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In many universities there are honors math classes. For example, instead of having five "mixed" Calculus I sections they arrange one "honors" class and four "ordinary" classes.

How effective is this? What are the arguments for and against such arrangements? Is there any research showing the effectiveness of such an approach?

My personal view (from teaching in a regional university) is generally against the practice. I prefer all "mixed" sections, thinking that the presence of well-prepared and gifted students among the general group of students helps both. I also think honors credit are better to be given to students who work on a project with a professor in a personal setting.

Edit:

In USA the terminology of "honors" is used to refer to classes of well-prepared and motivated students who take the same course at a higher level. There are no standard terminology for what I referred to as "mixed" or "ordinary". Typically no adjectives are used, or they may be referred to as "non-honors". Sometimes terminology such as "STEM College Algebra", or " Calculus for majors" etc. may be used to note a course with higher standards than the average.

The expectations from an honors class varies very widely.

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    $\begingroup$ There is lots of research at K12 levels that math works better if you find find a way to effectively teach students of different levels together. But I think it's different once you hit college. $\endgroup$
    – Sue VanHattum
    Commented Mar 20 at 22:01
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    $\begingroup$ Would you say that the presence of mediocre and disengaged students among the general group helps both? $\endgroup$
    – Sneftel
    Commented Mar 21 at 10:53
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    $\begingroup$ @SueVanHattum How does that (the K12 part of your comment) work e.g. if you have some 12-year-olds still counting on their fingers while others have discovered and are working their way through their parents' boxed set of Newman's World of Mathematics? $\endgroup$
    – shoover
    Commented Mar 21 at 16:12
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    $\begingroup$ @SueVanHattum a lot of these kinds of studies test the higher-level students on grade-level math after combining the classes, which completely negates the point they're trying to make. Of course if good students are tested on easier material, they'll do better. That doesn't mean they' $\endgroup$
    – Esther
    Commented Mar 21 at 22:14
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    $\begingroup$ @SueVanHattum Finger-counting doesn't mean you can't do maths: it's just a sign you're struggling with addition tables. $\endgroup$
    – wizzwizz4
    Commented Mar 21 at 22:24

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My undergrad university had an honors math program and the point was to prepare students for top graduate programs in math. The content was far more rigorous/intense than the regular classes, which I don't think would have provided adequate preparation.

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    $\begingroup$ I agree. I graduated well before the whole 'honors' thing started. Both freshman first-semester physics and linear algebra math had advanced sections of courses explicitly intended to break away from the other sections after the midterm to head on their own, more rigorous way for the rest of the year. You signed up for a different course number initially, and if you were not doing well by the midterm you were switched into the regular section. Basically everyone who ended up majoring in physics was in that section, so it worked as designed. $\endgroup$
    – Jon Custer
    Commented Mar 20 at 17:22
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    $\begingroup$ Yes, these are different courses with different goals, so it doesn't make sense to combine them. Although we could debate whether "honors calculus" is the right name for it. $\endgroup$ Commented Mar 20 at 17:38
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    $\begingroup$ I was in an honors math program at U of Mich in the 70s. (There were actually 2 levels of honors! I was in the top level.) I was doing analysis in my very first calculus course. (I was very good at math, but certainly unprepared for that. I was the only one in my class who hadn't taken calculus in hs.) But yes, preparation for top graduate programs. $\endgroup$
    – Sue VanHattum
    Commented Mar 20 at 18:09
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    $\begingroup$ I took honors calc I for the first semester because I had gotten AP credit for the first semester of calc and didn't feel like jumping right to the second semester (and/or maybe it wasn't offered in the fall; it was a small school). Honors calc worked through Spivak and did a lot of theory and proofs. Regular calc, whose book I have forgotten but may have been Thomas, covered the mechanics of the calculations to prep us for engineering classes. $\endgroup$
    – shoover
    Commented Mar 21 at 16:17
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This can play out very differently depending on the setup of the educational system that you're in. When I was an undergrad at Berkeley in the 80's, honors classes were much more rigorous than ordinary classes, they were only intended for students intending to major in the subject, there was an entrance exam to get into them, and they used a different textbook. But at the California community college from which I recently retired, honors classes were expected to use the same book and cover the same material as non-honors classes. Any student could enroll in an honors class, and there were explicit rules saying that an honors class was not supposed to have more work or more difficult work.

