Knobless Cylinders and the mathematical mind

Bee Pape is the expert here, and a search for her documents is well worth the time. She has extensive extensions with the knobless cylinders. She's been teaching for a long time in a Montessori school, done research with early number development. She also knows the work of Piaget and Constance Kamii. Many of my own thoughts are generated from a workshop of hers that I attended combined with my own research. I found her delightful, because I've read and understood so much of the same material.

Jean Piaget was 26 years younger than Maria Montessori. His developmental theories were also based on the observation of children and many interesting and experiments. These included conversations with the children about what they were thinking. He was familiar with the ideas of Maria Montessori and was at one point president of the Swiss Montessori society. One of his fundamental beliefs is that children construct their own knowledge through interaction with the world. In many ways, this is similar to Maria Montessori (and many current educational theories) that young children learn best through concrete materials and actual experiences.

Jean Piaget observed an interesting phenomenon called "conservation." There is "Conservation of liquid," "conservation of number" and several more. These are a developmental stage that children go through where they can only hold one attribute of a physical material in their mind at a time. For example, in "conservation of liquid" once a child agrees that the liquid in two equal shaped glasses is equal, if one glass is poured into either a wide/short glass or a taller/thinner glass the amount of liquid is no longer the same. The same thing is true with "conservation of number." If a child agrees that two rows of similar materials (poker chips) have exactly the same amount, if you stretch out one line – then they are no longer the same. There are many reasons for this. Sometimes the child will say that the longer line has more, sometimes the shorter, sometimes they will provide an answer that is difficult to explain. The result is consistent and has been shown many times. Young children do not recognize that quantity is consistent regardless of spatial arrangement.

This developmental understanding of number only comes with actual experience (in many forms) with things to manipulate. Experience with objects to count, adding or subtracting one, matching, all promote learning about equality, greater than and less than. What do you do if there is one more doll than you have a bed for?

If you look at current educational research and activities for children, you see seriation, ordering, pattering, matching, and measurement. Maria Montessori was brilliant. How many people remember their training that a child should truly spend time with and master many of the sensorial materials before being introduced to mathematics? In my training, the knobless cylinders were one of the last materials that were introduced to a child because there is no control of error except visual. The knobless cylinders are a set of 40 with 4 subsets in different colors. You have a few pieces that are actually identical. A child can sort them by color. A child can sort them by size. A child can sort them by height. If one is missing, because they are sitting on it – they have a problem to solve and a interesting learning lesson about quantity. They can explore relationships. You can even set up logic puzzles that are designed for the youngest child and visually based. Find me a piece that is this height and this color (or not this color…)

100 Words

There is a Montessori group that is having a discussion recently. How would you describe Montessori in 100 words or less. I wrote two different versions from two perspectives in my own head. One is that of a director of a school and meant at least at some level either as recruitment or advertising. The other is more personal from my experiences as a teacher in some poor schools and my experiences as a parent with my children in high quality Montessori schools. Neither one is truly adequate. It turns out that 100 words does not offer much scope for the depth of what Montessori is and what I truly believe about it. It is about the individual in so many ways and allowing a child to explore their own interests. I have a child right now that is my own 12 year old's terms is "the most off-task." He's brilliant. I've evaluated him. He has conservation of number at age 4. Rare, and I've done the research! He turned a wooden cylinder which was meant as a sewing job into a yo-yo. He came up to me one day and said that he wanted to write a book. There weren't enough books… and he started dictating the entire story of the three little pigs- a story that we have not read in class. He also recently went up to the school office and started to touch everything to the point the director wanted to know who this child was! A teacher should know better, but I love this child. I fear for him. He will either find the teachers that will nurture him, or he will find the teachers that will destroy him. Is it really my place to say that he "must" study this now, when his brain is concrete and he's fascinated by how things move? No- that's for some "other" form of education. Montessori finds a way to accept where he is and still teach him everything that he needs to know to move forward.

#1

An individual has their own unique interests, talents, skills and obsessions. When learning comes from an individual's own interest it is joyful and will be deeper and better retained. Ideas and questions will be generated and facts related to current knowledge. Projects may be designed and undertaken with because of sheer captivation in learning with great enthusiasm. Montessori education is designed to spark a child's individual imagination and allow them to direct their own learning process within guidelines while developing specific skills. Children are nurtured for the wonderful individuals they are with unlimited potential.

#2

Montessori education aims to first teach the child independence. A child develops a sense of organization as a prerequisite to successful work. This forms the basis of further work with language, mathematics and the development of the senses. Each child is able to work at their own individual pace exploring in depth what skills they are currently motivated to learn. For the young child this may be learning to use a spoon or scissors or writing about sharks! It is not a mandatory curriculum where everyone completes the same task at the same time.

Montessori something....

I think that my frustration level is building. I don't want this current job if the temporary position is offered as a permanent position. You can't truly teach with Montessori philosophy when you're forced into nonsense and theme of the week.

Stupid workbooks. One page per day. I have a child that just doesn't get them. We need to work on rhyming words. Other children that need to work on basic sounds. Most of them need to explore numbers. Stupid workbooks so they can make progress. Make consistent progress talking to each other, getting help from the teachers just to read the picture understand the sound in the first place. Some of these children can't even write there their own name! There sense of accomplishment is in finishing a worksheet.

