Multiples in Mathematics

Lower elementary multiples example - this is typically first year in lower elementary if the child has had a decent amount of primary mathematics. It could happen in later first year if the child is brand new to Montessori altogether. 

each number is circled according to its multiples
see my son's key in the upper corner :)
who says Montessori children don't learn how to use legends?
right here it is - they just don't have an album page for it ;) 

my sample for how to highlight a particular number
again - using a key ;) 

Embellish your work!

Sample from other work I've seen
Borders around the page or around sections/problems
Separators between sections
Just to fill in some space

In Montessori we say, "DOODLE!"

Ok, so we don't say that. We DO say, "Embellish!"

We want the children to cherish their work, so we allow them the opportunity to personalize it, fulfilling the fundamental human needs towards vanitas: to embellish oneself and one's environment.

All Montessori parents and teachers reading this, raise your hand if you have NEVER been tempted to embellish the children's environment on some level or another - to the point of being TOO much?

Not ONE hand should be raised! ;)

We want the children to learn balance, so we allow embellishment from the beginning, integrating art and daily work, even when the subject matter is not necessarily art.

But isn't it all art? Math is art, language is art, music is art, history is full of art and those timelines are works of art, geometry is pure art in my mind, geography is an art --- all these things have grace, beauty, boundaries, creativity ---- ART.

So why isn't this information available online anywhere, on all those Montessori blogs and all those Montessori albums that provide samples!? I don't know! It's a bit frustrating, because it is so basic to Montessori, especially elementary Montessori. But it starts in primary with the stamp game in particular - embellishing the row between two problems.

My son's sample today. He chose to use markers
for the first time. Please note: Markers do not mix well with
colored pencils. Any work done in pencil should be
embellished with color pencils, NOT markers.
Use markers as a medium to themselves on their
own separate sheet of paper, if at all. 

My son has always done embellishment of some kind and he LOVES it. The work means something to him and he's perfecting his art skills.

 I will show you an example of one I'm not proud of because he rushed it. It was not meaningful and he even said so. We discussed the reality that if he doesn't WANT to embellish, he doesn't have to.

And we discussed the reality that markers and writing pencils and colored pencils don't mix well. The aesthetics are lost.

Therefore, all is not lost, because this experience prompted discussion on balance, necessity and art media usage.


What about work plans and work contracts? Well, these should generally not be embellished - perhaps very lightly (instead of a checkmark, a creative child could use another symbol) - now the work journal could be a place for art, if you use one for the child to document his or work! The work journal can be embellished in any way the situation allows. While it is not the place for the child to store finished work (only a place to record time spent and what was done; perhaps answer a follow-up question), many children prefer to draw their work into these journals or separate entries with fancy designs.

Playing with Language

You know when you've been doing Montessori very well in your home, specifically the grammar boxes and sentence analysis, when your child speaks in transposed sentences without even thinking about it - no hesitation. Just as natural as can be.

"Son, what are you doing right now?"

"Apple cutting core and it Mommy peeling an." With a big bright smile :)

Tessellation Patterns

While there is no specifically AMI Montessori album page for tessellations - they are just plain fun! And when presented right, with the right materials, they fit right in with Montessori.

While we do not encourage the children to create images (we want them to explore the shape, and the function of each shape - not be focused on creating boats and flowers) - elementary children do utilize them to create images. I encourage the exploration of shape and function and steer them away from creating images until it is inevitable.

Some samples of our work:

a friend's toes

exploring different kinds of flowers

what if we just use hexagons? what would happen?
note: this particular mode teaches far more
 more than creating a flower teaches a child ;) 

she was exploring pure shape; filling in gaps with other shapes,
seeing where it would lead (this is a transition stage into
creating intentional images)

Purely exploring with shapes - and look at the beautiful pattern emerging!

Long Division - Final Results

I did the writing, because he insisted on doing it with colored
pencils, which would have been too faint for our camera.
And I wanted him to focus on the step at hand: multiplication.
He has not yet embellished this page because he wants
to do more long division problems on it first. 

My son is just at the final stages of long division - writing without the beads. It is a work we started quite a while ago; he has flown through it relatively speaking, however he will continue to utilize the beads and boards for quite a while to come. (this material starts in year 1 of the elementary Montessori mathematics album - and claims to be done by age 8 - well, that's the presentations being done at age 8 - the work will continue into upper elementary --- this expensive piece of material (approximately $100 after shipping with IFit) is worth EVERY PENNY).

This is sort of bitter-sweet! My just-turned-8-year old is doing long division on paper!

Tomorrow, we will go over the basic steps of estimating the answer, then multiplying AND checking with the beads to be sure we get each step correct.

After that, the last step is using trickier numbers and estimating.

This material is beyond amazing! Once the process is mastered, the understanding just FLOWS. I recall a woman in my primary training crying when she finally understood division - and that was the same material just doing SHORT division - as an adult she was able to comprehend how the long division would work; but as a child she had been terrorized by what is now a joyful experience for her.


I strongly encourage the use of graph paper in doing mathematics. The children can embellish their work in so many ways with graph paper, plus it keeps their numbers lined up and organized. The more complicated the operation at hand for that particular child, the more you want graph paper (ie stamp game for a 5 year old is complicated; stamp game for a 8 year old maybe not so much; long division - get the graph paper!).