Sadly, this is not what we instill in our students. By and large, the kids are firmly convinced that easy questions are the only ones worth answering. In fact, in the face of anything that resembles a difficult problem, the kids tend to give up and wait for someone to show them the answer, regardless of whether or not the answer is accompanied by an explanation. This is the common experience of pretty much any high school teacher. It is not, however, the experience of anyone who has regular contact with younger children. And by "younger", I'm not talking about much younger than high school age; my 6th grader is a shining example of this. These children are never satisfied with only knowing an answer, and anyone who tries to simply give them an answer to a question without explaining why the answer is true is in for it.
I'm not going to attempt to delve into why it is or when it is that this thirst for an answer to the question "Why?" is killed off. Instead, this has to do with an observation I made in my classes this week. In one of my classes we are in the middle of the material on trigonometry. The way we organized this part of the course was to spend the first unit on sine and cosine exclusively, from the basics inside a right triangle to the graphs to the identities. We then repeat the process with the tangent and secant functions in the second unit and the cotangent and cosecant functions in the third, while still holding on to the material from the previous units. This term, the kids did reasonably well with the first unit and notably worse in the second unit. I sort of saw this coming because in discussing the exercises in the first unit the kids were running through the "easy" exercises without really delving into the "why" of them. Since the main topics were sine and cosine, topics with which they were already quite familiar from the last several math courses they have taken, they were able to still do well on the first test. However, coming into my course they are not as familiar with the tangent and secant functions, so when they weren't bothering to slow down and discuss the "why" of the easy questions, I sort of knew that they would struggle with the "more difficult" questions late in the unit, which they did. However, rather than putting forth the effort to answer the more difficult questions, and rather than going back and looking at the easy questions a second time to see what they may have missed, the kids tried to fall back to their old habit of memorizing everything for the test. This, of course, did not serve them well, since the tests are designed to make evident whether or not they actually understand the material.
On the flip side of this, the first unit also includes the material about ellipses, and the second unit contains the material about hyperbolas. Overwhelmingly, the kids did well on the questions about these topics on the tests. In my opinion, the reason they did so well is because they did not come in to the course feeling completely comfortable with these topics, and spent the time necessary with each exercise, not seeing any of the questions as easy, and really trying to understand the content. In doing so, they did not resort to simply memorizing the material, but focused on understanding it instead. In other words, none of the questions was easy, so from the beginning they were focused, as opposed to what happened with some of the trigonometry questions where the easy questions were effectively ignored and the important content from them was missed. On the positive side, in neither instance did the kids turn and run from the more difficult questions, nor did they wait for me to simply give them the answers. Both of these are very good things.
The moral of the story is this: there are no easy questions. There are foundational questions, the answers to and the content of which are more readily understood at a surface level, but when it comes to really understanding something, there are no easy questions. The obvious must be discussed and thoroughly understood for there to even be the possibility of grasping the more difficult material. The kids will get another opportunity at this in the third unit, and I plan to explicitly tell them to slow down and make sure they really understand the answers to the "easy" questions...questions which technically don't exist.
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