From: Bob Hayden (email@example.com)
Date: 18 Nov 1995 19:52:46 -0500
Subject: Are humans... (a better way to teach stats?)
This was posted on EdStat-L but it reminded me of some of the things Johnny Hamilton has said here.
> From firstname.lastname@example.org Sat Nov 18 02:55:51 1995
> From: email@example.com (Duane Griffin)
> :It's the awful way it's taught.
> : uuuurk -- we upchuck it and then
> : blank out the experience.
> : I had exactly this experience.
> As did I. From first grade through high school, I was given ample reason
> to loathe all things mathematical. Never was I shown, in any meaningful
> way, WHY any of this was important or useful. I came to hate math with the
> sort of passion that only an adolescent can muster. My obligatory intro
> stats course in college did nothing to change my attitudes. I survived it
> through sheer force of will, dutifully upchucked and blanked.
> It wasn't until I began taking graduate courses in environmental sciences,
> where we posed questions about the world, went out and gathered data, and
> analyzed them using the statistical methods I'd so abhorred, that I began to
> understand what I'd supposedly learned in the course and to see true
> purpose and power in things mathematical (beyond simple counting and
> arithmetic). Three years of intense catch up work followed (thank
> heavens for the Schaum's outline series), during which I kicked myself daily
> for being such a stubborn ass for so long.
> I find myself now in the position of teaching assistant for the same course I
> hated, perpetuating the same experience in my students. Now that I understand
> the material, the professor's lectures seem brilliantly straightforward and
> clear, and I find myself frustrated with how little the students seem to
> get. Only when I make a conscious effort to take myself back to those
> painful days can I begin to appreciate just how opaque, complex, and
> meaningless even the most basic stats concepts can be. Add to this the
> frustrations engendered by unhelpful (and uninspiring) textbooks, minor
> typos and mistakes that creep into homework assignments and handouts, the
> fact that many, if not most, of the students have forgotten what algebra
> they might have ever known, and the not-unfounded perception that statistics
> are just a fancier way of lying, and POOF! Is it any wonder that their
> enthusiasm has to be scraped off of the floor?
> I do what I can to try to offset the tendency towards "survivalism" among my
> students. I've found that they show far more interest when I use examples
> from my own research, but most of these I contrive for their benefit. I
> looked through the past three volumes of our discipline's flagship journal
> for examples to have them read, but couldn't find a single instance of anybody
> using anything as basic as the stuff we cover (we rarely make it past ANOVA,
> and that's crammed into the last week), which reinforces the notion that
> stats is something that they'll never be able to use.
> There has to be a better way. My suggestion (completely untested) is to try
> to duplicate the route that I (and many of my fellow graduate students) have
> taken. Somehow we have to start with questions that need answering (real
> questions, not examples using cards or dice or defective light bulbs...
> UUUUUURRRKKKK) and can be answered using statistics, teach students enough
> to get them started, and then backtrack to fill in the blanks. Why not
> start by having students collect data, or at least find datasets that they
> are interested in, and spend a couple of weeks just playing around with
> them, graphing this or that, calculating descriptive stats and learning
> about them? Then why not jump right into regression or ANOVA, teaching them
> enough to make sense of their results, and spend the rest of the semester
> understanding why and how it works or doesn't work?
> I can see how this might be less of a problem in other fields, business or
> engineering for example, where the motivation and background already exist.
> But most students, at least in my experience, are taking the course because
> it's required (like I did). Some will eventually come around, but most
> won't. By failing to reach this latter group, I'm afraid we're doing both
> them and ourselves a great disservice.
> Just a few thoughts.
> Duane Griffin
> UW Geography Department
Bob, thanks for the quote. My reactions go in several different directions. Two years ago when I was on a sabbatical I took an undergrad stats class. I saw what your quote says. However, to make this short I will respond this way: "when the horse gets thirsty, he will drink."
I will make it even shorter. Most of the horses, say about ninety percent or so, never return to a sour well, but look for another. You and I know that there is but one well, but they don't. Not providing the majority of people in this country a solid mathematical base to work from is destructive. Waiting for the country to see it the right way is wasteful. Making beautiful math sour is distasteful especially when it can be so sweet.
