Montessori Elementary Mathematics - Multiplication Checkerboard

Montessori Elementary Mathematics: Multiplication Checkerboard

Materials: white number tiles 0-9, gray numbers tiles 0-9, checkerboard[1], checkerboard bead bar box (only has bead bars 1-9)

Introduction:

            This is the checkerboard. We are going to learn how to do multiplication on the checkerboard.

·   Numbers: We have these numbers across the bottom (read and point to each number).

·   Multiplicand and Multiplier: This ridge is where we place our multiplicand and multiplier (indicate the ridge at the bottom and to the right, respectively). We also have some numbers along the right side (read and point to each one). We will put the multiplier here.

·   Squares – color and value: Then we have all of these colored squares. Here are the units (point to the lower left corner – green). Tens are here and here (point to each of the blue squares touching the unit square – 2). All of these are hundreds (indicate the diagonal 3 reds). Here we have thousands (indicate these 4). Here are the ten-thousands (indicate these 4). Continue with each category.

·   Squares – value of beads: What is interesting about the checkerboard is that the value of the bead bar is determined by which square upon which it is placed. I have a 3-bar; if I put it here (place on the units square), it is three units. If I put it here (place on a hundred square), it is three hundred; if I put it here (another hundred), it is still 300. What if I put it here (place on a 100,000 square). We don’t have to stick with one bead bar. If I have a 3-bar here (on tens) and a 5-bar here (on hundreds), we have 530. What if I put it here? (slide each over one to show 5,300). What if I put it here (slide up and right one place – diagonally)? It is still 5,300. Repeat until the concept is clear and the children can readily read the numbers on the board – invite them to place beads and you read the number; and to place beads to read the numbers themselves.

Exercise 1: representing each multiplication with the bead bars

Prerequisites: knowledge of the process of multiplication; Introduction to the checker-board; Ability to read hierarchical numbers; (technically the children can do the 1^st exercise without knowing the multiplication facts); Can precede work on the large bead frame (this exercise only)

Notes: If the children choose very large digits, they will have a ton of bead bars in each square that will overflow into other squares. In that situation, after they have done the multiplications, exchange within each row before sliding diagonally; then finish the exchanging to reach the final answer. 

                This work can help the children learn their multiplication because they are represented by the quantities they are putting down each time. If they do know their facts and they’ve caught on to the procedure of exercise 1, move them right on to exercise 2 on another day.

Exercise 2: Using the Multiplication Facts

Purpose: Further experience in long multiplication. Indirect preparation for category multiplication.

---

[1] Boards: current manufacturing only go to units of millions; felt version and the presentation in AMI training goes to hundred-millions on the horizontal

why does the checkerboard have more than 3 spaces per grouping? e.g. multiples of thousands beyond hundred thousand?

Each square on the checkerboard represents the multiplication of the bottom and right-hand values. So a board that is 9 squares along the bottom and 4 squares along the side - that last upper left hand square will represent 100,000,000 taken 1,000 times. 
The checkerboard does a few things for the children - it helps them work with VERY large numbers (now we can go into billions! - and it helps them see the "why" behind shifting the numbers one over when we multiply on paper (which the large bead frame starts to do) as well as the combinations of the quantities (so goes beyond the large bead frame in these concepts). I start the children with small numbers and they challenge themselves when they are ready to do the larger numbers.

Not to be confused with the Decimal Checkerboard (some tidbits found at Montessori Trails - a Montessori Nugget will be posted soon).

Elementary Montessori Materials Continue into Adolescence

Montessori materials truly go DEEP. Perhaps Montessori didn't have time to develop every detail from birth to college, but she left us an outline and some people have been able to follow Dr. Montessori's trailblazing style. 

