Algebraic Thinking with Boom Cards

Teaching Algebraic thinking early and often is a core feature of the Common Core math standards (and standards with similar underlying foundations such as TEKS). Algebraic thinking is the ability to recognize and analyze patterns and relationships in a mathematical context. The Common Core/TEKS approach replaces an elementary curriculum focused primarily on calculation with a model more like that used elsewhere in the world (yes Canada you beat us too it!). We still have some time in the U.S. before we’ll see the adoption of the Common Core methodology influencing high school math scores: the first cohort to have Common Core standards from Kindergarten is just now entering 5th grade.

Newer research casts significant doubt on earlier findings that students are not developmentally ready for algebra until a certain age, suggesting that those finding were are a function of instructional shortcomings, not neurological limitations. Neuroscience has shown that introducing concepts such as variable notation (representing numbers with letters or shapes) is within the reach of even lower primary-grade students.

Did you know?

When performing mathematical thinking, our brains activate the occpital lobes, the frontal cortex, and our parietal lobes. The parietal lobes help us find our way home. They combine mental maps with proprioceptive feedback to perform real world geometry and trigonometry. In the educational context, the parietal system helps us transform sequential information into quasi-spatial information, transcending order to find meaning. It is used to comprehend spoken language, perceive melody, and perform mathematical reasoning. If your students have basic navigational skills or can hold a melody, they have the baseline for algebraic thinking.

Introducing Algebraic Thinking, a pathway to success

Students begin developing algebraic thinking when they learn to decompose numbers in kindergarten. By first grade, they are ready to begin working with variable notation when solving basic addition and subtraction problems. They are also introduced to the first set of abstract rules that help them decipher relationships: the commutative and associative properties of addition. They also explore the relationships between addition and subtraction and the meaning of the equal sign. Patterns of equal groups of objects are then introduced to lay the foundation for reasoning about multiplication.

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The foundations laid with patterns of repeated addition in second grade, expand in upper elementary into manipulations with multiplication and division. The commutative, associative and identity properties of multiplication are introduced. Prior work with variable notation sets students up for success in understanding division as an unknown-factor problem. Complexity accelerates into fourth and fifth grade, adding fractions and decimals into the mix. Students use algebraic thinking to explore comparative relationships and practice writing equations using variable notation. Pattern and relationship work continues, with a focus on the patterns of factors.

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With these solid foundations, students are ready to continue developing their skills into middle school, high school and beyond, working with expressions, equation, inequalities, and when ready, working with polynomials, rational functions, and more.

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Resources for Deeper Investigation

Bárbara M. Brizuela, Maria Blanton, Katharine Sawrey, Ashley Newman-Owens & Angela Murphy Gardiner, Children’s Use of Variables and Variable Notation to Represent Their Algebraic Ideas, Mathematical Thinking and Learning, Vol. 17, Iss. 1, 2015.

Carolyn Kieran (Editor), Teaching and Learning Algebraic Thinking with 5-to 12-Year-Olds: The Global Evolution of an Emerging Field of Resarch and Practice, Springer International Publishing AG 2018.

Cathy Seely, A Journey in Algebraic Thinking, NCTM News Bulletin, Sept. 2004.

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