Our university has a math sequence for elementary education majors. The courses mostly take a very deep and comprehensive look at the underlying mathematics involved in about K-6 education. Since we have a number of mathematics education professors in the department, the courses are extraordinarily well designed. It meets in a room with large hexagonal tables where students can sit together in groups of six, and we have two large cabinets filled with all kinds of wonderful manipulatives which are used in many class activities. (A manipulative is any sort of physical object which can be manipulated to learn math. We have various types of colored chips, geometric shapes, and other cool toys for demonstrating mathematical concepts.) Many of the activities are actually similar to and based on activities which could be used to introduce concepts to elementary students, although of course the college students are expected to go a little deeper and are asked to do some things that we don't ask elementary students to do. (For example, we have the students in the course perform various operations in bases other than ten to emphasize the basics of a place value system. No one teaches base four or base twelve to elementary students anymore.)
It's actually a really fun class, and full of all sorts of wonderful discoveries waiting to be made. I personally find myself fascinated by the fact that in many cases, the way we explain a concept to our students parallels the abstract definitions which can be used to define that concept in advanced mathematics. So whereas in class we may use groups of red and yellow counters to define the integers, a mathematician might start tossing around scary sounding phrases like "sets of ordered pairs" and "equivalence classes", but ultimately mean pretty much the same thing. I personally found the demonstrations hugely enlightening the first time I did the class. It provided me with very concrete way to think about and explain concepts such as why a negative times a negative is a positive and why dividing by a fraction is done by multiplying by the reciprocal. The idea of the class of course is to provide our future elementary educators with similar insights.
Unfortunately, the class is always a struggle to one degree or another. One particular point which the students never seem to get (no matter how often they are told) is that this is not a class in elementary school mathematics. We obviously expect them to have already learned how to do things like add and subtract integers and fractions, how to multiply and divide multi-digit numbers, and the like. After all, they were supposed to have mastered these topics in grade school. (Except of course we know many of them actually can't do these things reliably, so the course also helps back up these concepts. But I digress.)
As a result, the students sometimes ignore instructions on how to complete an activity. For example, they are supposed to learn how to represent integers with sets of colored counters and then use the counters to add and subtract integers. (This is actually a really cool activity; I'll have to write about it sometime.) But since they know what 7+(-4) is, and following the directions to make representations of the numbers using the colored chips seems complicated, they instead just write down "7+(-4) = 3" and explain to me that "the model was too hard, so we just did it." Since they feel the class is about (or should be about) learning to add, subtract, multiply, and divide just like they did in grade school, there is no need to learn anything else about these topics as long as they know what the right answer is. They sometimes fail to understand that the colored counter model they have been asked to use is, in essence, the content of this course: we want them to learn to use a physical model which represents basic operations on integers, and to use that model to derive various known properties of addition and subtraction with integers. This issue is usually an uphill fight all semester with the students.
But this semester I'm getting even another argument from some students in one class. With almost every activity we do and with almost every mathematical model we describe and learn to use, the students complain to me that "this is too hard for any little kid to understand!" Which is completely irrelevant, since I'm not asking any little kids to do this work, I'm asking my class full of college students to do this work. I've told them I don't address the issue of how to teach their future students, but rather just teach them mathematics. I leave it to other people to teach them how to teach math. This doesn't sway the students.
I tell my students, "I'm not asking your students to to this."
The students respond, "Yes you are!" against all evidence to the contrary.
My students somehow feel that any topic which they consider to hard for a third grader should be too hard to ask a college student to do either. I suppose they want a refresher of third grade math without any of the "hard stuff." Remarkably, I seem to have little success with convincing the students that they are not, in fact, third graders. Do they really think that in a college math class they should learn nothing more than what grade school students are expected to learn?
But do you want to know what the worst part is? Most of the activities actually aren't beyond the grasp of moderately intelligent third graders. I consider it the dirty little secret of the course. Granted, it would take more time, but grade school students could certainly be taught rules for representing integers with colored chips. With practice, they could learn techniques for adding and subtracting with the colored counters and even explain how it works. Eventually they would find patterns in what happened when you add and subtract integers. The same is true for almost every other topic we discuss, from the most basic (addition of whole numbers), to the most advanced (division with fractions, perhaps). You couldn't do all of K-6 in a semester obviously, and children may not make as many connections as a college student ought to be able to, but they could do almost every activity we do in the college course, and learn a lot.
I don't even bother to argue the point with my students 'though, because whether grade school students could do what we do or not is entirely beside the point. My class isn't filled with grade school students. It's supposedly filled with college students. College students who want to be elementary teachers. The same teachers that will lay the next generations mathematical foundations. Which will, in another ten to fifteen years or so, become our next generation of college students sitting in my college classes. And that thought usually fills me with the urge to go lie down for a while.