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I've heard the phrase "we passed that course already!" too many times when asking for e.g. the derivative of a simple rational function or a simple integral, getting blank stares, and digging deeper. This in a second-year discrete mathematics course, after they took at least an introductory calculus course. The gist is that students consider a course passed as subject matter to be forgotten as soon as possible.

Two, probably related questions: How can I make them aware of the mathematical prerequisites, in a concrete way? Adding a line in the course description saying you shouldn't take course XYZ-203 if you haven't passed ABC-101 clearly doesn't work. I don't have the time to struggle with such trivial stuff in class.

The wider question is how to string courses (particularly courses from different areas, here common Math and Discrete Math as a specialty course for Computer Science) together? How can we convince our students that what they are being taught is a series building on previous courses, not just a set of scattered, unconnected subjects?

[Yes, I know this is very broad. If somebody has specific suggestions for a Discrete Math course, covering proof techniques (again, should be covered in the first math course, but isn't really), a glimpse of generating functions (thus at least the mechanics of derivation/integration and power series), and a bit of number theory (using a bit of groups/rings) to give some understanding on modern cryptgraphy, I'd be thrilled.]

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    $\begingroup$ Hand out a syllabus that mentions some of the expected background for students; use the first class meeting (and assignment) for review - even if the questions asked don't really seem related to one another. Maybe spend two class meetings (and two assignments...) on this. Suggest students take time to review material if they are rusty, and make the requirements known (as best you can) so that they can drop the class if feasible. Tell them upfront that the class requires hard work. (That said: If the course description just says that introductory Calculus is required, then I... [cont'd] $\endgroup$ – Benjamin Dickman Mar 30 '14 at 6:28
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    $\begingroup$ That seems like a rather strange perspective on any subject, to be honest. A music major shouldn't be surprised when composition theory comes up in a musical history course. Is there time to have an expository lecture on higher math that emphasizes interdisciplinary problems? I.e. can you dedicate time to convince them with prominent examples that such intermingling is actually the normal state of things? $\endgroup$ – Robert Mastragostino Mar 30 '14 at 8:06
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    $\begingroup$ I have to confess that here in the US I've never heard a student say "But we passed that course already!". More common in my experience is "That was a long time ago." In other words students acknowledge that they ought to remember something from a previous class but appeal to the passage of time to justify their forgetfulness. I wonder if this is a cultural difference between USA and Germany? $\endgroup$ – mweiss Mar 30 '14 at 16:19
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    $\begingroup$ To me, this seems like a problem in the way the student is approaching their education. If a student sees their education as a goal in itself, who is studying because they love the subject matter or want to better themselves, it wouldn't make sense for them to say this. To me this indicates that the student is either (1) studying just to "have a degree", or (2) are not interested in their math classes specifically (perhaps they're programming majors that feel they're being forced to take unnecessary math courses). $\endgroup$ – Jack M Apr 17 '14 at 23:11
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    $\begingroup$ I have been somewhat surprised that I don't get asked this question too often. I guess it's because I largely teach freshmen, where we do need to review a lot of high school math anyway to make sure they are on the same page. Anyway, I reply with an analogy. "When you were learning a foreign language in school you didn't expect to be allowed to forget the basic vocabulary you learned in grade three, when you were on grade seven. It's the same way with math." $\endgroup$ – Jyrki Lahtonen Apr 19 '14 at 8:26
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Sorry for the necromancy but I had a slant to add and too long for comment.

Viewpoint: It's not reasonable for the students to eschew previous techniques. But it's ALSO not reasonable for you to be so super surprised that they lack perfect recall of previous material. Instead of this all-or-nothing attitude, why not a more sympathetic approach: "yeah, I know partial fractions integration was a while ago and it was at the end of a long course...but we actually need that thing now...don't worry, we won't rehash every tough integration technique but this snake needs to come out of its hole again...no sweat, let's walk through the problem slowly and show the steps we use."

Course design (you asked): I'm not sure that proofs is the key thing to working on with a course where you are trying to give "exposure". For someone who uses a technique a proof is really more like a motivation for why you can use the technique. But the technique is the important thing, not the proof.

Course structure: I would also suggest designing the course to follow a simple text, not as an ad hoc set of subjects or something that requires following lecture notes or stitched together handouts. This gives the students some supporting scaffolding.

