The Curriculum of the Steiner School - Class 2

Notes and Lesson Plans

Column Algorithms (Place Value)
updated September 10, 2020


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Column Algorithms (Place Value)
for Class 2



Pinterest - Renee Schwartz
My curated collection of visuals! Browse sample main lesson book pages, watercolor paintings, chalkboard drawings, etc. for Column Algorithms.

Note: Of all the math blocks that we teach over the years, I think this one is the most important to do with the Montessori materials, to help children understand place value, carrying, and borrowing in a deeply hands-on way. I do think in this case that they are absolutely essential. Get the hands-on materials, not just the place value stamps.

Montessori materials that I see as compatible with the Waldorf Column Algorithms block:

    Golden Bead Material

    Unit Beads

    Ten Bars

    Hundred Squares

    Thousand Cubes

    Number Cards, 1 to 9000

    Stamp Game

    There are lots of YouTube videos made by Montessori schools which show how to present and use these materials.


For Montessorians: If you do this block first, and then the Shapes & Number Patterns block (where I recommend using the Colored Bead Material), students will be in a good position to then move into the Golden Mat and Checker Board work.


Other Helpful Links


Booklist


Activity One - Gems / Pouches / Basket

    Many years ago I was at a homeschool training at Barbara Dewey's and a woman named Misty shared with me what she had done to introduce place value. A group of families worked together to set this up for the children in their homeschool group. They laid out 100 individual gems in a long line on the ground. The children counted as they picked them up: 1, 2, 3, etc. When they got to gem 9 their hands were beginning to get full. Right beside the tenth gem was a small pouch. They made a group of ten gems in a pouch, then picked up the eleventh gem, etc. After nine pouches (each of ten) and nine individual gems, they finally reached gem 100! For this they were provided the tenth pouch and a basket to put all ten pouches in.

    It was a lovely idea and she said they always understood it thereafter.

    To do this in a way that is compatible with the Montessori color coding for the digits, use green glass gems, small blue pouches for the 10, and a large red basket for the 100.


    1 lb. green glass gems (100+)


    for small blue pouches of 10:

    Etsy white linen favor gift bag
    beautifully made 100% white linen and so easy to dye blue

    LinenForest sells these as a set of 10 (and it would be fun to dye these with your child first and keep a mystery what you were going to use them for...)



    red willow basket with folding handles


    To introduce this place value activity with a story you could use The White Snake from Grimm's (organizing the grains of millet into sacks)


Activity Two - Place Value Color Coding

    Practice writing numbers to the hundreds place with the digits in the correct color

    units - green
    tens - blue
    hundreds - red


Activity Three - Introduce Golden Bead Material

    Read Redwoods by Jason Chin

    Some useful numbers you can make from this story are
    2,000 ; 200 ; 29 ; 2 ; 1,600 ; 370 ; 379

    It is important to see that the digit 2 in different places represents different things. As one litte girl remarked to me, "It's still two but they count in a different way."

    For an SWI connection, you can show the base < Titan > and the word sum for < Titanic >
    Titan + ic

    Introduce Small Wooden Number Cards with Box
    (if they ask why the thousands place is green, you can simply say "hmmm, I wonder...")

    Introduce unit beads, ten bars, hundred squares, and thousand cubes


    Small Wooden Number Cards with Box


    100 Golden Unit Beads


    45 Golden Bead Bars of 10 with Box


    10 Wooden Hundred Squares


    9 Wooden Thousand Cubes


    100% Cotton Small Working Mat

    this is placed under your work as you build and work with the Number Cards and Golden Bead material; using a mat to delineate your work space is an important part of the Montessori philosophy and teaches the student to show care and respect for the materials

    Practice making a quantity with the number cards and then representing it with the beads

    Stack the number cards to show standard form; slide them apart to show expanded form and to make it easier for your child to understand and make the given quantity

    Note: Writing a number in color and building the quantity vs. building a quantity and then writing the number in color vs. saying a number aloud and then writing it in color and making it... these are all very different skills so do many variations. You can also have them say a number and you write it in color and they tell you whether the colors are correct!

    A simple story that can give you lots of inspiration for making and writing numbers (how many animals hopped onto that raft? 47? 1,210?) is Bear Came Along by Richard Morris. I also chose this for more practice with SWI, and we did the SWI Investigation of < water >.


