### Exam notes: qualitative

- Despite the fact that marks-wise, I was most disappointed by the statistics students (several of them
*can’t plug numbers into their calculators!*I gave them a formula sheet! They copied the correct formula down correctly! Wrote down the correct numbers for the variables! And got the wrong result!), their papers were the most enjoyable to grade, because the statistics students tended far moreso than the students in my other classes to leave me little notes about how they felt about the material. For instance, I had a probability question about a die-rolling game, which one girl opted to inform me was the “stupidest game EVER.” Another student (correctly) subjected a manufacturer’s claim about its products to statistical analysis before stating her conclusion, as instructed, “in [her] own words”: “It’s TOTAL BS!” Also, my attempt to make mathematics relevant to the lives of my students succeeded in the mind of one student: I included a question in which students were to test a hypothethesis that women generally underreport their weights, which prompted one pupil to write sic, “OMG THIS IS SO TRUE, I DO THIS ALL THE TIME.” (She did the question incorrectly.) - Aside to all students reading this: if you are ever asked to define a term on a test, and you don’t remember what that term means in the subject - for instance, math - that the test is about, try analyzing the language in the term. I mention this because hardly any of my students were able to tell me what mutually exclusive events were (most gave me the definition of independent events), even though realizing that “exclusive” and “exclude” contain the same root just might be enough to jog the memory. The only student to attempt a linguistic desconstruction was an Asian girl whose English is shaky. “Mutually exclusive events,” she wrote, “are events which are very special…”
- Back to the die-rolling game: when my contract expires at the end of the month and I am without regular income, would it be unethical to engage some of my former students in various games for money? I have certain students in mind, such as the guy who wrote that if you play one hundred rounds of the game that costs you $4 per round and that pays you the dollar value of the number appearing on the top face of the die, then you can expect to make $2000.
- The college where I work prides itself as being “student-centred”. Part of what this means is that the many, many student lounges are located “centrally”, like right outside classrooms and adjacent to them. I submit that if there was ever a good reason to revoke an architecture degree, then surely this design is it. Precalculus was held in a lounge-adjacent room, as was the statistics exam. During both, students were in the lounge, chatting at perfectly normal volumes, which was enough to create a distraction for me and my students. On approximately fifty occasions this term, I had to enter the lounge to request quiet, because a dozen students talking quietly results in a din that is not quiet. On forty-eight of these occasions, students were apologetic and lowered their voices accordingly, which helped things for the next twenty minutes until either a) they got excited about something else, or b) they were replaced by an uninitiated group of students. (Large signs in the lounge requesting quiet do not help.)
The other two times, I had the pleasure of dealing with one girl who asserted her rights to sit at the lounge and talk because “this is the only place where we can do this”, where (judging from the lack of books or anything else on the table where she and her friends were sitting) “this” consisted of sitting and talking with two friends in the lounge outside my classroom. She told me that this was “total bullshit” and that “maybe I should use a different classroom”, as though it was my decision to teach and hold exams in a room that is a ten minute walk from my office and is located right next to a student meeting hub that has the acoustics of a cathedral. When I quietly explained that although perhaps the lounge was the only place where she could do “this”, it was even more certain that the adjacent classroom was

*actually*the only place where my students could write their exam for the next twenty-five minutes, she declared that this was TOTAL BULLSHIT. WE ARE TALKING AT A NORMAL VOLUME, which at that point was no longer true. Grasping for common ground, I started to tell her that I was in agreement with the bullshit assessment, and reached into my pocket to call campus security on my cell phone when her friend - bless his heart - said, “Come on, we can go somewhere else for the next half hour…there’s an exam, she’s asking us nicely.” And that, apart from an evil eye directed my way (me without my camera, I tell you), was the end of that. - Often I get teaching advice from my readers. I should do dimensional analysis! I should make math relevant to their lives! I should relate the difficult concepts they’re studying now to the easy concepts they’re familiar with!
Generally I smile and nod, because I think that few of my readers know what I’m dealing with. Allow me, then, to present an example of what I am dealing with:

