Sunday, April 5, 2020

Concrete Math Materials for Assessing & Learning

For the past three weeks, I’ve been away from my Montessori learning environment (and my students) due to the COVID-19 pandemic. Since I am feeling nostalgic for the hands-on didactic manipulatives that help children learn without abstract formulas, I want to describe why these materials are so wonderful. (I last wrote about Montessori math materials on this blog over seven years ago, in case you want to search under the Math tab on the right hand side to learn more!) The following are Montessori math materials most relevant to the youngest learners in a lower elementary classroom, the 6-year-old first graders.

100 board
This material is often introduced in primary (age 3-6) Montessori learning environments, and I also find it a helpful assessment of number sense for first grade students and children new to the Montessori lower elementary classroom. The hundred wooden tiles sit randomly in a two-column rectangular box and fit neatly on a board with a hundred blue grid boxes. Children often enjoy doing this work in partners, although it can also be done independently. One way of giving this lesson is to ask the child to find all of the numerals that end in zero, then all the numerals that end in five – and lay them out on the board first, followed by other numbers per teen.

I have observed some very linear children like to lay them out in order, some children group them by teen, and some children who enjoy just taking their time with it choose tiles at random and find where they go. There is no need for an adult to intervene – by showing or asking the child where the tiles go. This is an intuitive experience that is only disturbed by interruptions and interventions. All Montessori materials are naturally self-correcting.

Golden Beads numeral composition
Another Montessori math material used frequently in primary classes that is also very useful as an assessment for new lower elementary students is the set known as the Golden Beads. One of each – a unit bead, a golden ten bar, a golden (or wooden) hundred square, and a golden (or wooden) thousand cube – sit in a rectangular presentation tray. Montessori guides (teachers) lay out a Golden Mat (actually usually colored green with place value columns) and write, on a board or piece of paper (or using wooden numeral tiles), a four-digit numeral. Children take an empty tray to the shelf where the Golden Beads are stored and retrieve the correct amount, sometimes all at once, and sometimes very slowly, one place value at a time. There is even a little golden dish for the unit beads to sit in.

The child places the Golden Beads in the correct place value column and reads the numeral aloud when they are ready. This lesson assesses number sense and place value sequencing, as well as demonstrates the child’s spatial balance and math-language abilities.

Infinity Street
The first official Math lesson I do with first graders and children new to the Montessori lower elementary environment also reinforces place value understanding. Infinity Street is not an original lesson created by Maria Montessori in 1907, however it was introduced in my Montessori training. I have used it with children for 15 years with great success. It takes awhile to do, mainly because it has a few steps, a lot of coloring, and requires some fine motor skills.

Infinity Street is a basket containing twelve yellow houses that are in ascending order of size, much like a two-dimensional set of nesting dolls. Each house comes with a label (Simple, Thousand, Million, etc. up to Decillion, the largest house in the basket – not the largest possible number family!) and a yellow “mailbox” comma to put between each house. There are three “doors” on the front of each house – green for units, blue for tens, and red for hundreds – as each house contains these three place values. In the first lesson, the Montessori guide tells the story of Infinity Street and invites children to help read or lay out the labels and commas.

In the first extension (maybe a day or two later), the children work in partners to lay out the houses, labels, and commas in order. Size is the control of error on the houses, and the labels and commas have number keys on the reverse if children need to check their work. In the second extension, children lay out their house and trace it onto a long swath of butcher paper. They write the label name for the number family (such as the “surname” Trillion), and color the house yellow and each of the doors green, blue, and red. They even draw the comma “mailbox” between their house and the next child’s. At the end, we display Infinity Street on a wall so all children can access it for reading large place value numerals!