Looking at the Elementary and/or Primary Materials used into Montessori Adolescent Algebra

• Geometry Sticks
• Fraction Circles (metal works fine, but the plastic materials with the multiplies of each fraction family from 1 to 1/10 – you do NOT need or want the children to use fraction segments beyond 1/10)
• Bead bars, squares and cubes from the bead cabinet (“complete bead material”)
• Wooden cubing material
• Power of 2 (Power of 3 if you purchased that too – or just make one)
• Pythagorean Insets
• Binomial Cube
• Multiplication Checkerboard
• Pegboard (ensure it is sized to your geometry sticks – so they all align properly – this generally means holes that are 1 cm apart from one another)
• Yellow Area Material
• Large and small geometric solids
• Number/operations tiles – with some additions, including adding in the fraction labels as well (called Special Math Symbols on the adolescent album)
• Set of 12 blue right-angled triangles

For review purposes:

• Decanomial bead bar box (called “large bead bar box” in the album)
• Constructive triangles

Longevity of Montessori: Mathematics

The Montessori approach meets the needs of children where they are at - both collectively and individually. Therefore, it is an approach that meets the needs of ALL children. The only limitation is the preparedness of the adult to meet those needs ;)

A recent post at MariaMontessori.com introduces this concept as it applies to primary (3-6) and elementary (6-12). Montessori is Developmental

Even in our material,


Let's look at math specifically - just some highlights: 

With infants and toddlers, we do a lot of natural one-to-one correspondence. Few toys, each that belongs in a particular place. Matching activities in sizes, shapes, colors. Helping to set the meal-table using a diagram of what goes where.

We can also give the language of numbers (counting), and children love language at this age, so most do pick up on counting numbers, although they typically skip a few numbers or repeat a few sequences ;)

Primary Mathematics:
While we don't typically start math in primary until around age 4, we begin with a few materials that extend into primary mathematics as well as into use at elementary and adolescence. Sensorial and Mathematics materials are both noted here:

• Red rods - extend into the number rods 
• All the groups of ten we have extend into the decimal system
• Pink Tower and Brown Stair can be used in geometry at elementary
• Binomial Cube and Trinomial Cube (elementary and adolescence)
• Geometry Cabinet and Solids (elementary)
• Golden beads (elementary and adolescence)
• Bead Cabinet and contents (elementary and portions in adolescence)
• Snake games (if you purchase the negative snake game, it includes all you need for primary as well as elementary and adolescence)
• Decanomial bead bar box (elementary and adolescence)
• Stamp game (elementary and adolescence)
• Short Division with Racks and Tubes becomes Long Division with Racks and Tubes (elementary)

Common Threads: 

• Place value color-coding remains consistent throughout all levels - into the checkerboards that are the visualization of the multiplication process, the bank game (just numbered cards, no beads), and more.
• The bead cabinet colors also remain consistent through all levels - even into the solid wood blocks of the cubing material that is used in elementary and adolescence. 

Additional posts of interest:

Toddler Exercises of Practical Life

All Montessori Trails posts on Mathematics

Mathematics Logic Game from Wff'n Proof

Review post on Adolescent Algebra Album

And that, dear friends, is today's show ;) 

Snake Games, Part 2

There are only THREE snake games:

As started in first post, when it says gray with red binding,
actually DO that. Everything shows up nicely on GRAY. 

1. Addition Snake Game
2. Subtraction Snake Game
3. Negative Snake Game

(for the first Montessori Nugget on snake games, visit this Montessori Nugget from last June)

Addition and Subtraction are done in primary or very early elementary (can be continued through lower elementary as practice is needed).

Negative Snake Game is an elementary material when we are preparing the children to work with truly negative numbers (in a way, subtraction is already "adding a negative" but with the negative snake game, we can actually get an answer below zero).

Each Snake Game has verification steps.

The verification step 2 of the addition snake game has the child grouping his original snake by lengths of bead bars and multiplying each group to see if his golden snake matches. (details are in the album)

Presumably, you can do that work with all of the snake games in the appropriate places.

Some people are calling this a separate snake game: Multiplication Snake Game. It's not a different snake game. It's a verification method on the addition (and again, presumably the other snake games).


:)

Dot Game in Primary Montessori Mathematics

The dot game is a primary Montessori experience that parallels work with the stamp game, but comes after the child has done several addition problems with the stamp game.

