habitat of animals or the

preservation of our woodlands. He does this so he can make a

profit of $20.

What do you think of this way of making a living?

Topic for class participation after answering the question:

How did the birds and squirrels feel as the logger cut down their homes?

(There are no wrong answers.)

6. Teaching Math In 2008:Un maderero vende un camión de madera de construcción para $100. Su

coste de producción es $80. Cuántos de su familia pueden usted alimentar

desde los $20 beneficios?

## Re: (Score:3, Interesting)

Not quite the same thing as here, but standards, for Maths A-levels at least, had toughened between the 70's and 80's.

## Re: (Score:3, Insightful)

Education is turning into almost a two tier system. There are those kids which are pushed by their parents and aim to succeed and then there is everyone else.

The kids who push hard all fight over a small handful of places in top schools fighting off with multiple public and private schools (who often are rubbing the Uni's asses).

It does amuse me that we have these moral panics about exam difficulty without really addressing the key question - Does it tea

## More From Ontario, Canada (Score:5, Informative)

When I went through the Ontario system (1986), the requirement for engineering was 3 high schoool math courses in: Calculus, Algebra, and "Functions and Relations".

I did some Calculus T.A. work, and the new students are missing certain critical concepts. The new curriculum has eliminated Integration from High School Calculus. It is actually lucky that the students get any Calculus in High School at all. One of the original proposals for the new curriculum recommended eliminating Calculus entirely. The Engineering schools fought hard to keep Calculus in High School.

Some of the first year engineering students have not seen key trignometric functions like the sine function. Other students have not seen Sigma notation, which is used for for finite and infinite series. Almost all of the students struggle with the university Algebra course, which makes me suspect the high school introduction to vectors and matrix algebra was been watered down.

Reducing the high school requirement from three to two high school math courses hurts the undergraduate engineering students. Further, a subject like Calculus benefits from repeat exposures over a number of years. The students would benefit from an introductory Calculus course in Grade 11, a deeper course in Grade 12, and then the 4 more courses in first and second year university. That way, the students have had 4 years Calculus experience before they need to apply the hard stuff in 3rd and 4th year engineering. As it is, students might only see Calculus for 2 years at university, and I'm not sure if this is enough time to really absorb the subject.

As for the quality of the students themselves, the students from the new curriculum are different.They are very fast (faster than me) at solving problems with known forms. On structured problems, similar to ones they have seen before, they are very fast. Unfortunately, they are very poor at solving unstructured problems, and problems where they have not seen the solution technique in advance. It is like someone has beaten the creativity out of the students. They can write tests really well, but they can't do original math. I imagine the students will pick up the creativity as they gain experience. It is just that someone has removed the fun advanced questions that really get the students thinking from the curriculum. The high schools are somehow creating students that can do simple stuff, but lack deeper insights into what they are doing. The students haven't been allowed to try, fail, and sometimes succeed at solving the harder mathemetical questions.