a calculus professor's email exchange with a parent

Number 2 Pencil links to this exchange of emails between a math professor and one of his students, who is flunking calculus.

The professor is using a pedagogy known as Process Guided-Inquiry Learning, or POGIL:

I mentioned in an earlier post that one of the more controversial -- and to me, appealing -- aspects of POGIL instruction is that the instructor is not seen as a source of knowledge but rather as a facilitator of learning. In non-eduspeak, this means that the instructor is there to observe and to guide, rather than to tell students what to do or think.

I also mentioned, and one commenter pointed out, that students HATE this (at least at first). Typically, students -- even upper-level students in the major who have been around the block -- just want to get the darn problem done, and when people like me insist on students asking the right questions rather than just forking over the answers, things can get heated.

…when Pat would ask me a question such as, "Can you tell me how to do problem 7?", I would say: Let's start by asking the right questions. What are you being asked to do in this problem? What information is given to you in the problem statement? And what do you know from the course, your reading, or your work on other exercises that will help get you to the goal? I made it a point to NEVER give Pat explicit help on content unless it was a last resort -- Pat absolutely HAD to cut the apron-strings from me an learn how to approach, analyze, and solve a problem alone, or else Pat's chances for success in a future career or even making it through college didn't look good.

Finally 'Pat' sends an email explaining that he requires direct instruction in order to learn.

The professor tells him he is wrong.

Pat sent me an email just after midterms that said something like: I now understand why I am not doing well in your class. My learning style is such that you have to show me exactly what to do, or else I can't do it. But you always answer my questions with more questions, which isn't showing me exactly what to do. So from now on, please show me exactly what to do first, and then I should be able to do it. My response was something like: Pat, we've been doing this every day in class -- I work a few problems at the board all the way through during lecture, and then I give you exercises that are based on the stuff you've seen. So you are seeing me show you what to do, and yet you're still having difficulty solving problems on your own. So perhaps your assessment wasn't quite right, and we should be working on your problem-solving skills in office hours.

(via email): I know [Pat] tried to explain to you that when [Pat] asks questions [Pat] needs answers not another question. We had [Pat] tested at [a local university] in January through the suggestion of [an academic counselor at my college].

During this testing we found out [Pat] has a learning disability. [Pat] does better with visual explanations then being asked another question. [Pat] needs to see how to physically work a problem so he can comprehend it. [Pat] is a slow reader which also frustrates [Pat]. If it is a word problem [Pat] has problems figuring out what are the essential parts of the question to find the answer.

Pat fails the class.

The commenters are united in their view that Pat is a lucky guy to have experienced POGIL calculus, and he had no business hosing the course.

Memo to self: the time to begin instilling core take no course from professor who blogs principle in 10-year old son is now.

POGIL

This does not sound good, POGIL.

I should reserve judgment.

I should, but every one of the little-red-light thingies on my Constructivist Nightmare Detector is flashing wildly, and all the sirens are going off—

So I’m not doing a very good job of reserving judgment.

POGIL is a student-centered method of instruction that is based on recent developments in cognitive learning theory and results from classroom research that suggest [sic] most students experience improved learning when they are actively engaged, working together, and given the opportunity to construct their own understanding. POGIL emphasizes that learning is an interactive process of thinking carefully, discussing ideas, refining understanding, practicing skills, reflecting on progress, and assessing performance. In a POGIL classroom or laboratory, students work on specially designed guided-inquiry materials in small self-managed groups. The instructor serves as facilitator of learning rather than as a source of information. The objective is to develop skills as well as mastery of discipline-specific content simultaneously. (Emphasis added)

homeschool mom with common sense-y

I do want to read them.

But in the meantime, there's one homeschooling mom on the thread (son has LD) whose comments make sense to me:

Pat says clearly that your Socratic style doesn't work for [him]. Why do you then believe that it does work? You (rightly) want [him] to learn problem solving, but just because your method of teaching ... works for others doesn't mean it works for [him]. Maybe [he] needs repetition, repetition, repetition of the underlying content before [he's] ready for process. Other students may grasp the process after going through the underlying solutions three times, or six times, but maybe [he] needs thirty times.

You model the problem solving that you want [him] to do-- Where have you seen a problem like this? What rule did we use?-- but in a sense that's no better than modeling the actual rule for Problem 13. You want [him] to intuit that [he] is supposed to be asking [himself] those questions. But what if, as seems to be the case, [he] isn't intuiting that? Then [he's] not learning anything.

Even worse, math professors are attending them.

inflexible knowledge, flexible knowledge, and expertise

That option probably isn't on the menu.

According to Daniel Willingham, knowledge is always inflexible before it's flexible. You can't hopscotch over the inflexible stage by teaching process, or asking students to discover addition.

Problem solving and critical thinking seem to grow out of extensive practice of surface, shallow, inflexible knowledge.

I’d like to know more about how this happens.

At a minimum I’d like to know what cognitive psychologists (psych department cognitive psychologists, I mean) understand about the process at this point.


And I hope that Robert, who writes the brightMystery blog, will join us at KTM once in awhile to think about these things.

update

Back to main page.

Comments

This sounds a bit like "reform calculus" that is being taught on various campuses. To use the vernacular, it sucks. One of the places where it was pioneered, I'm sorry to say, is my Alma Mater U of Michigan. One of the people involved in its promotion is a woman (instructor not professor) named Pat Shure, who is buddies with Glenda Lappan of Michigan State U, who was one of the big forces behind Connected Math (CMP). I communicated with Pat Shure briefly, before I knew about the math wars. I was asking about how certain aspects of calculus were being taught, and her answer was snippy and downright insulting. Now I know why. She thought I was snooping around, trying to undermine the reform calculus program.

On a slightly related but admittedly tangential topic, if there are lawyers out there, can you put in your two cents regarding whether any of this, whether at college level, but particularly at K-12, constitutes educational malpractice? I am writing an op-ed conjecturing on this possibility, and the more facts I have, the more dangerous that makes me.

-- BarryGarelick - 29 Jun 2005

Barry, I'll email you the name of the man who is suing the state of MA.

-- CatherineJohnson - 29 Jun 2005

I can state with certainty that the majority of math professors don't know squat about learning disabilities.

Unless they've been made to do seminars since my time. But the thing about having tenure is that there really isn't that much that you can make a tenured professor do.

I know I've said this before. But I still wish I knew, back when I was teaching, what I know now about learning disabilities.

-- CarolynJohnston - 29 Jun 2005

I read Robert's post, and Number 2 pencil. Here is what I posted at Number 2 Pencil:

Robert's interaction with his LD student reminds me so much of my own ignorant interactions with mine, ten years ago when I was a college mathematics professor. Reading what he wrote makes me practically ache with guilt.

