What Do Grades Have To Do With It?

Once again, I rant about meaningless grades and courses.

I was sitting in a meeting in which we were discussing some students that were going to participate in a small research program, and as part of the conversation, we reviewed the courses and grades for the students. The students had taken or were taking advanced courses, honors version, etc., and had made high grades in the courses that they had completed.

And, either out of arrogance or ignorance, my internal response, was ‘So what?’ My experience with similar courses led me to believe the content and the practice of those courses did not have any relevance for the program we were considering (except possibly some help with the vocabulary; but that can be taught quickly). The grades only meant that they were successful students, but probably didn’t really tell me anything about the student’s intellectual ability or his or her capacity for research. I’m sure these are fine students and that they will do well in the program, and it’s not their fault that we have incomplete or useless information about them. So who or what can we blame?

I’d like to blame grading and grades and all the things that are decided by them. Do you want be part of some special opportunity?, then What’s your GPA? How do we find the best student?,  let’s Look at the averages. How well do you know topic X?, becomes What did you score on the final? I understand all the reasons that we reduce complex information down to this one number, but I keep running into situations where it is no good. Having this one mystical number define so much of a student’s academic life naturally drives them to figure out ways to get good scores, which often becomes separated from actual learning. So we drift off into some form of holistic review, or standards based grading (SBG) or something more ‘complete’, increasing someone’s workload, but most likely at the end reducing a student back down to a single number.

I keep wanting to write out a solution, but I always see it sliding back into one final number. For example, suppose we keep track of how a student does on every activity, including early drafts, corrections, in-class participation, test, etc. and then for any query we have about the student, we have some way (a computer avatar?) to use that information to evaluate that student (think of those evaluation forms where they ask how this student would be as a teacher). We have retained some flexibility as we can evaluate against different criteria, but in the end, we will come up with a score, like Ability to Do X: 9/10, Ability to Do Y: 8/10, and someone is going to add them up or average them, because that’s what we do with a bunch of numbers, and because we know that all 10/10s is better than some 9/10s.

Maybe our only hope is to find a way to stop asking for this type of final or overall evaluation and make everyone who wants to know about our student, read or listen to a long description of what he or she has done.

Hypocritical Postscript: soon after the meeting that prompted this rant, I was looking at my child’s grades and was pleased that one teacher seemed to be providing lots of opportunities to earn grade points through low-stakes, participation-like activities, thus providing a means for my child to get a high grade. Shame on me.


Which Students Do You Remember?

Celebrating 30 years in the classroom, I reflect on past students.

First off, I’m not that old, I just happened to have a chance to be a TA while I was an undergrad.

As another generation of students leaves campus and the next group starts, I’m tempted by the thought of what makes a student memorable.  And thus, I’ve gone into the archives and have remembered these: (BTW, I’m not using names, not out of privacy, but just that I’m horrible at remembering names and too lazy to go look them up; same for the dates.  Also, I’m not including anyone from the last 5 or so years as I may remember them, but they haven’t had enough time to become memorable).

1983 – Calculus recitation – The only student I’ve ever had that bombed the first test (she got a D) and then put in the work needed to pull it up to a B by the end of the semester, without really much help from me. I thought that this would happen again, but it really hasn’t.

1985ish – Precalculus – An older student (probably about my age at the time) returning to school to get a degree.  He passed the course, but what is more memorable is that I continued to see him around campus as he finished his undergraduate degree in engineering, and then began work on a graduate degree. I didn’t expect a student starting in precalc to end up at that level.

1993 – Some PDE Course – A graduating senior, super bright, and searching for a non-academic job. He was my first encounter with a really bright and mature student (outside of my fellow graduate students). He came by to chat often, especially about his job search which included Microsoft and whatever passed for Bell Labs at the time. He’s memorable because of this and that I remember writing in a letter of recommendation that he was (truthfully) the brightest student I had taught so far.

