Logo Vignettes
of Teachable Moments
Wallace Feurzeig |
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| Wally
Feurzeig was the project manager at BBN Technologies
and one of the driving forces, along with
Seymour Papert, behind Logo. Here he recalls
early experiences with the Logo language. |
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| Vignette 1:
I was interested in showing that young children
– seven year olds – could learn
to use Logo to write elementary mathematical
procedures requiring logical thinking. This
was at a time when there was real scepticism
among some Piagetians about whether children
that young could exhibit formal stage thinking.
In 1969, I sought to show that young children
could indeed demonstrate formal logical thinking
skills when given appropriate interactive
language. I worked with second-graders of
average mathematics ability in a traditional
public school in Massachusetts. After introducing
them to the elements of Logo, I turned them
loose on tasks like writing a procedure for
Countdown (counting down integers in steps
of 1 from 10 to 0 followed by “Blastoff!”).
After the first moon shot, kids everywhere
in the US were joyously mouthing this down-counting
sequence as a mantra – it rivalled hula
hoops in popularity! Using Logo, some kids
were able not only to develop a Countdown
procedure but also to extend it in various
ways, for example, to writing a countdown
from 10 in decrements of 3 (this necessitated
finding and fixing a bug which called for
stopping at 0 in the original version) and
to write a procedure for counting down from
an arbitrary integer and then counting back
up to it. |
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| Vignette 2.
In 1972 BBN engineer Paul Wexelblat built
the first wireless Logo-controlled floor turtle,
dubbed Irving. Irving moved freely under Logo
commands via a radio transceiver attached
to a teletype terminal connected to a remote
computer. We were interested in seeing whether
elementary and junior high school students
could control Irving to do simple AI tasks
such as trips involving circumnavigating an
area and avoiding obstacles. In its initial
implementation, Irving lacked touch sensors.
(In fact, we explored using antennas as sensors
before we settled on bumpers.) During this
phase we asked ourselves what kind of AI task
could be designed for a robot without sensory
feedback. On the face of it this seemed impossible.
I responded by designing the path reversal
task (another trip task.) The challenge was
to move Irving from the centre of a room to
an adjoining room where it was no longer visible.
This typically required a sequence of about
twenty move and turn commands. After Irving
was out of view in the adjoining room, the
challenge was to bring Irving back home, to
its starting point in the original room. This
had to be done only through Logo commands,
without leaving the room to peek around the
doorway! The student had a complete record
of the sequence of commands she had used since
each command she had typed was listed on the
console printer. |
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| The task was
fascinating and, for all but the most sophisticated
children, initially quite difficult. The notion
that there is an algorithm for accomplishing
it was not at all obvious. They knew that
Forward and Back were inverse operations,
as are RightTurn and LeftTurn and PenUp and
PenDown. But they didn’t know how to
use this knowledge for reversing Irving’s
path. Even bright students were initially
perplexed. However, once students were asked
how to make Irving undo its last move so as
to get where it had been the step before the
last one, most children had a rapid “Aha”
experience and saw how to complete the entire
path reversal. They found the procedure easy
to implement in Logo. (Some students were
able to generalise the procedure to paths
that included Logo procedures as components,
not just primitive turtle commands.) The algorithm-performing
the inverse operations of the ones that moved
Irving away, but performing them in the reverse
order – is an example of a mathematical
idea of considerable power and simplicity,
and one that has a wide range of application.
The use of the turtle made it accessible to
beginning students. This activity directly
paved the way for developing a procedure for
solving linear equations in algebra. Fifth-grade
students were delighted to find that the algorithm
for solving a linear equation is to so the
inverse of the operations that generated the
equation, but in the reverse order –
the same algorithm as for the path reversal. |
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| As an example, consider
the turtle path reversal problem (Feurzeig
and Lukas, 1972). This problem introduces
elementary students in an engaging way, to
the algorithm for inverting a functional chain.
This is a powerful algorithm – it is
the basis for matrix inversion and the solution
to linear systems. It has important applications
in thermodynamics and system design. It ought
to be given a memorable name, such as “the
theorem of return”. The task is easily
stated and grasped. The student is first asked
to direct the Logo turtle out of the room
and around the corner somewhere so that it
is no longer visible. He is then asked to
bring the turtle back, i.e. to issue the Logo
commands that will return it home. |
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| Students are typically
taken aback by this because they can no longer
see the turtle. How can one control what one
cannot see? (The task thus provides a concrete
metaphor for scientific inference, by introducing
the idea that the mind’s eye can transcend
the limitations of the senses.) The students
have the complete list of the Logo commands
that took the turtle from home to its current
invisible resting place. But even a bright
student often lacks the initia; insight of
how to use this information. However, once
students are asked how they might bring the
turtle back to where it was “the time
before last”, they often get a flash
of understanding. If the last command was
to move the turtle forward some number of
units, they can undo that by moving it backwards
the same amount; if it was to turn right some
number of degrees, they can undo this by turning
the turtle left the same amount. They see
that the complete plan for returning the turtle
home is to undo the sequence of actions that
the turtle took in proceeding to its final
location and that this is accomplished by
performing the opposite actions in the reverse
order. |
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| After they implement the
sequence of commands that return the turtle
home, students are given the task of generalising
their solution. They are asked to write a
Logo program to reverse any path defined by
a sequence of turtle commands. Then, generalising
this, they are asked to describe a plan for
a program that reverses turtle paths generated
by arbitrary Logo programs, comprised not
only of turtle commands but also of Logo programs
that may themselves contain turtle commands.
Recursive generalisation tasks like this occur
naturally in Logo programming work of all
kinds and at all levels. Powerful mathematical
skills are fostered in the course of developing
rich mathematical content. |
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