Monday, December 14, 2009

Math in Traditional School

As a disclaimer - if you are unfamiliar with or new to Montessori, please understand that the following is not a representation of Montessori math and none of the activities shown are part of the Montessori curriculum. This is a chronicle of my attempt to help a group of students deepen their understanding of some math concepts by bringing a tiny sliver of what I have learned as a Montessori teacher into a very traditional setting.

As I mentioned in previous posts, I have suddenly found myself teaching 4th grade, not just in a traditional school, but in a traditional school on a military base! Yikes! ; ) Needless to say, it is different.

Dr. Montessori's math curriculum is one of the most beautiful things in the world as far as I'm concerned, so it hurts my heart to teach math from a textbook.

I was just wrapping up a chapter on multiplication with my class. I had used a modified version of Multiplication with the Golden Beads to introduce multiplication to begin with. I felt the best way to begin would be to demystify the operation itself. Multiplication can intimidate 4th graders (it still blows my mind that we're waiting until 3rd or 4th grade to introduce it for the first time, but that's beside the point), so I felt that introducing it as we introduce it in the early childhood classroom might help to create a clearer sense of what the operation of multiplication really is - simply taking the same number multiple times.

I set up an "Operations Table" and a "Bank," and designated a banker for our first lessons. The only difference is that the group was much larger than I would have liked, and we were using bright orange plastic "base ten cubes" instead of lovely golden beads. This did seem to help and the majority of the students seemed to really understand what multiplication was before we dove into the text. This chapter introduced a lot of terms, and some students were able to easily memorize all of them, but as we neared the end of the unit and prepared to test I observed that many students still didn't have the foundation necessary to understand the terms that were presented (and even those who had memorized them had done just that -memorize. They didn't necessarily have a concrete sense of the concepts taught). For example, we learned about Prime and Composite Numbers, and many students weren't even totally clear on the difference between odd and even! I wish I didn't have a textbook at all, but I do; and I wish I could do everything hands on all the time, but I can't. I decided that some work with concrete materials was absolutely necessary at this point, however.

I also wish I had enough materials and space to allow each student to work individually, but instead I set up the following in small groups and we worked through the following activities over the last week.

Odd and Even
To make sure that every student really understood the concept of odd and even numbers, I used the materials we have to give every student the opportunity to do the Cards and Counters as we do in the Montessori Early Childhood classroom. Unfortunately, I can't do one on one lessons in this environment, and the students are trained to do everything with worksheets, so I had to set the activity up with written instructions and a follow-up sheet for the students to fill out.

Below, you can see the instructions, the counters (I sure wish they were all one color! They're read on one side and yellow on the other), and the "Digit Deck" which contains the numerals 1-9.

Here's a close-up of the directions:

Another way in which all of this was beneficial was as an exercise in following directions. It was surprisingly difficult for some students to complete activities step-by-step. They often got ahead of themselves and became very confused about what was supposed to be next. I may need to reword my description of the layout if I do this again as well, as most students were confused as to what I meant for them to do with the last counter for each odd number.

After completing the activity, students had this to complete:

Not very Montessori at all, obviously, but I had to work within the system we're in.

Square Numbers

Prime and Composite Numbers
I am not aware of a Montessori material for teaching these concepts, although I've never worked in a school that had an elementary program, so maybe there is one. I tried to use concrete materials to demonstrate Prime and Composite numbers as they are defined by the students' textbook. The key difference here is that the concept had already been introduced, and only after the students learned it on the surface, were they given a concrete experience with the concept. If I were not just in this position temporarily, I would definitely change the way this was done for the future. Anyway, again, instructions, materials. Not a Montessori activity, but an attempt at using Montessori principles to better teach the material this class is working on.

The Properties of Multiplication
This one is really not very concrete, and it's not really Montessori at all either, but I'll add it in anyway.

So that's what I've been up to. After 6 years of teaching Montessori 3-6 this is really an odd and trying experience, but I love the challenge. I have never seen attempts at bringing Montessori principles into a totally traditional classroom and I have no real expertise in this area - I'm just trying what makes sense to me and modifying based on what I see working for my students. If anyone has suggestions or experience with such things - I would love to hear them! I'm going to spend Christmas break doing a lot of planning for our next chapter, which will be division. Hopefully I can bring in things like this earlier!

Wednesday, December 9, 2009


I have been struggling to apply all I know and believe from working in Montessori to the somewhat strained fourth grade DoD school situation I suddenly found myself in the middle of. I adore this group of children, and I want them to learn as much as they possibly can, and I know and have seen how children learn best. It's really tough to know that the methods I'm using are not the best methods I could use, and to keep using them anyway. I do my best, they do their best, and I'm enjoying it and taking it day by day, but oh how I want to do more! I came up with some exciting ideas this week, finally, that pull from my Montessori background and I have put them into action. It has been fun! I took quite a few photos today and I'm excited to get some time to upload them and tell you all about it this weekend!

Oh, and I'm 27 weeks pregnant today -- it's a girl!! : ) Hard to believe ...

