logo Your Math Help is on the Way!

More Math Help

Algebraic Symmetries
Radical Expressions and Equation
The Exponential Function
Math 1010-3 Exam #3 Review Guide
Rational Numbers Worksheet
Are You Ready for Math 65?
Solving Simultaneous Equations Using the TI-89
Number Theory: Fermat's Last Theorem
Course Syllabus for Intermediate Algebra
Solving Inequalities with Logarithms and Exponents
Introduction to Algebra Concepts and Skills
Other Miscellaneous Problems
Syllabus for Calculus
Elementary Linear Algebra
Adding and Subtracting Fractions without a Common Denominator
Pre-Algebra and Algebra Instruction and Assessments
Mathstar Research Lesson Plan
Least Common Multiple
Division of Polynomials
Counting Factors,Greatest Common Factor,and Least Common Multiple
Real Numbers, Exponents and Radicals
Math 115 Final Exam Review
Root Finding and Nonlinear Sets of Equations
Math 201-1 Final Review Sheet
Powers of Ten and Calculations
Solving Radical Equations
Factoring Polynomials
Section 8
Declining Price, Profits and Graphing
Arithmetic and Algebraic Structures
Locally Adjusted Robust Regression
Topics in Mathematics
Syllabus for Mathematics
The Quest To Learn The Universal Arithmetic
Solving Linear Equations in One Variable
Examples of direct proof and disproof
Algebra I
Quadratic Functions and Concavity
More on Equivalence Relations
Solve Quadratic Equations by the Quadratic Formula
Solving Equations and Inequaliti
MATH 120 Exam 3 Information
Rational Number Ideas and Symbols
Math Review Sheet for Exam 3
Linear Algebra Notes
Factoring Trinomials
Math 097 Test 2
Intermediate Algebra Syllabus
How to Graphically Interpret the Complex Roots of a Quadratic Equation
The General, Linear Equation
Written Dialog for Problem Solving
Radian,Arc Length,and Area of a Sector
Internet Intermediate Algebra
End Behavior for linear and Quadratic Functions
Division of Mathematics
161 Practice Exam 2
General linear equations
Algebraic Symmetries
Math 20A Final Review Outline
Description of Mathematics
Math 150 Lecture Notes for Chapter 2 Equations and Inequalities
Course Syllabus for Prealgebra
Basic Operations with Decimals: Division
Mathematics Content Expectations
Academic Systems Algebra Scope and Sequence
Syllabus for Introduction to Algebra
Syllabus for Elementary Algebra
Environmental Algebra
More Math Practice Problems
Intermediate Algebra
Syllabus for Linear Algebra and Differential Equations
Intermediate Algebra
Rational Expressions and Their Simplification
Course Syllabus for Intermediate Algebra
GRE Review - Algebra
Foundations of Analysis
Finding Real Zeros of Polynomial Functions
Model Academic Standards for Mathematics
Study Guide for Math 101 Chapter 3
Real Numbers
Math 9, Fall 2009, Calendar
Final Review Solutions
Exponential and Logarithmic Functions

Try the Free Math Solver or Scroll down to Tutorials!












Please use this form if you would like
to have this math solver on your website,
free of charge.

Algorithms in Everyday Mathematics

  An algorithm is a step-by-step procedure designed to achieve a certain objective in a finite time,
often with several steps that repeat or “loop” as many times as necessary. The most familiar
algorithms are the elementary school procedures for adding, subtracting, multiplying, and
dividing, but there are many other algorithms in mathematics.

Algorithms in School Mathematics

  The place of algorithms in school mathematics is changing. One reason is the widespread
availability of calculators and computers outside of school. Before such machines were invented,
the preparation of workers who could carry out complicated computations by hand was an
important goal of school mathematics. Today, being able to mimic a $5 calculator is not enough:
Employers want workers who can think mathematically. How the school mathematics
curriculum should adapt to this new reality is an open question, but it is clear that proficiency at
complicated paper-and-pencil computations is far less important outside of school today than in
the past. It is also clear that the time saved by reducing attention to such computations in school
can be put to better use on such topics as problem solving, estimation, mental arithmetic,
geometry, and data analysis (NCTM, 1989).

