Posted by
Antonio Cangiano in

Essential Math,

Math Education on
May 26th, 2007 |

17 responses

Last week “Digg effect” had quite an impact on my hosting provider. So much so that they kindly (sic) pulled the site off the web after 10 minutes of Digg love, without even bothering to send me a warning or any notification. When I complained they told me, “let us know when you are off digg”. Yes, we all know that the Tera bytes of traffic that they promise you are fictitious, but I was naively expecting better customer care, especially after having referred about a hundred clients to them. By the way, my programming blog that’s hosted with them as well, has been previously on the frontpage of Reddit, Del.icio.us, and even Slashdot and I’ve never experienced any problems (caching does wonders). But the traffic load generated by Digg was too “fast and furious” for them to cope with.

It’s not all bad though; in fact I was already planning to switch to a more serious provider with VPS or dedicated server plans. After endless research I picked Liquid Web which is fully managed, provides root access and support 24/7 on the phone. So far so good, they worked quite a bit on tweaking my server and I’ve actually called them at 3 and 5 AM and got a person on the phone in less than 30 seconds. I don’t know how well they will be able to cope with the “Digg effect” though, only time will tell.

With my “Refresh your High School Math” article on the front page of several social websites, the amount of feedback received has been terrific. It also allowed me to confirm a theory that I’ve always thought about: there is little consistency and standardization in the teaching of mathematics. I say this because the reactions to my basic math test were very highly varied. Many people said they were not able to solve the problems. That’s sad given the admittedly basic nature of the questions, but it wasn’t a big surprise. You could consider it the FizzBuzz of mathematics. What strikes me the most though, is how many people considered the test to be rightfully “middle school” material and far too basic (except for some parts of it), while others argued that it was way too advanced and too difficult for high school mathematics. This variety of reactions shows that the topics and depth of coverage in math classes in high school are quite different throughout the world.

So I’d like to explicitly ask a question to my readers, what topics did you cover while studying mathematics in high school and in what state/country were you? Please use the comment section to answer, thank you.

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In my high school (private Mexican high school) we covered a wide variety of topics,but many of them are not on the official study plan. We covered all of the analytic geometry and precalculus and calculus stuff that pretty much everyone has to go through. In particular we covered everything necessary to solve the questions in your “Refresh your High School Math” article. However we also covered topics like set theory, combinatorics and probability, linear programming and even Newton-Raphson approximations and Taylor series.

Personally, I think teaching all of those topics was a very good idea since we got a glimpse at what Mathematics is all about. After high school most people tend to think that the job of a Mathematician is to solve equations, factor polynomials, and find derivatives or integrals. In reality however this is far from true, and in fact, compared to engineers or physicists, Mathematicians rarely do any of those things.

Thanks for your comment, neop. That’s the kind of response I was hoping and looking for.

Hi, I studied at the Lycee Francais in Mexico City, many years ago. When I was 15, I could solve all of the problems you posted, and some more. The math proficency of the group of students I was in was considered above average, and that involved lots of intellectual as well as emotional pressure . Hence, even if back then as a High School student I was able to solve a few math problems, I was terrified. This was a french school, and in France, if you wish to succeed professionally, you must pass a selection based on math problems solving skills. This selection starts quite early in life… I finally ended up hating the subject, and opted for a “non-math-degree” in my final High School year (Philosophy). Then I became a Psychologist. Now, at 45, my math skills are lousy… You see, I do not think the only question here is “what kind of math did they teach you”, but also “how did they teach you”: According too which pedagogical standards? Our teacher might have been a great mathematician, but he was an awfully bad teacher… I think this can be often the case while teaching science: You can have a great scientist in front of the classroom, but a terrible teacher indeed… When I wanted to be a language teacher, I was asked to learn “how to teach languages” (Teacher´s Diploma Course), INDIPENDENTLY of my own language skills. Is there such a “teaching degree” for math teachers, focused on developing “math teaching skills” ?

When I was in high school in Ontario, Canada, all the materials required to solve your problems, plus some calculus (mainly how to do derivatives and integration), were part of the province’s curriculum. The only problem was, the math courses were optional beyond grade 10/11. On the bright side, the grade 13 calculus course was almost always required for admission to university programs, including many business-related programs such as accounting, economics, and commerce.

After the province scrapped grade 13 and totally revamped the high school curriculum a few years back, I’m not sure about the current state of affairs, but calculus is still taught at high school level, I think.

My high school teacher totally ruled. He taught up through complex numbers and matrices in my junior year. My senior year he let me teach myself calculus and even a little set theory and propositional logic…

He was so great.

It’s been so long since I took math courses, the memory is hazy. During elementary school in the early 1960s, in El Dorado, Kansas, I remember teachers receiving training in something called “New Math.” Far as I knew, 1-plus-1 still equaled 2.