The question, which is a bit unclear in the image, reads “Solve for x: 4

^{x+2}(1/2)^{-2x}= 8^{x+2}.” The student’s work, which follows, resembles the sort of thing junkies might hallucinate if LSD induced images of equations. Notice how the variable*x*goes missing in the very first line of this “solution”, leaving an equation that reduces to 0=0, which is then tortured until it finally gives up the ghost - though not before confessing, in its dying breath, that*x*(remember*x*? it’s back!) equals 50. Note also how the declaration that we “can’t have negative numbers” (surely a surprise to those of us who have endured frigid Canadian winters during which the temperature dipped to…oh, CRAP) apparently allows us to do away with them altogether, on our own terms, and at our leisure. (Ernie has taken the time to transcribe this work of art, in toto.)No, this corruption of reason is not representative of a typical paper in my precalculus class. However, it

*is*representative of a typical page of this student’s exam (she got a 3% on it, and this was with me assigning marks to everything that even vaguely resembled mathematics - by the way, she sat in the exam room for an hour and a bloody HALF tormenting her paper; I left out the question in which she concluded that a $1000 principal invested at 4.8% per year compounded continuously yields $240,000 after five years, and I also left out around ten others that I could easily have posted instead of the one above), and I have had, on average, two students like her in every class I have taught in the past five years. I guess I could have spent more time on dimensional analysis and such, but somehow I don’t think it would have helped that much.(ETA: By popular request (okay, one person asked, but apparently he speaks for all), here’s more of the exam.)

- One of my students missed an exam because of a dire family situation, which now requires me to navigate that murky territory between being sympathetic and being a sucker. I don’t want to get into too much detail, but suffice it to say that Distressed Student is both suffering a terrible crisis, and attempting to play the system like it’s going out of style. I’ve had to outline explicitly what I am and am not willing to do to accommodate her. For example: I am willing to allow her to write the exam a month from now, but I am not willing to just assign her a final grade that assumes that her exam grade would have surpassed the class average by nearly 10% when her term test grades consistently fell far short of it. As evidence that this student’s family crisis is not the only thing standing between her and such success on the final exam, I present the email she sent me the other day, which read, in part,
*Since I am going thru such difficulties would you be able to just make my term mark my final grade? Right now I have 41/65 on the quizzes and tests. I have been meeting with a math tutor for the past few weeks and he told me that this is a 63%, which is the mark I need in this course.*

The grad student here who teaches the class I TA for has a simple policy, which may be university-wide: the student must get a note from the “emergency dean” explaining and excusing the absence. I don’t know if such deans exist universally (they also existed in some form at MIT), but maybe your school has something like that?

Not that it’ll do you much good since you’re leaving soon. But if it exists, it allows you to shift the burden to weed through it all to someone else — who has more experience with that.

Are there any qualifying examinations that students must pass in order to be permitted to enroll in these classes? I’m not at all familiar with the protocol.

I would think that students would need to demonstrate a grasp of basic math concepts before they enroll in statistics or precalculus.

Dan - this student has documentation accounting for her absence. Like I said, I know that her excuse is legit; it’s just that in addition to her excuse being legit, she’s being so overwhelmingly manipulative that I just want to smack her.

Edwonk - oh, you’re so naive, it’s really cute. Technically, students need to have passed grade 12 math, or attained marks of C+ or higher in grade 11 math before they can take any of the classes I teach. This would be a perfectly reasonable requirement, if students actually learned anything in high school math, which apparently they don’t. (Well, they do: they learn how to graph functions on their graphing calculators.) What makes this even worse is that the adult education math class counts as a prereq - the older students who are fresh out of the class are by far the weakest. This is the course that purports to teach, in a single year, three years’ worth of high school math to students who have been out of school for three decades; 80% of graduates get A’s in that class.

The student had to ask a tutor to find out that 41/65 is 63% ???

