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Algorithms in Everyday Mathematics
Algorithms in Everyday Mathematics
Everyday Mathematics includes a comprehensive
treatment of computation. Students learn to
compute mentally, with paper and pencil, and by machine; they learn to find both
exact and
approximate results; and, most importantly, they learn what computations to make
and how to
interpret their answers. The following sections describe in general terms how
Everyday
Mathematics approaches exact paperandpencil methods for basic operations with
whole
numbers. For details about particular algorithms and for information about how
the program
teaches mental arithmetic, estimation, and computation with decimals and
fractions, see the
Everyday Mathematics Teacher’s Reference Manual.
In Everyday Mathematics, computational proficiency
develops gradually. In the beginning,
before they have learned formal procedures, students use what they know to solve
problems .
They use their common sense and their informal knowledge of mathematics to
devise their own
procedures for adding, subtracting, and so on. As students describe, compare ,
and refine their
approaches, several alternative methods are identified. Some of these
alternatives are based on
students’ own ideas; others are introduced by the teacher or in the materials.
For each basic
operation, students are expected to become proficient at one or more of these
alternative
methods.
The materials also identify one of the alternative
algorithms for each operation as a focus
algorithm. The purpose of the focus algorithms is twofold: (i) to provide
backup methods for
those students who do not achieve proficiency using other algorithms, and (ii)
to provide a
common basis for further work. All students are expected to learn the focus
algorithms at some
point, though, as always in Everyday Mathematics, students are encouraged to use
whatever
method they prefer when they solve problems.
The following sections describe this process in
more detail. Note, however, that although the
basic approach is similar across all four operations, the emphasis varies from
operation to
operation because of differences among the operations and differences in
students’ background
knowledge. For example, it is easier to invent efficient procedures for addition
than for division.
There is, accordingly, less expectation that students will devise efficient
procedures for solving
multidigit long division problems than that they will succeed in finding their
own good ways to
solve multidigit addition problems.
Invented Procedures
When they are first learning an operation, Everyday
Mathematics students are asked to solve
problems involving the operation before they have developed or learned
systematic procedures
for solving such problems. In second grade, for example, students are asked to
solve multidigit
subtraction problems. They might solve such problems by counting up from the
smaller to the
larger number, or by using tools such as number grids or base10 blocks, or they
may use some
other strategy that makes sense to them. This stage of algorithm development may
be called the
invented procedures phase.
To succeed in devising effective procedures,
students must have a good background in the
following areas:
• Our system for writing numbers. In particular,
students need to understand place value.
• Basic facts. To be successful at carrying out multistep computational
procedures,students need proficiency with the basic arithmetic facts.
• The meanings of the operations and the relationships among operations.
To solve 37 –
25, for example, a student might reason, “What number must I add to 25 to get
37?”
Research indicates that students can succeed in
inventing their own methods for solving basic
computational problems (Madell, 1985; Kamii & Joseph, 1988; Cobb & Merkel, 1989;
Resnick,
Lesgold, & Bill, 1990; Carpenter, Fennema, & Franke, 1992). Inventing procedures
flourishes
when:
• the classroom environment is accepting and
supportive;
• adequate time for experimentation is allotted;
• computational tasks are embedded in real life contexts; and
• students discuss their solution strategies with the teacher and with one
another.
The discussion of students’ methods is especially
important. Through classroom discussion,
teachers gain valuable insight into students’ thinking and progress, while
students become more
skilled at communicating mathematics and at understanding and critiquing others’
ideas and
methods. Talking about why a method works, whether a method will work in every
case, which
method is most efficient, and so on, helps students understand that mathematics
is based on
common sense and objective reason, not the teacher’s whim. Such discussions lay
the
foundations for later formal work with proof.
The inventedprocedures approach to algorithm development has many advantages:
• Students who invent their own methods learn that
their intuitive methods are valid and that
mathematics makes sense.