The dot game is only for addition.

We don't use color. WHY NOT!? We've been teaching our children the colors for the place values - and elementary materials continue the colors, so why don't we have color here???

1) The "color" that is visible focuses on the carrying over. This is becoming less concrete and more symbolic - so we want to reinforce the creation of ten-bars, carrying and....
2) the comma.
3) Most importantly though: this is where the children can really start seeing the place values specifically because of their place - not by their colors (which they will get again in future materials - it's not lost!), not by colored columns some people use for the stamp game paper (color also not necessary there since the color is in the squares and we want to focus on the zeroes for that work) and not by the shape of the objects contained in that place (i.e. the golden bead units, bars, squares, cubes). Right now we are looking at value based on place - only.

One baby step along the way. We neither need to hand the child everything on a tri-colored platter, nor do we ignore their needs for visual aids.

We give them just what they need when they need it.

Wooden Hierarchical Material - Number Cards

This material is a clear indicator of the differences between primary and elementary, even with the same material. It is also a perfect example of how the outward appearance of Montessori changes with each plane of development - so for those who jump ship after primary because they instinctively know that the primary experience is not appropriate for their second plane child, here is just one teeny tiny example of how Montessori addresses those very needs. 

At primary, the children can receive this presentation – it uses place-value colored number cards with spaces between the hierarchies, NO commas. These cards are available for download in a number of locations – free or for purchase – or can easily be handmade.
DIRECT AIM: focus on place value and the new hierarchy (millions) - counting the zeroes. This age does not need the comma for this presentation. (they do get the comma with the dot game and can use the comma with the division with racks and tubes work)

At elementary, the child should get this work again (or for the first time), because now, there is a new component: The Comma. The primary level colored cards can be used, just add a green comma after each number, such that when the cards are stacked in number formation, you have the following number:

1, 111, 111 

Those of you who have been following the recent blog posts and various discussions may recall one more aspect: some elementary albums describe the elementary number cards as being written in black (and still with a comma).

FINAL RESULT in the Real World: 

• Primary: colored, no commas 
• Elementary: colored OR black numerals, with commas to match (either green or black)

Homeschool option:
Take the cards from the Elementary Bank Game cards if they are laminated, use a dry-erase marker to write in the comma for this presentation; then remove it for Bank Game work. 

Great Lessons - Materials

Within AMI (American Montessori Internationale), the following is a description of the materials for the 5 core Great Lessons to establish Cosmic Education in the elementary Montessori environment.


God with No Hands

• 4 impressionistic charts
• materials for 6 demonstrations
• NOTE: There used to be a 5th chart (solar system) which has been pulled to use at another time (a follow-up telling of the story)
• NOTE: There used to be far more demonstrations (experiments is a misnomer in this case) which were pulled out to use as follow-ups, thus allowing the story to focus on the core keys, with plenty left for re-tellings and review - going deeper and encouraging a return to the Great Lesson. 

Coming of Life

• 2 impressionistic charts
• Timeline of Life (a mute version of the Timeline of Life is a follow-up work)

Coming of Human Beings

• No materials
• The focus is on the human connection itself - particularly on love; the children can look at their own arms and legs to see that they do not have specialized protection as animals do. 
• Note: The two timelines called "First Timeline of Human Beings" and "Second Timeline of Human Beings" are follow-up work in the history album. They are not part of this Third Great Lesson. 

Communication in Signs

• 8 impressionistic charts showing the changes of writing styles through time and civilizations

Numbers

• 8 impressionistic charts showing the changes of numbers through time and civilizations

For further Montessori Nuggets on the Great Lessons:

• General search for Great Lessons on Montessori Nuggets
• God With No Hands
• Coming of Life
• Coming of Human Beings
• Communication in Signs
• Numbers

Montessori Snake Games

What are all those snake games in Montessori !? Children are playing with SNAKES!?

