Wednesday, November 9, 2011

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

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


  1. I strongly agree with this finding, as I see it happen in my own teaching of students. In a Piagetian sense, the formalism of a formal algorithm requires a level of abstract thinking that most young children do not (yet) exhibit. Some of them do, but most do not, until as you say, about 4th grade.

    One thing that concerns me about math education is how we ignore the idea that different students progress at different rates, and I think this often contributes to the problems students have learning mathematics. Should we teach formal algorithms to that 40% of students who are not yet ready for them? Can we find a way to introduce topics in a class to some students, and not to others?

  2. Nicely put, Nev. I completely agree. Much of what we have perpetuated in mathematics education is due to conveniences of the printing press rather than what makes sense to the developing mathematical mind, e.g., standard long division vs. galley division. Even with paper and pencil, the galley division method makes just as much sense as, if not more than, the standard algorithm.

  3. Hi David – and thanks for your comment.
    I think you have identified one of the real challenges for maths education – the “one size fits all” approach that clearly is ineffective. I note on twitter that you are active on #mathchat and are looking for a source of open-ended activities. I think this approach has potential in addressing the “sameness” of maths teaching. Let’s hope that the collective wisdom of maths educators around the globe can come up with some valid alternatives.
    And here is some mathematical heresy – should we teach students mathematical processes that we can’t make meaningful and relevant outside the classroom world?

  4. Hi Adam,
    Hi – and thanks for your comment.
    I think your comment about the printing press is an astute one – I sometimes wonder if one of the obstacles to effective teaching is the photocopier. Without this effortless method of reproducing “worksheets” teachers might have to reflect a little more on their practice and the progress of their students – rather than simply “teaching” the next page or chapter of the book.

    Again, thanks for your comment.

  5. I love your thought about the photocopier. Young children need hands-on experiences with math to get the "feel of it." In language arts, young children are surrounded by language and stories and writing which aids their reading and writing development. Most children, however,are unaware that they are also surrounded by math everyday. We need to help them see math in the world around them.

  6. Hi Denise,
    Thanks for your comment.
    I think you make a really good point when you contrast the approaches we take when teaching mathematics with how we teach language related activities. Related to this is I think is the fact that language is easy for teachers to make “fun” (think poems and stories). However, teachers who may lack confidence in mathematics tend to retreat to schemes and worksheets. This has the impact of defining mathematics in a very narrow way. Effective teachers try to make language related tasks engaging and purposeful – our mathematics teaching needs to be the same.
    Again, thank you for your comment.