[IAEP] reconstructed maths

[Bill Kerr]

> We can safely ignore the C and D words in favor of the E and T words (or
the M and S words)

hi alan,

thanks for a very valuable post on the nature of educational surveys - and
your reasons for not using the C (constructivism) or D (direct instruction)
words

unfortunately your final sentence threw me, what are the E,T, M and S words?

cheers,
- Bill



On Wed, Jul 23, 2008 at 6:34 AM, Alan Kay <alan.nemo at yahoo.com> wrote:

> Hi Bill --
>
> Suppose we take as a premise that the following results of surveys over the
> last 20 years were gathered well enough to form a real generalization:
> - only 20% of American adults can read a well written important ideas essay
> (they used Tom Paine's "Common Sense"), and understand it well enough to
> discuss it, write about it, criticize it, advocate it, etc. (National
> Literacy Foundation)
> - only 5% of American adults are "literate/fluent" enough in math & science
> to deal with mainstream ideas, have an extended conversation with a
> mathematician or scientist, be operational enough to put a mathematical map
> on a set of ideas and do something with them, etc.
>
> If we weed a few more artifacts out of this survey (and the surveyors did
> some of this already -- such as not counting those who were not educated in
> American schools, etc.), then we can pretty reasonably conclude that the
> schemes of education employed in the US have failed (miserably) to meet the
> goals of education in America.
>
> The reason I say "suppose ... premise" above is that we have to be very
> careful about "scientific studies" outside of "science", because sometimes
> only the trappings and not the substance of "science" remains. In the above
> case, it looks as though (especially) the NLF did a comprehensive job of
> sampling and concluding.
>
> However, this is not often the case in most studies of educational methods
> and results. And, it is very difficult to separate out and test any method
> from the testing that is also going on of evaluating how well randomly
> chosen teachers in America can teach anything in any style. I would posit
> that trying to do this in single trials is essentially intractable.
>
> I've mentioned before that just validating a piece of our curriculum
> requires three years of doing it with a teacher in a classroom before enough
> of the artifacts and distracters can be nudged out to get even a qualitative
> judgment. (In our case, this has very often been a "no, this is not a good
> way to approach this that will get more than 90% of the children above a
> real fluency threshold" -- i.e. failure.) No-one wants to pay for these
> extra years, so we use our discretionary research budget to support the
> extra costs and time.
>
> A much more important kind of investigation in education is a "transfer
> study", which is all about whether enough of both practical and abstract
> understanding is retained and operationalized enough to be applied by the
> learner in later contexts (both related to the original learning and in
> areas where there could be very fruitful analogies). For example, in the 70s
> Adele Goldberg and I designed an extensive transfer study to see if
> "powerful ideas in programming/active-math" could be foundations for more
> powerful learning and thinking in other fields (we chose major parts of
> Biology). Because of the kind of setups and testing needed, we thought that
> at least 7 years should be devoted to this. For example, there were several
> overlapping three year implementations in the programming/active-math ideas,
> and the kind of testing we had already been doing, and then there would have
> to be another series of these in the later experimental and control classes
> when the children started learning about Biology.
>
> Needless to say NSF turned this down flat, and turned down several
> subsequent requests we made.
>
> However, even if they had funded the study, we realized that it would be
> adding more of the largest problem of doing anything in a school with math
> or science, which is working with teachers who don't remotely understand
> their subjects -- and (even in the case of reading and writing) don't do the
> activities with the children (when was the last time you saw a 5th grade
> teacher assign a composition to the students and then let the students pick
> a topic for them and write an essay along with the students?). (Actually,
> given the excellence of your blog, you might be an exception!)
>
> This, along with many other reasons, is why I don't worry about the "C"
> word or the "D" word, or any other simple scheme. As Marvin Minsky once
> pointed out, every educational method works for some students. This is
> because another deeply important factor is that children in a single
> classroom exhibit a wide variation in motivations, knowledge, skills,
> maturity and "wiring". Different children need different approaches. A
> classroom is a tough place to learn anything (as an orchestra is a tough
> place to learn how to play an instrument). The US factory approach to
> education was hoping for economies of scales via method, but it forgot that
> it wasn't about just turning out Model-T's, but every kind and variation of
> vehicle using every kind and variation of materials and design.
>
> Long (very long) ago I was a professional musician (jazz guitar) and also
> taught guitar for a few years. The basics for musical learning are rather