Let's try to evaluate Pat The Student fairly. Did he genuinely try to learn the stuff? Was he putting in the time, and failing anyway? Was he able to learn when the material was approached in the way that he and his mother suggested? Did Robert fail to even give their suggestions a try?

I had a few LD students like this. Some passed my course in the end, with the help of tutors (not mine, for the most part, I'm sad to say); some couldn't, even with the help of tutors.

I wish I knew then what I know now. And I wish Robert knew now what I know now.

-- CarolynJohnston - 30 Jun 2005

I know I've said this before. But I still wish I knew, back when I was teaching, what I know now about learning disabilities.

You're definitely going to have to fill us in as you go along.

I wish I knew more.

-- CatherineJohnson - 30 Jun 2005

Wow--Carolyn--you ROCK.

I'm so glad you posted that.

I had a strongly negative reaction to the whole saga...but I just couldn't quite tell whether I was on target or not.

On the one hand, by the time you get to college, you do have to be 'responsible' for your learning.

On the other hand, this professor was utterly unbending and frankly self-congratulatory.

Did you read the homeschool mom's comments?

Every one was good.

-- CatherineJohnson - 30 Jun 2005

The mom is 'Cardinal Fang.'

I made a post to the Mystery blog, but he hasn't let it through yet. (I assume he will--I posted Willingham's article link.)

-- CatherineJohnson - 30 Jun 2005

Oh good for you! I wanted to do that -- but didn't know the syntax (speaking of inflexible knowledge...)

I will post on this at length in time, though it's sort of a sore spot for me. I really do feel guilty about it.

Professors in math -- probably in all the other fields as well -- often don't know a thing about learning disabilities, and can't tell the difference between someone with an LD and someone who is just not trying. I am afraid I just didn't really believe in LDs, even in those kids whom I knew were trying hard and not getting it.

Never mind LDs, on some level I just didn't really believe in hidden disabilities. And I imagine I failed to hide my attitude very well.

So of course I get a kid with a serious hidden disability. Honestly, I've had to re-evaluate my beliefs and eat my words so often in my life, I think the fates must be using me as some kind of poster child.

-- CarolynJohnston - 30 Jun 2005

This is Robert from brightMystery. First of all, thanks for reading the blog and commenting. Second of all, I apologize to Catherine about the comment not getting through -- I've been having unfathomable technical issues with spam-blocking software (trying to stem the tide of trackback spam) that have caused legitimate comments to get blocked or even despammed. I'll get it all straightened out this evening now that my daughter is down to bed.

I only just now read the two comments left at my blog by Carolyn and Catherine, so I haven't read the Willingham (sp?) article you linked at my site, but I intend to do so. From the brief summary given here, I don't necessarily think that POGIL methodology is totally at odds with Willingham's ideas. At least, I don't disagree at all with the summary you gave. Perhaps that problem is that I haven't been clear enough on my blog on the role of content expertise in the framework of a POGIL classroom. I'll keep my comments short here, but I think there could be a useful discussion about all this in the offing. At least, I hope so.

The thing that attracted me to attend the POGIL workshop and give it a try in my classes was actually my complete disillusionment with all prior forms of constructivism and "reform" calculus I'd used in the past. None of that stuff worked to any effect, in my opinion, and I'd been to all the workshops and read all the research. The very problem I had with it, in fact, was that this form of math instruction DE-emphasized the role of content mastery... it was something to be sacrificed in favor of teaching some more squishy, touchy-feely version of math that I couldn't even recognize AS math any more. This was the kind of stuff I tried during maybe the first 3-4 years of my career. Having gotten nowhere with it, I went back to trying to provide a solid, traditional classroom experience.

But the thing is, that didn't work either. I was producing students who had content knowledge but no problem solving skills, and the kind of pedagogy I was employing frankly didn't fit with the kind of institution that employed me -- a liberal arts college, where the thrust is to develop the broad intellectual skills common to educated people. My students could take derivatives all day long, but put them in a position where they had to work with poorly-formed or ambiguous tasks, find optimal solutions to complex problems, or analyze problems in a systematic way, and many of them were out to sea.

To make a long story short, what I'm finding with POGIL is a methodology that works on both ends -- content is emphasized and process is emphasized, in a context that prevents it from becoming an either/or situation. We do emphasize content in the calculus class -- roughly three out of every four class meetings are actually fairly traditional lectures with times built in for students to get on-the-spot practice with the calculations and techniques of calculus, and over 50% of the points in the course come from individual timed assessments like quizzes and tests which largely cover traditional kinds of exercises and problems. I can post some of these if you like. What's different is the semi-Socratic teaching style I employ. There seems to be a basic misconception here. The kind of questioning I am using with "Pat" is not something separate from or prior to content mastery; rather, it makes Pat go back and review his/her basic content knowledge in order to answer my questions. This is why I suspect that Willingham and I will actually be in agreement -- students need to get experience with content, but from that point forward they have to develop the skill of putting that content to use, and they have to do that by DOING... having me lecture to them about how to solve a problem simply won't work, any more than lecturing on how to play the guitar or throw a football will make the listener an expert there either. So content is there. But I am forcing my students to connect the dots when it comes to problem solving, because my experience with simply showing and telling what to do makes me think that it makes students happy in the short term but ill-prepared in the long term.

A few other points to address: (1) I am actually not tenured. I teach the way I do not because I have tenure to protect me from negative reactions but because I believe in it firmly (although I am willing to listen to opposing viewpoints). (2) I think the characterization of my interactions with LD students as "ignorant" is unfair and inaccurate. As I mentioned in the post, I have years of experience (prior to full time teaching) tutoring high school students with ADD/ADHD as well as teaching basic literacy to adults with profound brain injuries. I have also worked very closely with our Teaching and Learning Center to accomodate students with LD's in whatever way they need. (3) The pedagogy I was employing in Pat's case was not POGIL. I had actually not even heard of POGIL at the time of Pat's course. I actually think a true POGIL setup in that class would have helped matters out quite a bit. I can go into that more if anybody is interested.

Thanks again for the interest.

-- RobertTalbert - 30 Jun 2005

"I haven't been clear enough on my blog on the role of content expertise in the framework of a POGIL classroom."

Versus...

"POGIL is a student-centered method of instruction that is based on recent developments in cognitive learning theory and results from classroom research that suggest [sic] most students experience improved learning when they are actively engaged, working together, and given the opportunity to construct their own understanding. "

Students are expected to create their own "content expertise"? Perhaps what you do is quite different than what is described here. Perhaps you don't understand how these concepts are applied in K-8. Perhaps you don't understand how the above POGIL description causes many of us to run for our hip boots.

