Back when I was teaching one of may favorite things was to answer questions. Sometimes you can just get by with the facts. But a lot of the time, when you're working on getting someone to understand some mathematical concept (or any other concept, I'm sure), you need to nudge them in a way that works for them. And people think differently. So to help them understand, you have to answer the same question different ways for different people.
Sometimes it might require giving them a way to visual the concept, for those that are visually oriented. Other times it might require getting up and moving about, when a person thinks more kinesthetically (with their body). And some people are able to grasp abstractions with ease. And a class can be filled with people with each of these ways of thinking. Or even more. And all are valid ways to learn.
And ask questions. And asking questions is the most important thing there is. And just because the same question is asked by different people, it doesn't mean they have to use the same way to answer the question.
Take as an example the question of whether there are stars, other than our Sun, that have planets. Several years back they answered the question “yes” by looking for stars that wobbled as a result of planets pulling them ever so slightly. And they found planets. The looked at piles of data and found the wobbles they were looking for. And the planets the found were strange. Big, usually much bigger than Jupiter, and hot, circling close in to their stars.
But that makes perfect sense. The method is biased towards find planets that pull hard on their stars. And those would be big and close in. And all by itself, bias isn't a problem, though it requires a person to acknowledge that their understanding is incomplete if they use a biased method to answer their questions.
But that's not the only method to look for planets around other stars. I recently became aware of another that also uses gravity. But it is able to see the tiny bending of light of one star passing behind another one. This is referred to as lensing. (Lensing was one of the first predictions of Einstein's General Theory of Relativity to be confirmed.) And then if there are planets circling the star, we get a tiny bit of extra lensing.
So new measurements were made, and what happened? They found planets. And just as the wobble technique was biased to large planets close in, the new method actually was better suited to finding small planets, and ones further out.
Small planets? Yes, small rocky planets like the Earth. Or Mercury. Or Mars. And they're out there.
Using statistical techniques they found one in six stars has a planet like Jupiter. But two-thirds have planets like the Earth in size. Calculating away, just about every star in the galaxy is likely to have a planet of some kind.
And the Milky Way Galaxy in which we live has 200 to 400 billion stars. And if most of them have at least one planet, there are truly billions and billions of worlds out there.
So what do we want to ask next?