Inverting the transition-to-proofs class

When I see the first back-to-school sales, I know it’s time, like it or not, to start prepping classes for the fall. This fall I am teaching two courses: a second-semester discrete math course for computer science majors and then two sections of “Communicating in Mathematics” (MTH 210). I’ve written about MTH 210 before when I taught it last fall. This fall, it’s going to be rather different, because I’m designing my sections as inverted or “flipped” classes.

If you’ve read this blog for any length of time, you know I’ve worked with the inverted classroom before (here, here, here, etc.). But except for a few test cases, I haven’t done anything with this design since coming to GVSU. I decided to take a year off from doing anything inverted last year so I could get to know the students and the courses at GVSU and how everything fits together. But now that I have the lay of the land, I think it’s time to start implementing.

For context: MTH 210 is a transition-to-proof class that, roughly speaking, is supposed to be taken between Calculus 2 and Linear Algebra (although that’s not a rule). It’s required for such upper-level mathematics courses as modern algebra and discrete mathematics, and it also serves as a “Supplemental Writing Skills” course for GVSU students. Here is a list of learning objectives [PDF] that I drafted for the course, and this should give you a sense of what we cover in it. There is a lot of content and a lot of process to learn.

MTH 210 is a big course in the math major here. We usually enroll about 80-100 students per semester in it and we even run a section in the summertime. But it’s also a course with some issues. I know you can’t put too much validity in course grades, but when we faculty received some grading data his past year that for the MTH 210 sections combined throughout the 2010-2011 academic year, some points really jumped out at me:

  • 25% of students enrolled in the course made an A or A- in the course. That’s pretty good, BUT:
  • 13% of students made a grade of F in the course, and 20% made either D or F. And,
  • We don’t know how many of the “A” students were repeating the course. It’s not uncommon to see students take MTH 210 twice or even three or four times before passing. So perhaps that 25% “A” rate isn’t what it looks like.

The concern here isn’t that grades aren’t sufficiently inflated. It’s that a significant portion of students aren’t learning what they need to the first time, and this leads to all kinds of problems — from delays in graduation times to problems downstream in modern algebra and discrete where students sometimes fail to show mastery of the mathematical skills they need for those courses.

So why am I flipping this class? I want to make clear that it’s not because I need to maintain my bona fides as a flipped classroom person or because I feel the need to show how clever I am at course design. I pass on flipped designs more often than not, because many times they are not the best choice for students. But here, with this course, I think the flipped design can work extremely well. Here’s why:

  • You really can’t learn proofs without doing lots of proofs. No more than I can learn how to be a great quarterback by watching Andrew Luck clips on YouTube. You have to get your hands dirty, and the flipped design is all about using time wisely to develop hands-on skills with the guidance of a coach (me).
  • The class is already mostly flipped as it is. We use a book written by my colleague Ted Sundstrom which is one of the few mathematics books I’ve seen that is really designed to be worked on prior to class. Each section is set up with two “Preview Activities” and fortified with “Progress Checks”; it’s historically common for MTH 210 profs to give the Preview Activities as pre-class work and then work on the Progress Checks in class and not lecture so much. My plans just consist of structuring the work students do outside of class a little more than we do already, and creating a bunch of videos to provide additional insight and examples for students as they read.
  • Having curated video resources will help students later on. In bouncing this idea around to my colleagues, one of the biggest benefits they see of creating the screencasts is so students can refer back to them easily in later courses. If you’re taking discrete math and you can’t remember anything about induction, for example, just go dial up the video(s) on that subject and have a quick refresher. For that matter, any person with an internet connection can get such a review.
  • This course is all about becoming independent as a learner. I think the most important thing students pick up in a class like this is the ability to sit down with new technical material and make sense out of it. You exercise that independence in a number of ways — evaluating your writing, instantiating new definitions, seeking out resources to help you understand a concept, and so on. If students can show ongoing evidence that they are independent learners, I consider the class to have been successful.

More on that last point. As much as college students want to be independent, becoming an independent learner is very tough for a lot of them. In fact, I tend to think that the cause of the 13% failure rate has a strong connection to whether students choose to move toward independence or not. Students who are coming out of the calculus sequence (or come in with AP credit) who want to remain in passive-observer mode when learning tend not to do well in MTH 210; students who come into the class with an open mind and a willingness to learn on their own tend to do well. Or so we think; we do not have data on this, which gives another angle on what I’m doing with MTH 210 this fall and I’ll tell you more about that later.

