This is my construction, traced onto heavy paper. Other object-to-picture works could be made by the adult tracing a 3-year-old's construction, labeled with the name dictated by the child. 4 and 5-year-olds could trace their own work, and write their own name. 5 and 6-year-olds can write a story.

Copies of the drawing can be used to do logical-mathematical analysis. I'll show a picture of that when I've done one.

For Goose, I challenged myself to use only the Box 8S pieces ( I had to put the 7's out of reach, because I kept finding them in my hand! I allowed one Box 9 point to be the eye). Then I duplicated my construction on art paper  by tracing some blocks, then when that was slow, I switched to a clear ruler. Using the ruler gave me the idea to add the dotted and straight lines. After making the drawing in pencil, I went over it with colored pencil.

Math analysis for Goose: how many 6" lines, how many 5", etc.  I haven't done the math yet, because we had a party.

For children under 2, it's best to use this as a suggestion for how adults can present the Box 1 stringed balls as a puppet show, rather than to use it as a movie for them to watch. The oral storytelling aspect is a valuable addition to reading picture books - both before the child has seen and heard the book, and afterwards. Quoting books the child knows while away from the book itself brings art to life. 

For children age 3-6, this movie is an introduction to how literature can be presented in a different medium. It also subtly shows an attentive audience, who each see themselves reflected in the theatrical production. Teachers might also find it useful to discuss how the "audience" are seated in an order conducive to appropriate behavior - because red is next to orange, it's easier to be quietly attentive, rather than the bright high contrast of red next to purple, even though red and purple are good friends who like each other very much!

Finally, this was VERY easy to make, using the video function of my iPhone. Parents and teachers could make a version featuring their own voices, or a child's voice. I'm planning to record this again for YouTube, with a native Spanish speaker, and in other languages as I find native speakers of German, Mandarin, etc.

My own memories of elementary school math are painful. In third grade, I knew I wasn’t learning math, and I just couldn’t believe it. It simply couldn’t be possible that I wasn’t smart! 
Today, we believe everyone can learn to read. However, we think only the gifted can learn math. We are comfortable making jokes about our own math lacks. Looking back, I see my not unusual experience as typical for the 60’s, when I was a child. At the time, my math struggles plunged me into a state of hesitancy and self-doubt that came to permeate my psyche. I still felt competent reading and writing, and I’ve always enjoyed occasional bursts of belief in my own genius. But the failure of my education to succeed in making me math literate felt horrible. Standing at the chalkboard, feeling scornful eyes on my back, the division problem too high and too close, it felt like the marrow in my bones was being slowly frozen.

Eight years old, I kept trying to figure out when and how I’d fallen behind. I also tried to catch up, doing flashcards at the wood-and-formica dining room table. Could I have missed six weeks of school when everyone else learned the multiplication tables? It didn’t occur to me to ask my teachers for help. If they knew my grades, and weren’t helping, it was because it was too late to put a life vest on a person who’d already fallen in the water. Math class was like a river in the Grand Canyon. Everyone else was together on a raft, being steered by the competent hands of Mrs. Gioux at the front, and the California Board of Education at the back. I had fallen off about a mile back, and was now holding onto a log mentally composed of only the 2’s , 3’s, and 5’s times tables.

When we moved to Rochester, New York, in the alternative high school, I by choice took algebra three times, got D’s three times, and finally gave up.
As a preschool Montessori teacher, the logical thinking, logical quantification, and association of numeral to quantity was math I loved and could handle. Then at an American Montessori Society conference, I encountered Mortensen math, a technique that finally, to my amazement, let me understand algebra. Alas, it disappeared from the market instead of spreading fruitfully across public and private education as I had deeply hoped.

Math is a language for understanding and expressing relationships. Like me, many adults had trouble learning math because they were taught only techniques of calculation, and that only with chalk and a chalkboard, paper and pencil. Too abstract, and too meaningless: a recipe for a dish we never get to taste.
Using objects that can be held in the hand, and learning to understand fundamental relationships, is the key to math literacy. A brain-friendly environment is also essential.  This is why I'm passionate about the Froebel Gifts - they let students use their own constructions to gradually build math skills.

Do you have memories of your childhood experiences with math? I'd love to hear them, whether positive or negative.

Here's a graph that arose when I decided to stack the Box 7 squares. It was natural to stack them by color, and structurally more sound to put stacks together into cubes. I also decided to organize the colors to match the organization of Box 8S (the smallest sticks came from the factory in two baggies per size: black, white, orange, and yellow  together, and purple, blue, green, and red are together.) I had decanted them into their wooden compartments, and didn't think about it further.

For about 35 play sessions, I only thought of those compartments as useful for telling the colors apart. I did all tallying naming individual colors, which made for some long x axis.
Then, it struck me that it's simple and useful to tally using those two color pallets - which I began doing, but only when using Box 8S. After 5 more times playing, I now see that Box 7 can also be studied using the A and B color sets.

It's a good thing I like to learn slowly!

Open-Ended Survey Question:
​What are your memories of being taught math?

Let preschool children create pictures with the blocks, and then make another version to be a "documentation". As soon as the child is old enough, it should draw or paint a version of what it built. Equally important, as the student collects pieces to put away, it is natural and inviting to count the blocks, and begin to consider them logically and mathematically.

April Showers, my piece below, could be tallied by different colors, and by different shapes of blocks, for a simple math activity.  Tallied amounts could also be added together to give the total number of blocks used.

Building on a grid, such as the gingham cloth I've used, helps with precision in building, and then in drawing a version. Don't insist that referring to the grid be part of the child's process of building. Let the student become interested in greater precision gradually and on their own. Truly, the blocks themselves direct the builder toward being more exact - you won't need to say much about this.

​To refer to the grid when drawing is a high-level skill - save it for (way) down the road. Some students (of all ages) can be very critical of their art: if necessary, be clear with the child that drawing isn't expected to be identical to the block construction. Different mediums produce different results.

At The Montessori Center, the 3-6 year olds liked the song "Friends Are Like Flowers". We would sing it at our graduation ceremony, with sign language. While we were learning it in the spring, I would carry a clipboard and take a survey, asking whether they considered themselves a daisy, a rose, or a dandelion - and found they always knew the answer easily. They could also tell what their parents were.

Survey Question:
Are you a daisy, a rose, or a dandelion?

This is pattern #8: Mosaic of Subcultures, and #13: Subculture Boundary. The colored containers with points (Box 9) of the same color represent distinctive subcultures within a city, separated by the various colors of sticks (Box 8S). Green sticks = parks, black and white = work spaces, orange = public buildings, and blue = water.

The math analysis I did on this piece is an estimate of the percentage of available Box 9 points used, followed by determining the exact percentage.

I'm interested in getting a total tally of all the blocks in all the boxes, and dividing by the cost of the set, so that I could work up a pretend "cost of materials used" for future constructions.