Look at this shape. Is it a rhombus or a square?
What would your students say?
Well, it depends. It depends on whether you use exclusive or inclusive definitions:
According to the exclusive definitions, the shape is a square and that’s it. According to inclusive definitions, in which some categories remain within others, the shape is a very specific type of rhombus, called a square.
Traditionally, in school geometry, the vocabulary, definitions and, therefore, the classifications, were exclusive. Let’s take an example: the equilateral triangle is the one that has all sides equal, while the isosceles triangle is the one with two equal sides and the third is different.
Another example: the square has four equal sides and four right angles, while the rectangle has two sides equal to each other and the other two equal to each other, but each pair is different from the other, and four right angles. Do you see? These specifications, in italics, exclude definitions. So, an equilateral triangle is NOT a specific type of isosceles triangle. A square NOT a specific type of rectangle, they are shapes that exclude each other.
However, today most of us in mathematics education see it differently. “Why always change things!” , some will exclaim, more concerned with not having to learn anything new than with the coherence and clarity of what they transmit to their students. Well, lack of coherence is a problem that causes confusion in the long term. Test it out. Ask anyone to explain to you what a trapezium, or a parallelogram, is. You will see the confusion! All of this encourages an arbitrary and opaque picture of mathematics that we should avoid.
What do we propose then? Opt for inclusive definitions, whenever possible! Inclusive definitions generate classifications in which some shapes remain within others, just like with Russian dolls. Does this way of ordering the world sound familiar to you? Exactly! Biologists, who I’m sure will forgive me for the simplification, are very clear about it: within the classification of animals there are vertebrates, invertebrates, etc. Within vertebrates, we have mammals, birds, fish, etc. Within mammals there are primates, canines, marsupials, etc. Biology uses inclusive definitions. But we didn’t even have to look outside of mathematics! If we look at the classification of numbers, we will see that it has always been inclusive:
Everything fits. In fact, this way of looking at geometry is not new: in the 1940s, the renowned Italian mathematician and pedagogue Emma Castelnuovo (who deserves her own article) already proposed this type of inclusive definition and classification in her famous book Intuitive Geometry.
The Inclusive Classification of Quadrilaterals
Let’s try it with plane geometry. Of all the closed plane figures, those defined by straight sides are called polygons. Within polygons, depending on the number of sides, we have triangles, quadrilaterals, pentagons, hexagons, etc. And it doesn’t end there: because we come across them often, we have spent more effort classifying triangles and quadrilaterals, being a little more precise than with the others. Within quadrilaterals, we have the trapezoids (two parallel sides); and within them, we have the parallelograms (two pairs of parallel sides, different to each other); and within them, the diamonds (four equal sides) and the rectangles (four right angles); and at the intersection of the diamonds and the rectangles we have the squares. Therefore, a square is both a rectangle, a rhombus, a trapezium, a parallelogram, a quadrilateral and a polygon, just as the number 5 is a natural number and at the same time an integer, rational and a real number.
The key, if we opt for inclusive definitions, is to convey to students that when we want to refer to any shape, we always use the most restrictive category possible. In this way of ordering the world, there is no room for words that end in -oid, such as rhomboid or trapezoid. This suffix, comes from Greek and means “similar to” or “in the form of”, and often has derogatory connotations, it takes away from the exactitude of the classification and only creates confusion.
Let’s look at the diagram that comes up in Challenge 4 of the 5th grade Adventures, in which the children are asked to find all the different possible quadrilaterals in a 3 × 3 geoboard and then classify them. If you’ve never faced this challenge before, watch out for spoilers, stop and look for quadrilaterals before continuing!
Have you found them all yet? Are you sure? And how would you classify them? Let’s see!
Any complete activity is like a tree, with a main trunk and many branches that we can explore. In this case, a branch would consist of exploring the area of quadrilaterals (for which no formula is necessary, but rather taking the squares that define the points of the geoboard as a reference) or finding their perimeter (which, until we know the Pythagorean theorem, will depend on measurement with a ruler). But let’s go back to the trunk and look at the Toolbox, the sticker album that we suggest completing throughout the 5th and 6th grade Adventures. This is what the classification of quadrilaterals looks like, with the visual representation of the properties that allow them to be classified:
The Inclusive Classification of Triangles
Any article that addresses inclusive classifications from the perspective of plane geometry cannot end without talking about triangles. How do we classify them? The first thing to remember is that historically, there are two main criteria to classify triangles: by angles and by sides. And it is precisely this “double” way of looking at them that offers us an excellent opportunity to work on double-entry tables in the classroom.
In challenges 15 and 16 of the 4th grade Adventures, we asked students to find all the different possible triangles in a 3 × 3 geoboard and classify them in a double-entry table. The most interesting thing about this classification is that we understand that equilateral triangles (three equal sides) are a particular case of isosceles (at least 2 equal sides) and do not need a category for themselves. Finally, let’s see what this table looks like in the Toolbox: