Are My Classes Too Hard?

My exams are scheduled for this Thursday and Friday and suddenly—quite, quite suddenly—my office hours are filled with activity. In a hurry today.

Perhaps you, my dear readers, can help me. I have been approached by the leader of the math tutoring squad (for the second time) with the complaint that my introductory classes are so hard, that even the tutors cannot figure out what I am asking in my homework. Here is a question that was said to be the extremely difficult:

I teach a class called Statistics 101. There are 36 students signed up for this class. Before I come to class each day, I guess how many students will actually show up (I know you will be shocked, but some people actually miss class!). Obviously, I do not know for certain, the exact number. How do I express my uncertainty in this number? What is everything that can happen in this case? Which probability distribution would best represent my uncertainty?

Understand that this comes in a chapter which describes the binomial distribution and how to work it. It also comes after several lectures in which I describe—endlessly, to my mind—how to recognize a binomial, how to set it up, and how to calculate it.

The answers are, in this order: Using a probability distribution; 0 students show, 1 student shows, 2 students show, …, 36 students show; binomial distribution.

This question flummoxed many great minds, so perhaps it is I that am at fault. Can you suggest ways in which I might lighten the load of my students?

Before you answer, consider that one complaint about that question was that, “I couldn’t figure out how to calculate the answer.” It is so that there is nothing in this problem that needs calculating; but it is also true that I later offer questions solely for the enjoyment of those who like plugging numbers into binomial equations.

I told the tutoring chief that I ask questions like this because this question is just like the way problems come at you in real life: you do not know which distribution to use, you have to infer it. I admitted that this was more difficult that in other statistics classes.

For my Algebra Sans Algebra class, the most frequent complaint is that I do not allow calculators. But I also do not require exact calculations. If the answer works out to be, say, 2304/3208, then all the student need do is to write “2304/3208” and leave it like that.

Nearly every single time we do arrive at an answer like this, I get the questions, “But is that the way you want us to write it on the test?”, “If we don’t have calculators, how can we figure it out?”, and “Can we leave it like that?” Every time—and I am using the exact definition of the word every—I tell them, “Leave it like that. Or simplify if you can.”

I frequently tell them that anybody can plug that kind of number into a calculator, and that what is important is that they arrive at the right answer, or the right form of the answer. Understanding why the answer is what it is is vastly more important than figuring it out to the third decimal.

Now, I think I would be have been OK, except that I sometimes—for fun—offer methods that allow approximation of answers. These methods do not use calculators. They involve such things as writing numbers in scientific notation, using logarithms, finding answers to the nearest order of magnitude, and such forth. I say not only do you not need a calculator, but you are learning more math this way.

But this approach, I am told, is confusing. “Do they need a calculator or don’t they?” is what I was asked, yet again, today. Apparently, my explicit statement, in the syllabus and often in class, “No calculators are allowed” is insufficient.

Since this is not the first time this complaint has arisen, I am obviously at fault for not making my wishes clear. Can anybody suggest a way that I might let students know that calculators are not allowed, nor needed?

After the exams, I’ll post the questions I asked in Statistics so you all can see how difficult they are.

Update: I have taught the introductory statistics course many times, including three other times at this very university. I have not changed the course, but I have never had as much difficulty as I am having this time. One reason might be that the university where I am visiting has a record enrollment. They even ran out of room at the dormitories.

If you haven’t already, read this first and second conversation with myself about teaching.

54 Comments

  1. It’s fairly obvious that students are only worried about their grade. Make sure they know that in order to get a good grade, all they need to do on these types of problems is show some common sense and critical thinking. Also, I’m guessing that the last math class these students had probably consisted of problems similar to simplifying 2304/3208. I can only imagine the extent of their consternation at being told those types of “math problems” are trivial. Hehe.

  2. Sir, perhaps you should go where the market for your talents may exist and do a Matt Briggs “Maths for those who do care and do want to learn” set of courses for fees.

    In my 40’s I finally confronted my difficulty with math. I took a set of Kumon tests and then began at the beginning with Kumon classes. In moving around a lot as a child, I had pretty much missed division with fractions.

    Once that problem was identified and worked on, things did become easier.

    Now I’m a retired fogey. I still like to do maths and simple algebra every now and then to keep my one remaining wit as usable as possible.

  3. Well no matter their ages you obviously are still effectively dealing with “children”, so repetition is always appropriate, but stark object lessons also work well. Why not first day of class haul out a calculator and hammer and right in front of them smash the thing to bits, take a picture of the wreckage and display a blow-up of the photo for all to see on subsequent class days? It might also help if you use a few unsimplified answers in your examples.

    Still waiting for the WMB online store so I can procure a proper “approved” pocket square

  4. William,

    I don’t know if your classes are too hard; but the question you posed is quite easy for someone who is on the path to understand probability. And I suspect the math tutoring squad is having trouble remembering what they hopefully learned in their own probability course.

    I say this from the perspective if taking one graduate level course in probability (in 1974). I was completely confused by the whole thing and I still am learning. It was as confusing as my course in Soil Mechanics where also one did not (at least at that time) learn much about computing soil mechanics as one had to learn the “art” of soil mechanics.

    I guess that’s your student’s learning opportunity–to learn the “art” of your subject.