At this point, I have made up a lengthy list of highly-granulated learning objectives for the course that will be served out in small, frequent doses to students as they prepare for class — to let them know what, eventually, they must master. And I have made out a list of screencasts to prepare to augment the book. That list is currently at over 100 distinct videos, each of which will be short (they have to be!). Note that I will not be putting “lectures” in the video because I consider our book to do a good job at conveying basic information. The screencasts are going to be examples upon examples — and unlike some major purveyors of tutorial videos, I intend for those screencasts to be not only mechanical but also to illustrate the concepts and expert-learned decision-making processes that students can then begin to model. I hope to start work on those screencasts next week and finish half of them by the time school starts.

Any thoughts or suggestions here are welcome, and stay tuned for more.


17 thoughts on “Inverting the transition-to-proofs class

  1. wansort says:

    The results of your survey remind me of a similar phenomenon I have observed when examining reliability scores of state-wide holisitic rating of student writing at several grade levels (5, 8 and 11). The domain or performance area where the greatest variance in scoring occurs is conventions (a domain including spelling, punctuation, usage and grammar).  Even when papers are machine-scored, those several companies operating such programs report the same kind of lowered reliability co-efficients for the domain  of conventions. Surely, there is a dissertation topic here.


  2. what4 says:

    With English becoming an international language, it’s time to agree on a second English language where grammar, usage, spelling, and style that are governed by what is clear and understandable, not by what fits some rule that experts can argue over. 

    By this standard, a number of distinctions can be dispensed with — such as who and whom (and the objective case in general), affect and effect, hopefully, subject-verb disagreements where the meaning is clear, and others.

    If experts don’t embrace a streamlined international English, it will be forced upon us by a billion new users trying to be understood.

    Now’s your chance. Make a simpler English or get used to reading someone else’s version of it.


    1. yabba says:

      English isn’t becoming an international language. It’s already been one for quite a while. And it has remained remarkably consistent, at least in terms of grammar. When it comes to vocabulary, the core of what people need for international communication has been fairly well identified (e.g. the “Oxford 3000”). The most important simplification is probably to use less idioms.

      If experts don’t embrace a streamlined international English, it will be forced upon us by a billion new users trying to be understood.
      But if they are using simpler English, surely we will be able to understand them? The risk is the other way round: if native speakers don’t learn to tone down their English, they are the ones who will be poorly understood. But millions of them already have learned to do that fairly well.


  3. grupenhoff says:

    What I find most interesting is Lucy’s usage, in the very first sentence of this article, of “a few of the actual sentences…” and in the third sentence, “I actually thought….” I for one am tired of the overuse of the words “actual” and “actually.” They add nothing to the sentences they appear in.
    “…a few of the actual sentences” could just as easily be read as “…a few of the sentences,” and “I actually thought,” should be “I thought….” It seems to me that “actual” and “actually” are overused these days to add emphasis or stress to sentences that don’t need it.


  4. anon1972 says:

    I think your diagnosis of “personal taste” is a  little off the mark.  From the results you cite, it looks more to me as though people were inclined to object more vociferously to sentences where meaning was more compromised.  The “beg the question” example, because it involves a misuse (albeit a common one) of the phrase, invokes the specter of a meaning that it does not intend.  By contrast, the dangling modifier “Poring over the argument transcript and the briefs,…” since it’s clear who the implicit subject is (and the main clause of the sentence does not introduce another personal subject who would conflict with the implicit one) is grammatically incorrect — and I would always flag it in a student essay, to help the student learn what a dangling modifier is — but does not impair the meaning of the sentence.  Most of us have had the experience, when speaking informally, of heading into a sentence with a phrase like “As a child,” only to find that we then want to continue with something like, “…my favorite toy was….”  On such occasions, where the grammar is wrong but the meaning is plain, an audience is usually willing to forgive (or mentally correct) the grammatical misstep in the interest of focusing on the content.

