One of the most touted reasons for the necessity of every student undertaking mathematics courses in school (regardless of their ability or interest level in the subject) is that math teaches students critical thinking skills. I hate to burst bubbles here (actually I don’t) but this claim is completely and utterly false. Critical thinking in mathematics is a myth.
To “think critically” is by definition “to be critical of thoughts” or in other words “to critique ideas” as they arise rather than accepting or rejecting them blindly. Critical thinking means thinking rationally and reasoning through arguments with care and consideration of the options on the table. This, of course, sounds all well and good until we as educators stop and actually consider the reality of the situation in the math classroom.
A critique can only be meaningful if you already have a standard by which to judge something. I can’t critique an argument as “true” unless I know what it means for something to be “true.” I can’t critique a painting as “beautiful” unless I know hat it means to be “beautiful.” I can’t critique an act of charity as “good” unless I know what it means to be “good.”
There is an underlying standard of judgment that is being instilled in students (whether knowingly or unknowingly) and this is the root of what is occurring in mathematics classrooms. The way in which students implement their “critical thinking skills” upon the completion of their mathematics courses is simply a symptom of a deeper reality that is being formed within them. That reality can be formed from a Christian perspective where it is God who sets the absolute standards for Truth, Beauty and Goodness, or it can be formed by very humanistic standards of relative truth, subjective beauty, and goodness defined by utility.
I submit to you that there is an opportunity present before us as math educators to impact the very standards by which students judge their thinking. This is the goal we should ultimately be aiming for. Don’t settle for simply teaching “critical thinking” skills.
They are just a myth.
GOD & MATH: Thinking Christianly About Math Education 
These are interesting reflections. Please write more on this idea. I hope I am understanding correctly that you are saying that students cannot actually engage in “critical” thinking if they do not have the foundations to judge mathematics critically. As a Christian educator, I prefer the term “higher-order” thinking, though this is problematic as well. I agree that we need to teach students the foundations of mathematics in order that they can truly think critically about it.
But I believe math already holds a standard of “truth” which is known and (hopefully) practiced by professionals mathematicians, and is tested by the sciences. That “truth” being anything which is the result of a deductive argument. To think critically under this definition is to criticize anything against this standard in which case one certainly cannot conclude it is a myth. As such I have to wonder how one would investigate the relation between the truth of deductive reasoning and that “deeper reality” formed by God. If I interpret your meaning of “relative truth” correctly, deductive reasoning, as opposed to inductive, is by no means relative. I might even argue that deductive reasoning is the highest standard of truth and any higher one is knowable only by enlightenment.
While I would disagree with teaching students the foundations of math to think critically with my standard, with Josh’s I think I might see how it would work. Err, actually no I mostly don’t. This has been a bit of a tough one. To reduce math down to its foundations means one would have to have a way to rederive (or discover) what has been reduced. Interpreting the foundations as the basis of truth (so that we may think critically) means that the exploration of this “deeper reality”, which should be governed by truth, admits deductive reasoning as this is how we deemed the foundations to be the foundations. This all seems sorta problematic so I’ll leave that there. Again I think that you must make clear of the relation between deductive reasoning and the truth set forth by God.
Under your standard of truth Josh I think that generic understanding of how math works as well as results such as Godel’s Incompleteness Theorems would teach more “critical thinking” than any other subject in particular.
R.A., I agree, math does have its own standard of truth as established/practiced by mathematicians. In fact, the point of my post was that this standard is implicitly taught in our classrooms without students (and even teachers) being aware of it. But this standard of truth (which is best categorized as logical and rational) allows for no divine mystery. This standard of truth is completely dependent on the minds of men and gives no consideration to one who created those minds. God sets the standard for truth. My point is that anything labeled as “true” must be measured against that standard, and we as math teachers should make students explicitly aware of that concept.