Yesterday, I looked up a former classmate of mine, B. We were in high school together; I haven’t really thought about her since we graduated. She is now a pediatrician back in my home country.
When we were in high school, she was what we used to call (loosely translated) a “crammer.” A crammer would be someone who crams, someone who works really hard at memorization, who just stuffs information into their head, without necessarily asking why or how any of it fits together.
The system I went through was different than in the US. In high school, you’d choose a profile (mine was math and natural sciences), and once you’d chosen a profile, your curriculum was set. In math and physics, we had tests, but we also worked on the board in front of the class. I distinctly remember her (and several other people, of both genders) really struggling to set up and solve problems. I don’t think she was dumb, but she put in enormous amounts of effort and still could never get an A in either math or physics, even though she had all other As. I also remember a couple of other people whose GPAs were not very high, but who were good at reasoning through math or physics problems.
Why did I think of B?
Because I had the first midterm a couple of days ago in my large undergraduate class. Usually, I hold all-day office hours before the exam, so I got to see a number of students and work through problems with them. And a few reminded me of B struggling on the board.
I am not sure what to do for these kids. They all came in with high GPAs and were among the best students in their high schools, but a good 60% should not be in this major or any physical science major at all. Yet, here they are, and they are not going anywhere, because their tuition dollars are coveted. I have been trying to figure out how to describe what I see, and it’s not easy. It is a perfect storm of being sloppy, having inadequate math background despite having cleared all the calculus courses (whenever I see people trying to take the curl of a scalar field, or they tell me that the divergence of a vector field is a vector field, I die a little inside), and just not being willing or able to think things through. All this manifests itself as overarching shallowness. I see these students trying to just somehow quickly tunnel through problems, pulling out equations from the book or memory that make no sense while on the surface seem like they do (i.e., they involve many of the right letters), but the actual thinking is completely absent from the process. There are no words that I say more often in office hours than, “Stop. Don’t rush. Think. What is going on here?” I spend a lot of time in class and discussion setting up each problem: drawing, describing the framework out loud, explaining what each step means and and how it translates to math, then solving the resulting equations. I know I try my best and I assume most of my colleagues in the courses before mine do their best, too. And for many students it all clicks and they do great. But for 60% it doesn’t seem to, almost as if they don’t want it to. They come to office hours not to learn, but to do well on the test. I post practice problems and they don’t even attempt to do any of those unless they have solutions. These kids just speed through everything and just want to be done. As if they deliberately don’t want to retain anything, or as if turning on the brain requires too much energy.
(An aside: It blows my mind how bad people are at drawing. I am not talking about becoming a comic-book artist here. I am talking about sketching simple geometric objects better than my 5-year-old: a circle or an ellipse; a disc, a cylinder, a sphere; a cube — oh, my God, the crimes against art committed on the cube! Very few students can sketch what looks even remotely like a cube in perspective, rather than like something that Picasso vomited. A surprisingly high fraction never resort to sketching anything, even though, in many courses, visualization is immensely beneficial to solving problems.)
Our colleagues in the humanities often argue that the humanities courses are necessary in order to instill critical thinking. I feel like we in STEM really try to do the same, but with the help of math. I don’t think we are particularly successful; it’s just that our STEM crammers come out with degrees that on the surface look more employable, even though these degrees — sadly — don’t guarantee that their holders can actually apply reasoning to problems within their discipline.
xykademiqz 
One common math major joke is that if we could draw, we would have gone into physics.
(That said, I am TERRIBLE at art, but I am extremely good at drawing 3-d geometric figures. I have terrible handwriting, but my numbers and proofs are clean and beautiful. I did what I had to do, and math was a priority for me so I cleaned things up.)
I have, in the past, done the cramming thing the first time around but then really gotten it later. I saw this in calculus where I had a terrible calc 1 prof and an amazing calc 2 prof (and great profs for a baby real analysis class and an actual real analysis class). I got by through memorization and pattern matching the first time, but later actually understood what I was doing and how it all fit together. By the third time I’d seen epsilon delta proofs I could even explain them to people who were stuck. In my high level probability class I got overwhelmed and lost somewhere around midterms (got an A, again through memorization and pattern matching), but sat down with the TA at the start of statistics and actually got random variables so that the second semester of the sequence was a breeze.
