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OCR for page 239

,`~..~..~.r.. I A.
MARCY'S DOTS
Participant Handout
A pattern of clots is shown below. At each step, more clots are aclclecl to the pattern. The number of clots aclclecl at
each step is more than the number aclclecl in the previous step. The pattern continues infinitely.
( 1 St step)
(2n~ step)
(try step)
2 Dots
·
·
·
6 Dots
1 2 Dots
Marcy has to determine the number of clots in the 20th step, but she does not want to clraw all 20 pictures and then count them.
Explain or show how she could clo this and give the answer that Marcy should get for the number of clots.
Dicl you use the calculator on this question?
~ Yes ~ No
SOURCE: National Assessment of Educational Progress (NAEP), 1992 Mathematics Assessment

OCR for page 239

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APPE N DIX 4
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OCR for page 239

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MARCY'S DOT ACTIVITY
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OCR for page 239

FaciIitator's guicIe for the Marcy's Dots activity
Marcy's Dots
1. To get participants involved in
thinking about content and learning,
have them solve Marcy's Dots, a prob-
lem in the content area of algebra and
function from the 1992 National Assess-
ment of Educational Progress (NAEP)
Grade ~ Test. About 20% of the test for
grade ~ assessed algebra and function,
and this is Apical of an extended con-
structed-response question.
The following information might be
shared after or at an appropriate time
during the discussion. Do not start off
with this but use it after people have
become engaged and come to some
conclusions about the problem and
about the ways students found their
solutions. Only 6% of the students
provided a satisfactory or better re-
sponse; 6% provided a generalization;
10% made some attempt at a pattern;
63% provided inaccurate or irrelevant
information; and 16% did not respond.
(Remember that this test is a no stakes
test for students.)
2. Investigate Student responses.
Points that might be made in the
.. .
cl~scuss~on:
Wrong answers usually occurred
because they had the wrong notion of
the pattern.
APPE N AX 4
Student ~ alternated between adding
4 and adding 6.
Student 5 multiplied by 3 then by 2 as
the pattern.
Student 7 multiplied by 3 then by 2.
Student 12 increased by multiplying
by2.
No recognition of pattern:
· Student 9 added the numbers in the
problems.
· Student 3 added 6, the last set of dots.
Right Answers:
· Student 2 used the pattern within the
pattern and the picture; wrote out all
steps (recursion).
· Student 4 used relation between rows
and columns and was able to general-
ize to a rule.
.
Student 6 used relation between rows
and columns from picture and was
able to generalize to a rule.
· Student ~ used the pattern within the
pattern: recursively adding two more
each time (4, 6, 8, ...)
Student 10 focused on relationship
between rows and columns, wrote out
all steps (recursion).
· Student 11 used relationship numbers
and was able to generalize to a rule.
.

OCR for page 239

3. Discuss the problem from the
perspective of its role in the middle
grades mathematics curriculum. How
does the plenary session on content and
learning relate to the problem? How
does the problem fit into the larger
picture of algebra and algebraic reason-
ing? Why is it important for students to
recognize and be able to work with
patterns?
4. Hand out the excerpts from the
draft section of the middle grades
algebra section from "Principles and
Standards for School Mathematics:
Discussion Draft," the Standards 2000
MARCY'S DOT ACTIVITY
draft being prepared for dissemination,
comment and input this fall. Provide a
few minutes for people to read the
excerpts. How does the mathematics in
this task relate to the discussion of
algebra in the document?
Resources for Marcy's Dots
Dossey, I.A, Mullis, I.V.S., & [ones, C.O. (1993~.
Can students do mathematical problem solving?
resavltsirom constra~cted-response questions in
NAEP's 1992 Mathematics Assessment.
Washington, DC: National Center for Educa-
tion Statistics.
Kenny, P.A, Zawojewski, J.S., & Silver, E.A
(1998~. Marcy's Dot Problem. Mathematics
Teaching in the Middle School, 367}, 474-477.

OCR for page 239

TABLE 2. ~ ~ National Results for Demographic Subgroups for the Extenclecl-Response Task,
Marcy's Dot Pattern, Gracle 8
No Satisfactory
Response Incorrect Minimal Partial Satisfactory Extended or Better
Nation 16(1.0) 63(1.3) 10(0.7) 6(0.7) 1 (0.2) 5(0.6) 6(0.7)
Northeast 18(3.2) 61 (3.2) 10(1.9) 4(0.7) 2(0.5) 6(1.8) 8(1.6)
Southeast 20(2.0) 6A (2.2) 9(1.5) 3 (07) 1 (oh) ~ (1.1) ~ (1.3)
Central 10(1.5) 65(2.1) 10(1.~) 8(1.~) 1 (oh) 6(1.1) 7(1.~)
West 16(2.0) 62(2.8) 10(1.1) 7(1.8) 0(0.2) ~ (1.1) ~ (1.1)
White 12(1.1) 63(1.5) 11 (0.8) 7(0.8) 1 (0.2) 6(0.8) 8(0.9)
Black 2A (2.9) 67(2 9) 6 (1.6) 2 (0.9) 0 (0.0) 1 (o.5) 1 (o.5)
Hispanic 28(2.8) 61 (3.1) 7(2.0) 3(1.2) 0(00) 1 (o.5) 1 (o.5)
Male 19(1.5) 63(2.2) 8(1.0) 5(0.9) 1 (0.2) 5(0.9) 5(09)
Female 13(1.2) 63(1.6) 12(1.1) 6(1.0) 1 (o.3) 5(0.8) 6(0.9)
Advantaged
Urban 8(2.9) 62(5.1) 10(1.9) 6(1.6) 1 (0.6) 11 (2.5) 13(2.6)
Disadvantaged
Urban 32 (3.9) 59 (A 7) ~ (1.3) ~ (1.9) 1 (0.6) 1 (o.5) 1 (o.7)
Extreme Rural 16(2.9) 69(3.6) 8(2.3) 2(1.1) 1 (o.7) A(2.0) 5(2.3)
Other 15 (1.3) 62 (1.5) 11 (0.9) 6 (0.9) 1 (0.2) ~ (0.7) 5 (0.7)
Public 16(1.2) 64(1.4) 9(0.8) 6(0~7) ~ (0.2) 4(0.6) 5(0.6)
Catholic and
Other Private 11 (1.7) 56(2.7) 12(1.6) 7(1.2) 2(0.9) 10(2.2) 13(2.0)
The standard errors of the estimated percentages appear in parentheses. It can be said with about 95 percent certainty
that for each population of interest, the value for the whole population is within plus or minus two standard errors of the
estimate for the sample. In comparing two estimates, one must use the standard error of the difference (see Appendix for
details). When the proportion of students is either 0 percent or 100 percent, the standard error is inestimable. However,
percentages 99.5 percent and greater were rounded to 100 percent and percentages 0.5 percent or less were rounded to 0
percent. Percentages may not total 100 percent due to rounding error.
SOURCE: National Assessment of Educational Progress (NAEP), 1992 Mathematics Assessment
APPE N DIX 4