The Habit of Observation for Math
The habit of observation radically changed the way we did math. One day in my struggle to teach math, Sonya at Arithmophobia No More finally made things plain to me.
She explained Gattegno's instructions to observe and notice the relationships in math is very much like Charlotte Mason's habit of observation training.
Of course, this makes sense. Math is a study of observing quantitative relationships.
How did I miss this important application of habit training?
Well, it could be that Charlotte Mason didn’t really make such a connection. Yet, mathematics is a great place to develop this habit.
Let’s look at how Charlotte Mason trains the habit of observation.
Charlotte Mason gave educators a guide for developing the habit of observation in the context of nature. This habit of observation prepares the student to explore, manipulate and extract useful information from their environment.
The Habit of Observation in Nature
Charlotte Mason describes a mother in a field telling her children the names of things found in their surroundings. After much time is given to naming things and naming things in relation to other things, the mother then sends her children out alone to go observe for themselves.
The mother requires the children to return to her to tell her what they saw. She presses them to return to the place of observation to look a little harder and come back with more details.
There is an emphasis on observing relationships.
"Tell me the relationship of the valley to the cottage. Tell me the relationship of the trees to the open fields. Where is the snail found? Near a tree, a bush or up high in the tree?"
In connecting children to their environment, we always begin with describing a state of being.
"Here is an apple. This is your toe. That is a tree. This is a leaf. There is a snail."
Then we name things in relation to other things.
"The snail is smaller than the tree. The tree is left of the cottage. The bat comes out in the evening. The flowers bloom in the spring."
The Habit of Observation in Mathematics
When we introduce quantity, we stick to describing it as relationships between objects.
Using Cuisenaire rods, Gattegno begins the introduction of mathematics with naming the objects.
First, the student is asked to build long trains of rods and describe them.
"Red, white, red, white, red."
Then the student is asked to build a train the same length with different rods.
"Red, yellow, white."
Now a relationship is born, and the teacher describes it.
"Red, white, red, white, red is the same length as red, yellow, white." The trains have a relationship of equality.
Through varying presentations of the rods numerous relationships are described to the child and the child begins to perceive and look for similar relationships. (See the following guides on addition, subtraction, and fractions to see presentations.)
In nature, the child perceives the connections of the trees to the birds to the insects to the vegetable garden to the rodents to the foxes to the caves. Many relationships exist at one time and open a door to curious wonder.
Much like nature, many mathematical relationships exist at one time. It is the educator's job to open the student's perception of these relationships. This begins by describing a plethora of relationships that one quantity has to all surrounding quantities.
This is the feast of mathematics that we must lay before the child to create in them a delight and wonder for mathematics.
The Feast of Mathematics
But what does this feast look like?
Examine the exhaustive relationships observed in the decomposition of a number.
A Pure Mathematical Environment for the Habit of Observation
This is just one feast, but how does such a feast unfold? The observation of relationships in mathematics must be explored in context just like nature must be explored in context.
Now these mathematical relationships exist naturally all around us, but the noise of all sorts of other curious things often crowd out the opportunities to describe math.
The noise also tends to mislead the child, so it is best to create an environment to pursue math solely.
Gattegno perceived Cuisenaire rods to be an excellent tool for creating a pure mathematical environment. Cuisenaire rods allow the student to perceive relationships between quantities and to easily generalize the rules for establishing these relationships.
The Habit of Generalizations in Observation
Wait? Can kids generalize? Yes, they can. In fact, they generalize as early as toddlers and generalize very well. Gattegno begins with general ideas that move students to specific ideas a similar principle found in Charlotte Mason's philosophy.
Take for example the rule of not touching plug outlets, after a few (or many times) of telling the toddler not to touch this outlet or that, they begin to generalize that touching any plug outlet is not acceptable. You soon find yourself never mentioning the rule again.
The ability to generalize in mathematics allows students to extract information easily and apply information more fluidly in different context.
When I taught math using traditional methods, I found myself frustrated. My children could not see the connection between one concept to the next. There was too much noise. The presentations were also too disconnected for them to generalize the information and apply it to a new context. Nor could they apply new information to old information.
To lay the groundwork for generalization, the student must see at the simplest level how all the operators (+, -, /, squares, etc.) are interconnected. They must also observe key properties that are always true like the commutative property, the additive property, the distributive property, associative property, etc. Then there is the connection between the properties and the operators.
What a change there is after this groundwork is laid. It's a change that Gattegno, Goutard, Powell, and others observed, and the results are always astounding.