Pedagogy: The surprise at imperfect students and the attention to proofs makes me suspect someone who is good at understanding MATH but weak on understanding STUDENTS. The math is half the problem. But understanding students and how to improve them is PART of the problem. Yes, absolutely, it is! You can't ignore it, just as you can't ignore other independent variables in multivariable calculus, just as you can't ignore interactions and higher terms in stats problems that are nonlinear.

Course analysis: Take a look at your course and take a look at the previous calc course (or even algebra). Figure out the key concepts that usually trip students up and make a little cheat sheet or refresher formula list or even list of key sections in the actual text used most commonly in your uni (or in nearby high school). [This would be given to the students on day one.] But note, the point here is to be thoughtful and analytical and find the MINIMUM that they need to know. Force yourself to list what is needed and not needed. And then within the needed, list the things that typically confound students, not the ones that are usually no problem. So, instead of saying "you should know that whole calc book"....well, sure they SHOULD. But we are dealing with imperfect beings. Not angels. Not robots. So if they need integration by parts (and it usually is an issue) but they don't need partial fractions, list parts integration and don't list partial fractions. Figuring this out is part of your JOB. And note that it is a relative thing (very different from Euclean proofs). This is about max gain for min pain (AND with imperfect information). At the end of the day, teaching math is more like science, engineering, business (practical activities using preponderance of evidence) than it is like math itself.

Coverage of old material: I would take a middle road when using topics that students are "supposed to know" but in reality often don't remember. Don't just run remedial sessions and reteach the topic in general form. Don't just expect students to have ready facility.

Instead, something like (class before, end of class): "OK, for the next class...we are going to need to use partial fractions...I know that was a frustrating topic for many people in calc class and you may not have remembered it, so please take a look back at it tonight, because we will have to use it next class."

Then when you do whatever derivation you are doing and the partial fractions come in..."here are those darned partial fractions coming back. I know it has been a while, so let's walk through the algebra carefully here." [Then do show all the steps...and for the problem of interest, not a general partial fractions review.]

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  • $\begingroup$ +1 for stuff in sentence with just as you can't ignore other independent variables in as well as for (but not limited to) But we are dealing with imperfect beings. Not angels. Not robots. $\endgroup$ – Dave L Renfro Apr 22 '18 at 8:11
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I've heard the phrase "we passed that course already!" too many times when asking for e.g. the derivative of a simple rational function or a simple integral, getting blank stares, and digging deeper.

In my opinion, the only appropriate response to a statement like that is "Excellent, then you should have no trouble answering this!"

Then — assuming, of course, that the student in question does in fact have trouble answering the question — you can take some time to explain to your students that, yes, they are going to need the skills they learned in earlier courses if they want to keep up with this one, and that, yes, this was explained in the course description, and finally that, if they've forgotten those skills, now would be a very good time to pick up their old textbooks and refresh their memory.

This is how you convince your students that the courses they're taking really do build upon each other. Nothing teaches a lesson quite as well as experience, and the realization that all the stuff they crammed and forgot last semester is actually something they'll need to re-learn now will hopefully teach them to pay at least a little better attention to future courses.

Hopefully, this is something they'll encounter fairly early, preferably during their first year at college, or at least at the beginning of the second year. It won't be easy for many of them — let's face it, this habit of study-and-forget can become very strongly ingrained during secondary school, where much of the material taught is something the students can forget after the exam without any immediate and concrete consequences — and some of them will complain, but they'll also gradually learn better.

In any case, you'll probably have to incorporate some remedial teaching and refreshing of earlier material into your course, if you want to ensure that most of your students will be able to follow along. One option can be to prepare and hand out an extra set of optional "warm-up" exercises that briefly cover the things you're expecting the students to already be able to do. (If your course covers a broad set of topics, you may want to do this several times during the course, as new prerequisites become relevant.) It could also be a good idea to encourage the students to help each other, perhaps even by offering extra points to students who volunteer to help others catch up.

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    $\begingroup$ You are certainly right, but the trouble is that they take this as a justification to never have to do it again. $\endgroup$ – vonbrand Apr 18 '14 at 13:48
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One thing I tend to do is to mention fairly often in class when students will need current material in future courses of various types and also how we are relying on previous material in the current course. This doesn't completely ameloriate the problem, but I hope it raises awareness of the often vertical structure of mathematics and the fact that the courses are taught in a certain order for a reason.

