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Inspired by Tutoring is depressing because my students are struggling too much with my exercises. What do I do?

The text of that question says:

I am a tutor for first year math majors at a European university. By "tutor," I mean that I have office hours that all math students can attend. I am paid by the university.

...

Now, this is not the case anymore. Exercises consist simply of paraphrasing the definitions, and almost all the students are struggling pretty hard on these very simple exercises. Typically, they think that solving Ax = 0 when A is a 3×3 matrix is too hard. They have very little intuition about what is going on, especially in algebra. When they ask me questions, I try to answer so they see a way of thinking that can be generalized. But sometimes they just look at me as if I was an alien.

This is very surprising because I cannot imagine doing a math major without knowing algebra or matrix multiplication. Since they're commonly taught in high school, one would naively expect college-level math majors to not only know them, but to know them well. Without them, one cannot start learning the "real stuff". The obvious path of action seems to be to deny the student admission - if not to the college then at least to the major - due to the prerequisites not being met. But in this case, the linked question would be moot because it doesn't arise in the first place. Why doesn't this happen?

The only answer I can think of is that the students pay money to study math and the colleges are happy to take that money - but that sounds very cynical.

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    "The only answer I can think of is that the students pay money to study math and the colleges are happy to take that money - but that sounds very cynical." If that can alleviate your cynicism, universities in France are funded directly by the state (as opposed to paid for by the students, like in the US). Yet we have the exact same problem in France.
    – Stef
    Dec 9 '21 at 15:57
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    Why is everyone insisting in the answers that the question is about US community colleges. The OP quotes a question that begins "I am a tutor for first year math majors at a European university". Have i missed something? Dec 11 '21 at 21:24
  • Comments are not for extended discussion; this conversation has been moved to chat.
    – Massimo Ortolano
    Dec 12 '21 at 6:41
  • @IanSudbery the question changed substantially in an edit. Most of the answers about US community colleges are based on the version before the edit. I still learned things from those answers though, so I don't mind.
    – Allure
    Dec 13 '21 at 3:15

12 Answers 12

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The only answer I can think of is that the students pay money to study math and the colleges are happy to take that money - but that sounds very cynical.

I think this is effectively right, but it's maybe not as cynical as you would suggest. Consider: Bob is a student who has learning disabilities and spent most of high school experimenting with drugs. At the age of 27, Bob has gotten clean, gotten therapy, and wants to pursue a career as a programmer. He decides to start by studying math at a community college. At first, he is way behind and has to take remedial math. But over time, he builds his skills and eventually gets an associates degree in math. From there, he is admitted to a software engineering program at a reputable, four-year college. And after that, Bob earns a master's degree and a high-paying job.

I think lots of people -- from instructors to administrators -- pursue teaching less-qualified students because they want to help people like Bob. The problem, of course, is that students like Bob are vanishingly rare. In practice, most students taking remedial-level math courses will never achieve proficiency in basic math, and only a few will achieve anything useful (e.g., a better job) from their studies. And the college cannot help people like Bob if it goes bankrupt, so it's easy to rationalize difficult choices in the name of "we have to keep the lights on so we can help people like Bob."

So: is it exploitive that colleges take money from desperate people who (statistically) have little chance of getting a return on their investment? Or is it a good thing that colleges make opportunities available to students who would otherwise have no opportunity of pursuing higher education at all? Or are the students and colleges merely symptoms of a deeper problem? Opinions will vary.

