ترجم ما يلى للعربية::Note that many of the steps of the cycle can be mapped to the sources of validity
evidence from the Standards (AERA et al., 2014) discussed earlier. For example, “Analyze
Logical Structure of the Domain,” which is contained within box 1 of Figure 1, maps to evidence
based on test content. “Collect/Analyze Evidence of Student Thinking,” (box 4 of Figure 1) maps
to evidence based on response processes. “Fit and Compare Psychometric Models” (box 9 of
Figure 1) and “Examine Empirical Recovery” (box 10 of Figure 1) map to evidence based on
internal structure. “Compare Student Classifications to Other Measures” (box 12 of Figure 1)
maps to evidence based on relations to other variables. And “Evaluate Instructional Efficacy”
(box 15 of Figure 1) maps to evidence based on consequences. These examples can be captured
through a mapping; other steps in the cycle may be related to multiple sources of evidence.
With respect to constructing validity arguments, Kane and Wools (2020) distinguished
between the functional perspective and the measurement perspective, and argued that for
classroom assessment, the functional perspective is most salient. The functional perspective is
concerned with whether an assessment is successful in achieving its intended goal, whereas the
measurement perspective is concerned with whether it is accurate and reliable. The functional
perspective was introduced by Cronbach (1988), who expressed that “The literature on validation
has concentrated on the truthfulness of test interpretations, but the functionalist is more
concerned with worth than truth.” (p. 5) In other words, Kane and Wools made the case that with
respect to classroom assessment, the focus of validation should be on the value for supporting
learning, with the measurement perspective playing a supporting role towards this end. Since
LPs are intended to support learning, the last step in the validation cycle, “Evaluate Instructional
Efficacy” (box 15 of Figure 1) is arguably also the most important; it also certainly addresses
validity from a functional perspective. Starting with the fourth step, each step of the cycle in
17
Figure 1 is labeled with “F” or “M” to indicate whether it corresponds to the functional or
measurement perspective of validation. Note that while the majority of steps correspond to the
measurement perspective, these steps serve to support steps 13, 14, and 15, all of which
correspond to the functional perspective.
Initially, we attempted to indicate many of the possible subcycles of version 2.0 within
the main cycle. But the diagram quickly became overcrowded, because one might return to many
of the preceding steps following any step in the cycle. To indicate this, we filled the center of the
diagram with a crosshatched pattern to indicate the many possible connections.
An Application of Part of the Cycle to an LP for the Concept of Function
A Collaborative Effort
We next turn to an example of how the first six steps of the cycle in Figure 2 might be
applied. This work represents a collaborative effort among staff at Educational Testing Service
(ETS), The Algebra Project, Southern Illinois University Edwardsville (SIUE), and the Young
People’s Project (YPP). The organizations are working together to build and validate an LPbased assessment for the concept of function, for students characterized as underserved in
mathematics.
The Algebra Project’s mission is to prepare students from historically marginalized
school communities for college-level work in mathematics. YPP staff include college and high
school students who mentor their peers in mathematics after school and during the summer, with
the goal of supporting mathematics literacy. The context of this work is the inequities in our
society that produce segregated schools in which students’ race/ethnicity and income status are
linked to inadequate mathematics education (Moses & Cobb, 2001).
18
As part of theory development (box 1 of Figure 1), the first step in this work was to
develop a three-stranded LP for the concept of function. Two of the strands are aligned with
Algebra Project curriculum modules, while the third strand is aligned with a traditional
curriculum. While there is overlap in the ideas expressed in the three strands, there are also ideas
that are unique to each strand. Nevertheless, we included general descriptions that link the three
strands. The LP is consistent with the Algebra Project’s five-step teaching and learning process,
which assumes a gradual introduction to abstract mathematical ideas beginning with events
common to everyday experience and culminating in symbolic representations (Moses, Kamii,
Swap, & Howard, 1989, p. 433).
APOS Theory
Each of the strands of the LP is grounded in APOS Theory (Dubinsky & Harel, 1992;
Dubinsky & Wilson, 2013). APOS Theory consists of four stages: action, process, object, and
schema. The theory is sufficiently general that it has been applied to characterize the
development of understanding across many domains in mathematics, including the concept of
function (Dubinsky & Harel, 1992). When instantiated for the concept of function, Dubinsky and
Harel used an additional stage, prefunction, at which students do not yet possess the concept. At
the action stage, students generate outputs from inputs, but they carry this out one input at a time
(for example, imagine a student generating an input/output table, one input at a time, or plotting
points on a graph, one point at a time). By the process stage, students can imagine that outputs
are generated from inputs simultaneously. At the object stage, students can treat functions as
entities in their own right, to be further operated on. The schema in APOS Theory is not so much
a stage as it is a structure into which action, process, and object conceptualizations are organized.
Arnon et al. (2014) also proposed the addition of a totality stage in response to the observation
19
that some students seem to fall between a process and an object stage, at least based on studies of
the concept of infinity.