Heck, I didn’t get calc-based mechanics until I took E&M with a far better teacher (who also had the benefit of not being a sexist jackhat who couldn’t keep his hands off female students). But memorization and pattern matching didn’t really save me there because I didn’t learn vectors until I took Calc III the summer between the two classes.
So I guess I’m saying don’t give up on the students. Some of them will get it later even if they don’t now. And some of them are missing something that they’ll pick up eventually.
When I teach public finance, I have a math recitation section where we go over the calculus that they need for the students who are rusty. It seems to help a lot– with the basic tools they get the economics better. We also have a math bootcamp for incoming first years so that they don’t freak out when they see a summation sign in stats. That has also helped a lot.
The worst thing to me is when physics majors can’t go step-by-step. I’ve gone so far as to do group activities with worksheets. (I swear that I’m not a touchy-feely liberal, but if I give them group activities with worksheets then I am deemed to not be responsible for their failure to learn.) They’ll set something up because on the worksheet I tell them to. But then they try to jump three steps ahead by guessing, rather than systematically figuring out which terms are zero, which terms add, which cancel, etc. It drives me up the wall.
The other thing they’ll do is use the more complicated formula when the less complicated one would suffice (and be equivalent). For instance, I derived in class a formula that relates the kinetic energy in an arbitrary reference frame to the kinetic energy in the center of mass reference frame. They think this means that they have to compute all energies in the CM frame, then translate. Which means they’d have to transform all of the velocities to the CM frame, even though they were given the energies in another frame. I had to pull teeth to get them to see that they can do it either way and since they happen to have in hand the velocities in some other frame they should probably just compute 0.5*m*v^2 rather than transforming v to the CM frame.
I think the point you make about slowing down and thinking through is really important (in the Humanities, too! And arts!). It’s about understanding the process and using the process, rather than focusing on the answer. If you don’t get the process, the answer really isn’t going to be useful at some point.
But I think it takes either special teaching at some point in the past, or a degree of maturity to get that. So don’t give up on them, because they may get it at some point!
I think this is the crux of the problem: “I post practice problems and they don’t even attempt to do any of those unless they have solutions.”
This is not how learning works. Learning is hard work that requires constant struggle. I would argue that if you are not feeling a struggle then you are not performing at your best, even if you are doing well. I try to tell the students that physical science/mathematics is, above all else, a skill, similar a sport. As such, it requires honest and hard practice just like any other sport. In this analogy, I, the professor, am their coach. I give them practice problems and I nudge them in the right direction, I hold practice for them, but if they don’t put in the effort, they will never be good players.
In contrast, what they seem to want is for me give them all the plays and a little theory behind how the game works. When we think about sports, it seems obvious that could never work. I don’t know why students think it works in science.
cmt: “…if you are not feeling a struggle then you are not performing at your best, even if you are doing well.”
Yes! For many of these kids, college is the first time they have their butt kicked a little bit, and instead of saying, “Cool! Challenge, finally!” they feel like I am meting cruel and unusual punishment.
In relationship to sports or even art, we as a society accept that some people have a talent and others don’t, and that even those who do must put in a tremendous amount of dedication and hard work in order to get any good.
In academic subjects, however, we are not supposed to talk about talent, but it also seems like we’re not supposed to talk about hard work either! So many kids feel entitled to a high grade after having barely lifted a finger and then feel viscerally wronged upon being shown that their sloppy work and just plain wrong answers are nowhere near an A.
I’m open to the possibility that most people could become good at science. What I’m not open to is the possibility that most people who were never before challenged will become good at science within the time frame of a four year degree unless either they push themselves to do insane amounts of practice or we require them to do huge amounts of practice. The problem with the first option is human nature. The problem with the second option is that we’d have to either assign a gazillion homework problems (and then hold firm and give lots of F’s when most of them don’t do it) or else turn a 3-4 unit class into 3 hours of lecture and about 12 hours of discussion/practice sessions PLUS a lot of homework. And administrators would never agree to the last option, because they’d either have to pay us more, give us more TAs, or reduce our other duties in exchange for a gazillion hours of discussion/recitation time.
