Category Archives: Mathematics


BW working out the laws of division. “Mhhhhh. I wonder if there is a way that you can know if this really giant number is divisible by four without having to divide it alllll the way out?”

Just a reminder to myself:  

The process defines who we are.  It isn’t the product that defines.  It is the journey that guides us and allows us to guide others.  It is the way we approach the difficult, the impossible, the unknown that determines who we are in the end and what we understand once we have reached the product.

How we approach the weak, the needy, the hopeless shows the depth of the understanding of ourselves and our own poverty of soul.  It is not the giving of charity as a product but rather the recognition of the deepest needs in ourselves and overcoming those needs to aid another person in their pilgrimage through this mortal plane. When it is in “pardoning that we are pardoned” then we are aware of the importance of the work.  This is process.

How we sit with the suffering in silence; how we laugh when laughing is easy; how we find hope in the time when laughter is impossible;  these are process.  This is what matters.

What my hand, mind, mouth, soul makes is not as important as the process by which it is envisioned, spoken, pursued, constructed, and loved.



Filed under BW, Educational Philosophy, Mathematics, Montessori, Moral Compas, Operations, Practical Life - Elementary, Students

BW – The whole story first

BW has been working out division using the stamp game. If the math was not too complicated he could work out one digit divisor problems (ie: 452 divided by 5).  So, in the fashion for BW to settle into deeper learning, I pulled out the racks and tubes and we began two unit divisors (ie: 45672 divided by 46).  He is overwhelmed.  This is good.  He needs to be beyond his on logic to not be distracted from his work.

(He is cleaning up the work here. I couldn’t grab the camera until he had completed his problem.)

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Filed under BW, Mathematics, Racks and Tubes - division, Students

Beginning Division in Elementary

BW is beginning division.  Teaching this mathematical operation often begins in Casa (3-6 classroom) using golden beads and a special division board.  Because of BW’s attention difficulties and his more practical life focused guide, BW did not receive many lessons that his intellect and tendencies would lead one to expect him to have been challenged with.  So here we are.  We are skipping all the Casa lessons and the division charts (which are in my Casa manuals but I’ve never seen given until lower).  BW will find the division charts interesting and we may make our own in the process of abstracting the division problems.

Anyway, we jumped straight into Stamp Game Division.  I found it most interesting that BW NEEEEEEDED to put all the stamps into neat rows even though I gave him permission to clump them.  I’ll be interested to see how long that lasts.



Filed under BW, Mathematics, Stamp Game, Students

Negative Numbers and Montessori

Montessori kids handle negative numbers from around age six.  They enter the fun of the inverse rules without knowing all the details.  In upper the oddities of subtraction of negatives is taught.   Today MMcC laid out the bead bars. She first learned their pattern back when she was in Casa. Now she added to them the negatives so she could see the negatives numbers.

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Filed under Algebra, Bead Boxes, Mathematics, MMcC, Montessori, Students

Inverse Numbers and the Snake Games

When I taught a large lower elementary, one of my favorite first year works was negative snake game.  The boys, yes it was most often boys who loved this work, would stretch a row of positive and negative beads across the floor.  They would then scramble along looking for ones that were opposites (+4 and -4) to remove first since they canceled each other out and the problem would be much simpler to work without all those guys.  (See the history of negative numbers – cool side trip.)

Today I began working with inverses in algebra with my older kids.  I realized that I should have taught the term “inverse” in the process of working with the negative snakes.  We had to back fill a bit about inverse numbers and their relationship to the way positive and negative integers work in algebra problems.


Filed under Algebra, AV, DW, JV, Mathematics, Snake Game Addition, Students

BW – Abstraction of Multiplication

BW has begun abstraction of the checkerboard.  It is an exciting week in our life here.  He decided he didn’t need get the material out today and so he and his pencil work on two problems.

The red circle is to denote moving up to the tens place when multiplying the multiplicand.

We are also reviewing addition by adding his answers together to find out how many watermelons and edamame pods he has.

Just so we are on the same page.  BW is seven. So not the expectations of 7 year olds in our area.


Filed under BW, Checkerboard, Mathematics, Multiples and Factoring, Students

Working on Skip Counting…

otherwise known as memorizing multiplication tables

BW is working on learning his multiplication tables.  In Montessori lessons, there must be at least two purposes.  This expectation would nix the use of rote memorization.  The second reason we don’t work on the flash card system of learning the multiplication tables is that it leads to quick recall but no understanding.  I had a mom stop by the other day for me to show her daughter that 5 x 3 and 3 x 5 were the same thing.  Wrote memory works wonders for speed but fails as the need to manipulate numbers grows.

So, beyond the chains and skip counting them how can we reinforce the multiples.  One way that has worked wonders for our very active BW is tossing a ball back and forth while calling out the skip counting numbers.  We’ve tried this with several different children and all of the children have stopped at one point or another to write down the number pattern 3, 6, 9, _2, _5, _8,  _1, _4, 7, _8, _0, _3, _6….   This has led to great conversations and more practice.

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Filed under BW, Educational Philosophy, Mathematics, Montessori, Multiples and Factoring, Students