The Flip Side of Logo Maths

Originally published in the September 2000 issue of Australia’s Hotsource online newsletter

LogoWriter and MicroWorlds have done so much for interdisciplinary projects that it is useful to remember that MicroWorlds can play a major role in the development of mathematical knowledge. This issue and next will explore the numerical side of MicroWorlds.

First the Boring Stuff
MicroWorlds procedures come in two categories, commands and procedures. Most Logo-users are quite comfortable with commands such as CG, FD, RT and SETC. Commands may or may not take inputs and they always produce an action. Every Logo expression (line of code) must begin with a command. This is why typing HEADING in the command centre produces the error message I don’t know what to do with HEADING. SHOW HEADING, FD HEADING, RT HEADING * 2 will all work because HEADING reports the turtle’s current orientation and hopes a command is listening. Commands may have any number of hoppers, but they never have a spout. REPEAT is an example of a two input (hopper) command.

Every one input command beginning with the prefix, SET, has a corresponding reporter with no inputs. For example:

At the core of it all
Reporters are procedures that may or may not take an input, but they always output a result. Reporters are also known as functions or operations. Reporters are absolutely essential for most mathematical and interactive MicroWorlds projects. They pass information that can be used by other procedures or turtles. Reporters may have any number of hoppers, but they always have just one spout.

It’s your call
You can write your own reporters if you remember one simple rule. Every reporter procedure contains one output. When Logo encounters the OUTPUT reporter, the procedure is terminated. To create a new reporter you need to remember the rule about OUTPUT and decide how many inputs the reporter needs. For example, if we wanted to write a procedure to double a number, we would only need one input.

to double :number
output :number * 2
end

or

to double :number
output :number + :number
end

Type: DOUBLE 45 in the command centre and see what happens? Why did you receive an error message?

Many people who wish to double a number would write the following procedure.

To dumb.double :number
show :number * 2
end

Then if they type, DUMB.DOUBLE 45 in the command centre they will get what they think is the desired result. This is the result they need only if they want to see the number 90 appear in the command centre.

Try typing the following instructions in the command centre:
FD DUMB.DOUBLE
45 DUMB.DOUBLE DUMB.DOUBLE 45

Now try typing:

SHOW DOUBLE 45
FD DOUBLE 45 RT DOUBLE 45
SHOW DOUBLE DOUBLE DOUBLE 45

Our DOUBLE procedure is much more flexible and versatile than DUMB.DOUBLE.

They can speak to each other
Reporters can perform a manipulation/operation on an input and then report that result to another reporter. Logo (MicroWorlds) reads reporters from right to left since you can’t type from top to bottom. The following graphic illustrates FD ADD5 DOUBLE DOUBLE 5.

Logo is a prefix language. That means that inputs always follow the procedures. Since humans like the standard arithmetic operators (+-*/), Logo will tolerate them, but often requires parentheses for grouping. These infix reporters tend to give the turtle indigestion. Logo much prefers PRODUCT 3 4 to 3 * 4. See how SHOW DOUBLE DOUBLE 3 + 4 behaves if you add parentheses, like SHOW (DOUBLE DOUBLE 3) + 4.

Make it simple
Young children can use similar simple arithmetic reporters to leverage their own turtle graphics. For example, a child incapable of calculating twice the distance for the turtle travel could use a DOUBLE or TWICE reporter and operate algorithmically. These procedures could be written by a teacher ahead of time or by the student herself.

Operation of fractions may also be explored with simple reporters.

To 3fourths :number
output :number * 3 / 4
end

to 2thirds :number
output :number * 2 / 3
end

to 1half :number
output :number / 2
end

To “play with” multiplication of fractions, try typing:
SHOW 3fourths 100
SHOW 1half 100
SHOW 2thirds 3fourths 100

You may of course use these fractional reporters to command the turtle. Type the following BAR procedure on the procedure page.

To bar :height
pd repeat 2 [fd :height rt 90 FD 25 rt 90]
pu rt 90 FD 35 lt 90
end

See what happens if you type the following in the command centre.

BAR 100
BAR 3fourths
100 BAR 1half 100
BAR 2thirds 3fourths 100

Battle of the Functions
You can make a game out of all these arithmetic reporters. Put kids in groups of four or five and have them each contribute one new arithmetic procedure in the style of DOUBLE. They may use their own imprecise names for the reporters if they wish (as long as they can explain its function to their peers). Each kid takes turns inventing a number problem consisting of stacked-up reporters and one numerical input. The object of the game is to invent a problem that is difficult, but not impossible to solve in one’s head. Wiseguys are penalized by the rules of the game.

Gary Stager

Veteran educator Gary Stager, Ph.D. is the author of Twenty Things to Do with a Computer – Forward 50, co-author of Invent To Learn — Making, Tinkering, and Engineering in the Classroom, publisher at Constructing Modern Knowledge Press, and the founder of the Constructing Modern Knowledge summer institute. He led professional development in the world’s first 1:1 laptop schools thirty years ago and designed one of the oldest online graduate school programs. Gary is also the curator of The Seymour Papert archives at DailyPapert.com. Learn more about Gary here.