Almost all math problems call for the student to assume one thing or another. Without assumptions — definitions, postulates, axioms, common notions, or whatever you want to call them — mathematics of any kind is impossible. Tony at Pencils Down (who plans to be a math teacher when he grows up) reminds us that, necessary though it may be, we are stepping on dangerous ground when we assume:
Meanwhile, I read some interesting things in the world of ScienceDaily:
Physicists Tackle Knotty Puzzle
Knot theory for the real world. Now if only they could tell us how to un-tangle those knots!Chimpanzees, Unlike Humans, Apply Economic Principles…
A game theory application: Could the universal childhood cry of “That’s not fair!” be related to what the Bible calls the “image of God,” the quality (whatever it is) that sets us humans apart from other animals?Nanotechnology Surges Into Health And Fitness Products
Anyone want a cup of Nanotea?
Denise Gaskins' Let's Play Math 
It has been interesting to me to see older math books begin sentences with “We assume that…”
I should open up my modern algebra texts and see what sort of terminology is used. I recently found an article in an older newsletter of the AMA written as an OP/Ed piece saying that we need to spend more time discussing with students what axioms are and why they are needed. Even in courses in which some subject matter is developed axiomatically (I guess Euclidean geometry is all that’s left) they claim is that the role of the axioms themselves is ignored.
This morning I was flipping through one of the older Dolicanis and they stated two axioms (to be assumed) and yet the second axiom was easily derivable from the first one.
Even in elementary school, I encourage my students to notice what assumptions are inherent in their word problems. I have heard teachers tell students to always ignore the word “if” in word problems, but to me, that is one of the most important words. It is laying out the foundation, the assumption, that the rest of the problem is based on.
I copied off the beginnings of Euclid for my geometry students, and they gave me the “You’re out of your mind!” look. A point is what?! I have managed to draw a little bit of axiomatic reasoning from them, with plenty of coaching, but most of the time they leap to assuming the conclusion and using it to justify their intermediate steps.