[Sur] los lenguajes de programación dificultan el aprendizaje de algebra
Daniel Ajoy
da.ajoy en gmail.com
Sab Mar 13 13:31:20 EST 2010
interesantes ideas sobre porqué los lenguajes de programación procedimientales (TurtleArt) dificultan el aprendizaje de algebra.
(Las cuales no comparto)
Digest from
http://mail.geneseo.edu/pipermail/math-thinking-l/2010-February/date.html
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Date: Fri, 19 Feb 2010 13:06:06 -0800
From: kirby.urner at gmail.com (kirby urner)
Subject: "No time in my schedule for Computer Science"
On Fri, Feb 19, 2010 at 5:48 AM, Shriram Krishnamurthi <sk at cs.brown.edu> wrote:
<< SNIP >>
> The responses to our article mostly missed the central point, which is
> that current curricula do not have room to accommodate computer
> science (as we understand it), and the only CS programs that are going
> to succeed are those that tie to critical skills that students are
> tested on -- in particular, math. ?This ties back to using programming
> and mathematics to reinforce one another (ideally without using
> "variables" to mean two different things, ahem).
>
Still trying to get clear, in part so I might report back to potential funders
and sponsors...
>>> turtle = Turtle( )
>>> turtle.forward ( 10 )
>>> turtle.pos
(10, 0)
The above would be a case of a mutable variable (turtle is the name of
an object that changes state through its various methods).
As such, it would be inappropriate in math class, anti-mathematical.
We know from Seymour Papert that he doesn't have a problem with an
"explicit receiver" i.e. sending messages to a turtle object ala Smalltalk
makes sense to him, even though the original LOGO turtle was
contextual, i.e. just say FORWARD 10, and the turtle would move.
With an explicit receiver, you may have multiple turtles.
Alan Kay mentioned this was Papert's view at our workshop in London,
hosted by Shuttleworth Foundation. This plan to fold computer topics
into basic math was already well along in South Africa. The USA is not
a market leader on this one, except maybe in some niches.
So projects like Turtle Art by the One Laptop per Child group, other
such initiatives, are considered unworthy and ruinous to the mathematical
comprehension of children, including middle and high schoolers.
Named objects of any kind, that change state, are bad. The way
OO people think is wrong.
Is this too simple a characterization? I want to be somewhat accurate
in my reports.
Using for-loops to explain what Sigma means is waaay too deep,
requires a math degree right off the bat.
I've already reported on that one.
...
Kirby
------------------------------
Date: Fri, 19 Feb 2010 20:21:10 -0500
From: schanzer at bootstrapworld.org (Emmanuel Schanzer)
Subject: "No time in my schedule for Computer Science"
On Feb 19, 2010, at 4:06 PM, kirby urner wrote:
> So projects like Turtle Art by the One Laptop per Child group, other
> such initiatives, are considered unworthy and ruinous to the mathematical
> comprehension of children, including middle and high schoolers.
...
> Is this too simple a characterization? I want to be somewhat accurate
> in my reports.
Speaking only as someone who has taught algebra to actual middle and high schoolers, I can honestly say that the Turtle Art project IS ruinous to children who are approaching algebraic functions and variables for the first time. I work with kids in the "behind" zip codes, and I spend an enormous amount of time and energy working to undo some of the bad *procedural* habits kids have picked up from earlier grades. Reinforcing those habits with programming that involves mutable variables and procedures makes my job much, much harder.
(FWIW: If I could go back in time and teach all of these kids arithmetic without PEMDAS and infix notation, I would weep with joy)
Emmanuel
------------------------------
Date: Fri, 19 Feb 2010 18:09:36 -0800
From: joshua.zucker at gmail.com (Joshua Zucker)
Subject: "No time in my schedule for Computer Science"
On Fri, Feb 19, 2010 at 5:21 PM, Emmanuel Schanzer
<schanzer at bootstrapworld.org> wrote:
> Speaking only as someone who has taught algebra to actual middle and high schoolers, I can honestly say that the Turtle Art project IS ruinous to children who are approaching algebraic functions and variables for the first time. I work with kids in the "behind" zip codes, and I spend an enormous amount of time and energy working to undo some of the bad *procedural* habits kids have picked up from earlier grades. Reinforcing those habits with programming that involves mutable variables and procedures makes my job much, much harder.