Haha - well maybe they are! ;)

What follows is how AMI organizes the snake games - I would love to hear how other albums/trainings name them or organize them:

when it says gray with red binding - don't use red ;)
all the colors show up nicely on GRAY (even the gray!)
not so much on red - too distracting

Addition Snake Game

• first presented at the beginning of the addition memorization sequence, but then is to run concurrent with the addition strip board and memorization charts. The albums seem to make it look like it should be finished before starting those other works; nope - concurrent ;) Hm. Perhaps AMI albums could be more clear in stating this concurrency; that the snake game should be introduced but not completed. I'll go have a talk with myself on that one ;)  
• Materials: 3 boxes - One red box/lid with five colored bead stairs of 1-9 (total of 45 bead bars); one gold box/lid of 25 ten-bars; one white box ( black lid) with the black and white bead stair (beads 1-5 on each bar are black, 6-9 white); tray for the three boxes; small notched card (bridge) with its own container; piece of gray felt with red binding (32x18 inches)

Subtraction Snake Game

• first presented at the beginning of the subtraction memorization sequence, then runs concurrent with the subtraction strip board and memorization charts. The addition snake game should be mostly mastered before starting subtraction, but this can vary on the child.
• Materials: 4 boxes - One green box/lid of five colored bead stairs; one gold box of 25 ten-bars; one gray box with the gray bead stair (beads 1-5 on each bar are light gray, 6-9 dark gray); one black and white box with the black and white bead stair (beads 1-5 on each bar are black, 6-9 white); tray for the four boxes; small notched card (bridge) with its own container; piece of gray felt with green binding (32x18 inches)
• NOTE: If you are doing all one set and pulling just the boxes you need; the red box in addition IS the green box in subtraction (red for addition, green for subtraction). Use felt for this portion - or don't worry about it. The children in a homeschool setting don't need the cue as much as children in a school setting where they have a separate addition snake game from the subtraction snake game. OR if you have bought this set and have a white box with red lid for the red/white bead stair - you could use that for addition, then swap the colored bars to the green box for subtraction; since that red/white won't be used again until you are doing the full set-up, at which point the colored bead bars are in the green box anyway ;)

Elementary Negative Snake Game

• typically an early upper elementary work, but with interest or gifted children can certainly be done in lower elementary. Pre-requisites listed are "Knowledge of addition and subtraction facts; preparatory work with the ideas of negative numbers; addition and subtraction snake games in the primary class." Later notes in the album pages say "usually age 8". Thus interest or local educational standards are the driving forces here. 
• Materials: 6 boxes - colored bead bars, gray/white bead bars, golden ten bars, black/white bead stair, 25 gray ten bars, 1 red/white stair 

Some albums seem to list a middle-ground negative snake game; it is even mentioned in the elementary AMI math album because some children will come in with experience with it; but it is not found in the AMI albums. From what I can tell (please provide information in the comments if you know about this one!) it seems to be the Subtraction Snake Game with Negative Snake Game terminology added in. ???
The main difference: with the "real" Negative Snake Game, you can actually GET negative answers because you have the bead bars to do so.

And some people just call the Subtraction Snake Game a Negative Snake Game. It does the same thing, just is visually different, and doesn't get into the terminology of negatives. 

UPDATE 3/1/2014: For a second Montessori Nugget on Snake Games - including a note on the Multiplication Snake Game, please visit this Montessori Nugget.

UPDATE 1/22/2016: Adding the following somewhat more detailed pictures: 

All boxes for the Elementary Negative Snake Game
for 4-5 year old children, remove the unneeded ones

Addition Snake Game needs
3 boxes (contents listed above)

Subtraction Snake Game - same as the addition
adding the gray bars of 1-9 - 5 sets

Elementary Negative Snake Game - now we go to negative numbers
and need the red/white negative place holders and gray negative 10-bars. 

All boxes, opened, to show the tops with the bottoms (if you choose to go with bottoms in a homemade set)
If using tacklebox storage - good to have a felt bottom with the sides lined with the "lid color"

Montessori Cube Primer

There are many cubes in the Montessori environments :)


Found in Primary (3-6):

Geometric Solids

Bead Cabinet: cube for each number 1-10

Golden Beads: cubes of 1,000

Wooden Hierarchical Material: cubes of 1,000 and 1,000,000

Binomial Cube

Trinomial Cube


Each of those items are found in both primary and elementary.


