Trinomial Cube - LONG Life

I would like to introduce you to a long-time companion of mine - the Trinomial Cube:

The "big" cube in primary!

The Binomial Cube
The small guy in primary

He has a few buddies: 

The Algebraic Cube - Elementary

Power of Two Cube - Elementary

Binomial and Trinomial are beautiful puzzles in the primary environment. But the Trinomial - he goes places! While the binomial is also useful in the elementary class, the trinomial does most of the work. 

(a + b + c)^3 

The children are learning this sequence, sensorially at age 3 1/2, and it builds and grows from there. 

The children continue to use the material sensorially in elementary, then they start doing some squaring work with the lid (the lid has (a + b + c)^2). 

And one day, they hear of the Story of the Three Kings. Oh, no,  not THOSE 3 Kings - the OTHER Three Kings. These Kings: A-Cubed, B-Cubed, and C-Cubed. They were marching along happily in procession - and... there was a REVOLT! That revolt overturned the order of the procession! It was chaos! It was order! And the children learn algebra! And cubing!

And the Trinomial Cube has a new lease on life! 

And the children work with the trinomial cube; then expand into the algebraic cube; finally they work with the power of two cube - it all began with the trinomial cube and his little brother the binomial cube.

But the children have moved on. And the trinomial cube appears done. A wash-out. Used up. It's all over for him. 

----------------------------------------------------------------------------------

I recently read through an older NAMTA journal, on the topic of teaching geometry in adolescence, using Euclid's Book of the Elements (all 13 volumes are currently available in a set of 3 books).  

Guess who makes an appearance! The trinomial cube is truly an ever-present material for the children in Montessori environments. And a credence to the AMI stance that we don't need "more" materials - we need to go DEEP with the materials!
ETA: The adolescent Algebra Album available through NAMTA - lists it as one of the necessary materials ;) NICE! 

Just looking at it, consider for yourself: how deep does this material go for you? What uses could you see for your children, even if you don't have all the album pages for all the levels? What could your children experience with just this one piece of Montessori equipment? 

Montessori Scope and Sequence: Elementary

The following links take you to a general scope and sequence for AMI elementary albums.

These Montessori scope and sequences -

• follow ages 6-12 
• do not include remedial (bridge) work
• are not set in stone - follow the children before you!
• are general (they do not include every single exercise with an album page that might start in year 1 and extend to year 4 or 5) - so when you see a range of 1-3, figure that the album page probably starts in year 1 (or 2 at the latest) and should generally be done by year 3. It could be earlier, but it shouldn't be much later, because the age ranges are pretty generous already. If you want the detailed version that contains every exercise and follow-up, in both Word and Excel formats (easily adapted for personal use), you will find them at Garden of Francis (for elementary) and Garden of Francis (for primary)

The years you see noted in "start" and "end" refer to the year in Montessori elementary - year 1 is the equivalent of age 6, through year 6 which is the equivalent of age 11-12. 

The last math work may or may not be completed in elementary and may extend into early adolescence. The other albums are typically easily completed in a 6 year elementary cycle. 

• Mathematics Elementary Scope and Sequence
• Geometry Elementary Scope and Sequence
• Geography Elementary Scope and Sequence
• History Elementary Scope and Sequence
• Biology Elementary Scope and Sequence
• Language Elementary Scope and Sequence

Looking for detailed scope and sequence that includes every single exercise of the album page and more specific guidelines and ages? Check with your album provider. Or join us at Keys of the Universe. 

Or looking for primary Montessori album scope and sequences? Those are trickier. Presentations that are introduced at age 3, might have extensions that go through age 4, 5, and 6 - so you'll want a very detailed one that includes all exercises for each album page. Ask your album provider for a scope and sequence to correspond with that album set.

Or some general ages are included here in various Montessori Nuggets.