-- SteveH - 30 Jun 2005

The quote above is actually from the official POGIL website (www.pogil.org), not from me. It doesn't say "construct their own content expertise", it says "construct their own understanding". That's actually two quite different goals. There's no room for differences of opinion on what the definition of the derivative is or the use of the Product Rule ("content expertise"); but there's plenty of room for differences in understanding what a derivative tells you and hwo you might interpret the result ("understanding"). You don't get a choice in deciding what the Quotient Rule says; but there is latitude for interpreting the meaning of the statement f'(2) = 4. Being able to compute derivatives is a lot different than understanding the meaning of a derivative, and yet both things are equally important in a college-level calculus class. So we need to have a pedagogy that treats both as important and puts them in proper context. Whether that's POGIL or not is immaterial.

In my original posting about POGIL on my blog, you'll find the same kind of skepticism about the methodology that you're expressing. I think I said there that this sounds like the same quasi-new-age crap that I'd tried all too many times before. Even now my BS meters peg into the red when I read it. And yet I gave it a hearing, and I've found it to be a pretty adaptable and workable framework for instruction so far. You're right in guessing that I don't follow the POGIL "rules" precisely.

Keep in mind too that I am teaching in a liberal arts college, which has a particular educational mission of equipping students with broad intellectual skills that make them into lifelong learners. This is somewhat different than a K-12 setting, and in fact I would absolutely not recommend POGIL be employed in the K-8 grades, and only sparingly in the 9-12 grades (e.g. AP classes would probably be OK). But I think it's entirely appropriate to at least think about it on the college level.

-- RobertTalbert - 30 Jun 2005

I should also add to my previous post that some students' interpretations about what a derivative means will be flat-out wrong. But they have to see WHY those interpretations are wrong. And the best way to do this might not always be by the instructor saying, "No that's not right".

-- RobertTalbert - 30 Jun 2005

This sounds a bit like the Moore method that R. L. Moore used for his grad students in topology at U of Texas, except Moore's was much more intense than POGIL. Moore himself said what Robert did about not employing POGIL in K-8, except Moore said he wouldn't employ it below the grad level. He hand picked his students and picked only those who showed great promise. Among his students were Ed Moise, R. H. Bing, Mary Ellen Rudin, Ben Fitzpatrick and others. Robert makes it sounds like POGIL is the "guide on the side". Something is not quite right.

-- BarryGarelick - 30 Jun 2005

Hi Robert, thank you for registering and posting!

I commend you for getting more involved with LD students than I did as a professor, and for being interested in pedagogy. I know there's a possibility that Pat The Student was working the system. But your post doesn't answer that question, so I ask again:

Did he genuinely try to learn the stuff?

Was he putting in the time, and failing anyway?

Was he able to learn when the material was approached in the way that he and his mother suggested?

Did you give their suggestions a try?

What I've learned since I taught as a professor is this: that people need to learn in different ways. No one method works for everyone.

The other thing I learned is that students' learning is more important, in the scheme of things, than my idea of how to teach.

-- CarolynJohnston - 30 Jun 2005

Barry - I'm all too familiar with the Moore method, since I had a year of Advanced Calculus when I was a senior in college, taught Moore-method-style by a prof who employed the Moore method because she herself had no clue about the material. I'd put up proofs that were correct and she would rip into them... only to later have to correct herself. It was a truly awful experience. Even with profs like these factored out, though, the Moore method has some pretty serious flaws IMHO, the biggest of which being that it seems to favor only the best and the brightest in a class. You hear all these rave reviews about the Moore method- but they're all from accomplished PhD^?'s. They were the survivors, in other words. You never hear from anybody else, it seems.

Moore method and POGIL are similar in some ways but very very different in some key ways too. The biggest way being probably that POGIL insists that students work in groups (at least for classroom activities), whereas the Moore method insists that they don't (Prof. Moore himself used to threaten students to within an inch of their lives if they shared any information about a problem); and the POGIL framework tries to set up a classroom environment in whcih students can feel comfortable maing mistakes early so that they'll be more inclined to fix them, whereas Moore method tends to be pretty stringent about getting things right the first time. So I think it would be wrong to conflate the two.

Carolyn -- I'm going to try to answer your questions as best I can a little later today. I regret that I never got around to answering Cardinal Fang's similar questions in the comments on my blog, because I do have answers for them.

-- RobertTalbert - 30 Jun 2005

Robert,

I appreciate your comments on Moore's method. I'm not a fan of it, by the way, in case you were thinking I was.

I have to say that I also am not a fan of working in groups and the various "guided discovery" methods that go under different names. Sorry, but that's my opinion. Prof. Ken Bogart of Dartmouth, a combinatorist (recently deceased in a bike accident I'm sorry to say), received a grant from NSF and wrote a "guided discovery" text on combinatorics, which is available at http://www.math.dartmouth.edu/~kpbogart/ComboNotes3-20-05.pdf.

I downloaded it and went through it, but found myself answering questions correctly but not understanding very well what it was I was supposed to have discovered. At one point after having answered some questions, the text announced: "You have just proved the binomial theorem" to which I thought "I did?"

Coincidentally, one of his PhD^? students, Joe Bonin, is a combinatorist who teaches at George Washington U. I took a class in combinatorics from him earlier this year. Same material was covered as that in Bogart's book, but I found I like to be told what it is I'm learning. Ultimately,learning comes from some discovery by the student, but the 20 questions type approach is in my opinion inefficient and also in Dr. Bonin's opinion, who though he liked and admired Dr. Bogart very much, did not care for the text he wrote.

Also, an expectation that students must understand EVERYTHING is unrealistic. In lower grades, understanding why algorithms work is a nice goal, but a bigger goal should learning how to operate with them. If a student doesn't understand why multiplying fractions is done by multiplying numerators and denominators, at the very least they should still know that's how you multiply fractions. They can be "retrofit" later. Same with division by fractions. I've worked with kids who know how to work with algorithms and those who don't. Those in the first group may not understand how and why they work, but are in a better position to learn than those in the second group. I would argue the same for understanding some of the concepts of calculus. It wasn't until I got into the more advanced calc classes and analysis that I really started to understand what was going on. But I had enough understanding to get me there, and certainly a working knowledge of how to solve problems. Speaking for myself, I don't think I would do well in a POGIL structured class.

As far as guided discovery and how it works, an extremely well written and illuminating article is one by Alan Siegel of NYU's Courant institute, describing how math is taught (geometry is used in the paper) in Japanese classrooms. What's important is he brings out that some people see what is going on in Japaneese classrooms as pure "discovery" method based on kids working in groups. His observations, which are very well documented, show this not to be the case, and the learning comes from very judicious guidance from well prepared instructors. But students are not allowed to flounder. If they can't get the answer, they ultimately are instructed what the solution is. The paper is Siegel, Alan R. Telling Lessons from the TIMSS Videotape: remarkable teaching practices as recorded from eighth-grade mathematics classes in Japan, Germany and the US. Chapter 5 in ``Testing Student Learning, Evaluating Teaching Effectiveness,'' Williamson M. Evers and Herbert J. Walberg, Eds., Hoover Institution Press, May, 2004, pp. 161-194. It can be found at: http://www.cs.nyu.edu/faculty/siegel/ST11.pdf .

