Thursday, November 5, 2015

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)

            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 1st 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).

Friday, October 23, 2015

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 rom 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

Thursday, October 1, 2015

Montessori Phonograms - Key Sounds

In the English language, there are 16 "key" phonograms: 

ai, er, ie, ee, or, ue, oo, qu, th, ch, oy, sh, ou, oa, ar, au

All others are variations on these ;) 

The sandpaper letter phonograms

The sandpaper letters phonograms should be introduced at the same time as the individual sandpaper letters - learn 3-5 individuals, then introduce a phonogram; continue introducing one new one at a time with individuals in between such that all sandpaper letters, individual and phonogram, are finished up at about the same time - typically 3 weeks after beginning the first sandpaper letters ;)

In this way, the children can immediately write anything with the movable alphabet, without needing prompts or guides as to what they can write versus what they have not yet learned - they'll have learned it all! 

Wednesday, September 16, 2015

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 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 ;) 

Sunday, September 6, 2015

Montessori Scope and Sequence Options

UPDATE: I have set up a dedicated page for this post: 
I aim to keep that one up-to-date.

Montessori Scope and Sequence Options

All of these are free options! And there are many more out there, also free.
(paid variations are listed further below)

Benefits of a scope/sequence - give you some idea of what is developmentally appropriate within a range of years.

Each scope and sequence option below is in a different format, with different a different focus. Some are based on AMI, some are AMS, some are a compilation of several, and some are just straight-up independent. Look through them, see which styles speak to you. Hopefully, this will show you that there is no one right to get the job done, except this way: follow the needs of the child!

Albums or manuals or videos or online resources will help flesh out what each activity is, how to use the materials and ensure Montessori experiences that don't require materials. The options currently available to Montessori Homeschoolers are listed here at Montessori Nuggets - Montessori Albums.

Infant through Age 12 (could include Adolescence): 
Montessori Works (free for first 5 students listed)

Infant and/or Toddler (0-3)
(the ones listed above mostly include 0-3 as well)

Primary (ages 3-6)
The Helpful Garden (incomplete - does not include the kindergarten year)
Montessori Print Shop (visit the "overview" for each of the subject areas)
Montessori Story typing up the Gettman periods
Keys of the World (word .doc download) - link to it is here: Montessori Trails - Keys Albums
What Did We Do All Day? (mathematics only)
(and see the ones in the first listing above)

Elementary (ages 6-12)
Hilltop Montessori - upper elementary only
Montessori World - lower elementary only
What Did We Do All Day? (mathematics only)
(and see the ones in the first listing above)

Adolescence (12-18)

Pay-For Variations: 

Infant through Age 12 (could include Adolescence): 
Montessori Works (free for first 5 students listed)

Infant and/or Toddler (0-3)
Montessori For Everyone - Toddlers
(the ones listed above mostly include 0-3 as well)

Primary (ages 3-6)
Montessori for Everyone - Primary (3-6)
Montessori Story - typed up the Gettman book "Basic Montessori"
AMI & Keys of the World - available at Garden of Francis
(and see the ones in the first listing above)

Elementary (ages 6-12)
Montessori for Everyone - Lower Elementary - Upper Elementary
AMI & Keys of the Universe - available at Garden of Francis

(and see the ones in the first listing above)

Adolescence (12-18)

Sunday, July 5, 2015

Montessori and Fairy Tales

What did Montessori actually speak about Fairy Tales:

"Now, everyone who knows my name says that I am against fairy tales. Apparently I say they are dangerous to a children's mind. 
But you must know that I never assert anything that I have merely reasoned out, because if I did that it would just be a theory of no importance. It would be just a matter of opinion and therefore not a serious statement. Serious statements must come from observation - this is the truth. I have never before given an opinion on this subject. 
So, if I were against fairy tales, it would not be because of a capricious idea of mine but because of certain facts, facts observed many times. These facts have come from the children themselves and not from my own reasoning. (she goes on to describe following the children's truest, deepest interests and choices - choices for valuable work that led to a leaving behind of certain negative qualitites) The great love of fairy tales disappears too. (continues with the children walking away from fairy tale stories) This shows that children listen, or at least the older ones do, but inside they have more important urges of nature. So we do not see complete interest. If they are free to do something else, children will choose something that is more important to their development.  
(The 1946 London Lectures Lecture 26: Truth and Fairy Tales)

Monday, June 1, 2015

Montessori Cursive - Original Cursive Materials

Building on a previous Montessori Nugget: Which comes first - cursive or print - writing or reading?

From Dr. Montessori's Own Handbook (the link is to the free edition on - a great historical piece, NOT where I would recommend anyone *start* their Montessori reading, but for historical value, this book and The Montessori Method are interesting.

Scroll down the section entitled:

Exercises for the Writing of Alphabetical Signs

For the section on the cursive letters used at the time. (the method/experience itself was tweaked over the years to respond the needs of the universal child - and greatly simplified). 

So the children ONLY learned cursive. Yet they could read this: