The responses to Kelly and Steve's interviews were as follows:
Interviewer: Rob Landley Date: Sun, Sep 14. Subject ID: Kelly Age: 49 years, 11.05 months. Part A The S classified by color. Named the two groups red and blue. Part B The S classified by shape. Named the two groups squares and octagons. Part C The S classified by "Things with more than four sides, things with four or fewer sides." Part D Added a pair of scissors. When opened up, it had more than four sides, and was grouped accordingly. Part E Next classification was "things with acute angles and things without acute angles". (Also "things with all equal sides, things without equal sides", "things where the angles add up to 360 degrees, and things where they don't"...) Response category: 5
Interviewer: Rob Landley Date: Sun, Sep 14. Subject ID: Kelly Age: 49 years, 11.05 months. Part A Yes, subject drew front view correctly. Part B Yes, subject drew side vertical view correctly. Reason: "It's round, it's going to look that way from all directions, it's a cy-lin-der." Part C Yes. (Subject jumped the gun and drew a front view of the tilted dowel while interviewer was still reading the question. At the end of the question, correctly drew side view.) Reason: "It would look shorter since it's tilted, and because it's tilted you would be able to see part of the top." Part D Yes, subject drew side horizontal view correctly. Reason: "Because I'd be looking at the little end of it, and it's a circle because it's a cylinder." (Went on to elaborate by drawing in the upper right hand corner. The eraser in question does, in fact, have a happy face built into it.) Response category: 7
Interviewer: Rob Landley Date: Sun, Sep 14.
Subject ID: Kelly Age: 49 years, 11.05 months.
Part A: point in two dimensions.
Subject measured from nearest corner with string, reproduced angle
on second piece of paper by eyeball. When prompted for another
method, made a y-shape in the string to measure both dimensions
at once.
Response to question "Is it in exactly the same place?":
"Molecular level, no. Gross level, plus or minus half a centimeter."
Paused a moment. "Of course they're not in the same place, it's on a
separate sheet of paper!" (Note, point is, in fact, in right place.)
Part B: point in three dimensions.
Subject wanted to triangulate from three or four different angles,
adjusting each time. Alternate method involved calculus. (Explicitly
refused to measure at right angles. Mentioned a high school
project she did on extending analytic geometry to the fourth dimension.
Volunteered to draw a Tesseract.)
Response category: 10.
Interviewer: Rob Landley Date: Sun, Sep 14. Subject ID: Steve Age: 57 years, 5 months. Part A S classified into "null group" (empty) and "the rest". First non-sarcastic attempt classified by shapes. (Squares and not squares.) Part B Classified by color, "Red and Blue". Part C Fewer corners group, greater corners group. (Then classified the colors by wavelength to sort the visible spectrum into high and low halves.) Part C The S classified by "Things with more than four sides, things with four or fewer sides." Part D Given a pink pencil, he continued to sort by wavelength. Part E Subject had, in fact managed "squares and not squares" in part A. Continued to list additional categories for some time anyway. Response category: 5
Interviewer: Rob Landley Date: Sun, Sep 14. Subject ID: Steve Age: 57 years, 5 months. Part A Yes, subject drew front view correctly. Part B Yes, subject drew side vertical view correctly. Reason: "That's what it looks like." Part C Yes, subject drew side tilted view correctly. Reason: "Perspective." Part D Yes, subject drew side horizontal view correctly. Reason: Showed instructor the end of the eraser and said "there". Response category: 7
Interviewer: Rob Landley Date: Sun, Sep 14. Subject ID: Steve Age: 57 years, 5 months. Part A: point in two dimensions. Subject measured from two nearest corners with string, marking a line each time, which formed an x. Put dot in middle of X. Response to question "is it in exactly the same place": "Yes." Part B: point in three dimensions. Subject suggested measuring X, Y, and Z co-ordinates. (By name.) Response category: 11.
I suspected Kelly and Steve might understand how to do all the tasks. I met them through Mensa.
I delayed handing this in because the subjects I interviewed were too old. This assigment really seems aimed towards kindergarten and grade school teachers, and the only people I know under the age of 18 are outside of texas. That said, these subjects are represenatative of the community college classes I've taught; many of the subjects are adults returning for continuing education. The introductory courses have a higher proportion of retirees, and the evening courses tend to be full of full-time employees whose employers are paying for their courses as "professional development". Perhaps as much as half the students are young adults, but I do not believe that any of the students in the last four courses I taught were under the age of 18 years.