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    This is important. Even in the absence of a drug problem and subsequent recovery from that drug problem, a sufficiently determined student can catch up, in the course of a bachelor's programme, from (say) being unable to add fractions and apply multiplication tables to being reasonably competent in university-level mathematics - yes, including matrix operations and calculus. I've seen it happen quite a few times. Dec 9 '21 at 15:38
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    That story is great, but makes it sound as if the "remedial math" that Bob studies is appropriate for his level in maths? The main issue raised in the question is specifically that universities' curriculum have a prerequisite, and that many (most?) of the students don't have this prerequisite. But the teachers don't have a lot of freedom and have to teach the curriculum anyway. So, tutors are stuck trying to teach "relatively advanced maths" to students who would actually need "basic maths" courses, and it's a waste of time for everyone.
    – Stef
    Dec 9 '21 at 16:01
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    @Stef Perhaps you aren't American, but there is a very important distinction between "community colleges" and "universities." Dec 9 '21 at 16:39
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    I went to uni in the UK in the mid-90s. Unis there got course fees from the government based on entry numbers, not on who completed the course (nor even the first year). There were many "access" type students who dropped out in that first year, around 10% of initial intake, all of whom didn't have the skills in the first place and should never have been there. With UK unis now charging tuition fees as much as the US or even more, I would be amazed if the problem wasn't even worse now, and I would be amazed if the same wasn't true in the US too.
    – Graham
    Dec 10 '21 at 1:01
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    Australia too, until they close this up. In Australia a student may enroll all over town and the Universities will get paid, whilst the student racks up a debt they may never repay. The attrition rate is very high in early semesters. Occasionally one or two lesser qualified students actually excel. I remember too that I wrote more in the first three weeks of Maths 110 than my final two years of high school.
    – mckenzm
    Dec 10 '21 at 2:33
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My experience is outdated but the core idea is probably still valid about 60 years later.

There is tremendous variability in the quality of secondary education in the US. Some high schools teach a lot (lot) of college level subjects. Others none at all. Philips Academy Andover (no connection) is one of the outstanding secondary schools. But I went to a very academically poor school. This was before the AP program existed, of course, but I was deficient in many subjects.

In particular, I knew nothing of trigonometry, much less Calculus or matrices when I graduated. I was fascinated by geometry (my first true educational success, actually). Some of the math courses were taught by a golf and football coach.

I did well enough that I was accepted into a very good, but small, teaching focused university. The first course for majors was Calculus and while I was permitted to start, I was also required to take a makeup trig course. I worked hard and was able to pass out of that by exam, freeing up some time for calc.

Eventually I wound up with a math doctorate from a State University (R1) institution that had some of the top professors in the world in a number of fields.

But, in my university and graduate education, luckily, none of my professors (few anyway) had the attitude that "here it is, take it or leave it". The best and those I followed as mentors, were available for questions and would occasionally even give unsolicited advice.

The why of it isn't surprising. For those interested in education, it is a mission to nurture those who show promise, even if it is largely unrealized at the time. For state universities, such as my graduate program, it is an economic imperative to educate the citizenry for the future. So, even at top R1 institutions, a lot of effort goes into building an undergraduate program that is accessible to most people willing to put in the effort. Not everyone has the Philips Andover advantage (nor the money to attend, of course).

Sadly, secondary education isn't improving fast enough. There is tremendous reluctance to fund it properly. Thus, the variability in quality is probably worse now than in my youth. There are too many forces that don't seem to recognize any value in the "common good" (and not just in education funding).

Currently, my school taxes are very high and I don't have kids in the system anymore. But the future depends on my willingness to pay them so that today's youngsters can have good teachers and teachers get both a decent salary and the ability to increase their skills.

And it also depends on the willingness of colleges and universities to deal with the realities. I am thankful that it still seems to be the case.

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    What did you study in secondary school math classes if you didn't study calculus/matrices/trigonometry? Did you have math standardized exams that were required for admission into a math major at university? If so, how did you pass those without these topics?
    – Allure
    Dec 10 '21 at 2:50
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    Maybe if the maths department was funded as well as the football team the quality of the education would be better.
    – Neil Meyer
    Dec 10 '21 at 11:29
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First of all, these students are almost all not math majors. In the US, college education includes a lot of general education, because we think all adults should have a broad range of knowledge.

Isn't adding fractions a basic life skill that everyone needs? If you are in court, wouldn't you want the people in the jury deciding your case to know how to add fractions? (Most cases do rely on some basic mathematics!)