-- BarryGarelick - 30 Jun 2005

I have never taught in a classroom, but I tutor both LD kids and kids who are having trouble in math. What I find that they have in common which restricts their ability to do the problems is accuracy in calculation and attention to detail in solving the problem. When problems become complicated and multistep as in calculus, their calculation errors and their inability to lay out every detail of every step makes it very difficult for them to make sense of the problem. When a professor does a problem on the board in front of the whole class, a student may think he understands it. When he gets home and looks over his notes and tries to recreate the professor's solution, he finds he is missing little details that connects one step to the other. Some students can fill in this knowledge by themselves. Some students cannot. These details may involve gaps in the student's knowledge that he doesn't even know he has. My job as a tutor for the upper level classes has involved teaching to the gaps in knowledge and then helping the student organize the details in the problem and solution. Then, they are usually able to solve problems on their own.

I also think that professors have to realize how intimidating it is to go in and ask for help. First, you usually have to wait on line. Then, you know the professor only has so much time, so you get right to the point and ask for help in a particular area. Now, imagine that your professor starts shooting questions at you. Let me describe it to you in better words than mine. This quote is from 'Never Too Late' by John Holt. He is describing a music lesson with his teacher. Remember he is an adult and this happened to him. "He (the teacher) told me to play a piece I was supposed to have worked on, and was exasperated to find that I made so little progress....As teachers to often do, he decided that by golly he was going to make me learn that piece...I began to make mistakes....A pressure began to build up inside my head....Some kind of noise, other than my own miserable playing, was in my ears. Suddenly something popped loose in my mind and the written music before me lost all meaning....Those black and white marks in front of me were completely disconnected from all previous experience....Since then, quite a few professional musicians have told me that they have had the same experience at a lesson or rehersal working on a difficult piece....". I have even had this experience with my son. After going through a very difficult (for him) multistep problem, the last step may be to calculate 12-6. And he just can't do it. He can't access this knowledge because everything has been diconnected. And I have to let him go out and play until everything has reconnected or reset itself in the brain. I believe that this is part of the 'here today, gone tomorrow' problem in LD kids.

-- AnneDwyer - 30 Jun 2005

Barry - Thanks for the link to the combinatorics text. I definitely want to take a look at that at some point.

The way I see it, guided discovery or "guided inquiry" -- the "GI" in POGIL -- is a means to an end, that end being for students to have a solid understanding of key concepts in whatever course the student is taking. Not EVERY concept -- just the ones that benefit most from students interacting with them. I think that a big mistake of earlier "reform" math efforts is that the pedagogy became an end in itself; the collaborative learning or whatever was being employed was being done jsut so professors could say they had an innovative classroom. If you read my posts at my blog you will find my distaste for this approach in no uncertain terms.

Here's how I see guided inquiry fitting in, in a college calculus classroom.

First, we start with a question, usually motivated by an application from real life. For example, suppose you have a function that models the profitability of your business as a function of time. You would like to know the answer to the following question: How can I pinpoint the times where my business' profit is at its highest point and at its lowest point? Everybody agrees this is an important question and one worth investigating. We are now motivated to find an answer.

Second, we look at a particular example or "model" to try and get a handle on this answer. For example, I could hand out a table of data that gives the monthly profits of the business over a five-year period, and the students use a spreadsheet to create a data plot for it; or maybe I just give a graph or a formula. It's easy enough to see where the high and low points are, but the next question is -- How can we pinpoint the exact location of these points? We've answered the initial question on one level, but there is a new question -- and a new motivation -- on another level.

Third, we start to bring the content of the course into the picture. I might ask students, "Look at this point where the graph hits a low point. What can you tell me about the function at this point?" Almost always, some student will eventually -- usually immediately -- comment that the derivative of the function is equal to zero there. A quick look through the rest of the graph confirms that there is a pattern at work -- where the high and low points of the graph happen, the derivative is zero. (Experienced calculus people also know that the derivative could be undefined at those points too, but we can save that subtlety for later.) In other words, we have begun to answer not only the question about our particular model but the BIG question at hand; I might say, "What do you think would be a way of locating these extreme points on a graph?" It's pretty obvious from the model that the answer will be to look for places where the derivative is zero.

From there, we can look at other examples and try this method out. There will be things to clean up, as you know -- a zero derivative doesn't always yield an extreme value; an extreme value can occur where the derivative is undefined; if the function is continuous on a closed interval, the extreme values might happen at the endpoints of the interval and not at the zero-derivative points. But the main idea has been discovered and isolated, and we have something to work with. At this point I would have students do some individual and group practice with a variety of different exercises all involving finding critical numbers and locating extreme values. The book has plenty of those, and that's ultimately where they're headed for their homework anyhow.

What makes this approach more conducive (IMO) for long-term learning than, say, just lecturing to students that in order to find extreme values, you have to take the derivative and set it equal to zero and so forth?

First of all, students now have a direct experience with a concrete example to which we can all refer later when working on another problem. When a student is working on exercise 21 from the book that's asking for the extreme values of f(x) on the closed interval [-1,3], I can say: "We worked out an example like this in class... how did that example go then?" This is forcing the student to connect the stuff he or she has learned in the class meetings (see Anne's comment) and extrapolate from a previously-solved problem to a new problem. Not only is this a skill students HAVE to have when they graduate, it's also a skill they have to have in order to pass tests in the short term.

Second, this approach portrays calculus as it was intended to be portrayed: As a system of concepts and techniques for posing and answering certain kinds of questions about change, and not just a collection of algebraic tricks that score points on tests. The real-life applications of calculus almost never fall under the rubric of what's in the "Applications of Calculus" sections in a textbook; in order for a person to have a realistic shot at really applying Calculus to problems they will actually encounter in a real job, they have to understand not only the techniques of calculus but also the kinds of questions calculus was invented to answer.

Third, students forget things they are told. They are less likely to forget things that they do. I have taught calculus many times where I just lecture on the "Closed Interval Method" (which is what the example above describes) and what invariably happens is that students hear it and follow my examples fine, but when it comes time to do it on their own, they don't know where to start because all they've gotten is a lecture, with no direct experience. Many of them actually claim that they've never heard of any such thing as the Closed Interval Method, even though I clearly labelled it as such on the board and in the notes prior to the examples. This illustrates an important point -- "teaching by telling" is so ineffective for many students that they not only forget what they were told, they forget the very fact that someone told it to them!

So straight lecturing is certainly "efficient" in a sort of signal-to-noise-ratio sense. It gets a lot of information out there in a short period of time. But I have doubts about how effective it is in terms of equipping students to be lifelong learners or even just to get them ready to do problems on their own on a test.

