from pattern-block play to logo programming

From Pattern-Block Play to Logo ProgrammingAuthor(s): John E. NewtonSource: The Arithmetic Teacher, Vol. 35, No. 9 (May 1988), pp. 6-9Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41193404 .

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From Pattern-Block Play to Logo Programming

By John E. Newton

Pattern blocks are the most seduc- tive materials in my mathematics lab. All I need to do is dump them in front of children and interesting things hap- pen. Ideas are explored, enhanced, and transformed until satiation; then other ideas arise. Children seem to have a continuous source of inspira- tion. My role changes from curmud- geonly taskmaster to welcomed facilitator.

This article describes some devel- opmental changes in children's free creations with pattern blocks and demonstrates how to build on this natural interest to teach Logo programming.

Spontaneous Creations When allowed to play with the pattern blocks, preschool and primary-grade children tend to make pictures or symbolize action (see fig. 1). I have seen trucks driving over hills, dragons breathing fire, forests with identical trees, and a Japanese garden. A group of third-grade girls treated them as building blocks and made three-di- mensional monsters. (This is mathematics class!)

At about the age of eight, children begin making abstract designs. Older children do this almost exclusively and rarely make pictures. The abstract works are either tessellations or symmetrical designs (see fig. 2). Ra- dial symmetry is more common than bilateral symmetry.

Pattern blocks create a dynamic so-

John Newton is a remedial mathematics spe- cialist in grades K-12 for Central Schools, Warsaw, NY 14569. His teaching is based on problem solving and manipulative s .

Fig. 1 Students' pattern-block pictures

6 Arithmetic Teacher



ciai environment. Because of the lim- ited number of blocks, children exper- iment with various means of acquisition, including collaborating, sharing, trading, flattering, and fight- ing. Mathematics is indeed part of the real world.

Nobody better wreck this

Young children are perfectly happy making something and then destroying it so something else can be made. By the age of seven, children are interested in saving their creations. At this stage a facilitator can suggest various means of preservation.

Elementary-grade children want their work preserved "as is." "No- body better wreck this." "This better be here tomorrow when I come." Of course, when the next day arrives, they delight in destroying any extant design that would limit their play.

Older children, however, want some lasting memory of what was created. Some transfer their designs to paper by using pattern-block templates, such as those available from Creative Publications. They carefully line up the next shape to be traced by referring to their original design. They often color the drawing to match the original design when they are done. Besides requiring more precision and persistence than many students want to devote, the templates give results that are not as pleasing as the actual pattern blocks.

Some designs are preserved as pho- tographs, but two major problems arise with this approach: it is too easy and too expensive. More can be learned if greater effort is expended in preserving the designs. That's where Logo is valuable.

Logo Programming Reproducing a pattern-block design with Logo practically compels top- down programming because the final product is the whole design, yet the discrete parts are readily discernible. Top-down programming is a hierar- chical method in which the largest goal is defined first, then the subgoals to support it, then the sub-subgoals, and so forth, ending with the writing of code for the simplest ones. Stu-

Fig. 2 Tessellation, radial and bilateral symmetry

Radial symmetry Bilateral symmetry (symmetrical around a center)

Tessellation

May 1988 7



dents can quickly see the importance of writing code, a short program, for the individual pattern pieces. They want the computer to draw an isomorphic representation of the blocks that they used to produce their design.

As students try to program a spe- cific piece, they discover how the piece can be reduced to its component angles and sides. Much of the code is written at this level, using Logo prim- itives like FD, RT, and REPEAT. See figure 3 for an example.

These primitive commands are then organized as a Logo procedure and given an assigned name, the first actual programming. Once the programs for the individual pieces are done, they can be linked into more complex procedures. This linking of the parts constitutes a major programming task. More and more procedures are built and linked as the students work up the programming hierarchy toward the goal of the completed design.

Degrees = days Although many preschool children can recognize, name, and draw the basic shapes, programming these shapes is a daunting task even for junior high school students. The problem arises because the essence or Pla- tonic idea of the shapes, the lengths and their angular relationships, must be abstracted from the real objects. These abstractions are then translated into the Logo programming language. Elementary school children are quite familiar with length, but an extensive lesson on angles is usually needed. Understanding how angles are mea- sured is difficult, so this topic should not be rushed.

One approach is to set a model of the sun, such as a light bulb, in the center of a circle of children. Each student represents a place in the earth's orbit. As the children pass a globe around, explain that it takes 365 days for the earth to circle the sun and return to the same location in space.

Next, each child can tie a string near the sun. The strings can show the angle made for two consecutive daily positions of the earth (see fig. 4). One degree is defined as the amount of sweep, or turn, that the earth makes

Fig. 3 Student's flowchart and Logo procedure !