As for why these people are admitted to college, what's the alternative? Remember that, for the most part, it's not that they don't know how to add fractions but are particularly adept at something else. Most of them probably could not handle the drive through line at a busy Macdonalds. (That's not an easy job!) What do we do? Put them on disability for the rest of their lives? It's not like there are lots of jobs out there that they can do.

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    What a curious answer. How do you know they aren't particularly adept at something else? Let alone know that they could not handle a drive through line?
    – Johan
    Dec 9 '21 at 17:26
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    This is an important answer. Once people graduate high school, what choices do they have to pick up the skills they failed to gain when they were young (and perhaps less motivated or disciplined)? Self-study? That very rarely works for people. College seems to be the most obvious location where adults could remediate their knowledge, and they'd not have many options otherwise. There should be less shame in learning a subject later than the "normal" age.
    – Drake P
    Dec 9 '21 at 17:46
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    @Johan Obviously there is some number of spectacular novelists who never learned to add fractions. But there is a far, far larger number of people who never learned to add fractions, or much of anything else. All intellectual skills are very highly correlated. I wouldn't necessarily defend the drive-through line jibe myself, though; or at least, I'm doubtful whether college math classes make a person much more likely to succeed at such a job. Dec 10 '21 at 1:01
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    Taking an order over the headphones from one customer while simultaneously handling cash from a second customer and communicating with your coworkers about a third order, repeatedly and accurately over the course of several hours, is not an easy task. When the manager isn't careful to make sure one of the people scheduled on a busy shift can handle this task, it's quite noticeable. I certainly think the ability to succeed at this task is correlated with the ability to succeed in college math classes. Dec 10 '21 at 1:51
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    @VladimirF - In the US, there is no secondary school final exam. The requirements for graduating from secondary school can be very low, particularly in schools that don't have enough resource to really educate their students. Dec 10 '21 at 6:33
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@cag51's answer is excellent, but there's an additional issue (UK, or more specifically England-and-Wales, perspective, but may apply more widely): how would the university know, at admissions stage, that the students lack those prerequisite skills? In my experience, the students who do lack the skills @DanielR.Collins lists have, nevertheless, almost always passed the public examination (GCSE Mathematics) whose officially-listed assessed learning outcomes include those skills. A few of the higher-ranked universities have flirted with working around this by setting their own entrance examinations and/or using admissions interviews, but it turns out to be extremely challenging to set this up in a way that doesn't introduce unacceptable social-class, ethnic, and gender bias into the admissions process.

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    I have seen a number of cases where students lack the required ESOL skills, either due to incompetent examiners or some sort of fraudulently obtained certification. There's no way for the college to know this beforehand. Dec 9 '21 at 23:59
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    How does one pass GCSE mathematics without knowing basic math skills like the ones listed by Daniel R. Collins?
    – Allure
    Dec 10 '21 at 2:52
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    @Allure Two hypotheses. 1. GCSE exam is not a valid assessment of purported learning outcomes of GCSE course (perhaps due to too few examiners trying to assess too many candidates on too tight a timescale). 2. Students do have those skills at the moment of sitting the GCSE exam, but learn them in a manner that is not sticky, and have thoroughly lost the skills by the time they arrive at university. Dec 10 '21 at 11:35
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    "but it turns out to be extremely challenging to set this up in a way that doesn't introduce unacceptable social-class, ethnic, and gender bias into the admissions process." No; it's challenging to avoid people claiming that such biases were introduced. After all, if the universities can't easily write unbiased entrance questions, how are they writing unbiased problem sets and assignments? But there will always be someone around to claim that disparate statistical outcomes somehow prove a bias on the part of the university. Dec 11 '21 at 1:08
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    @KarlKnechtel The University doesn't have to be biased in order to introduce bias. If the university sets its own entrance exam, even if that entrance exam is, in itself, highly valid and reliable, it will have a particular style and format. Applicants with well-resourced schools and/or parents will be able to receive specific coaching for that style and format as well as being prepared for the public examinations... Dec 11 '21 at 1:52
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The OP quotes Daniel R. Collins saying:

most students entering U.S. community colleges do not have 8th grade algebra skills, nor even 6th grade arithmetic skills (e.g., fractions, proportions, negatives, estimations, times tables).

