Apply modern technology to learning and assessment Use aesthetics in mathematics Broaden the view of mathematics appropriate for 4th grade Use geometry in a visually appealing way |
Suggested time allotment
Three class periods
Student social organization
Students working alone
Task
Assumed background: There are two principal assumption about the students' preparation. First, children should have had many and varied experiences with manipulating geometric objects to create patterns of various kinds. This includes experience with geometric transformations and symmetries. Second, since the computer software around which the task is built runs on Macintosh computers, the students should be familiar with the standard conventions that are typical of Macintosh programs.
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Measuring Up: Prototypes for Mathematics Assessment
The Quilt Designer
Apply modern technology to learning and assessment
Use aesthetics in mathematics
Broaden the view of mathematics appropriate for 4th grade
Use geometry in a visually appealing way
Suggested time allotment
Three class periods
Student social organization
Students working alone
Task
Assumed background: There are two principal assumption about the students' preparation. First, children should have had many and varied experiences with manipulating geometric objects to create patterns of various kinds. This includes experience with geometric transformations and symmetries. Second, since the computer software around which the task is built runs on Macintosh computers, the students should be familiar with the standard conventions that are typical of Macintosh programs.
OCR for page 85
Measuring Up: Prototypes for Mathematics Assessment
A third assumption is that there are as many computers available as there are students who are being assessed at any one time.
Presenting the task: The teacher describes a software package that enables users to design quilts on the computer. Giving each student a handout showing the screens below, the teacher should guide the group through the first example of a sequence of designs, with students using their own computers to reproduce the example. Note: a limited number of copies of the Quilt Designer software are available for $20 (prepaid) by writing to MSEB, 2101 Constitution Avenue, NW, HA 476, Washington, DC 20418.
Essentially, the Quilt Designer program allows a child to create a ''quilt" of 64 squares by starting with a basic square of his or her own design. This starting square is acted upon by a sequence of three 2 × 2 designs, each entry of which is a rotation through 0°, 90°, 180°, or 270°. Thus the initial basic square design becomes a 2 × 2 quilt, each square of which is a congruent (and perhaps rotated) copy of the initial design; the rotations are specified by the first pattern of arrows. The second pattern of arrows turns that into a 4 × 4 quilt, and the third pattern of arrows results in an 8 × 8 quilt.
When the Quilt Designer program starts, the user can retrieve work that is already in progress, or start a new quilt. In the latter case, the first screen that appears allows one to design the basic unit square.
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Measuring Up: Prototypes for Mathematics Assessment
The space on the right is reserved for a greatly expanded picture of the unit square, which can be designed pixel by pixel. The tools shown along the left edge of the screen are the standard Macintosh drawing tools — the pencil (for changing individual pixels), the paint bucket (for filling in entire areas), the eraser, the line-draw tool, and two kinds of rectangle-drawing options, filled and unfilled. Various kinds of shading are also available. Another option superimposes a rectangular grid as an aid to drawing.
Once the child has created a unit square, he clicks "Done"; the program immediately goes to the screen on which the quilt will now be built. The unit square that was just created is reproduced on this screen, along with three sets of arrows. Initially, all the arrows are pointing up, as in the figure below.
The user can click on any arrow at any time; each click rotates the arrow 90° counterclockwise. When the child wants to construct the 2 × 2 design, he clicks on the box under the first configuration of arrows. Immediately the 2 × 2 pattern is created.
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Measuring Up: Prototypes for Mathematics Assessment
The 4 × 4 designs and the final 8 × 8 quilt are created in a similar way, each building on its predecessor, as shown below.
Even after the 8 × 8 design has been completed, the user can change any or all of the arrows that specified the rotations. All the drawings that are affected are instantly updated, so that the effects of different patterns of rotations can be seen quickly. The user can also edit the unit square itself and see the effects on whatever designs (2 × 2, 4 × 4, or 8 × 8) have been created so far.
Once the quilt is completed, it can be saved under a name chosen by the user. This is accomplished via a pull-down menu. Using the other options from the menu, one can open two (or more) quilts at the same time, so that one can be compared with another. A "Print" option allows the quilt to be printed on paper.
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Measuring Up: Prototypes for Mathematics Assessment
Student assessment activity: A suggested schedule for the assessment tasks is outlined below.
Days 1 and 2: Students are given the opportunity to explore other examples on their own. The teacher should encourage students to experiment with positions of the arrows and to try to predict how a change in a single arrow will affect the appearance of the 8 × 8 quilt. This is complicated, because the effect will depend not only on the symmetries (if any) of the unit square itself, but also on the positions of arrows in a preceding pattern.
Similarly, students should be encouraged to experiment with changing the order of the arrows. For example, the teacher could say: "For what kinds of unit squares will these two sets of arrows produce the same final quilt?"
Another set of investigations involves working backwards. The teacher could ask: "Given a final 8 × 8 quilt (which the teacher can create and print out), what initial unit square and three arrow patterns could have produced it?" "Is there more than one possibility for any of them?"
Day 3: Students are asked to write a description of their favorite quilt, explain what features make it especially attractive or interesting, and then describe how they went about creating it.