Home page - http://www.istar.com/constructpress/page1.html
Why do I have the distinct feeling that if someone HAD taken MR. Griffin out to collect real world data when he was in high school, and maybe into those first few years of college, he would have screamed about the choice of data? One mans "real world experience" is another man's mindless drivel.
He found a shortcut to avoid really learning the material that someone, I suspect, tried very hard to make meaningful to him, and in the end it was essential to his career. Essential in a different way from that of a future carpenter, or a future chemical engineer, yet it is the same data, and the STUPID teachers didn't realize that they should construct their classes to promote the development of a single interest (his) over all the other applied areas they could use. Poor idiots, they fell back on cards and dice and defective light bulbs (which NO ONE is interested in). Things that were part of the experience of a wide group of students, and had a wide range of applications.
To all the teachers out there, press on, or as the gyoji says:
Seems to me in the letter that Duane blamed himself for his survivalistic tactics and he even went on to say that after trying to teach the subject that his teacher's work seemed brilliant. I got the distinct impression that this letter was not an attack on teachers at all. It was an attempt to say that there must be a better way. Duane's letter did not seem to be a call for individualized instruction relevant to only one group or another.
I do relate to Duane having to relearn math when he found how it applied to his work. Most people will find relevance for the math they did not learn in school. Most, however, do not return to the well, but continue to work and live without the proper math base. I think that there is a single interest group that is being served by the way math is taught-math majors. I don't think it is intentional or anything of the sort, but there just too many intelligent people out here who are math phobic.
From this forum and the nctm-l list, I see that among these phobics are many elementary teachers and teachers in other disciplines. I can't believe that all of these people were poor or non-caring students in school. As Shapiro notes, geometry and visual art are intertwined, yet most artists, commercial and fine, usually have poor mathematical geometry skills. I have heard from many engineers that they did not really begin to enjoy math until they got into engineering school and had something to relate it to.
I agree that teachers should press on. I know personally that teachers work very hard. We work very hard at our jobs too, but we could be working smarter if we understood how mathematics relates to our work. I do not think that the time to understand the relationship is after school. In reading Steve's post about the projects that students developed for relating things that interested them to geometry, I have hope. I would much rather see the geometer tie universal math into practical work than it be left the practitioner to tie work to limited math knowledge.
If I have misread the young man, I apologize to him, but I suggest that if you reread the first few lines of the article you will have a difficult time ascribing even a neutral attitude to math teaching to the author.
I too think people in the trades, arts, or engineering could work smarter if they understood how mathematics relates to their work, but that does not indicate that I agree that students will learn MORE usable math if the emphasis is on Applications. A student excited about Chaos, or Hypercubes, or any other mathematically deep topic will widen his mathematical experience and pleasure. My disagreement is how much of this should be "classwork". Good math teachers have nurtured the interests of students with challenging problems, references to interesting resources and the like as a regular part of teaching, and then given them a full background of theory in the classroom. I don't know what vocations my students may take up, but I know that a core of mathematical knowledge will prepare them for ANY applied area, and a student who truly understands the mathematical relationships will see them when they appear in his job.
I support good student projects, and independent study into areas of interest to them, but I just disagree with you about how much that should be the emphasis in the classroom. Certainly the emerging media of e-mail and www pages offer the potential to nourish a really wide open exposure to their interest areas, but again, they need math foundation to perceive the patterns they will find.
This raises once again what I think is a very important point. There seems to be a lot of pressure from everyone for "real-world" applications. I think this is because of the perennial student's question "But what is all this good for?", which makes teachers feel that they must respond with ways that mathematics actually is used in the real world.
The trouble with this is frankly that lots of these real-world (or "real-world" - I don't think it makes much difference!) problems, are, frankly, BORING. As Pat Ballew says, no student is actually interested in problems about defective light-bulbs, even though one or two of the people who make light-bulbs might be very interested in the answers.
I saw this in action at a meeting of school-teachers. They themselves pressed for real-world applications, but when the speaker gave them one, they all started to yawn, whereas in the previous part of the talk they'd been very lively.
