Tackling Misconceptions in Primary Mathematics

Preventing, identifying and addressing children's errors

Kieran Mackle

• Publisher: Routledge
• Year: 2016
• ISBN:
  9781317443353
• Edition: 1
• Topic: Education
• Subtopic: Education General

Did you know that a circle has more than one side? Are you aware of the difference between 1: 2 and ½? Could you spot when a 2D shape is actually 3D?

Tackling Misconceptions in Primary Mathematics is a practical guide based on the principles that sound subject knowledge is key to fostering understanding, and addressing misconceptions is central to pupil progress. With an emphasis on preventing as well as unpicking misconceptions in the classroom, it offers trainee and practising teachers clear explanations, practical strategies, and examples of the classroom language and dialogue that will help pupils successfully navigate tricky topics.

The book demonstrates the importance of preventing misconceptions through what is said, done and presented to children, giving a variety of examples of common misconceptions and exploring how they can be addressed in a classroom environment. Proper intervention at the point of misconception is regarded as a key skill for any outstanding classroom practitioner and the author stresses the value in understanding how the pupil got there and explaining that it's okay to make mistakes. Misconceptions are only one step away from correctly formed concepts if harnessed with care and skill.

This comprehensive text is designed to be read as either a short course introduction, or dipped into as a guide to assist teaching. It is essential reading for trainee primary school teachers on all routes to QTS, as well as mathematics subject leaders and practising teachers looking to inspire the next generation of confident and inquisitive mathematicians.

Part I
Common misconceptions

1 Common misconceptions

Diamonds are forever unclear

With hundreds of millions of card decks sold each year there’s a fairly good chance you’ve seen 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 it is I am referring. You see when I use the term it is my intention to describe the playing cards used in games that are known colloquially as hearts or chase the lady, solitaire or patience, rummy or gin rummy and even the more competitive poker and blackjack, sometimes known as pontoon or 21. Usually they come in standard 52 card decks, of which approximately one quarter, not counting joker and instruction cards of course, 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 defined 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. It is worth noting a square is also a non-self-intersecting quadrilateral with equal sides and that the word diamond is frowned upon as a common misconception at Key Stage 3.

All of this information is easily accessible from a well-known online encyclopaedia but that isn’t even remotely the point.¹ If you do one thing other than read this book I urge you to search for the forum post ‘Diamonds are forever unclear’.² A selection of posts regarding the illustrious nature of the lesser spotted rhombus, it provides an example of how interesting and absorbing shape, space and measure can be as a branch of mathematics. The problem is too many professionals, myself included for a long time, consider shape and space to be a list of general knowledge facts, of little mathematical importance that are to be taught with little to no forethought and planning.

How wrong we all are, or were, I can’t quite be certain of the tense in which this most common of foibles exists! 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 shape and space. In real terms, shape and space is the Steve McQueen of mathematical content areas. We’ve all heard of ‘Bullet’, we all know a car went fast down a hill, 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 can be said for shape and space. Do you honestly believe you can extend and deepen your pupils’ understanding, as per the statutory requirements of the 2014 National Curriculum, in the same way you could in number? Do you feel confident enough to extend your most able mathematicians beyond the scope of your year group? Do you believe your teaching of shape and space is engaging and inspiring? If the answer to any of these questions is yes then I doff my proverbial hat.

If you answered no to any of the questions then now is the time to take action. The National Centre for Excellence in the Teaching of Mathematics, NCETM, and nrich.maths.org have both taken great strides towards the provision of greater depth and challenge in light of ever changing demands of the education system. Access to either of their websites will put you a few clicks away from the acquisition of the knowledge and guidance you need to truly rejuvenate your mathematics teaching.³ Now there’s a very good chance you are at the beginning of your teaching career and haven’t had the opportunity to reach this level of regret at opportunities missed in the past. You are in the fortuitous position of having the opportunity to write your own story from scratch. Why not champion shape and space? Take the bull by the horns and inspire those around you to do the same.

When considering the impact you can have I strongly recommend you consider the real life application of the content you wish to disseminate, in this particular case what is generally known as shape and space. As the aims of the 2014 National Curriculum explain reasonably clearly and succinctly, mathematics is an interconnected subject in which pupils need to be able to move fluently between representations of mathematical ideas. This fluency will only come about with proper planning and consideration of the wider connotations of this interconnectivity and the role it has to play in our lives. 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 even be none, but you, as facilitator of learning, should make it your priority to ensure that you have given sufficient consideration to the possibility.

This opening section is brimming with some of my favourite mathematical controversies. Feel free to borrow them but most importantly take on the mantle and find your own, develop them and mould the curriculum into something that will allow you to inspire in the way you’ve always dreamed you could.

How many sides does a circle have?

With that in mind I think we should look at one of my personal favourites and a guaranteed mind-blower, particularly with children aged 9–11. Technically the information I plan to impart is still up for debate but the most philosophical of mathematical problems can be the most interesting, diverse and inspirational problems to be found in our universe. The circle is up there with 1 = 0.9999999 … in the upper echelons of mathematical problems. If we could personify them, or even deify them, they’d be Zeus and Orpheus. Such is the regal lineage of these factoids.

It must be noted that I’ve not found any practical application for this in the classroom other than encouraging children to challenge the mathematical knowledge handed down to them but it’s definitely worthy of an appearance on QI. A typical conversation might go something like this …

Adult A: How many sides does a circle have?

Adult B: That’s easy. One curved side. (Folds arms defiantly)

Adult A: Try again.

Adult B: Look, here is a circle, it has one curved side.

