In a school where kids learn problem solving math, there will be an understanding that nothing new can ever be learned in a vacuum! One can only hope to build on whatever understanding already exists. In other words, what experience the child already has, will to a large degree, determine where you can take he or she from there. This is where a child's developmental readiness also comes in. It is for instance why you'll have trouble teaching a 1st grader subtraction and a second grader division. Developmentally they are just not ready. In fact there are 4 developmentally significant stages that a child needs to learn in before real understanding of math concepts and equations can take place, according to Dr.'s Piel and Green of Charlotte's University of North Carolina. They are as follows: The Concrete, the Representational, the Transitional and finally the traditional or Symbolic.
If you are like my daughter Amanda and me, any visit to any mathematics stage, needs to be preceeded by a visit to Davidson Chocolate!
Now strictly speaking the child probably needs to visit all 4 stages before it can be said that they've learned a new math concept. The Concrete stage will involve some sort of manipulative objects that help hands on learning of the concepts desired and are the basic tools responsible for say, 2nd and 3rd graders executing pre-algebra problem solving!
The Representational stage is simply a bridge between being able to handle and manipulate objects in order to develop conceptual strategies, and solving only pencil and paper math equations. Representation of objects perhaps with pictures of the same manipulates (but not necessarily the same ones) just used, is the norm during this stage of learning. Pictures, diagrams and drawings help further develop conceptual and number oriented understanding.
Transitional. Typically this stage give students an alternative method of solving a math problem, to the traditional. After all, most of our traditional methods for solving, for instance, multiplication or division problems, involve tricks or short cuts, that don't divulge the actual math involved, and therefor require the child to memorize steps and procedures that have no relationship to their conceptual understanding of the pre-requisite ancestors of addition and subtraction!
Last but not least is the Symbolic or Traditional stage. What most learning systems consider the final destination. But what if you get to this "holy grail" of solving equations as they are written in every text book, with the yoke of having memorized a lot of neat tricks and procedures (that are based on good math, the student never gets to see unless he/she takes apart the process - this never happens!). What if! - well, all but the strongest math minds will be cut off from the learning tree when they are maxed out in terms of mastering and retaining, not to mention regurgitating, memorized formulae and their excess baggage. This is why a high number of students leave math completely once they leave their high schools for the ivy walls of university or college.
Now strictly speaking the child probably needs to visit all 4 stages before it can be said that they've learned a new math concept. The Concrete stage will involve some sort of manipulative objects that help hands on learning of the concepts desired and are the basic tools responsible for say, 2nd and 3rd graders executing pre-algebra problem solving!
The Representational stage is simply a bridge between being able to handle and manipulate objects in order to develop conceptual strategies, and solving only pencil and paper math equations. Representation of objects perhaps with pictures of the same manipulates (but not necessarily the same ones) just used, is the norm during this stage of learning. Pictures, diagrams and drawings help further develop conceptual and number oriented understanding.
Transitional. Typically this stage give students an alternative method of solving a math problem, to the traditional. After all, most of our traditional methods for solving, for instance, multiplication or division problems, involve tricks or short cuts, that don't divulge the actual math involved, and therefor require the child to memorize steps and procedures that have no relationship to their conceptual understanding of the pre-requisite ancestors of addition and subtraction!
Last but not least is the Symbolic or Traditional stage. What most learning systems consider the final destination. But what if you get to this "holy grail" of solving equations as they are written in every text book, with the yoke of having memorized a lot of neat tricks and procedures (that are based on good math, the student never gets to see unless he/she takes apart the process - this never happens!). What if! - well, all but the strongest math minds will be cut off from the learning tree when they are maxed out in terms of mastering and retaining, not to mention regurgitating, memorized formulae and their excess baggage. This is why a high number of students leave math completely once they leave their high schools for the ivy walls of university or college.
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