ByZandra De Araujo, Barbara J. Dougherty, and Fay Zenigami
When students move from the elementary
to the middle grades, they progress from arithmetic to algebra, shifting their
attention from known, specific quantities to unknown, often generalized ones.
To succeed in this transition—and to develop a robust foundation for the idea
of a function—students must develop a rich understanding of the meaning of a
variable, an expression, and an equation.
This work builds on students’ earlier
work in the elementary grades, where experiences with variables typically include
solving problems like 3 + ? = 7 and ? – 2 = 6. But this early work may also
leave students with incomplete ideas or misconceptions—a shaky foundation on
which to build important concepts for algebra.
Middle-grades students may suppose,
for example, that a variable is always replaceable by one, and only one, value,
leaving them puzzled about what a
stands for in an equation like 3(a
+ 6) = 3a + 18.
Likewise, students’ early work with
the equals sign may limit their ability to interpret the meaning of the symbol
=, which many students read either as a command to perform a computation in a
“problem” that they expect to find on the symbol’s left or as a signal indicating
that “the answer comes next,” on its right. Often, students’ early experiences do
not prepare them to think of “=” as an indicator of a relationship of
equivalence—or to work with and solve inequalities.
Putting Essential Understanding of Expressions and Equations
into Practice in Grades 6–8 focuses on common
misconceptions about variables, expressions, equations, and functions that
students bring with them to middle school. The authorspresent tasks given to middle-grades students and examine samples
of students’ responses to them to show how middle-grades teachers can use these
and other, similar tasks to bring misconceptions to the surface, where teachers
and students alike can inspect them, recognize their errors and limitations, and
dispel and replace faulty thinking with more robust ideas. Throughout,
the authors include questions for the reader’s reflection, and the last chapter
provides a look back and ahead at related learning in earlier and later grades.
Three appendixes provide extra resources for teachers, and reproducible
activity pages for all tasks discussed in the book are available at
This book is the eleventh volume in NCTM’s Putting Essential
Understanding into Practice Series, which builds on the earlier Essential
Understanding Series. The original series detailed the content knowledge that
teachers need to teach challenging topics in school mathematics. The successor
series takes the next step, focusing on the related pedagogical knowledge that
teachers must have for success in the classroom as they help students develop
ideas that are essential for their mathematical growth.