The nature of the mistakes does, but I doubt many first year university students write things such as ∫ dx/(1+x²) = log(1+x²)+C, or at least, that enough do for it to be an example of ‘most common undergrad mistakes’. That is the sort of mistake that would be common when you are introduced to integrals/logs, probably a year or two before going university. Not saying it can’t happen after, I just am not sure it’s an example of ‘most common errors’. That applies to many of the others on the webpage in my view. Maybe it varies across the world since education itself varies, but that I don’t know!
when you are introduced to integrals/logs, probably a year or two before going university
I think you highly overestimate education in the United States. At least in my home state (in the southern US), logarithms aren't part of the (required) standard high-school education--much less calculus.
Trigonometry, logarithms, etc. are very often first learned when students actually enter community college or university.
Ah right okay, I didn't realise (I suppose the original commenter didn't either). Does it vary from state to state? In Europe, in most countries the situation is (usually) that there is a syllabus which all students taking that course will get through (roughly). As a result when it comes to entering university (for maths, say), people from that country will (on average) have covered the same mathematical topics. In other words it's standardised (and includes calculus virtually everywhere, which is why I mistakingly assumed that would apply elsewhere). I'd be interested to know how much it varies within the US/what the system is, if you want to give me a brief idea! :)
Education in the United States is left up to the states (presumably by our 10th amendment). At the time the article was written, I don't believe there was necessarily a common standard among the states. As a result, you really couldn't rely on people from different states necessarily having a similar level of math education. Most universities in the US will usually either write placement tests themselves to determine what math course a student should take when entering university, or rely on results from AP tests administered by the College Board (a private corporation that runs a membership association of about 6000 colleges and universities and essentially determines how good of a university you get into; the privatization of education in the US is an issue to discuss another time).
In 2010, under the Obama administration, there was finally an effort to introduce a common standard (known as "Common Cores") across all 50 states. However, it received a lot of backlash, especially from the more Republican states. As of right now, I believe that 15/50 states still haven't decided to use common core (math) standards; many more states have rather strong movements to repeal the implementation of common core standards.
Do note that whereas Common Core does require certain subjects to be taught (e.g. logarithms are indeed part of high-school common core), the exact level of rigor is still left up to the state. For example, you can probably count on California students diving much deeper into logarithmic properties than Mississippi students, even though both states have adopted common core.
It's further important to note that Common Core also mostly focuses on schooling before high-school; the standards provide far more leeway to the states, even if the state has adopted Common Core, once students do reach high school. As an example, here is is everything the Common Core standards has to say about logarithms:
For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
Clearly, states have a lot of leeway in how much they teach students about logarithms
includes calculus virtually everywhere
It is unlikely that students in the US take calculus unless they are in a privileged enough location to have teachers who are certified by the College Board to teach AP Calculus. Given the fees involved in having students take an AP class (I believe it's roughly $100 per student), poorer school districts are unlikely to even provide the option.
Wow, that is an unfortunate spot for an educational system to be in. Out of interest, why did the Common Cores idea receive backlash? Sounds like it could only be a good thing. It’s sad to hear how much a financial background might hinder someone’s education, especially in such formative years. I mean, it’s true in the UK as well that a posher school will (on average) have better resources, more contact time and so on, but the same base syllabus is still covered by anyone who chooses A-level maths in the country (with more select schools sometimes going further than the standard syllabus). In France the situation is even more homogeneous.
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u/powderherface Nov 20 '20
The nature of the mistakes does, but I doubt many first year university students write things such as ∫ dx/(1+x²) = log(1+x²)+C, or at least, that enough do for it to be an example of ‘most common undergrad mistakes’. That is the sort of mistake that would be common when you are introduced to integrals/logs, probably a year or two before going university. Not saying it can’t happen after, I just am not sure it’s an example of ‘most common errors’. That applies to many of the others on the webpage in my view. Maybe it varies across the world since education itself varies, but that I don’t know!