1998ish – Graduate Numerical Course – He was an undergraduate in a graduate course (which was less common then than now), and he dominated. In the second semester, he barely passed as his girlfriend of 3+ years had broken up with him over Christmas break. He was another super bright, mature student. He went on to a big name graduate program, and did super well (Ph.D., I think). He’s memorable because of the quality and the dynamics of the year I knew him, and his eventual success.

1998ish – Modeling Course – Same year as above, and actually a friend of the student above, but this guy was not the same type of student. He’s memorable because he was the first student I ever saw get really excited about doing mathematics (from a modeling project) and then choosing to go to graduate school. He got into a good school, and I think got a Masters.  By that time I’d seen too many students, mostly undergrads but some grads, just going through the motion, and it was energizing to see a student get excited about math, make some serious changes to pursue it further, and be successful.

At this time I should start listing some memorable graduate students, but I remember most of them as good people and good students, and as we recruit many students like that, none stand out above the rest.  I remember them and am glad I knew/know them.

Of course there are more recent students, but as I wrote above, they need time to see if the become memorable.

So, what can I conclude from this list? I’ve enjoyed working with many bright, hard-working, successful students, but that typically does not make a student memorable. I probably can’t tell you who made the higher grades in my classes. But students that exceeded expectations, ones that pushed out on their own and achieved something, and those that had life transformations and shared them with me, those are memorable. And so, when I’m advising a student as they prepare for preparing for life past my little corner of academia, I always tell them to be known by someone(s) (faculty) on campus; to share their experiences and dreams. That will make them memorable.

Teaching Proofs

How I teach students to do proofs (and it’s not how I do proofs).

As in many math departments we have a ‘transitions’ course in which math majors are introduced to logic and mathematical proofs. The key feature of such courses is the students get to do lots of proofs and, hopefully, are prepared to do abstract math/proofs in the rest of their advanced math courses. For my own sake mostly, here’s a description of how I approach teaching students to do proofs.

We start with proofs after we’ve done some work with logic: truth tables, or, and, negative, statements, variable statements, contrapositives, etc., and discussed definitions. We’ve actually done some proofs already via truth tables, but those are so exceptional that I don’t really count them as proofs. Here are the main ideas/steps I use, in approximately the order we work through them.

Step 1: A proof is an ordered sequence of true statements (definition). You might be expecting more, but that’s where I start, as the first point I want to make is what it takes to not be a proof. One false statement, and you are out. I also make the point that every proof, proves something, just maybe not what you want it to prove. So now I’ve set the stage for reading and evaluating proofs: (1) is every statement true? and (2) what does the proof prove?

Step 2: Every statement in a proof is true for a reason. Initially I have them use two column proofs with every statement (column 1) having a reason (column 2). These are simply proofs with sets or basic number theory.  You’d be surprised how hard it can be to come up with reasons (or at least names for reasons) some times. As the frustration with the tedium of 2-column proofs builds, we begin discussing the purpose of a written proof (to convince the reader of the truthfulness of some statement) and how different communities establish the standards for a complete proof. We soon move to paragraph proofs, but the students know that they have to be able to support every statement they make.

Step 3: A proof communicates its results clearly. As we move to paragraph proofs, the emphasis shifts to clear communication. At times I’ve used a 3 part rubric with one part being purely on the quality of the communication, grammar, etc.  I’ve also had them read different proofs and evaluate and discuss their clarity. For practice, they can also rewrite some of their two-column proofs in excellent paragraph style.

Step 4: The structure of the proof reflects the statement being proved and vice-versa. By this time we’ve moved past sets to field axioms, functions, and other topics and so they are seeing many different types of statements to prove and many different types of proofs. So I start asking them to look for the connection between the statement, the definition involved and the form of the proof, and I expect them to make the match. For example, when we do proofs for equality of sets, we know, by definition, that two sets are equal if each is a subset of the other. Thus every ‘set equality proof’ is structured as two subset proofs (‘subset-subset’), and a subset proof is a class if-then proof, which has it’s own specific structure. In the big picture, this is a fairly limited idea, but for the proofs we do, it is adequate, and, in any case, it is always a reasonable place to start.  It also gives us a new way to look a definitions, to try to determine what proof structure the definition evokes.