-- RobertTalbert - 30 Jun 2005

I meant to also say, Barry, that I agree 100% that closure is a very important part of learning. I would end off the extreme value example I outlined above by taking 5 minutes to debrief the class -- something like stopping and saying, "Now, let's outline a general method for finding the extreme values of a continuous function on a closed interval" or something like that and then help the students put together the Closed Interval Method. I don't think it would be enough to stop after they've done some examples and say "Congratulations! You just discovered the Closed Interval Method." I think the instructor's role here is to make sure that the students are aware that they now have a general method for answering a certain kind of question, coordinate the formation of what that method says, and give more practice so that I can be sure they really know how to use it.

I bring this example up because it's exactly what I am about to go do in my 9:30 calculus class this morning! I have an activity written up and I'd be happy to share it if anybody is interested.

-- RobertTalbert - 30 Jun 2005

Anne - I think tutors such as yourself play an extremely important role in making college education actually work. And your points about students with certain kinds of LD's having difficulty connecting information are certainly in agreement with my own experience with tutoring LD students. I think that the teaching I am describing here actually can HELP students in this situation because it is training them to ask questions of themselves that are fairly simple and easy to invsetigate (e.g. "Where have I seen a problem like this before?") and then identify and connect the past examples and so forth that they've seen.

The point you raise about content deficiencies getting amplified in multi-step problems is certainly not confined just to students with learning disabilities. This is true for everybody. One of the big issues in teaching calculus is that the content deficiencies can come on so many different levels -- from not understanding the Chain Rule all the way to not knowing how to add fractions.

As for office hours, I can assure you that what you describe about office hours is not the case here where I work, except for possibly the part about being intimidated. We here at Franklin love to work with students and open up all kinds of extra time to work with them, and we make this very clear to the students. My course Blackboard site has a discussion board on which students can post questions anonymously if they are really intimidated (although it's in the student's long term interest to work through that intimidation). And unfortunately, there are rarely lines for people trying to get into office hours. If anything, there is some kind of cultural element among the students here that office hours is the education equivalent of dialing 911 -- something to be done as an absolute last resort rather than as a first line of defense.

-- RobertTalbert - 30 Jun 2005

Hi Robert!

Thanks for coming over!

(I'll read carefully later on---thank you!)

-- CatherineJohnson - 30 Jun 2005

And by the way, Robert, I DO want to apologize for being flip about your class & your efforts.

Flip accounts of other people's blogs are fun to write, and probably reasonably fun to read, but in fact these are serious & real questions, which have been raised by you in a real and serious manner.

So thanks for being a mensch.

-- CatherineJohnson - 30 Jun 2005

"The quote above is actually from the official POGIL website (www.pogil.org), not from me. It doesn't say "construct their own content expertise", it says "construct their own understanding"."

This was exactly my point. It sounds like you are doing something other than POGIL.

As you know, POGIL sets off all sorts of crap detectors in those of us concerned about the low expectations and non-existent content in grades K-8. Being an NSF funded project increases my suspicions. There is big money in teaching/learning pedagogy as long as you are "on the bus", as Ken Kesey said.

From the www.pogil.org web site under Background Information:

"Recent developments in cognitive learning theory as well as results of classroom research suggest that most students experience improved learning when they are actively engaged and when they are given the opportunity to construct their own knowledge. These results counter the widespread misapprehension that effective teaching must be instructor-centered, involving the transfer of content directly from the expert (professor) to the novice (student). More "student-centered" approaches to learning are based on the premises that students will learn better when: they are actively engaged and thinking in class; they construct knowledge and draw conclusions by analyzing data and discussing ideas; they learn how to work together to understand concepts and solve problems; and the instructor serves as a facilitator to assist students in the learning process."

So, the instructor does not transfer content to the student. The student has to construct his/her own knowledge. A classic NSF project. They have been trying to bury direct teacher instruction for a long time. This philosophy is a major problem in K-8 education and is the reason why kids get to 6th grade without knowing their times tables, get to 8th grade without knowing much about fractions, and only get into college prep math if they have outside help. This is not because the methodology is done incorrectly (it is, however, in many cases), but due to the vast amounts of wasted time and no enforcement of timely mastery of the basics. I have already commented about how these people feel that there is no linkage between understanding and skills.

From the POGIL News Page, we have:

"Science and Math Education for the 21st Century: Scholarship and Innovation"

Where are my hip boots. New and Improved! 21st Century. Throw out that old periodic table. NO, better yet, discover it!

How you describe your sample profit versus time example doesn't tell me what I need to know. It sounds like the kids are supposed to know something about calculus before they start the problem. What POGIL states, however, doesn't mean that at all.

I apologize for being redundant on this web site, but everyone has to state clearly whether they are using a top down approach to learning the material or a bottom up approach. It seems clear to me that POGIL is a top down approach where they start with a real life problem (without any previous knowledge) and expect that the students will discover, construct, and work their way down to the basics with the teacher as only a guide on the side. Nothing is said, however, about mastery of the basic skills, just understanding. This is opposed to the bottom up approach where the students learn and master basic skills and then apply them to more complex problems.

The old joke is that the traditional approach is all skills and no understanding and that the modern, "reform" approach is all understanding and no skills. As Barry said in a previous post, you can fix the former, but you can't do anything about the latter. Understanding without skills is nothing.

However, in your profit versus time example, it sounds like you have already gone through a certain abount of learning about calculus. Teacher-directed, I assume? What you are doing sounds like a perfect add-on to a bottom up, teacher-centered classroom. You must realize that this is NOT what NSF is after. It also doesn't seem like you go very far with this profit example related to understanding. It sounds like it is an easy add-on to a bottom-up, direct instruction course. You could go further and help the students understand the different categories of problems: M equals N, M is less than N, and M is greater than N and that the one you are describing is where you have more unknowns than equations. You could describe the whole world of non-linear optimiztion and a few basic techniques about finding local or global minima or maxima. You could describe measures of merit and defining objective functions. Perhaps the students would wonder where you got the profit versus time equation. This is not a criticism of your example. I am just trying to make a point that a whole world of understanding is only possible IF you have some mastery of the basics. Otherwise, the understanding will be quite shallow. Besides, group learning and constructivism is neither necessary or sufficient to understanding. Some of the best learning experiences I ever had were when I was directly taught by a well-prepared teacher who really knew the material.

I taught college math and CS for many years and I appreciate your problem with Pat. I was always pragmatic and would try almost anything to get the student further ahead. Unfortunately (being college, not grade school), I would not chase after the students who were at risk of failure. I would hold extra group study sessions and offer (quite encouraging) office hours help, but I couldn't force them to come. I would tell them that I was on their side and would do anything I can to get them through the course, but their tests and homework would decide whether or not they passed. Perhaps some feel that you were being too dogmatic and unwilling to alter your approach to help Pat. I didn't get that impression.