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Measuring Up: Prototypes for Mathematics Assessment
Name ________________________________________ Date _____________
Of all the quilts you have made so far, which one is your favorite? Write its file name here: __________________________. Als PRINT it, and attach it to this paper.
Why do you especially like this quilt? What features or parts of the design make it interesting, or attractive, or unusual?
Please explain as well as you can the steps that you went through to create this quilt. What did you start with? How did you decide what to do next?
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Measuring Up: Prototypes for Mathematics Assessment
Rationale for the mathematics education community
The Quilt Designer task has some unusual features that distinguish it from some of the other tasks in this collection: (a) it illustrates connections between mathematics and another discipline — in this case, art; (b) it allows children to express their own creativity in a way that is not ordinarily associated with the study of mathematics — in part, students decide what is "right" based on their own aesthetic sense; and (c) it uses modern technology and sophisticated software to enable students to explore mathematical concepts heretofore inaccessible through traditional means. A related issue for tomorrow's mathematics students is how well and how quickly they can learn to use novel computer programs flexibly and effectively.
Moreover, the task incorporates important mathematical skills (e.g., prediction, spatial visualization) and concepts (e.g., symmetry, rigid transformations, the composite of two transformations) that are not often seen in the fourth grade curriculum. In the process, it gently introduces some fundamental algebraic and geometric ideas.
The Quilt Designer has potential as an extended classroom activity as well. The choice of a favorite quilt does not require a strictly mathematical explanation (see the protorubric section). However, with additional classroom time spent with the program, instructional questions could be posed to students to motivate them toward mathematically interesting reasons for preferring a design.
The Quilt Designer program, in fact, captured the imagination of many students in the pilot; they asked that the software be made available for future experimentation.
Task design considerations: The assessment task suggested here for Day 3 (as well as the introductory activities) emphasize the children's aesthetic sense, their abilities to use the software to create designs or patterns that they find interesting or appealing, and their abilities to explain what they have done and why. Clearly one could ask questions that more narrowly focus on particular skills or abilities, but one of the purposes of including the Quilt Designer is to illustrate assessment tasks that are more open-ended and creative.
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In situations in which many Macintosh computers are tied together into a local area network, the child might fill out the assessment sheet electronically. In that case the quilt would not be printed out on paper at all; the teacher (or other assessor) would simply access the file that the child designated.
Design of the software. Whenever questions of style arose in the programming of The Quilt Designer, the standard conventions of Macintosh programming were used (even though in some cases it might have been desirable to alter them a bit). As a result, anyone who is familiar with other Macintosh applications will find the mechanics of The Quilt Designer to be perfectly straightforward; and if The Quilt Designer is someone's first experience with Macintosh computers, it will not interfere with subsequent encounters with Macintosh programs.
One important design question as the software was written was what symbol to use in the patterns of arrows. This version of the program uses an arrow pointing straight up, to the left, down, and to the right; this is intended to be reminiscent of a pointer on a dial. The arrow shows the result of the rotation, not the rotation itself. While an advantage of the dial-pointer notation is that it helps in visualizing the effects of a rotation on a given unit square, a disadvantage is that it will no longer be usable if reflections are used in addition to rotations. (See the discussion of extensions of the Quilt Designer, below.)
Variants and extensions: As suggested above, there are many variants of assessment tasks that would tap more explicitly and directly the child's ability to visualize the geometric transformations involved in creating quilts. For example, at a very basic level, one could give the child the 4 × 4 stage of a quilt, together with the final quilt, and ask what final pattern of arrows would produce it. If the 4 × 4 design, as a whole, has axes of symmetry there will be more than one set of arrows that will give the final quilt.
A natural extension of the Quilt Designer would be to allow reflections ("flips") as well as rotations. This would necessitate eight distinct symbols; one for each of the eight rigid motions of the square into itself. One would have to experiment to see if symbols that suggest the results of the transformations would be more or less easily understood than symbols that suggest the transformations themselves.
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Measuring Up: Prototypes for Mathematics Assessment
Protorubric
Note that the descriptions of the high, medium, and low responses below pertain to all three days of the Quilt Designer activity, not just to the final responses about the "favorite quilt."
Characteristics of the high response:
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Measuring Up: Prototypes for Mathematics Assessment
The child explores the software productively, creating interesting patterns. Trial-and-error is used systematically to see how changing the initial design and the rotations in the arrows affect the final quilt. The effects of varying the order in which the arrows are used are explored, and the child keeps track of intermediate results in a sensible way (by saving files appropriately). The child can explain clearly why or how the final "favorite quilt" is attractive and how it was produced.
Characteristics of the medium response:
The child understands how the software operates and can create interesting quilts with it. There is not, however, a systematic approach used in exploring the effects on the final quilt of individual changes in the arrows. The idea of working backwards is difficult, and often the predicted arrows do not produce the desired result. The explanations of why the final quilt is interesting and the steps that were used to create it are not completely clear.
Characteristics of the low response:
The child can make quilts using the software, but they are more the result of random trials than any sort of thoughtful planning. Intermediate results are not recorded in any systematic way, so that comparisons of quilts made with the same arrows in different orders, for example, cannot be made. The final quilt of Day 3 shows little sense of balance or form.
Reference
Children's Computer Workshop (formerly a component of Children's Television Workshop) developed a version of this software to run on a Commodore 64 computer.