    As for “If anybody was going to be shy … it was him,” the sentence is not only clear but would be perfectly grammatical in, say, French (where the verb “to be” does not have to take a nominative complement).  Many of us who are particularly sensitive to/well-trained in grammar also have some facility with other languages, and can therefore recognize not only the logic of the English rule, but also its arbitrariness; besides which, “It’s me” has been accepted in the vernacular for centuries.   The “whom” example (“whom” as subject of a verb) is different because it is just flat-out incorrect — no mitigating parallels from other languages or from commonly accepted colloquial usage can be adduced here.


  5. eeels says:

    Am I the only one who was bothered, in the first quoted sentence, by “the tech industry is again on the verge of another bubble popping”? Can one thing be on the verge of something happening to another thing?  “Another popping bubble,”perhaps; the industry could be on the verge of a bubble that happens to be popping, but surely not on the verge of a popping that is not its own.


  6. yabba says:

    it’s no wonder that our students, bouncing from one of us to the next, sometimes feel the game is rigged against them.
    That’s not necessarily very different to many other subjects where one might think the factual substrate should be in less dispute. One example I experienced was organic chemistry, where there are different conceptual approaches, different ways of simplifying what is a very complex subject for beginners, and different naming systems for the compounds – all mixed up with generational differences.

    The responsibility of the teacher, I guess, would have three components:
    1)  to make sure that a good proportion of what they represent as being reliable content does actually have a decent foundation.
    2) to equip the students to research issues and make up their own minds for good reasons
    3) not necessarily to strip the presentaion of a personal approach but to be upfront about it

    And I personally would be in favor of neither condemning nor approving things; deciding what words to use is about practical judgements of what will do the job, and not moral ones about good and evil.


  7. quasihumanist says:

    1) I don’t like your long list of “learning objectives” – though that may be because I’m not seeing the context of how you’re thinking of it or how the students are asked to look at it.  In my opinion, this course really has only one goal: for the student to learn to read and write proofs.  Everything is filler towards this goal.  Concentrate too much on the filler (as this long list seems to indicate) and the overall goal can be lost; a student can memorize the definitions, memorize the proofs that are presented, never figure out how to write a proof for himself or herself, and miss the point of the class.  Even worse is if this student gets an A on the basis that he or she has learned all the material presented!

    2) I’m wondering how you work with the cognitive challenges your students undoubtedly have.  It takes a good deal of mental effort for many of my students (no wonder – they haven’t had the practice) to keep three distinct ideas in their head at the same time.  To successfully understand and write proofs, one has to keep at least three ideas in one’s head – most implications have at two premises and a conclusion, and that’s only the simple ones – repeatedly over the length of an argument.  Even for a short proof of three or four lines, students can find that quite challenging purely on a cognitive level, never mind the mathematics.  What do you do for these students (who may very well be most of your class)?  How do you acknowledge these limitations (and you have to in order to work with them) without being discouraging?


    1. Robert Talbert says:

      You’re not the first person to have made a similar comment about my learning goals list, and I would just say two things about that. First, the context you don’t see in the class includes the overall high-level learning objectives for the course that will appear in the syllabus. Those would include being able to write clearly, to assimilate new technical information with a minimum of dependence on another person, and to make observations of mathematical phenomena and develop reasonable conjectures based on those observations. The learning objectives list is just the ingredients, not the meal. 

      Second, you say that the course “has only one goal: for the student to learn to read and write proofs”. But reading and writing proofs isn’t a goal, it’s a project — the end result of dozens of smaller goals that lead up to the ability to read and write proofs. I wouldn’t consider the smaller goals to be “filler” but rather tributaries. Even the idea of “proofs” is a multi-headed monster. We say, correctly, that we want students to be able to “write proofs”, but there are direct proofs of conditional statements, proofs by contrapositives, indirect proofs, proofs using cases, proofs by induction (and there are at least three distinct forms of induction we touch on), and so on. To say that we want students to “write proofs” artificially simplifies the process, kind of like saying we want students in calculus to be able to “do calculus”. There’s a lot under the hood there, and the learning objectives list shows students all the parts. 

      Your second point is well taken. I might have to go into more detail on that later but I will say that inverting the classroom sets aside the entire class session for students to work and for me to observe their work, allowing me to give different kinds of instruction to students struggling with different things. The short answer to the question of “what do you do for those students” is: give them clear objectives (hence the list), an overarching sense of where this all fits, and them put them to work and coach them as they work.