    I don’t understand the repeated complaints. They are, in my mind, unfounded. My hunch is at your university there are few courses for these students where they learn the “art” vs. learning how to answer questions. I watch my sons (1st year university and the other at secondary school) and education is about getting top scores on exams. Life, I guess.

    Re the calculator question. Instead of “No calculator are allowed”, say “Electronic calculators and slide rules are not needed, not required, not expected, and are not allowed.”

  5. In good computer (programming) courses, tests cover definitions (what is a loop?), facts (what is the set of operators supported by this language) and pseudo code that demonstrates understanding (how would you find the highest, lowest and average value in a list of numbers).

    Requiring something as nit-picky as writing and debugging an actual program would make the instructor’s job easier (does it run and give the right answer?) but wouldn’t accurately measure the student’s knowledge and understanding of the subject. Your questions require understanding. Calculating 2304/3208 = 0.718204 does not.

  6. Just give them the test answers before hand. Just the answers, not the questions they go to. Then you can sit back and watch their little brains implode on test day, guilt free.

    I like 49erDweet’s calculator smashing suggestion. Theatrics are always a big hit in the classroom. Especially if a big wooden mallet is used.

  7. I think 49erDweet is on to something, and I’m being quite serious. I don’t know that you need a photo representation all year, but smashing a calculator the first day of class should do it. What a great object lesson, and they’ll remember, guaranteed!

  8. One problem for the students is that any introductory course has an incredible amount of content. From the student’s perspective, despite how easy this seems to the instructor, it is very difficult to sort out the important concepts from all of the superficial details of the course.

    Make sure that the students are given a clear set of learning goals at the beginning of the course. These should be in the form of statements that indicate what the students should be able to do. (i.e. Students will be able to identify examples of binomial, gaussian, and Poisson distributions.). Discuss each goal with the students and indicate precisely how it is going to be evaluated on exams and homeworks, as well as how they should practice learning the required task.

    Once this list has been written up and distributed at the beginning of the course, do not try to evaluate them on *any* other things. Let them focus on mastering the list. You may want to make broad, vague tasks like ‘thinking like a mathematician’ a part of your course, but teaching a course effectively is largely about choosing your battles from the outset. One can’t create an expert mathematician in one term.

  9. It is math instructors like you who annoy us physics instructors no end. My students often leave their answers un-simplified; both algebraically and numerically. This is annoying for two reasons. Internal cancellation is missed that greatly simplifies the result and I would prefer an answer that looks like the one I calculated. I keep telling them that the Physics is in the simplification and can easily be missed in the convoluted messy form. It is a constant battle against the mathematicians and their corruption of young minds. 😉

  10. Maybe the tutors have to have calculators, too. You may be dealing with an entire culture.

    Once, when our little family of four were at a Chinese restaurant, my ten year old son got really frustrated trying to learn how to use chop sticks. I told him he would have no problem if chop sticks were the only way he could eat. He jumped at the idea of this little game, and was using chop sticks in no time at all.

    When my wife and I were in college, we would grocery shop together, and I would keep a running total of our purchases by adding each item mentally, and not rounding off. I was surprised that I could get within a few cents of the final bill by just doing mental arithmetic. It was no game, then. We had a budget, and running out of money was a weekly concern.

    So, if the Ruskies exploded a series of nukes in the atmosphere and the resulting EMP’s wiped-out our electronics, how in the world would you get anything done? How could you operate a corner grocery store? How could you keep a balanced check book?

    You have probably tried various games, though.

    If a student can’t figure out WHEN a calculator is needed, there is a more fundamental problem.

  11. Just let your students bring and even use a calculator if they think it will help. Heck, let them use a GPS device too if they want to, or perhaps a crescent wrench.

  12. I’m with Jerry. Let them bring the calculator, then only accept answers expressed in fractions. Hell, go nuts and make them express in prime factored fractions.

  13. I certainly agree that this is the way things come at you in the real world. A tutor wants to be able to show the student how to get “the answer” though. Although you ask “which distribution,” you really mean “which type of distribution,” since you don’t mean for them to estimate p. Also, students thinking about this problem may consider that some students are more likely to attend class than others, in which case a binomial may not be the appropriate model.

  14. Maybe a better textbook is in order.

    My favorite statistics book is Probability and its Engineering Uses by Thornton C. Fry. Copyrights, 1928, 1965

    The reason I like this book so much is that the author is always dragging the reader back to the real world. He, like you, wants the reader to do a lot of thinking rather than blindly plugging in a formula. OK so I am not sure the book is even in print but you can find one in the library and copy if necessary.

    I opened the book randomly and pulled the following quote from p. 137:

    “We must remember, however, that the probability measures the importance of our state of ignorance, as we said in Chapter I. Only confusion can result – and all too frequently has resulted – from applying a result that was valid for one situation to another in which the known facts were significantly different … ”

    One possibility that you may consider: Give the students a set of circumstances and ask them which of five formulas represent the circumstances best. The formulas don’t need to be solved, just recognized. The upside for you is that marking is easy because it is multiple choice. The downside is that it takes forever to create the multiple choice questions.

  15. All,

    See the update, which provides information I should have given you first.

    William Sears,

    Ah, but I agree. That is why I show them how to simply without a calculator. Doing this lets them know when an answer punched into a calculator is flat out wrong. Besides, nothing beats being able to do back-of-the-envelope calculations.

