Algebra is a useful tool for connecting mathematics and the real world as well as for sense making across all areas of mathematics. This standard focuses on the study of patterns and functions, and representing and modeling mathematical and everyday or imaginary contexts (38). Supporting learning in this domain engages learners in the activities of noticing, extending, conjecturing, proving, and generalizing patterns, functions, and change. Developing these algebraic habits of mind is critical for future mathematics learning (39).

Elementary Mathematics Specialists (EMSs) create opportunities for learners to engage in algebraic thinking by asking purposeful questions and the use of intentional task design. They know that with exposure over time to algebraic habits of mind such as seeking and identifying patterns, making connections, justifying reasoning, etc., learners begin asking “I wonder” and “what if” questions prompting student-directed exploration of relationships and eventually, making generalizations that were historically teacher-led. As students’ experiences across the grades engage them in developing a deep understanding of mathematics, these experiences simultaneously strengthen their mathematics identity and this strengthened identity, in turn, encourages them to explore mathematics deeply.

EMSs understand that patterns can take multiple forms (e.g., growing and repeating patterns, symbolic and visual patterns), model various kinds of change (e.g., constant, linear, nonlinear, periodic), and can be represented in multiple connected ways (i.e., words, tables, symbols, graphs). They understand that when dealing with a function or rule, an input (x-value) corresponds to exactly one output (y-value) and that some functions have “rules” and some do not (i.e., some can be represented with an equation and some tell a more complicated story, such as melting ice or running a race). EMSs know how to support learners with seeing and describing change or covariance, specifically, how change in one variable (e.g., number of children, an independent variable) results in a predictable change in a related variable (e.g., number of ears, a dependent variable). EMS professionals understand that moving toward generalization requires opportunities for learners to: (1) explore a mathematical situation; (2) develop a conjecture; (3) test the conjecture to determine if it is true or false; (4) if the conjecture is not true, revise and test again; and (5) generalize–if the conjecture is true and justified with evidence (41).

In addition, EMSs understand the characteristics of “simple” patterns that represent proportional relationships and the characteristics of more complex patterns (e.g., non-zero start value or non-proportional linear, nonlinear). They recognize the natural progression of describing relationships with words before expressing them with symbols. They understand the tendency of noticing recursive patterns before attending to the relationships between corresponding values. EMSs use their knowledge to develop intentional instructional moves that build upon recursive pattern observations in support of moving toward explicit rules that are possible when attending to correspondence. They use their “algebra eyes and ears”(42) to influence task selection, design, and implementation with the intent of developing algebraic thinking over time.

C.3.b. Representing functions with visual models and contextual situations

Critical characteristics of functions show up in a variety of representations and features within each representation (i.e., contexts, words, tables, symbols, graphs) suggest functional growth and change that is linear or nonlinear (e.g., quadratic, exponential).  Relatedly, contextualizing and decontextualizing patterns involves moving between abstract representations (i.e., symbols, graphs, tables) and potentially more relatable representations (i.e., visual and physical models, real world contexts) and vice versa. Taking time to translate between mathematics and the real world and look for connections within and across representations supports a deepened understanding of: (a) the function itself, (b) its defining characteristics, and (c) ways those characteristics show up in different representations.

Early work with functions often includes hands-on experiences such as using concrete objects or manipulatives. For example, consider the visual model using toothpicks to form a pattern with squares in a row (i.e., arrangement 1 has 1 square, arrangement 2 has 2 squares, arrangement 3 has 3 squares). One student wrote numeric expressions to communicate the growth and change in the toothpick squares as one square was added (see figure 5). Specifically, the student saw the pattern as adding 3 toothpicks each time after the initial 4 toothpicks–translating across representations from physical or visual to symbolic. Another student saw the pattern in a slightly different way, with 3 toothpicks for every square with one more to close the final square. This student recognized the equivalence across the two expressions stating, “3 + 1 at the end is the same as adding 4 at the beginning”–translating within representations from symbolic to symbolic.