Interesting conversation on Twitter today with Bryan Meyer, Denise Gaskins and Justin Lanier. It began with these tweets on my part, the result of grading some student work.
Oh dear. Class inclusion is giving some of my elementary kiddos fits. Do we have any post-Piagetian research on this?
— Christopher (@Trianglemancsd) February 7, 2014
I suspect some of it is linguistic and probably appears not just in their math work but in their writing too.
— Christopher (@Trianglemancsd) February 7, 2014
Things quickly got too nuanced for Twitter.
An example of something my students struggle with is answering a question such as, Is a square a rectangle?
This type of question asks about class inclusion. Is an element of a subset also an element of the larger set?
Many useful and interesting questions in geometry have to do with whether one class is a subset of another class. Do all isosceles triangles have a pair of congruent angles? Are all quadrilaterals formed by connecting midpoints of other quadrilaterals parallelograms? Are all Stacys concave?
I am trying to sort out the extent to which my students’ struggles with questions of this sort are linguistic, and the extent to which they are about struggles with the idea of class inclusion.
Justin suggested this wording, which I will investigate:
Is a square an example of a rectangle?
Or, more generally:
Is an X an example of a Y?
My suspicion is that this will be helpful for some students when asked in this direction. But I also suspect that asking it in the other direction will be problematic.
Is a rectangle an example of a square?
See, part of what I wonder about is whether class inclusion—and the fact that it doesn’t have to be symmetric—is at the heart of a particular kind of struggle in geometry, and whether this is also related to the ways students think about and use language.
I hope these three (and others) will weigh in here where we have more space to work than we do on Twitter. The ideas are really useful. If you’d like to follow the prior discussion, you can follow this link.
Overthinking my teaching 
I view this difficulty as being included in the class of difficulties with quantification. For instance, students have difficulty with quantification in the vertical line test — is it every vertical line that needs to touch a curve in at most one point? Is it some line? It seems to me when students internalize concepts, it often doesn’t naturally come to them that quantifiers need to be internalized too. I feel I internalize these verbally, and if others do too then it is definitely a linguistic issue.
I teach fourth grade and have noticed a similar difficulty with questions posed about class inclusion in geometry (but also generalizations in other content areas, and, specifically, with the concept of metaphor in literature, or the ability to form general ideas from particular experience while writing.)
I”m not sure if this is helpful, but I’ll offer what I’ve been thinking about. I wonder if some of the issue might be the difference between inductive and deductive thinking and how general ideas get formed from particular experience. It seems to me that it takes a lot of induction (and revision of whatever overarching thesis has been developed) before the reasoning can go the other way. Much of life is led, it seems, using “good enough” kind of thinking, as in, “I do this and that’s good enough to approximate the result I desire.” Can I develop a theory of action that will result in the same outcome in ALL cases? That kind of move toward general concepts is a different kind of thinking than “good enough”, and might require learners to develop an appreciation for the beauty of simple, overarching concept formation at the same time they are developing the concepts themselves.
As I’ve tried to teach things like class inclusion (or anything that uses a general idea to reason about a particular instance) I’ve become more conscious about spending a lot of time with an inductive stance, the point being to explore overarching concepts through various iterations and “If…then…” thought experiments, and then figuring out the most inclusive (and shortest) way to say what you mean.
Maybe all of this is to say I think some of the difficulty might stem from a thinking issue (which is, of course, linguistically related, but also related to how many discrete ideas we can hold in our working memory at any one time as we consider inclusion), and how little experience we have with deductive-type reasoning.
By the way, I’ve followed your blog for quite awhile and really enjoy reading what you are thinking about. Thanks for sharing your thoughts and questions.
My Grade 6 students have a similar hangup, but I can usually get them past it quickly by inserting the word “always.” If I ask “Is a square always a rectangle?” they have an easier time seeing that a more exhaustive method is needed. It also leads into great discussion about the burden of proof, proofs via exhaustion, and the ability to disprove something with a single counter-example. I even talked to one class about Fermat’s last theorem, which is such a tangible example for kids that understand the basics of algebra and exponents (probably a little beyond your 4th graders).
Also, the Always-Something-Never activity I read about on Fawn’s blog seems like a great way to facilitate this kind of activity. I’ve yet to try it myself.
I like Tommy’s idea of adding the word “always.” It seems to me that at least part of the confusion (aside from my little kids, who are still learning vocabulary) is that a student’s answer can change depending on how he interprets the word “is.”
“Is a rectangle a square?” could mean:
Can we think of any rectangle that is a square?
Will every rectangle always also be a square?
If it’s a rectangle, is it allowed to be a square? (That is, is a square a rectangle?)
Does a rectangle have squarish properties?
Etc.
“See, part of what I wonder about is whether class inclusion—and the fact that it doesn’t have to be symmetric—is at the heart of a particular kind of struggle in geometry, and whether this is also related to the ways students think about and use language.”
I think you’re right that this problem is related to language, especially to the difference between informal language and formal usage. It reminds me of the problems logic students have with OR, and even more with implication. Our intuition is shaped by a lifetime of informal usage, and it’s hard to break through to a new way of seeing.