Just a random rant and rave. There is only so much time in the classroom and in life. Especially with the commute. I know what most of the schools near me are like though. I already decided that my real priority is to open my own school. I'm just not sure if I want to NOT accept any position, not remodel the kitchen (possibly indefinitely) in order to have one year to fully develop a functional business plan and location options. CA is expensive and you need to cover the cost of the real estate.

My first step is a complete mission statement. The second would be an employee manual that would help to match it. I also need the financial breakdown and break even points. Those are the steps that I can do at home.

Thinking about the Hundred Board

“I want to make this clear what ever instruction is necessary must be given. We are guided by the children’s development. So when we say ‘follow the child’ I do not suppose the child to be a responsible being able to tell us what they should do, what I mean is; the child possesses characteristics which serve as our guide in giving instruction according to their developmental stage.”
- Maria Montessori, AMS Triangle Magazine 1946/or 1947 (located via internet)

What happened to follow the child in the political correctness of giving the lesson the absolutely right way? What happened to scaffolding and the concept of the zone of proximal development? The idea that there is a place between what you know and what you don't know that is ideal for learning, particularly if it is structured to provide just the ideal amount of support. This makes sense - nothing too easy and nothing so challenging as to be impossible or perpetually frustrating.

These concepts are inherent in Montessori philosophy. The materials are designed to move from one to the next in a progression of concepts, building confidence and support as needed. They aren't perfect though. Sometimes a child just isn't interested in the "right" material, sometimes it needs to be demonstrated in a different way. Sometimes... whatever.

The Hundred Board.
In my training, I learned a presentation method where each group of ten numbers was in it's own box and the child worked with ten numbers at a time. There was no control chart. They learned how to carefully put it away for the next child.

At the next school that I worked with the child had the option of working on a hundred board that was preprinted as a matching or a blank board. They also had the option of completing a paper extension. (In that school it seemed as if the paper extension was the true goal of the children!)

I've heard and seen of schools where the hundred board had all 100 tiles in a box and randomly spread on the rug. I've also seen where the first 40 tiles where in one box and the remaining 60 in a second box. (I don't know the logic of that division...)

Yet children approach the hundred board from all different levels and can learn from it. The other day I watched two girls absolutely euphoric about going to the calendar realizing they needed "a one and a three" and finding "31" for "13" and continuing all through the teens. When they were done, I had them actually write the numbers that they might have seen if they had a control chart and they were still excited. (They were doing this job because they "had a lesson, and were volunteering to clean it up.)

Some children need to match the numbers and can find the hundred board and opportunity for learning numbers and the patterns inherent in building the hundred board. Some children can work independently from a control chart. Many children when they first begin the hundred board will have difficulty with a hundred tiles spread on the rug and will need to find a way to organize their work. "Scaffolding" may include sorting the pieces by the tens, but I've never heard of a formal training presentations that include that. I do, the children need it. It's no different than organizing the pieces of the trinomial cube.

Order is fundamental in many math aspects.

New Curriculum

Montessori education is designed around the materials and the ability to follow the child. This means providing the child with the right materials at the time for what they are interested in.

Yet, the environment is not static. Changes are made to reflect the season, to try to regenerate interest in a material that is not being used. Changes are often made to reflect current holidays or teaching themes.

The more connections that you can make between current knowledge and new knowledge there will be a higher possibility of longer term retention. (It was only recently that I actually learned the dimensions of the knobless cylinders, but I was willing to guess they were in metric units!)

If we ARE teaching to current holidays, perhaps it is the ideal time to incorporate the geographic history and related significant figures over time. If we ARE teaching about the rain forest, perhaps we should also teach about the people in the rain forest- not just the animals. If we ARE teaching about dinosaurs, perhaps we should look up significant locations of fossil remains and teach about geography.

What would you change about the curriculum if you could start from scratch?

Wild and untamed!

I picked this video up on one of the Montessori groups.

"My heart is still racing thinking of what this boy has accomplished. What was he thinking? THAT he was thinking is the key. Push the boundaries, turn over new ground - make, create, innovate. I'll bet this kid wasn't put on Ritalin.
- Mark"

I too am amazed at where he is right now and the concentration that he showed. He broke rules. He probably wasn't always easy to manage. We've all had the children in our classes that when very young march to their own drum - rather insistently. They can usually steal our hearts. For me this is a reminder of what they may accomplish!

Online and thinking again.

I'm online again!
Comcast has now replaced everything from the pole on the street to the junction at my computer and we replaced the cable modem as well.

I need to fix the picture, and I won an award! Thanks.
In the meantime, I've started a temporary job...
I want to make a new section on the blog - one that is a resource for known albums. It is a useful reference, even though I will not vouch for the quality.

Once again I'm thinking about a more coherent curriculum for preschool children rather than the entire theme based approach that many schools use and I want to investigate a new book on teaching science to preschoolers. It's based on research by Rutgers. Science is one of the areas that I do not feel Maria Montessori truly provided enough for children. They can do more, and many schools actually demand more. It's all about taking the philosophy and the right approach and the correct implementation.