Papers in the following categories were defined as "successful" in the quantitative analysis.
A. Unresponsive. Papers which did not demonstrate an attempt to answer the questions about the individual digits were classified as Unresponsive. Unresponsive papers included those with no response, or, more frequently a written response of "I don't know." Sometimes these were accompanied by pictures colored all one color. There were two further types of papers which we placed into this category. Some students were quite articulate about explaining how they counted or computed the 35 objects in the picture, but they did not make any reference to the meanings of the individual digits. Others explained that both digits are needed to make 35: "If it did not have the 5 it would only be 3."
B. Five objects, three objects. Student papers in this category interpreted "5" as representing five objects and "3" as representing three objects in the picture. No attempt is evidenced to account for the remaining objects in the picture.
C. Invented quantitative meanings for the digits. The whole quantity is preserved; all 35 objects are accounted for in the invented meanings for the digits. This category included those papers where the student interpreted the "5" (in the 5 x 7 rectangular array task, as representing "5 in each row", or "the top row." These papers may or may not have described an invented meaning for the "3." Another popular invented response was phrased as "counting by 5s" and "counting by 3s." Some students even discussed the remainder you get when counting by 3s.
Papers that partitioned the whole set in some manner (e.g.. 20 and 15, or in half), were also placed in this category.
D. Verbal Knowledge. At least one of the individual digits is described in tens or ones language. The accompanying picture might be colored but does not show a five and thirty partitioning. No picture or other evidence of 30 is present. For example, "the five is the one colum. The tree is the first digit in the number so its the tens [sic]."
No quantities are indicated, or those indicated are invented, as above. For example, "3" is described verbally as "the tens" and a quantitative interpretation is provided for the "5". (Five is interpreted as a column of five squares or beans, five rows, or the number you multiply by.) Papers in the following categories were defined as "successful" in the quantitative analysis.
E. Probable Understanding but Evidence is Not Entirely Convincing. Papers in this category belong to students who may well understand that "5" represents five objects and "3" represents 30 objects, but in the paper being scored the evidence is skimpy, ambiguous, or includes extraneous additional interpretations of the digits. Either the picture or the written explanation is satisfactory, but not both. The picture may be correctly partitioned into a 30 and a 5 OR the written explanation discusses 5 and 30 but the pictorial evidence is missing or ambiguous. In the classroom a teacher would most likely want to talk to students whose papers fall in this category to get more information about the student's understanding.
Convincing Correct Responses. Pictures and language are used to explain that the "5" represents five of the pictured objects and the "3" represents the remaining thirty objects. In each, the picture clearly shows a partitioning of the 35 into a five and a 30. The written language used distinguished the following subcategories.
F. The written explanation uses both tens and ones language AND five and "30." OR The written portion describes five as 5 ones, 3 as 30.
G. The written explanation uses "5" and "30" (or "thirty"). Neither tens nor ones is mentioned.
H. The written explanation uses tens and ones language but not "thirty."
I. Five and three sets of ten. The picture is partitioned into three sets of ten and either a set of five, or five sets of one. The written language connects the "5" with the set of five and the "3" with the three sets of ten. "Thirty" may or may not be explicitly mentioned.
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