      EDIT: I’ll also say that Ted’s book does a really good job of helping students develop the ability to manage the cognitive processes of proofs. It starts simple and then adds complexity gradually, and the writing is very clear and the examples well-selected. I hope my screencasts for the course turn out similarly helpful.


  8. quasihumanist says:

    On reflection, my real underlying question is the following:

    What are realistic objectives for this course for the student who are simply too far behind (cognitively or mathematically) to develop within the next four or five years the aptitude to start a PhD program in mathematics?

    Notes on the phrasing of the question:

    a) “A PhD program” most emphatically does NOT mean a top 20 or even top 75 program.
    b) Many students will have the aptitude but not the inclination for graduate studies in mathematics.  This question is not about them.
    c) Four or five years is to allow the possibility of first enrolling in a Masters program to further develop skills.

    Of course one can ask the same question for the math major itself, but because this course focuses so much on learning skills rather than information (or mechanical procedures), and specifically on skills particular to doing mathematics, it brings this question of objectives into focus.


    Perhaps here is why I am uneasy about the long list of learning objectives.  The long list makes it seem like learning 2/3 of the list is some kind of partial success, when the student who has learned 2/3 of the list will be almost as completely lost in a real analysis class as a student who has learned none of the list.


    1. Robert Talbert says:

      It seems like you really have two questions. One is the rephrased question you gave, and the other has to do with preparation for graduate study. 

      We don’t expect students with math degrees to enter PhD programs. Some of our graduates do, but most do not — typically about 2/3 of our math majors are pre-service teachers, either elementary or secondary. So what we are trying to do with this course is prepare them for upper-level mathematics, to get them to the point where they can reason mathematically with skill, and to teach good writing and communication practice. Graduate study is not especially on the radar screen for most of these students, and so preparing them for grad school is not the dominant concern when designing the course.
      So this fact mitigates your rephrased question somewhat. We don’t look at students and say “you’re too far behind to start a PhD program anytime soon”. 

      I should point out that the individual skills listed are fairly typical for transition-to-proof courses wherever they are taught. This isn’t an especially overstuffed or overly-ambitious course.


      1. quasihumanist says:

        Here is where I am confused:  I don’t really get what “can reason mathematically with skill and has good writing and communication practice” means if it does not mean “is ready for graduate study in mathematics”.  Could you give me some examples that illustrate the difference?  Some skill necessary to be ready for graduate school but not necessary to be considered able to reason mathematically with skill?  (I don’t want examples mostly to do with content (and I realize there is no sharp distinction between skill and content) because most of this content is not part of a transition course.)

        (Also, I certainly believe that, at least in an ideal world, a secondary school teacher in mathematics should be adequately prepared for graduate study in mathematics, although that is not what they are choosing to do!)

        Background: As an undergraduate, I went to an elite SLAC where essentially every graduate is ready for graduate study (at least at a lower-tier program) in their major (and frequently in several other fields as well), even though almost all graduates choose other paths.  I now teach at a place where becoming ready for graduate study is simply unrealistic for most students.  I don’t understand what my job is!!  I don’t know what I should aim to teach my students!!


      2. Robert Talbert says:

        I’d say that the difference is that to be ready for graduate school, skill with mathematical reasoning and facility with communication are necessary but not sufficient conditions. There also has to be some experience with doing mathematical research (IMO, at least) and above all a desire or interest in going farther than the undergraduate degree. Not all students have those, and that’s OK, but we still insist that students earning a mathematics degree show evidence of skill in basic mathematics methodology, regardless of their destinations. What we’re aiming for are competent practitioners of math, not necessarily creators of new mathematics. 

        I think having said that, maybe the notion of competence at creating mathematics is the link between basic skill and readiness for grad school. We do not, in this course at least, engage much in the creation of mathematics. That’s left for senior theses, REU experiences, etc. (which are sometimes expository projects but always creative in nature). 

        Does that make sense? 


  9. Sebrinap789 says:

    I agree that the math course should be restructured. Maybe a great technique would be to form a collaboration group with collegues and feedback from previous students will help with the transition. Networking is a great idea. At times, we as people believe that what we say and do is what its all about. Truly its not and without involving others thought process and techniques the underlying message of the teaching and learning process will not occur. It is all about the students succeeding in the eduation field. I do believe a positive step was made when the problem was recognized.


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