Texas Mathematics Teacher Spring/Summer 2021 - 25

Constructing a Firm Foundation for Multiplication
Curricular materials such as the Introduction to
Multiplication and Division: Working with Equal Groups
Unit available through Achieve the Core (n.d.), provide
opportunities for students to create representations of
equal-sized groups to model the connections between
addition and multiplication as they develop fluency. The
unit is available for download at
https://achievethecore.org.

Figure 2. The Very Hungry Caterpillar task adapted from
Illustrative Mathematics (Illustrative Mathematics, n.d.).
In these formative years, building connections between
these representations is obvious as students are
introduced to new mathematical symbols and develop an
understanding of the language of mathematics. By the end
of second grade, there is an expectation that students have
developed fluency with addition and subtraction. The
foundation should have been laid and the focus shifted
from the development of conceptual understanding to
establishing procedural fluency.
Building Onto the Foundation
For third graders, operational concept expectations pivot
from addition and subtraction to multiplication. Educators
are left at a crossroads: Students have developed fluency
with addition. They understand that mathematics is a
language with conventions, concepts, and skills. We ask,
is this foundation enough to introduce multiplication as
a procedural extension of addition? The short answer is
" no. " This quick transition is akin to thinking a house is
finished once the exterior is complete without completing
the hard work of adding drywall and flooring; the interior
is not fully developed. Similarly, teachers need to engage
students in the important, challenging work of building a
framework of understanding around multiplication. This
means " build[ing] procedural fluency from conceptual
understanding, " (NCTM, 2014, p. 42) recognizing
that the increased time for development of conceptual
understanding is an investment. Specifically, " effective
teaching of mathematics builds fluency with procedures
on a foundation of conceptual understanding so that
students, over time, become skillful in using procedures
flexibly as they solve contextual and mathematical
problems " (NCTM, 2014, p. 42).
A meaningful introduction to multiplication builds on the
same models that served to establish addition fluency.
Children can utilize the same tools (e.g., counters, number
lines, and contextual stories) to build multiplication
models such as arrays, equal-sized groups, equal-sized
jumps on a number line, and area tiling. As with addition,
connections between these representations should be
made, as appropriate (see Figure 3).
www.tctmonline.org

Figure 3. Students build fluency with multiplication facts
by modeling with area model, equal-sized jumps on the
number line and equal-sized groups.
Over time the focus of instruction should shift from
conceptual understanding to procedural fluency as
students develop proficiency and efficiency with
multiplication of single digit factors. In grade four,
students utilize the flexibility developed through the
meaningful development of fluency with multiplication
as they develop fluency with multiplying multi-digit
numbers. Like before, models are essential. For example,
the area model in Figure 4 can be used by students to
develop fluency for multiplication facts transfers to multidigit multiplication and provides a connection to the
partial products algorithm. Students build connections
between the area model for multi-digit multiplication and
multiplication algorithms as they work toward fluency
and automaticity with multi-digit multiplication.

Spring/Summer 2021

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https://achievethecore.org https://tasks.illustrativemathematics.org/content-standards/tasks/1150 https://www.tctmonline.org/

Texas Mathematics Teacher Spring/Summer 2021

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http://www.brightcopy.net/allen/txmt/68-02
http://www.brightcopy.net/allen/txmt/68-01
http://www.brightcopy.net/allen/txmt/67-01
http://www.brightcopy.net/allen/txmt/66-02
http://www.brightcopy.net/allen/txmt/66-01
http://www.brightcopy.net/allen/txmt/65-02
http://www.brightcopy.net/allen/txmt/65-01
http://www.brightcopy.net/allen/txmt/64-02
https://www.nxtbook.com/allen/txmt/64-1
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