Column Algorithms (Place Value)
updated September 8, 2024
Recorded here is my own personal collection of articles, resources, favorite links, teaching ideas, and lesson plans. It encompasses many years, from the very beginning of my experience studying and learning about Waldorf to the present time. People from all around the world visit my site and recommend it to others. Welcome!
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!
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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.
How to clean block beeswax crayons blog post - Celebrate the Rhythm of Life
I use Sweet Orange essential oil and a soft cloth. Children love this job!
Our World as Math Lesson Plans (PDF)
first part for Grade 2, second part for Grade 3
by Laura Kryger and Dr. Nancy Smith
Booklist
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
that page
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.
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...)
If you prefer to purchase blue pouches, I suggest these
A red box, bin, pie plate, platter, Christmas cookie tin, or any other container, would work just as well. The color is what's most important.
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 100s place.
*NEW* Just found a new possibility for the number 100, particularly for the idea of skip counting by tens. It is also highlights the color red AND
is a true story!
The Firehouse Light by Janet Nolan (for SWI, this book would also work well for the < light > family).
Activity Two - Place Value Color Coding
Practice writing numbers to the hundreds place with the digits in the correct color
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
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
A really nice story to introduce this is The Autumn Calf by Jill Haukos, where they are weighing
the bison calf to see if she has reached 200 pounds before winter
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
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.
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!
We present multiplication as repeat addition. For the problem 215 x 3, you would build the number 215 three times, and then add it all up (start
with the units, in case there's trading).
For children who forget to slow down at 10 and make the trade, I've started calling it a "Ten Timeout" and making the timeout
symbol with my hand. You got to Ten. Timeout! Make a Trade. The alliteration and the physical movement of making
the T with your hands both help to lock in the habit.
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.
Thu Oct 20 - read The Devoted
Friend (PDF) by Oscar Wilde, introduce the Equals Sign (the handles of the wheelbarrow) and Ghostie, create and solve simple Ghostie Number problems
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