Adolescence: Adult Training

Barring being able to attend NAMTA/AMI's Orientation to Montessori Adolescent Studies, the primary foundation that the adult needs for working with the adolescent age:

• elementary training/experience of some sort because of the continuation of studies which adolescence builds upon
• primary training/experience of some sort because of the parallels of the first and third planes of development

At bare minimum:

• fully fleshed out theory albums at both levels
• cosmic education access at elementary
• read Childhood to Adolescence
• read all required primary and elementary reading selections
• observation at both earlier levels in a full Montessori environment
• Reflection on one's own adolescent experience and what could have been improved; what good should be maintained

In addition: 

• NAMTA's adolescent reading list - found at the bottom of Montessori Nuggets Reading List Page

Cosmic Education in the Third Plane - Adolescence

Margaret Stephenson:

For the third plane, the exploration is even wider, encompassing the farm and the community of the rural area.

It echoes what the children explored at the second plane: civilization and how it came about.

But now the exploration takes place in reality because the adolescents are actually doing it.

Cooperation with the land, cooperation in commerce, and cooperation in the cultural life of the rural society touch materially the things studied in the second plane and afford the adolescent the opportunity to see his or her place in society. 

Cosmic Education at all Planes: An Aid to Life

Margaret Stephenson as quoted in NAMTA Journal 23:3 Summer 1997:

Underlying the Montessori idea of "an aid to life" is her vision of the child as the cosmic agent of humanity and its survival. The theme that runs through the four planes is putting the child in touch with the cosmic task of the world and its inhabitants in a manner that is fitted to the psychology of each discrete plane. If a child does not get the connection in one plane, he or she cannot move on to the next plane fully prepared. 

(end quote)

The child does move on to the next plane, just not fully prepared.

What is given at each plane:

1. This connection is made subconsciously yet concretely through the materials. 
2. This connection is made more consciously, through approaching Cosmic Education as a drama, not as a lifeless curriculum. This drama is the vision of Montessori. If this drama inspires the child of the second plane...,
3. then when that child reaches the third plane, he or she will be able to ask, "What is my role in this drama?"

Ms. Stephenson again:

Going Out is also important to give the Elementary child a complete vision of the universe, and a particularly valuable aspect of Going Out is community service. This is the basis for the service we've been talking about for the adolescents; it needs to begin earlier than adolescence (in order to reach its fullest fruits). 

Album Pages are Incomplete - Part 2

No album page or lesson plan or curriculum can plan for every interest! 

ETA: No album page SHOULD be complete, because of this very reason given just above. Album pages are designed to introduce a concept, introduce a material, pique an interest -- guiding the adult in doing each of these things. Then the adult and child together, more and more the child as the years go on, come up with "what next". Will it be some of the suggested follow-ups on the album page? Another idea? What local resources do you have? What educational resources are on hand or can be made accessible. Only YOU the local person can determine YOUR local situation. 

Any album that suggests that it will provide everything for your child's complete education, implying that your child will not need ANY other resource besides these albums and materials presented to him by you... is NOT a Montessori album. 

Montessori albums are guides. They are your structure and your foundation for a life-long learning, a cosmic education. 

If an alleged Montessori album provides every last detail and instruction, then  it becomes your slave-driver; rather than following the child and the unique situation of life in front of you, you are obeying a merciless master who will make you feel inadequate when things are not done at the proper time in the proper order - and your children will not be learning how to learn, enjoying learning, establishing cosmic education within their own souls. 

Now, that does not mean we introduce unnecessary novelties; it does not mean that we do not go as deep as possible with the materials at hand before bringing in other resources. 

It simply means that we present the album presentation, pique the child's interests, cull the depths of the material at hand with resources at hand, then work with the child to see where to go next. Encourage the child to create the materials he needs whenever possible, when it is an interest of his. 

As the children get older, they can be involved in assisting to procure even the core materials - creating, helping with the ordering process, unpacking, cleaning up, repairing - whatever is needed. This sort of activity goes beyond the album page without introducing unnecessaries. 