    Commie Bob,

    I never (well, exceedingly rarely) given multiple choice questions. Real life almost never gives you options. Not in probability, anyway.

  16. Jerry has the right approach. You obviously don’t care about the result of the division. Then why be so uptight about letting them calculate it? You still get want you want (the division) and the students get what they want (a little comfort, or maybe just putting aside that completion urge that we all have). Everybody’s happy in the best of worlds!

  17. David,

    Well, not quite. They are not allowed to have notes, or cheat sheets, which means no information stored in a calculator, or calculator “app” (many want to use iPhones etc.). I encourage them to use the methods of simplification to arrive at an order of magnitude answer, which I again say is more valuable a skill than being able to hit keys on a calculator.

    And consider: if they can arrive at $500(1.0425)^10 (interest of 4.25% on $500 compounded annually), this is the answer. Having a calculator to go that last step on a test adds very little (they can of course use one on homework). And wouldn’t it be more fun to learn to approximate that so that you didn’t need a calculator?

  18. I don’t know how I can help you exactly. Perhaps, ask the questions differently?! For example,
    with your scenario described in this post,

    Could a binominal distribution be employed to model the number of absent students in a class of 36 students? Explain (since we can also argue why the binomial distribution might not be appropriate).

    Yes, this may appear that you are making the problem easier, but it might serve the same purpose, i.e., knowing when to use a binomial distribution.

    Anyway, some students have annoying habits of complaining and asking questions without thinking first. And, of course, a popular reason they don’t understand is either the class or the teacher is too hard. Those students need a reason to make themselves feel better.

  19. Who uses a calculator? Calculators are obsolete. There are still a few people around who use the HP12-C with reverse polish notation (sounds racist). It is unchanged in 30 years. But, it is an exception in the continuous march of technology.

    2 is 2 it is not 6.28.

    Students know how to follow instructions. Math education tends to focus on the process, and forgets to teach when to apply it.

    I used to tutor high school students, and would frequently have approaches to problems that were radically different from what the students were taught. Is it better to teach to the book and improve their skills with a limited set of tools or to try to offer more tools to the kit? I always thought that having more tools was better. The teachers didn’t like that.

  20. My goodness. I got those answers. And the grand total of my prob-stat training is two “one semester” courses in introductory prob-stat, taken back-to-back in the two summer sessions between my sophomore and junior years of college (thus compressed into 10 weeks, total). Almost 30 years ago!

    I made the mistake of taking the intro courses for stats majors, rather than the variants aimed at engineers or social scientists. This had the unfortunate result of packing my brain full of theory, while leaving the useful application of that theory to later classes (which I never took, because I wasn’t a stats major). Need I point out that I promptly forgot the bulk of the theory, because (as presented) it had no useful connection with anything I actually did? So I remember some key ideas (such as the answers to this sample question), but come up short when I actually need to *apply* some statistics (sigh). It’s my own %$#%$#^$% fault for taking the wrong course.

    49erDweet and Rob Schneider offer some good ideas w.r.t. to the calculator thing. Randy Pausch used to open his class on user interface design by demonstrating just how bad the UI was on a cheap VCR, and then telling the students: “I will now ensure that you never, EVER forget this lesson!” He’d grab a sledge hammer from behind the podium, and *vigorously* apply it to the VCR, while telling the students that they must “NEVER, EVER PRODUCE a user interface THAT BAD.” A similar stunt with a calculator should make a lasting impression.

    Meanwhile, please continue asking questions that require actual thought. There’s no replacement for making the students think. And far too few classes actually succeed at it.

  21. Briggs says,

    Well, not quite. They are not allowed to have notes, or cheat sheets, which means no information stored in a calculator, or calculator “app” (many want to use iPhones etc.). I encourage them to use the methods of simplification to arrive at an order of magnitude answer, which I again say is more valuable a skill than being able to hit keys on a calculator.

    I preffered the proffessors who said, feel free to use any resource available to you. It is the way the “real world” works.

  22. what a shame! i wish there was some way we could pool all the teachers like you and that i could have gone to that school.

    i say don’t sweat it. what you’re doing is right, the students are doing it wrong. simple.

  23. You didn’t include the chance of people attending who weren’t meant to…

    I wouldn’t have used the binomial, anyway – I’d have answered that it depends on each student’s individual chance of attending, and more deeply on the fact that these are not mutually exclusive. Since these probabilities are all unknowable anyway, you’re going to have to work them out through observation, and you might as well forget the model and just tune a simpler distribution shape, say uniform from X to Y – I doubt you’d get enough data over the course of a term to tell the difference anyway.

    My experience of first year maths at university involved a fair amount of learning new ways to look at solutions I already knew. Partly things like extending applications of Newtonian physics to more dimensions – 2D did come up in a limited form at A level I think, but going to university and working everything through in terms of vectors rather than trying to resolve them to cartesian forms suddenly made everything incredibly obvious. Beyond that, though, stepping away from some of the formality – or even out-modedness – of lower grade teaching, and seeing the bigger picture. Such as chucking traditional statistics textbooks out the window and embracing distributions rather than parameterizations.

    The same pattern repeats at every stage of education – at A level you can look back at GCSE problems and they’re completely obvious, not because you’ve studied the methods more carefully (which is how they’re approached the first time around) but because you have a better background knowledge so can absorb the whole problem in one go. At university, A level is obvious, and in later years at university the first year is obvious.