Step 5: A proof is read and judged within a community with its own standards. This first came up in Step 2, but I bring it back towards the end and spend more class time discussing all the aspects that we’ve covered and what we as a class (our community) has set as the standards for proofs. This step helps them begin the movement from highly technical and verbose proofs, to more concise proofs that tell the true (no pun intended) story. I try to move out of my position as ‘arbitrator of truthfulness’, but am not always that successful at it.

Step 6: A perfect proof is a thing of beauty and you should be proud of having created it. The last part of the course includes each student creating a portfolio of 20 or so, ‘perfect proofs’. The perfection reflects what we’ve done in class in clarity of composition, correctness, and the telling of a good story. They can submit versions of a proof until they get it right, and, at the first, I am very picky about even the tiniest of punctuation or grammar errors. The reward for a complete proof is it gets stamped ‘Completed’ and initialed and it goes in their notebook. There is often much celebrating when the 2nd (or 3rd or ..) draft gets approved.

I do other things, and I have ideas for the next time I teach the course of other things I want to try (like including more experimentation and conjecturing for forming statements), but this is the essence of my course.


Technology in the Classroom Survey

A recent Chronicle of Higher Ed article on technology has me thinking….

In the CHE article Tools That College Students Wish Their Instructors Used Either More or Less, 2013 they report on a survey on the topic of the very long title. The results have prompted some questions, but first some concerns about the survey:

  • The option of ‘same’ is confusing as we don’t know what current practice is.  Even ‘more’ and ‘less’ without a basis, is difficult to interpret. It would be more meaningful if the results were measured in frequency of use, or in amount of time used (as a percentage of class time). I suspect the intent is that we interpret ‘more’ as they like it, ‘less’ as they don’t, and ‘same’ as undecided.
  • Some of the areas are relevant to the entire course (e-book or not, LMS or not, capture or not), and some are for the day-to-day activities (integrated laptop, cellphone, tablet use), so it seems unreasonable to have them on the same chart. It also makes me wonder what basis a student used to answer the survey. For example for the LMS question, would a student interpret it as more classes using the LMS, or a particular class using the LMS more?
  • The type of use of each kind of technology probably also makes a big difference, but is probably beyond the scope of this survey. (This may be the most important thought, as using technology works when it is used where it provides a real advantage; if it is just a fancy update to something that can be done the same or better by non-high tech means, then it really should be used less).

Now, my questions/thoughts:

  • Why is cellphone and tablet use wanted less, compared to laptops? (Want Less Usage: 30.8%, 28.4%, 18.4% respectively) I’d think that laptops and tablets would score about the same as they would/could be used similarly, while cellphones probably are used as clicker substitutes and the results may be a comment on clicker use in general. It could be an issue of wrong use of technology, like using a cellphone to read material or take online quizzes, or using a tablet to take extensive notes. Or it could be a matter of choice, where students have to use a certain personal or class device, and they just aren’t comfortable with it. If its a BYOD class and the material supports all types of devices, would students disfavor the use of any of the technology as high?
  • Why all the hate for e-books? (Okay not really hate, but not a lot of love: Want More: 47.1%, Same: 25.3%)  e-books typically cost less (but not as much less as you’d think, plus there’s no resale market, so the net price is probably higher), are easier to port around (assuming one has a smart phone, laptop, e-reader, etc.), can be searched, and usually have extra interactive features, so they seem better than regular books. Plus, at least from a math perspective, most students use the book primarily for the homework problems (with examples probably second), and e-books make it somewhat easier to do this.  I know there is personal preference, but I wonder if students just aren’t used to or comfortable enough with using e-books to really take advantage of them, but I really don’t know.
  • And finally, looking at the big picture, as students are big users of technology in general, have they basically separated their lives into academic and non-academic parts, and then separated their use and preference for technology into ‘more’ for non-academic and ‘less’ for academic? A student might text and tweet prolifically outside of class, but not want to do either for a class.  A student might spend lots of time online reading webpages, chatting with friends, updating facebook and pinterest pages, etc., but have no interest in reading or creating course material online. Will this change as more K-12 schools are adopting technology for daily classroom use?


