Sequential and simultaneous modes of awareness
The most interesting part in Ted Chiang's "Story of your life" is the parallel of the causal and teleological explanation with a sequential and simultaneous mode of awareness.
Fermat's principle of least time can be interpreted in terms of cause and effect: a difference in the index of refraction caused the light ray to change direction when it hit the surface of the water. This is most intuitive to us humans.
It can also be interpreted teleologically: the ray of light has to know where its destination is in order to compute the path of least time. This is more intuitive to the heptapods.
The parallel to the causal explanation is a sequential mode of awareness: experiencing events in order, and perceiving their relationship as cause and effect. This is how humans experience things. We don't know the future and are therefore able to exercise free will.
The parallel to the teleological explanation is a simultaneous mode of awareness: experiencing events all at once, and perceiving a purpose underlying them all. This is how heptapods experience. They already know the future, so freedom is meaningless and every act is performative*.
If you have free will, it's impossible to know about the future because you could change it. On the other side, if you know the future you cannot act freely anymore. (as in the example of the book of ages).
Sequential and simultaneous modes of awareness are like the optical illusion of the old and young lady. Both are valid but you can't see them at the same time.
*Performative language: Saying equals doing.
Example: At a wedding ceremony everybody knows that at the end the pastor will pronounce the couple husband and wife but it doesn't count until he actually says it.