The Linear Functions Learning Progression

The current version of the CBAL learning progression for linear functions (Arieli-Attali, Wylie & Bauer, 2012; Graf & Arieli-Attali, 2014) was influenced by the work of Sfard (1991, 1992), Kieran (1993), Kalchman, Moss and Case (2001) and Vinner and Dreyfus (1989). A table that summarizes

Table 9.1 Levels of the Linear Functions Learning Progression



Aspect of change



Nonlinear functions

Changing change

Can examine rates of change in nonlinear functions (e.g., in polynomial or exponential functions)


More than one linear function

Comparing rates of change

Can compare rates of change (slopes) of linear functions


Linear function

Constant change

Can recognize constant change in a linear function


Coordinate plane

Mutual change

Can recognize dependence between variables


Separate representations (numeric, spatial, symbolic)



Can extend a pattern or sequence

Adapted from “Advisory Panel Presentation,” by M. Arieli-Attali, June 2011, IES project of Middle School Developmental Models.

key features of different levels of the linear functions learning progression (especially with respect to slope) was developed by Arieli-Attali (2011); an adaptation is shown in Table 9.1. A complete description of the levels is given in Arieli-Attali et al. (2012).

A key transition is posited to occur between Level 2 and Level 3. At Level 2, the student recognizes that as one variable changes, the other variable also changes, but is not yet able to characterize the nature of that change as constant. By Level 3, the student recognizes that linear functions are characterized by constant change. An item shell that is specifically designed to target this transition might specify that a function is linear, and then ask, “If the [dependent variable] [increases, decreases] by [Ду] when the [independent variable] increases from [x] to [x + Дх], then how much does the [dependent variable] [increase, decrease] when the [independent variable] increases from [x + a] to [x + a + гДх]?” Individual instances of this item shell are generated by replacing each variable with an integer, with Дx, Ду, and r positive.

While a student at Level 2 is unlikely to answer items based on this shell correctly, a student at Level 3 is able to identify the change as the quantity гДу, even if [x + a] and/or [x + a + ^x] are too far out on the scale to be represented in the stem (e.g., in a graph or table).

Table 9.1 focuses on how student understanding of slope develops. Another conceptual shift is hypothesized to occur between Levels 2 and 3, however—namely, that students learn to work with alternate but equivalent representations of linear functions and can translate among them. For example, a student at Level 2 may be able to manipulate equations, but might not recognize the equation for a linear function and its graph as alternate representations of the same model. Items that require students to translate among alternate but equivalent representations might identify this shift in understanding.

Student placement at a level of a learning progression has implications for instruction. For example, for a student at Level 2 of the linear functions learning progression, it is presumed that the most useful activities are those that emphasize the pattern of constant change and that encourage students to translate among equivalent representations. Since these activities require an investment of student and teacher time that might be directed elsewhere, their selection carries consequences—motivating the need to validate the learning progression.

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