Preface
A few weeks ago a friend and colleague from another school asked if they could observe one of my Maths lessons. As I still teach this is not an uncommon request and of course the flattery and the massaging of the ego that attend it are most welcome. So it was that we ended up in our year three class teaching the children the finer points of fractions. Drawing the fraction ½ on the board we embarked on a discussion about what the two numbers stood for. I was initially content with the answer that; the two was how many bits you split it into whilst the top number stood for how many pizzas you had altogether always aware that reasoning and dialogue lie at the heart of all good Maths teaching and that quality discussion is able to tease out misconceptions such as these.
However within a few minutes the lesson had started to nose dive, in hindsight I probably allowed the misconception to remain for too long and this led the children into a morass of confused thinking. Over the next few minutes I realised to my horror that the lesson was in complete free fall, misconceptions were articulated with seemingly great authority by the children and I sensed that the misperceptions picked up earlier in the lesson were become further ingrained in the children’s thinking. It was at this point in the lesson that I heard one of the year four girls announce to her friend in a voice audible enough for myself and my (no doubt bewildered) colleague to hear; “I’m really confused”
With my reputation as a Maths teacher of any standing in complete tatters, and only ten minutes of the lesson left I discussed with the classteacher (an NQT, just to add insult to injury) whether we could fold the lesson and whether she would graciously do something else with the class before they went to lunch. Her reply surprised me, as she claimed that the children were actually developing a measure of understanding and encouraged me to continue. We pressed on throwing further fractions at children, and amazingly the children did seem to be appearing uo from the mire that was my lesson and gaining a grasp of some of the concepts. Towards the end of the lesson we asked them to discuss whether 4/4 was a fraction; to my surprise they were not daunted by this or by the concept of 6/4 which most readily drew in symbols as six quarters.
My own confusion was only accentuated by dropping into my year 5 class and posing them the same question. Only half the children believed that 6/4 was a true fraction, the rest were either confused by the seeming numerical discrepancy or claimed that you could not have a numerator larger than a denominator. Initially I was encouraged that half had obviously grasped the notion of improper fractions, but even this bubble was burst when most of the children disclosed that they believed it was a fraction because although they were totally confused by the numbers it looked like a fraction because it had a line in the middle. My misery was made complete when a new child to the school informed me that whilst he did not understand it, he knew it was a fraction because his last teacher had told him it was. Only three children had any notion that the fraction related to a number larger than a whole. As the year 5 teacher is one of the school’s most competent mathematicians on the staff it might appear to the untrained eye that the issue relates to the teaching, in the sense that the poorer it is the more progress the children make and the more qualified the teacher and the more excellent the teaching the more the children are thwarted in their learning. With these counter intuitive thoughts buzzing in my head I decided to delve into this further.
Reflecting on the Lesson further
Not only is it counter intuitive to believe poor teaching aids learning it also goes against all known research so what factor was it that led the children to make so much progress in a lesson that many would have deemed to have been so poor?
What struck me in the early part of my reflections was the simple fact that where the teacher is not driving the learning then the children themselves need to be taking up the slack. I do believe this goes some way to answering the question but that still leaves us with the conclusion that poorer teaching leads to stronger learning and this cannot be the case.
I then started to reflect on how we teach Mathematics and the role of the teacher within it. Our default method of teaching is that we take a concept and then seek to explore this with children in the most linear and simple way possible so that the children don’t get confused. So, staying with the concept of fractions, I might decide to teach the simple concept of fractions using the following diagrams:
The reasoning is that fractions are a hard concept to comprehend so the simpler we make it for children the better for them it will be. Therefore the only variable that has changed has been the denominator. The shape is the same, the colour is the same, and even the position of the shaded numerator is the same (in the top right of the shape) everything is in place for the child to fully understand the principle without the clutter of having to take on any other distracting factors. We have made the path of learning uncluttered and straightforward for them, which surely is what teachers ought to do.
I have shown the following diagram (right) to various groups of teachers always posing the same question: What fraction is being shown? Interestingly the answer is always the same; ¾ (although some question whether the fraction is the missing ¼ or the ¾ that are shaded). However, neither of these answers is correct. When one looks more closely at the shape we realise that of course the fraction is 3/3. So how come so many of us make the mistake of thinking that the shape has anything to do with quarters? I was once posed the question whilst on a Mathematics course; Is it possible to teach fractions without using the word - Pizza” The comment was tongue in cheek but demonstrated the problem with so much of our Maths teaching. In our desire to make the complex accessible for children we seek to oversimplify the teaching and this often leaves children with to a very narrow and shallow understanding of a given concept. The reason that so many of us see the diagram above as quarters is because your mind has developed a schema that readily accepts that the concept of the “whole” as being the whole circle (or pizza!). Your brain therefore sub-consciously fills in the missing circumference thus compounding and cementing the error of your thinking.
