Is it a good thing as a teacher to declare things like "Good! Things are becoming clearer already."

Question

Is it fine for a teacher who makes a topic look easy by saying things like:

"Good!"

or

"Easy!"

or

"Everything becomes perfectly clear now!"

Things I can think of in favor of doing this is that it might reduce students' concerns and will create an environment where nothing is too complicated.

On the other hand, I'm afraid it can also create an environment that if for some students the topic is not clear, they will feel worse now as the teacher said that "it's supposed to be easy" and they might even be afraid to ask to not show that they couldn't understand something that supposed to be "easy".

Answer 1

If a teacher is speaking to one student and has evidence that the student now grasps something, then is fine. Otherwise, I'd worry about such things.

Every student is different and some will fail to grasp an important point for various reasons. Sometimes it is something that the teacher said that the student misinterpreted and was misled.

Those of us who teach sometimes fall into the trap that what we say is said in the most perfect way possible. But we have a long history that leads us to speak in a certain way about our subject and the students don't share that history. The human languages we use aren't perfect and often don't map words and phrases one-to-one with the ideas we try to convey. Language is seldom exactly precise, especially for complex things.

Saying "All is clear now" to a group is, therefore, probably not true. Those for whom it is not clear will be frustrated and feel frustrated and even angry.

Asking "Is it clear now?" is a better approach. And if you then get questions, don't use exactly the same words over again to try to make it clear. Those words may be the problem.

Many things need to be explained in several ways before a group of people will reach a common understanding. Even some mathematical proofs are suspect because they leave out "obvious" or "trivial" steps that a reader doesn't grasp without a lot of work.

Answer 2

I started my academic career as a high school teacher. As part of earning the credential to teach high school, I had to go through a semester-long "internship" as a student teacher---I taught in a classroom under the close supervision of a veteran instructor. One of the first bits of feedback I got was that I had a tendency to say that a problem was "easy" or "clear". In my own head, it was in the context of a universe of possible problems, e.g. "This problem is easy as compared to this other problem on a similar topic," or "Let's start with an easy example before moving onto harder examples." My supervisor encouraged me to avoid this kind of language, because

1. no problem is easy and nothing is clear, and
2. telling a student that a problem is easy destroys morale when they can't figure it out.

No problem is easy

There is a great Saturday Morning Breakfast Cereal comic which describes a view of how ideas come about in mathematics, and how we disseminate those ideas. Essentially, modern mathematics is the distillation of several thousand years of thought. The ideas and results were hard-won, yet we somehow expect students to master them in very little time.

I imagine that this observation is true across disciplines. For example, I can't imagine trying to give a one-hour lecture summarizing Lewis Henry Morgan and the modern study of kinship, nor can I imagine how clear the ideas would be to students after such a lecture.

Ultimately, I think that this is a form of the curse of knowledge: as instructors, we might think that the ideas are easy because we are so familiar with them, or perhaps because we never had difficulty with them ourselves, but that doesn't mean that they are actually easy. We have to figure out how to put ourselves back into the shoes of a novice, and remember how difficult the ideas were in the first place.

The impact on morale

Because mathematics is so often stigmatized^[1], I think that mathematicians are (or, at least, should be) particularly sensitive to things which are likely to further drive people away or cause them to shut down. However, I think that all instructors across all fields should be aware of how their style impacts morale, so this isn't really a math-specific observation.

If you tell a student that something is "easy" or "clear" and that student is still confused, you are implicitly suggesting that the student has failed to grasp an obvious concept. This tells the student that if they admit to their confusion, they are admitting to being inferior, and that they might become an object of ridicule. The student understands the failure to be theirs, and theirs alone, rather than a failure on the part of the instructor, or a blameless failure in the process of communication. Such a student is not going to ask a question, is not going to seek clarification, and is likely to withdraw from the topic.

As such, I think that it is imperative that we avoid saying things that trivialize the work that we and our students have to do. It is bad form to tell a student that a concept is clear, or that a problem is easy.

What might we do?

In the question, it is suggested that an instructor who makes a topic look easy might note how easy things have become. I think that the reality is quite different: the teachers who make things look easy are those who are most sensitive to how difficult things really are. They lean into that difficulty, and explain clearly to students where the problems lie. It is then up to students to come out the other end and say to themselves "Wow! I thought that idea was hard, but the professor really made it easy to understand!"

I don't think that there is a one-size-fits-all solution to this, but there are a few things that I do in my own practice which, I think, help:

• Ask Questions: I don't assert that ideas are clear; I ask if they are clear: "Is this clear now?", "Do you all understand?", "Do any of you have any clarifying questions you'd like to ask?", "Can we move on, or would you like to work through another example?", and so on. Indeed, I often ask "Is this clear?" in a situation when I know that it should still be quite unclear to students. Done often and early enough, this gives students permission to admit that things are confusing.

• Provide Context: To the extent possible, I like to give students some historical context for the development of ideas. For example, we might spend as little as a week defining the derivative of a function. This short period of time belies the enormous intellectual achievement it represents: the ancient Greeks were the first to spot a problem (Zeno's paradoxes), Newton and Leibniz (both of whom "stood on the shoulders of giants") first put the ideas together, Euler (and Fourier and Cauchy) introduced the modern notion of "function", and late 19th/early 20th century mathematicians (Hilbert, among others) nailed down the formalism. If the derivative looks easy, it is only because it took such a long time to refine and distill the ideas. It is actually a remarkably hard concept.