>
> (FWIW: If I could go back in time and teach all of these kids arithmetic without PEMDAS and infix notation, I would weep with joy)
Yeah, one of the big misconceptions I run into is "=" as an
imperative, meaning "evaluate the expression on the right!", as
opposed to as a relation.
--Joshua Zucker
------------------------------
Date: Fri, 19 Feb 2010 23:33:15 -0500
From: droujkova at gmail.com (Maria Droujkova)
Subject: "No time in my schedule for Computer Science"
On Fri, Feb 19, 2010 at 8:21 PM, Emmanuel Schanzer <
schanzer at bootstrapworld.org> wrote:
>
>
> (FWIW: If I could go back in time and teach all of these kids arithmetic
> without PEMDAS and infix notation, I would weep with joy)
>
> Emmanuel
>
I have a feeling it's something I should know already, but I will ask
anyway: what is bad about infix?
Cheers,
Maria Droujkova
------------------------------
Date: Sat, 20 Feb 2010 00:04:37 -0500
From: schanzer at bootstrapworld.org (Emmanuel Schanzer)
Subject: "No time in my schedule for Computer Science"
> I have a feeling it's something I should know already, but I will ask anyway: what is bad about infix?
Well, infix isn't bad...but as an algebra teacher, I find it painfully counterproductive. My objections boil down to two things:
1) It's ambiguous! What operations comes first in "4+3*5"? Who knows? To fix this, we have to invent arbitrary precedence rules like PEMDAS, which have nothing to do with the underlying mathematics involved. I might be okay with this if it were taught as a lesson in ambiguity and notation...but I've never heard of that happening. Instead, it's just this list of rules that kids need to memorize in order to "do math". And then kids starting thinking this IS math.
2) It's incredibly difficult to get kids to re-conceptualize mathematics as properties involving function objects. They're not used to the idea of a mathematical object that consumes inputs and produces an output, or one that can be composed with other objects. They freak out when they see f(g(x)), or z(x, y). And of course, "f(x)" happens to be an entirely new piece of notation that kids need to learn. I see kids struggle with these issues all the time, even though they've actually been working with function composition and multi-input functions since elementary school:
4+3 can be thought of as ADD(4, 3). This notation matches "f(x)" perfectly.
4+3*5 can be rewritten as ADD(4, MULTIPLY(3, 5)). This is function composition, plain and simple.
I know we're not going to give up infix notation in elementary school, but I think that infix creates problems for kids that algebra classes don't address. The first step towards recovery is admitting you have a problem -- and when I started devoting time to helping my students make this transition I've made that time back a hundredfold.
Emmanuel
------------------------------
Date: Sat, 20 Feb 2010 00:27:28 -0800
From: kirby.urner at gmail.com (kirby urner)
Subject: "No time in my schedule for Computer Science"
On Fri, Feb 19, 2010 at 5:21 PM, Emmanuel Schanzer
<schanzer at bootstrapworld.org> wrote:
> Speaking only as someone who has taught algebra to actual middle and high schoolers, I can honestly say that the Turtle Art project IS ruinous to children who are approaching algebraic functions and variables for the first time. I work with kids in the "behind" zip codes, and I spend an enormous amount of time and energy working to undo some of the bad *procedural* habits kids have picked up from earlier grades. Reinforcing those habits with programming that involves mutable variables and procedures makes my job much, much harder.
>
OK Emmanuel, thanks for the feedback.
I have not used the OLPC Turtle Art project at work, have not had the
privilege of using an XO in any classroom context, even though I have
two of them.
Ed Cherlin on edu-sig is a contributor to that project.
Gregor Lindl in Vienna maintains the turtle module for the Python
standard library.
I'm grateful to have publicly accessible archives where these
viewpoints are considered by experts.
Many educators have not stopped to consider that Turtle Art might be
harmful to those attempting to learn mathematics.
In the curriculum I favor, a turtle is sometimes used to draw plane
nets for geometric shapes we fold up.
Controlling the turtle requires knowing about the angles, edge lengths
and so forth.
Using a turtle is just one skill among many.
Controlling robots and/or devices is simply part of what we do.
For example we have this T-module.
60 left handed and 60 right handed T-modules assemble into a rhombic
triacontahedron (where the T comes from, not Turtle).
