Suppose you are introducing a new concept, and you start with (what you think is) a toy example. Is it okay to refer to your example as a "trivial problem?"
Does this depend on the field and/or level of the class you are teaching?
asked Jan 24 '15 at 8:06
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8There is a question What does it really mean for something to be "trivial"? on Math SE which is not directly related to your question, but I find it interesting to read myself. – scaaahu Jan 24 '15 at 8:44
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3The word 'trivial' also has other meanings in maths. On a different note, if you're introducing something to a lower level class, they probably won't think it's trivial. – Jessica B Jan 24 '15 at 9:58
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4@scaaahu That's a great link which leads to another great link. In short, tt all depends on your audience. If you are not sure it is easy for everyone in your audience, then don't say it is easy simply because it is easy for you. – earthling Jan 24 '15 at 9:58
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1See also matheducators.stackexchange.com/questions/1896/… on Math Educators SE. – J W Jan 24 '15 at 12:47
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1I didn't see that in any of the answers nor the links, so one more piece: in math (I don't know about other domains) some objects are defined to be trivial, e.g. trivial group or trivial topology, etc. So, if you are teaching group theory and showing the one-element group among your first examples, then you should use the word trivial in that particular case. – dtldarek Jan 26 '15 at 10:49
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"Trivial" often means "too simple to be a real problem". For example, in an class on optimisation methods years ago, the lecturer said "for our first example, we're going to study the problem of maximising the number of ones in a binary string of length n". My thought for most of the class was "well, that's bloody obvious—you just write n ones in a row—why are we looking at this". I eventually realised that it was being used because it made it easy to describe how the algorithms worked, not because it was a challenging problem in any way.
I think that's a good example of where the word "trivial" is good shorthand—the problem isn't of any importance, but it is being used to illustrate a point. This is similar to how trivial is used in a lot of pure mathematics, where it often means "the simplest example that satisfies the definition" (like "the trivial group"). As with all such jargon, though, it is worth explaining to students what you are going to be meaning by the word "trivial" and why trivial examples are worth looking at at all.
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It has importance as a "teaching tool," but is 100% useless in real life. We do this to build to an actual example. I had a math teacher that did that, and his success rate was astounding-- most students picked up a concept in just days instead of weeks. – phyrfox Jan 26 '15 at 12:54
Context matters. If you say, "Let's start with a trivial example" and then you do start with a trivial example, that's good. You've put your class on notice that you starting with something that's stripped down and you're going to build on it. If you say the same thing but immediately introduce something that's moderately complex (even if it has been simplified as much as possible) then "trivial" was the wrong word.
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23Sometimes the professor (an expert) might assume something is trivial, but see looks of confusion in the eyes of the students. In that case, yes, perhaps the wrong word was used, but the instructor can easily "recover" by saying something like, "Oh, maybe that's not quite so trivial after all," thereby shifting the blame to himself for using the wrong word, as opposed to on the students for not understanding something so "simple"). – J.R. Jan 24 '15 at 11:44
I personally would not use the word "trivial" to describe a simple example. Instead, I would call it a "simple" or "toy" example.
The reason is that, to me at least, calling something trivial means it is not just simple, but degenerate in some way, and that there is therefore no insight to be gained from it. The examples that one uses in teaching are often very simple, so that they are able to be worked in real-time, but anything but trivial: instead, a good example is typically very carefully crafted to reveal particular interesting aspects of the material from as simple a situation as possible.
It is useful to recall what "trivial" means. It derives from the seven medieval liberal arts, and refers to the Trivium, or the "lower three" arts: grammar, logic and rhetoric. After the Trivium, students would study the Quadrivium, or the "upper four" arts: arithmetic, geometry, music, and astronomy.
The idea was that the Trivium provided the student with the tools needed to address the Quadrivium.
In this context, calling something "trivial" essentially means that this was either already addressed in a lower-level course, or can easily be understood using tools from that lower-level course. It does not mean that the "trivial" problem is easy, or boring, or useless.
Keeping this bit of etymology in mind, I'll happily call something "trivial" that fits this definition.
answered Jan 25 '15 at 20:38
No. The other posters have based their responses primarily on the meaning of the word. Mine is based on the impact on students. Students may be offended or hurt if you call something "trivial" which they do not feel is trivial. Students may not view the word with the connotations you, or other posters on stack exchange, anticipate. In addition, you cannot anticipate in complete detail what academic backgrounds your students might have. Maybe they were sick the day they should have learned the basis for your "trivial" concept. I consider the word unwise to use unless given a specific meaning first.
answered Jan 25 '15 at 20:48
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1So true. When I was in undergrad not too long ago, I honestly thought trivial was just a professional and academic way of saying 'stupidly easy'. On the day when the instructor is most likely to use the word trivial --the first day of class-- the students are not used to their tone of voice, their hand writing, and they may be just trying to decide if this class is for them. On top of that, they're still dusting off the cobwebs from the knowledge they required from an earlier term. Then they hear the words 'trivial example' and see the instructor write an equation that spans two boards O.O – NauticalMile Jan 26 '15 at 0:54