Follow-Ups for every album page: 

• Child-planned Goings Out
• Invite a visitor on a related topic
• A timeline or chart or drawing or diagram
• Original artwork
• Write a story about this topic
• Teach this topic to another child
• Repeat however long necessary and desired to learn the concept at hand

What other follow-ups can you add to this list that would be present for ALL album pages? 

Normalization and Following the Child's Interests

Justification for "follow the child" - how and when to follow the child.

We can guide the child who is not yet normalized by controlling the environment and our interactions.

We can enhance the normalized child's experience by continuing to control the environment and our interactions.



From NAMTA Journal 23:3 Summer 1997

Maria Montessori did not base her goals on the expectations of parents, of society, or of the government. She looked at the children without compromise and strove for the best realization of their potential. 

To observe children successfully, you need a criterion for what is progress and what is not. Montessori found this criterion in the phenomenon of normalization, the primary mark of which is polarized, concentrated work. 

At the beginning, we make a lot of decisions for the children, but after they are normalized, their likes and dislikes can be trusted to reflect their developmental needs. 

(emphasis mine)

Cosmic Education: When to Present the Great Lessons

Margaret Stephenson:

Too often I ask teachers, "Have you done the Great Lessons?" and they respond, "No, not yet, because they can't read and write." I say, "Have you ever thought that maybe they only need ears to listen with?" How can we open for the child these doors into the cosmos if we don't give the Great Lessons until the year is half over?

(end quote)

 Main points:

• children do NOT need to read and write - they only need to have ears (or ability to read sign language) and an imagination
• Margaret Stephenson: "The cosmic plan can not be fully explored at the second plane. It is too vast. That is why Montessori tells us to sow seeds. And after these seeds are sown, there has to be a period of silence, darkness, and rest, during which the roots grow deeper, before any shoots appear above the ground."
• There will be some results from the Great Lessons - subject areas open up, interests are enticed, the children receive a REASON for reading and writing --- but the most obvious signs will be seen in adolescence when they start asking of themselves their own place in this Cosmic Plan. 

So tell those stories early in the year! 

     The 5 lessons should be told by the 8th week of the school year at the latest; preferably 5th or 6th. 

Tell them EVERY year! 

    Throughout elementary, they will hear something new; invite them to pick a new point and explore it. 

Do NOT give the children everything! Many of the original demonstrations from the first Great Lesson have been separated out from it into separate album pages for follow-ups; so the children have more opportunities for follow-up work, inviting them to delve deeper into the story with each new demonstration they do. 

Entice their imaginations and enthusiasm for exploration - again don't give them *everything* - but do give them enough to get going on their own. 

Have adolescents new to Montessori? Tell them the stories too! Have them help create some of the charts after they've heard the story without it; invite them to help prepare materials and demonstrations for the youngest children; and work on helping them answer their own questions of "where do I fit in?"

Keys at Every Plane of Development

Consider how the following words apply to EVERY plane of development. 

John McNamara as quoted in NAMTA Journal 23:3 Summer 1997:

Like I did when I taught at the Elementary level, I am still trying to help the students' passage to abstraction, to stimulate their interest, to capture their imagination, to meet their need for group work, to respect their growing moral awareness, and to give precise keys so they can proceed to make discoveries on their own. 

He refers to the adolescent age, but how does this apply to every age?

Social Studies in Montessori

Where is "social studies" in Montessori? 

Montessori sure does shake things up! History is obvious, but let's discuss the other albums first. 

Our geography at primary is what we expect (maps, countries, capitals, flags), along with a great deal of cultural experiences. 

Our geography at elementary is what most schools consider "earth science" - physics, geology, chemistry and the like - including climactic zones (and which people and animals live where). AND it then adds in Economic Geography - how people connect with one another, most particularly in regards to the earth itself: source of our food, economic interactions and taxes. Within economic geography is where we get into community helpers that many schools make their entire 1st grade curriculum. 

