Science
Science: what it is
Science comes from the Latin word "scientia," which translates to "knowledge." Despite this, science isn't knowledge. Science isn't technology, either. Science is actually a way of searching for answers.
Specifically, science is looking for answers in a way that tries to remove personal bias, and that can be repeated by someone else who wants to make sure you got the right answer. When you find something you're not sure about, you can frequently make a guess about what the answer is. That doesn't mean it's right, though, and that's where science comes in. Actually, it starts with coming up with the question, but it's starting with this next step that science is special. After you've come up with what you think the answer is (also known as your hypothesis), you test it, usually against another idea called the null hypothesis.
To test your idea against the null hypothesis, you observe whatever it is you're interested and see what it does. If you're lucky, the thing you're interested in is something you can interact with, so you get to make two groups of it. One is the control group, which you don't touch at all, and the other is the testing group, which you change in some way. If your idea is right, you should see the changes you expect in the test group but not the control group, and so should anybody else who repeats your experiment. That's the real point of science: doing something in a way that someone else can try the same thing to make sure that you got it right, and that it wasn't fake data or a broken instrument.
How we teach it
Lower Grades
We don't expect kindergartners to immediately understand the idea of testing a hypothesis (or at least not most of them), so we start off by exploring natural history. We cover a lot of the same things you'd expect to see in regular science classes (simple anatomy, light and sound waves, etc.), but we get them used to asking "why" from the very start, not that we expect we'll have a lot of problems getting five-year-olds to do that. They get a broad overview of discoveries in natural history while we subtly introduce the idea of testing a hypothesis.
Once we've formally introduced the idea of testing a hypothesis, we change the whole game. Instead of us just telling them about something and showing them a demonstration that shows how it works, we start bringing up questions and asking them how to figure out what the answer is and having them actually test their hypotheses (that's the plural of hypothesis, since each kid will come up with their own). Just to make it perfectly clear, they will be designing and carrying out their own experiments (under adult supervision), not reading the results out of a textbook.
Unfortunately, repeating every experiment ever done to independently verify all scientific knowledge is pretty inefficient and insanely expensive, so we continue exploring natural history even as they conduct their own experiments. We're aided in this exploration by the wealth of scientific artifacts and equipment from the collections of the Academy founders, Lou Ellen Kay and Vince and David Gutschick, as also by the generous donation of 60 university-level microscopes.
Higher grades
We still have the students conduct their own experiments, but as we said, it rapidly becomes an impossible task to replicate every experiment ever done (especially as the science becomes more advanced). We compromise by doing longer, more involved experiments interspersed with research involving more books and presentations but fewer microscopes.
It will be a few years before our students learn calculus and statistics, but once they've reached a sufficient level of math, they'll start learning calculus-based physics. We're some ways from figuring out the order of the physics curriculum, but we do know that we'll spend some time focusing on the physical principles that chemistry is based on. Teaching physics first, we'll be able to cover chemistry without glossing over all the reasons acids are acids, bases are bases, and atoms in general act the way they do. Next, we'll take the knowledge they gained in chemistry and apply it to biology. DNA, metabolism, and basically everything else in biology makes much more sense with some background knowledge of the chemicals that go into making a living organism. This isn't the order physics, chemistry, and biology are taught in most schools, but we think it's pretty evident that it's a much more reasonable way to approach them if you can teach calculus early enough.