As I mentioned on another thread, I had one professor, who was the king of direct instruction, (homework was used for discovery with some time in class for questions/answers), but when you went into his office for help, he would have you go up to the board and he would start asking 20 questions about what it was we wanted to do. If it seemed that we didn't understand one of the basics, he would suggest that we go back and study a particular chapter in the book. Keep in mind that this is college, not grade school.

"The objective is to develop skills as well as mastery of discipline-specific content simultaneously."

According to your blog, this is what you were looking for. I could say that skills and content are not the same as understanding. If you are actually saying that you want to develop skills, content knowledge and understanding at the same time, then I would still be confused as to the details of how this was done. You were worried about your MOPS course being a mile wide and an inch deep. Perhaps you are expecting too much from one course? For many of my computer science courses, I had to really resist the urge to keep adding more and more material that I thought was really important.

-- SteveH - 30 Jun 2005

Hi Robert, I would be interested in the exercise that you wrote up for your calculus class.

I can tell that you really care about teaching. As far as Pat is concerned -- the only thing I would hope for is that you ask yourself whether you really did your best for him. At this point, if I had an LD student with a good attitude (as Pat apparently did), I would have done what he asked me to do (within reason, that is! 7.5 office hours a week is enough). If that had failed, I would have tried a bunch of other approaches.

But I have a fairly unique perspective at this point, having a kid with high-functioning autism and a fantastic memory. Over and over, I've seen his rote style of learning blossom into fully generalized understanding, almost of its own accord once the procedure is understood. I don't fully understand it, but I've learned that people don't always learn the way you expect them to.

-- CarolynJohnston - 30 Jun 2005

Steve-

Let me respond to your comments starting from the end of your post. (A "bottom-up" approach. :) )

Unfortunately the MOPS situation is forced upon me by the way our curriculum was structured before I was hired here. The course was actually planned and approved before I was hired, and then I was hired to implement the plan. I have actually been arguing for a couple of years now that the content level of the MOPS course needs to be cut roughly in half, so that we can have more room for teaching process, because MOPS is basically a course about process. I do think the course is way too ambitious in terms of content, and I hope one day I'll be allowed to make appropriate changes in that regard. But this is an ongoing disagreement between myself and the people who hold the power to make the changes happen.

The example I drew up about profit and time does presuppose certain knowledge, and we do cover that, and yes it's in the form of a short lecture. This morning when I ran an activity very similar to the example I mentioned, I prefaced the activity with about 15 minutes of lecture on what an absolute minimum value is, what a local minimum value is, the differences between the two, and ended with a statement and explanation of the Extreme Value Theorem. Then the students were given the activity. (If I can figure out how to attach a file here, I'll do so.) So yeah, there is quite a bit of direct instruction on my part for the prerequisite mechanical skills that students need to go on and do a guided inquiry activity. I can tell you with certainty that despite the way the POGIL language might read, nobody who actually practices POGIL really feels any differently. The guys at Stony Brook University who came up with this approach only use POGIL in a once-a-week recitation section; the rest of their courses are interactive lecture.

You say that "group learning and constructivism is neither necessary or sufficient to understanding". This is quite true. It's also true for traditional pedagogical forms like lecturing too, though. "Understanding" is a highly complex cognitive state that isn't attained by one pedagogical approach. The key thing is balance. Lecturing is not BAD inherently; it's a pedagogical tool like any other, with its strengths and weaknesses. But to use it all the time for everything is a little like using a power drill for all household tasks. It's appropriate for some things and highly inappropriate for others. I want to balance out direct instruction with giving students the chance to practice, explore concepts, screw up and fix things, and generally get their feet nice and wet with a good deal of supervision by me. If you read the email I sent to Pat's mom I basically say exactly this. I also say that if I tell her kid how to do everything, then s/he has basically zero chance of performing on tests, and not much better prospects in a future job where s/he really will be expected to learn on his/her own -- in other words, no education will have taken place.

Let me reiterate that to the best of my knowledge, nothing about POGIL or the NSF grant that funds the web site and workshops has anything to do with K-8 education. The fact that the language of higher ed ("professors" rather than "teachers", for instance) ought to be a clear indicator of that. Several of the comments here are directing an overall negative feeling towards POGIL based on what's happening in K-8 these days. This seems illogical given that the K-8 and college audiences are so radically different in cognitive development. If a constructivist approach fails miserably in K-8 (which I don't doubt that it would), it doesn't mean that it should be guilty by association or dismissed out of hand for college teaching.

Finally, given that POGIL was developed for use in lab science courses and has only recently begun to be used in other areas, I guess I have no choice but to do something other than modify the "classic" POGIL framework. But my clarification about "constructing content knowledge" versus "constructing understanding" was not an indication that I am not doing POGIL. It's a clarification that POGIL doesn't intend to have students construct all content knowledge. The point here is to build an internally strong understanding of content knowledge -- to be able to explain with confidence what a derivative means, not to stumble upon how to compute one.

-- RobertTalbert - 30 Jun 2005

All- I just attempted to post a PDF file of the calculus activity I've referenced here. I hope I succeeded! It's called ge_4-1.pdf.

Some notes about how this activity functions:

(1) The exercise was preceded by a 20-minute lecture covering definitions and examples of absolute and local extreme values of functions, as well as the Extreme Value Theorem.

(2) Students were put into teams of 3 or 4. One person is designated as the "manager" of the team, another is the technology/math fact checker, and the third is the recorder and spokesperson. In the case of a 4-person group, the spokesperson and recorder roles are split up.

(3) Teams were given ten minutes to examine the model and work through the Key Questions. At the end of ten minutes, the spokespersons for the teams came to the board and wrote up their answers (one team did questions 1 and 2, another did 3 and 4, etc.) Then the entire class looked over the work and discussed the results.

(4) Teams were given an extra 15 minutes to work through the exercises. At the end of 15 minutes, teams again reported on their results, and discussed the answers.

(5) The problems at the end are intended for out-of-class work. Problem 1 will be presented at the board by a randomly-selected student on Tuesday. PRoblem 2 will be turned in for credit at a later date.

(6) Also on Tuesday, we will take 5 minutes to do a short quiz covering the main parts of the activity -- mostly true/false question to check for understanding of the main topics, and a little calculation to do. Teams will be awarded class participation credit in the amount equal to the average of their team's score, so it's in each individual's interest to make sure each person on the team has understood everything.

The activity was wrapped up by a discussion in which we distilled the overall method they had discovered for finding absolute extreme values. This method was written up on the board for future refence purposes.

There were some typos I found on the activity, which you'll probably see yourself if you read it. Also, the graph in the model is supposed to come to a cusp at x = 3, although the computer printout doesn't show that very clearly.