  Another reason the role of algorithms is changing is that researchers have identified a number of
serious problems with the traditional approach to teaching computation. One problem is that the
traditional approach fails with a large number of students. Despite heavy emphasis on paper-andpencil
computation, many students never become proficient in carrying out algorithms for the
basic operations. In one study, only 60 percent of U.S. ten-year-olds achieved mastery of
subtraction using the standard “borrowing” algorithm. A Japanese study found that only 56
percent of third graders and 74 percent of fifth graders achieved mastery of this algorithm. A
principal cause for such failures is an overemphasis on procedural proficiency with insufficient
attention to the conceptual basis for the procedures. This unbalanced approach produces students
who are plagued by “bugs,” such as always taking the smaller digit from the larger in
subtraction, because they are trying to carry out imperfectly understood procedures.
An even more serious problem with the traditional approach to teaching computation is that it
engenders beliefs about mathematics that impede further learning. Research indicates that these
beliefs begin to be formed during the elementary school years when the focus is on mastery of
standard algorithms (Hiebert, 1984; Cobb, 1985; Baroody & Ginsburg, 1986). The traditional,
rote approach to teaching algorithms fosters beliefs such as the following:
 • mathematics consists mostly of symbols on paper;
 • following the rules for manipulating those symbols is of prime importance;
 • mathematics is mostly memorization;
 • mathematics problems can be solved in no more than 10 minutes — or else they cannot
be solved at all;
 • speed and accuracy are more important in mathematics than understanding;
 • there is one right way to solve any problem;
 • different (correct) methods of solution sometimes yield contradictory results; and
 • mathematics symbols and rules have little to do with common sense, intuition, or the
real world.

  These inaccurate beliefs lead to negative attitudes. The prevalence of math phobia, the social
acceptability of mathematical incompetence, and the avoidance of mathematics in high school
and beyond indicate that many people feel that mathematics is difficult and unpleasant.
Researchers suggest that these attitudes begin to be formed when students are taught the standard
algorithms in the primary grades. Hiebert (1984) writes, “Most children enter school with
reasonably good problem-solving strategies. A significant feature of these strategies is that they
reflect a careful analysis of the problems to which they are applied. However, after several years
many children abandon their analytic approach and solve problems by selecting a memorized
algorithm based on a relatively superficial reading of the problem.” By third or fourth grade,
according to Hiebert, “many students see little connection between the procedures they use and
the understandings that support them. This is true even for students who demonstrate in concrete
contexts that they do possess important understandings.” Baroody and Ginsburg (1986) make a
similar claim: “For most children, school mathematics involves the mechanical learning and the
mechanical use of facts — adaptations to a system that are unencumbered by the demands of
consistency or even common sense.”

  A third major reason for changes in the treatment of algorithms in school mathematics is that a
better approach exists. Instead of suppressing children’s natural problem-solving strategies, this
new approach builds on them (Hiebert, 1984; Cobb, 1985; Baroody & Ginsburg, 1986; Resnick,
Lesgold, & Bill, 1990). For example, young children often use counting strategies to solve
problems. By encouraging the use of such strategies and by teaching even more sophisticated
counting techniques, the new approach helps children become proficient at computation while
also preserving their belief that mathematics makes sense. This new approach to computation is
described in more detail below.

  Reducing the emphasis on complicated paper-and-pencil computations does not mean that paperand-
pencil arithmetic should be eliminated from the school curriculum. Paper-and-pencil skills
are practical in certain situations, are not necessarily hard to acquire, and are widely expected as
an outcome of elementary education. If taught properly, with understanding but without demands
for “mastery” by all students by some fixed time, paper-and-pencil algorithms can reinforce
students’ understanding of our number system and of the operations themselves. Exploring
algorithms can also build estimation and mental arithmetic skills and help students see
mathematics as a meaningful and creative subject.