Technology boost. What is the benefit and any strategies for maximizing its effect?

How acquiring technologies works

First of all, it's important to understand that you do not get technologies by the number of turns you invest, but by the number of science points you have, just like it works for culture policies. It's just hard to see when looking at the technology graph because it only lists the amount of turns it will take; but each technology does, in fact, have a cost in science points. You can see that cost by opening the Technology's Civilopedia page (this can be quickly done by right-clicking the technology in the graph).

This is the reason, by the way, that you sometimes get technologies earlier or later than what is listed in the graph - because the estimation is done based on your current amount of science per turn, and that can change.

Also, be aware science points overflow - if you completed a technology, all left-over points will be immediately put to use on your next technology being researched. This applies regardless of the source of science points, so it will also work with a science boost (see below).

How the science boost works

So, a science boost is given in science points, not in number of turns. If I understand correctly, a Science Boost works this way:

1. Make a list of all the technologies you can currently research (that you have all the prerequisites for).
2. Find the cost for each of these technologies.
3. Find the median value of the above costs.
4. Multiply it by 0.5. This value gets boosted by 0.25 if you started Rationalism, 0.25 if you have the Porcelain Tower and 0.5 if you both started Rationalism and have the Porcelain Tower.

This number is the number of science points you get.

For example, say you can currently research Steam Power, Biology and Fertilizer. Steam Power and Biology cost 2350 Science, while Fertilizer costs 1600 science. The median of the set {1600, 2350, 2350} is 2350. Let's also assume you have the Porcelain Tower. The number of Science points you will get is 2350 * 0.75 = 1762.5.

By the way, on a personal note, this is a much better system than the previous one, in which you just got the cheapest technology available for free... although admittedly this is also much more tedious to calculate. As a rule of thumb, this will usually advance you by "half a circle" in your current research.

How to maximize the effects of a science boost

As for maximizing the effects, there are 3 basic ways to make it better:

1. Build the Porcelain Tower, which is a great Wonder even without this boost, on account of the free great scientist.
2. Start Rationalism. That's a great policy anyway for getting more science.
3. Make more research agreements! Remember you can make a lot of research agreements simultaneously, as long as they are not with the same civ. Also remember that the AI civs love their research agreements, so if they won't make them with you they'll make them with the other civs - meaning two other civilizations get stronger and you get nothing. It's better for you to be one of the two.