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# Algebraic Thinking

Algebraic Thinking in the early elementary school should focus on two key areas. The first is Generalized Arithmetic, which involves building generalizations about operations. The two key ideas within generalized arithmetic are (a) understanding the equal sign and (b) understanding and applying properties of operations, such as the commutative and associative properties and the inverse relation between addition and subtraction. Both of the ideas are part of the Common Core Standards for first grade, but many older children will continue to struggle with them. To help children learn about these ideas, teachers should encourage students to reflect on their arithmetic work and look for patterns and general rules. Often, these opportunities are missed in the curriculum. More problematic is that children often infer incorrect ideas from doing traditional arithmetic practice. For more information on Generalized Arithmetic, click here.

The second area is Functional Thinking, which is identifying numerical and geometric patterns and describing their functional relationships – how quantities vary in relation to each other. In pre-K and Kindergarten, students should work with repeating patterns (e.g., ??•??•), with focus on the pattern unit (the part that repeats over and over). In 1st grade and above, students should work with growing patterns and both variables should be explicit so that children can reason about how the two vary together. Teachers should ask students to identify patterns and describe how quantities vary in relation to each other. Simply asking children “what comes next” in a pattern does not support functional thinking and is not doing early algebra. For more information on Functional Thinking, click here.

Supporting both of these topics is a central way to support the Mathematical Practice Standard “Look for and make use of structure.” For more information on this Practice Standard, click here.

Teachers must possess the mathematical knowledge to understand how students learn mathematics as well as an understanding of the trajectory of mathematics content. Understanding the trajectory enables us to effectively plan and make instructional decisions to impact important mathematical learning. This continuum contains K-3 Number and Operations in Base Ten standards, Number and Operations – Fractions standards for grade 3, Counting and Cardinality standards for K, and Number and Operations and Algebraic Thinking standards K-3.

The second area is Functional Thinking, which is identifying numerical and geometric patterns and describing their functional relationships – how quantities vary in relation to each other. In pre-K and Kindergarten, students should work with repeating patterns (e.g., ??•??•), with focus on the pattern unit (the part that repeats over and over). In 1st grade and above, students should work with growing patterns and both variables should be explicit so that children can reason about how the two vary together. Teachers should ask students to identify patterns and describe how quantities vary in relation to each other. Simply asking children “what comes next” in a pattern does not support functional thinking and is not doing early algebra. For more information on Functional Thinking, click here.

Supporting both of these topics is a central way to support the Mathematical Practice Standard “Look for and make use of structure.” For more information on this Practice Standard, click here.

Teachers must possess the mathematical knowledge to understand how students learn mathematics as well as an understanding of the trajectory of mathematics content. Understanding the trajectory enables us to effectively plan and make instructional decisions to impact important mathematical learning. This continuum contains K-3 Number and Operations in Base Ten standards, Number and Operations – Fractions standards for grade 3, Counting and Cardinality standards for K, and Number and Operations and Algebraic Thinking standards K-3.