> similar to sports learning, and they involve rather different approaches and
> mixes of processes than in formal schools. (Of course, they might be so
> different from learning math that no analogies will hold -- but let's
> pretend that they aren't so different.)
>
> The goal in music-sports is fluent playing. It is not known how to do this
> without having the learners undergo a lot of "doing of playing". However,
> there is not a lot of discovery to be done early on that is going to help
> and not hinder later on (i.e. most ideas are mediocre down to bad -- this is
> why good ideas are so rare and precious). But, as Tim Gallwey the great
> tennis teacher says, "The problem with most theories of learning is that the
> parts of your body that need to learn, don't understand English!" Saying it
> a different way, the parts of our mind that do understand natural language,
> aren't often able to do other subjects well. We can see this is also true
> for math and science -- otherwise we could just write the best expositional
> essay on each subject (called "great books") and just get the learners to
> read them! And, imagine how easy it would be to teach teen-agers to drive a
> car! Obviously, other elements are vital.
>
> If we combine a few ideas -- e.g. discovery is really difficult, it's hard
> to learn via language, we have limited capacity for dealing with ideas at
> one time (7+-2 according to George Miller), etc. -- then we can see that
> Jerry Bruner's notion of "scaffolding" starts to come front and center as a
> way to devise strategies for learning sequences. For example, a teacher can
> set things up so that only a few degrees of freedom remain, and now there is
> a much higher chance of actual discovery, or homing in on what is best to
> concentrate on. This is done all the time in music-sports.
>
> For example, Ted Williams introduced the batting tee into professional
> baseball and was pooh-poohed for "silly, unmanly, etc.". But he was the
> greatest hitter of his day (and one of the greatest of all time) so
> gradually others began to surreptitiously practice. His idea was that it was
> almost impossible even getting the muscular feeling and memory for a level
> swing if you are going against a moving target of "round thing against round
> thing". Now the batting tee is found in every training facility for all
> levels of baseball and there is even a league for very young players.
>
> Scaffolding has to be carefully vetted. For example, short skis really seem
> to work for learning beginning skiing, but putting frets on a violin doesn't
> (even though they seem to help in the beginning - then they hurt badly).
> However, "multiperson African Drumming" really does help all aspects of
> music learning, including classical music.
>
> Showing" often helps. If you can't feel the phrasing of a musical sequence,
> sometimes it's just best for the teacher to play various phrasings to be
> judged. Or to get you to watch them serve (the flip side of this is that the
> top tennis pros have rather different strokes and serves -- i.e. personal
> wirings and idiosyncrasies have to be tolerated -- it is very difficult to
> learn exactly what someone else does -- but one can learn "just as well
> though a little differently").
>
> This hurts badly in school when the teachers don't know enough math or
> science to be flexible about perspectives, etc. We would be surprised if our
> music or tennis teacher weren't fluent and refused to play with us (for one
> thing, that's the best way for them to assess where their students are) --
> we would doubtless drop a "non-doing" teacher. But the opposite is
> egregiously true for most school teachers, most are not and have never been
> practitioners. However, we only see a few parents take their kids out of
> public school for such reasons.
>
> We could well imagine that one form of instruction might score better than
> another if teachers are not up to snuff (however, as mentioned above, the
> "better" is not nearly good enough to get the eventual American adults above
> any reasonable threshold). If we are going for "evidence" and "scientific
> evaluation", then we have to include getting to real thresholds, not just
> relative differences. Here, all methods currently fail -- and probably will
> until better conceptions and thresholds are created for teachers.
>
> Gallwey again: "You still have to hit thousands of balls to learn tennis,
> the difference is what you are thinking about and how you are focusing while
> doing".
>
> This is as good a key to progress as any.
>
> An interesting paper by one of your countrymen that Mark Guzdial pointed me
> to (After the Gold Rush: Toward Sustainable Scholarship in Computing, by
> Raymond Lister, University of Technology, Sydney, Australia --
> http://crpit.com/confpapers/CRPITV78Lister.pdf ) shows some of the
> difficulties of dealing with this very complex area. I don't know quite how
> to do justice to a counter argument in a very short space here, but I think
> there are real parallels with what happens with learning programming (he
> gives his POV as a college teacher of programming) to what happens with
> learning music, sports, and even driving a car, if the learners don't do
> enough of the actual processes. For example, he makes the (to me) astounding
> statement that:
>
> I taught a first semester programming subject, where the final exam
> consisted entirely of multiple-choice questions (Lister & Leaney, 2003a&b;