Come high school, I took a couple of algebra courses, geometry, trigonometry and college algebra. The last two were electives, I think, about 35 years ago. Calculus had to wait until college.

I wnet through a state high school in New Zealand in the early 1980s. Memory is a bit hazy but everything you mentioned in the Refresher was covered.

Here are the few distinct memories I have of maths back then:

1) In 3rd form / year 9 we were introduced to graphs and did some simple algebra (slopes of graphs, crossing points of axes, etc). We’d never been taught anything about graphs prior to that. Basic algebra had been introduced in primary school (year 7 or 8, something like that). I remember that we already knew E = 1/2 mv^2 and E = mgh (from primary school science) before we got to high school.

2) In year 10 we were introduced to logs and calculations using log tables. Even back then log tables were old hat but it was thought useful to give us a brief introduction to such things. We also used slide rules briefly (once again, old hat even at that time). An aside: Calculators were not allowed at school until 6th form / year 12. We did mental maths (simple calculations in our heads, no writing allowed) for 5 or 10 minutes in each class.

3) Calculus was introduced in year 11, I think (possibly year 12 but I think it was earlier).

4) By year 12 we’d come across matrices. I remember doing translations and scaling involving matrices that year.

5) In year 13 maths was split into two subjects – pure and applied maths. Applied maths included statistics and probability, as well as some computer science. I think some of the basics were introduced earlier – I think we first came across normal distributions in year 10. I remember doing regression. In pure maths we were doing double derivatives and integration. I vaguely remember the chain rule for derivatives and something similar for integration (integration by parts?). I think they may have been introduced in year 12.

By the way, I was surprised you said that calculus isn’t high school material. What year groups do high schools in the US cover? In NZ they’re 3rd form to 7th form, years 9 – 13.

In addition to the subjects mentioned above there was plenty of other stuff, such as trigonometry and geometry, but I can’t remember exactly when we got into that.

Cheers

Simon

Public high school in suburban Philadelphia (Class of ’98):

Freshman: “Geometry” and “Finite Math” – geometry is self-explanatory. Finite math taught algebra-based applications

in statistics, probability, matrices, trigonometry, and sequence and series

Sophomore: “Algebra II” – just rounding out the algebra taught in middle school

Junior: “Pre-calculus BC” – polynomial and rational functions, graphing techniques, conic sections, exponential and logarithmic functions, sequences and series, limits, and basic trigonometry

Senior: “Calculus BC” – Equivalent to two semesters of college calculus–that is, differentiation and integration of a single variable

If I recall correctly, this was about all the math my high school offered at the time. If I had wanted to take more math, I’d have had to go to Villanova University down the street for stats, computer science, linear algebra, etc. I regret not. Nowadays, I believe they offer substantially more mathematics at my alma mater.

I go to school right now in Minnesota and it doesn’t seem as though we cover near as much as the others in these comments. If you’re one of the smart ones in the class (the top third or half) you take:

Algebra 1 8th grade

Geometry 9th

Algebra 2

Pre-Calculus

AB Calculus

Everyone else is a class or two behind but in the same order. In fact, you could even start pre-algebra in 9th grade and only take Geometry by graduation while fulfilling the 3 required math courses.

I was lucky enough to be able to work independently and finish Algebra 2 and Pre-Calculus in one year and take Calculus my junior year. Our calculus class is nothing to brag about, though. On the national AP exam, no one aside from foreign exchange students has gotten anything but a 1 in years (hopefully our class changed that this year).

I was able to figure out (almost) all of the questions on the quiz but I’m sure that over half the people leaving out school couldn’t. We also are never taught most of the terms in questions 8 and 9. We learned how to find the vertex of parabolas in A2, but none of the others.

My high school in north central Texas had a sequence similar to Cade’s. Importantly, though, there were massive personnel problems, and in the end we were advised not to waste our money on the Calculus AP Exam. Sadly, I can also echo Cade’s experience with regard to the Exam – a foreign exchange student was the only person who decided to take it that year, and she of course received a 5. None of us would have fared as well.

Also, now – a full 10 years after Algebra II and Geometry – I’m learning that my sequence (the “advanced” one!) somehow skipped both geometric proofs and matrices. Which is a shame, since I would have gotten interested in math far sooner, especially if proofs had been a major component of my high school program.

Antofagasta, Chile, class 2000: Matrices, analytic geometry, complex nubers, calculus (derivatives and a little of integration).

I must say that my teacher did so well that I wanted to keep studying math after high school, so I got a BS 4 years later

Ah! and basic trigonometry

I went through the pre-Harris Ontario curriculum.