    So part of the problem for your freshers may be that they don’t really have the right tools to approach your problems yet, or they may just not be comfortable with them. Sure, they know, or should know, what a binomial distribution is, but maybe they’ve learned it in the wrong form. I didn’t study much statistics at the level you’re teaching, but I would have thought the fact that it’s binomial ought to be pretty irrelevant by that point? Focusing on a few idealistic distributions seems too restrictive at that level.

    Another problem is that your questions are more like the kind of thing I’d expect to deal with in a tutorial (you do still have those I hope?) rather than as homework. The students might also be taking the questions too seriously, too traditionally. They might be looking for too deep an answer, based on expectations from past exams or homework questions, when you’re just after something superficial.

    On that note, writing “this question is obvious!” might help some people be satisfied with just writing a one line answer with no actual “working out” in it. It might even help your colleagues recognise that what you’re asking is not rocket science, and that you don’t expect a detailed response. Maybe that’s what they mean when they say they don’t know how to answer the question.

  24. Sorry to double up after such a long post already, but maybe you could consider making the question look simpler by making it multiple choice, but with all the options wrong.

    What distribution?
    A) Normal
    B) Random
    C) Poisson
    D) Quantum
    E) Other, please specify: _______________

  25. Nice question, I’m S-O-O-O stealing it!

    I learned several years ago that most modern students need to be herded into doing any sort of abstract thinking that gets them beyond the “plug and chug” stage of math or statistics–and some of the math majors are the worst offenders. Part of my herding process is to step away from the textbook occasionally and provide an overview or “underview” lecture. Next week’s overview lecture is “Which model do I use?” for which I’m swiping your question. Today’s underview lecture was “Linear Interpolation” wherein my students finally found out what the slope of a line is good for, and why “y = mx+b” ain’t all it’s cracked up to be.

    I’m a “cheat sheet” kind of guy, but I always warn my students, “The answer isn’t on your cheat sheet.” Occasionally some genius photoreduces an entire semester’s worth of handouts to a 3×5 card for the Ultimate Cheat Sheet and hilarity ensues during the final. So far I have not fallen on the floor laughing, but it gets harder every year.

    Fractions vs. decimals: I’m down with fractions, or even unevaluated combinations, e.g. C(6,4)*C(48,2)/C(54,6); these get full marks on an exam. On homeworks, I want to see the decimals, and GOD HELP YOU IF YOU MISPLACE THE DECIMAL POINT. This is one of the most common errors in drug overdoses, and I’m not enabling some pre-med/pre-nursing bozo to give someone 1 gram of morphine instead of 100 mg.

    Don’t worry about those whiners at the tutoring center. I get this gripe every semester from student tutors who pulled an A- in some chump section of a stats course by reverse-engineering the answers in the back of the book. I tell my students to get an A and go replace those loser tutors. I also get attaboys from the tenured professors who see my students in upper-division courses; one senior prof even calls them “Andersonoids.”

  26. George,

    No, I can’t. I only teach two distributions in this introductory class: binomial and normal. The students had a choice on that question which was best.

    Anyway, since no evidence is given about the probability parameter in the binomial, assuming it is equal for all is certainly plausible. But I don’t want to get into a long discussion here of “ignorance” priors and so forth.

  27. I hope you consider this. When I read a problem like you described or when faced with a problem I try to picture what is being asked or what the question actually is. My attempt is biased by who is asking or under what circumstances the real life problem presents itself. By this I mean if my mother asked the same question I would know she wasn’t asking about a binomial distribution etc. And my effort is biased by what I know and my life experiences. In a question all you have is what is stated. If it doesn’t fit a model you are familiar with then you can go down many paths. IF most of your students failed to understand a question then I would say the fault is yours. Either you did NOT pose a good question (enough information and a clear request) OR you have never taught this model in the classroom. Clearly if in the class or in the homework you had posed a similar question then everyone would have had “the model” in mind and could have applied it this time. Having said that I must add that if you wanted to test critical thinking skills then it is fine to test them with a question you have never covered before. However in that event you should expect most people to fail. I’m going to guess here that you want your students to have critical thinkng skills and may even believe a test of that skill might encourage it. I would disagree. It is likely that none of your students have been taught critical thinking skills and simply testing them won’t teach them. So if that is your goal you need to teach that as well as teach statistics. I have an MBA and I can count the number of great teachers I have had in my entire life on the fingers of one hand. Not simply blaming teachers, my point is something is seriously wrong with a system that is failing that badly.

  28. If you are a perfectionist and have a tendency towards impatience, you’ll find teaching very frustrating. Students will make mistakes, and they will upset your plans and your good intentions.

    However, teaching will get easier if you are willing to try new things. It also gets easier as you gain more experience. When a student says, “thank you, I’ve learned a lot from you,” then it’s all worth it. Dr. Briggs, I see a good teacher in you.

  29. I’ve often found it harder to set good questions than to work out the answers. Often, it’s not that they don’t understand the subject, but that because you knew what you meant when you wrote it, the ambiguities in ordinary language don’t stand out for you.

    “How do I express my uncertainty in this number?”

    Obvious answers are variance, standard deviation, 95% interval, the distribution (probability function or cumulative density function), a plus or minus number, or not at all. (i.e. when you make your guess, you don’t give any error estimate with it. I know you will be shocked, but some lecturers don’t!)

