Inclusive and exclusive definitions... again!

[Allan Turton]

After wrestling with definitions and hierarchies for quadrilaterals yet again, I'd like to get some other opinions on the matter. I adopt inclusive definitions for the good reasons offered by John Conway, Walter Whiteley, et al. To avoid all the arguments about what is included in an inclusive definition of quadrilaterals, I've included a link to a picture that should show it clearly enough: http://www.origoeducation.com/classification-charts-2d-shapes/ A word of warning: it is written for UK and US audiences so UK trapezium = US trapezoid. As a refresher to the inclusive definition of trapezoid, it means that it has *at least* (not *exactly*) one pair of parallel sides.

What I think compounds the problem of teaching an inclusive approach is what to call the "endpoints". For example, following a line of descent down from trapezoids, through parallelograms then rectangles, we come to two end points - either a "square" or a "non-square rectangle". Now, as John Conway noted, "oblong" is a perfectly acceptable term to use instead of "non-square rectangle". So, we now have two very clear terms to describe related (but different) shapes that we can say are both rectangles.

But what names do we have for the other end points? Looking at the class of shapes we call rhombuses, we have endpoints of "squares" and "non-square rhombuses". The latter term is a bit of a mouthful and the only alternatives I've found are "oblique rhombus", "lozenge" and "diamond". You'll see in the diagram that it is just labelled "other". The term "diamond" has become dreadfully unfashionable for reasons I haven't been able to fathom, yet in everyday use it almost always refers to a non-square rhombus. It seems perfectly suited to the job! Anyone know of the sources that put the death warrant on "diamond"?

Stepping up a level to look at parallelograms, we have ones that are either rhombuses, rectangles or "non-rhombic, non-rectangular parallelograms". Awful. A suitable replacement is "rhomboid" which is underused.

John Conway, in looking at the class of kites, suggested that kites could be divided into the end points of "rhombuses" and "strombuses", "strombus" being used for the toy shape (two adjacent short sides, two adjacent long sides). He mentioned that he hoped to include the term in books he was writing - does anyone know if this happened?

As for trapezoids, well, what should we call "non-isosceles trapezoids"? I know there are "right trapezoids", but what about the ones that are non-right and non-isosceles? "Scalene trapezoid" is about the best I can think of, but even this isn't truly accurate for the whole shape, just the "sides".

Isosceles trapezoids branch into "rectangles" (then onto "squares" or "oblongs") and "non-rectangular isosceles trapezoids". I am at a complete loss as to what to call the latter. This naming problem would provide the only reason I'd have to exclude trapezoids from having more than one set of parallel sides. What do we call it apart from non-rectangular? For such an omnipresent shape (kids see them all the time in pattern blocks) it is no wonder that it is often seen as the only representative of the term "trapezoid" - there simply doesn't appear to be any other name for it!

In outside-school use, and especially in the early years of school, we use the "end points" more often than the "classes" - they have more utility. I think that once there are names established for the end points then the higher classes of shapes can be more easily introduced. For example, using "rhomboid" to label that particular type of shape would have to be better than labelling it with the higher order name of "parallelogram", then trying to convince kids later that sometimes when we talk about "parallelograms" we're actually meaning a whole bunch of shapes, some equiangular, some equilateral, some both and some neither!

I hope I've made sense. Can anyone provide some words of wisdom apart from "relax"?

[Allan Turton]

Thanks Walter. It would be nice if everyone would start off with the more general classes as we seem to do it with everything else when kids are young. With two children under 4 in my house I am constantly explaining general classes and introducing specifics only as the need arises: “This is a dog. This is another type of dog. This is yet another dog and another. That small one is called a Chihuahua - they look kind of like big rats, don’t they? But they’re actually dogs.” They get it pretty easily, just as Doug Clements suggests.

But when it comes to geometric shapes in classrooms or in story books we’re rather let down. The “end-points” are nearly always used - I’ve yet to see a page in a story book that has “rhombus” at the top of the page and pictures of squares and non-rhombuses underneath. They’re using the class name for only part of the class of “rhombus”. Therefore, only non-square rhombuses get attached to the label.

I just can’t help thinking that if we had “end-point” names then we could at least work with whatever approach seemed best. If the research tells us to start with specifics and work towards generalities then we could do so without any confusion. I’ve done this with my 4 year old - she understands at a basic level that squares and oblongs are both types of rectangles because the corners are all the same size and that’s what makes rectangles special. Similarly, if the research tells us to start with generalities and work towards specifics we could do that too, knowing that once we hit the “end-points” we can say, without any conflict, that “this type of rectangle is an oblong” and “this type of rectangle is a square”.

Thanks for reminding me of the circular frames for orientation - such a simple idea to implement. I think it’s a related issue but only part of the problem. Orientation and prototypical examples are easily fixed, but if you have an imprecise label for multiple examples you’re still stuck when it comes to naming things. At the other extreme is over-specification as Michael De Villiers showed in his “Extended Classification of Quadrilaterals” - I certainly don’t expect us to get to that level of labelling for kids aged 3-15, but at least it shows that it is possible to assign labels to very specific end-points.

Allan

[Allan Turton]

Thanks for replying Walter. The diagram I linked to in my original post comes from a book that also has the quadrilaterals set out according to symmetry - a way of organising them I'd never thought of until I'd seen your work. I'd forgotten about your label for the non-rectangular isosceles trapezoid - "butterfly" - so I'll make a note as it's a useful label that nicely links to the reflective symmetry of the shape.