Grade 9 and 10:

- Quadratic equations

- Factoring

- Plotting

Grade 11 and 12:

- Geometric proofs

- Trigonometry

- Logarithms

Grade 13 Calculus:

- Differentiation

- Integration

- Max/Min

Grade 13 Finite:

- Matrices

- Systems of linear equations

- Combinatronics

- Probability

Grade 13 Algebra & Geometry:

- Systems of linear equations

- Geometric proofs

- Hyperbolas and ellipses

I think that the new curriculum introduces things faster rather than slower. My girlfriend teaches grades 7 & 8 and they do things I didn’t get to touch until grade 9.

Cheers,

Leo

I went to school in western Pennsylvania in the late 60′s. We got algebra in the 9th and 10th grade, geometry in 11th grade and trigonometry in 12th grade (last year of high school). I did poorly in arithmetic and algebra because I was too lazy to check my work. But I fell in love with the elegance of geometry and did well. That continued into trigonometry. Some of the better students were offered the chance to take calculus in our 12th grade but I was not one of them. When I finally took calculus in college (1969), everyone else had already had it and I was the “dummy” of the class. I never recovered after that. I still love math but have no confidence now.

I am currently in high school in Colorado as a senior(12th year), and compared to what the rest of you are saying, my school is doing quite poorly(or at least, it’s worse than it was in 1960. Heh).

For normal students, it is usually Algebra 1-2, Geometry, then Integrated Algebra 3-4(basically a re-hash of the last 2 classes…) and then 12th year is optional(The students usually don’t take any).

I was in the normal classes until last year, because they’ve been too easy and repetitive, so I am starting to take calculus on my own(and getting my own books, the school textbooks and classes are at best mediocre, and at worst downright pitiful..).

I love math, and I’d like to thank you for your book recommendations. I’m going to buy some of those books myself(unless I can find them at the library, but the good ones usually aren’t there. Usually they have ones that were used for school.. You know, the sad ones that make your eyes bleed.).

My school has omitted a large amount of information… In the refresher, I was only able to do 1,3 and 10, so I’m going to get a real precalculus book(How do schools manage to get such shitty textbooks and teaching?! Agh!), and actually learn this.

(I’d write something about how much schools suck nowadays(at least in America), but that’s for other sites)

Thanks for the links and books, I’ve put your site on my favorites.

~John(Colorado, America. 17 years old in 12th grade(senior))

I attended a school in a suburb outside of Detroit Michigan, and am currently a middle school math teacher.

These are the following courses I took while in school:

9th Grade- Honors Algebra

10th Grade- Honors Geometry/ Honors Trigonometry

11th Grade- Honors Pre-Calculus

12th Grade- AP Calculus

I took your refresh your math skills quiz, and because I don’t use all of these skills now, there were a few questions I was unable to answer. However, if I looked over a book to refresh my skills, I know I wouldn’t have any problem answering them.

In this article you mentioned that some people believe that these skills are middle school level, and others believed they were above high school. Being a middle school teacher, I know that many and almost MOST of these concepts are above the level of all my students. (I teach grade level, and no advanced courses). I think that all of these concepts are suited for high school students.

Something that might be useful to know, every child in the state of Michigan must take Algebra I, Algebra II, Geometry, and another elective math course to graduate from high school!

I’ve enjoyed reading a lot of your blogs!

I went to a private middle and elementary school near Cleveland, Ohio before transferring to public school in 8th grade. My math education was as follows:

7: Algebra – didn’t really understand it or get much out of it, did badly in the class

8: Algebra – this time I did very well

9: Geometry (learned a few theorems, did some “proofs”) and Honors Algebra II (basic linear algebra and analytic geometry)

10: Honors FST (Functions, Statistics, Trigonometry)

11: Honors Precalculus (basic logic/set theory, proofs, basic differentiation/integration), AP Statistics

12: AP BC Calculus (more differentiation/integration, infinite series) – I also borrowed my teacher’s book for a few weeks at the end of the course to teach myself multivariable calculus

aaand in college so far (started in engineering, switched in my second year to applied math/physics):

13: Calculus AGAIN (had to do it, despite a 5 on the AP BC exam, but on the bright side, did more multivariable/vector calculus), set theory/combinatorics, number theory (self-taught)

14: Probability theory, differential equations, linear algebra, complex analysis, more combinatorics (discrete mathematical structures)

15 (anticipated): Partial differential equations and boundary value problems, abstract algebra, more linear algebra, more combinatorics (discrete mathematical models), more statistics

16 (anticipated): Numerical analysis

That just about wraps up my undergraduate maths education – not very much math my senior year, since I am taking a bunch of physics and general education required classes.

I was asked to compare the elementary and high school mathematics curricula of the Philippines and Singapore, and the result is the following paper:

http://www.scribd.com/doc/21869627/A-Comparison-of-the-Content-of-the-Primary-and-Secondary-School-Mathematics-Curricula-of-the-Philippines-and-Singapore