I suspect there is likely to be less confusion when working out the fraction involved in the two shapes to the right. This is because the schema is broken and the presentation of these shapes in a format that is not well established enables you to see clearly that the shape is divided into three clear sections and therefore assisted you in your ability to calculate the fraction accurately.
This led me to think that whilst all concepts need to be scaffolded for children and structured in a manner that makes grasping the concept possible, it should not be done in a manner that constricts the option for children to undertake some deep learning on the way. I have come to see that much of our teaching of Maths has the potential to become so narrow and structured that it drives erroneous beliefs deeper. It is not that we are teaching direct error such as when some teachers tell children that to multiply by 10 they should just “add a zero” This is plainly false as anyone who has sought to multiply 1.8 x 10 will readily evidence. However sometimes we teach in a manner that is so narrow that it allows misconceptions to take root alongside the true mathematical concepts we are seeking to instil. I am minded of the oft quoted “All that is necessary for evil to triumph is for good men to do nothing”; so in this case all that is necessary for misconception to prevail is for the teacher to be negligent in presenting a concept in its fullest form.
For instance I wonder if the fractions above would be better taught in the following conceptual framework;
But surely this will be more confusing for children? Initially this may well be the case but it will shift the focus in the lesson away from the teacher driving the agenda and leading the children along a narrow path of restricted learning, often driven, in many classes, by the end goal of worksheet completion. Instead children will be forced to look afresh at the concepts they thought they understood and which they now being asked to apply in a fresh context. It is this element of apparent “confusion” that would appear to enhance the learning. It drives a deeper engagement from the learner and delivers a richer understanding of the underlying mathematical concepts being explored.
Dylan Wiliam illustrated this wonderfully when he asked delegates at a conference of teachers to draw an upside down triangle. As you can imagine the majority of those present drew something similar to the shape on the right. Yet when we stop and think about it there can be no “upside down” triangle in the truest mathematical sense of the word, a triangle is simply a shape with three angles and therefore three sides whichever way up it is. Why are we happy to draw the triangle “upside down?” It is simply because our schema for the properties of the shape have been driven by narrow teaching that has allowed us to develop the idea that any “proper” triangle should be presented with the widest horizontal line on its base. This is further compounded when one comes to work out the area of the triangle because we have all been told that the formula is ½ base x height. But on our triangle (above) where is the base? Or are we assuming that there is no time in their life when children will need to work out the area of a triangle that is not set on its “base?” It might make working out the area of this building a little interesting, as well as dangerous if you are the engineer or the quantity surveyor using the base to measure the area!
It is why the following question in one of the recent SAT papers becomes a challenge for many children because “everyone knows” that a square is the shape you can put a door and four windows in and if you just add a triangular roof you will have a perfect formed house. This single faceted approach leads children to develop a very narrow schema for the concept of a square, therefore when asked to; Draw two lines to complete the square? They fail to see that the shape can be successfully completed by drawing a “square at 45○” as many children would articulate it. Indeed children lower down in the school have often told me that the completed shape is not a square – again confirming their narrow schema view. Instead they proudly declare that the shape is in fact a “diamond”, it would have been better if they had said it was a “kite” as at least that is a shape with some mathematical content to it, but a diamond is simply a “precious mineral formed from a metastable allotrope of carbon” and has nothing to do with mathematics whatsoever!
Can confusion aid the Learning Process?
When children (and adults for that matter) engage in a piece of learning they come to the table with underpinning preconceptions related to the task in hand. For progress to be made the children need to go through a process that social psychologists term “Cognitive Dissonance”. Our experience of life causes us to construct a cognitive framework that enables us to make sense of the world. These constructs become our norm and form a set of core beliefs about the way we perceive the world. We don’t stop to question whether the sun will rise in the morning or whether apples fall downwards when we drop them because we consider that “we all know them to be true”. In the majority of cases these paradigms serve us well and release our mind to focus on higher order thinking, however if they are narrow in substance in certain areas then all future learning that seeks to build on them will be flawed. The upside down triangle is a case in point. When a child is first presented with such a drawing their original construct of what a triangle is comes under challenge. There are elements of the shape which might confirm it as a triangle but if we have always come to believe that triangles have “a base” then we find our view challenged by this new representation of a shape we thought we understood. This is the process of “Cognitive Dissonance”. The mind is challenged and confused by the new perception and must engage afresh with a concept it thought it understood.
As Guy Claxton says; “If you are not in a fog you are not learning” and the fog he is referring to is simply a metaphor for the process of cognitive dissonance. As the brain always seeks a harmonious and consistent belief system there is therefore a deep motivational drive to reduce dissonance within the mind. This provides a strong motivation for children to engage with the learning and to enter the process of either altering existing cognitions, add new ones to create consistency or reduce the importance of one of the dissonant elements.