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## Measuring Up: Prototypes for Mathematics Assessment (1993) Mathematical Sciences Education Board (MSEB)

### Citation Manager

. "The Towers Problem." Measuring Up: Prototypes for Mathematics Assessment. Washington, DC: The National Academies Press, 1993.

 Page 133

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Measuring Up: Prototypes for Mathematics Assessment

### The Towers Problem

 Use manipulatives in problem solving Apply exhaustive thinking to create a convincing argument Communicate results to others, including work in small groups

Suggested time allotment

One class period

Student social organization

Small group work followed by individual work

Assumed background: This task requires children to enumerate in some systematic fashion all possible ways of constructing towers of blocks under certain constraints, and then to explain convincingly that all the possibilities have been found. Hence the task assumes that children have had prior experiences with combinatorial situations, as well as with explaining clearly how one can be sure that all the possibilities have been determined.

 Page 133
 Front Matter (R1-R10) Introduction (1-3) The Challenge (4-4) The Criteria (5-6) The Caveats (7-7) The Audience (8-8) The Prototypes (9-11) The Tryouts (12-12) The Format (13-13) The Protorubrics (14-15) The Standards (16-18) The Future (19-20) The Prototypes (21-22) Mystery Graphs (23-30) The Checkers Tournament (31-42) Bridges (43-52) Hexarights (53-64) Bowl-A-Fact (65-74) Point of View (75-84) The Quilt Designer (85-94) How Many Buttons? (95-100) The Taxman (101-114) Lightning Strikes Again (115-124) Comparing Grizzly Bears and Black Bears (125-132) The Towers Problem (133-140) The Hog Game (141-156) Resources (157-160) Mathematical Sciences Education Board (161-164) Credits (165-166)

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OCR for page 133
Measuring Up: Prototypes for Mathematics Assessment The Towers Problem Use manipulatives in problem solving Apply exhaustive thinking to create a convincing argument Communicate results to others, including work in small groups Suggested time allotment One class period Student social organization Small group work followed by individual work Task Assumed background: This task requires children to enumerate in some systematic fashion all possible ways of constructing towers of blocks under certain constraints, and then to explain convincingly that all the possibilities have been found. Hence the task assumes that children have had prior experiences with combinatorial situations, as well as with explaining clearly how one can be sure that all the possibilities have been determined.

OCR for page 134

OCR for page 135
Measuring Up: Prototypes for Mathematics Assessment Name________________________________________ Date _____________ 1. Please send a letter to a student who is ill and unable to come to school. Describe all of the different towers that you have built that are three cubes tall, when you had two colors available to work with. Why were you sure that you had made every possible tower and had not left any out?

OCR for page 136

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Measuring Up: Prototypes for Mathematics Assessment for the task-writer to be sensitive to the differences that the wording can make. The instructions deliberately say "Please send a letter to a student …." instead of "Please write a letter to a student …." Drawing pictures or tables or charts is a perfectly fine way to communicate results in this case; the aim is to avoid giving the impression that only "writing" is acceptable. For the same reason there are no lines on which to write — just blank space that the student can use as he or she wants to. The directions for the teacher specify that Unifix® cubes be used. Other kinds of colored cubes are often used in elementary school classrooms, but one should be aware that certain brands of cubes can snap together on their sides, so that L-shaped towers can be built. As a result, these cubes are not appropriate for this task unless the students understand that only three-in-a-row towers are to be counted. Variants and extensions: This task lends itself well to simple alterations of the numbers: One can change the height of the towers or the number of different colors that are available. Moreover, one can vary the difficulty of the task by changing the rules that determine what towers are allowable. For example, how many towers five blocks high can be made from red or blue blocks if no pair of blue blocks can touch each other? One can vary the whole context as well, using something other than towers of blocks. Care must be taken to ensure that the mathematics of the situation is still what is intended. Consider, for instance, the problem of creating rows of plants in a garden. Blue-flowered plants and red-flowered plants are available. How many different rows of three plants are possible? This is not the same as the towers problem because a garden row can be viewed from either side; R-R-B is the same as B-R-R.

OCR for page 138
Measuring Up: Prototypes for Mathematics Assessment Protorubric Characteristics of the high response: The high response shows recognition of the need for a systematic scheme to keep track of "all possibilities" in a way that supports a conclusion that there could not be any other towers of height three. The student reasoning does not rely on the argument that "I cannot think of any others," but instead presents some reasonable scheme that is potentially exhaustive. Among the arguments that children invented in the pilot are these three: Proof by cases. There is only one tower that has zero blues. There are three towers with exactly one blue (in the bottom, middle, or top positions in the tower). There are three towers with exactly two blues (there is usually some weakness in the argument at this point). And there is one tower with three blues. Total: 8 different towers. Improved proof by cases. Same as above, but the troublesome "exactly two blues" is handled by arguing that two blues implies exactly one red, which is easy to keep track of: bottom, middle, or top. Proof by induction. There are four different towers that are two cubes tall — BB, BR, RB, and RR. Atop each of these can go either a blue or a red. The resulting towers are all different because they differ either in their top color or in the color of one of the lower blocks.

OCR for page 139
Measuring Up: Prototypes for Mathematics Assessment Characteristics of the medium response: The response shows some suggestion of a method for being exhaustive, but shows no recognition that this feature is present or that it is needed. There may also be explicit statements to the effect that "I couldn't find any more." An answer qualifies as medium if it presents a proof of some important part of the problem — for example, that the number of towers must be even because every tower has exactly one "opposite" by interchanging the colors. Characteristics of the low response: The letter describes one or more methods for generating new towers, but fails to deal with the question of devising a method that will exhaustively produce all possible towers, and shows no recognition of the need for such a method.

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Measuring Up: Prototypes for Mathematics Assessment This page in the original is blank.

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