Students with bogus excuses for not attending class or missing exams are, I think, usually found out soon enough. Such behavior is often part and parcel of a dysfunctional approach to being a student. Students who abuse legitimate excuses, on the other hand, are a peculiarly trying problem. I have one student who missed Tuesday’s algebra exam because of illness. He promptly e-mailed me concerning his condition (good boy!) and I made arrangements for him to take the exam in the testing room of our Math Learning Center on Wednesday. Had he returned on Wednesday, all would have been well, because he could have taken the exam while his classmates and I were going over the results (I graded the exam Tuesday afternoon & evening). But now he tells me he won’t be able to take the exam till maybe Saturday (our math lab is open then). But all of his classmates have had the corrected exam problems in their possession since Wednesday morning. That exam is well and truly burned, but he expects to take it this weekend. Now I have to go swap it out for something that everyone hasn’t seen. Ugh.

On second thought, I could decree that this is the exam that will be dropped from his course grade. I drop their low score for the semester and this gives me a possible escape route. Hmm.

By the way, I love the second problem on MS’s scanned page. It appears that the student has invented a new way to deal with exponential functions: Just divide 3^(x+1) by (x + 1) and it will cancel the exponent! The student’s solution trails off the image at that point, but I’m pretty sure that’s what was done.

TonyB, it’s possible that the student just plain wasn’t healthy by Wednesday. What I’m dealing with is far more sophisticated: Distressed Student asked me to exempt her from the final. She made the same request of department head, in a separate email. She didn’t tell me she was contacting him, or vice versa. In other words, she was hoping that one of us would grant her request, and then she could go with whichever one of us that was. Because, of course, I’m not going to talk to the department head about a minor matter of university policy! Anyway, this is still bush-league stuff, though it brings me back: like every other kid who grew up in a two-parent family, I’m sure, I knew how to play Mom and Dad off of one another in exactly this manner.

What made this particularly obnoxious is that the story she told the department head was quite different from the one she told me. Department head was told that she felt that she knew the material decently enough, but that her life has been so difficult that she physically cannot be on campus to write it. However, but for her legitimate, difficult family situation, she’d be writing the exam and doing just fine, so couldn’t we just give her her term mark now since it’s what she’d have gotten if thing were going fine in her personal life? Naturally, she didn’t tell

methat story, because she knew that I, having interacted with her for the past three months, knew that it was bullshit: the only reason she’s doing as well as she is is because the third test I gave was too easy (my fault), she’s done OK on the weekly test-the-basics quizzes, and she took advantage of the opportunity I gave the students to submit corrections for their two first, abysmally-done tests. Which is fine, and more power to her for taking advantage of [above]- my point is that she’s very weak when it comes to putting all of the concepts together on a major test, let alone an exam. Oh, and she’d also told me that her life has been so crazy that she hadn’t done any math for the past three weeks, so I’m inclined to think that her email to the deparment head claiming that her mathematical weakness had nothing to do with her not wanting to write the exam was more than a shade disingenuous.In re: #3 - go for it! Wouldn’t that be extending relevant education outside the classroom, part of our mandate?

in re: #4 - I’m all for crimes-against-humanity trials for architecture. Architecture is all too often a form, so far as I can tell (and I have a Ph.D. in art history), of human experimentation. Architects build these buildings for a few years and then there is NO FOLLOW UP. Nor are the offending buildings torn down and replaced (I know, it would be impractical, but if they promise a “result” from their scattering of student lounges around the building wouldn’t it be nice if someone looked and said “oh! It didn’t work!”).

Holy Crap this is better than General Hospital, Days of Our Lives, and One Life to Live PUT TOGETHER!!!

Thank you very much for taking the time to share.

What a great method to solve equations! Just eliminate the variable, put all the numbers on one side of the equation and (nothing up my sleeve…) and PRESTO the unknown magically appears. Full marks. (S)he basically understands the material, right?

Oh, that paper is beautiful. It’s modern art — mysterious and magical. I smell a whiff of burning herbs from a neopagan ritual. It is truly awe-ful, which gives us the idea of the etymology of our watered down word “awful”.

