- Montessori Mathematics
article - American International Montessori School
Article about why Waldorf math is fantastic:
- Waldorf's Brilliant Approach to Teaching Math
blog post - Root Parenting
They are both wonderful but they each have their particular strengths.
Waldorf and Montessori were both developed in the early part of the last century and have been time-tested for generations in Europe. Many people don't realize that Maria Montessori was the first woman doctor in Italy and designed her classroom activities from a pediatrician's point of view. Montessori math materials are hands-down the best in the world. BUT you must be trained in how to use them. Because they are hands-on materials, they just look like a box of colorful pieces. Because the lesson is given hands-on, and the value of the lesson is in the interaction between the teacher and the child, they aren't intuitive. Because the child works on them independently and they are designed to be self-correcting, there aren't any obvious worksheets or answer keys. Because the teacher is trained by receiving the lesson hands-on as if he/she were the student and then creates his/her own binders of how the lessons are presented (like making a Waldorf main lesson book, actually), there aren't any obvious teacher guides.
So a link to a picture of a Montessori material isn't all that helpful! Pictures or videos of a child working with a material are somewhat more helpful. But the only thing that will really help you understand what's going on is if your child gives you a lesson in the material and then you get the manipulatives in your hand and start solving problems with them yourself.
Maria and her son Mario Montessori color-coded place value AND color-coded the numerals themselves. So we began with learning the color-coding for the numerals. My first Montessori math purchase was the colored bead bars (I got the set for the Checker Board so we have them when I get that later on) and I supplemented them with the colored bead bar stamps and colored pencils. Nienhuis is the only way to go in my mind for the Math. Because Montessori is engineered to be so mathematically precise, DIY Montessori isn't feasible and lower-quality brands cut corners. Sloppy materials don't help you to see the patterns and rhythms and beauty in mathematics.
Montessori materials allow you in a hands-on way to do things like multiply to the billions place WITH manipulatives. I remember when I got my M.S. in Curriculum & Instruction, one of my classmates was making a dismissive (and very loud) comment about how hands-on materials weren't possible with larger numbers because they obviously wouldn't fit in the classroom. I couldn't believe it. My mind was numb on so many levels. 1) his dismissive attitude that hands-on math is only for little kids. 2) his complete lack of education in progressive educational philosophies such as Montessori yet he was taking master's level classes. 3) his lack of imagination and creativity... he could only imagine hands-on math with large numbers as being done with large bulky materials that couldn't possibly be moved around. It's so clear to me how these things can be done because I've seen it and done it, but he couldn't fathom it at all.
Geometric Hierarchy of Number
There are materials TO SCALE up to a million in the Montessori classroom (pictured above... which show the color coding for the place value as well as the logarithmic progression in size) but there are also in between steps that slowly take you more and more into the abstract. This is the really brilliant thing about Montessori materials. They go in many little stair-steps between the concrete and the abstract so that the child has lots of time to internalize the concepts. Then when a child is in 3rd grade or so, he or she is ready to move into completely pencil-and-paper math and what is recorded on the paper will MAKE SENSE because the child has worked with the concepts of quantities and operations so many times and SEEN what is happening by manipulating the materials, that the black and white marks on paper are simply the most abstract representation of the math problem. And the child is fully ready for that at that time.
THIS IS THE KEY: If you just jump to abstraction (anyone remember having a terrible time learning long multiplication????) math just seems like nothing but tips & tricks.
How many adults do you know who can't tell you WHY long division is done the way that it is. They were told to just memorize the steps. And that doesn't actually teach anybody math! It infuriates me when people don't take the time to teach a child.
So mathematically and philosophically precise and respectful of child development... that's Montessori. Where does Waldorf come in?
Well, Waldorf emphasizes creativity, art, and storytelling and incorporates mathematics into that approach as well. In Waldorf first grade education, for example, students hear stories about squirrels collecting nuts for the winter or gnomes digging up jewels in their quest for treasure. The squirrels or gnomes are required to find a certain quantity and each character in the story (addition, subtraction, multiplication, and division) has a different way of going about it and has a hard time in some way, either having too much or too little, and then having to adjust their quantity to the right amount. You can use this storytelling approach with Montessori too. And the colored bead bars and bead chains (these are color-coded quantities of beads strung on wire and hinged so that you can fold them to literally make the square and cube of a number) can be made of glass -- as Maria Montessori first had them, since she was in Italy where there were beautiful glass beads -- if you're looking for all natural materials.