Montessori, utilizing the precepts of cosmic education, emphasizes the link between the land (geography) and the people - who lives where, how they live and interact with one another and their environmental surroundings and the changes that take place... starting to sound like ecosystems in Biology, huh? ;) 

Within the language album, we look at the history and development of languages, literature, and modes of communication. We look at the biographies of authors, their times and lives, to provide us deeper insight into their choice topics, their writing styles, their beliefs - their inner souls. 

Within Mathematics and Geometry, we'll have the usual mathematical skills, but also have the Story of Numbers and various Geometry stories. These stories place mathematics and geometry in the context of history and within daily life - why are these subjects necessary and useful? How can we utilize these skills to be service to *people.*

Art and music are relatively obvious - these areas are not just about skills, but appreciation for the work of others. We study the history of artists and musicians in the same way we study authors - in the same way we study all of history: as a gift. 

History is the most obvious as far as social studies. Montessori provides a framework within which to study history, not from any one resource (i.e. a textbook), but from a variety of sources utilizing one's own developing reasoning skills to sort bias from fact and create one's own view of history, within the framework of gratitude for the gifts given us by those who came before us and responsibility to those who are with us now and those who will come after us - how shall we hold dear and develop the gifts we have been given? How shall we then pass on these gifts that are a living heritage from ancient times to the present. 

So you can see that social studies - the study of society, its people, its functionings and even the study of ourselves, truly is cosmic education, utilizing every aspect of a child's life, even outside these academic albums. 

Album Pages are Incomplete

   At all ages provide appropriate activities according to the ideas presented in the environment as the children’s interests are piqued. 

Plant those seeds of interest and they will sprout when properly fertilized. 

As much as possible follow the children’s interests in developing follow-up activities (without becoming "cheap" - provide high quality experiences - and invite the children to utilize basic supplies to create their own follow-ups). 

No album page or lesson plan or curriculum can plan for every interest! 


(and no album page or lesson plan or curriculum SHOULD!)

Spiritual Preparation of the Adult: Ownership of the Environment

This Montessori Nugget applies to all environments - not just Montessori ones.

Who owns the environment?

Adults? Children? Parents? Teachers? Do any of us? Do any of us truly own the environment?

The adult prepares the environment, but does so while considering the needs and characteristics of the children who will be present. The adult also looks at the bigger situation: is this is a school environment, a part-time co-op, a homeschool, an after-school, an impoverished area, an affluent area. All of these considerations and so many more dictate that the adult is not going to come in and set something up in a vacuum.

The children utilize the environment, take into themselves and construct themselves through their experiences.

This whole concept is a tough one for us adults, and there will be many more Montessori Nuggets on the subject.

For our Montessori environments:

Trust the materials described in the albums - go deep with them. There is a LOT that can be culled from just the basic materials. Then, if a child has found a true deep interest that even the depths of the material can't provide, then look at your situation, clarify the needs and fulfill them with appropriate further basic material. 


Anything beyond basic: let the child create - it's his interest, let him own it. 


You the adult provide the keys; you provide the environment; you provide a structure within which the child knows and can fulfill the requirements. Fully formed people at every plane are able to take responsibility for themselves through having access to the right tools. Provide those tools and disengage your own emotions, except to be proud of the wonderful creation in front of you :) 

History Timelines

When it comes to extra timelines needed to fulfill interests OR other educational requirements:

          Equal weight has to be given to allow the children to explore freely whatever they wish to explore and to develop the sense of responsibility to society.  Cosmic education was first fully developed in 1944 by Dr. Montessori and her son Mario while they were living in India.  When they arrived back in Europe in 1945, they began to present this idea in training courses.  In one lecture, Mario said, “Regarding the memorization of historical elements, a sort of game has been devised.  The teacher has to first give the lesson, then by way of testing the memory, cards and charts are prepared.”