As I hovered around from group to group, I was struck by the degree to which students could correct their own mistakes. For example, a guy on one team thought that there wasn't a local minimum at x=3 because of the cusp and seemed pretty sure. But then the whole team, including him, jumped in and started using the definition of local minimum to say that it really was a local min. So they USED the content knowledge that came across in the lecture, but constructed an UNDERSTANDING of that knowledge that has probably got a much longer shelf-life than if I had just thrown up an example of the exact same thing myself.

It wasn't without its problems, and I'd phrase some of the questions better next time. We'll find out how successful it really was when I quiz them on it Tuesday.

-- RobertTalbert - 30 Jun 2005

Hi Robert! I just had a look at your POGIL exercise that you attached below. This exercise gives a good idea of what POGIL is about.

I can definitely see, too, that this style of instruction would give some learners trouble, and here is the precise category of learners that I think you'll have trouble with: people with non-verbal learning disabilities or Asperger's Syndrome, and other people with real disabilities in organizing knowledge.

People with these disorders don't organize their own content knowledge well; it's not that they can't do it at all, but they can't organize it and access it. It helps them enormously to have help with laying out a set of rules.

One way you can modify the POGIL concept so it might work for such students is this: say to them, let's work out a list of things to check for absolute and local extrema of functions. Add a little more structure to the outcome of the lesson, so that the kid ends up with a set of rules to follow -- because he needs rules to follow -- that he has worked out more or less himself, with your help.

Without the list, even if he got the basic idea when he first did it, he'll just be floundering when he has to try to apply his knowledge to a new problem.

-- CarolynJohnston - 30 Jun 2005

"One way you can modify the POGIL concept so it might work for such students is this: say to them, let's work out a list of things to check for absolute and local extrema of functions. Add a little more structure to the outcome of the lesson, so that the kid ends up with a set of rules to follow -- because he needs rules to follow -- that he has worked out more or less himself, with your help."

This is actually exactly what we did in the "closure" part of the activity I mentioned. What ended up on the board looked something like this:

--> for finding absolute extreme values of a continuous function f(x) on a closed interval [a,b]

(1) Find the x-values where f'(x) = 0.

(2) Find the x-values where f'(x) does not exist.

(3) Find the height/output values of the points from (1), the points from (2), and the endpoints x=a and x=b.

(4) The largest output value is the absolute maximum; the smallest is the absolute minimum.

The main thrust of the activity is to have the students understand WHY we are going through these steps. But even if the "why" of it all evades them for now, they've at least got a little recipe for finding the extreme values that they can use until the why part clicks. Probably on a test, I would make a point to ask student some kind of "why" question to ensure that they did at some point finally get around to understanding the concepts behind the method.

-- RobertTalbert - 30 Jun 2005

That's good. A student with NLD has a hard time extracting the Main Idea of a lesson -- he'll need it emphasized to him that this recipe is more important than all the other notes he took on the way to discovering that recipe. You could even suggest that he draw a big box around it with a highlighter.

I know all this sounds trivial, but that's the kind of cool thing about smart kids with LDs -- that such little details can make such a huge difference.

-- CarolynJohnston - 30 Jun 2005

Carolyn and others - I wanted to address those questions you asked earlier re: Pat.

> Did he genuinely try to learn the stuff?

This is a hard question to answer because Pat and I had a basic disagreement on what "learning the stuff" actually constituted.

Pat believed that a person has "learned calculus" if they can do simple, one- or two-step mechanical calculatons that look exactly like what is in the textbook, and any criterion for learning past that was optional. In a college setting though, "learning calculus" includes stuff like this but goes way beyond it. In a college setting, students need not only to master content (like mechanical calculations), they also need to show that they can apply this knowledge to new or different situations and explain the meanings of concepts in clear language. And there are other goals of the course besides that, not to mention mastery of more complicated mechanical material (like doing multiple-step calculation problems).

Pat worked quite hard to get the very basic material down, and I'd say he did OK in that regard. He could do simple mechanical stuff correctly about 70% of the time. But in his practice and exercises, I was struck by the fact that he never moved on from there. He never got to a point where he said, "OK, I'm doing alright with the basic stuff; let me try something a little further up the exercise list." Because of his idea of what "learning calculus" means, there was no need to do so. He had already "learned" it.

So from my point of view, Pat was spinning his wheels and not venturing out to try other things, and so from this standpoint he was expending a lot of energy to genuinely try and learn about 15% of the material he really needed to learn. Hence the approach I took with him -- try to get him to move on to more advanced exercises by deliberately tying them to the exercises he has done already, through the asking of questions that lead back to those exercises. And yes, I explained what I was doing and why I was doing it to him.

So I don't doubt his genuineness or the sincerity of his efforts, but he wasn't spending his efforts wisely.

> Was he putting in the time, and failing anyway?

I think I kind of answered that already. He was certainly putting in a lot of time, but it was more a question of quality rather than quantity. He spent lots of time doing something that he really should have spent far less time on than he did. And it wasn't because of a learning disability, it was because he had a bad conception of what the goals of the course were and just didn't want to change it.

One of the lessons everyone learns in college too is that the amount of time you spend on something doesn't entitle you to a good grade on it. In fact there really isn't even a reliable correlation between time spend and grade obtained. The only fair way to assess someone is on the quality of the product they produce, and if it takes one person five minutes and another person five days to produce something of the same quality, then that's life. You have to learn how to budget your time wisely, not just spend mass amounts of it.

> Was he able to learn when the material was approached in the way that he and his mother suggested? Did you give their suggestions a try?

As I mentioned in my email to Pat's mom, everything she suggested were things that were currently going on the class meetings. So yes, I had already tried what she suggested. And it got Pat nowhere, because the kinds of things he was being asked to do in the class again were outside his rubric of "success = simple mechanical proficiency". He had the content knowledge on a basic level; what he needed to learn was how to take it to the next level and solve problems, on his own without an expert guiding him through every step. What Pat's mom suggests, almost by definition, does not effectively address this issue. If his learning disability makes this more difficult, then it will require more resolve on his part and more time on mine, but I still think that training him to ask the right questions was the right approach if the goal here was to teach him how to be an independent problem-solver.

And I think if Pat had been more willing to open up his conceptions about learning calculus, he would have caught on better. If there's one thing I think I'd do differently in this situation again, it'd be to make very clear to Pat why I am not just giving straight answers to all his questions (particularly the ones like "Can you show me how to do #13?") and try different ways to get Pat to organize what he does know.

-- RobertTalbert - 30 Jun 2005

You lecture for 20 minutes first, but do they already know how to take derivatives? This is what I am talking about. Not whether they are prepared for 20 minutes.

"The guys at Stony Brook University who came up with this approach only use POGIL in a once-a-week recitation section; the rest of their courses are interactive lecture."

Once a week? This is not the impression one gets from reading the POGIL web site. Are they using all of this NSF-speak just to make them happy?