Adult A: In actual fact a circle has an infinite number of straight sides.

Adult B: (Slumps to the floor following brief but intense brain explosion)

Adult A: (Goes off to find a mop and bucket)

Alternatively Adult B could express disagreement with your statement through the application of a range of gestures or use of the local vernacular but there are a number of reasons why a circle is considered to have an infinite number of straight lines. Alas, such a rare creature is the mathematics graduate who has chosen to pursue a career in primary education that it may be detrimental to my intentions to provide a purely mathematical explanation. Allow me, however, to try and clarify my proposition.

A circle is made of an infinite number of points. A line or a side is made by joining 2 points. If there are infinite points there will be infinite lines that can be made by joining 2 points. A circle, thus, has an infinite number of sides.

Granted, the relevance of this information is miniscule in relation to other topics covered in this book but if not to prepare our pupils for a future in mathematics it can be used to encourage mathematical reasoning and deduction. Try the conversation out on a group of children and allow the pupils to guide the discussion. I have no doubt you’ll be amazed where you end up. After all, my intention is to inspire greater understanding and as we all know dialogue and investigation are cornerstones in the development of relational understanding.

I must also add that there is a lot of debate surrounding this as a circle is not officially classed as a polygon but there is a school of thought which believes that as the number of sides approaches infinity the polygon becomes closer to looking like a circle. If you’re not working during your summer holidays you may even want to search for information on notions such as the Jordan Curve Theorem, the area and perimeter of polygons with an infinite number of sides or you may wish to spend time with your family, the choice is yours.

2-D shapes are not 3-D shapes

It may seem blatantly obvious and I expect you’d like to see my credentials at this point but, trust me; all is not what it seems. I dare say in every school in the country there sits at least one tray of resources, hidden in plain sight and utterly brazen in their defiance of mathematical convention, bearing the label ‘2-D shapes’. Their use is widespread and I have no doubt they were labelled with the best of intentions but when all is said and done 2-D shapes have two dimensions. They are two dimensional. They cannot be picked up. They cannot be collected in boxes and used as a practical resource and, if you want to be extremely pedantic, they cannot be drawn on an interactive white board, such are the geometrical properties of the surface.

This, though ostensibly difficult to comprehend, is a very simple premise which defines their very being and lays the foundations of our understanding of the universe. 2-D shapes are two dimensional 2-D = 2 dimensions. Yet this apparently elementary principle is seemingly overlooked in primary classrooms the world over for reasons unknown. Your job now is to find your maths coordinator, hunt down every mislabelled box or tray and prevent an easily avoidable misconception in the making. Call them 3 dimensional representations of 2 dimensional shapes, 3-D shapes, 3 dimensional shapes, call them whatever you like (except 2-D shapes of course) but please do not actively seek to immerse children in an inaccurate environment with incorrect vocabulary which will only have a lasting effect on their concept development for years to come.

Why should we worry about this? What harm does it do?

Well, in the more immediate future, those children will be in your class and one or more will look admiringly towards you holding a 3-D shape with properties covering all three dimensions. Yet with which they have attached a two-dimensional meaning. For example ...

Child: Look, a rectangle.

Adult: That’s not a rectangle, that’s a cuboid.

Child: No it’s not. It looks like a rectangle. Look! (Child points to rectangular face)

Adult: This shape has depth, it cannot be a rectangle.

Child: But it was in the 2-D shape tray.

Adult: ……………….

… and because there is nothing you can say in retort, the child has you hand over fist and you only have yourself to blame.

In the longer term, these children will be in charge of your twilight years. A terrifying thought indeed. All being well they’ll develop an advanced understanding of shape and space, honing the basic conceptual understanding you fostered, and will eventually apply their knowledge of shape and space to the buildings you inhabit and the structures you use on a daily basis. If you want your walls to stay upright, your bridges to reach from one piece of land to the other and support the weight of hundreds of cars I suggest you get a move on with that relabelling.

Something I’ve noticed since taking the leap into fatherhood, an event which perhaps unwisely coincided with the writing of this book, is the plethora of material available online which reinforces this most torrid of misconceptions, creeping into our homes at night and misinforming our children right under our noses. If you have embraced, or wish to embrace, technology then I suggest you treat it as you would treat any resource in your classroom. Ensure that you have prepared thoroughly beforehand, understand the content that will be delivered via the chosen medium and make certain that there isn’t a cylinder jumping up and down purporting to be a circle. If you don’t not only will you have shot yourself in the foot but you’ll pre-empt any potential domestic disputes with your significant other as to the relevance of your argument and insistence that the cylinder be exterminated post-haste.

A triangle can’t be upside down

Now, as we all know, the internal angles of a triangle total 180 degrees, they are three in total, as are the sides and the corners too. This is the definition. Orientation, however, unless you are trying to get to grips with bottom up leadership models, is never mentioned. Yet there are children who believe that a right-angled triangle as shown in Figure 1.1 or the equilateral triangle shown in Figure 1.2 are the one true representation of each. How can it be? What is happening that this, in the 21st century, is the case?

The mathematics, in this particular example, comes a mere second to the questions raised about provision of opportunity and pedagogical perspective/outlook by this occurrence. We must ask ourselves why so many children are allowed the time to cement such misinformed conclusions, why they are not corrected and what has happened in the intervening period between the introduction of the triangle and this realisation of the limited scope possessed by the children in your care.

They should, using accurate two-dimensional representations, have been given the opportunity to explore shape and develop an enquiry based approach to new learn...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Prologue
• Introduction
• Part I Common misconceptions
• Part II Preventing, addressing and identifying children’s errors
• Index