Re: Programming languages for the very young

From: Brian Mastenbrook (NOSPAMbmastenbNOSPAM_at_cs.indiana.edu)
Date: 01/19/04 

Date: Sun, 18 Jan 2004 22:41:31 -0500

In article <pan.2004.01.19.02.22.39.751055@yahoo.com>, Samuel Walters
<swalters_usenet@yahoo.com> wrote:

> First off, Thank you for writing a well-considered response. I'm glad to
> find someone intelligent of opposing views. One cannot, after all, trust
> one's own views until they are tested against someone who can
> intelligently criticize them.

Agreed. That's what Usenet is for, right? :-)

> Now, I'll put my experience in this matter into context. I have worked
> for many years, with a couple of breaks, as a math tutor at the university
> and community college level. The community college is in rural Florida
> and connected to a magnet high school where they lump the troubled and
> exceptionally bright students together and treat them with a bit more
> respect than is afforded them at public schools. I have helped some with
> these students, and have worked with many students who couldn't add or
> subtract. In a strange coincidence, it seems that almost every girl I've
> dated seriously has been from a family deeply involved in education. This
> has given me some insight into what grade school and middle school
> teachers worry about.

To provide some context myself: Before I went off to grad school, I
tutored algebra at a local middle school. I occasionally tutor someone
on the collegiate level, but in my experience college students have far
different mathematics troubles than those on the middle school level.

> I also spent a good deal of my time during high school assistant coaching
> P.E. and teaching reading to Kindergarten and First grades. I worked
> closely with the head start program. That was, to say the least, an
> eye-opening experience. It's the sort of thing that makes you want to
> side with Communist China and require people to be licensed to have
> children. I won't go that far, but it does make one reconsider the idea.

Perhaps everyone could agree on an alternative - requiring the smart
and capable parents to breed more :-)

> After several times of being bitten by the career bug, I keep finding
> myself back here, living in poverty and loving every minute of it. It's
> time to turn teaching into my career.

Ah, that would be nice, but I have multiple passions, and the level of
headache involved in dealing with teaching - from the required classes
to obtain a teacher's certificate, to the bureaucracy at the school
district, to the required union (does anyone else hate this concept?) -
would just be too high. If we as a nation want well-qualified teachers,
we ought to do everything in our power to ensure that it is pleasant to
go into education, not a major hassle.

> The instructors here have come up with a clever way around this. They
> have two portions of the test, each on a different colored sheet of paper.
> One is sans-calculator. The different paper allows one to see who's
> working on which portion of the test so that the no calculator section is
> enforcible. Trig is just the first place

I think your sentence got chopped off here. My personal experience is
that classes who do mixed calculator / no calculator aren't often as
easy for the students to cope with. Many people simply don't have the
discipline to force themselves to practice without a calculator, and so
when it comes time to take the no-calculator portion of the test, are
unprepared. I think after the initial frustration sets in, most
students can get more out of a class where all tests are no-calculator.
Perhaps there are ways of alternating so that students find it easier
to work practice problems without a calculator.

> I agree that one cannot effectively learn algebra if one is still baffled
> by basic issues of arithmetic. I, however, think that addition,
> subtraction, multiplication and division are much more than skills learned
> by rote.

Of course they are - that's why I used the word "intuitive". Intuition
is more than just rote knowledge; it's having your brain signal you
when looking at a problem and say "hint hint - this might be part of
the puzzle!". But you can't look at a complex problem, and realize that
all you need to do is multiply some lengths and widths and then sum an
area, unless you are so familiar with the concept that it has become
intuitive to you. Watch as you're working with a student, and they come
to a problem that they have no clue where to begin on. How is it that
you know where to start? Is it because you sat and consciously thought
of every similar problem, until you came on one that would help you
with this? Did you have a script? Or did you just kind of know where to
begin investigating? This is intuition in action. It doesn't happen
overnight though, and programs like Everyday Mathematics which
sacrifice practice in favor of introducing bits and pieces of algebra
in the third grade (no joke) quite simply are counterproductive. Of
course, I'm not of the opinion that Everyday Mathematics was designed
to be productive in the first place.

> I actually believe that numeracy *is* intuitive. I think that the proper
> mental models are not being provided. Again, more in my response to Tim
> Hayes.

I haven't read your response yet, but I think we need to clarify our
definitions of intuitive before we go any further. I'm using it to
refer to skills or knowledge which are known so well that they directly
influence how we solve and think about problems. Everyday Mathematics
focuses on developing a "toolbox" of problem solving strategies, and
students actually maintain such a written toolbox. But I carry a
toolbox around in my head that's learned so well that it just comes to
me when I need to solve a problem. It's the result of years of
mathematics practice. (FYI, my undergrad majors were CS and
Mathematics.)

> If it is as you describe it, then I would agree with you. I do not have
> any experience inside any large education college. My experience is
> distinctly small-town. I cannot assert anything about this without more
> research.