    Of course, all those are answers to the question “how might I express my uncertainty”. How you actually did express it is… uncertain.

    “What is everything that can happen in this case?”

    Well, maybe there will be an traffic jam, and nobody will be able to get in, or maybe a flu epidemic, so a lot of the class will be off sick, or maybe somebody who doesn’t belong there will turn up by mistake, or maybe…

    Oh. You didn’t mean that, did you?

    You meant “What are all the possible values this variable can have?” Right?

    “Which probability distribution would best represent my uncertainty?”

    Do you mean your Bayesian belief? Well, that depends. Or did you mean what distribution would it actually be? Well, unless the probability of each student turning up is independent and identical, there’s no reason to assume a binomial distribution. If it depends mainly on the traffic that day, you’re going to get a bimodal all-or-nothing sort of distribution, for example.

    It must be a trick question, right? After going on and on about the conditions in which Binomial distributions occur, he’s given us an example where they obviously don’t, to see if we’re stupid enough to think “it’s in the chapter on binomials, the answer is bound to be binomial” and write it down without thinking. I don’t want him to give me that withering look. I’ll be honest, and say I don’t know.

    In the real world, you usually don’t know what the distributions are. The number of times I’ve caught people assuming independent, identically distributed Gaussians, because that’s all they’ve been taught to expect… (Anybody who tells me ‘2-sigma’ means 95% gets their head smashed in with a brick.) The usual state of affairs is “I don’t know” and “the question is ambiguous” and “it depends on what you assume” so they had might as well get used to it, and being honest about it.

    It was an amusing article, though.

  30. ‘Obviously, I do not know for certain, the exact number. How do I express my uncertainty in this number? What is everything that can happen in this case? Which probability distribution would best represent my uncertainty?’

    Without knowing what language (i.e. words) you use in the classroom, I find it difficult to guess what might be happening BUT the question ‘Which probability distribution would best represent my uncertainty?’ is potentially confusing. It is the phrase ‘represent my uncertainty’ which is the problem as this is a somewhat unconventional expression (I know, not to you).

    I would find ‘Which probability distribution would/might best represent/describe/model this situation?’ clearer and simpler to understand.

    The information that you only teach two distributions is critical because, for the students, it should come down to a choice (or guess) between these two but it doesn’t seem to.

    The issue of calculators is different. First, you are moving them out of their comfort zone (by removing a tool) and, second, you are teaching a math course in which you don’t want them to exactly work out the ‘correct’ answer. Many will find the latter very confusing because it will be counter to their experience and expectations.

  31. ‘This question flummoxed many great minds, so perhaps it is I that am at fault. Can you suggest ways in which I might lighten the load of my students?’

    I wish I could.

    I also teach an intro stats course and, in my experience, one of the most difficult things is helping the students learn how to extract/identify the essential features of a question/problem/situation so that they can decide how to go about addressing it.

    (In fact, I got an email half an hour ago saying ‘I don’t know what to do for problem 5.’)

    The underlying problem is, I think, that the students are being asked to think about things in a way which is very different to what they have had to do in the past and this is a difficult thing for most to do.

  32. I have not read all the comments above, and I suspect there will be some better than mine, but I suspect the problem may be that the students lack the self confidence required to adopt the statistical model required to answer the question. It appears to me that it would require him to regard all the students as being the same in their probability of missing any given class in order to arrive at a binomial distribution. The student, realizing that his classmates vary interest, diligence, etc., will be reluctant to assume that you are looking for such a simple model, unless you have presented them with a large enough body of comparable examples that they will confidently embark on such a simplification.

  33. We know calculators are not allowed. A student fully aware of this restriction may simply be asking do I have the skills to perform without a calculator. So perhaps an answer like hell yes you need a calculator– you couldn’t think your way out of a room with 4 open doors–and given the fact calculators are prohibited when taking the test it most likely means you’re screwed.

  34. I think the calculator request might be a way for them to feel they are doing something – protesting about something: they’re concerned about the exam, they’re not sure they’ll be able to handle the questions, so they complain about something, and the calculator is the easiest thing. If you allowed it, they would probably ask about the precision and so on.

    You mentioned the iPhones – what if you only allowed the simple calculators (not scientific because some of those have interesting memory features that can be used to cheat)? I suspect many of them wouldn’t care about calculators anymore (at least in my class they wouldn’t have).

    We had several courses where we were allowed to use the course book during exams, and in my opinion that was a good thing. Those who didn’t study had little help from that, because they didn’t know what to use and where to find it and those of us who studied focused more on the algorithm and less on the individual formula (I remember one subject that asked for the approximation of something based on different criteria: it was important to know what method of calculation to choose and apply it, not the numbers that were used in its formula). This depends of course on the complexity of the course, but there were as many failed as for the ‘difficult’ exams, if not more.

  35. All,

    I think many of us are forgetting my (belatedly written) evidence that I have taught the statistics course many times, even at this same university, and have never had such difficulty before. The upper-level calculus course goes well, and the Algebra Sans Algebra badly. You should have heard them complain, even angrily, when I had the audacity to assign homework last Friday. They would have to work over the weekend!