• Inventing procedures promotes conceptual understanding of the operations
and of base10
placevalue numeration. When students build their own procedures on their prior
mathematical knowledge and common sense, new knowledge is integrated into a
meaningful
network so that it is understood better and retained more easily.
• Inventing procedures promotes proficiency with mental arithmetic. Many
techniques that
students invent are much more effective for mental arithmetic than standard
paperandpencil
algorithms. Students develop a broad repertoire of computational methods and the
flexibility
to choose whichever procedure is most appropriate in any particular situation.
• Inventing procedures involves solving problems that the students do not
already know how to
solve, so they gain valuable experience with nonroutine problems. They must
learn to
manage their resources: How long will this take? Am I wasting my time with this
approach?
Is there a better way? Such resource management is especially important in
complex
problem solving. As students devise their own methods, they also develop
persistence and
confidence in dealing with difficult problems.
• Students are more motivated when they don’t have to learn standard
paperandpencil
algorithms by rote. People are more interested in what they can understand, and
students
generally understand their own methods.
• Students become adept at changing the representations of ideas and
problems, translating
readily among manipulatives, words, pictures, and symbols . The ability to
represent a
problem in more than one way is important in problem solving. Students also
develop the
ability to transform any given problem into an equivalent , easier problem. For
example, 32 –
17 can be transformed to 35 – 20 by adding 3 to both numbers.
Another argument in favor of the
inventedprocedures approach is that learning a single standard
algorithm for each operation, especially at an early stage, may actually inhibit
the development
of students’ mathematical understanding. Premature teaching of standard
paperandpencil
algorithms can foster persistent errors and buggy algorithms and can lead
students to use the
algorithms as substitutes for thinking and common sense.
Alternative Algorithms
Over the centuries, people have invented many
algorithms for the basic arithmetic operations.
Each of these historical algorithms was developed in some context. For example,
one does not
need to know the multiplication tables to do “Russian Peasant Multiplication” —
all that is
required is doubling, halving, and adding. Many historical algorithms were
“standard” at some
time and place, and some are used to this day. The current “European” method of
subtraction, for
example, is not the same as the method most Americans learned in school.
The U.S. standard algorithms—those that have been
most widely taught in this country in the
past 100 years—are highly efficient for paperandpencil computation, but that
does not
necessarily make them the best choice for school mathematics today. The best
algorithm for one
purpose may not be the best algorithm for another purpose. The most efficient
algorithm for
paperandpencil computation is not likely to be the best algorithm for helping
students
understand the operation, nor is it likely to be the best algorithm for mental
arithmetic and
estimation. Moreover, if efficiency is the goal, in most situations it is
unlikely that any paperand
pencil algorithm will be superior to mental arithmetic or a calculator.
If paperandpencil computation is to continue to
be part of the elementary school mathematics
curriculum, as the authors of Everyday Mathematics believe it should, then
alternatives to the
U.S. standard algorithms should be considered. Such alternatives may have better
costbenefit
ratios than the standard algorithms. Historical algorithms are one source of
alternatives. Studentinvented
procedures are another rich source. A third source is mathematicians and
mathematics
educators who are devising new methods that are well adapted to our needs today.
The Everyday
Mathematics approach to computation uses alternative algorithms from all these
sources.
In Everyday Mathematics, as students explain,
compare, and contrast their own invented
procedures, several common alternative methods are identified. Often these are
formalizations of
approaches that students have devised. The columnaddition method, for example,
was shown
and explained to the Everyday Mathematics authors by a first grader. Other
alternative
algorithms, including both historical and new algorithms, are introduced by the
teacher or the
materials. The partialquotients method, for example, first appeared in print in
Isaac
Greenwood’s Arithmeticks in 1729.
Many alternative algorithms, whether based on student methods or
introduced by the teacher, are
highly efficient and easier to understand and learn than the U.S. traditional
algorithms. For
example, lattice multiplication requires only a knowledge of basic
multiplication facts and the
ability to add strings of singledigit numbers, and yet it is more efficient
than the traditional long
multiplication algorithm for all but the simplest multidigit problems. Students
are urged to
experiment with various methods for each operation in order to become proficient
at using at
least one alternative.
