Article written by Javier Bernabeu. Wonderful educator whom I had the pleasure of meeting a few years ago and to whom I owe my reconciliation with mathematics.
What if the almost general problem of learning mathematics was found in numbers and symbols in general?
What a truism! Well, where is the problem going to be with maths, if maths are numbers and more numbers?
And this is where this story begins. The story that the problem with math is not in the numbers but in the stories. Or, rather, in the absence of these. In the absence of stories.
We are used to relating mathematics with numbers, symbols, expressions that are used to solve anything. That’s the problem! Let’s focus on that “whatever.” What does this mean? What is this story about?
In the expression, 48: 3 – 2, the important thing should not be to have learned by heart a series of rules to know in which order to solve the expression. Not even knowing how to correctly divide 48: 3 and take 2 from the result.
That expression is the way to write a meaningful story in mathematical language. It is the way to translate
a symbol, for example, this story:
Julieta had 48 candies and distributed them into 3 jars. Afterwards, he decided to eat 2 candies.
At the moment when the expression stops reading like this: “forty-eight over three minus two” and begins to read like this: “Julieta divided her 48 candies into three jars equally, but then, as she became hungry, she decided eat 2 candies” the thing makes sense.
We begin to understand that symbols are nothing more than socially and mathematically accepted drawings to “universalize” history (so that, in any corner of the world, we can express ourselves mathematically and be understood). BUT, and here comes the problem (in my opinion), many times, we reverse the process. That is to say, we start from the symbols, from the abstract when, in the “concrete – pictorial – abstract” process, the start should always be from manipulation and, only, when the time is right, will we move on to symbolic representation.
It is almost always (if not always) a good idea to eliminate numbers in a math problem. It is common for symbols to condition us. We see numbers and… let’s operate! Losing ourselves, in this way, the entire process of deep understanding. If we eliminate numbers we can find out if our child can establish relationships (from manipulation and experience). We’ll see how we do with the numbers. That is another topic, but if the objective is to check if the child can establish the necessary relationships that allow me to check if he really understands the situation and if his hands carry out appropriate tasks that, later, can be specified in the mathematical language of one or the other. In any case, we will have achieved the purpose.
Let’s imagine the previous problem without numbers. Even without drawings (without pictorial phase). Only with the pure manipulation of materials.
Today’s story, the problem to be solved is like this:
Julieta had a few candies. He distributed them equally in three boats. Profeeee!! But how many candies does Julieta have? The ones you want. Decide! Today, polycubes are going to be candy (as long as there is a minimum symbolic game that allows us to give significance to the materials).
Here our observation task begins: the child has picked up some polycubes (candy). The first thing you have done is check if you can distribute them equally between the three containers. This will lead you, to begin with, to modify (if necessary) the amount of candy you took. The starting situation is created. The child built it.
Obviously, he does not know that in order to start the situation we need to choose a quantity of candies that is divisible by 3 because otherwise, he would not be able to distribute them equally among 3. But, without a doubt, since we did not put a condition on how many “candies ” take but acted by “trial and error” has found a solution to his problem and has done so from the most important point in all of this: from his autonomy. He has decided, he has acted and he has resolved the small problems that arise in order to be able to face a larger situation.
Once I, by myself, have created the situation, I can do with it whatever I want, because the situation is mine. What should I eat some? Well, I decide how many and from where. We have just ensured that there is understanding. The translation into mathematical language, the writing of the symbol is another phase.
For now, the important and fundamental thing to take the next step is to understand.