In Waldorf students also march to the rhythm of poems or verses that have an emphasis on different syllables, thereby mimicking the skip counting. There's no reason why you couldn't do that before, or after, or while, doing a bead chain. In the Montessori bead chain work, you lay out a chain of the colored beads (the short chains are the square of the number, the long chains are the cube of the number) and are given a whole little set of tickets, each with a multiple of the number chain you are working with. The student lays the chain out carefully on a mat (the thousand chain, the cube of 10, is usually done in a hallway), carefully counts the beads, and puts each ticket down at the appropriate bead. So the short chain for three would have tickets for beads #1, 2, and 3, then 6, and 9. The long chain for three would have tickets for beads #1, 2, and 3, then 6, 9, 12, 15, 18, 21, 24, and 27. This is an early introduction to multiplication.
Short Bead Chains and the Squares
Montessori has students work with large problems early on, since Maria Montessori felt that students were fascinated with large numbers and big ideas. Waldorf tends to be smaller because you are trying to count out all those acorns! The colored bead bars are a simple material that can be used for small quantities as well as large ones.
In our classroom, we pretended that each of the colored bead bars was a collection of different gems. One is red so that is obviously rubies. Two is green so that is emeralds or jade. Etc. You can have Waldorf gnomes gathering gems and still be doing math with colored bead bars. What could be simpler than that?
As we continue to find connections between Waldorf and Montessori, I will share them. I know these two philosophies are considered to be worlds apart but I think it is important to bring the best things you can find to children, and to be trained in a variety of approaches and be able to have enough flexibility to teach a child in the way he or she learns best. Teachers should be trusted and treated like we treat our first responders. Highly trained before they begin, provided with continuous ongoing training in their profession, respected, and fully allowed to assess a situation in real time, exercise their judgement, and do what they think is needed.
Ok, soapbox moment aside.
In the first few days of this week, I showed the students the straw bale cold frames I had built over the weekend. I've planted green and red lettuces, golden beets, carrots, and radishes. We also cut down the remaining weeds, laid down cardboard weed-blocking paths in the garden, watered the seeds, checked for germination, and monitored the temperature outside to see if the windows needed to be put in place (not yet). We continued our read aloud chapter book, a long fable from China, and learned a haiku as our poem. We practiced skip counting by 2, 3, 5, and 6. We had a lesson on our final philosopher for Nature, Baruch Spinoza, and had a stimulating philosophical discussion about his idea that everything in the universe is one substance which manifests itself in an infinite variety of ways! We also brainstormed fundraiser ideas for our endangered animals. I demonstrated how to use the salt tray for penmanship practice as well as meditation. We debated whether we should have a classroom job chart. Handwork continued with lots of sewing, weaving, finger knitting, and knitting. One older student did a difficult form drawing exercise.
In Physics, my older students explored the Sources of Sound. Our study of Physics is a phenomenological approach, which focuses on demonstrations and experiments and observations in lieu of formulas. After each lesson the students are to complete an "exit slip" with one question they still have, inspired by our lesson. Our focus is on the habits of mind which make a person into a scientist. (A cat is a cat. But you would observe it differently depending on the lens / academic subject you were using. If you were writing a poem about a cat, measuring a cat, or doing science experiments on a cat, you would be doing very different things. You would look differently, question differently, write differently. There is only one world and the "subjects" are really just different ways of looking at it.) They also sat in on the biography of Leonardo da Vinci, since our main lesson books this block are inspired by his notebooks.
Physics is Fun! A Sourcebook for Teachers
In Great Inventors, my younger students heard several stories including Beautiful Oops!, the tiny book-cover-inside-flap biography of the author of Girls Think of Everything and her "Swim Fishies" invention (and the students gave some thought to what they would like to invent), and Neo Leo, a book about our first inventor, Leonardo da Vinci. They also did a measurement activity to see if his observations on the proportions of the human body (discovered through many countless hours of dissection of corpses in his quest to understand human anatomy) proved accurate. Ms. Sherri took them across the street with chalk and meter sticks and rulers to lay on the blacktop and trace one another and measure. Did you know that the length from your elbow to the tip of your longest finger (the Egyptian cubit) is 1/4 of your height?