Marcus' mental mathematics mistake

In 2009 Marcus du Sautoy, Professor of Mathematics at Oxford University and Simonyi Professor for the Public Understanding of Science, teamed up with an unlikely co-presenter,  comedian / actor Alan Davies, to present an episode of Horizon dealing with the joys or otherwise of mathematics - “Alan and Marcus go forth and multiply”. In one part of the documentary the two participate in an experiment to see if the brain of a member of the public who dislikes maths (Davies) and the brain of one of the most recognisable mathematicians on the planet (du Sautoy) process mathematics differently.  However, the interesting thing for me was this clip extracted from the session.

Play this embedded clip and pay particular attention to du Sautoy’s answer to the third question.

I need to make a clear statement here - it is not my intention to take a cheap shot at Marcus du Sautoy - just the opposite in fact.  Later du Sautoy easily answers more complex questions before I, in the comfort of my lounge chair,  have even processed the information required. He admits  that he was nervous about getting simple calculations wrong as it would be embarrassing for a man in his position to do so.  We can also assume that du Sautoy could have insisted that the error be edited out of the program - but clearly he didn’t and this reflects positively on him.

The interesting thing to me was that when nervous or under stress even one of the most prominent minds in the field of mathematics can make a simple errors.  If this holds true for an expert practitioner such as du Sautoy how much more significant might it be for students in school?

This may prompt some to review their teaching practice in mathematics classrooms. Reducing  stressful situations (and for some students stress may result from  something as simple as being required  to provide an answer publicly in class) would surely make the process more enjoyable for all - and it may just remove a barrier to learning.

What can QWERTY teach us about education?

 History has virtually forgotten Frank E. McGurrin – which is perhaps a pity given his  unintended but very real impact upon virtually every  person living in the developed world.   It could be argued that it was McGurrin, as winner of the first ever typing contest, that cemented the place of the QWERTY keyboard as the dominant key board layout for typewriters – which evolved (or perhaps mutated) into the computer keyboard.

The survival of the QWERTY keyboard is worth contemplating as it is truly a bizarre story.  The layout of keys with which we are so familiar was actually designed to make typists less efficient – or at least slower.  Early typewriters were mechanically cumbersome and the mechanisms jammed if typists struck the keys too quickly. So,  keys were arranged in such a way as them harder to press quickly – letters that are used more often were moved so that weaker fingers were used to type them or to places where fingers would need to be moved more in order to press them – in other words to slow down the user.  There was considerable experimentation to find the right balance and produce a keyboard that was efficient enough to be better than writing by hand but not so easy as to result in jammed machines.    It is possible to write 70% of all English words using the letters   D H I A T E N S O R  - yet few of them are in favoured positions on the keyboard.   Amazingly, salesmen also apparently had input into the QWERTY design – they wanted to be able to create the word TYPEWRITER  using letters on one line of the keyboard to demonstrate the ease of use of the new machine – and they got their wish.

McGurrin, was employed by a Ms. Longley, also largely forgotten by history despite being the founder of the Shorthand and Typewriter Institute in Cincinnati.  A rival company, using a different configuration of letters on their machine, challenged Longley to a contest to see which company was producing the best typists – McGurrin was nominated to represent his employer and duly won the contest.  This was seen as endorsement of the superiority of QWERTY over other layouts and helped propel the configuration to undisputed domination of the market. (An interesting footnote to this anecdote is that McGurrin was apparently the first typist to memorise the keyboard – and it may well have been the simple fact that he may have been the first touch typist in history rather than any design feature of the keyboard that enabled him to win the competition.)