What is just in elementary?

Power of 2 Cube

Hierarchical Trinomial Cube (also called "Arithmetic Trinomial Cube" and "Algebraic Trinomial Cube" - all three are the SAME thing) --- there is a story called "The Three Kings" that connects the Trinomial into this new Trinomial with new colors and new concepts.



But no matter the environment, the Trinomial Cube cannot be substituted for the Hierarchical Trinomial Cube - because they teach different things and are colored differently. Sorry!   The good news, the Hierarchical Trinomial Cube isn't just TOO expensive. ;)

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.

Fractions Material in Montessori

In Montessori, we have several sets of fractions materials:

The earliest material we give the children consist of circles cut from 1-whole through 10-tenths. They can use these primary as well as elementary for sensorial purposes (experiencing equivalencies) and design work purposes (tracing and creating figures).

In primary, they can begin simple operations - no changing of the denominator or going over 1-whole. They can experience all 4 basic operations in this manner. 


In elementary, they continue (sometimes begin) this work with operations and go much further with it, dividing and multiplying fractions by fractions, converting into decimal fractions, and going far beyond simple operations through the use of a plastic material of multiples of each fraction circles (lots of unit-wholes, halves, thirds, etc. through lots and lots of tenths). There is so much they can do with this portion alone, without even going into different denominators.

Why did the Montessoris only go to 10 with this material? Probably because it is the basis of our number system, and because it directly leads into the use of decimal fractions with the yellow board and all those little colored cubes. 

The fact is, there is even more fraction material in the traditional albums:

• Metal squares cut into further squares and triangles, with total pieces in some squares totalling more than 10 (here the children sensorially experience division of more than 10) - while not specifically a fraction work (it is used in geometry), the material still provides a fraction experience. 
• Fraction skittles for use in dividing by fractions
• Clear plastic overlays to help the children abstract beyond denominations of 10; you provide 2 rectangles with lines cutting into 3 and into 4, along with blank transparencies for them to make their own divisions for their own purposes. 
• And as previously mentioned, the decimal fraction material with the yellow board and cubes to represent each decimal place - along with the decimal checkerboard for continuing one's work from the multiplication checkerboard. 

Use caution when utilizing other fraction materials. Consider if it is truly useful or a distraction from the foundation they are building now (going ahead too soon; not fast enough; getting off the path of delving deep); be sure it does not cause them to become too dependent upon the materials such that they later cannot get away from them. 

Many children develop an intense interest in fractions at age 5 or 6; they work with it a LOT (hence the fractions are found in the primary class), and then they step away for a time; and when they come back, they are ready for more. They were building a foundation that, if not filled with distractions, is truly full and strong. 

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. 

Remedial Mathematics and Remedial Language

Montessori remediation is a bridge. 

In our society, we think of remediation in a negative light. The child is deficient in some way, his previous experience was impoverished, whatever conclusion one reaches - it is always negative.



In Montessori, it is a simply a bridge to assist the children getting from where they are now to where they need to be at a particular plane of development.

Children new to Montessori at elementary and in need of reading and math skills? They need some mathematics and language skills to entirely gain from cosmic education, but they should still participate in all other aspects of the environment, including the Great Lessons. They should NOT be held back from any other area just because math, reading and writing skills aren't where they could be.

Many times those other experiences are exactly the path they need back into mathematics and language arts.

Give them cosmic education. We fill their other needs so that they are not paralyzed, and we provide experiences that provide an impetus for utilizing those mathematics and language skills.

Meet them where they are.

Follow the child.

Know where the child is and should be (plane of development characteristics, human needs and tendencies - other theoretical aspects) - so that you can follow the child appropriately.

Montessori remediation is a bridge.



