Skills & Strategies #4  Visualizing and Understanding Square Roots Using a Multiplication Table4/10/2017
Square roots can be a confusing concept. They can be especially tricky for younger students (e.g., middle schoolers) who may not be developmentally ready for the concept, for students who have difficulty making connections between concrete and abstract concepts, and for students who have difficulty visualizing information.
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 fillin 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. Some upper elementary, middle, and high school students do not know their basic multiplication facts. These students not only struggle with using multiplication to find products, but they also have difficulty with other types of math problems and mathematical thinking that require an understanding of the quantitative relationships between specific factors and products, and between specific dividends and quotients. A multiplication table has become an essential tool for my students who lack mastery in multiplication facts. Once students become familiar and comfortable with using a multiplication table (stay tuned for more on to strategies to achieve this), it becomes an ally and a tool that can support their mathematical thinking in ways that cannot be achieved with a calculator. Here are four examples of nonmultiplication problems for which a multiplication table can help a student who struggles with his multiplication facts.
The ChallengeMany students do math very quickly. For some, a quick pace might be appropriate if they are cognitively engaged and monitoring their thinking at this speed. However, for most students, doing math too quickly means rushing through problems without fully engaging their minds, impulsively writing down answers before they have thought them through, and completing problems on autopilot without taking thought as to what they are really doing. Speedy math often leads to underlearning the material, creating sloppy and often illegible work, and of course, making preventable mistakes.

SOLVING MATH PROBLEMS BLOGBlending her backgrounds in mathematics education and educational/school psychology, Adena offers an integrated perspective to understanding and supporting students who struggle with math.
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