    The reason, by their own admission, that many are in this class is that all the other sections of the easier math course (which is required to graduate) were full. This class also starts at 8 am, and I’d say about 1/4 to 1/3 skip on any given day, more on Fridays. Be sure to read those two conversations with myself. One student came to my office complaining that she just could not understand how to solve for t in the equation FV = P(1+rt). This is halfway through a class with a stated prerequisite of being able to solve for t in equations like that.

    Ron DeWitt,

    By that point, they have indeed been led through many binomial examples.

    That awkward student,

    The language I use is the same I have always used, and it is, I think, proper. We do not, at this level, engage in advanced metaphysical discussion of why some people (incorrectly) think that all models are wrong, etc. I give them two: normals and binomials. I explain thoroughly, and endlessly, why normals are no damn good for most things they are used for, but that we have to learn them because they are ubiquitous.

    Essentially, every question is a multiple choice, where the answer is either binomial or normal.

    GoneWithTheWind,

    Could be me, it’s true. But why have I not had the same difficulties before? And why have I not had them at another, let us say, better university? It’s not like I’ve grown a mustache or have taken to wearing distracting articles of clothing, like jeans.

    The kids who are having the troubles are all Freshmen, in their first math class at college. In a university that, I suspect to cover a budget shortfall, enrolled a record number of students.

    Incidentally, the stats class, God help us, starts at 5pm. I don’t think that’s helping.

  36. We’re kind of in the same boat.

    We have a math test that we have given to our entering freshmen every year since long before I got here. The scores have been steadily slipping. The skills of the entering students are becoming measurably worse. Our choice was to keep teaching the way we were and lose more than 50% of the students at Christmas, or change the way we teach. I described what we had been doing as filtering rather than educating.

    My approach has been to have on-line (multiple choice) quizzes based on the required reading and in-class quizzes based on the previous lecture. For the in-class quizzes, we use clickers. In both cases, the marking is automated. The result is that the students keep up with the reading assignments and attend class. They know quickly if they are out of their depth and can drop the class or get a tutor (or knuckle down and work harder).

    If the students understand the basics they have a much easier time developing a deep understanding. In our case, the students demonstrate that understanding in the design labs.

  37. I’m glad you are teaching some approximation. I think it’s a pity you’re teaching it so late in life.

    Recently I looked over my fiance’s daughter’s maths homework (the girl is 10 years old). The homework comprised about 50 questions, all of similar form (mostly multiplication [but not all “tiimes tables” questions] and some addition and subtraction). Enough homework to put me off maths for life.
    I guess about a quarter of the questions were “times tables” which she should have known by heart (but I doubt she does, as maths is so boring). Of the others, about half were right (so I guess she got at least 65%; I should follow up and ask to see the marked version). On looking over her work, I could clearly distinguish questions she got wrong through arithmetic errors from questions she had decided to guess instead of work out.

    And what struck me was how good her guesses were. Yet every one of them would be marked wrong.

    So she learns she is no good at maths, and does not learn how good she is at estimation (and good estimators earn great salaries [of course, they also need to be good at basic maths]).

  38. “I only teach two distributions in this introductory class: binomial and normal.” What a pity that they’ll have to do without the horse-kicks example. You could have had the fun of explaining what a horse is. And demonstrating what a kick is.

  39. William

    I am horrified but I would have done badly at your questions .
    My answers as I would have written them for you EXACTLY would have been :

    – standard deviation
    – small variations around an average near 36 . Dramatical excursions far below or above 36 are theoretically possible but practically have a probability of 0 .
    – I don’t know . Perhaps a Gaussian around the average . When one has no clue one always takes a Gaussian .

    If you read that , you fail me , right ?

    Admittedly I answered before reading the hint that you were pumping in their heads binomial night and day .
    If I knew that and was a student , then I would still think what I wrote above but would answer binomial on all 3 because I would have been 99,99% sure that this is the answer you expect .

  40. I’m with TomVonk. The question “How do I express my uncertainty in this number?” is rather vague and I also would have answered “standard deviation”.

    Please tell me that this university is not Cornell. If it is, I will need to send my degree back (or at least hide it where the kids can’t find it).

  41. George:
    In my experience, tutorials are not common in American Higher Ed. even at Ivy League Schools. In fact it is rare for students, outside of their honors thesis, to work directly with Professors.

    Matt:
    There is something here that illustrates how our, or at least some of our, minds work. My wife rattles on about English grammar. I nod at the appropriate times, but I would fail any pop quiz she gave me. The arcanae of the grammar simply does not interest me – so I do not engage. I can think of no other circumstance where I would pay $3000 or more to learn something in which I am truly uninterested. The system that produces such requirements and depends on paying consumers who freely submit to such requirements is as bizarre as the Thurber story you relate in your next post.

  42. When simple protocols are at issue (no calculator, just write down the fraction & don’t worry about calculating it, etc.) here’s what should work:

    Pass out a sample test with the answers worked out as you’d like it. Go over it with the class. An into speech to the effect that due to school priorities & scheduling you’re doing something you abhor–giving out “secret” info. Tap into human nature’s inclination to be “in on” something.

    I.E. show’m, give’m a reference, and bait the whole process with sinister secrecy. If that doesn’t work, either disregard them & grade them accordingly.

    If you’ve got the time, equipment & inclination you could put a “plant” in your class and/or surrepticiously videotape the whole thing…or do so openly with a banner headline on the board (which must be captured on film). Then splice the pieces together that show you repeatedly telling them the same thing vs. those asking the same now-stupid questions repeatedly.