Remediation album pages are found at the end of the appropriate albums so that the teachers and parents don't START there; but if you find a child isn't able to keep up with the early mathematics work or is not reading or is not writing well, then you refer to the remedial mathematics or the remedial language sections for JUST and ONLY what the child needs. BUT KEEP COSMIC EDUCATION GOING.
(so many teachers keep the child separated, so this last line cannot be over-emphasized!).

Remediation = Bridge.

Cosmic education - happens for the 2nd plane child no matter what.

:)

Numerical Values of Squares by Montessori

Colors of the Numbers
1 - red
2 - green
3 - pink
4 -  yellow
5 - light blue
6 - purple
7 - white
8 - brown
9 - dark blue
10 - gold

Each number corresponds to the bead bars - there are boxes and sets of bead bars for various materials and works.


The numerical value of the squares: 
1, 4, 9, 16, 25, 36, 49, 64, 81, 100

Each square has a corresponding SHORT chain of that number of bead bars linked together to fold up to make the square. (so 8 bead bars are linked together so that the fold up to a square)
The short chain lays flat on a horizontal plane.


The numerical value of the cubes:


Each cube has a corresponding LONG chain of that number of bead bards, linked together to fold up to make a row of squares, with larger links to indicate the end of each square formation.
Each cube also has a corresponding number of squares which, when stacked, form the cube. These lay flat, behind the sections of the short chain laying down.


So when they hang in the bead cabinet, you see the squares of which the cube is composed, hanging down below that cube. When those squares are stacked up, you have the cube.

You can see the squares from across the room.
What a BEAUTIFUL visual! and preparation for later work. 

Practice on Mathematics Problems

Montessori builds repetition into the materials without requiring tedium.

When a child plays a 3-period-lesson game; or language games with the sensorial materials, with the language materials, with the interconnectedness of every area of the classroom and life. From laying out a Far Eastern place setting to the continent folders of that region - in primary; from all the timelines of elementary to all the peoples and events throughout history - OUR history.

Everyone seems to accept these sorts of connections and repetitions and reinforcements with ease and peace.


But what about mathematics?

Golden bead material:
Addition - when a child is adding 2957 + 3495 - he is really doing 4 addition problems. More when you consider the carrying.
Subtraction? Same deal. 4824 - 3604 - the child has 4 subtraction problems in ONE.
Multiplication? We start with numbers such as 1242 taken twice. And quickly move into 4958 taken 4 times. LOTS of carrying.
Division? Same idea.

These are 4 year olds.

As they work with the materials, they get the patterns because they're not just writing it out, they're DOING it. Then they have the stamp game, bead frames, racks and tubes for short division and other tools to continue working with BIG numbers; and the memorization boards to start memorizing the basic math facts in each operation.

Let's consider a short division problem: 5,496,284 divided by 2. There are SEVEN division problems in that one problem alone.

And that is primary level. 


By elementary, they are getting into long division, intense fractions work, decimal fractions, squaring and cubing - and again, we are working with real numbers with real materials so that the child, even by manipulating the material, gains an understanding of what is going on.

If a child starts in the middle of the elementary with no Montessori background, he still gains from the experience, because he is still working with the material, surrounded by children who already "get it" and who are joyful to help him just where he is, without doing the work for him. Bank game, long division, and fractions are fantastic materials for new-to-Montessori elementary children.


Repetition is built in. Use the materials; use language and other games to get the materials out and put them away; provide real math problems; invite the children to think of their own (they always want to work with something harder than we provide anyway - LET them!). And they have joy in their work.

Binomial Cube and Trinomial Cube

What do these represent?

First, let's consider the lid of the binomial:
Look at each side and label each section a or b - so the length of each side is a + b
To find the area of the whole thing, we would multiply the length times the height (a + b)*(a+b)
We can also write that as (a+b)squared

Ok, so let's multiply that out - we end up with these pieces:
a-squared + ab + ab + b-squared

And that is what we have: a red square, 2 rectangles representing a times b, and a blue square.


But we have a CUBE.
So we multiply length by height by depth - (a + b)*(a+b)*(a+b)
Multiplying this out, we end up with a component for each prism within the binomial cube box.