> (FWIW: If I could go back in time and teach all of these kids arithmetic without PEMDAS and infix notation, I would weep with joy)
>
> Emmanuel
>
For those who don't know what PEMDAS means, here's a link:
http://www.purplemath.com/modules/orderops.htm
Thanks again for sharing your viewpoint on a publicly archived list.
Many subcultures have input into K12 mathematics. Airplane pilots for example.
Intelligent policy / funding doesn't happen in a vacuum and journals
are somewhat slow to turn things around.
Time is of the essence these days. Students shouldn't have to wait
for their new kind of math class.
We should already be airing some recruiting commercials, don't you
think? For their sake.
We move forward more quickly when we have shared access to the
relevant discussions.
Kirby
------------------------------
Date: Sat, 20 Feb 2010 09:38:07 -0500
From: schanzer at bootstrapworld.org (Emmanuel Schanzer)
Subject: "No time in my schedule for Computer Science"
> Many educators have not stopped to consider that Turtle Art might be
> harmful to those attempting to learn mathematics.
Just to be really clear about my viewpoint here: I think it's harmful to the learning of Algebra.
> In the curriculum I favor, a turtle is sometimes used to draw plane
> nets for geometric shapes we fold up.
> Controlling the turtle requires knowing about the angles, edge lengths
> and so forth.
Yes, I can totally see this being great for Geometry.
> We should already be airing some recruiting commercials, don't you
> think? For their sake.
I think the key is to start identifying the areas where CS and Math overlap, based on real cognitive science research. Far too many people have naively asserted that CS == Math, and give programming a bad name in the eyes of math teachers. Turtles might be great a geometry class, but don't belong in algebra IMHO. Functional programming has its place to, and its limits.
E
------------------------------
Date: Sun, 21 Feb 2010 07:44:29 -0500
From: droujkova at gmail.com (Maria Droujkova)
Subject: "No time in my schedule for Computer Science"
On Fri, Feb 19, 2010 at 9:09 PM, Joshua Zucker <joshua.zucker at gmail.com>wrote:
> On Fri, Feb 19, 2010 at 5:21 PM, Emmanuel Schanzer
> <schanzer at bootstrapworld.org> wrote:
> > Speaking only as someone who has taught algebra to actual middle and high
> schoolers, I can honestly say that the Turtle Art project IS ruinous to
> children who are approaching algebraic functions and variables for the first
> time. I work with kids in the "behind" zip codes, and I spend an enormous
> amount of time and energy working to undo some of the bad *procedural*
> habits kids have picked up from earlier grades. Reinforcing those habits
> with programming that involves mutable variables and procedures makes my job
> much, much harder.
> >
> > (FWIW: If I could go back in time and teach all of these kids arithmetic
> without PEMDAS and infix notation, I would weep with joy)
>
> Yeah, one of the big misconceptions I run into is "=" as an
> imperative, meaning "evaluate the expression on the right!", as
> opposed to as a relation.
>
Do you see kids doing things like:
23+5=28+10=38-7=31...
It drives me nuts!
However, there was a Keith Devlin's column, "What is conceptual
understanding?" where he makes fun of the "balance" model for equations,
especially weights and balloons, and to this day I have no idea why:
http://www.maa.org/devlin/devlin_09_07.html
Here is what he wrote: "I'll give you one example of just how bad school
textbooks can be. I was visiting some leading math ed specialists in
Vancouver a few months ago, and we got to talking about elementary school
textbooks. One of the math ed folks explained to me that teachers often
explain whole number equations by asking the pupils to imagine objects
placed on either side of a balance. Add equal numbers to both sides of an
already balanced pairing and the balance is maintained, she explained. The
problem then is how do you handle subtraction, including cases where the
result is negative? I jumped in with what I thought was an amusing quip.
"Well," I said with a huge grin, "you could always ask the children to
imagine helium balloons attached to either side!" At which point my math ed
colleagues told me the awful truth. "That's exactly how many elementary
school textbooks do it," one said. Seeing my incredulity, another added,
"They actually have diagrams with colored helium balloons gaily floating
above balances." "Now you know what we are up against," chimed in a third. I
did indeed."
Cheers,
Maria Droujkova
http://www.naturalmath.com
Make math your own, to make your own math.
------------------------------
End of Digest
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