Place-Value as Soulcraft in Montessori
Place value is complex. It’s a form of compression that itself relies on iterated multiplication. The relative position of a number determines its order of magnitude. In Montessori math, the target of mastery is decimal place value.
The whole Montessori math curriculum is geared towards developing a cognitive, habitual, almost physical understanding that going past 9 means that something changes.
From 1 to 9 you are piling things. But add one more and you get something new, a new form, a new unity, and new “one”.
In the words of Montessori,
The key to the decimal system lies in the final play between nine and ten. This is the key that opens the door to the organization of the various groups of units into a highly interesting systematic framework. Indeed, exceeding nine by only one, there are no digits to represent the new group that is formed: one has to go back to the beginning again… 
Consider how you interact with Montessori’s Golden Beads:
- Pile single beads until you get to ten, then replace them with one ten bar. The ten bar is a new thing. It’s composed of ten individual beads, but they are wired together into a new unity, a line.
- Or, pile ten bars until you get to ten tens, and then swap them for a hundred square.
- Or, ten hundred squares get swapped for a thousand cube.
- Then the whole thing recurses: ten thousands cubes become a line, a ten thousand bar. Ten ten thousand bars become a hundred thousand square, and ten of those squares becomes a million cube.
This is true not just of the golden beads (pictured above) and associated work, but, in different ways mutatis mutandes, of every arithmetic Montessori material, including the classic sensorial materials (pink tower, red rods, etc.). They are organized around tens.
When children learn column addition and multiplication, they understand why it works, because they understand—because it’s been part of their physical environment and activity since they were 2—what place value is.
Why does any of this matter? It adds up.
Is your mind is made of routines that are alien incantations that you mysteriously “work”? Or is it understood, made up of independently cognized algorithms, which you can mull and interrelate, and in which you have earned confidence?
It matters for math—for when you hit a wall in your learning.
It matters for your relationship with the world, which is increasingly mathematical. Even if your life’s work isn’t particularly mathematical, being alienated from math means being alienated from progress. Montessori writes of the increasing importance math, in a passage that could have been written today:
Mathematics are necessary because intelligence today is no longer natural but mathematical, and without developing an education in mathematics it is impossible to understand or take any part in the special forms of progress characteristic of our times period a person without mathematical training today is like in illiterate in the times when everything depended on literary culture. But even in the natural state human mind has a mathematical bent, tending to be exact, to take measurements and make comparisons, and to use its limited powers to discover the nature of the various “effects” that nature presents to man while she conceals from him the world of causes. Because of this vital importance of mathematics, the school must use special methods for teaching it, and make clear and comprehensible its elements with the help of plenty of apparatus that demonstrates the “materialized abstractions” of mathematics. 
It also matters for your soul.
Math is the realm of precision, exactitude, quantity, measurement, logic. If that’s the realm you’ve populated with secondhand incantations, that will invariably transfer to areas of life in which those things are cognate. (Which is every area.) The exactitude of the mind, the quality of judgment of the mind, and the independence of the mind are interrelated. Montessori again:
When you say “There goes a man of vague mentality. He is clever but indefinite,” you're hinting at a mind with plenty of ideas, but lacking in the clarity which comes from order period of another you might say, “He has a mind like a map. His judgments will be sound” in our work, therefore, we have given a name to this part of the mind, which is built up with exactitude, and we call it “the mathematical mind.” I take the term from Pascal, the French philosopher, physicist and mathematician, who said that man's mind was mathematical by nature and that knowledge and progress came from accurate observation. 
This is why Montessori’s main works on math are called Psychoarithmetic and Psychogeometry. It’s math—learned in a way so that it has a deep, positive impact on your psyche.
Place value, early and materialized in a dozen ways, is an important component of a great education.
1. Montessori, M. (1934), Psychoarithmetic.
2. Montessori, M (1923), From Childhood to Adolescence B.
3. Montessori, M. (1952) The Absorbent Mind.
Dr. Matt Bateman
Dr. Matt Bateman earned his Ph.D. in Philosophy in 2012 from the University of Pennsylvania. He taught and continued his research at Franklin and Marshall College in the Department of Psychology, on topics ranging from neuroscience to evolutionary theory to philosophy, before joining the LePort Schools as Director of Curriculum and Pedagogy in 2014.
In 2016, Dr. Matt Bateman became a founding member of Higher Ground Education. He is now Vice President of Pedagogy for Higher Ground and the Executive Director of Montessorium.