> Lister, 2005). I adopted that style of exam because it was clear to me
> that many students could not write code by the end of first semester, and
> I was tired of setting and marking exams where I pretended that students
> could write code.
>
> They couldn't write code at the end of a semester? Was the course about
> anything else? What kind of grades could he be giving? "Air guitar" grades
> for "air programming"?
>
> Much other of interest will be found herein.
>
> There aren't enough details in the paper to comment on his teaching style
> or to guess why his students couldn't program at the end of a semester.
> (This is not the only such story that has been written up over the years.)
> In some of the latter cases, I knew some of the instructors and they were
> not dunderheads by any means. So we could certainly give Mr. Lister the
> benefit of the doubt, and wonder instead about the processes in his class
> and in universities in general.
>
> Now, if we do the (so far unwarranted) act of substituting music, sports or
> even driving a car, we might guess that the main reason the students were in
> this unhappy state at the end of a semester is because of the pace, depth
> and amount (if not also the nature) of the doing experience.
>
> Another unwarranted comparison is to the way programmers were created in
> the military services in the 50s and 60s. Virtually all participants were
> enlisted personnel without college educations and some without high school.
> Programming was needed, but was not glamorous enough to be within the ken of
> the college educated officers.
>
> Prospects (in this case, the Air Force) were given a short aptitude test
> (about 45 minutes) made up by IBM that essentially assessed interests and
> latent abilities in patterns of various kinds. Only people who got through
> this went to the next state -- which was a one week wall-to-wall (40 hours
> of class plus lots of assignments) of instruction in how to program a
> computer. This was also conducted by IBM, and in my memory was just about
> perfect in the balance of description, advice, examples, and many doings
> with one's own code. (I had similar favorable impressions with the rest of
> the training I got while in the military -- the only thing left out was
> "education", meaning that "theory" was scant -- every other aspect could not
> have been better thought through and presented.)
>
> One hectic week later, one knew the machine code and assembler and could
> write many programs for the real computer that was back on base, and that
> was what we did to other's goals for several months. This was intensive and
> literally "on the job training". One thing that people find unusual today,
> was that not only was there no interactive programming (punched cards were
> submitted for a batch run), but one was allowed a maximum of five minutes
> actual contact each day, not with the machine, but via an operator who ran
> the machine, could punch in addresses, etc. One had one's listing draped
> over the card reader and was kept well away from the console. Basically, the
> only way you could get a program to run was to have it be "almost perfect"
> before testing. This was accomplished via another developed skill called
> "desk checking" (Don Knuth attributes his facility with programming to this
> quaint process as well.)
>
> Then there was another intensive week of wall-to-wall "Advanced
> Programming" in which one learned a little more architecture and how to use
> the extensive macro facility in the assembler, etc. I will only compare the
> first intensive week and month or so which resulted in real programming
> skills to Lister's very different experience in university.
>
> The point here is that the armed services scheme had almost no failures,
> everyone who went through it was successful. The instructors weren't any
> better than the college professors, but the process really was. And the
> goals were very different. There wasn't any class to pass, no multiple
> choice tests to take, no grading on the curve, only a few hours of "lecture"
> (and just when needed), and (no small matter) there was nothing to do but to
> learn programming that "semester". The basic idea here in 1961 (I think) was
> that if you can think a little, then a "summer music camp" approach is the
> best way to really get going on something. If you can't think a little (play
> a musical instrument a little) then you should get across this threshold and
> then go to summer camp.
>
> (Way afield, CMU did something quite similar and very wonderful and
> successful for their incoming CS grad students wrt CS at CMU.)
>
> Again, this successful scheme doesn't necessarily generalize to every
> subject. But it's strong enough to be worth considering in areas where
> "doing skills" are an important part of the subject. (One problem with
> "math" in the US is that it isn't actually "math" but only simple
> calculation skills. This isn't enough to help with actual math thinking
> (which is a special skill all its own that can indeed be taught, but isn't.)
>
> An important aspect of this approach is that it nicely avoids having to
> categorize methods: it is really about a somewhat vague but readily
> understandable approach in which the only real goal is to help the learner
> achieve fluency in "something that is done". We can safely ignore the C and
> D words in favor of the E and T words (or the M and S words).
>

