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.
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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 133
Measuring Up: Prototypes for Mathematics Assessment
Presenting the task: Initially the class is to work in groups of three. Each group should have a supply of Unifix® cubes of two different colors — about 40 cubes of each color. The teacher should explain that the task is to build towers of Unifix® cubes, saying something like this:
"Each tower is to be three cubes tall. You may use the cubes on your table, which include cubes of two different colors. Please build as many different towers as possible.
"Besides building the towers, please explain your work to the other students at your table, to convince them that you have not left any out, and that you have no duplicates. Please make only towers that are right-side up, like this:
and do not make any "upside down" towers, like this:
Then the teacher should pass out copies of the student sheet and read through the directions to be sure that everyone understands the task.
Student assessment activity: See the next page.
OCR for page 133
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?
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Measuring Up: Prototypes for Mathematics Assessment
Rationale for the mathematics education community
Problems or situations that involve systematic counting of the number of ways that something can be done provide good opportunities to students for problem solving, reasoning, and communicating their results to others. Although such problems from discrete mathematics are not explicitly called for in the K-4 Standards, they are described in the standards for the upper grades. The foundation for discrete mathematics can be laid early, particularly through the use of manipulatives.
An analysis of videotapes of the pilot tests on this task suggested that fourth graders' oral explanations in small groups often were much more detailed and sophisticated than their written explanations. That is, the "letters" that they write to their sick classmates often do not capture their full insight into the task. One would expect the discrepancy between written and oral explanations to diminish as students get more experience with the kinds of mathematical communication emphasized in the Standards.
The discrepancy may arise partly because the students know that their small-group colleagues will not accept inexact or unclear oral explanations, whereas a written letter provides no immediate feedback. This lack must be addressed, however, because the development of students' communication skills is an important goal of reform.
Task design considerations: This is an excellent task to illustrate the importance of the precise wording of questions. It is tempting to say "using two colors" instead of "when you had two colors available to work with." But some children will (correctly) interpret "using two colors" to mean that both colors must be used in each tower, and conclude that there are only six 3-block towers that use exactly two colors. There's certainly nothing wrong with the task of determining the number of towers that use exactly two colors, but it is not the same as the task of finding the number of towers that use no more than two colors. The essential point here is that small changes in the wording of questions can have significant and often unintended consequences. Ordinarily, it is not the aim of the task to have children make these subtle distinctions, so it is important
OCR for page 133
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 133
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.
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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
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