As students begin to work with functions, sometimes as early as 6th grade, x/y function tables become a standard way to organize information and derive “y” for given values of “x.” Students are asked to "find" different values of "y" given different values of "x," and to fill-in these values on a table.
Completing a function table requires multiple steps of thinking, which students may or may not be able to complete effectively in their heads. Whereas advanced students and math teachers are able to select an "x" value, create a numerical equation using that "x" value, hold that numerical equation in their heads, and perform one or more calculations to determine the value of the numerical equation, younger students and students with math learning disabilities may become confused, lost, and/or make calculation mistakes trying to maintain the high cognitive demand that this task requires.
Expanded function tables are a great tool for scaffolding and supporting students' thinking as they plug-in different values of "x" to solve for "y." An expanded table gives students a space to write down their thinking for each step of the process. This is beneficial because it takes away the demand of having to hold information in their working memory, which decreases their cognitive load and gives their brain more space to think. It also helps students to keep track of where they are in the problem solving process. For students who have a hard time remembering the sequence of steps, an expanded table also provides a guide to what step comes next. And for students with calculation difficulties, a 2-step expanded table provides space to write out any calculations that they need to solve on paper (i.e., "Side Math"), and it allows students to focus their whole attention on doing this calculation because the table is acting as their working memory and is also saving their place in the problem solving process for them.
Here are some ways to integrate expanded function tables into your teaching:
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Blending her backgrounds in mathematics education and educational/school psychology, Adena offers an integrated perspective to understanding and supporting students who struggle with math.