It goes without saying that, if at all possible, children should enjoy their experience of learning mathematics. But there are different levels of enjoyment.
Human beings are (at least for the most part), built to enjoy the aha experience of insight. That means that math can be full of enjoyment. The insight method of teaching math hence is intrinsically enjoyable, allowing students to experience the real enjoyment of really doing math.
As an aside, it seems that many problems are constructed by people who do not enjoy math. They just aren't as enjoyable as they should be.
As another aside, if your only goal is to teach math, not enjoyment, you still need a secondary goal of the student's enjoying it. One reason is so that they will continue with it. The insight method actually requires them to become absorbed in the problem. Also, enjoyment is important feedback. If the students aren't enjoying themselves, that means that they aren't getting insights and the learning isn't occurring.
An Enjoyable Milieu
The normal milieu for math is that there is a problem, the problem has a right answer, the student's goal is to solve the problem, and the teacher gives feedback as to whether the answer is right or wrong. (It's hard to imagine anything different, right?)
This works fine for learning. It is just not as enjoyable as other learning environments you can embed your math in. And because enjoyment is important for motivating the student and increasing absorbtion, the other learning environments are probably slightly more effective.
So, as much as possible, you want a good milieu. This includes using story problems rather than abstract math, using a milieu more like skiing or surfing, and asking for opinions and preferences (when feasible) rather than right answers.
I have a math problem I want to give my student. It could be about anything. My student likes faeries. So I make the problem be about faeries. Other problems are about her favorite doll who comes to our math sessions.
This is just "wrapping" the math problem around a topic that is likely to interest the student. It just makes sense -- even if enjoyment isn't an important goal to you, why not make the math as enjoyable as possible.
It turns out that you can add enjoyable things to the math experience, things that have nothing to do with math. Jokes. Enjoyable social interactions. Exciting competition.
I am not dissing these. When I start out teaching a math phobe one-on-one, I can immediately start with math problems, and I expect to immediately engage that student in math. The reason I can do that is that I am very good at creating a fun social interaction.
But it's not Just Wrappers and Gum Drops
I try to make math enjoyable. But everyone does that. The problem is, everyone but me seems to think that the enjoyment in math is in the wrapper and the gum drops.
So let me be clear. Yes, I try to use enjoyable wrappers. Yes, I use gum drops. But in addition to that, I try to help the student find the enjoyment in doing math -- the insight.