I’m amazed there was one correct step in there (1st line to 2nd line). Because these other operations are =so= wrong, I’m amazed she knew that 8^2 = 64 (I suppose they could use standard calculators, but still — that she knew which button to push).

What really gets me is how 50 appears at the end. I know she added 2 to 48, but I’m wondering how she got from 1/2 to a negative power to 2.

Negative numbers may not exist in her world, but the numbers that do exist have some interesting properties.

It’s obviously a hoax. Reject.Moebius Stripper’s student wins the lifetime achievement award for Worst Math Answer EVAR. Solve for x: a. (4 marks) 4x 2(1/2)-2x = 8x 2 42(1/2)-2 = 82 16(1/2)-2 = 64 x = 64 - 16(1/2)-2 x = 48(1/2)-2

I think I speak for one and all here that:

WE MUST SEE THE WHOLE EXAM!!!

I live to please, RR.

The whole exam now lives in the department head’s office, alas. I took some photos of other pages, but they didn’t come out as well. I’ll link to the larger versions of what I have - they’re blurry but (I think) readable. Let me know if you want any interpretation:

How do we solve x^(2/3)-5x^(1/3)+4=0? By turning the exponents into coefficients!

Finding the equation of the line passing through (4,2) and perpendicular to x-3y=7, or something

The function g(x)=4-x^2 is found not to conform to my student’s syntax

The rest of the one TonyB was impressed with

Who has more money after five years? She could be wrong.

Commentary for a question that required the student to find two people’s ages, given two relationships between them

Enjoy; someone might as well.

That little gem recalled Clack (or was it Click?) saying “It only works if the number… is… two.”

Miss Needs-A-B

Can’t do math to save her life

“Welcome to Wal-Mart”?

(just looked at the exam photos…. fuzzy but hilarious. I would have flunked 7th grade algebra had I done that badly.)

Your commentary on these exams is both hilarious and painful. I teach college math and I have seen these same mistakes so many times it’s not even funny. But I’m too chicken to post stuff like this on my own blog — we’re supposed to be “student sensitive” too.

Regarding the last image linked above (”Commentary for a question that required the student to find two people’s ages, given two relationships between them”), I love it when students make hash of the problem and then get to the end and say “I can’t REMEMBER the solution”, as if the point of the test was to memorize all the solutions to problems assigned and then spit them out. You know - not all test questions are just reiterations of homework problems, and at some point in the future you might just be asked to solve a problem you’ve never seen before! It’s a frickin’ math test, not a memory test

That’s a good exam question, MS.

Tests a student’s ability to

(a) use exponents in multiplication/division

(b) use logarithms to simplify

(c) use algebra on a simplified equation

and, most importantly,

(d) not panicking at the sight of a variable in the exponent

BTW, am I right in saying that x=4 is the correct solution?

I have to stop reading your blog at work. I burst out laughing at those fuzzy photos of the exam, which was pretty embarrassing in our library-quiet office. I basically lost control at “I could be wrong.” ^.^

Can I invest money in this woman’s bank?

karrde: it only works if the number is two. ;-)

4^(x+2)* (1/2)^(-2x) = 8^(x+2)

4^(x+2) * 2^(2x) = 4^(x+2) * 2^(x+2)

2^(2x) = 2^(x+2)

2x = x+2

x = 2

That’s a mind twister, all right.

MS, the ones who suggest things to you seem to think that you are not experienced or smart enough to have tried those tactics. Having been where you are, I wisely refrain from suggesting anything because I KNOW how many things you’ve tried. I used every trick in the book and still some Chem I student woud ask me why an atom couldn’t weigh 50 kg like he’d just caculated. Didn’t help that we Yanks use the English system, but still, I brought in a 1 kg barbell and a paper clip to show how much a kg and a gram really weigh. Be glad that you don’t have to worry about unit cancellation follies on top of the math errors. I really wish I’d thought of the bingo idea when I was teaching.

Well, sometimes bad arithmetic works. How do we know that 16/64 = 1/4? Cancel the 6, of course.

