Just now I'm thinking a crazy reversed idea: Is it good to teach math to elementary students without starting from its real world motivation or making it intuitive first, but rather by realizing that it is a game with some rules which revealed later to have real uses?
I am thinking that if we start with the intuitive ideas of some concepts, then we are required to make almost every concept we introduce intuitive..which is not quite obvious for a lot of students. Thus, sometimes we have to skip introducing some concepts intuitively, but we still introduce only some easily intuitive concepts. Doesn't this affect students' motivation, making them ask: "I can't grasp this. What's the real world example?" when there is not an easy connection..
I mean, since math is abstract, although many ideas in it are intuitive, (many are not)..if we start by introducing a concept as a game with rules and students have to master this game, how would this impact the learning process?
Example: When teacher teach about fraction, they usually start with dividing cake or pizza or something, and then put it in the mathematical notation of fraction that we know well. I used to (and probably still) think this is a reasonably good way to teach math. However, as implied in my question, can the teacher not start with this real world motivation, but instead start immediately by introducing the notation for fraction, and give some of the rules of summing, multiplying, etc, and the challenge for the students is to apply this rule..which sounds pretty much like playing a game of fraction?
Hence, I don't mean that we use games when teaching math, but I mean math itself is made as collection of games. Then, at some point, the teacher relates that fraction to real world uses, if any,..such as dividing pizza, etc.
I don't assume this idea is new. In fact, I ask whether some have considered this method or not. I just think that this way, students won't rely too much on real world uses of math before studying it, hence not being overly dependent on that..since some math concepts such as negative times negative, or even the whole integers, or square root, or even some geometrical objects, are quite abstract in the minds of elementary students, right?
I am asking this only relating to elementary math, not higher math, since I know the higher we go, we have motivation before abstraction..and at some point math students will learn that..but can we start by giving "the rules of the game" and "play" for elementary students..