If there's a teacher college you can visit, go visit the bookstore and
see what the required class texts are. I think it's a good way to get
an idea of what a program's flavor is.

> I agree with each point here, save perhaps one. That is your definition
> of "philosophically-driven curriculums." This "cargo-cult reading" you
> describe is a poor replacement to phonics if only for breaking the maxim
> that learning is not a spectator sport, if not on more fronts I don't
> intend to dig up at the moment.

But it works on one philosophical front - it does not view the teacher
as a "font of knowledge", as you put it. The drive towards
student-directed learning is a very powerful one in educational
philosophy, and the influence of the free schoolers has been
tremendous. The study of totalitarianism from a constructivist
framework has replaced the study of how to effectively guide student
learning in a classroom.

> I'm not sure our districts are *anything* alike now.

I'm beginning to see that.

> When I see students
> entering college, they have a strong memorization of their times tables,
> but no earthly clue what multiplication means. For instance, I'll be
> damned if I can find more than a handful of freshmen able to tell me why
> multiplying length and width gives us area of a rectangle. The ones I
> *do* find are probably going to be resident aliens or immigrants. I'm not
> asking for deep understanding here. I just want a student to understand
> that counting by five is the same as the fives table in a multiplication
> table. It's truly baffling how little understanding they possess.

If I had the opportunity to trade students and situations, I probably
would. If anyone wants to understand where the drive towards grade
inflation comes from, just try telling a seventh or eighth grader that
they need to practice their addition and subtraction. It's hard. When
you watch them counting on their fingers when trying to solve a linear
equation it breaks your heart. I agree that we need to make sure our
students understand the actual meaning of what they're doing, but I'm
not sure how to translate that understanding into the use of the tool
when they haven't practiced it enough to recognize when they need it.

> It seems you were using a technical term whose mundane meaning was coopted
> by a theory. I was reading it as the colloquial sense. It is obvious that
> you are far better read on theories of education than I am. I hope this
> won't diminish your view of my ideas. :-P

Of course not. It just means I need to try to communicate better myself.

> Just beware, if you use a technical term with a common usage, I'll
> probably read it with the common meaning. Hopefully, though, I will spot
> those landmines of miscommunication since I know they're there now.

If I'm using a new term in quotes, it's probably for that reason.

> To this, and the last point: I think that there is reasonably more
> exploration to addition and subtraction than it first appears. This
> assertion will be elaborated, you guessed it, in the next post.

I think that when I write a reply to someone, I have a tendency to yank
in one direction as hard as I can. I don't mean to do that. There is
much wonder and excitement in developing an intuitive understanding of
mathematics, including understanding how to break more difficult
problems down into easier ones and recognizing patterns. One of my
idols, the late great Richard Feynman, was a master at this. His mental
arithmetic could beat a master of the abacus. It worked because he was
so familiar with numbers, that he could instantly decompose harder
problems into ones he already knew. But just telling someone how to do
this, and giving them a few examples to practice on, and requiring them
to write out the method they used from start to finish on every
problem, just hinders the practice it takes to become truly familiar
with the world of numbers.

> *chuckle* Well, that certainly is a colorful way of putting it. And here
> I thought that explaining to statistics students the reason statistics was
> required "is not so that you could find the standard deviation of a data
> set ten years down the road, but so you can call 'bullshit' when a
> consultant tries to pull the wool over your eyes with bogus statistics," was a
> brazen thing to say.

This is Usenet, is it not? :-)

> All I have to say is that there are effective ways to conduct peer review
> and there are horrifyingly poor ways to conduct it. While I support the
> former, sadly bureaucracy favors the latter.

The world would be a much better place if bureaucracy didn't exist, and
people of all professions were truly excited about what they were
doing. Needless to say I don't think that's going to happen anytime
soon. Unfortunately a lot of teachers I have met are not truly
passionate about the material they are teaching - if anything, they
have a bit of trouble with it themselves. I can't say I trust people
like this to review their efficacy in teaching it. It takes people who
are truly passionate about it to spot the problems.

> I agree wholeheartedly with your point that standardized testing loses all
> effectiveness when tied to funding. In Florida right now, there's a
> disturbing system that ties school funding, individual teachers jobs. The
> problem comes from two fronts. One is that the large number of senior
> citizens will never vote for an increase in funding for schools. The
> other is our "education governor." You know him as Dubya's brother, Jeb
> Bush. Big headlines came recently when Bush announced an 8% increase in
> funding for Community Colleges. After the 25% cutback at the beginning of
> his last term, an 8% increase still leaves us with a 19% deficit. It
> seems he's not only hoping to keep the children from learning arithmetic,
> but he's hoping that the general population is bad at it too. (I leave it
> as an exercise to the reader to connect this with the last presidential
> election.)

You forgot the effects of inflation / deflation, but I'll let that one
slide. Yes, I do think that the inability of most of the populace to
understand the electoral college system is sad.