    Then show that to the class & ask them what they recommend. Or just have a special session with the tutoree’s when they come for Q&A & show them their idiocy & ask them why they still don’t get it.

    You might have a reputation for being a hard grader or whatever…but what you need is to develop a reputation for really making anyone being stupid-by-choice regret it so much that nobody wants to get themselves into that position.

  43. TomVonk (stans),

    Yes, you would have done horrible coming in cold. But you would have aced it had you read the text and attended my lectures, where the answers follow in a litany that you have to work at not memorizing.

    dearime,

    Amen, brother.

  44. You only cover the normal and binomial distribution?

    It seems to me that the Pareto distribution pops up most frequently. And, too many people don’t get it.

  45. Your question was:

    “I teach a class called Statistics 101. There are 36 students signed up for this class. Before I come to class each day, I guess how many students will actually show up (I know you will be shocked, but some people actually miss class!). Obviously, I do not know for certain, the exact number. How do I express my uncertainty in this number? What is everything that can happen in this case? Which probability distribution would best represent my uncertainty?”

    I must admit, as someone who took a few statistics classes 40 years ago only because it was required for cladistics in my second-year ancillary subject of botany, I had no clue what you wanted. Incidentally, I’m currently involved in education and have an Mphil in that area, as well as professional teaching qualifications. I also spent many years as a professional computer development person.

    With the benefit of looking at the answers, I hope I understood them and how they relate to the questions you asked. I could probably have given them correctly myself had I been sure what you actually wanted.

    It’s not that what you want is difficult; more, I’d say, that how you ask for it is difficult. Indeed, what you want probably seems absurdly easy, but you’ve made it an exercise in divination of your intent. It’s perfectly obvious to you what you want, of course, and probably perfectly obvious to anyone else once they see the answer.

    IMO, the problem is in your framing of the question. It’s irrelevant that in the past it has caused fewer problems. Times change, the ways people are taught change, and they arrive at your doorstep accustomed to certain ways of doing things, and with certain expectations. At some point, maybe something gets noticeable, and doubtless a PhD could be written to analyse precisely why and when something has changed.

    Have students actually become more stupid, or is it more a matter of the periodic necessity to change modes of presentation and testing? I doubt the former. Maybe if there’s been a push in recruiting from a different demographic, there’s been an influx of students who begin with a different mindset – but that does not mean that they’re stupid.

    FWIW, one way I would have better understood the requirements would have been if you had said:

    I teach a class called Statistics 101. There are 36 students signed up for this class. Before I come to class each day, obviously, I do not know for certain the exact number that will turn up.

    Q1. For each class, I record the number of students who have attended. State how many different integers could I record for any given class, and list the lowest three and the highest three. (A: 37 [0,1,2…34,35,36]).

    (Not: “What is everything that can happen in this case?” – “Everything” is a big set and maybe students will start bringing in considerations you don’t want them to be concerned with. Such as, “Hmm… well, maybe 30 will turn up, 16 men and 14 women, and – oh wait, maybe one of the women will be pregnant, so does the baby count – and perhaps seven of the men will have beards…”. They might be worrying that what you want is something frightfully complicated, when it isn’t.)

    Q2. Name the general term for a distribution that one uses to represent uncertainties in such things as the number of students attending class. (A: Probability)

    (IMO, your question: “How do I express my uncertainty in this number?” is problematic. Particularly that word “express”. One of its colloquial meanings is “describe in words”, which isn’t what you really want. You want a precise naming of a specific and relevant term.)

    Q3. Name the specific term for the distribution that you think would best represent my uncertainty about the number of students attending class. (A: Binomial)

    Notice also that I have changed the order of the questions, as I believe they lead more naturally from one to the next. This allows a natural internal dialogue to develop in the mind of the student.

    Also, notice that you use the term “probability distribution” in your original third question. At the same time, that term is actually the answer to your first question. As a student, even if I had the right answer for Q1 in mind, I might be asking myself could it possibly be right bearing in mind he’s already named the answer?

    Then again, if you ask three distinct questions, I think you should make each stand out and call for three distinct answers. Otherwise, because they run on sequentially with a somewhat rhetorical flourish, they could be read as indicating there is actually only one answer that could satisfy all three.

    I don’t think you realise just how confusing your seemingly simple test question might be. It could actually cause the more reflective and analytical students to tie themselves in knots fretting over what you want. I think you have an obligation as educator to be as precise and unambiguous as possible about that. (BTW, I’m not saying my effort is the best rendition of your intent there could possibly be – I acknowledge I’m not a subject matter expert; I’m just hoping that even if my expertise is limited, and even if it displays ignorance, that you are going to accept the possibility of valid criticism).

    The questions you pose might actually be much more exercises in comprehension and interpretation than anything else. In order that assessments be totally reliable, questions must exclude any kind of ambiguity or fuzziness and not confuse, even if inadvertently.

    It’s my experience that tertiary level educators often don’t have a formal educational qualification in teaching (as is required for primary and secondary education, at least in my native Britain), and haven’t studied the theoretical underpinnings of assessment techniques and strategies. Incidentally, with my ex-computer-developer hat on, they also usually don’t have the formal training in coding and systems analysis and design that would be of benefit in such areas as climate modeling! :-). Unfortunately, a PhD doesn’t automatically confer the knowledge or tools that make for a competent educator (or programmer/analyst for that matter).

