Educating Awareness of the Necessary

One of Gattegno’s teaching strategies that we discussed at the recent Gattegno Conference was providing the arbitrary and not the necessary.  The arbitrary being the conventions that society has decided upon to relate an idea.

For example, the student can discover the train of three yellow Cuisenaire rods that is three yellow rods lined up end to end.  As a teacher, I would provide the arbitrary convention of how this train is represented with signs and symbols which is y + y + y or 3y. The necessary would be what the student could discover for themselves, like 3y is the same length as b + d.  The student would then be provided with the convention of the equal sign.  Thus providing the arbitrary 3y = b + d.

Deciding what is arbitrary and what is not can be the most difficult for educators.  For we as educators are in the habit of making everything arbitrary.  Instead of allowing the child to discover why five plus four equals nine, we make it an arbitrary piece of information that must be memorized.  Of course, what are the consequences of making mathematics arbitrary and what are the benefits of making certain distinctions between teaching the arbitrary and allowing students to discover the necessary?   Even the more pressing question is what does it really look like to allow a student to find the necessary for themselves?

When trying to find more information to make this clear for myself, I came across an article written by Dave Hewitt where he makes the distinctions clearer.

“Learning should not only take us somewhere; it should allow us later to go further more easily.” -J.S. Bruner.  The Process of Education

Memorization serves a purpose and can’t be done away with completely in the realm of education, but certainly it is a difficult thing to hold everything in one’s mind.  This is especially true for children.  The arbitrary should be memorized for it helps everyone to be on the same page and eventually many things that are necessary do (because of frequency) become memorized.  But Hewitt discusses that it is not important to memorize the necessary if a student’s awareness is developed, and this is what a teacher must facilitate.

Hewitt further points out that memorization cannot provide a student with the ability to tackle a new situation where as awareness can help to make informed decisions and to know how to act based on their awareness. In Hewitt’s view, developing awareness is key to developing the mathematician within us.

This of course I can agree with but how does one develop awareness?  For me, I know that strategic tasks given to a student can raise awareness and bring about discovery of the necessary.  I have seen this with my own children.

My personal struggle is I have an over eagerness for them to make the discovery I want them to make. My children like to make discoveries for a particular task that I have purposed to lead to a different discover.  It is easy for me to fall back to the old me and point them to discovery I would like them to make, but then I am not submitting myself to their timeline of discovery.  By not submitting, I am not validating their discover as important and I believe validating a child’s thinking is necessary to developing their growth mindset.

This brings me back to something Hewitt said.  Hewitt believes when you educate awareness you should realize that each student becomes aware about certain things at different times.  No one student will make the same discovery of the necessary at the same time.  This forces the teacher to be adaptable and to recognize the validity of each student’s discovery.

Hewitt gives an example of this using an exercise of shading for fraction equivalency.  Two students provide two different answers, and neither is wrong.  It is merely that each is aware of equivalency at different levels.  This demonstrates that tasks based activities can provide a vast array of different awareness and we need to be attentive to each student’s level of awareness.

Group activities and discussions help to raise awareness for everyone and I think mixed age groups is advantageous for such discussions.  As my children complete the same tasks, much discussion goes into the difference of their discoveries and the difference in their interpretation of the tasks.  It has shown me how impossible it is to grade a student on their mathematical understanding.

Mathematical awareness is such a deep and vast hole where location of individual knowledge inside the hole cannot really be nailed down.  Instead, measuring it is better based on how much you have filled your hole.  How aware are you? And the validity of one person’s awareness certainly can’t be diminished merely because it differs from another’s awareness.  No the difference should be in volume, and what test out there could manage such a task?

As I play with Cuisenaire blocks, I realize my hole of mathematical awareness was quite empty.  I mean I just discovered the difference between the squares of one number to the next consecutive number is always an odd number and with each consecutive square the difference goes up to the next consecutive odd number.

Hopefully that made sense, and some fancy mathematician could give me the arbitrary information that would allow me to express my idea in a way that is acceptable amongst the math community. Until then, at least I know that my awareness has grown.

As I practice providing only the arbitrary to my children in math, the task has made me more self aware of what is arbitrary and necessary in other subjects like science, writing and even Bible study. Deeply rich experiences provide the best window for discovering the necessary so in all we do, adding rich experiences is going to be paramount to our daily schooling.

That sounds like more field trips.  How about you? Comment below and let me know how the necessary and arbitrary are changing the way you think.

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