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We tell undergraduate students that they should study two to three hours for every hour they spend in class. We know that many students don't follow through with this nearly to the degree that they should.

In response to this, faculty I know do everything from constructing elaborate daily homework exercises graded with various degrees of rigor, to simply giving students weekly quizzes and midterm exams...since if the students aren't going to take responsibility for the feedback given to them, there really is no point in sinking many, many extra hours grading homework. In fact, many mathematics faculty report very little gain from grading homework, which may be related to the sort of damping phenomenon found in certain answers to this helpful question on grading homework.

Of course, depending on the type of institution, the answers to my following question will differ. That said, there must be some research to help us guide our efforts to build a sense of responsibility and autonomy of study in our undergraduate students.

Question: What is an appropriate amount of responsibility for an undergraduate student in mathematics courses, roughly classified by student year (Freshman, Sophomore, Junior and Senior). Please discuss in particular Freshman service courses like Calculus and Finite Mathematics. What I mean by responsibility is, essentially, how much and what sort of professor feedback foster optimal student performance? What, according to mathematics education research, is an appropriate amount of responsibility to put on undergraduate students in mathematics courses?

The purpose of this question is, roughly, to address the difficulty that comes from giving either too much or too little feedback to one's students. Qualitatively, colleagues have noticed that when faculty over-organize student work, students continue to expect this and don't take on adequate responsibility for their mastery in later coursework, whereas too little guidance in early courses simply leads to disastrous outcomes since students seem to leave high school with very little sense of how to appropriately study.

The purpose of this question is to provide some sort of standard guidelines for determining when it is best to "remove training wheels" etc. for undergraduate students.

I personally try to use in Inquiry Based Learning approach as early on as possible that puts a great deal of responsibility on the students, but this may not be best, scientifically. EDIT: I should flesh this out a bit. Students seem to come to college without the idea that they both need to teach themselves AND we need to teach them. The IBL approach puts a ton of responsibility on students to create knowledge for themselves, but requires a lot of energy from the instructor to guide that knowledge creation. The general idea is that student homework is the content that drives an IBL course, and the professor evaluates this content production. This, in contrast to an instructor meticulously grading homework and passing it back in order for it to be glanced at and filed away or thrown in the bin. IBL puts enormous responsibility on the student, and a lot of work on the professor. I'm wondering how much of such responsibility is best, according to research.

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    $\begingroup$ Note that in some countries there is no tradition of homework -- no feedback along the course -- and possibly just a final exam: if a student isn't able to organize their own study, they fail the exam. That's it. $\endgroup$ – Massimo Ortolano Jan 2 '17 at 9:06
  • $\begingroup$ @MassimoOrtolano: Indeed, I tell my freshmen this every year... $\endgroup$ – Jon Bannon Jan 2 '17 at 12:44
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    $\begingroup$ In addition to the national-culture differences Massimo Ortolano describes, there are local influences. Sometimes there is a sort of arms race phenomenon, where students have Prof. A grading biweekly homework that amounts to 25% of their grade, and Prof. B who doesn't collect homework papers, so they respond by not doing any work in B's course, and they fail B's course. There can also be a race to the bottom. In the US, students look at ratemyprofessor and myedu and decide which professor's class to enroll in based on who assigns less work. $\endgroup$ – Ben Crowell Jan 2 '17 at 15:02
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    $\begingroup$ @BenCrowell: I have some evidence that this can even happen between two college departments. One department decides to grade homework and provide more "student support", and students will come to expect this support and will do poorly in another department that does not provide commensurate support. This might be especially pernicious in certain "student centered" environments. My concern is, though, despite the resulting market forces, providing higher levels of student support may actually hurt students in the long run. Hence this question... $\endgroup$ – Jon Bannon Jan 2 '17 at 16:48
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    $\begingroup$ One challenge is that the level of mathematical skill in prerequisite material varies greatly in most courses. A very well-prepared student might well be very successful in the course putting in 2-3 hours of focused work outside of class each week, while a very ill-prepared student might struggle to merely pass even after putting in 15+ hours of focused work outside of class each week. These are extreme cases, but it indicates how hard it is to give a "one size fits all" answer to this question. (I assume the course evaluation is primarily based on mastery of course material, not effort.) $\endgroup$ – Michael Joyce Jan 5 '17 at 17:40
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Here's another too-long comment posted as an answer:

We tell undergraduate students that they should study two to three hours for every hour they spend in class. We know that many students don't follow through with this nearly to the degree that they should.