Geography Puzzle Maps in Elementary

Especially for homeschoolers, there are certain materials that always provoke questions: are they useful or not and at which ages?

The puzzle maps are one such material. Large puzzles of the world and each continent, with each continent or country its own pieces (and knobs placed at the capitals as much as possible).

The official AMI statement is that elementary children are too old for this material, even if they are all new to Montessori. Experiences by so many parents and teachers show that they most certainly CAN be useful, although they are likely to be used in different ways than primary.

In any area, the main points to consider when choosing a material that is "out of age range" are the following:

• is this material FEEDING a child's needs?
• is it useful?
• how long will it be useful for?
• Do you have money and space for it? Or do you need an alternative?
• Is the child just going to hide behind this work? (it is ok for a 4 year old to spend 3 hours with the Africa map, just reconstructing it; but it is not ok for a typically developing 8 year old to do so - the 8 year old should be labeling the countries, listing the capitals, creating a booklet or chart of information, creating a new control map, etc - anything that is an extension and using his imagination to come up with new extensions). 
• is there a more appropriate material for the proper age range? if so, hone in on that material instead. 

For the Geography Puzzle Maps: 

• Elementary children should not be working them "just as puzzles" if they work with them. 
• The outlines are not the most accurate. They are fine for the primary age as a sensorial impression. This is a good conversation starter at elementary and gets the children analyzing. 
• The ideal piece of material are pin maps, but if you have puzzle maps it makes sense to use them for as long as possible. The elementary children can ultimately create pin-maps themselves out of cardboard and pins - or skip the pins and just create their own maps. 

Think through each material carefully and make the decision that is right for YOUR situation. 

Environment Not Working

Always keep in mind the psychological characteristics of the children and teach to them. 


Anytime there are bored or frustrated children in a Montessori environment, the adult must ask, 

“Where are the needs of their psychological characteristics not being met? What should be done differently within the environment to change their attitudes?” 

It is the adult’s role to find those solutions.

Living History

From AMI History album: 

          There is a story that comes from Leicester in England, the name of which is derived from the Latin castra, meaning “camp”.  There is a museum in this area with lots of things that have been dug up from the Roman times onward.  There was a little account in the newspaper told by the curator of the museum:


       Every day before he goes to work, he goes down to the basement and picks up a brick and holds it.  That brick takes him back to the time of the Romans being in the city of Leicester because in the brick is the little print of a child’s foot and lines that must have been drawn with a stick, a child of about two years old.  He was probably amongst workmen making bricks and came upon the brick before it was dried.  He probably took a stick, as tiny children do, and began to draw lines on it.  Some adult must have come along and shooed him off, and he went running, leaving his foot imprint on the brick.

         It is stories like this that make history come alive; this little boy playing at the time of the Romans.  Now that brick is there in the museum, a display-piece of the lives and times of these people.  

History needs to be made visible.  

It is not a record to be memorized.  

It is a life to be lived and to be studied.

Adolescence - Third Plane of Development

There are many questions about how to apply the Montessori method to the third plane of development.

There are no solid answers, since every child is unique as is every situation.

Some universal and general principles to consider:

• the Montessori approach is about respect for life, meeting the needs and tendencies as differently expressed at every stage of life, and a balance of freedom and responsibility
• at every age, we want connection with real life, a building upon the past plane and a foundation for the next - while fulfilling needs and tendencies NOW, at THIS age. 
• preparation for adolescence (in the primary and elementary years) will include fulfilling both the first plane and second plane of development as much as possible; providing skills and intellectual knowledge at the times appropriate for maximize the child's potentials. Art skills, language skills, history studies, cosmic education. 
• The adolescent is looking for his place in the grand scheme of things; has ideas that are grand; yet has crazy things going on with bodily development, so less mental capacity for continued learning at the same rate as elementary. 

Please contribute your ideas! 