As a CalcI student, I would like to appologise to my teacher, who I’m sure has, on atleast on occasion, looked at my tests and said, however it goes in Polish, “WTF…”. Moebius, you have my sympathy.

And to other students: Study, and get sleep. Your brain doesn’t function right on

Robert -

It’s a frickin’ math test, not a memory test- oh, god, yes, yes, a thousand times yes. But I can’t for the life of me get that through to my students. I give them a carefully-designed assignment consisting of ten questions that all reinforce the same concept, and they see it as ten different concepts, so that when I present a different question thattests the same concepton the exam, I get complaints that that question wasn’t fair, because it was different from all the others! (And sometimes I get solutions to question 5 on the test that look suspciously like solutions to question 9 in the homework, whichwastotally different except that, for instance, it also was about boats.)John - exactly. My lessons about doing “sanity checks” on one’s work (does it make sense that you’d have $250000 after 5 years? Wolfangel, if you can’t invest in that bank, I could give you several other student banks that are almost as good) fall on deaf ears. I even make public my policy of marking very generously if a student gives an unreasonable answer and writes something along the lines of “I know that this can’t possibly be the answer, because it’s too big/too small/negative/the square root of a negative - but I don’t know quite what I did wrong.” I think that in my entire teaching career to date, five students have taken advantage of this. The others just claimed that, say, the area of the farmer’s pasture was the square root of negative 50000000, and moved on to the next question.

Heh. I do that, too: Tell students to write me a note if an answer is impossible and they recognize it as such. They almost

neverdo! I give mercy points for perceptiveness in such cases, though I guess no one wants them. It appears that ‘fessing up that an answer is clearly wrong is more painful to them than the loss of points from noodling along as if all were well. I don’t get it, even though I see it all the time.Thinking about the answers must be too difficult. That must surely be the reason that a couple of my students believe that Griselda can sweep 2000 sq yd of sidewalk in 12 minutes but 2500 sq yd will take only 3 minutes. Yeah, that makes good sense.

The other thing about that compound interest problem….she ends up claiming that 30000 x 5 = 250000 (wrong) while 48000 X 5 = 240000 (look, she got something right!). But wouldn’t a simple sanity check suggest that a small number times 5 should give a smaller result than a bigger number times 5? Clearly, calling this student a mathematical illiterate is too generous.

In the only defense I will ever offer of this student, that was a 50000, not a 30000.

Your last point, though, still stands.

Just out of curiosity, what is compounding interest “continuously”? I’ve never heard that phrase before. Is that the same as compounding daily?

If I was solving such a problem, I would take “continuously” to mean “with infinite smoothness”, though compounding to the second would probably be indistinguishable.

Continuous compounding is actually easier to calculate; that’s what the e^x key is for.

Continuous compounding is more often than daily; it’s more often than hourly, and it’s more often than minutely, than secondly….

The formula for the amount, $A that an investment of $P grows to after t years at an interest rate of r compounded n times per year is A=P(1+r/n)^(nt). For yearly compounding, n=1; for monthly, n=12; for weekly, n=52…

So, for continuous compounding, we take the limit as n goes to infinity. As Engineer-Poet says, this limit gives an amount A of P*e^(rt). Try both formulas on your calculator - the first one for large n (n=1000, 10000, 1000000…); the result it gives is very close to the result given by the second.

Engineer-Poet wrote:

it only works if the number is two.Student replies, “So when the number is two I do it this way?”

“It only works if the number… is… two” is a reference to a Car Talk puzzler which got Ray Magliozzi (one of the two wits behind the NPR program “Car Talk”) in a bit of mathematical hot water.

What kind of mathematical hot water, Engineer-Poet?I’m glad you asked me that…

Ray’s “Puzzler” one week involved a fruit tree (I want to say pomegranate) belonging to the king, a set of walls around the tree (memory seems to say it was 4), a set of guards of negotiable loyalty but perfect honesty (one at the door at each wall), and a hypothetically mathematically literate (and conniving, and hungry) thief. The thief found the walls impenetrable, but exploiting “social engineering” he negotiated a deal with the first guard: if the guard would let him in he would give the guard half of the pomegranates he came out with, but the guard would then have to give him one back. He negotiated the same deal with the other three guards. How many pomegranates did the thief pick?