Each piece represents a mathematical principle. 




The same concept applies for the trinomial cube, except there are 3 factors: a, b, c


As the children proceed through other mathematical experiences, they will begin seeing these same images in SO many other places. Because these principles surround us.

And most of you only remember those formulas from high school algebra, right!? ;)

Calculators in Montessori

What calculators do we use in Montessori?

We have a lot - at various ages - but here there are all lumped together:

• bead bars
• someone else's eyes
• built-in control of error (anything where the pieces won't finish going in if one answer is wrong)
• figuring the problem two ways
• the human brain (which is mathematically designed)
• experience
• intuition developed through experience
• studying mathematics in history and culture
• graphing calculator - middle school; could be introduced as a future tool in upper elementary

Montessori mathematics develops perception and intuition; the children learn their facts through the consistent use of the Montessori mathematics materials. The fact is, they don't *need* a calculator. 

Any child or adult with experience in mathematics can figure out how to USE one. So it's not about "teaching them the tools they'll need in the future" - they'll figure it out when they get there. 

Graphing calculators are a handy tool - but to gain the maximum benefit, sufficient understanding of the concepts must be applied. Graphing calculators can be useful in middle and high school. Perhaps introduce in upper elementary, but it's not necessary. They have so much else to learn, why hinder them with using a tool that just gets in the way? 

Montessori Mathematics is Incomplete

Or so the story goes ;)
(just at elementary)

So is the language album - and history - and geography....

It is true that the traditional Montessori elementary level mathematics album (and the others listed) does not contain some information that is required by every state in the US and probably by most standard school requirements around the world's first-world countries.

But let's think about cosmic education.

Mathematics is not a stand-alone subject by itself.

Mathematical principles are embedded in the very life around us; the very things that support that life.

Mathematics as a human language has come down to us through history as a gift from our ancestors. We have built upon it, used it, expanded it, created a supranature with it -- we live it every day and we pass it on to those who come after us.

We humans by nature have a mathematical mind (bear with me on this one - future Montessori Nugget!).

So let's think about this.

What is "missing" from the mathematics elementary album? 
--graphing: look in geography (economic geography with production and consumption); also see botany and zoology studies; it is also included in math, but not as a stand-alone album page; be sure you have an album with ALL the exercises. Specifically designated coordinate planes are not included.
--estimation: used in practical situations all over the place, most notably in geography (economics), fractions, long division
--statistics and probability: really get into it with economic geography; also covered in history studies
--comparisons (greater, equal): exercises within math presentations
--measurement: may be introduced in math, but heavily utilized in every other area of the room in practical life situations --- check Geometry ;)
--money: practical life (Goings Out, literal practical life); extension of various decimal fraction album pages and experience is assumed when children are working on divisibility (early math)


What is "missing" in  language?
--Research skills: no formal album page exists, but exercises, presentations and almost all follow-up work requires it. The entire elementary sequence focuses on research - show the children how to use tools as they come across them; formally these things are introduced in economic geography (upper elementary), but the children are very likely to know it all by then, if they've truly been exploring their own interests already.
--telling time: see the history album
--5 paragraph essays: they're not necessary.


"Missing" in history: 
--local history - timelines and charts: the children build these themselves according to the local educational requirements and their own interests (no album can cover every single "local history" requirement - so guidelines are given to create your own)
--the album just seems too short: because the history album provides the framework and the highlights - the children's interests and the local educational requirements dictate the details; encouraging the use of many different sources for information, rather than depending on just one source that may or may not be accurate or non-bias, or even cover everything.


"Missing" in geography: 
--everything that should have been done in primary, can be taught quickly in elementary with similar materials - so elementary geography "officially" speaking does not include functional geography. This is a research area for the children - as they study cultures, climatic zones, peoples of various places and times, they will be learning political geography at the same time.
--basic map skills: Goings Out - maps of museums, bus routes, maps of the city, etc.