• Employ Appropriate Comparatives: It is, I think, entirely reasonable to assert that one example is less difficult than another. Logically, this is the same as asserting that one example is easier than another. However, when one says that a problem is "less difficult" it could still be hard, whereas if one asserts that a problem is "easier" then a student might only hear the "easy". I don't start with "easy" or "simple" examples; I start with "less difficult" or "illuminating" examples.

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[1] Speaking from the perspective of an American, it seems that no one has a problem with a person saying "I'm just not any good at math!" or "I'm not a math person!" Fear and loathing of mathematics is considered appropriate, and mathematicians are seen either as god-like geniuses, or asocial nerds (or both, I suppose). It is considered okay for a "normal" person to be mathematically illiterate. A seemingly equivalent admission of illiteracy ("I'm just not any good at reading!") would be shocking.

Answer 3

I agree with the other answers here that such statements are problematic because they can both demoralise students and also make it difficult for them to ask for clarification or a better description.

However, I would also, in general, heavily advise against using questions such as Is that clear? Before I became a TEFL teacher (English language teacher for non-native learners), I used to use such questions all the time in my teaching. It may surprise readers to learn that within that field such questions are considered a cardinal sin of the highest order. I believe this evaluation to be correct, and have used this insight to improve my teaching elsewhere. However good or bad my teaching may be, there is no doubt it is better for avoiding this pitfall—painful though it has been to take this on board.

The problem is that questions such as Do you understand?, Is that clear? or even Have I explained that clearly? put students in pretty much the same kind of situation as when a teacher says And suddenly it all becomes clear. The reason is that students don't like to say No! to question like these. Even if they are not scared of offending their tutor, they will be unlikely to want to show themselves up (as they will see it) in font of their peers, and they certainly will not want to risk looking slow or stupid. If tutors ask questions like these, their students will almost definitely be chasing their friends in private to help them better understand the material afterwards. It really is neither here nor there if occasionally one has experienced students saying "Actually, no that isn't at all clear". It's not what most students will normally do.

How to get round this problem, then? Well, one way is to prepare, decide what the key points that your students need to understand are, and also note what some predictable misunderstandings may be. Then check your students' understanding by asking a very few, very simple questions. Ideally a majority of these, but not all of them, will require your students to disagree with some presupposition in the question or to give the answer No!. This is just because otherwise the students will just say "yes" repeatedly. There are very many ways to go about asking/presenting such questions and soliciting answers. (For example you could ask them as a group directly, you could get them to anonymously fill out a three of four question mini-quizz before leaving and stick their answers in a box on the way out. They could ask their peers questions from a slide and so forth). It doesn't matter how simple the questions seem to you.

To reduce this to a ridiculously simple example, from which one can extrapolate out to physics, some point of law, or propositional logic: Suppose I've tried to teach my beginner English students the item rice amongst some other vocabulary. My two main concerns are that they understand what the word refers to and that they know that this word is uncountable in English. I might say,

• Is it meat?
• Is it black?
• Is this good or bad: "I ate a lot of rice"
• Is this good or bad: "I ate many rices with my chicken"

Of course, in this situation the students won't give just yes or no answers, but will supply a lot of extra information. Compare this with:

• So then, rice is that white stuff that you eat with curry or with chicken. Rice is uncountable in English. You can't usually say two rices. Do you understand?

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^{Yes, of course, given the appropriate resources one could clear up half of that by using a photo of some rice. But for the purposes of the example, let's suppose such resources aren't available!}

Answer 4

I've often rated the topic at the start of class: "a lot of people think this next thing is hard. I'm going to go over the problem it solves, then hopefully the rules will make sense. There are only 2, but they're weird". In that context when I say something is easy I'm saying that compared to the class so far, it won't be as bad as you were expecting.

But that thing about "...who makes a topic look easy by saying "Good!" / "Easy!" / "Everything becomes perfectly clear now!" seems like a nervous habit. Maybe that's their way of saying "and that's the end of explanation" -- "and now everything is clear!". That seems fine. I used to say things like "So...was that a minute longer than it needed to be? Let's do some problems". If someone is sincerely taking extra time to say "my explanation is so good that you understand it perfectly now", that's just weird.

Answer 5

I do think this is a poor piece of rhetoric.

It shares shades of writing that includes a lot of "obviously", "of course", "trivial to see", etc. This tends to signal a place where the writer has a weak explanation and is trying to get over that hump by using a proof by intimidation.

As an example, when I reviewed Sebastian Thrun's Udacity Statistics 101 course, this is one of the criticisms I made (see section 9, "Hucksterism"). In those presentations he had an immense number of, e.g., "You now know a lot about scatter plots!"”. "Isn't this a lot of fun?", "You are a very capable statistician at this point!", "smile and say you took Sebastian's Stats 101 and you understand!" -- which was pretty thin paper over lectures he was clearly ad-libbing with no prep or editing, late at night after his day job.

It's not the biggest issue in the world. But I do think it's a clear mistake -- and usually symptom of other weaknesses in the presentation.