Uncomplicating Algebra to Meet Common Core Standards in Math, K–8
reviewed by Carol Buckley - January 13, 2016
Title: Uncomplicating Algebra to Meet Common Core Standards in Math, K–8
Author(s): Marian Small
Publisher: Teachers College Press, New York
ISBN: 0807755176, Pages: 165, Year: 2014
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Success in algebra seems to be the proverbial gatekeeper for students, dividing them into two groups: those who can understand algebra, and those who cannot. The Common Core places a heavy emphasis on algebra and successful completion of this subject as measured by a standardized assessment is a graduation requirement in many states. Furthermore, successful completion of algebra in high school is often required before a student is considered for certain disciplines or areas of study at the college level. In Uncomplicating Algebra to Meet Common Core Standards in Math, K8, Marian Small looks at the bigger picture, then brilliantly offers suggestions for instructional techniques that will promote a more complete understanding of algebra for K8 students, setting the stage for success in this vitally important subject both in high school and beyond.
Before moving forward to a more complete understanding of algebra by the masses, Small believes that mathematics educators must first understand some of the obstacles which have kept many from achieving this goal with traditional instruction. Small points out that arithmetic and algebra are inseparable. She contends that only after teachers facilitate lessons that lead students to recognize this connection can these learners achieve success in algebra. Specifically, students must see the relationships between values rather than focusing on correct calculations. It is also critically important for students to understand the inverse relationships between operations. Teaching through problem solving is essential, and students must learn to represent problems in a variety of ways. For example, the equal sign should be seen to represent two equivalent expressions that are balanced, rather than simply a means of achieving a correct answer.
In Uncomplicating Algebra, Small attempts to define what algebra is, and points out the relationship between this definition and that of the Common Core State Standards, which is different. She addresses the Standards of Mathematical Practice and provides a sampling of ways in which teachers across the K8 grade span can establish a mathematical environment that will promote mathematical thinking and conceptual understanding, rather than one which strictly values a correct answer. Several specific types of algebra errors made by students are delineated in this book. Small points out that all of these errors are typically rooted in one of three mathematical misunderstandings: failure to understand the role of the equal sign, an incomplete understanding of variables, and inappropriate generalizations of mathematical concepts.
Emphasizing the most important underlying ideas at each grade level as they relate to early algebraic concepts, Small unpacks the Common Core State Standards at each stage to form a solid base of mathematical comprehension, ultimately leading to a conceptual understanding in algebra. Here is a brief summary of each level.
The decomposition of numbers supports the use of addition and subtraction equations. Kindergarten students need to understand the idea of balance when thinking about an equation to give evidence of a solid start in the development of algebraic thinking.
Teachers need to build upon the idea of an equation being balanced and transition students to recognize the relationship between both sides of the equation. Teachers should use equations as means for modeling problems. Students at this age are still using concrete objects to represent and solve problems. Students should be reading equations in more meaningful and deeper ways. In other words, students in first grade should begin to realize that the parts of an equation have meaning and represent something. This is the start of abstract reasoning where equations may begin to have unknown values, sometimes in the form of a box.
Students continue to refine their use of equations to model problems and begin to answer questions based on an equation. At this grade level, students are developing the ability to differentiate between even and odd, and are beginning to represent problems symbolically with Math tools such as a 100-chart, ten-frames, or an array.
Instruction should focus on the use of equations to represent and solve problems involving multiplication and division. Students should begin to understand the relationship between multiplication and division, and solve problems involving the four operations. Students will also identify and explain patterns in arithmetic. Letters will be introduced to represent variables at this level, and students will see an increasing variety of ways to present equations. Students are encouraged to use invented strategies for increasingly complex problems.
Students should be progressively comfortable with making arithmetic generalizations and solve problems with whole numbers using the four operations and generate equivalent fractions. Here students will have a more complete understanding of the importance of estimating and interpreting remainders.
Students begin to analyze numeric patterns and relationships as well as expand previous knowledge of equations. The relationship between algebra and geometry begins to evolve, with the use of a coordinate plane, measurement of three-dimensional shapes, and plotting linear equations.
Grade five concepts are further developed in grade six with proportional reasoning, equivalent expressions, and inequalities involving rational numbers added to previous learning. Function tables are created and plotted on a coordinate plane. Distances on the plane are measured, and the use of an algebraic/math language is in full motion.
Proportions, rational numbers, equivalent expressions, solving linear equations and inequalities, and solving measurement problems are key concepts unpacked in seventh grade. Exponent conventions and properties, finding equations of lines, solving more complex linear equations, solving two equations in two unknowns, exploration of linear functions, discovering the Pythagorean Theorem, and linear trends are all concepts for eighth grade.
Throughout the book, Small makes a clear connection between the progression of skills at each grade level, and how they lead to a methodical uncomplicating of algebra until students are exposed to more complex algebraic concepts by eighth or ninth grade. She recognizes that teachers may be teaching content beyond their comfort level. With each concept discussed, Small offers instructional recommendations about how teachers could engage the students in the learning. In addition, she provides sample questions teachers might use to promote deeper critical thinking on the part of the students while interacting with the new content.
In summary, Uncomplicating Algebra to Meet Common Core Standards in Math, K8 is a well-written book that systematically unpacks K-8 algebraic concepts. This book would be ideal for all math coaches and curriculum directors as a basis for professional development of K8 classroom teachers. Smalls writing is clear and coherent. She focuses on instruction to make a difference in student success with algebraic concepts. As a result, her book supports teachers by demonstrating a progression of skills by thoroughly describing the components of those skills, and modeling higher order thinking questions to promote a deeper conceptual understanding.