For all its flaws, the van Hiele model seems useful for understanding this kind of thing. (Not really post-Piagetian, though.) Most children operate at the lowest van Hiele level where “square” and “rectangle” are defined by visual prototypes which look quite different. It does not make sense to children to say they are the same thing until they become comfortable enough with the various properties of each that they can begin to reason: “Since all of the properties of a rectangle are also properties of a square, a square must be a type of rectangle.” Until that point, most children will probably continue to depend on a visual prototype, or a combination of prototype and some properties, which means asking them to understand that one is a type of the other will not make much sense.
Scott,
Your reply certainly resonates with what I’ve seen in the classroom. When I taught third grade I noticed that I could magically transform a square into a “diamond” simply by turning it 45o. Noticing this, I stood in front of the classroom, slowly turned a square, and asked the kids to tell me the exact moment that the transformation occurred. We then talked about what properties a diamond had and what properties a square had and how they knew when one became the other. It was only after we had created a “problem” with the visual representation (and how perspective fits into that) that we could get to other properties that might apply to all orientations, and did not require magical transformations, which the children were willing to give up for the sake of consistency.
Steve,
I have never thought to rotate a square and ask children when it magically transforms from a diamond to a square. That is definitely worth trying!
Agreed Scott. It seems to me that these visual prototypes, which are hammered home at very early ages, contribute to the confusion. I wonder what would happen if the actual shape names were withheld until children were more comfortable with naming and describing attributes.
Also, what about teaching some basic principles of set theory? Perhaps this might give kids a useful way to organize their thinking.
I wonder if Lusto’s musings on the Strawberry would be applicable here: http://linesoftangency.wordpress.com/
While I am hardly qualified to post here, I am eager to share my layman knowledge amongst so many seasoned professionals. To demonstrate my novicity (I made that word up) Your use of “class inclusion” in the opening paragraph confused me, as I could not figure out what including special needs students into the general classroom had to do with learning classifications of geometric shapes.
Now that my bona fides are out of the way…. As a future teacher, and peer, I am going to share my opinion. I am struggling with my math certification for teachers 2 course. I am reasonably intelligent. I earned (with much frustration) a “b” in the previous course. And managed a high “B” in college algebra. At this time my name resides on the “deans list” and I don’t intend to let it move. None of this is to brag, and none of it is to waste your time. It is simply to let you know that intelligent ADULTS struggle with the concepts that elementary aged students struggle with.
Is it linguistic? That’s an ironic line. I sat in class 3 days ago and though to myself “they need a class called “Math as a second language” or “MSL” for short”. It was a joke born of frustration. Unless there is real time dedicated to the discussion of the different geometry slang so that each student actually uncovers its truth it quickly becomes overwhelming. It is easy to understand what a median is, or what attributes a kite has, or why is a rectangle a square but a square not a rectangle… for a minute or a day. it is easy to temporarily memorize a fact. But without true understanding of the concept those “definitions” fade. If the foundation of truly understanding is not there to begin with then there is little hope for any true scaffolding and even less chance of any true learning.
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I just wrapped up a geometry unit with my 9th grade students on this very thing not so long ago. We devoted a LOT of time to defining, to classification and hierarchies. We looked at using dichotomous keys (stolen from the biology teacher!) vs. Venn diagrams to illustrate the relationships between categories of things – starting with animals & footwear and working up to quadrilaterals. We talked about how a flip-flop was always classified as a shoe, but not every shoe was a flip-flop; how every lion is a feline is a mammal, and how we have groups and sub-groups; (etc) before we moved on to mathematical beasts. The idea of class inclusion was pretty solid in these non-mathematical venues (although my assessment of this was very informal).
We also did a lot of sometimes/always/never questions, drawing examples & counterexamples and making deductive arguments. Given a quadrilateral with certain properties is it S/A/N a parallelogram/ rectangle/ kite/ rhombus/ square? For all the time devoted, I’m not sure that all my students really understood class inclusion at the level I hoped for. One real benefit is the understanding that any properties of the larger set are also properties of the subset.
I’m not convinced the problem is as much linguistic as it is about their fluency with deductive reasoning. The understanding that a conditional statement is NOT equivalent to the converse is really quite shaky… but maybe the heart of that _is_ linguistic? Part of me thinks that it’s because so much of the work I have done with them to this point involves statements that _do_ have converses that are also true…
I’ve been more aware of linguistic difficulties in geometry this year than I have in the past. While we’re on the subject, can I lament the fact that “corresponding angles” means one thing when we are talking about congruent triangles and another when we are talking about parallel lines? Some of my students very diligently read definitions from their textbooks and aren’t flexible in the way they think of “corresponding”… this isn’t the only word with this issue but it’s the worst offender that comes to mind at the moment.
I think there is a part of this problem related to rectangles and squares in particular.
I find students are much more comfortable accepting that a square is a parallelogram or rhombus, but rectangles are a completely separate issue. Ask students to define rectangle on their own and they all want to but some form of “one side of the rectangle is longer than the other” into their definitions
I just recently had my students go through classifying quadrilaterals. By asking the students “Is a square a parallelogram?” and then asking “Is a square was a rhombus?” and finally is asking “Is a square a rectangle?” I was amazed at how quickly and comfortably students accepted this concept in that structure.
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