The OP says:

The obvious path of action seems to be to deny the student admission - if not to the college then at least to the major - due to the prerequisites not being met.

So it sounds like the OP isn't from the US and doesn't know what a community college is or how it works.

Community colleges don't just provide instruction in traditional college subjects, they also provide vocational education. They don't want to exclude someone who wants to study welding or cosmetology just because that person's algebra skills are nonexistent. They also serve the function of helping people to remedy their inadequate high school educations, so that they are not just excluded forever from any further educational progress. I don't know about other states, but in my state, California, CC admissions are open to anyone "able to profit from instruction," i.e., anyone with a pulse and respiration. Selective admissions are part of the missions of the Cal State and UC systems, not the California community colleges.

Excluding someone from majoring in math at a community college would be sort of silly. The type of student who wants to major in math is one who intends to transfer to a four-year school and get a four-year degree in math. That student does not need to declare a major while at the CC, nor do they need a two-year degree in order to transfer. Some CC students do care about their AA degree, especially those who are the first in their families ever to attend college.

However, declaring a major at a CC and getting an AA degree in an academic field are both basically a joke. At the CC where I taught before retiring, students would make a final visit to a counselor before transferring, and the counselor would have been told to make sure to get the student to apply for as many AA degrees as possible. This is because the number of degrees awarded is one of the criteria for state funding. The requirements for an AA in math are a proper subset of the requirements for a physics degree, which in turn are a proper subset of those for an engineering degree. Therefore every engineering student would walk out the door with at least three AA degrees. It wasn't uncommon to see students getting four or five of them.

One could exclude students from calculus if they didn't have appropriate arithmetic and algebra skills. My school used to do this by administering a placement test when the student was admitted. This worked very badly. You could get a student who was 35 years old, returning to school after having had a job and kids. She doesn't remember her algebra on the day she walks up to the window at Admissions and Records, but that doesn't mean she needs to languish in remedial courses for years. Remedial math education simply doesn't work. The success rates are nearly zero. That's why my state abandoned it.

What would make more sense would be for community colleges to get tougher with students who don't make appropriate academic progress or who take the same class over and over without ever passing. California did this to a slight extent ca. 2010 by instituting a limit of 3 repetitions of a class; this was a response to the overloading of the CC system during the economic crisis following 2008. There are genuine abuses, such as students who will habitually drop a class at the 13th week, even if their grade is a B, because they want to retake it for an A. In general, though, the system is run by people with bleeding hearts, and the school also benefits in terms of state funding by having as many butts in seats as possible.

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    I think the OP is not suggesting excluding someone from a Welding course if they can't do algebra, as Algebra is not a prerequisite. However this same person could be excluded from a Physics course until they have reached the required Algebra level. Presumably the college would also offer remedial Algebra course for this purpose... Dec 9 '21 at 23:57
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    Thanks for quoting me. :-) As a fellow community college teacher, I agree with 80% of what you've said. A thing I'd differ with would be: "Excluding someone from majoring in math at a community college would be sort of silly." Now, we absolutely get students who are witless enough as to lack 4th-grade arithmetic skills and still register as a math major. In fact, admin has forced us to mangle our offerings to guarantee a pathway for such students to a math degree in 2 years. It's greatly damaged scheduling and resources for our department; not excluding such students has caused a lot of harm. Dec 11 '21 at 17:05
  • Also, I've never seen the multiple-AA's racket, so that may be unique to your institution. At my school students do need to declare a single major, and matriculation to the connected senior colleges is only guaranteed with the degree. Dec 12 '21 at 14:54
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The question is about teaching math in U.S. universities. However the situation with English/communications and other subjects that people from other countries expect to be covered in secondary school is similar.