LL

s. FD 40 (First command)

! LT60 ^sj Alt 120

i Start хчх

TO TRAPEZOID (Resulting procedure) FD40LT120 FD 20 LT 60 FD 20 LT 60 FD20LT120

END

Fig. 4 How degrees are derived

Earth's position (day 2) О

/ О Earth's position / / (day1)

/ 7 Day's change = 1 degree

э Sun

8 Arithmetic Teacher



each day from the viewpoint of the sun. (Don't worry about elliptical orbits.)

Ancient people in Egypt and Babylonia discovered the same important idea a few thousand years ago (Asimov 1965). They studied the rela- tionship of the earth to the stars and found that a cycle for the earth around the sun took 365 days.

The Babylonians tried using the number 365 to divide circles and found that it was not a very friendly number, since its only factors were 1 , 5, 73, and 365. They decided to "cheat" and use 360 because its factors were 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, and 360. Therefore, circles are cut into 360 pieces.

Programming the shapes Seymour Papert (1982), the developer of Logo, has stressed the importance of being physically involved with geometry. One approach is to have students create the shape by walking it out, something they tend to do any- way. A protractor can be introduced as one way to measure the degrees, or amount of turn, at the corners of the walked shape. We use AngleMaster protractors, available from Creative Publications. Each protractor is a full circle and has an arrow that can be turned to the desired direction. The angle of turn is then represented on the protractor.

Walking a shape only approximates the angles, and students will quickly prefer the more precise method of measuring angles directly from pattern blocks. They will need guidance in distinguishing exterior from interior angles.

The rough programming code for the shape can then be completely written before the computer is turned on. Thinking before acting is an important habit to develop.

Figure 3 shows the primitive flowcharting my students use. First they draw the shape and then start walking it. The walking helps them to figure out the direction and degrees of the turn. When they finish a turn, they write the Logo command at the proper place on their drawings. The exact angles are determined through

debugging after the program is run on the computer. The lengths are usually guessed at and refined when the program is run. With a flowcharting pro- cess directly tied to physical move- ment, the children quickly learn concepts like complementary and supplementary angles, parallel and equivalent lengths, and the "total turtle trip" of 360°.

Linking shapes Once the shapes are programmed and debugged, students can begin linking them. Again, it helps to walk out the shapes and then decide where the turtle has to be and what orientation it requires in order to start the next shape. These linking steps can then be written as code.

Standardization quickly becomes important. Students will recognize that the sides of the various shapes must have equal lengths or multiples of the lengths if they are to fit together neatly. They also begin to understand why the various shapes have either equal or complementary angles. The basic steps in sophisticated programming have been taken. The harmony of geometry that the ancient Greeks revered is expressed by children to- day as "truly awesome!"

Color printing of Logo The design is saved with the standard SAVEPICT command when using MIT/Terrapin Logo. The Koala Graphics Exhibitor is then booted and used to print the picture. It accepts a Logo file with no modifications. We use Apple lie or Ilgs computers with an Imagewriter II printer and needed an updated version of Graphics Ex- hibitor that Koala supplied at no charge with an exchange.

Using the Koala disk allows colors to be altered without changing the original program. Before printing from the Graphics Exhibitor, the color bar can be chosen and different colors swapped for the originals.

One program enhancement is fre- quently desired: the single lines of color are often not saturated enough for the student when the design is printed. The original program could be altered by students so that slightly smaller shapes are drawn inside the

original shape, thereby doubling or tripling the thickness of the lines, but doing this is difficult. The saturated colors of the computer monitor re- main the preferred medium.

Extensions Students readily create new design and programming problems for them- selves and then solve the problems. They enhance their original designs, create decorations for report covers, or write books about how it was all done.

Students also begin thinking in a new way. As their ideas become tan- gible, they are better able to think about those ideas. Thinking about their thinking, metacognition, is an important problem-solving skill to develop. This skill leads to the level of formal operations described by Piaget.

Summary My students had been familiar with Logo, but I was disappointed in their lack of persistence and imagination. They had been entrenched members of the RT 10, FD 6000 club, using any large number to fill the screen quickly.

Pattern blocks are a natural medium for stimulating spatial and geometric thinking. The blocks seem to resonate with children's ideas. With Logo, those same ideas can appear in bright colors on a computer screen or print- out.

The abstract, formal notions of mathematics and programming grow quite naturally when rooted in the fertile ground of manipulatives. Piaget spent years trying to teach us this.

Bibliography Asimov, Issac. The Clock We Live On. New

York: Harper & Row, 1965. Billstein, Rick, and Johnny W. Lott. "The

Turtle Deserves a Star." Arithmetic Teacher 33 (March 1986): 14 -16.

Clithero, Dale. "Learning with Logo 'Instant- ly.'

" Arithmetic Teacher 34 (January 1987): 12-15.

Papert, Seymour. Mindstorms: Children, Com- puters, and Powerful Ideas. New York: Ba- sic Books, 1982.

Thompson, Charles S., and John Van de Walle. "Let's Do It: Patterns and Geometry with Logo." Arithmetic Teacher 32 (March 1985):6-13. m

May 1988 9



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