Alternative Algorithms Expand Critical Thinking

Collection of images for math algorithms

Let's start with a definition, an "algorithm" is simply a procedure or set of step-by-step instructions on how to perform a task.  In mathematics education, we use the term algorithm to refer to the set of steps we teach to students in order to perform calculations.  In elementary grades in the United States, these algorithms are rather standardized for each of the four main operations:  addition, subtraction, multiplication, and division.  In other words, almost every school-age child in the country has learned the same set of steps for performing these calculations in exactly the same way.  We do this for a number of reasons including efficiency and consistency.

Math History Moment: Origin of 'algorithm'

The word derives from the name of the mathematician, Mohammed ibn-Musa al-Khwarizmi, who was part of the royal court in Baghdad and who lived from about 780 to 850. Al-Khwarizmi's work is the likely source for the word 'algebra' as well.

Often, however, the most efficient algorithm is NOT the one that is most conducive for deep mathematical understanding of the underlying structure of the operation being taught.   In fact, in the case of the long division algorithm especially, sometimes the algorithm itself can cause or perpetuate errors in mathematical thinking.  Research in how students learn mathematics has shown that often misunderstandings in place value can be traced to being taught standard algorithms as a rote process rather than developing the algorithms through methods that make sense to the child and make evident the relationships that are important.

An additional unintended consequence of teaching traditional algorithms as the sole means to perform computations is that it gives students the erroneous belief that there is "only one way" to add (or subtract, or multiply, or divide).  This is a particularly harmful belief to have if the person ever travels outside of the U.S. since the algorithms we teach here are unique to us.  In other words, a teacher taught and trained in this country without exposure to alternative algorithms will have no idea that students in other countries perform these operations in completely different ways than we do.  When training pre-service teachers, this was the biggest shock they told me that they felt as they were amazed to hear that students all over the world do not multiply (or add or divide or subtract) the same way we do!

Besides the aforementioned dangers of teaching only standard algorithms, I  contend that there are additional benefits that arise when students are exposed to and asked to create their own alternative algorithms.  As you may have already deduced, alternative algorithms are simply those algorithms that are not the standard, or traditional, ones we use.  

This video, showing an alternative algorithms for multiplication used in China, demonstrates a wonderful technique for children who are not good at memorizing multiplication tables (since that is not required in this algorithm):

There are many alternative algorithms I demonstrate for teachers I work with, depending upon their goals.  Often I will suggest an alternative algorithm for long division (that I've dubbed the "Price is Right" algorithm) as a solution for the child who is having trouble with the estimation step in the traditional algorithm.  

Teachers have also reported much success with an Alternative Algorithm Assignment I recommended in which the students get to create their own method for computing.  This is especially effective when combined with some history of mathematics connections and research.  I usually advise that teachers then name the process after the person who invented it.  It strengthens classroom bonding and the feel of an educational community when "Albert's Method" and "Joanne's Way" become the language used to describe a particular algorithm developed by individual students.  If you thought putting their names up in lights (a suggestion from a previous blog) was effective, just wait until you see how your students level of engagement increases when they realize they can have an algorithm named after them!

There are myriad alternative algorithms and sometimes show up under different names, but below I give you a list of suggestions to get you started.  As always, you may email me for additional information or to set up a consultation to learn more.

Alternative Algorithms for Addition:

  • Partial Sums
  • Cashier's Method

Alternative Algorithms for Subtraction:

  • Austrian Subtraction
  • Partial Differences

Alternative Algorithms for Multiplication:

  • Lattice
  • Partial Products
  • Russian Farmers
  • Egyptian Multiplication

Alternative Algorithms for Division:

  • "Price is Right" (scaffolding)
  • Partial quotients
  • Egyptian Division

References:

Math History Moment quote came from here

Graphic created by me from a Google search of images related to "math algorithm"

Excellent resource from Everyday Mathematics on alternative algorithms

Nice PDF of several types of alternative algorithms

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The Solver Blog

Author:  Dr. Diana S. Perdue

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