Our Philosophy for the Teaching and Learning of Mathematics
Barbara Topiol, Principal
Tim Kaltenecker,
District Director of Curriculum

It is our belief that skill development and procedural knowledge should go hand-in-hand with
conceptual knowledge. At Wampus, students engage in activities that lead to an understanding
of underlying concepts and show how mathematical ideas relate to one another. Skills are not
taught out of context and without meaning; instead, skills grow from an understanding of the
big ideas and are taught through hands-on experiences.

To this end, students engage in activities of inquiry and problem-solving. By doing this, they
draw meaning from their observations, making sense of the mathematics and how it relates to
previously learned ideas. Students are encouraged to invent their own methods to solve
problems (Cobb, 1991.) This approach allows each student the opportunity to engage in the
material at her own level of understanding; and from this place of understanding, she will move
forward to learn new concepts and skills with confidence.

This does not mean, however, that skills and procedures are never “taught” by the classroom
teacher. It means that the student understands the concepts and the meaning behind the skills
before learning procedures. Thus, the procedures are taught within the context of meaning. At
defined points in the curriculum, students are required to use the most efficient methods for
computation and problem-solving. But efficiency (traditional procedures) is not to be used at
the expense of understanding. By learning procedures in the context of meaning, students are
likely to learn and understand concepts more easily in the upper grades than if they were to
learn skills and procedures in isolation without understanding the underlying ideas.\

Standards-Based Mathematics

The mathematics program at Wampus, Growing with Math, is a standards-based spiraled
program and is a continuation from Coman Hill’s K – 2 program. The spiral approach meets the
cognitive needs of students at this grade level. There are some topics in the math program that
are meant for exposure only. That is, students are not expected to master these concepts; but
instead, they are exposed to them in a developmentally appropriate way. Later in the program,
the students will encounter this content again with the expectation of mastery. The program is
carefully designed to take advantage of children’s cognitive ability and curiosity in an ageappropriate

Growing with Math has been carefully aligned to the New York State Learning Standards to
ensure that students reach mastery as per the state requirements. More often than not, the
math program at Wampus includes more content and more thinking skills than required by the
state standards. We value high standards and believe our expectations are realistic for all

Using Language in Mathematics

The math program requires students to talk and write about mathematics on a regular basis. By
doing this students internalize the math concepts and become proficient at communicating their
understanding. This language-based approach is developmentally appropriate and researchbased;
students learn concepts through exploring, writing and speaking mathematics (Berk,
2000.) Furthermore, students develop a strong number sense through speaking about math and
through estimation. This is done through number sense activities each day. These activities
provide opportunities for students to understand big ideas within number sense and reasoning.
As students engage in these discussions they develop good estimation and reasoning skills. This
should be supported outside the classroom as well.

The Parent’s Role

Parents play an important role in their child’s mathematical development. It is crucial that
parents support the math work that is coming home and that they provide opportunities for
their child to develop a positive attitude toward mathematics. First, parents should use the
vocabulary used in the classroom and support the approach developed by teachers. For
example, we no longer use the term “borrow” when referring to subtraction. Instead, students
“regroup” numbers. This is not a fastidious technicality. Instead, it is grounded in the conceptual
nature of the mathematics. To borrow has no conceptual meaning. However, students regroup
numbers in various ways as they develop flexibility with mathematical representation. For
example, the number 27 means two “tens” and 7 “ones.” Students can also represent 27 as one
“ten” and 17 “ones.” Thus, they regrouped 20 + 7 as 10 + 17. This flexible thinking about
numbers – promoting a solid understanding of place value – benefits students as they learn
subtraction and multiplication. Parents must support this thinking so their children develop a
sound mathematical foundation in a consistent manner.

Second, parents need to promote a positive climate for mathematics at home. Parents should
never tell their child that they did poorly in math as fear of math is learned from one’s
environment (Fiore 1999.) Instead, parents should find opportunities for their children to
explore mathematical thinking informally. Many board games promote good problem solving.
Also, children can estimate during a visit to the grocery store, for example. The Parent
Resource Guide offers suggestions on ways parents can support a positive mathematical
environment at home.

The Facts are the Facts

Finally, students will learn facts and procedures. It is readily misunderstood that a standardsbased
math program is “fuzzy” math. This is not true with our math program. Understanding
concepts and big ideas is not easy; in fact, most students find this aspect of the curriculum most
challenging. Memorizing procedures for multiplying two-digit numbers, for example, is not
difficult; most students can do this, and this is what we traditionally view as mathematics.
However, mathematical thinking – observing what happens when you multiply consecutively by
2, for example, and describing and generalizing the pattern – is much more challenging for
students. But this type of thinking benefits students’ mathematical development more than
memorizing a procedure (Thornton, 1990; Isaacs and Carroll, 1999.) That said, our curriculum
necessitates a strong background with basic facts and skills, and these items are identified
throughout our program. For example, all students learn their 10 x 10 multiplication facts in
grade 3, and this is reinforced and extended in grade 4. The approach, however, is not purely
memorization but instead built from students’ previous work in k – 2 with skip counting. Skip
counting in the lower elementary grades teaches students about patterns; this helps with the
memorization of facts later. Again, the math content is learned through meaning and context,
an approach that supports conceptual understanding. Parents can be confident that facts and
procedures will be taught, albeit, not the same way they learned it in school. This is not to be
feared; the methodology is research-based, and moreover, consistent from grade to grade.

Wampus Elementary School | 41 Wampus Avenue | Armonk, NY 10504 | (914) 273-4190