I've become pretty unconvinced that raw amount of study time is a significant factor to mastery of mathematical material. Frankly, some students just "get it" on the first presentation in class, while others do not. If the mean amount of study time is 2-3 hours per week, that could be true; but the variation by student is enormous, between 0 and 6 hours per credit being likely (and sharply right-skewed).

I keep a rather sick level of data on all my courses, including pre-course diagnostics and mid-semester study surveys of my design (none of it published or IRB'd). The self-reported average in my college algebra courses is about 1 hour per week per credit, in statistics about 2 hours per week per credit (N ~ 100 and 200). A few students have said they're studying 20 hours per week, and one reported 30 at one point.

But the correlation of study time with final results (score on final exam, and quality points from letter grade in the course) is effectively zero in all of my courses. I don't think that's a novel observation, because there's even an exercise in the statistics book I use that expresses that. On the other hand, Likert responses to "Agree that you've done all the homework" are significantly correlated with final results (R^2 ~ 0.15 in college algebra for me).

So I'm wary of berating students that they should be studying a particular amount of time each week; the variance is so high that 68% of the class may think I'm full of BS because it obviously takes much less, while 7% thinks I'm totally unrealistic because it obviously takes much more (numbers pulled from my statistics class self-responses on study time). And I don't see any way to fix that right-skewed distribution of study time without creating distinct assignments for every individual which (without any graders or TA's at my institution) seems logistically and grading-wise infeasible.

I think I even knew this for myself as an undergraduate/early graduate student; if I just paid rapt attention during a lecture, and avoided taking notes, I could basically digest the material in one go.

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    $\begingroup$ This is interesting! It seems to me that it might be worth building stamina in these courses more than worrying about content, then... $\endgroup$ – Jon Bannon Jan 5 '17 at 22:43
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    $\begingroup$ @JonBannon: Yes, I guess there's something of a "teaching grit" movement nowadays? nbcnews.com/video/rock-center/49202614 $\endgroup$ – Daniel R. Collins Jan 6 '17 at 2:07
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I think that your point here is spot on: "Qualitatively, colleagues have noticed that when faculty over-organize student work, students continue to expect this and don't take on adequate responsibility for their mastery in later coursework, whereas too little guidance in early courses simply leads to disastrous outcomes since students seem to leave high school with very little sense of how to appropriately study."

I teach high school mathematics, and I adjunct for lower level math classes at a Junior College. It is my opinion that as soon as students enter the collegiate level, they are primarily responsible for their learning. We must maintain rigor, or a college degree will become essentially worthless. Students should be held to a high level of accountability. I do not think that professors should ponder this question because we are not here to coddle students. Every professor is different in the "load" they give, but our primary responsibility is to teach our subject matter at the collegiate level. Learning at this level is totally dependent on the student. I do not hold this same view in teaching high schoolers, but college is a different game.

This would serve better as a comment. I understand I've not referenced any mathematics education research, but the comment box would not hold this response.

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(I thought of answering your question, but then I realized that it was more of a comment. As I began typing, it turned into be too big to fit into the comment section, so take this as a grain of salt.)

My first thought was that this needs to be solved piecewise. STEM majors are very different from Liberal Arts majors; students at four year Universities are different from students working on an Associate's degree. However, quantifying your question with the phrase "Freshman service courses like Calculus..." probably means you are interested in those on the STEM track, so I'm assuming that's what you're interested in.

