By NCTM
President Linda M. Gojak

NCTM *Summing Up*, October 3, 2013

One of the
most memorable moments I had in teaching mathematics occurred in a fifth-grade
class. We began the year using rectangular arrays as a model to develop the
concept of prime and composite numbers. We hung student-made posters of the
numbers from 1 to 100 with representations of arrays and lists of factors for
each number around the room. By the end of that unit all my students had
mastered multiplication facts and could factor with facility as we began our
work with fractions. The connections among concepts and the use of concrete
representations certainly led to deeper understanding. Later that year,
students worked with a variety of models to find area and perimeter of
rectangles and extended that experience to find the areas of triangles, parallelograms,
and trapezoids. Most students were able to generalize a formula, albeit not
always the most efficient, for each polygon. One day, a student commented that
this was just like what they had studied at the beginning of the year. When I
gave a puzzled look, the class pointed to the posters still on the wall from
our first unit of study and said, “You know, that factor and multiple stuff.” I
had a new appreciation for the power of providing experiences that enable
students to make connections among mathematical ideas. My students remembered
and understood the mathematics that we had studied months earlier!

Since
that experience I have given much thought to the Process Standards in *Principles and Standards for School
Mathematics*, and their impact on teaching. With the current focus on
progressions and trajectories of content standards, the potential of the Connection
Standard (NCTM, 2000) continues to pique my interest. It’s a powerful standard:

Instructional programs from prekindergarten through
grade 12 should enable all students to—

- recognize
and use connections among mathematical ideas;
- understand
how mathematical ideas interconnect and build on one another to produce a
coherent whole;
- recognize
and apply mathematics in contexts outside of mathematics.

Too often, rather than making sense of mathematical
ideas, students focus on remembering procedures or tricks. For example, how
many students learn “flip and multiply” to divide fractions but have no idea
why it works? Often those who understand why the procedure works struggle to
apply it in problem situations. The procedure alone often leads to
misconceptions. Students who work from rote memory often invert the wrong
fraction, forget to change operations, or even apply the rule when multiplying
two fractions. The meaning of operations doesn’t change from whole numbers to
fractions. For example, in the early grades, the understanding that students develop
of division of whole numbers often rests on the idea that “9 ÷ 3,” for example,
asks how many groups of 3 are in 9. As students move to fractions, it is
important to provide them with experiences that connect this whole-number understanding
to similar examples with fractions: “9/16 ÷ 3/16,” for example, asks how many
groups of 3/16 are in 9/16. In this way, students gain a deeper understanding rather
than depending on a memorized procedure and can apply division of fractions to
a variety of problem-solving situations and real-world applications.

Many teachers
use manipulative materials to introduce a new concept. Manipulatives themselves,
however, do not ensure understanding. We must provide experiences that support
students’ efforts to make connections between what they are doing with the
materials and the mathematical ideas that they represent. This takes time and teacher
expertise. Algebra tiles are not an end to teaching basic algebra concepts—when
used appropriately, they provide students with opportunities to connect their work
to the concepts. And it is these connections that enable students to make sense
of the abstract representations.

Although it is important to think about the
connections among concepts within the grade level or courses that we teach, it
is also important to reflect on the connections across grade levels. This work involves
thoughtful discussions with other colleagues about the way that concepts are
taught and the potential linkages among those ideas. Many of us learned mathematics
as isolated pieces of information. Taking a mathematical concept and
considering how it originates, extends, and connects with other concepts across
the grades will help teachers to develop a deeper understanding. It is then
that we can plan instruction that ensures that our students regularly make
connections to help them make sense of the mathematics they are learning.