If one was to measure the effectiveness of such a program, how would it be defined? By whether students gain self-esteem and have transcripts that look more impressive? By how they do on a standardized test? By how many of them end up having research careers?

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    $\begingroup$ What differentiated an honors class from the other sections? $\endgroup$
    – Sue VanHattum
    Commented Mar 20 at 22:00
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    $\begingroup$ @SueVanHattum The honors section could differentiate and the other section couldn't. $\endgroup$
    – Thierry
    Commented Mar 20 at 22:06
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    $\begingroup$ @Thierry: The honors section could differentiate and the other section couldn't. --- Although I'm sure this was intended as a bit of hyperbole humor, a very slight revision is probably true in many U.S. colleges/universities: "The honors section could differentiate from first principles and the other section couldn't." (By "first principles", I mean by using the limit definition of a derivative.) $\endgroup$ Commented Mar 21 at 11:11
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How effective is this?

Mathematicians have difficulty understanding how the practice of requiring nearly everyone to take calculus leads inevitably to disastrous results. In fact, they have difficulty even realizing this is happening, after seeing it every day for over half a century.

Since this kind of subject matter is being taught to students who are not there out of any desire to understand this subject, but who are there in order to get an impressive grade, the result is that mathematicians have given up on the impossible and undesirable task of coercing people to understand, and instead they teach them to execute algorithms with no understanding. And it is those who work the hardest at that, who end up in "honors" courses. The mathematicians who organize these see that those students aren't really better than the others and the whole thing isn't working, and at least have enough sense to abolish such "honors" courses.

As for gifted students, they learned calculus before their first year at university.

It is unethical to design the curriculum for the purpose of extracting the few lumps of gold from a few tons of dust. If mathematics is to be taught to broad masses whose interests are in other subjects, it should be designed to teach them things they will understand and things they will use. Otherwise those broad masses of students are being abused and defrauded.

There are some universities at which the "honors" classes may work they way those who design them hope they will work. And it is very difficult to tell which universities those are without talking to people at each university and knowing which questions to ask. You can't tell from university web sites or any sort of published information.

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  • $\begingroup$ mathematicians have given up on the impossible and undesirable task of coercing people to understand, and instead they teach them to execute algorithms with no understanding. Disagree. The task is neither impossible, nor undesirable. It is just damn difficult, especially in the so called "mixed" class. Honors classes where the students are pre-selected according to their abilities (even if those are just the abilities to work hard and nothing beyond that) are easier and more pleasant to teach and you can go both deeper and further in them. $\endgroup$
    – fedja
    Commented Mar 30 at 21:35
  • $\begingroup$ @fedja : When they are selected for their abilities, it is easier and more pleasant and more honest. But students in honors classes have often been selected for their grades in prerequisite courses, earned by working hard and meticulously following instructions, rather than for their ability to understand. But in the more pleasant course in which students are selected for their abilities, one need not COERCE them to understand. And those who have to be COERCED to do things end up badly misunderstanding the subject. $\endgroup$ Commented Apr 3 at 22:57
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    $\begingroup$ The only way to coerce students to understand I know is to give them assignments that are impossible to accomplish without understanding the subject matter. They don't need to be technically complicated, quite the contrary, but I agree that designing such a problem set takes more effort than choosing a few random problem numbers in a textbook :-) Also, I usually try to make sure that those who aren't really ready for the course exercise their option to withdraw without penalty as early as possible. $\endgroup$
    – fedja
    Commented Apr 3 at 23:31
  • $\begingroup$ Having students who aren't ready for calculus take calculus is a huge problem. Specifically, they don't know that each day in mathematics you're trying to understand why things work the way they do, rather than merely learning dogmas, and they don't know that with math problems you should expect to be given enough information to figure out the answer, but you should not generally expect to have been given an algorithm that will give you the answer. Students like that bring in tuition money, so they are told to take calculus. It would be possible to design other courses suitable for$\,\ldots\,$ $\endgroup$ Commented Apr 3 at 23:43
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    $\begingroup$ Are you assuming the "clearly better alternative" would consist of some other form of calculus course, rather than some other kind of math course? That depends on the major: for engineering and physics, I would stick with calculus, but for, say, English literature I'm all in favor of switching to combinatorics, discrete math, and similar courses. $\endgroup$
    – fedja
    Commented Apr 9 at 20:38

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