Criterion D: Results
Type I, Mathematical Investigation: Generalization
Achievement level
0 The student does not produce any general statement consistent with the patterns and/or structures generated.
1
The student attempts to produce a general statement that is consistent
with the patterns and/or structures generated.
2
The student correctly produces a general statement that is consistent
with the patterns and/or structures generated.
3 The student expresses the correct general statement in appropriate mathematical terminology.
4 The student correctly states the scope or limitations of the general statement.
5 The student gives a correct, informal justification of the general statement.
D0
Type I The student does not produce any general statement
consistent with the patterns and/or structures generated.
This could only happen with an assignment that was largely incomplete. Happily, I didn't have any examples to show.
D1 Type I
The student attempts to produce a general statement that is consistent
with the patterns and/or structures generated.
Here
the student has attempted a generalization, but it's not complete. His
table indicates that he hasn't gotten enough of the pattern down -- he
ignores the values from Pascal's triangle, and the probabilities on the
side don't clearly represent anything in particular. You may also
notice the messiness of the work. His handwriting is naturally
atrocious, but on this page, he has actually made an effort to be
legible. It doesn't remain this easy to read later in the response.
D2 Type I
The student correctly produces a general statement that is consistent
with the patterns and/or structures generated.
Student 1
Earlier in the response, this student has gotten a correct general statement for the coin-tossing process. At this point, he tries to extend that to the general statement, but as you can see, he really hasn't done it. In fact, he succeeds mainly in being confusing.
Student 2
This example shows an incomplete general statement. He has sort of
implied
how a general statement would be constructed, but he never gets there.
The exponents are missing from the description of how to do the
computation (and a formula would be much easier to follow); he only
really says how to get the coefficient from the calculator.
D3 Type I The student expresses the correct general statement in appropriate mathematical terminology.
The conclusion here is incomplete. There has been a general statement in a previous part, but this conclusion does not have sufficient scope to constitute the
general statement. It's an important distinction. Later in the
student's work, he implies a fuller conclusion, but he never states it
explicitly, which means that the "appropriate mathematical terminology"
requirement is not met, either.
D4 Type I The student correctly states the scope or limitations of the general statement.
This student did get level 4. He has stated what values of r and n
make sense here, and in other places has addressed the numerical
relationship of the probabilities of success and failure. What he does not do is explain why
his process should produce the correct probabilities. This is why he
doesn't earn level 5. You can see that in my last comment on the page I
point out that he hasn't said why those powers produce the desired
results.
D5 Type I The student gives a correct, informal justification of the general statement.
These
excerpts show how the student has explained why her processes work in
two different parts of the assignment. Note that she's not just stating what she has done -- she's saying why it
works mathematically. In the second example, she has a couple of
notation problems, but she's done a good job of making clear why each
of the computations is done. When she gives the general statement in
generalized terms soon after, it is clear what she means, since she has
worked through the reasoning here.
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