While reasoning and proofs are an ultimate goal, my concern is mostly at the very basic level. There are many problems with labelling shapes too early as it can lead to poor understanding. For example, attaching the label "rectangle" to only shapes that are oblongs often results in students being unable to identify squares as rectangles too. This occurs from age 3 until even age 12 in some cases (sometimes beyond unfortunately!). So it seems important that we should use labels that always make it clear when we are talking about general cases ("rectangle") and when we are talking about specific cases ("oblong" and "square"). Imagine if we showed kids pictures of Saint Bernards exclusively with the label "dog" for the first 7 years of their lives. Then trying to convince them that a Chihuahua is a dog too will be a difficult process! (I'm still not convinced either, but for different reasons.)

For better or worse, the "end-points" are where kids begin learning about shapes. The standard set of pattern blocks is a classic example of where the problems creep in. There's nothing wrong with the shapes, just the labels. If we accept for a moment that we can use 2D names for them, we have an orange square, a green triangle, a yellow hexagon, and then... well, this is where the problem occurs. The blue shape is a rhombus, but it is a non-square rhombus, just as the pale rhombus is too. But if we attach the label "rhombus" only to non-square rhombuses, should we wonder that kids have difficulty later with understanding that a square is also a rhombus? Similarly with the red shape, usually called the trapezoid. Again, it is a special trapezoid (non-rectangular) but giving it the class name of "trapezoid" causes confusion all through childhood and into adulthood as people struggle to see how oblongs and squares could possibly be included in the class of trapezoids.

Whether the chart I linked to is used, or the chart you present with Lily Moshe is used, we still have "name gaps". There is no exclusive name for a kite that is not a rhombus. There is no exclusive name for an isosceles trapezoid that is not a rectangle. There is no exclusive name for a rhombus that is not a square. Putting aside higher-order thinking about the relationships between shapes, when I have the humble task of asking a young child to draw a non-square rhombus, it would be really handy to have a label for it apart from "non-square rhombus".


Allan Turton


P.S. I came across the Shreddies ad just a few days ago - very cute, but I agree, it does reinforce the idea that a diamond is based as much on orientation as on shape.

[Allan Turton]

>From Walter Whiteley:

I don't have young children around (yet - maybe grandchildren in the future). Overall, children as simply amazing. I wish we connected to more of what they are capable of.

I am not sure I have linked out to the wiki pages of a working group looking at geometry in the early years (pre-school up to grade 3). We have decided to work on a thread of symmetry all the way through to the situations where it is unavoidable - i.e. engineering, chemistry, biology, and sometimes math.

There are some links starting at:
http://wiki.math.yorku.ca/index.php/CMEF_Geometry_Curriculum
and proceeding to some thoughts on early years, etc.

Walter

[Allan Turton]

*Due to trouble sending this dialogue around the email list, I'm posting it here in the Math Forum board*

>From Walter Whiteley:

An interesting chart - and a topic worth continuing conversations.

I have a alternatives to how the classification is done - and therefore what is worth naming, and how the names are done.
Two perspectives lead to some different classes:
(a) if we classify quadrilaterals by symmetries, then some distinctions don't matter so much. On the the other hand, a kite (with a mirror through two vertices) can be non-convex. By the way, in this classification, parallelogram is the class with half-turn symmetry.
(b) If we think about classifying on the sphere - where there is duality between angles and lengths, then some of the alternatives you have get 'paired up'. Interestingly, these pairings carry on into the plane under polarity about a circle - between shapes with four vertices on the circle, and shapes with four edges tangent to the circle.

One version of this is linked at the Geometer Sketchpad Users Group site:
http://www.dynamicgeometry.com/General_Resources/User_Groups/JMM_2006.html

Note that (a) and even (b) actually work well with the names for triangles, and we don't really try to capture the comparable analysis for 5 or more sides. Also, in 3-space, with skew quadrilaterals, there is a further set of connections. In the end - 3-D reasoning is a key goal, so I am happy to do a bit less in the plane if the larger vision opens up (see the link above).

These perspectives do come from some types of reasoning one wants to do - and I think naming is best developed to help cue some reasoning / connections etc. So classifying parallelograms by half-turn symmetry, cues us to the fact that most proofs for parallelograms implicitly use this property - or would be easier if we do use this property. For example, when the proof uses a diagonal and that cites ' congruent triangles' - the congruence actually is a half-turn symmetry! Much of the symmetry analysis becomes evident / even essential, when we observe which isometry is used for the 'congruence'.
I have some other charts etc. for some of this. On of the criterion: How well does it generalize' is useful, as well as what reasoning / connections does it afford?

In terms of the dislike for 'diamond' - many people (including many students) would certainly consider a 'square' oriented with vertices up and down (45 degree angle to the 'standard' orientation' as a diamond. There is a even a commercial in North America which plays on this for a cereal (Shreddies) has a square shape (see en.wikipedia.org/wiki/Shreddies).

Walter Whiteley

[Michael de Villiers]

Rather than using terminology such as "non-square rectangle" or "non-square rhombuses", etc. I generally prefer to call these shapes 'general' rectangles, 'general' rhombuses, etc. (to distinguish them from the special cases of the squares). In the end it's ALL about communication and ensuring that one's students understands which figures one is talking about.

[Michael de Villiers]

Just recently experienced another area with an assignment for some teachers where a hierarchical classification of quadrilaterals is also important. For example, asking them which are necessary & sufficient conditions by which to define certain quadrilaterals.

Many got right answers for some questions, but did so by incorrect reasoning and providing incorrect counter-examples! For example, all of them correctly said that 'equal diagonals' is a necessary, but not sufficient condition for a quadrilateral to be rectangle. However, all of them gave as 'counter-example' (to show it is false) a square, but did not realize this was an INVALID counter-example as a square IS a rectangle! They didn't realize they had to produce an example of a quadrilateral that has equal diagonals, but is NOT a rectangle, for example, a general isosceles trapezoid to show that the condition is insufficient.