"The key thing is balance."

Oh No! balance is another code word for arguing with generalities, but controlling the details. Who can be against balance? Balanced Literacy and Balanced Math, but just let them decide what balance means.

"Lecturing is not BAD inherently."

Don't tell that to NSF or NCTM!

Direct instruction or lecturing does not preclude the active involvement of students, although some would have you think so. I have had numerous classes where the teacher asks questions and expects all students to participate. Many in the education community don't want ANY "sage on the stage" or "passive learning". These pejorative statements underlie their fundamental, pedagogocal dislike for anything related to direct instruction. (A preparatory lecture to the student groups is OK, however.) However, there is no such thing as passive learning. All classes require students to pay attention in class, read on their own, do homework problems, and prepare for tests. Why is it that some think that true learning can only be done in student-centered groups IN CLASS!?! Why do some feel that balance only means balance for class time?

I remember one computer science lecture where I created an algorithm (a variation of a balanced tree instertion) in class without having done it beforehand. I thought out loud and wrote it down (mistakes and backtracks) on the board so the students got to see my thinking process. They could ask questions and tell me what to do, but I would try to show them how I could think ahead and forsee problems. I could have broken them into groups to do this by themselves in class (I would be "on the side"), but I knew that this would be a waste of precious class time and wouldn't be as effective for what I wanted them to know. Besides, this group stuff is what they did with the homework. That was the purpose of homework. I suppose you could add a calculus lab to the course where POGIL is done, but I wouldn't use it to replace precious class time.

"I want to balance out direct instruction with giving students the chance to practice, explore concepts, screw up and fix things, and generally get their feet nice and wet with a good deal of supervision by me."

Isn't this what homework and projects are for? It may not be quite as interactive as if it's done in class, but it doesn't waste class time and you do have office hours.

"If a constructivist approach fails miserably in K-8 (which I don't doubt that it would), it doesn't mean that it should be guilty by association or dismissed out of hand for college teaching."

Perhaps you don't know how constructivism is applied in K-8. It surely isn't once a week and they do require kids to construct their own solutions without prior content knowledge or skills. As for college, I did notice that it was designed for college courses, but I still question the wasted time factor. POGIL talks about generalities, but the devil is in the details. I see no neat pedagogical solutions.

-- SteveH - 30 Jun 2005

> You lecture for 20 minutes first, but do they already know how to take derivatives?

Yes. If you read the activity posted below, you'll see all the skills listed in the Prerequisites section. In terms of the textbook, this activity corresponds to Chapter 4, section 1; Chapter 3 is all about derivative-taking techniques like the Product Rule and the Quotient Rule. So we've been doing derivatives for two weeks now, which is like three and a half weeks in regular-semester time.

I think that if you had observed how well my students progressed through the activity this morning, you might reconsider thinking about active learning activities as "wasting time". And the minilecture plus the activity took right at 50 minutes total, which is exactly the same amount of time that I used to spend lecturing on the stuff in the past, only to have students literally swear to me that they had never seen any such thing as the Closed Interval Method before. It seems to me that lecturing is in that sense at least as likely to "waste time" as a well-desgined group activity.

> Isn't this what homework and projects are for? It may not be quite as interactive as if it's done in class, but it doesn't waste class time and you do have office hours.

Class time is for learning the basic ideas, "basic" in the linear algebra sense -- the ideas out of which all the other applications and concepts are built. We do that through a combination of different approaches, as I've described. Homework and projects are for taking those basic ideas and taking the time outside of class to do more complex things with them. And again, I would disagree that group work wastes time whereas not doing it doesn't waste time. Either approach can be wasteful if the approach doesn't fit the instructional goal you have for the day or the students you have in your class.

> POGIL talks about generalities, but the devil is in the details. I see no neat pedagogical solutions.

It goes without saying that neat pedagogical solutions do not exist under any framework, POGIL or otherwise. Having actually taken a minicourse in POGIL, I can assure you that it goes a lot further than just rhapsodies about generalities. I wouldn't touch it if it didn't.

Look, everybody - POGIL is just an idea, a way of teaching that can be accepted totally, partially, or not at all. It's not a religion, and it doesn't purport to cure all pedagogical problems, and it's possible to salvage the best ideas from this framework and throw away the others. In the few months that I have actually used it in the classroom, it has worked not flawlessly but very well. I think it holds a lot of promise, way moreso than other so-called "reform" pedagogies, and it ought to be considered objectively without prejudice toward failed pedagogies we've had bad experiences with or terms that got used one time in a way that we don't like.

I get the sense from some of the comments here that some folks have been burned so badly by nontraditional pedagogies in the past, or indoctrinated so fully against anything nontraditional, that anything that has the remotest connection to constructivism or whatever is automatically bunk. Something like that may well be bunk. But let's not reject new ideas out of hand, but rather consider what's good about them, so we can use it to help math instruction finally make some kind of tentative step forward in this country.

-- RobertTalbert - 30 Jun 2005

I don't want to reject new ideas, and I don't think anyone here does either. What I don't want to see, however, is an outright dismissal of methods that HAVE worked in the past. It just seems too easy to say, "well we all know how calculus is being taught isn't working." Do we? There's a lot going on here. Math ed in K-12 is in a very bad state in this country, so if students aren't learning it well, shouldn't we look at what students are coming equipped with?

I don't teach college math, but I do tutor high school students. When they ask questions, I try to lead them to the answer, without telling them outright. But if they are confused and the guide on the side technique isn't working, I revert to more direct methods, and then reinforce it with other problems. Call me crazy.

-- BarryGarelick - 30 Jun 2005

I can definitely see, too, that this style of instruction would give some learners trouble, and here is the precise category of learners that I think you'll have trouble with...

Carolyn, are you implying that Robert needs to tailor his class materials for every type of learning disablity? I would guess that he has very few (if any)LD students in any one class, so it wouldn't make sense to make adapt his materials unless he knew that he needed to. By the way, nobody has mentioned that Robert didn't even know Pat was LD until 2/3 of the way through the class, and then only through unofficial channels (the mom).

I'm not trying to be a smart aleck with my question. I'm just trying to understand the sense of some of the discussion.

What I don't want to see, however, is an outright dismissal of methods that HAVE worked in the past.

Barry, can you define what you mean by "have worked in the past"? If every student is not "getting it" 100%, isn't it worth considering new approaches, to try to optimize the learning potential for everyone?

(Sorry about using "optimize" and "potential" in the same sentence)

-- DavidL - 01 Jul 2005

• ge_4-1.pdf: POGIL-style exercise on finding extreme values of functions

Attachment	Action	Size	Date	Who	Comment
ge_4-1.pdf	manage	77.3 K	30 Jun 2005 - 17:01	RobertTalbert	POGIL-style exercise on finding extreme values of functions