Lessons from Graduate School [30min]

Does the model for Graduate School inspire any changes to undergraduate education or even graduate education?

We just finished prelim week in our department, where graduate students, after a summer of intense studying, get 4 hours to complete 8-10 problems in some area, and demonstrate their readiness for further study. In graduate school, students take courses as they need some grades and a certain number of hours, but for at least some of the courses (2 in our case), the prelim exam is the real proof of achievement. As graduate students were very good undergraduate students, and the courses are not the critical part of the program, grades in the courses are usually quite high (mostly As, B+s, some Bs) with anything lower considered a strong signal that maybe graduate school isn’t right for the recipient.

This testing represents the ideal that success in graduate school is really determined by the final achievement (the dissertation), and the added tests are feedback, mostly to make sure the students are ready. The best part is that by decreasing the reliance on individual course success and the type of study habits that that encourages, is that is moves students into a focus on independent mastery.  This is all good and is pretty much the tradition in math graduate programs, but it has some flaws (for another post).

When this works, students learn the material, but they also learn to be better students, making connections, extensions, etc. We could probably do a better job helping and directly supporting this, but mostly students do all this on their own. Now for our undergraduate programs, graduate-level work is not the goal, but it would be an honorable goal to help students develop better learning skills and to take more ownership of their own learning. I wouldn’t do this with end of year or degree exams (although, from my understanding of the European models for undergrad, it can work), but I would look to a combination of a capstone course or experience, and an evaluated portfolio of work, to start prompting these changes in attitude and behavior. Another part would be to take the sequence of courses that students take, and work to incorporate supporting activities, like short independent projects, and direct discussions reflecting on learning.

One of the impacts of the graduate program structure is that students can really see that there’s a difference between undergrad and graduate studies. Incorporating more of these higher level activities would help distinguish undergrad from high school.


Stages of Teaching [30min]

A brief listing of the various stages a teacher goes through from beginner and on.

I wouldn’t say that every teacher goes through these stages (nor should), but here we go:

  1. Survivor – the goal is to survive, to get through the material for each day, avoid any disasters, and just make it to the end. Pedagogy, student learning outcomes, etc. are not under consideration.
  2. Lecturer – the focus is on the presentation, the quality of the graphics, examples, jokes, etc. Any failures (by student or teacher) usually causes the lecturer to just increase his or her effort and preparation. Sadly, this stage is often evaluated highly by some students and by most colleague during in-class reviews.
  3. Narcissist – a lecturer taking it to the next level, loving the sound of his or her own voice. This is much like the Lecturer, except that the students are often treated as a necessary nuisance (any failure is blamed on them). If the students figure this out, the evaluations drop, otherwise they accept the conclusion that the professor is so smart and they just aren’t good enough. Most colleagues view Narcissists as good teachers, although they might sense that something is wrong.
  4. Devotee – the focus is on the students, how they are managing the course, and their success. The course is often often highly organized with many extra resources for the student. Problems again tend to cause the lecturer to devote him or herself to developing more resources and holding more office hours, etc. Since students are often happier in such a class, they work a little harder, are successful, and evaluate the course positively. Unfortunately this stage creates a spiraling symbiotic relationship between the student and teacher (the happiness of the student for the positive reviews of the teacher) and learning often is diminished.
  5. Innovator – the focus is on learning but often through specific innovations and technology. There is more focus on learning theory based approaches and overall there’s a better match in this stage between what is done in class and what we want students to learn. Ideally the innovation supports the learning, but it can go where the point of the innovation is to innovate.
  6. Master Lecturer – the focus is on the instructor, but it is all optimized to support student learning. There are elements of the Devotee and Innovator in terms of course organization, innovation and technology. For most students this is what they expect of a great teacher.
  7. Mentor – the ying to the Master Lecture yang; focused on student learning, but now there’s more focus on student satisfaction and success. The Mentor knows how to encourage students without risking being trapped into depending on their happiness. Mentors end up being very popular for independent study, letters of recommendation, and graduation parties.
  8. Rebel – an almost insane devotion to doing whatever it takes to make learning happen, usually taking unusual approaches. All the ‘standards’ are fair game for modification or even avoidance. This is the teacher that everyone knows about, some students avoid, but most students who take the class, end up learning more than they ever expected.