Unfortunately, my only experience with the types of courses I think you're interested in is as a student, so I'm unqualified to answer your question. However, to aid anyone who is looking to answer, early Google Scholar searching yields one relevant book on Student Responsibility by Davis and Murrell (available for free download here). Though it is not mathematics specific, it might be a helpful starting point.

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  • $\begingroup$ The STEM track assumption is fine to make. Thanks for the comment! $\endgroup$ – Jon Bannon Jan 5 '17 at 8:27
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Gradient Limitations:

In the United States, scientists and engineers are trained in how to handle long-term projects as part of their technical writing classes, laboratory classes, design classes, and thesis-writing classes. They start with writing 1-5 page papers, with executive summaries. They work up to 20 page laboratory write-ups, design documents, and business plans. By their senior year of college, they write 30 - 50 page theses. Master's theses and Ph.D. theses are much longer.

Their math classes are supposed to provide a tool-kit, and lots of practice using the tool-kit. Their math classes are not expected to teach them how to break down projects that take longer than an hour.

My Experience:

I have taken math classes (and math-heavy classes) in a wide range of formats. These formats included independent study, large lectures with recitation sections, small seminars, and individual tutoring. I have taken classes in public schools, community colleges, MIT departments, and MIT's Experimental Study Group. Fortunately, I did not take any math classes from MIT's Math Department, though I knew many students who did. I personally have performed and published an applied math project that took several months -- but this project was not part of my undergraduate education.

Effective Freshman/Sophomore Format for Math Classes for Science and Engineering Students

In my undergraduate math courses (including Calculus through surface and volume integrals, Linear Algebra, Differential Equations, and Introductory Statistics), the most effective courses had the following expectations:

  • Classes met two or three times per week.
  • Classes were small enough that large colleges might call them "recitation sections" or "seminars".
  • Students doing passingly well were expected to spend about twice as much time on preparation as in class.
  • Students were expected to attend all or nearly all classes. (In California's community college system, this was the basis for the college's funding.)
  • Homework was assigned after each class, except there was not homework immediately before mid-terms.
  • Students were expected to perform a Check-By-Substitution at the end of each problem, to self-check whether they had gotten the answer correct.
  • Students could look up answers to their homework problems in the back of the textbook.
  • Students were expected to read the material for each class session ahead of time.
  • At the beginning of each class (other than mid-term sessions), students were expected to say which problems they had trouble with.
  • The first third of each class was dedicated to going through the homework problems that caused the most trouble for students, or the exam questions from the previous session that caused the most trouble for students.
  • The remainder of each class was used to cover that session's topic.
  • Mid-terms every couple of weeks, on coherent chunks of the course. Mid-terms were held during class time. Mid-term questions were similar to homework questions.
  • Different courses had different final exam formats. Merced College's math instructors used 50-question multiple choice questions graded by Scan-Tron. MIT's courses often had original questions that would be used as future homework questions. Both formats worked well.

Homework / Exam Summary:

  • Homework is for the benefit of the student. (And as a real-time feedback mechanism for the instructor.)
  • Students should check their own homework.
  • Homework difficulties provide "teachable moments".
  • Students should be learning from their textbook, in class, and while doing homework. Students should not be cramming for mid-terms.
  • Whether students have done their homework should be obvious to their classmates; failure to learn from homework will result in doing badly on mid-terms.
  • Frequent mid-terms can provide the bulk of the grading.
  • Final exams are a sanity check that the material was solidly learned earlier in the course. Students should not be cramming for finals.
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  • $\begingroup$ Totally gratuitous comment - many math theses are < 50 pages. $\endgroup$ – kcrisman Jan 9 '17 at 16:45
  • $\begingroup$ Less gratuitous comment - this answer does presuppose the "toolkit" point of view, and is very helpful for that viewpoint, but you might get pushback on some aspects of this from those coming from a viewpoint of trying to teach math qua math. $\endgroup$ – kcrisman Jan 9 '17 at 16:48

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