Over the years typewriters improved, non-jamming machines were created and more efficient keyboard layouts were devised. However, they were not widely adopted – despite being demonstrably more efficient. Even in the electronic age of computers QWERTY continues to rule supreme – despite there being absolutely no phsycial reason for it to do so. I’m writing this on a QWERTY keyboard, every computer in this building have one, all the  keyboards in my house are of that configuration, even my smart phone uses the layout. The  DVORAK keyboard, generally considered to be much more efficient than QWERTY, lies overlooked  in obscurity simply because we have become accustomed to QWERTY . In short, QWERTY is a survivor , it continues on and on and on – despite there being no compelling reason for it anymore. 

There are aspects of the education system that have a striking similarity to QWERTY  in that they continue long after their  usefulness has past. Examples include:

·         Starting and managing student progress through the layers of school according to the calendar rather than on progress, demonstrated learning or readiness.

·         “Streaming” students despite both social-constructivist theory and formal testing programs suggesting this does not enhance students.

·         Confusing factual recall with understanding.

·         Teaching “subjects” in silo-like isolation.

·         Clinging  to paper based practices when the world outside of the school ground  is essentially  digital.

·         Pretending that the school is still the primary source of information available to students.

·         Teaching all students more or less the same thing at more or less the same time in more or less the same way – regardless of interest, ability or appropriateness.

·         Spouting social-constructivist learning theories but practicing behaviourist techniques (seen a “star chart” lately?).

·         Providing A-E type feedback to students and parents.

·         Using formal testing data to evaluate school performance – regardless of the various social and economic factors known to impact on education.

·         The standard school day itself  is fixed and based upon the needs of a bygone era. (Why do some schools offer “before school”  and “after school” care? Why not just change the hours of operation of schools to reflect social needs?)

This list is of course far from complete...and is obviously open to addition, agreement or disagreement. In a sense agreement is not necessary here, it is discussion that is important. We as educators should question the fundamentals of our practice.  We need to examine our basic assumptions about what makes a good school, how we can best serve our students, what are we trying to achieve and how we are trying to achieve it. Then we need the courage and energy to act upon our reflections.

 If we don’t we are essentially engraving QWERTY on our educational practice.

Image:

http://www.slsa.sa.gov.au/exhibitions/boland/images/memorabilia/images/typewriter_jpg.jpg

Historical information about the surprisingly interesting history of the typewriter is sourced from Stephen Jay Gould’s book “Bully for Brontosaurus”, published by Penguin.  

When is a school a school?

I was privileged to be given a tour of a new school recently.  It was, without exception, the most exciting school building I have ever been into.

The design of the buildings was thoughtful and, dare I say it, “modern”.  ICT was present everywhere (but not always obvious), major walls were sound “proof” but flexible in that they could be opened (in effect removed) to increase teaching space and allow for co-operative learning, or inter-class interactions.  The school had avoided the trap of creating computer labs - instead each area had access to a half dozen or so desk top machines which were supplemented by a generous supply of lap tops via a booking system.  Wireless technology with fast connection speeds was ubiquitous and covered every area in the facility (or so we were told). Teachers had pleasant individual office space.  Each section of the school, or “pods” as they were called on site, contained significant art installations.  Large windows dominated the wall spaces filling the  facility with light and enhancing the impression of spaciousness. Classrooms seemed to have adequate resources. One of my fellow visitors commented that the staff room had the feel of an upmarket hotel rather than a school staff room.

Outside it was just as impressive - attractive from all angles, functional yet not overly institutional.  Play areas were clearly well used and functional.  It was, quite simply,  a wonderful building - but it wasn’t a school.

Then the siren went.

The empty rooms quickly filled with chatter, children moved to desks, teachers started explaining tasks, students started asking questions, books were opened, computer screens burst into life, the sounds of productive work became the back-ground audio-track for the visit.  Parent helpers slid quietly into a classroom to be greeted with smiles and accepting nods of “hello” from the students. A class of children walked past in an orderly but not overly regimented group. Seeing their teacher smiles burst onto the faces of the first few students as they entered the room.

Then it was a school.

Credits:
Image: http://www.wes3rdgrade.com/uploads/7/8/3/3/7833021/8576233.jpg?252

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