As an undergraduate, I did tutor remedial math when I was going to community college (math 001: addition, subtraction, multiplication, and division) for a student who had failed it four times already. I got her to pass, and Algebra I and II on top of that over the next two semesters, because the tutoring center was willing to pay me to devote ten hours a week to her education that first semester to finally unstick one of their repeat clients, and was happy to follow up once she showed progress. Mostly her problem was a lack of confidence and missing background information. (The New Jersey public school system is a glorified day-care system that socially promoted her to graduation without even caring whether she could add.)
Teaching at this level used the same general concepts as teaching at later levels: I broke my understanding of each topic down into bits, and simply kept going back until we found the things she already knew (correctly), and then guided her slowly and methodically through the missing bits, backing up and taking really really small steps at times. With a lot of analogies, real-world examples, and confidence-building exercises.
I'm not sure how much knowledge of Piaget's stages would have helped. It seemed to me that the impotant thing was to focus on the student rather than some abstract model of the student. If I wanted to know about her, I asked. I suppose that's what Piaget was trying to do.
There's a little bit of physics, geometry, or zoology in there. And the ability to group and handle abstractions is fairly central to math. Algebra, for instance.
My specialization is computer science, and I'm not sure how applicable Piaget's interviews are, even though he mentioned being influenced by data processing. Computers aren't really found in nature, and experience with the real world doesn't necessarily give you the background to understand how a computer works. (This is one reason so much emphasis is placed on math, which is closer to computer science than most physical phenomena, although it still not a very good match.) Then again, these days students often have background knowledge about computers due to a history of interaction with them, and finding out what they already know is the best way to begin teaching them. (At the college level, students are expected to select classes they have the appropriate background for, which takes some of this burden off the teacher. Then again, students reguarly get this wrong, and teachers need to be able to spot this.)
More generally, I suppose you could say that computer programming is all about taking what you know and explaining it to the world's most ignorant and literal-minded student. It's all about keeping track of your assumptions, and what you have and haven't said yet, and building on what you've already done in discrete steps without skipping any. And above all, being very, very explicit. Then the debugging stage is largely about finding and correcting mistaken assumptions, which can require constructing some fiendishly complicated questions to narrow down the problem.
Of course a computer is very different than a real student: computers never get bored, frustrated, scared, tired, depressed, or hungry. And they never have their own plans, while students usually do. (Also computers tell you when they have a problem; students often don't.)
The main lesson Piaget seems to be teaching is how to pay attention to the student, and ask the student questions rather than just lecture at them. If this was a major revalation to the educatinal establishment, I can only imagine how screwed up it must have been before Piaget.
But in terms of the teaching I've already done, the Piaget interview fails to address the real challenges I've experienced with group education. Teaching is easy when you have unlimited individual interaction with a motivated student. The real trick is keeping a group with varying abilities collectively interested in the material, confident in their ability to succeed, and challenged without getting too frustrated. If you go too slowly and repeat too much, you bore the fast learners until they stop paying attention. If you go too fast, some fall behind. If the class size is large enough, having both happen at the same time is almost inevitable; the best you can hope to do is teach at the middle of the class, and give everybody the chance to jump back aboard the train as much as possible.
Piaget would say the young ones score lower and the older ones score higher, Probably with some grouping according to the four developmental stages acting as plauteaus. It might also look like the datasets on pages 8, 16, 23, and 24 in the PDF file.
But it's not just age. I've found that there are plenty of adults who can't do this stuff. My mother was an art psychotherapist, she taught art classes to emotionally disturbed adolescents. Many of their _parents_ couldn't do this stuff.
I have yet to find a student "too dumb to learn the material". I regularly find students who have missing background information or shattered confidence. A teacher who does not understand the difference between ignorance and stupidity won't go very far: ignorance is nothing to be ashamed up and easy to deal with (if time consuming). Intelligence is largely just the rate at which somebody learns (as opposed to education, which is how much they already know). "Stupid" people are usually just slow learners; it doesn't mean they can't learn a given subject, they just need more time to do so. Some highly educated people are slow, plodding, and meticulous about it. (Some of these make great scientists.)
Willful ignorance is a third category, and there's not much you can do about that. And there are a number of problems like dyslexia, depression, or drug addiction that can't be reduced into a one-dimensional description. But for a student to make progress, the student's specific problems and needs must be addressed. You can't teach every student the same way because they're not the same. And Piaget was apparently one of the first people to point that out.