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 paper-and-pencil 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 two-fold: (i) to provide back-up 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 base-10 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 invented-procedures 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 base-10
place- value 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 paper-and-pencil
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 non-routine 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 paper-and-pencil
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 invented-procedures 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 paper-and-pencil
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 paper-and-pencil 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
paper-and-pencil 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 paper-and-pencil 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 cost-benefit
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 column-addition 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 partial-quotients 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 single-digit 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 alternative-algorithms 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 alternative-algorithms
approach to developing computational proficiency.
　• Providing several alternative algorithms for each operation affords flexibility. A one-sizefits-
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 invented-procedures/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 partial-sums; for
subtraction, trade-first; for multiplication, partial-products; and for division, partial-quotients.
(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 short-circuit the invented-procedures/alternative-algorithms 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 trade-first method so that he or she will have at
least one reliable way to subtract with paper and pencil. One aim of the focus-algorithm
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
focus-algorithm approach.