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How can I make them aware of the mathematical prerequisites, in a concrete way?

One way which has been proven effective time and time again would be to do a pretest, from that you can glean their experience in that level of mathematics as well as provide the student the knowledge of whether he/she SHOULD take that course. Require them to test somewhere around the halfway point of summer, long enough for them to forget everything they didn't learn, but with enough time to drop-out of the class or get tutoring if they need to.

The wider question is how to string courses ...

Perhaps the easiest way to get your students to really grasp the connections is to simply have them write, read, debug, modify, and just experiment with code, higher languages, assembly, as well as other interactive projects, anything that can get them closer to real-world applications.

Example: You could have an 'analyze linux day' where you decode exactly what those original 17 lines meant and how they function with respect to the conceptual foundations they've been learning. Have them write an essay that outlines the OS's limitations, algorithmic complexity, etc.

I'm assuming you probably only have a small block of time with each group of students, and I would encourage you, and any educator to create a blog/website/forum for the class (if they don't have one already). This will help ease the amount of work you need to do tremendously! It's also a good idea to record, or allow students to record, and post videos of your lectures onto said website for reference.

If you are still in contact with any of your old students, I would definitely talk to them and see what suggestions they might have, especially if they got a job in the CS field.

My final 2-penny thought: The easiest (IMO) and most effective method for learning is the 'keep them entertained' approach of most grade-schools. It's not going to get any better for classical educators with the amount of information vying for student's attention these days.

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    $\begingroup$ Welcome to the site! I think your answer is very good. May I suggest removing the last line about your background? The last meta discussion on whether student answers should be allowed concluded that answers should be judged on their own merits. $\endgroup$ – Chris Cunningham Mar 30 '14 at 17:57
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    $\begingroup$ At public schools, it may not be so easy to institute a requirement that students take a placement test. In California, for example, this kind of thing is regulated by the ed code and has also been the subject of some lawsuits that set legal precedents. $\endgroup$ – Ben Crowell Mar 30 '14 at 19:03
  • $\begingroup$ In my particular setting, my course is required. Students aren't really at liberty to take it or not. Most of our students live far away, so doing any sort of pretest during vacation is completely out of the question too. $\endgroup$ – vonbrand Mar 31 '14 at 11:08
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    $\begingroup$ @BenCrowell Well, the EASIEST way to make sure students are prepared is to work with the other teachers and make sure they understand exactly what they have to teach, but even then it's not always enough. $\endgroup$ – Woody Mar 31 '14 at 23:32
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I teach middle and high school math students. I have encountered the issue of students saying they have already covered the material on several occasions. As a result, I would give the students a pretest at the beginning of the academic year. This way I would have an idea of what the students know coming into my classroom, and it would also give the students a glance at what they would be learning. This by itself allowed the students to see that they would be 1) using previously learned information, and 2) learning more than what they already know. Another plus to giving a pretest is that if all the students are familiar with a topic, say solving one step variable equations, they you do not have to reteach the material, but rather give a review of the topic, which will allow you to focus more on the new material that hey students may have difficulties with. Because you have given a pretest, you can use that as your data for why you are not "teaching" the section.

A way you can make the point concrete that something is a prerequisite, is by giving them a problem that they should be able to solve by the end of the course, you could tell them that they have already learned some of the methods needed to solve the problem, and the course they are in will teach them the rest of the methods necessary to accomplish the task. You can also use this as a way of stringing other courses that the students have taken together, as a way of emphasizing that Math, as a whole, builds upon itself.

If you are talking more specifically about how you can show math is related to other courses, like engineering or computer science, you could give some examples from an engineering textbook, and emphasize the point that it is from another field of study, or state that the methods they are learning are also necessary for other fields of study while being specific as to what fields, courses, or particular practices.

To comment on your last note about some techniques for Discrete Math: I am currently teaching a student who is in Discrete Math, but we do not have a Discrete Math textbook. So as a result, I am using the Calculus, Algebra 2, and Applied Calculus textbooks for material. You could do the same or something similar as a way of showing your students that the courses are related, if you want. I find that the calculus book is a very good source for example and explanations of derivation/integration, power series, and number theory.