Students entering the first year of university in the U.S. can vary a lot in terms of math preparation, because many secondary schools really don't do a good job teaching the children that they eventually graduate.

Most U.S. universities administer some kind of math placement test to the incoming class. Depending on their results, a student may be placed in:

  • a "remedial" math class (not always called that, and sometimes non-credit) that covers very basics, like the number line, and adding fractions.

  • a "pre-calculus" class - definitions of functions...

  • "Calculus" - differentiation and integration...

  • A few students take "Advanced placement" (AP) calculus in high school and are allowed to take more advanced courses like differential equations right away.

Surprisingly many students need "remedial" math even at the most "prestigious" and exclusive universities.

Some universities have other verions of placemnent. Universities' requirements also differ in how many math is required for different majors. E.g., after a placement exam, non-math majors might take a "math for liberal arts" terminal class. However you can see how many students (not necessarily math majors) end up taking math courses that they don't want to take, but only need in order to graduate. They are not motivates. Often they barely got placed into this class, struggle, and would have been better off in remedial math.

Even among math majors, a university might have different concentration areas - "pure math" (who like proofs), "applied math" (don't care for proofs), and "math education" (want to become secondary school math teachers). It's common for some to start at a very basic level, even remedial.

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    Note that the current trend in U.S. academia is to eliminate the existence of placement tests and remedial courses. At my institution (community college within CUNY) , I just finished the last such class I'll ever teach, as they won't be offered any more. Dec 11 '21 at 16:55
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    Thanks for the update! I should note that the system worked better at other universities than at CUNY becasuse at CUNY remedial math classes had two types of people: 1 those who never learned how to add fractions in secondary school, and 2 people who did years ago, but forgot, or did not do well on the placement test for whateeve reason. The course covered way too much material too fats for the first type, so almost all of them failed repeatedly. The second type did well, but were angry that they had to take remedial class they did not really need. Dec 12 '21 at 2:31
  • I can respect your observations on that. IME the number of people who knew it years ago is vanishingly small (no more than 10%). I'd say a larger cohort in my remedial classes (at least half) have severe intellectual disabilities in math, who can't retain even the simplest arithmetic facts no matter how long it was practiced. Dec 12 '21 at 14:58
  • CUNY community colleges, thanks to almost-open admissions, get lots of incoming students who can't add fractions. However other more "prestigious / exclusive" universities also get, e.g. 18 year old athletes on sports scholarships and also returning students much older than 18 who haven't been to classroom in many years - working / armed forces / incarcerated. In my opnion, many other schools serve such students better in various ways, and in particular, their versions of remedial math usually takes 2 semesters to cover the material what CUNY crammed into 1. Dec 12 '21 at 17:38
  • Really? Because where I am in CUNY, we always had a 3-semester sequence of remedial math. (Prealgebra, elementary algebra, intermediate algebra.) I thought that was customary, e.g. looking at text offerings like OpenStax. Are you saying other colleges offer 6 semesters of remediation, or something else? Dec 13 '21 at 18:19
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Absolutely it's money. Re: "deny the student admission"; note that most students in the U.S. are attending open-admissions community colleges. There is simply no such thing as denial of student admission in those cases. And they can sign up for any major they like.

Full disclosure: I teach math at a U.S. community college that's won national awards for high quality. (I also wrote the selected answer in the motivating question linked by the OP.) My college is regularly announcing more initiatives to expand enrollment: (1) recruiting people who have failed out of other universities, (2) turning F's and incompletes into withdrawals so students never drop in GPA and can always register for more classes, (3) programs to invite intellectually disabled and autistic students into college courses, etc.

I just wrapped a basic algebra course where the students (N = 32) couldn't read, write, subtract, add negatives, identify written operations, tell if 0 = 100 was true or false, match a given answer to multiple-choice items, retain information from day to day, etc. Median score for all students for the semester: 3%. All of those students will presumably be registering for more courses next semester.

