Does a cylinder have edges? This is a question I have been asked before, and many more questions like it regarding shapes and objects. Once again, it was up for debate last week in a Facebook group of which I’m a member. It was encouraging to see so many teachers talking about mathematics.

It’s a great question, as are the others that regularly pop up such as;

- How many faces does a sphere have?
- Is a square a rectangle?
- Does a cone have an apex, a vertex or something else?
- What’s the difference between bases and faces?

Some of these questions are relatively simple to answer (and will be in a follow up blog!), others are a bit more complex.

For instance, it’s a square!

Even when its turned onto its vertex (or ‘corner’) it’s still a square. It’s not a diamond (no matter what Playschool says) or a rhombus. The orientation of a shape doesn’t change its properties or its name, just as a triangle on its ‘point’ doesn’t become something else. Speaking of squares, a square is also a rectangle, but a rectangle is not a square.

In trying to describe a square we may follow this path:

- It has four sides, this classifies the shape as a quadrilateral (at this point it could be many shapes, for example, a trapezium
^{1}, kite, rectangle etc) - It has opposite sides parallel (now we have ruled out the trapezium and kite, but it could still be other shapes, for example, a rectangle)
- It has four sides that are of equal length (it can no longer be a rectangle, but it still could be a rhombus)
- It also has four right angles (it can no longer be a rhombus, so it’s a square)

The Euclidian geometry system we base our mathematics teaching on is an inclusive classification model where the more properties in the description, the more specific we can be about the shape. This image visually depicts the relational system used to describe and classify shapes.

The question posed in the beginning, *does a cylinder have edges*? is a bit more complex. At face value (excuse the pun!) it may seem simple. No, it doesn’t have edges as edges must be straight. It also has no faces as faces must be a “flat surface with straight edges” (as stated in the mathematics syllabus language section in Stage 1 Three-Dimensional Space 1). Here’s the complexity, why did they call them straight edges? Haven’t we just established that edges are always straight? I know its semantics, but interesting none the less. So how do we describe a cylinder?

A cylinder has one curved surface and two circular surfaces with curved boundaries. It’s a mouthful but it is pretty descriptive. In Stage 1 of the NSW mathematics K-10 syllabus it talks about students learning to *“distinguish between ‘flat surfaces’ and ‘curved surfaces’ and between ‘flat surfaces’ and ‘faces.’”* Many people in the Facebook group provided great advice on the way to define a cylinder and reference was made to looking to the syllabus as a first port of call. This is sound advice as many other countries use different classification and category systems for geometry. It is great to have the language sections at the bottom of each page in the syllabus as these provide some good advice also, especially for curved objects.

There are always going to be areas of mathematics that are problematic and are cause for debate, this is healthy as it guides us to think and wonder, why does it matter?

It matters, as we as primary teachers need to think about what this knowledge is the foundation for? What lies ahead for our students in their mathematics learning expedition that they need to understand how to describe shapes and objects?

Another comment in the thread mentioned one of these reasons, Euler’s formula. We don’t classify a cylinder with prisms as it is not a polyhedron. Cylinders do not meet Euler’s rule for polyhedron, therefore we cannot compare them and shouldn’t try to list them together in any table of properties. We need to explore how to *use* the properties, not just list them. Making connections between concepts in geometry to other aspects of mathematics is also important.

Look for the relationships between shapes and how to find their areas, and between objects and finding their volumes. This is the case in Stage 4 in the mathematics K-10 syllabus where students are introduced to finding areas of shapes and volumes of solids (objects). In particular the syllabus talks about the link between geometry and multiplication and division. Understanding numbers as the product of two of its factors and how this knowledge is equivalent to forming a rectangle with the factors as side lengths. For example, if we take the number 12, I can make a rectangle with side lengths of 2 and 6.

This can also be applied in finding volumes of prisms where students express a number as the product of three of its factors. For example, a volume of 36 square units could have side lengths of 2, 3, 4.

There are also relationships between how to find the area of rectangles and triangles. These concept connections start much earlier and can be explored through showing the dynamic nature of shapes and their properties, check out my resource Stringy Shapes for classroom activities.

Mulligan, Woolcott, Mitchelmore and Davis (2018) state that mathematics learning is

“a complex, dynamic system of interconnected components, fundamentally dependent on spatial reasoning rather than upon the initial development of number concepts” (p.78).

They also highlight that number concepts are often spatial in origin, emphasising the need to make connections between number concepts such as multiplication and its visual spatial representation in geometry.

The questions we as teachers may have about the language of shapes and objects, are also the questions students may have. Use these as a starting point for mathematical investigations and transform them into wondering questions, or statements to be proved or disproved:

- A square is always a rectangle. True or False
- Does a sphere have a face?
- A rectangular prism and a triangular prism are similar because….
- I wonder how many acute angles a triangle must have?
- All quadrilaterals are parallelograms. True or False

The Quality Teaching Framework used in NSW discusses problematic knowledge:

“Students are encouraged to address multiple perspectives and/or solutions and to recognise that knowledge has been constructed and therefore is open to question.” (p.11)

Whether you still reference or use the framework or not, encouraging students to question knowledge is valuable and develops critical and creative thinking processes that are essential 21st century learning skills and underpin our current curriculum as one of the general capabilities.

^{1}Note: in some countries a trapezium is defined as having at least one set of parallel sides, not only one set of parallel sides- see, I told you its complex!

#### References

DET, N. (2003). Quality teaching in NSW public schools: Discussion paper. Sydney: Department of Education and Training.

NSW Education Standards Authority (NESA). (2012). Mathematics K-6 syllabus. Sydney, NSW.: Author.

Mulligan, J., Woolcott, G., Mitchelmore, M., & Davis, B. (2018). Connecting mathematics learning through spatial reasoning. Mathematics Education Research Journal, 30(1), 77-87. doi:10.1007/s13394-017-0210-x