ترجم للغة العربية:). 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 
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that some students seem to fall between a process and an object stage, at least based on studies of 
the concept of infinity. Arnon et al. were careful to note that the totality stage is tentative, and 
whether it is applicable to any particular mathematical domain needs to be examined. 
A Three-Stranded Progression 
The three strands of the LP that were developed include the Traditional strand, the 
Finite-to-Finite strand (see Eames, Graf, van Rijn, Budzban, & Voepel, this issue), and the 
Geometry strand. The Traditional strand assumes a brief introduction to the definition of function 
followed by a few examples of mappings from finite sets to finite sets, with fairly rapid 
immersion into the study of real-valued functions. It grew out of an earlier LP developed by Graf 
et al. (2019) for the Cognitively Based Assessment of, for, and as Learning (CBAL) research 
initiative (Bennett, 2010; Bennett & Gitomer, 2009), which was aligned with the Common Core 
State Standards for Mathematics (CCSSI; Common Core State Standards Initiative, 2010). It was 
subsequently revised following suggestions from an expert panel that reviewed all three strands 
together. Although we refer to this strand as Traditional, it is traditional only in the sense that it
mirrors the content of the traditional curriculum with its focus on real-valued functions; it was 
still used to develop a number of innovative response types, including expressions/equations, 
graphs, numeric entry, and text responses. It spans three themes, including Correspondence, 
Covariation, and Representations. The Correspondence and Covariation themes refer to Confrey 
and Smith’s (1994; 1995) distinction between these two views of a function: the first follows 
from the definition of a function as a mapping between sets, while the latter concerns how the 
dependent variable changes with constant change in the independent variable. Both views of 
function come into play with real-valued functions. The Representations theme focuses on 
working with alternate representations of functions and translating among them. The working 
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version of the Traditional strand is shown in Table 1. The first column gives the corresponding 
stage from APOS Theory, the second column gives a general description of each level which is 
common to all three strands, and the last column gives a level description specific to the 
Traditional strand.