Using formal algorithms too early - it doesn’t compute?

Picture an early childhood or middle  primary mathematics classroom.  What are the students doing? How are they recording their work?  Chances are, if you have a traditional view of effective mathematical teaching,  the students are using some form of formal algorithm.  This has been the case for many many years.  Yet, this entrenched  practice  may be actually reducing student understanding of mathematics.

According to Professor Doug Clarke of the Australian Catholic University, “The teaching of conventional written algorithms in primary schools dominates the (mathematical) curriculum with concerning effects on both student understanding and self-confidence.”  In his paper “Written algorithms in the primary years:Undoing the “good work”  Clarke challenges the effectiveness of teaching formal algorithms to students in the first five years of primary school.  His claim is based upon his research conducted with 572 students over two years.  It found that students who were “taught” mathematics by methods which required them to invent and use their own “informal” methods achieved more highly than those who were taught more formal algorithms. (Follow above link for details.)

Among reasons given for this is the notion that formal written algorithms do not match the way people naturally deal with numbers.  Formal calculations in primary school tend to operate from units, tens and into hundreds and so on - in other words, from right to left. However, people who are efficient users of mental calculations tend to operate from left to right - the opposite direction.  Thus the methods actually used by efficient mental calculators seem significantly different to those taught formally.  The introduction of formal algorithms also tends to encourage students to abandon their own intuitive methods of dealing with numbers - which in some cases has been shown to reduce the mathematical reasoning abilities of students.  (A useful overview of mental calculation and estimation techniques and its relationship to the teaching of formal algorithms can be found here.)


Clarke is not alone in his calls. One researcher has gone so far as to call formal algorithms in grade one and two “harmful” to understanding. Others, such as Kamii and Dominick, conclude that the teaching of algorithms too soon may “unteach” the child’s pre-existing understanding of place value and thus hinder development. *

So when should students be introduced to formal algorithms?  Clarke suggests this is appropriate when students are capable of mentally adding or subtracting two digit numbers.  Approximately 60% of students reach this stage by the end of grade four - but the obvious corollary of this is that nearly 40% of students do not.  The implication of this, if Clarke and the other researchers are correct,  is that large numbers of students are introduced to formal mathematical procedures before they are intellectually ready to benefit from them.

This calls for a wider discussion on the use of formal algorithms in education.  Clarke cites research by Northcote and McIntosh  who found that in one survey of mathematics use by adults only approximately 11% of calculations involved written calculations.  The same survey found that in around 60% of cases of adult mathematics use only a reasonable estimate was required. The conclusion drawn was that “It has become increasingly unusual for standard written algorithms to be used anywhere except in the mathematics classroom.”

The call to delay the teaching of formal algorithms should not be confused with a call to cease teaching methods of calculating or manipulating numbers. The opposite is the case.  This resource by Alistair McIntosh presents several significant methods for developing mental computation skills - and does so in a way that develops an understanding of the number system.

Clarke acknowledges that formal algorithms are have merit. He shares the views of others in observing that they are powerful procedures, particularly when dealing with large numbers, that they can allow for rapid computation, that they provide a written record of computation that enables error tracking (and correction), and that they are easy for teachers to manage. It’s just that they should not be introduced until children have internalised an understanding of numbers, place value and the specific concepts being addressed.  Repeating the information above - for many students, this readiness does not come before the end of grade four.

There is little doubt that these findings might come as a surprise to many parents - and possibly even a number of teachers. After all, according to John Van De Walle, lead author of “Elementary and Middle School Mathematics”, almost every commercial curriculum available teach using formal algorithms. He cites more than a century of tradition plus parental expectations as sources of pressure exerted on teachers to teach formal algorithms earlier than research would advise is appropriate. Van De Walle is a realist.  In view of the fact that students do not live in a vacuum it it probable that they will be exposed to formal algorithms outside of the school environment. His advice is to delay the teaching of formal algorithms in early grades if possible, but acknowledges that community and school expectations may make this difficult.

The issue then becomes, do we ignore the research (and there is much more than mentioned in this post)  and continue to teach “the traditional way”, or do we act upon it - in which case significant change is required in many classrooms?

A change of context might be useful here  - would we consult a doctor using established techniques practiced for over a century, or would we choose a doctor using newer treatments that have been found to be more effective?  When presented in medical terms  I suspect most would support research based practice. It is not so clear how people will respond to essentially the same issue when based in the educational realm.


Credits & references
Image:

http://extend.schoolwires.com/clipartgallery/images/19142777.jpg

Most sources cited in this post have an active link to the source. The exceptions are:

* Kamii & Dominick, 1997, “To teach or not to teach algorithms”, Journal of Mathematical Behaviour, vol 16, issue 1, 1977. (I have been unable to source a free electronic copy of this source - hence no direct link).

Van De Walle et al, 2010, “Elementary & Middle School Mathematics”, Pearson, Boston