If you have children ages 10 and up right NOW - we have an Adolescent Montessori discussion group starting at Keys of the Universe on June 21 - it is not too late to join. 

Later, we will add a discussion group for the general public. See this Montessori Trail post for more information. 

Spiritual Preparation of the Adult: Who Owns the Learning?

Who owns the learning that takes place in the environment?

We want the children to be responsible, to want to be responsible, for something which we put our heart into for their sakes - the environment and the learning that takes place within it.

It can be so hard to avoid turning into a dictator, even a loving, well-meaning one. When we see our children delve into an interest, most of us want to immediately own that interest, direct that interest, provide requirements for that interest. "Oh, my son, I see your passion for Ancient Egypt! I have this book I want you to read through, it has all sorts of Egyptian activities we can do together! We'll do these particular ones here and we'll have so much fun!"

That can turn out great - but it also takes away ownership of the interest from the child. What if the child really just wanted to study a particular area of the culture and create a timeline; but just wasn't yet interested in the art projects and activities? What if he would have gotten to it later in his journey of exploration and been thoroughly fascinated then?

There are times we provide requirements, and give new presentations regardless of interest (this is how a child discovers new interests), but if the passion is already there, we can strew materials, drop hints, provide ideas, and even minimal requirements, but the child should truly own this passion.

It is a tough balance. And there will be more Montessori Nuggets on this subject.

In the example above, we can make a discovery of the said book, "Look what I have found! I thought this might be of interest to you. I'd love to do some of the activities in it together if you'd like. Just let me know!" And let it go.

If it comes down an education requirement, then it should be discussed ahead of time as part of a work-plan, or just done as a group separate from the child's actual exploration and research.


It is the child's passion - let him own it. 

Montessori and the Special Child

Montessori and the Special Child by R.C. Orems is the first comprehensive guide to utilizing the Montessori Method with special needs children, written for the use of parents and anyone interested in this field.

Recall that Dr. Montessori started with special needs children; then worked with typically functioning children living in the inner-city slums.

Interesting that so many still think of Montessori as an upper-middle-class private school phenomenon.

The link is to a downloadable pdf file of the book as it is available on US Archive's website.

Art Presentations in Elementary (and all ages)

Provide art experiences in the basic skills at primary and early elementary. Nothing fancy, but use *real* materials: real watercolors with good brushes to make small designs on small watercolor paper; rubbings; outlining, etc. (in primary, some albums have art in the Exercises of Practical Life and some have it in "Culture").

"The best for the youngest." Give the children the best materials possible and show them how to utilize and care for the materials during this time of formation.

By providing basic skills and assuring they are well learned, with good materials, they can then apply those skills to their later studies, interests and passions.

When studying an ancient civilization or another modern culture, or finding patterns in geometry or mathematics, or just illustrating a work in language, they will have the skills they need to reproduce anything they see.

There is no need to present every art form, if you have provided the very basics.

So we might not have a presentation on mosaics, or on pottery, but they can re-create these things because of the experiences they have had utilizing non-drying clay, cutting and gluing. They can learn from a crafter or a book or a video the additional skills they desire to reproduce an art form for which they might not yet have the skills.

If they have basic folding skills (folding napkins in primary, leading to folding paper - very, very basic), we won't have to present origami to them before they read Sadako and the Thousand Paper Cranes and teach themselves how to make cranes.

And they can create pyramids from paper folding when studying Egypt.

Basic plant skills or other basic art skills: they can re-create their version of the Hanging Gardens when learning about the 7 Wonders of the Ancient World.

They can study artists and art periods under the guise of history, creating timelines, and creating in the style of various artists.

They will extend their art skills in day to day life and studies, without art skills being a separate study.

And they will be able to express themselves when words are not enough - now or in the future, as they prepare for adolescence.


How many pyramid paper patterns can be designed?

Let the children discover this! You've given them the tools: how to fold paper, all the sensorial materials; you've given them the structure, now let them explore! :)