Introduction

Welcome! Read this page to learn more about the two research-based Learning Progressions we have provided for the challenge.

Once you are familiar with the Learning Progressions and working on your application, you may find it useful to download a framework for thinking about how your project distinguishes the level of progression at which a student is performing. Submitting the tables in the framework is welcomed but optional.

For more information about Learning Trajectories in Mathematics, please see this informative document from CPRE. Appendix A has a long list of research-based learning trajectories in mathematics.

If you have questions about the Learning Progressions or other challenge details, please email us at gameschallenge@ets.org.

Equality and Variable: Equations and Expressions Model

At Level 1, students have a superficial understanding of the concept of variable that does not include the idea of variable representing an “unknown.” Instead, students may hold one or more beliefs about variable including, for example, that variables represent objects (e.g. that the variable b stands for “brownies” in a word problem because it starts with a “b”). Alternatively students may treat variables like individual digits of a number (if 2x = 24, x must equal 4), they may ignore variables in an equation only operating on the numbers, or separate variables from numbers in an equation or expression.

At this level students also have an operational sense of equal sign. Students with this type of understanding believe that the equal sign is a signal that indicates that there is a problem to be solved or computation expressed on the left of the equal sign and that the answer should be placed to the right of the equal sign.

Given these understandings of equality and variable, students should have trouble solving word problems, even with informal methods, algebra solving follows misconceptions as described in model of equality.

At Level 2 students still have an operational understanding of equality as in Level 1 above. Their understanding of variable is different. Students have an understanding of variable as a “specific unknown” in which letters stand for one and only one number. For example, at this level if asked “in the expression 2n + 3, can the variable n stand for the number 4?” the student will say “yes,” but if then asked if the variable n can stand for “37,” they will say “no.”

Students can apply informal methods to solve simple algebraic word problems, but will continue to display misconceptions when attempting to solve symbolic equations. Students can attempt to solve equations using guess-and-check strategy and other approaches that do not involve substantial symbolic manipulation

At Level 3, students have “specific unknown” level concept of variable as in Level 2. What’s different is their understanding of equality. They have a more sophisticated understanding of the equal sign as expressing that the quantity to the left of the equal sign has the same value or is in balance with the quantity expressed on the right of the equal sign (described sometimes as a relational understanding of equality). They also know some simple procedures for algebraic manipulation.

Students can solve problems with one variable represented in text symbolically using algebraic manipulation. Because students have few ways of understanding algebraic expressions and equations, they may apply less efficient methods to solve equations. Expressions are only understood correctly if they can take on a single numerical value (i.e., consistent with students’ understanding of variable as specific unknown).

At Level 4, students’ understanding of equality is richer than in Level 3. They understand relational equivalence in most contexts (meaning that they not only understand that both sides of an equation are equivalent, but they also understand that performing the same operation to both sides of the equation maintains the equivalence) and can readily use transformations to balance equations to solve for an unknown. Students still have the same understanding of variable as in Level 3 (i.e., the “specific unknown” level concept of variable).

Students do not yet have a “generalized number” concept but because of relational equivalence they can solve many equations with one unknown algebraically.

At Level 5, students have “generalized number” level concept of variable, in which a variable can function as a pattern generalizer for arithmetic, as in the statement a + b = b + a, or can express relations among sets of numbers as parameters and arguments as in the linear equation x = mx + b. Students have the same understanding of equality as in Level 4, i.e. they understand relational equivalence and can readily use transformations to balance equations to solve for an unknown.

Students can solve symbolic equations, parsing expressions to flexibly apply operations on to solve equations efficiently.

From: Arieli-Attali, M., Wylie, C., & Bauer, M. (2012). The use of three learning progressions in supporting formative assessment in middle school mathematics. Paper presented at the annual meeting of the American Educational Research Association, Vancouver, Canada. Copyright pending.

Functions and Linear Functions Model

At Level 1, students have separate numeric and spatial understandings.

At this level students can: complete patterns or sequences (numeric understanding), they can read bar-graphs and see differences in the sizes of bars (how tall they are)

At this level it will be hard for students to: quantify changes in a graph, and see the connection between those changes and changes in a numeric pattern.

Given these separate understandings, students will not be able to graph a pattern, or to infer about a pattern from a graph. They would only be able to graph discrete points (or bars) related to several [discrete] categories (categorical x-axis).

At Level 2 students have integrated operational one-to-one correspondence between numeric and spatial representations developing the concept of mutual change, connecting to the intuitive concept of rate, leading to the development of the concept of constant change, and slope (“rise over run”), and building the basis for understanding linear functions.

At this level students can: create a basic table from a linear equation of the form y = mx, complete tables for a given rule, interpolate and extrapolate in a table (fill-in missing values — without the rule), graph basic tables, graph basic word problems (rate), graph basic linear equation (given as a rule y = mx), interpolate in a line graph (extract the value of given y for a given x and vice versa). Success in this level depends on small positive numbers and familiar contexts (working in the first quadrant of the Cartesian grid).

At this level it will be hard for students to: work with y = mx + b, work with m negative, work outside the first quadrant, work with critical points, translate between scatter plot (point-wise “graph”) and line graph (may not see that connecting the points does not change the function), interpolate point-wise graph.

Given this integrated operational one-to-one numeric and spatial correspondence, the students are limited to basic operational match between “rule” – table – line-graph, within the range given without generalization.

At Level 3, students have generalized and over generalized operational correspondence with understanding of some of the characteristics of a specific linear function, i.e. critical points (the y-intercept, the x-intercept, the slope, and the kind of function you get for m = 0 or b = 0). In this stage students start to generalize the concepts, and overgeneralization (and potentially under applying) may occur. This is when linearity illusion can be detected (students apply linear relation to a situation that is not necessarily linear).

At this level students can: work with y = mx + b, where m and b are positive, work with word problem with baseline/ flat fee and rate, work with different forms of the equation, i.e., y = mx + b, work with critical points, translate between scatter plot (point-wise “graph”) and line graph (see that connecting the points does not change the function), interpolate point-wise graph.

At this level it will be hard for students to: see global features of graphs and work with families of functions (transform a function), work with qualitative graphs (graphs without values on the axes), work with different scales and/or changing of a scale, generate an equation for a graph or a table (beyond the over practiced ones like y = 2x and the like), coordinate information from different representations.

Given this generalized and over generalized operational numeric and spatial correspondence, the students are limited to basic operational match between “rule” – table – line-graph. This is the stage where most misconceptions occur, specifically the linearity illusion. Another misconception is that one needs three critical points to determine the function: y-intercept, x-intercept and the slope (the “3-slot-schema” misconception).

At Level 4, students have a well established concept of dependency. They have a good contextual understanding (when and where to apply which function), and can easily work with different representations (translate between representations).

At this level students can: detect global features of graphs and work with families of function (transform a function), work with qualitative graphs (graphs without values on the axis), work with different scales and/or changing of a scale, generate an equation for a graph or a table, coordinate information from different representations.

At Level 5, students have a “set theory” concept of function, viewing a function as an infinite matching between pairs. At this level students can work with different kinds of functions (non-linear, non-numeric, split domain, etc.) and with relations between functions (e.g., a function and its inverse). At this level the function is an abstract concept, although likely beyond most middle school students.

From: Arieli-Attali, M., Wylie, C., & Bauer, B. (2012). The use of three learning progressions in supporting formative assessment in middle school mathematics. Paper presented at the annual meeting of American Educational Research Association, Vancouver, Canada. Copyright pending.