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  • $\begingroup$ Take a peek at Lehman, Leighton, Meyer's "Mathematics for Computer Science", it covers a lot quite nicely. It is a set of lecture notes, roughly updated each term. $\endgroup$ – vonbrand Apr 17 '14 at 23:10
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I think one there is a bunch of problems. First, university curriculum tends to treat every topic exactly one time and then assumes people know how to deal with it. Second, as long as students may pass the second course although they forgot everything from the first (or maybe never attended it), other students will try the same. Third, they've passed it. Why are we giving students good grades who cannot do the basic stuff three months later? (I don't want to blame you for this, I believe this problem to be general.) I think there are even more problems involved.

So what to do? School's answer to the first problem is a curriculum which often treats roughly the same thing (slightly extended; spiral approach). I am not sure if that is good at university level. The second and third problem are connected since they are both on passing exams. The simple answer would be "be more demanding in your exams". However, this might go wrong if too many students would drop-out then.

So I don't have a clear answer but I have an idea in mind for some months now. What would happen if we can manage that students have to pass every exam twice. The better grade counts for their certificate. Most of them should try to improve their grade when they have their second exam so they would repeat all the stuff which is somehow spiral. You might then be able to raise standards a bit. In addition, they might believe that learning for understanding is more appropriate than learning everything off by heart, since they should not forget everything after exam 1.

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    $\begingroup$ I think, even at university level, you do repeat things often enough. Every time, you need to do something old or in many exercises; there are also some questions only for the sake of repetition. For example, I give them normally an "exercise sheet 0" with a big question like "Please look and explain the following list of known definitions and theorems and give examples" - and I think, I'm not the only one. $\endgroup$ – Markus Klein Mar 30 '14 at 8:31
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    $\begingroup$ On the one hand, I agree that students get many possibilities to check the basics. On the other hand, students are rarely induced to do so if they don't want. In addition, basics are often rather superficially checked (e.g. in terms of examples, but not in terms of handling definitions etc.). $\endgroup$ – Anschewski Mar 30 '14 at 9:43
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    $\begingroup$ That is the root of the problem: As a student, you can (at least most of the time) just cruise by without ever looking back to earlier courses. So, are they really necessary? If they are, how do we weave what was learned in them into later courses effectively? $\endgroup$ – vonbrand Mar 31 '14 at 11:06
  • $\begingroup$ You nailed the root of the problem: students are passed without really being able to exercise those skills three months later. $\endgroup$ – corsiKa Apr 16 '14 at 23:02
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My syllabus describes the prerequisites and usually elaborates on the importance of the prerequisites. And then in classes like precalculus, calc 1 and calc 2, I stubbornly give a quiz the first day over some of the simpler prerequisite material. Nothing enforces my prerequisite statements like a grade over earlier material!

I usually send out an email a day or two before the first class "reminding" students about the prerequisite material and alerting them to this first quiz. But regardless, I publicly insist that I gave a "nice" quiz over "easy" stuff and I move on. (Privately, in my office, I help a few students "remember" the prerequisite material.)

Getting on a soapbox -- I have colleagues who spend 2-3 weeks "reviewing" the prerequisite material. In my opinion that merely reinforces the student belief that we don't really expect people to remember the previous course. I don't do that -- I start fast in calc 2 with new material but then I occasionally stop for a moment (5-10 minutes of lecture) to "recall" some material "we all know." I find that these regular short reviews are helpful to the students but they don't give the impression that old material can be forgotten.

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  • $\begingroup$ I like this approach. $\endgroup$ – guest Apr 21 '18 at 15:42
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A tennis instructor once wrote that if a student insisted that s/he had mastered a technique even tough s/he had not, the instructor would hit shots to "pick on" the weakness in the technique, and thereby demonstrate that student actually had a lesser mastery of that "material" than of others.

So the teacher would give quizzes on the supposedly "earlier" material and demonstrate that the students actually score lower on it than more recent material.

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    $\begingroup$ -1, since the question is more about the attitude that "I've passed the class already, therefore I don't need to know it anymore." Not "I've passed the class already, therefore I mastered the material." Your suggestion would be good for the latter. See also OP's comment matheducators.stackexchange.com/questions/1028/… $\endgroup$ – Willie Wong Jun 16 '14 at 12:42
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    $\begingroup$ (Parenthetically, your answer is also already given as part of at least two of the answers from two months ago. Is it really necessary to post a new one?) $\endgroup$ – Willie Wong Jun 16 '14 at 12:44

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