Column Algorithms (Place Value)
updated April 12, 2021
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This site records my journey. I hope my honesty is encouraging and helps break down some barriers that may prevent people from trying Waldorf methods. Because this is an ongoing site documenting my curriculum planning and ideas, some materials are more Waldorf-y than others. Please feel free to take what you like and leave the rest.
This page has helpful links and LOADS of free resources to help you plan your second grade year. Enjoy!
Pinterest - Renee Schwartz My curated collection of visuals! Browse sample main lesson book pages, watercolor paintings, chalkboard drawings, etc. for
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
Number Cards, 1 to 9000
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.
I like to have a book which discusses trading for this block. In third grade there is an entire block on bartering
and the development of Currency (as part of the Maths of Practical Life) so you can look at
for suggestions of picture books you may be able to find at your library. We did this block
immediately following Native American Legends, so I used the historical fiction picture book
Journey to Cahokia: A Boy's Visit to the Great Mound City by Albert Lorenz.
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.
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)
You could also use The Hundred-Year Barn, a beautiful picture book by Patricia MacLachlan.
This story does a wonderful job of showing the passing of time. It is a lovely way to talk about the number 100 and, as a bonus,
the barn is red which is the color we use for the hundreds place.
Activity Two - Place Value Color Coding
Practice writing numbers to the hundreds place with the digits in the correct color
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
IF you have enough material (you need 45 of each), do the 45 Layout
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,
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.
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. The Story of Ping
by Marjorie Flack is a good one for this! How many ducks are in his family?
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!
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.
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.
Christmas Farm by Mary Lyn Ray is a perfect source for some real-life math problems!
For multiplication and division there are also materials and teaching techniques that come after the Stamp Game that help guide the child
gently towards abstraction.
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.
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