The alternativealgorithms phase of algorithm development has significant advantages :
• A key belief in Everyday Mathematics is that problems
can (and should) be solved in more
than one way. This belief in multiple solutions is supported by the
alternativealgorithms
approach to developing computational proficiency.
• Providing several alternative algorithms for each operation affords
flexibility. A onesizefits
all approach may work for many students, but the goal in Everyday Mathematics is
to
reach all students. One algorithm may work well for one student, but another
algorithm may
be better for another student.
• Different algorithms are often based on different concepts, so studying
several algorithms for
an operation can help students understand the operation.
• Presenting several alternative algorithms gives the message that mathematics
is a creative
field. In today’s rapidly changing world, people who can break away from
traditional ways of
thinking are especially valuable.
Teaching multiple algorithms for important operations is common in mathematics
outside the
elementary school. In computer science, for example, alternative algorithms for
fundamental
operations are always included in textbooks. An entire volume of Donald Knuth’s
monumental
work, The Art of Computer Programming (1998), is devoted exclusively to sorting
and
searching. Knuth presents many inefficient sorting algorithms because they are
instructive.
Focus Algorithms
The authors of Everyday Mathematics believe that the
inventedprocedures/alternativealgorithms
approach described above is a radical improvement over the traditional approach
to
developing computational proficiency. The Everyday Mathematics approach is based
on decades
of research and was refined during extensive fieldtesting. Student achievement
studies indicate,
moreover, that when the approach is properly implemented, students do achieve
high levels of
computational proficiency (Carroll, 1996, 1997; Carroll & Porter, 1997, 1998;
Fuson, Carroll, &
Drueck, 2000; Carroll, Fuson, & Diamond, 2000; Carroll & Isaacs, in press).
In the second edition of Everyday Mathematics, the
approach described above is extended in one
significant way: For each operation, one of the several alternative algorithms
is identified as a
focus algorithm. All students are expected to learn the focus algorithms
eventually, although, as
usual in Everyday Mathematics, proficiency is expected only after multiple
exposures over
several years. Students are also not required to use the focus algorithms in
solving problems if
they have alternatives they prefer. For addition, the focus algorithm is
partialsums; for
subtraction, tradefirst; for multiplication, partial products ; and for
division, partialquotients.
(See the Everyday Mathematics Teacher’s Reference Manual for details about these
and other
algorithms.)
The focus algorithms are powerful , relatively efficient,
and easy to understand and learn, but
they are not meant to shortcircuit the
inventedprocedures/alternativealgorithms approach
described above. Students still need to grapple with problems on their own and
explore
alternative algorithms. The focus algorithms have been introduced for two
specific reasons. One
is that they provide reliable alternatives for students who do not develop
effective procedures on
their own. The focus algorithm for subtraction, for example, is introduced in
second grade.
Second grade students are not expected to be proficient with the method, though
they are
expected to be able to solve multidigit subtraction problems in some way, by
using counting,
number grids, manipulatives, or some other method. A fourth grade student,
however, who does
not have a reliable method for subtraction despite several years of work with
invented procedures
and alternative algorithms should focus on the tradefirst method so that he or
she will have at
least one reliable way to subtract with paper and pencil. One aim of the
focusalgorithm
approach is to promote flexibility while ensuring that all students know at
least one reliable
method for each operation.
Another reason for introducing focus algorithms is
to provide a common ground for the further
development of mathematical ideas. Most algorithms for operations with whole
numbers, for
example, can be extended to decimals. This is easier to show in a class at least
one wholenumber
algorithm for each operation is known by every student. Focus algorithms provide
a
common language that facilitates classroom discussion.
Focus algorithms were introduced in response to
teachers’ concerns. However, a teacher who
has developed an effective strategy for teaching algorithms, and who feels that
the focusalgorithm
approach is unnecessary or compromises that strategy, is not obliged to adopt
the
focusalgorithm approach.
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