Here's the extended abstract for the paper. The Full Paper is under "Elementary Mathematics".
Extended Abstract
This study of 76 6th, 7th, and 8th grade students investigated students’ understanding of place value numerals. It was the research component of the Mollala Math Project in Oregon as generated by the opportunity to conduct embedded research into math development. The 11 teachers of the Molalla River District Math Leadership Team provided district coordination and surprising contributions to the conceptualization of place value understanding in students and the several years of oversight, editorial work, and field testing in converting the interviews into assessment tasks. Their primary purpose was assist teachers in advancing their understand of their students’ math development.
The
purpose of this research project was to investigate why math test scores
declined from 3rd to 10th grade. About 70% of 3rd
graders passed the Oregon state mathematics test. At 10th grade less
than a third passed. The decline accelerated after 5th grade. Our
investigation entailed three parts: the use of developmental theory to uncover
and describe the development of understanding in mathematics, the use of Rasch
modelling to verify and extend the developmental findings, and the integration
of the theory and the Rasch modelling to improve methods of differentiated
instruction that match teaching to each student’s needs.
The first investigation was immediately sidetracked when we could not find a valid definition
of place value numeration or its operational components in the literature or
instructional materials, This lack was also a complaint of teachers so we undertook
a project with the Math Leadership Team to abstract and define the necessary
and sufficient elements and operations that would specify the place value
system and distinguish it from the number system, a distinction surprisingly almost
never made in instructional materials or curriculum guidelines. Numerals are
the symbol systems for generating representations of numbers and all depend more
or less on some system for mathematically generating and transforming numerals.
The mathematical validity of place value system is exactly the same as
the criteria of any valid number systems since it has full group properties
including reversibility and conservation. We arrived at four defining elements and five basic operations. This
system of place value operations then makes possible two derivative operations.
These coordinate the movement of values across places within a numeral.
Next the committee formulated and refined five interviews organized
as Piagetian type of tasks. Each task posed a specific problem for students
which triggered reasoning we could follow by why and how questions directed to determining
the complete features of their understanding: The Place’s Values (What do the
digits in 16 mean?); Re-grouping of One’s Into Ten (How are place totals
re-grouped in 28 + 3?); Column Values in Long Division (Is place value
maintained in columns?); Conservation of Place Value (Is the numeral’s total
value maintained after borrowing?); and Naming of Transformed Numerals (What language
does the student use for meaning of a transformed numeral after borrowing?)
These results were condensed into the criteria that defined the stages: “The
intra phase leads to the discovery of a set of properties in objects and events
finding only local and particular explanations. The ‘reasons’ to be established
can thus be found only in the relations between objects which means that they
can be found only in ‘transformations’. These, by their nature, are
characteristic of the inter level. Once discovered, these transformations
require the establishment of relations between each other which leads to the
construction of ‘structures’ characteristic of the trans level.” We then had
descriptions of how students understood key aspects of place value numeration
at three stages. For this
place value study, only two of these stages were used, the intra-numeral stage
of understanding (stage 1), and the inter-numeral stage subdivided into two
levels, a beginning (stage 2) and a completed form (stage 3). The trans-numeral
stage was not used since it extends considerably beyond the elementary level in
complexity to the level of explicit, formal, abstract understanding of place
value as a system.
The
sample for these five studies was composed of two cohorts, the first was selected
as an epistemic sample designed to produce the widest possible range of place
value understanding. This teacher selected epistemic sample were students teachers
selected as high, medium, or low math ability. They selected 17 remedial 7th
and 8th grade students and 11 7th and 8th grade
students judged of high ability. The second cohort was a random sampling of 34
students from within three groups of 6th
grade students, a group of high (n = 12), medium (n = 11), and low (n = 11)
students identified according to their state test scores.
The results revealed a complexity of the place value
system that was mathematically beyond the understanding of most of these middle
school students. At the intra-figural stage 1, students made no distinction
between number and numeral, understood numerals as unitary figural symbols, and
used simple procedural notions in problem solving. For example, Chad, grade 8, showed 6 chips for
the 6 and 1 chip for the 1 in 16. When asked about the remaining 9 of his 16
chips, he said, “You don’t need the rest.” (But will we both have sixteen if
you don’t use those?) “I don’t know,” he replied.
At the beginning inter-figural stage 2, students used a
beginning but weak and unstable form of place value understanding that embedded
numerals with limited place value relationships. Ross, a grade 8 student, showed an early form
of stage 2 thinking in the 16’s task, “If I was explaining it to someone I
would say, I took out 6 and then ask him how many are left and say…” (indicates
two piles of five). “If he thought it was a 1, I’d show him it (the two piles
of 5) couldn’t be. There is 5 and 5 and the 10, and the 1 means a ten and not
just a plain 1.” He showed a pile of six and a pile of five and another pile of
five. Places for him had significance but only as empirically necessary to
account for all the chips, not because a base was explicitly involved.
At the completed inter-figural stage 3, the conservation
of place value appeared and students extended place value operations to new
problems with reasoning controlled by place value logic. For example, Konstantin in grade 7 was an unusual
case because English was not his native tongue. In #1, he showed 1 chip for the
1 in 16 but then almost immediately seemed to understand. He self-corrected
before being questioned and said “10.” From his performance on the other tasks,
this misstep was seen as one of miscommunication, not misunderstanding.
The combined
results of the three equal sized samples of high, medium, and low 6th
grade students showed about half the students had completed understanding on
the 16’s task. Since the borrowing problem depends on the conservation of place
value of a numeral, it is the best indicator of a completed conceptual
understanding of place value numerals, and least susceptible to learning
effects. In spite of the relative overweighting of the high student sample, the
results still showed only 13% of the combined samples had the completed
understanding of stage 3. Although appearing as rather simple, (and often
presented as simple in instructional demonstrations) place value numeration is,
in fact, a mathematically rich system whose complexity is not mastered by the
majority of students up through middle school.
In the second investigation, place value data were
combined with other data, and Rasch modeling was used to measure the difficulty
of the five place value tasks and locate them within a developmental math
continuum of 30 concepts. The interview data pool from 159 kindergarten through
8th grade students were coded into the 3 stages and used in the Rasch
modelling. With the conservation of number anchored to 2.0 logits, the place
value concept and volume both had a difficulty score of 4.5 logits with 14% of
the middle school cohort succeeding on the conservation of place value task.
The task fit scores confirmed the
validity and difficulty of the place value tasks (infit M = 1.01, S.D. = 0.2, t = – 0.02) and supported the conclusion
that these public school students did not understand the concept of place value
until well into the middle school years.
In the third part, individual mind maps showing each
concept’s stages, achieved and not yet achieved by the student, graphically
displayed a zone of development around each student’s developmental level. The
mind map method of reporting also showed teachers how quantitative test scores
could be diagnostically related to stage descriptions of actual understandings.
Combining developmental systems theory with Rasch modeling supports the
possibility for developing a continuous progress model of math instruction by
matching teaching and learning to each student’s specific developmental needs
and abilities.
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