> I despise the overkill manner in which mathematics has moved toward an
> emphasis on "applications." Mathematics involves a visceral, visual and
> tactile understanding that these methods obscure. Strange, you'd think
> that "applied" problems would be more grounded in experience, but they're
> not.

I can't comment on tactile understanding - I'm nothing like a tactile
learner. Most of the time I'm easily frustrated by objects anyway. My
brain does not seem wired up to deal with reality well :-)

The joy of mathematics is that by teaching an abstract skill, we can
actually teach critical thinking and problem solving. The skills
learned in mathematics leave their mark on the young person's mind, and
will encourage them to see all sorts of situations in new ways. In
effect, teaching mathematics makes people smarter. But it takes a level
of general familiarity with mathematics - an intuitive understanding of
what to do when looking at a problem, so deep that the right method of
inquiry can jump out at you - to realize these benefits. I'm a big
believer in the concept of intuition; I use it to guide my work in AI
as well. Intuition is learned, it's dynamic, and it's wonderful because
of the way it helps us solve problems, predict behaviors, and generally
allow us to think in higher-order ways about complex systems. My fear
is that these programs are destroying that in students for all sorts of
strange reasons.

> The true utility of mathematics is that it allows us to take small
> intuitive steps to discover surprising and often counter-intuitive truths.

If I could force a summary, it enables us to direct our conscious
thought in new ways, perhaps arriving at initially counterintuitive
truths. But how can you do that when you lack the intuition in the
first place? When you've been taught by a program that insists you have
to make a conscious decision at every step of solving a problem? And
you can't ever act on a hunch?

> Ah! Now you connect the dots. I can see why you were talking about
> fascism. I advocate computer programming as a required high school
> course. Perhaps there are bigger fish to fry at the moment, but an
> understanding of the nature of the beast is imperative for students to
> function in any future society. I'm not talking about web-surfing or
> using a word processor, though those are useful skills. I'm talking about
> the tools necessary to build a mental model of what a computer does. My
> personal political beliefs see the Internet as either the future
> choke-point for all information, or the liberating conduit that breaks the
> bonds of political spinsmen and double-speak. Education is the key to
> the outcome, as we both agree.

Believe it or not, every day I find out more information that's not on
the internet. It's out there! I just hope it won't be relegated to
inconsequence.

I think we can agree that teaching an understanding of the essential
nature of a computer is vastly important to the future of our society.
Of all the recent uses for technology in education, however, I have not
seen "programming" being one of them. That's sad. Programming is a
terrific way to teach mathematics, even on the elementary level. My
elementary math classroom had an Apple IIe, and the teacher would allow
me to program on it, and use it in solving problems. It was fun, and it
was another way to build my intuition - because when you program, you
have to have an idea of what statements do what, and what the
(intended) outcome of your program is. If I had my way, I'd throw out
all the iBooks and replace them with Apple IIes. (And I'd teach the
students to type, too. They don't do that anymore. Sad - it's an
essential motor skill that should be practiced early.)

> And my point was that the stewards of knowledge in this area are incapable
> of effectively educating students because they themselves do not
> understand that which they claim to teach.

If only it were easier for a specialist in a field to obtain a teaching
certificate, so us programmers could do this part-time. In our state,
two years of a teaching program is required for our alternative
certificates, at the insistence of the union - what was once a way for
a highly educated person to easily teach is now just another vehicle
for selling teacher training programs.

> I haven't read any of his other works.

I had the good fortune to see him talk recently. His application of
psychology to politics was fascinating, if a bit narrow-minded (or
two-sided). But I reserve judgment until science has caught up with
him.

> I found the first half of "WMCF"
> solidified a lot of ideas I had a subconscious grasp on. The ideas have
> proven very useful in my day-to-day experience. In the second half of the
> book, Lakoff and Nunez dig into some areas of mathematics that are clearly
> over their heads. This makes their assertions interesting, but not
> compelling. Overall, I felt the first half was well thought out and well
> researched. The second half seemed cobbled together at the last minute
> from loose notes. YMMV.

Well, if I chance upon a copy, I'll pick it up, but that doesn't sound
like a ringing endorsement. :-)

If you haven't already, you desperately need to buy this book:

http://www.amazon.com/exec/obidos/tg/detail/-/0393316041/qid=1074483340/
sr=1-1/ref=sr_1_1/103-3501597-4596624?v=glance&s=books

I offer every section of it a ringing endorsement, but the sections on
intuitive / mental mathematics and reviewing primary school textbooks
are probably the most relevant.

> P.S. I did not intend this as a "me-too" post, but I felt it was
> important to map out the areas we agreed on so that we can focus in on
> those where we disagree.

*Sigh* Only on Usenet. You do realize that most people would rather map
out the areas that they disagree on so they can focus on the areas
where they agree? :-) 

-- 
Brian Mastenbrook
http://www.cs.indiana.edu/~bmastenb/

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