Activity Four - 45 Layout


Activity Five - Crocodile Game

    Build both numbers, then compare them

    Go to the thousands place, using this as the opportunity to talk about why thousand is green (it is the Units place of the Thousands Family, thus the comma)

    For some children the comparison is easier to see with the Stamp Game

    You can also make the numbers with BOTH materials: this is 2712 compared to 2722


Activity Six - Dice Game

    Go to the millions place, and teach how to read large numbers

    Talk about the names of the families: the Millions Family, the Thousands Family, and the Simple Family (which has no last name)

    When you get to the comma, you say the last name of the family

    Definitely read A Million Dots by Andrew Clements


Activity Seven - Infinity Street

    A nice book to introduce this lesson is Is a Blue Whale the Biggest Thing There Is? by Robert E. Wells

    Here is a helpful blog post with photos for the Infinity Street lesson

    We have houses and mailboxes up to septillion, and slips of paper with the names of all of the families up to novemdecillion

    Simple
    Thousand
    Million
    Billion
    Trillion
    Quadrillion
    Quintillion
    Sextillion
    Septillion
    Octillion
    Nonillion
    Decillion
    Undecillion
    Duodecillion
    Tredecillion
    Quattuordecillion
    Quindecillion
    Sexdecillion
    Septendecillion
    Octodecillion
    Novemdecillion

    This is a fantastic extension for children who are learning about prefixes (which happens in Second Grade). I like to give them the slips of paper after septillion and have them figure out how to put them in order.

    Of course you can refer to the months of the year. I have also found Zin! Zin! Zin! A Violin by Lloyd Moss to be extremely useful in this lesson, especially for sliding under the radar the prefix < sex > as representing six (trio, quartet, quintet, sextet, septet, octet, nonet).

    Note: They see why September / October / November / December are months 9 / 10 / 11 / 12 and not months 7 / 8 / 9 / 10 (as would be more logical) next year in the Calendar study.


Activity Eight - Adding with Regrouping

    Read Journey to Cahokia: A Boy's Visit to the Great Mound City by Albert Lorenz, or another book that involves barter or trade (the trade must be a fair one, so don't choose "Hans in Luck")

    Here is a sample problem. Make and add with the GBM, 213 + 128

    Write the digits in color and arrange them vertically on the paper

    Have the child take ten individual unit beads back to the store and trade them for a ten bar. Tell them that you're trying to make the number in the simplest way possible. ALWAYS add beginning with the units place and then move to the left.

    You should then have 1 unit bead, 4 ten bars, and 3 hundred squares. Write the answer in color: 341. DO NOT show your child how to do this problem on paper. The paper is the physical representation of the active work.

    Other addition problems we've used (always write digits in color for problem & answer):
    679 + 120
    1214 + 65

    Play Crocodile Game using two addition problems; the crocodile eats the larger sum:
    236 + 23 ____ 242 + 7

    And don't forget to work with multiple addends. This is especially helpful if a child is trying to doing things with mental math and you want to bring him/her back to the material.

    Students need to have completely internalized the concepts through repeated experiences using the material, or you will find that they suddenly stumble when the problems get more difficult or you move to a new operation. If a child can carry with addition but can't figure out borrowing in subtraction, they never really understood regrouping in the first place!

    Crocodile Game with two addition problems with multiple addends:
    326 + 12 + 4 + 1 ____ 314 + 3 + 2 + 18

    It's also fun to build something with the golden bead material, like a castle or a mandala design, and then calculate its value. This often leads to practice in regrouping (10 beads is a ten bar, 10 ten bars is a hundred square, 10 hundred squares is a thousand cube)


Activity Nine - Subtracting with Regrouping

You can create practice problems with or without a story. Math problems can be found everywhere!


    The Titanic: Lost... and Found

    by Judy Donnelly

    page 18 static subtraction: 2,227 - 1,100

    page 37 dynamic subtraction: 2,227 - 705

    page 43 static subtraction: 1985 - 1912


Activity Ten - Multiplying with Regrouping


Activity Eleven - Dividing with Regrouping

    The whole idea behind division is "sharing out fairly." It's fun to read a story like The Little Mouse, The Red Ripe Strawberry, and The Big Hungry Bear by Don and Audrey Wood.

    Division is the only operation where we begin by sharing out the LARGEST pieces first (thousands), since to regroup here you have to go down to the next value piece. With all of the other operations you begin solving the problem at the units place.

    If you don't have the skittles from the Stamp Game to use for this you can also use peg dolls or even wooden animals. Do not go past dividing by 9 at this point. Two- and three-digit divisors are covered in the Stamp Game (Decurian Division and Centurian Devision).


Activity Twelve - Introduce Stamp Game

    Continue to work on daily math practice and adding / subtracting / multiplying / dividing with regrouping throughout the school year.

    This final activity, Introducing the Stamp Game, will come at some point in the school year and will vary per child depending on when he/she has fully internalized the Golden Bead material. Dr. Maria Montessori was famous for all of the in-between math materials she created to take the child sequentially on the journey from concrete to abstract. Do not just jump from the Golden Bead Material to paper & pencil. The Stamp Game is the next step.


    Stamp Game

    You can also make a Stamp Game yourself using small squares of colored construction paper. That would be completely fine (and when Maria Montessori invented it she did use actual paper stamps). Since it is the next step in abstraction, keep the color coding intact. And your child can also help make the squares.

    I do NOT suggest making your own Golden Bead Material. It needs to be perfectly proportionate, and I think that having it be store bought & special is important as well.


If you are a Waldorf homeschooler or classroom teacher interested in using Montessori materials in the Waldorf environment, and you have questions, I'm happy to schedule a phone or Zoom call! Click on Consulting to email me.


My blog posts from teaching this topic:



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