It's placing symbol fluency before number fluency. Symbol fluency is the student’s ability to move from one kind of relationship to the next and to know how to use symbols to articulate a specific relationship.
Moving from Symbol Fluency to Number Fluency
After describing nature and observing the relationships that exist in nature, the natural progression is to manipulate and replicate nature. For most, this is gardening, and for others, like George Washington Carver, it is laboratory work.
in the same way, students go from observing and describing mathematical relationships to operating on numbers to manipulate them for a desired result.
Therefore, we call the symbols (+, -, /, x, etc.) operators. They don't merely describe a relationship. They can also tell us how to operate on a quantity.
The Habit of Observation Leads to Individual Wonder
Just like gardening is a combination of observing and manipulating, so is mathematics. One never foregoes observing relationships even while operating on quantities.
As the student begins to operate on numbers, the student must continue to observe how operations impact expressions. We call this noticing. But noticing is not the end result. The habit of observation is meant to cultivate wonder in the student to develop their own explorations.
The Habit of Observations Respects the Individual
Such opportunities to wonder about observations allows the student to develop individual pursuits in mathematics. It respects the individual as a thinking human being who has interesting observations and questions worth pursuing.
Traditional mathematics doesn't make room for the student to observe, wonder and manipulate quantities. Traditional math doesn't recognize the ability of the student to generalize and thus, it overrides the students' thinking and demands the student to memorize in hopes that one day the student will connect the dots.
The reality is that most students never connect the dots in the context of disconnected ideas. They give up thinking they are no good at math.
Then there is the overemphasis on speed. This causes slow thinkers to assume mathematics is not in their future. This couldn't be further from the truth.
Many great mathematicians learned that it was their slow thinking that made them just as good if not better at mathematics. It is slow thinkers that take the time to observe and notice the curiosity of numbers. It leads them to wonder and take rabbit trails that lead to great thinking.
Traditional mathematics is also treated as an individual activity which leads me to the next point.
The Habit of Observation Refined in the Context of Community
A sole gardener is not able to observe everything let alone understand every plant. Neither are students able to do the same in the context of math. So, the habit of observation can be more deeply cultivated in a community of observers.
True mathematical exploration and development of understanding is found in community. As mathematicians share observations, understanding and talk through problems, great achievements are made.
Today, the classrooms see the value in group work to cultivate cooperation, refine observations, improve communication skills and develop patience in hearing a person's idea out. These skills are much needed in a world of general discord.
In the homeschool environment, there is this opportunity too. Yes, even in the homes of one child there is this opportunity. In the mathematical world, ages and experiences vary greatly and both the young and the old see the benefit of the other.
So, parents get in there and observe and wonder over mathematics with your child. Give your child space to share their ideas, refine their observations and work with you to solve problems.
Final Thoughts on the Habit of Observation
The habit of observation takes time. It is one that begins by offering a mathematical feast to a child by describing and naming the different parts of the feast.
No matter what curriculum you use, you can always pause and ask your student to just tell you what they see.
The first time you do this your child will see nothing. It is not that the child sees nothing. Often it is because the teaching methodology constrains the child to wonder what it is they are supposed to see. They sit there anxiously wondering what it is that they need to say to make you happy.
It should sadden us to see our children in this state. It is a revelation that we have not treated the child as a thinking human being with valid observations and valid questions. It is a revelation that the child believes that the only valid thought or observation is one that is given to them by the teacher.
The child believes that they just need to remember what is important to the teacher. Only there is so much to remember that the pressure is overwhelming to the child, so they freeze at such a question.
"What do you notice?"
Overcoming the Freeze to Cultivate the Habit of Observation
My suggestion is that you have the student create art out of Cuisenaire rods and then ask the question. It provides a non-threatening environment and the mathematical nature of the rods will eventually open the door for observing mathematics.
It also helps to develop your own habit of observation in the context of mathematics. So, observe your own art and take the time to observe the mathematical feast before you. How many relationships can you notice? Do you see a curious pattern? What do you wonder about this pattern?
Gattegno's textbooks carry this tone of notice and wonder throughout. He is not explicit in all that one should notice, and it is why Sonya and I worked hard to create Module 1 Handbook with activities to help you really cultivate the habit of observation using Cuisenaire Rods.
I have also created math journals for all my products to cultivate the habit of observation in various contexts. You can check out the free products below.
You can go check out Gattegno’s Textbooks for free.
If you want more great activities and a guide through the textbook, be sure to check out Module 1 with over 100 worksheets, activity mats and task cards to start developing the habit of observation.