That’s the crux of the problem: People come to us with inadequate preparation, and there’s no feasible path to getting most of them to do the gazillions of hours of practice. There’s no cheap path to mastery. Either the students push themselves, our employers give us the time, or they give us assistants to put in the time. They just want magic fixes, “This one simple trick!”, or “This scalable Best Practice!” There is no such thing. There’s practice and there’s charlatan bullshit. There’s nothing else.
“There’s practice and there’s charlatan bullshit.”
I will have this carved on my tombstone.
“3-4 unit class into 3 hours of lecture and about 12 hours of discussion/practice sessions PLUS a lot of homework.”
I think I am getting close to that. For this course, I teach 3x 50-min lecture per week, a 90-min discussion per week, and hold 6 hrs of office hours per week (3×2 hrs) plus additional 10 hrs the week of the midterm. I assign 15-20 problems per HW assignment (11 sets per semester). I think they do close to 200 problems for HW by the time the semester is done and they are blown away when I give them the number. I’m doing my part to have them get their tuition’s worth.
A 3–4 unit course is supposed to be 9–12 hours of work a week (including class time). You’ll be lucky to get your students doing half that.
In my 4-unit applied electronics course, the students have 3 ¼ hours of lectures, 3 ¼ hours of lab, homework, and design reports. Students do average the 12 hours they are supposed to put in.
Unfortunately, many of them are still in “answer-getting” mode, where they randomly apply vaguely related formulas rather than thinking through the problems. The number of formulas they can apply is tiny (Ohm’s Law, voltage dividers, complex impedance, negative-feedback amplifiers)—all the work is in applying them appropriately.
This year’s class is doing a bit better than previous ones—I think a third of the class is really getting the material, a third is struggling through, and only a third are still wandering around lost. The students who are most lost seem to be the last ones to ask for help from the undergraduate group tutors or me—often waiting until it is too late to do the design work for the labs in the remaining time, even if they grasped the material perfectly.
One of my former graduate students told me a story that I always think of in situations like this. He was tutoring football players at his undergraduate college who were frustrated that physics took so long to learn. He asked them how many hours they trained each day for football. And then asked them “Why do you think physics is easier?”
In my class, I’m a hard-ass about grades, but I also provide lots of support. I provide lots and lots of small homeworks that force the students to think about the problems. I find that the ones that become engaged do well, and the ones that don’t … well, I gave them a chance.
Part of the problem, I think, is that lots of students don’t understand that professors *are* interested in helping them learn, but need to ask for it. This year, I emphasized office hours over and over again in my “syllabus-day” on the first day of class. I told them stories about students crying in my office at the end of the semester because they were failing and it was too late to recover and about students who came to office hours and recovered. I practically got them to chant “That’s what office hours are for.” And to my great surprise, I have found that a large number of students are coming to office hours and participating in class discussions and being generally engaged in the class. More importantly, I can see clear improvement in the homeworks across the semester. Yes, they need to learn to think and to go through steps carefully and learn how to determine which formulae to use at each stage, but the key (IMHO) is convincing them that they can learn it if they only ask for help.
I also teach a physical science course that relies on problem solving skills, and I have the same experience with my students. Some of them are VERY hard workers, who spend all their time on things that will not help them learn the material. In their way, these students are more frustrating than the ones that just don’t work.
I have many students who have no idea how to problem solve, as in can’t even set up a simple problem even without any tricks or logical leaps. I also struggle to help them, though I try. It’s like they blank out as soon as they start reading the problem. They can’t apply concepts to concrete examples. Many of these are the same students that view prerequisites as boxes they need to check in order to get a degree rather than as indicating prior knowledge they need to do well in a particular course. Even if they learn the material rather than figure out how to memorize what they need, math stays in the math box, physics stays in the physics box, biology stays in the biology box and so on, which limits their ability to synthesize what they learn.
It has always annoyed me so much that people are proud of being innumerate, while being illiterate is considered the height of ignorance. Even in just the short time I have been at ProdigalU, I have seen a drop in students’ math skills. It is not surprising to me that my students struggle with calculus, when they can’t solve problems involving pre-calculus concepts such as logarithms.