On Sun, 13 Aug 1995, Andrei TOOM wrote: <stuff deleted> > > This time I definitely disagree with you, Narge. In this recommendation > you are mixing quite different levels of mathematical sophistication. > Students who can understand those logical reasonings, which you are > proposing, are so sophisticated that they have mastered fractions long ago. > On the other hand, students who need to study fractions, certainly > can not yet follow formal arguments. When teaching fractions, you should > appeal to common sense rather than to formal abstarct properties. > You should discuss such practical things as dollars, pizzas, apples etc. <more stuff deleted> > > Andrei Toom
Andrei seems to have a conception of math learning, and hence math education, as a linear process. Experience (mine, anyway) indicates that this is not the case. For example, some of my calculus students last year were quite capable of following formal arguments, but they were incapable of finding common denominators involving algebraic expressions. I'd say that these folks needed to study fractions.
Also, my understanding of what Andrei writes is that he sees early mathematical learning as well-grounded in "common sense," or what I would call everyday reality, whereas higher levels of math, in my understanding of Andrei's perspective, require students to leave behind the everyday realities and to begin working in a realm that is strictly mathematical in the sense of math-in-the-classroom.
An alternative perspective (and the one which I hold) is that students in the early grades, who use pizza and money to learn about fractions, can and should also explore deeper mathematical understandings of fractions. In other words, using pizza and money to introduce the need for fractions is fine and good, but at some point the fractions themselves, rather than simply the things to which the fractions refer, should become objects of discourse. [Incidentally, this shift characterizes reflective discourse as defined by Cobb, Boufi, McClain, and Whitenack and hearkens back to Ron Ward's question on the nature of reflection in a constructivist view of the classroom, be it math, math ed, or otherwise.]
Similarly, students at higher grade levels should always be able to "downshift," i.e., to relate the formal math that they do to everyday experience. This can be difficult, particularly with such things as n-dimensional spaces and abelian groups, but it should also be remembered that as the level of sophistication of the mathematics increases through the grades, so (usually) does the level of sophistication of students' everyday experiences. That is to say that what counts as an everyday mathematical experience for a fourth-grader, e.g., pizza and money, is probably different from what counts as an everyday mathematical experience for a 12th grader, e.g., integers and geometric shapes.
Kreg A. Sherbine | To doubt everything or to believe Graduate Student | everything are two equally convenient Vanderbilt University | solutions; both dispense with the email@example.com | necessity of reflection. -H. Poincare