Academic journal article Australian Primary Mathematics Classroom

Algorithms Are Useful: Understanding Them Is Even Better!

Academic journal article Australian Primary Mathematics Classroom

Algorithms Are Useful: Understanding Them Is Even Better!

Article excerpt

This is the first of two articles on the use of a written multiplication algorithm and the mathematics that underpins it. In this first article, we present a brief overview of research by mathematics educators and will then provide a small selection of some of the many student work samples we have collected during our research into multiplicative thinking. We contend that many primary-aged children are taught algorithms for multiplication and division without an appropriate understanding of the mathematical structure and concepts that underpin those algorithms. This is not about demeaning the use of standard algorithms. They have stood the test of time and can be elegant ways of getting a solution. However, imagine the power we give to students if we underpin the strength of algorithms with understanding! In the second article, we elaborate on what we believe are the key mathematical underpinnings of algorithms.

Introduction

Algorithms are very useful methods for calculation when numbers are too large to mentally calculate quickly or accurately. For multiplication, this is generally when there is a need to multiply numbers of two digits or more by another number of a similar magnitude. For example, when attempting to multiply a single-digit number by a double-digit number, students should be considering other strategies, such as applying the distributive property, and exercising their understanding of place value (e.g., 17 x 6 is 10 x 6 which is 60 and 7 x 6 which is 42 so 17 x 6 is 60 + 42 = 102), which allows them to complete these calculations mentally. However, where algorithms are deemed as necessary it would be preferable if the user of the algorithm had an understanding of not only what they were doing, but also, why they are doing it.

An algorithm can be defined as a step-by-step procedure used to solve a problem or complete a task (Anderson et al., 2007). The key here is the word 'procedure' and how this word can sometimes be interpreted. A generally accepted definition of procedure may be a series of actions carried out in a certain order, which seems innocuous enough. However, if the procedure is used without understanding, that is a different matter. Skemp's seminal article differentiated between relational and instrumental learning, with the latter largely equating to "rules without reasons" (1976, p. 20). One of the arguments for the efficacy of algorithms is that they save time and lessen the cognitive load on students, therefore allowing students more 'resources' for problem solving to occur (Merrienboer & Sweller, 2005).

This may be particularly so for students who are cognitively less efficient in mathematics. However, it is important that students do not become too reliant on procedures and algorithms but rather that they have the opportunity to be involved in productive struggle (Jonnson, Norvquist, Liljekvist & Lithner, 2014) to enhance the development of conceptual understanding (Hiebert & Grouws, 2007). It is the exploration of the mathematics behind the procedure that is important, not the uninformed use of the procedure. Whilst we encourage the use of algorithms to aid students in their mathematical development, the use of them without understanding may indeed be impeding that development.

We can choose to teach an algorithm as a purely mechanical way of reaching a solution, but if we do so, much of the potential power of the algorithm is lost. Brosseau (1997) stated that algorithms are designed to be efficient, and to avoid meaning. What he meant by this, was that you can focus on the mechanics without needing to understand what you are doing. For instance, when you "carry the one" you are dealing with it on its face value as being one, not the fact that you are actually renaming 10 ones as one 10, or 10 tens as one hundred etc. Students need to understand and be able to articulate this, which be facilitated by exposure to concrete materials to model the regrouping process. …

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