Everyone in the pipeline is telling students they need to go to college, and can do anything they want there. Our department's been forced to greatly mangle our sequence, being obligated to arrange it such that any student, even one entering without 4th-grade arithmetic skills, has a pathway to achieve a mathematics degree inside 2 years. Whether the student had legitimate skills or grades in high school is a complete nonissue to administrators, advisors, and thus students.

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Considering OP does not mention which country, I can explain the situation in India.

A large part of Indian education system (unfortunately) relies on learning by rote. While the top schools and universities in the country do their best to ensure that their students do not pass their exams by rote learning, this is not the case in most schools. This is not a requirement of the Education board as well.

For example, students in India give exams known commonly as "Board exams" to pass their Grade 10 and Grade 12. The exam is designed by the Education board (a famous one throughout India is the Central Board of Secondary Education) and applies to all students in the nation/state who register for the exam under that Education board. Even these "Board exams" rely mostly(1) on questions directly from the textbook, the reference book or previous year's exam questions. Students have access to all these through various sources, especially print media. A significant(2) number of students attempt to rote learn the answers and repeat them during the exam, thus, fulfilling the pre-requisite for the university course without gaining the skill to solve such problems in math. While the grade/marks vary from one Education board to another, scoring well above the passing mark/grade can easily be achieved through this rote learning.

(In my answer, I rely on relative quantifiers such as (1) and (2) because I am neither an educator nor an expert on the statistics. I am merely a recent, but former student of this system.

Also, this may not be the case for every Education board in India, but a majority of them.)

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Given that the quoted question talks about a "major european university", there is the very real possibility that the answer to the OP’s question simply is: “because they have no other choice. In various European countries (e.g. the Netherlands and ( parts of Germany). Successfully completing the appropriate high school exam gives access to any of the country’s universities. This means the universities do not have the option of not admitting these students. Moreover, one of the metrics that universities get judge upon is the percentage of students that make it to the finish line and get a degree. This means that there is a strong incentive for universities to teach the students any requisite skills they are missing.

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  • The Dutch situation is different since study in X may require that you passed the exams subjects Y, Z in "high school".
    – Deipatrous
    Dec 12 '21 at 9:14
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OP does not state which university it is, or which country. I can report from Germany, from Baden-Württemberg where I live.

If a person finishes Gymnasium (the academic stream of secondary school) with an Abitur at any level, that person can apply to any university to study any course. This is obviously absurd, but that's the law.

But some of the popular universities (e.g. Tübingen) are so overfilled that they have a limit of people for courses (e.g. German) and actually make applicants sit exams. This is a good thing. However, the "better" universities such as Freiburg have no restrictions whatsoever, so people sign on for Maths or Physics and flop at the first round of exams.

Of course they are pleased to get the money. They need it. But Tübingen seems to manage, because German etc is so popular. Maths, Physics, Chemistry are not so popular, and they can't afford to be choosy where their money comes from.

We all know stories of people who pulled themselves up by their bootlaces and managed to finish Engineering after a bad start. But we know many more who failed miserably. Universities should have minimum requirements for difficult subjects. There is no problem with the most popular "hard" subject of all, Medicine, which is always over-subscribed and thus can insist on high entrance standards.

However, I do not see that this is going to get better. B-W has a Green government now, and they are trying to stamp out selective secondary schools. Students of history can see that this always means falling standards, so in a decade or so B-W will no longer have one of the best education systems in the country.

Edit: Obviously absurd: Try studying Electrical Engineering if you don't know Kirchhoff's Law, or Maths if you don't know Algebra. This may not be a problem when studying Psychology or Politics or English, I don't know what they do all day.