>
> Best wishes,
>
> Alan
>
> ----- Original Message ----
> From: Bill Kerr <billkerr at gmail.com>
> To: Albert Cahalan <acahalan at gmail.com>
> Cc: its.an.education.project at tema.lo-res.org
> Sent: Monday, July 21, 2008 7:08:49 PM
> Subject: Re: [IAEP] reconstructed maths
>
> On Thu, Jul 17, 2008 at 7:16 PM, Albert Cahalan <acahalan at gmail.com>
> wrote:
>
>> Alan Kay writes:
>>
>> > Similarly, from the POV of a former guitar teacher and player,
>> > "playing guitar" has a threshold that excludes "air guitar",
>> > "Guitar Hero", certain kinds of noise, and "not enough fluency
>> > to make music yet". We can use "air guitar" as a metaphor for
>> > (a) taking such a small subset of an activity that only some
>> > form and essentially no important content is being done, and
>> > (b) for using form over content to fool ourselves that we are
>> > "players" and "part of the club".
>>
>> That works. I also like the term "math appreciation".
>> Like a music appreciation course, it doesn't get you
>> to be competent.
>>
>> > Two of Seymour Papert's most important insights about above
>> > threshold math-with-computers for children were to (a) find and
>> > use the real mathematical thinking that children could do at each
>> > stage of development, and (b) to both pick from the large body of
>> > existing mathematics and to invent new mathematics that embody the
>> > most "powerful ideas" that humans have come up with. One of many
>> > such examples is how to use the children's ability to add (and to
>> > think additively) and to physically move their bodies to make for
>> > them a powerful and valid version of Gauss' Differential Geometry
>> > which covers some of the most important parts of vector calculus
>> > in a way easily learnable by children.
>>
>> I do believe that many children can learn vector calculus, and that
>> this might have some value. However...
>>
>> When you put the cart before the horse, trying to skip all of the
>> arithmetic and such, you're teaching math appreciation. It's air math.
>>
>> An actual bake-off has been conducted. It was the largest educational
>> study ever done, covering 79000 children in 180 communities. I'm sure
>> you've heard of it: Project Follow Through.
>>
>> In that study, Direct Instruction (sage on the stage) trounced all
>> other programs in multiple ways. More here:
>>
>> http://www.jefflindsay.com/EducData.shtml
>> http://www.illinoisloop.org/oswegomath.html
>> http://www.heartland.org/Article.cfm?artId=19790
>>
>> That last article has a lovely graph of the results.
>>
>> One may question the ethics of such an experiment, since it does in
>> fact involve experimenting on children in a life-effecting way.
>> I see no alternative though, so I'm glad it was done. Now it is time
>> to accept the results, eat some humble pie as required, and teach
>> children math.
>>
>> It is unethical to endlessly repeat the experiment, hoping that you
>> will somehow get results that support a well-loved hypothesis.
>>
>> Vector calculus is a fine subject for children. Set high expectations
>> for daily progress, eliminate distractions, and soon enough most kids
>> will reach vector calculus. (for real, not vector calculus appreciation)
>>
>> High expectations go something like this:
>>
>> multi-digit add/subtract with traditional procedure: 1st year
>> multiplication table memorized: 1st year
>> long division: 2nd year
>> 2-step word problems: 2nd year
>> 4 basic operations on fractions: 3rd year
>> 4-step word problems: 4th year
>> order of operations: 4th year
>> algebra (with proofs): 5th year
>> geometry (with proofs) and trig: 6th year
>> regular calculus: 7th year
>> vector calculus: 8th year
>>
>> It's doable, but you won't get there if you waste time or if you use
>> educational methods that are proven to be horrible.
>>
>> In case anybody wants to look at current curriculum that work:
>>
>> The two best math programs, unfortunately subject to copyright, are
>> Saxon Math and Singapore Math. Saxon Math is better for the slower
>> students, particularly if students are missing school or transferring
>> in from places that use a different math program. Singapore Math is
>> better for the faster students. Both are available for purchase.
>
>
>
> hi albert,
>
> thanks for raising an issue sharply that does need to be discussed
>
> misunderstandings and misrepresentations aside, I'd raise this point in
> response - your assumption is that a large scale study is more important
> than the individual research and findings of one person
>
> I don't see why this assumption should necessarily be true - ie.
> historically it has been shown many times that lone individuals or small
> groups have turned out to be correct and the predominant or mainstream way
> of doing things has eventually been displaced - that is the nature of
> scientific revolutions
>
> it could be that whole systems have been built and maintained for
> generations on principles of direct instruction - that various challenges to
> this have arisen and been trialled, some good, some  not so good - but
> throughout this process the predominant form of teaching has remained direct
> instruction
>
> it seems to me that in a system that has evolved in that way, that due to
> forces of inertia and group think mainstream studies would tend to show that
> mainstream ways of doing things are the "best way"
>
> Piaget did many studies and wrote many books and papers based on the study
> of 3 children - that does not in itself make him wrong. He might be wrong
> but I can see many advantages of doing in depth studies based on a small
> group.
>
> I would like to discuss this issue more, just raising it here in simple
> form --> minority views are not wrong because they are minority views
>
> Of course your challenge still applies as a practical issue for those who
> want to go beyond direct instruction at least in some respects
>
> cheers,
> - Bill
>
>
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