The main thing to remember is that, since elementary Montessori is based on cosmic education, just picking and choosing one or two subjects won't cover all the concepts needed in that subject (ie mathematics - you'll need biology, geography, history and language to "complete" the math cycle).

This is where it is crucial to utilize the local educational requirements ("public school standards") in the 3rd and 6th years of elementary. These are a tool for the children to compare where they are with their locally schooled peers, check in on language usage, etc. It is a tool that builds personal responsibility as they do the comparing themselves, look for suitable tools to fill in, or in a few rare occasions the adult will provide the needed tool. 

Please reply back with anything that seems to still be missing. I'd like to make this post as comprehensive as possible, so we can honestly say which public school requirements are truly missing. 

Montessori Large Bead Frame

The small bead frame and large bead frames (sometimes called the "Montessori abacus", which is  not entirely accurate, but it sure looks like one!) are a necessary yet often overlooked piece of Montessori mathematics material.

Large Bead Frame -
goes up to 10 million

They each allow for addition, subtraction and long multiplication and are a step towards abstraction in the process of carrying into new categories and in exchanging.

While this material can be used for 3 operations, the main focus of this material is its teaching of *Long Multiplication* - this IS the Montessori material that teaches all the steps in proper order. Other materials either prepare for it, or expect a child to have this experience tucked away already.

The small bead frame has four bars, going into the thousands (with 10 thousand-beads, it can go to 10-thousand).

The large bead frame, with 7 bars, is the child's first experience going into the MILLIONS. And they LOVE it!

The children also utilize lightness of touch, visual organization, proper order of steps, which is preparation for the division with racks and tubes (test tube division) which starts with short division (1-divisor with a 4-7 digit dividend) and goes into long division (2, 3, or 4-digit divisor into a 4, 5, 6, or 7-digit dividend).

Many primary level children will get to this work and the short division with racks and tubes - generally between 5 1/2 and 6. Whether they get in primary or not, this material is heavily utilized and crucial in lower elementary on their work with long multiplication.

The main thing with this material is the connection to the writing. There is long bead frame paper that goes along with this material, for the children to begin their writing out of long multiplication - one sized for the short bead frame; the other sized for the large bead frame.

They are introduced to the bead frame with nomenclature first; then addition and subtraction just to get a feel for how it works and how the writing on the paper works.

Then they are introduced basic multiplication - 52 taken 3 times for example. They are writing the steps on paper and doing it on the bead frame, so they understand what they are writing and when they are exchanging, WHY they are exchanging or carrying. At this point, they might do a number such as 54 taken 3 times.

Once they are comfortable, they are given longer and longer multiplication examples, until they are making up their own (our goal is for them to create their own numbers, so that they own those numbers and will pay more attention to how they work, what are the patterns, what are the results).

And they begin to add zeroes into their writing.

Before you know it, they are writing out long multiplication problems without the use of the bead frame or a calculator.

AND they are prepared for the greater task of long division!



UPDATED TO ADD due to a small number of private messages received: Yes, there are a couple (?) of album sets you can purchase that do not introduce this material until the 3rd year of lower elementary and seem to indicate that it is an upper elementary work. AMI albums sit this material firmly at 5-8 years of age for normally functioning children, with the children working with it as long as they need (through any age). The fact is, upper elementary children are getting "less fiddly with the beads" (as quoted by one of my AMI trainers) and should really be far beyond the introductory stages to this material if they have had Montessori math since at least 5 1/2; still working with it as needed, but not JUST receiving it at age 8. If just getting it at age 8, they won't be doing LONG division until at least age 9, perhaps even 10, long after kids in the public schools have already started it; and Montessori has prepared them from the time they were 3 if they've been with Montessori that long - even starting at 5 1/2, there will be minimal slowing down in the early mathematics presentations. The big question is, if they're not doing long multiplication stages and long division stages at age 6 and 7, what have they been doing?
NEW children (starting Montessori after age 6 1/2 or 7) and children with severe learning impairments will follow a later schedule. THAT IS OKAY - but they are the exception and their needs should be considered when "Following the Child".