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    Medicine high academic standards have to me traditionally been more because a mediocre doctor takes his mistakes to the grave. It literally is a case of life an death.
    – Neil Meyer
    Dec 10 '21 at 11:36
  • 'We all know stories of people who pulled themselves up by their bootlaces and managed to finish Engineering after a bad start. But we know many more who failed miserably.' Actually, I've seen probably 3-4 times as many Engineering students succeed from a weak starting point as I've seen fail. Dec 10 '21 at 18:04
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    @NeilMeyer As the old and rather macabre joke goes, even a really incompetent surgeon can only kill half a dozen people in a day's work; a really incompetent engineer can kill ten thousand people in a day's work. Dec 10 '21 at 18:09
  • @DanielHatton: I upvoted your comment since I'm a fan of snarky jokes. :-) But to avoid that anybody takes this at face value, one should probably add that, in fact, even the most incompetent engineer typically has no chance to kill many people unless a lot of further engineers (and/or management guys) make various mistakes, too. Dec 11 '21 at 9:38
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    I downvoted for two reasons: (i) I can't see why the situation described in the second paragraph would be "obviously" absurd (I do recognize that this system has some disadvantages, but it has some advantages, too). (ii) Your last sentence presents a speculation as a fact (cloaked by a vague reference to history). Dec 11 '21 at 9:51
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It's certainly not an ideal situation that these students at university don't know the basic stuff that is taught in high school. However, one does not need to start at the basics and build up from there.

It's a bad practice in math education to stick to the dogma that one must first master the basics perfectly before being exposed to more advanced concepts. This is not how I learned many math and physics topics. Quite a few of the people I know who did well at university, learned a lot of math and physics themselves from advanced math and physics books long before they had mastered the preliminary stuff needed for a rigorous understanding.

Many math topics were in fact developed by mathematicians and physicists when they themselves didn't understand it properly. So, mathematics is a flexible topic, it allows people to step in and work with advanced concepts when they have not mastered the basics. It's then possible for them to master the basics later, simply by working their way through the course.

This is how I mastered complex analysis. I had mastered ordinary calculus at the age of 12 in a non rigorous way. At the age of 14, I browsed a book on complex analysis, it looked like a boring book containing lots of theorems that I was not interested in. So, I had no intention of reading the book. But I did see one interesting theorem in the middle of the book, it was the residue theorem. I could understand what it was saying, as I was able to figure out what was meant by the residue of a function.

I could then calculate contour integrals with the reside theorem, but I lacked a proper understanding of why the theorem is true. I only had a heuristic understanding, basically that the integral of 1/z is Log(z) and this should pick up a term 2 pi i if you go around the singularity. But this deficient background in the topic was good enough for met to start computing integrals using the residue theorem. Failure due to improper understanding, like in case of functions with branch point singularities motivated me to study the subject more properly later.

Six years later I was a third year theoretical physics student. We had to choose a few optional math courses given to math students and pass the exams. One of the subjects I chose was complex analysis. So, this was a proper, rigorously taught math course by a math professor for math students. I was familiar with most of what was taught, so I did not study much for the course. But I was a deficient student compared to the other math students who followed the course, because I had not studied measure theory and topology rigorously (I had vague familiarity with these topics, but physic students were given special math courses for physics students that did not include topology and measure theory).

The exam questions were mostly contour integration and summation questions. I made one minor mistake and scored 95%. One other student passed the course, he scored 70% out of 10. The rest, about ten other students all failed the course (they scored less than 60%).

I had a similar experience with other topics, I tended to do extremely well when I had studied in this sort of a non rigorous way years earlier. I ended up getting a more mediocre score when I had studied according to the curriculum, starting at the time the course was taught at university.

The way I studied math myself when I was in high school would not be considered to be a serious form of studying by educational experts. It's perhaps best described as playing. So, I think insisting on studying in the official way is not justified as it doesn't yield good results. We should be more open to alternative approached to learning math.

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  • +1 I'm not sure how this relates to the question, but I can relate to this anwser and it's an uncommon perspective.
    – Myridium
    Dec 12 '21 at 6:49
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Everyone has an area of struggle. Sometimes prerequisites are needed, sometimes it's just filler. I've seen students struggle with prerequisites, yet excel in the subject they came there for.

Also, on the opposite note. Why do some schools make people stay the whole semester even if they know all the coursework?

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