The beginning of "Tackling Misconceptions in Primary Mathematics" is essentially a consideration of some of the more common mathematical misconceptions held by, but not exclusive to, primary aged children. In actual fact, some of the misconceptions highlighted are often held by adults and one of my aims in writing the book was to try and encourage dialogue about what it is we are teaching, the language that we use and the impact this can have on our pupils as they progress through their education careers and beyond.
I'd like you to consider a typical deck of playing cards. With likely hundreds of millions of card decks sold each year there's a fairly good chance I can assume you've see the 14th Century French playing cards that have become somewhat commonplace, certainly in Western Europe, over the last seven or eight hundred years. Perhaps I'm being overly specific in my description but I want to be sure that when I talk about playing cards you know exactly to what is I refer. Usually they come in standard 52 card decks, of which approximately one quarter are adorned with what we are led to believe are two-dimensional representations of metastable allotropes of carbon.
Yet, no matter how much you want to believe the seemingly trustworthy merchants who brought the cards to our shores all those centuries ago, there is no officially recognised two-dimensional diamond. It doesn't quite have the same ring to it in a card playing context, but a rhombus on the other hand is described as a non-self-intersecting quadrilateral with sides of equal length. Where the angles are 60 degrees the shape may colloquially be known as a diamond and where 45 degrees, a lozenge but there is no standard.
If this shocks you, and I'm not sure it will. I urge you to investigate this discussion around the nature of 'diamonds', curated by "Dr. Maths". A selection of posts regarding the illustrious nature of the lesser spotted rhombus, it provides an example of how interesting and absorbing the properties of shape can be as a branch of mathematics. The problem too often, and I was guilty of this myself for a long time, is that it is posited that the properties of shape are nothing more than a list of general knowledge facts, of little mathematical importance, to be taught with little forethought and planning. This is not an attempt to undermine the importance of knowledge per se, not at all, and I write here about the essential role it plays in the development of our ability to think and act creatively. Rather, it is my intention to highlight how much more we are capable of with regards to the depths we can actually travel through this oft neglected content area.
This one section of conversation, aligned with possibly the greatest maths related blog post title in history, serves to provide evidence of the controversial and sometimes chequered past of the artist formerly known as shape and space. In real terms, it is the Steve McQueen of mathematical content areas. We've all heard of "Bullet", we all know a car raced down a hill reasonably fast, but we don't really know much else other than that and we've never really taken the time to get to know him. The same, in my experience, can regularly be said of the properties of shape.
The aims of the 2014 National Curriculum state that pupils need to be able to move fluently between representations of mathematical ideas. And rightly so. I believe every child has the right to achieve this aim. The expressed level of fluency, however, will only come about with proper planning and consideration of the core knowledge, hierarchical progression and the wider connotations of the inter-connectivity which run right the way through mathematics as a concept and as a subject.
Ask yourself, if, as an example, understanding the properties of a cube can serve a greater, more meaningful and relevant purpose, or is it a stepping stone to greater and deeper understanding, perhaps even a pivotal stopping point on the road map of mathematics education. It may be all of these things, it may be none, but we, as teachers, should make it our priority to ensure that we have given sufficient consideration to the possibility.
And if, in the end, you just can't bring yourself around to this way of thinking. Please remember just one thing. Diamonds are forever fictitious.