You DON'T really need to give your students real-world problems, even if they tell you that that's what they want. What you DO need to do, is give them interesting mathematics.
So what do you do when they ask that perennial question? I think, really, that you should train them NOT to ask it. I don't mean that you should have beaten them to a pulp the first time they asked - that would have been unkind, because they really didn't know - no, you should have told them that just about every interesting piece of mathematics has LOTS of applications (and named a few). Also, that applying the mathematics, though often very useful, can be a bit dull (say no more!), and that sometimes the closer you get to a real-life application, the harder the mathematics gets (it may either get more OR less interesting).
I remember that when someone asked for a real real-life problem, Johnny responded with a long thing, about something like redecorating a corridor that went for 53 feet in one direction, then turned through such and such an angle before going for 76 feet in another direction, the second part being smaller and having a guard-rail 2 feet 8 inches above the ground for 48 feet of its length. You had to work out how much paint ... .
I noticed no throng of eager solvers for Johnny's problem. That's often what real-world problems can be like - both complex and boring, unless perhaps you actually ARE decorating a corridor that's exactly like that. The only genuine real-world problems that kids should be asked to face are the ones that actually face them! By all means get your kids out surveying the school grounds and using their mathematics to plan or build something. But otherwise teach them stuff that's interesting rather than applicable, and TELL them that's what you're doing.
As I read this ongoing conversation, I keep coming up with two questions:
It seems to me that we diminish mathematics learning and teaching when we focus too much on pure mathematics or too much on its applications. First, it is diminished because we fail to respect student differences in interests and learning style that can only result in turning away more students from the math well than we have too. Second, pure and applied mathematics offer different concepts to students. Do we not have room in our mathematics classroom for Martin Gardner's logical puzzles and statistical applications to the social sciences?
It seems to me that this is one more reason to involve students in their learning as much as possible. Through student/teacher discussion, student/teacher reflection, and teacher observation, we may become more adept at adapting our curriculum to ensure we do not inadvertently alienate students like Bob Hayden describes. Shouldn't we endeavor to create a classroom that presents as many different aspects of mathematics as possible and thereby keep more students at the well (or at least nearby)?
I agree completely with your suggestions and ideas. Some of us interested in mathematics education would like to see how well we can tune into mathematical ideas, energize students, and help students to make math personally meaningful.
This is quite a challenge, it seems to me, because what excites or interests a person varies over time and from person to person. So this program would seem to call for lots of on-the-fly adaptation to the personal (individual) interests of students. A teacher adept at this would seem to need to be highly knowledgeable about various aspects of math that reflect and amplify a student's interests, and would also need considerable intuition and skill in understanding students and awakening their interest. The vision I am attempting to describe is mathematics as a part of and subservient to a liberal education, in the old sense of that phrase.
The short version of the preceding paragraphs is a response of "Amen!" to your posting. But I also wanted to make a plea for greater tolerance for a diversity of goals among all of us who are interested in mathematics education. We seem to be searching for the holy grail, the one truth around which all of us will be able to unite. I do not believe this can be accomplished. Rather, I believe the road for us to travel is to find people whose goals are close enough to our own, to band together to try to accomplish these goals as well as we can, and to celebrate the successes of other groups with different goals.
For example, people interested in this notion of a liberal education are, I believe, a minority among parents, administrators, and probably teachers as well. Of course, everyone will pay lip service to such American-sounding goals as "helping the student increasingly to take charge of her/his education," but how many will really be willing to gamble on this idea, to support it, monitor it, use it as the primary criterion for judging success? This takes courage, belief, and perseverance. Moreover, the path followed will not be the same, and will not accomplish the same things, as that taken by parents and teachers who take a prescribed (however broadly) curriculum to be the primary criterion. Instead of quarreling with these parents and teachers, I think we should wish them every success.
What I am objecting to is a lack of moral courage in taking some stand on goals. Instead we attempt to package everyone's goals in a national agenda. I believe such attempts are doomed to produce a bland pedagogy and curriculum whose great achievement is to pay lip service to the disparate goals of our society.
I think what is needed to achieve the diversity I am recommending is for respected bodies, like the NCTM, to encourage excellence in a number of diverse approaches to math education.
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