There are others, and variations of these. In any case, I hope you are always paying attention to your teaching and looking for ways to grow.

STEM, STEAM, etc. [30min]

Questions and issues with the current conversation about increasing STEM, etc. graduates.

Q1: Is there really a dire need for more workers with STEM backgrounds?

A1: Broadly speaking, there is a need, but it is not clear exactly what is needed. I suspect there really isn’t a general need for more PhDs or Masters or maybe even BS degrees in STEM areas, but there is a general need for employees to have more general STEM sense, with probably an emphasis on the technical (T) and quantity (M) side. The confusing part comes in considering specific needs. I suspect in this case that employers and, in some sense, society, has very particular, and almost impossible to satisfy, needs. For example someone with a unique combination of various technical (and soft) skills, with particular knowledge of some system or software.

Q2: Assuming there is a need, how should we go about meeting it?

A2: Because of 2 to 4 to 10 year delay between someone entering college and earning his or her final degree, and the amount of press on the STEM issue, I suspect that much of the added need is, or is going to soon be, met. I’m seeing more students who are interested in taking more math either to earn a minor, or to prepare for a post-BS role that requires more math. With that said, the STEM fields do have a need to keep looking for new opportunities to attract and serve new interests. There is probably a place for a general minor in engineering for students in business.  Or an interdisciplinary degree between the softer sciences/social sciences and math or physics.

Q3: What about some STEM training for all students?

A3: Most students get plenty of Science and Math in high school, but not much Technology or Engineering. Most universities continue this trend with their general education or core requirements. So if we do anything, we should look to integrate more T&E into the college curriculum, and in general try to have the science and math requirements prepare students for their possible futures. A little, but useful thing, would be to make sure that students are exposed to some practical statistics.

Q4: Why not more and more STEM?

A4: There’s a downside to this emphasis on STEM areas, as education is a zero-sum game in both time and money. A student taking more STEM classes takes less in other areas.  Money spent on developing, staffing, etc. STEM courses, means less money for other areas. We are seeing a variation of this problem as K-12 moves to the Common Core Curriculum, which, by testing, emphasizes math and language arts, leading to concerns that the rest of the curriculum will be marginalized. That’s part of why we are seeing initiatives like STEAM, where the A is for the arts. I think that, yes employers want employees with STEM knowledge and experience, but they also want employees with a broad background and with soft/people skills. There’s also the issue, going back to Q1, that the STEM areas traditionally have taught students to work in STEM careers, and haven’t always adapted to provide broad STEM skills for those not in STEM careers. If you tell a anthropology student who wants some additional but specific training in chemistry, that they need to take 4 or 6 courses before they can get that training, you’re not going to be adding much STEM to that workforce.

I think this will be an emphasis for several years, and lots of money will be thrown at it, with mostly minor results. We’ll hopefully see some expansion of programs and opportunities and some improvements in the quality and variety of instruction available to students.