Tom’s answer is quoted above… which is why I call the thief

hypotheticallymathematically literate. The number of two works; give half to each guard, get one back, leave each guard frustrated but honestly sticking to the deal. But that’s not the only possible answer; for 4 guards, the formula for the number P of fruits picked for a yield Y is P = 16Y - 30. Ergo, if the thief wants 3, he picks 18; if he wants 6, he picks 66.Generating the general formula for P given number of guards G and yield Y would be a worthwhile test of algebraic understanding, IMHO, as it demands proof by induction.

The Carnival Of Education: Week 12Welcome to the twelfth edition of The Carnival Of Education. Here we have assembled a variety of interesting and informative posts from around the ‘Sphere that have been submitted by various authors and readers. Those entries that were selected by us…

I would often write those types of answers. I was the bane of my math teachers.

My brain is simply not wired correctly for it. If I can jam in enough formulas and concepts, I can usually squeak out an answer, but it never comes naturally. It always…ALWAYS…involves going into my memory and asking ‘where have I seen this before?’ If I can’t remember a model I have to work out each problem as if I have never learned the subject at all. Some people are natural math people and others must work at it, so I sympathize with your poor student.

There needs to be a separate curriculum for those of us math illiterates. Anyone know of any good books doing it this way? I’d love to actually know the Calculus I took in college.

I don’t know of any good books (and I’m not sure that learning math from the ground up is best done in isolation with books, anyway), but I used to work at a Kumon centre, and I have heard some wonderful things about the JUMP program. I’m far from an expert on early math education and such, but everything I’ve read indicates that people who are weak and/or underconfident should start way back with the basics - in your case, many years before calculus. One thing I advise my weaker students is not just to go over the “how” of the problems they’re working on, but also the “why” - and not to put a problem aside once they’ve figured it out. Instead, go carefully over all of the steps, asking - why did I complete the square? Why did I move the variable over here? why did I use Formula A instead of Formula B?

Math is learned very differently from other subjects - done correctly, it involves a lot less memory work than many, if not most other subjects. The best exposition of learning math that I’ve seen is How To Read Mathematics by Shai Simonson and Fernando Gouvea (the latter of whom I’ve met - he is a master mathematician AND a master math educator), which outlines the sorts of things that students should be looking for when they study. It might be a bit out of reach for you at this point, but if you ever build up to calculus (and kudos for setting that goal - so many of my students are so happy just to pass and forget about it), it’s an invaluable resource.

I was lucky and only graded for my class versus actually teaching it. This was 451/551, a graduate level statistics class. I was called in by the prof teaching several times becuase the students had complained that I was “grading too hard.” I asked for an example and was presented with this: The question asked the AVERAGE of a list of numbers!!! I can’t remember off hand but they were between soemthing like 10 and 40, several student answers were in the hundreds. I was first appalled that the prof would ask such a rediculous jr.high question, I was shocked that the students in their fourth undergrad year and first year graduates could have no concept of such a jr.high question. Finally, I was mortified that I was being called on the carpet for not giving them PARTIAL CREDIT for such a rediculous answer!

Are you kidding me??!!??!!!

That’s college math? Looks like Algebra II to me, which is, um, a 10th grade class, age ~15 or so.

Alas, it is a college algebra class, which counts for college credit. And my students still whine that it’s too hard, and that I have no right to expect them to remember any math that they learned back in grade school or even high school. (One of my colleagues was explaining conic sections to a student - he mentioned that the asymptotes were straight lines, which she’d seen in Precalculus I. The student replied, completely serious, “But that was LAST term.”)

You should see the math classes for future elementary teachers - they’re even worse. My former officemate taught one. She was explicitly told on the first day of the term that since she was a math teacher, not an education teacher, she was under no circumstances to teach her students how to teach elementary school math. Her job was to teach them elementary school level math. I think they spent a week on multiplication.)