    I’m not taking potshots here, but hopefully, just stating facts well known to trained educators. All my very best teachers were in primary and secondary school. Good university teachers are rare as hen’s teeth. IMO, they need to be formally trained in the essentials of communication and assessment for educational purposes.

  46. If it is new this year, then there has clearly been a change, or addition, to the cohort of students in your class. There will always be some students who don’t have the background, skills or interest to do even moderately well. You appear to have more this year.

    ‘Incidentally, the stats class, God help us, starts at 5pm. I don’t think that’s helping.’

    It is a terrible time to run that sort of class. I’ve occasionally had my class scheduled at that time and it is just awful.

  47. Richard Feynman described a similar problem in a physics class. He had just finishing teaching a section about polarized light and as an aside asked a question about how light was reflecting off the ocean that was within sight of the school. He, of course, got the deer in the headlights look. What the heck does the ocean have to do with physics?

    Classes in math up through high school carefully show how to solve a particular calculation and then ask the students to show proficiency at doing that same calculation. That is what the students learn to deal with. Occasionally they are confronted with a “story problem” and even though it deals with exactly the same type of calculation, “story problems” are an anathema to many students. They don’t know how to find the variables in the story to plug into the calculation.

    Unfortunately, this doesn’t change in the real world. I was helping one of our production engineers figure out how adjust the settings on a machine to separate possible steel contamination from a metal powder. Any steel in the final product can cause a rust spot. Of course the metal powder has some trace amount of iron in the alloy but this is not a problem. It is single particles with high iron that need to eliminated. Our production engineer insists on using the chemical analysis of the lots to figure out if she is removing enough high iron material. The chemical analysis of course consists of dissolving a large number of particles in acid and then analyzing the solution. I simply cannot convince her that this analysis does not mean anything for this problem. The small mass dissolved in acid does not contain enough particles to have much probability of having a problem particle and a larger sample could not separate good material with some alloyed iron from a contaminated sample with less alloyed iron. She has been taught one method of handling a problem and has not been taught how to find the best method.

    Statistics classes I have seen tend to fall into the “here is a problem that is solved by this calculation, show that you can do this calculation” type of teaching. I don’t know how you teach the typical student how to find a good method until you have taught them a number of methods. Perhaps show two types of problems in each class, let them use any aid they want to do the calculations, but grill them on what is the best of the two methods for a new problem. Repeat this for two days and then quiz them on a new problem where they have now four methods to choose from. Repeat and then give a problem where none of these methods work and see if they can figure out that they don’t know how to solve the new problem.

  48. I think you are a right wing nut job when you post about politics and climate change. But you are spot on about this. Exam questions for stats need rigour and your question was fine. The problem of course relates to te fact students who have to pay for education see gem selves as consumers who have a right to get good grades. Of course the answer is to provide free state sponsored higher education but that’s another argument 😉

  49. Simon,

    Aren’t you a sweet talker. Who’s paying the State to give the students “free” education?

  50. “The language I use is the same I have always used, and it is, I think, proper. We do not, at this level, engage in advanced metaphysical discussion of…”

    I recall many examples from my college days of questions where I would sigh and roll my eyes at the sloppiness of the lecturer, but would play the game and answer the question they had meant to ask. Or, because I had a playfully rebellious streak, I would deliberately misinterpret in a (hopefully) mathematically entertaining way. I got something of a reputation for it, and while the teachers and lecturers all knew perfectly well that I could do the maths easily, many seemed to lack any sense of humour about my excursions into their own absurdities…

    Most of my fellow students kept their heads down and played the game all the time. They were not interested in exploring ambiguities, unstated assumptions, or the more advanced background to the material being presented. They wanted there to be a single, simple, obvious answer that would get them past the current assignment without getting into trouble. Dull and incurious, they would simply write down everything the lecturer wrote on the board with all the intelligence of a photocopier (you could have videotaped it once and saved a lot of expense), and it was considered socially unacceptable amongst students to ask the lecturer questions, especially near the end of the lecture. And the lecturers seemed to like it that way: whenever I wanted to divert from the prepared script, my suggestion usually got quickly shot down.

    But there was no doubt that students then were very skilled at guessing what the lecturer wanted and presenting it – “playing the game” as I called it just now. That’s often how they got to be top students. Now you have people who evidently don’t know the rules of that game. Or maybe the rules have changed.

    Sure, you’re undoubtedly right that they’re not just being playful like I was, and the reason they don’t understand is that they don’t have the talent or any liking for maths, but have been told they need it to pass the exams. The college sets them a target and conditionals for how to get it, like a big lump of cheese and a maze to run through. And then you’re surprised when they only want to know the quickest way through the maze.

    But in my view at least the questions you quoted were poorly worded; and the fact that the students were not saying that they were too hard, but that they didn’t understand what you wanted only confirms that. It is true, in real life the questions are usually poorly worded as well and yours were perhaps no worse than average, but it is surely the role of an educator and mathematician to do better. No offence intended, but you did ask, and I am aiming to be constructive.

  51. Simon:
    Methinks the evidence of who is or is not a “nut job” definitively exists in your last post. I am always amazed when someone believes that there is such a thing as a “free lunch.” In addition, when a consumer buys a hammer, there is no guarantee that the consumer will be able to drive a straight nail.

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