Wednesday, September 30, 2009

A shift

As any of my regular readers (if you're still out there despite my slow updates) will know, I'm taking a year off from teaching 3-6 due to a major location change. I began subbing at a K-5 school recently to keep myself occupied, and while I enjoy working with children, as always, it sure does make me miss Montessori! Having never worked in a traditional school, I underestimated how extreme the contrast would be between the two systems.

Anyway, back to the point of this post - as of now, it looks like I may be taking more than one year because I'm pregnant! Hooray! I'm just over 17 weeks along and due on March 10th (the ultrasound picture is two months old). I'm just beginning the exciting process of planning the nursery and I can't wait to have everything together so that I can set it up. I suppose the focus of my blogging may shift a bit more toward the birth-three world in the coming months, until baby is 3 and we both head back to school! I suppose we'll see. Any pearls of wisdom are welcome, as I'm not trained for this stage (not that a momma needs to be trained, but I'd still love to hear from people who are!). Give me your patterns, your favorite birth-3 quotes, books, and blogs. I want to see/hear it all!

Sunday, May 31, 2009

Another Year Over

It's hard to believe that yet another school year has passed. I hope that all of you had a year as fulfilling as mine, and I wish you all a lovely summer!

Now that the year has come to a close for me, I am in a place of uncertainty about my role in Montessori education.

I am moving to Guam next week, where it does not seem that any positions in Montessori are available. I am planning to go back to school, which makes me think that I may not be able to give a class of children all of me anyway. I am planning to approach the schools on the island and offer myself as a substitute. I am also planning on exploring the idea of providing an after school foreign language program at one or more Montessori schools - something I did in the past. Whatever I do, I hope that the schools on the island will be open to having me observe from time to time, as I can't imagine two years without being in the classroom!

I also hope to work toward completing a book about parenting with Montessori principles that I began a year and a half ago now. We'll see how that goes!

All of that said, my posts will probably be less frequent, and of a different nature. I may share observations, thoughts on philosophy, and other things I see about Montessori on the net. We shall see.

Enjoy your summers!

One final bit on color...

I wanted to share one more color extension with you. I created a paper extension (not an original idea of mine, but a good one, nonetheless), which you can see above, that is simply 7 blank color tablets in a row.

The original idea is for children to choose any color they would like and experiment with lightness of touch by using one single colored pencil to make 7 graded shades of the same color. However, as pictured above, this same paper extension can be used to experiment with creating shades or tints of various colors by the addition of black or white.

Wednesday, May 27, 2009

Still More Color Work

Above a four year old child has combined Color Box 3 with the Short Bead Stair.

Here, a six year old has traced the circle and triangle from the geometric presentation tray to
outline a color wheel. She traced one of our snack plates for the outer circle. She is in the process of tracing over her outline with a marker.

Here, she is mixing primary with secondary colors to paint her wheel.

She used the leftover paint for this painting, and gave me permission to photograph it.

My Classroom in Pictures

I am preparing for a big move, so I have been taking a great deal of pictures of my room. I'm going to miss it (but more so the children)!

I thought it would be fun to share them. The Practical Life and Art areas are missing. I don't know how I failed to photograph them, but I'm going to take some panoramic shots tomorrow before I start packing things up, so I'll post more later : )

If you have a classroom and want to share pictures of it, I would love to see them, too!

It is so inspiring to see the layouts and ideas other Montessorians have.




Cultural Studies

Lunch, some Art and EPL in the backround

The space is small, and I have so so many materials I have yet to make. It is the first year that I have been the sole Montessori teacher for a classroom, so I have been busy creating and I have quite a bit yet to do before I will feel that I have a truly complete set of materials. Then again, it will never be complete, as each group will be different.

Wednesday, May 20, 2009

More on Color

I finally snapped a picture of the color mixing work I currently have out, completed.

Here you can see it a bit more closely. I used colored Sharpies to add the control.

And, a shot of of the color wheel, built as part of a terminology lesson on primary, secondary, and tertiary colors. The triangle of primary colors (color box 1) is in the center. Around the outside of that is the wheel of primary and secondary colors (box 2), and the outermost ring includes the tertiary colors. Just to the right of that, you can see the bottles of color that the child who did this work used to make the tertiary colors, which she first mixed, and then selected a tablet from Color Box 3 to match.

Here are the tertiary colors after she mixed them all. After she finished, she used a paintbrush and applied these colors to a piece of easel paper, where she had drawn her own color wheel.

Math, Math, and more Math

I hated math for years. I struggled my way through all of my high school courses. I got the only D of my career as a student in high school Geometry. I finally passed college algebra on my third attempt. I was an A student in everything else, but math was the thorn in my side.

Montessori has changed all that. I now adore math and my understanding of it is infinitely better. The materials are brilliant and convey mathematical concepts in a way that just makes them seem perfectly natural. The materials link ideas together and help children (and adults!) connect math to all aspects of life in a concrete way. The curriculum is just absolutely brilliant.

Today, a five year old student who has recently learned the names of all of the polygons in the geometric cabinet and worked with all of the short chains, explored the connections between them.

She started by building a short bead chain stair (above).

Next, she used the chains to build a point, an angle, a triangle, a square, a pentagon, and so on. The 100 chain was in use by another student at the time.
In the background, you can see the arrows on another rug, and the book of polygons she had made earlier in the week and was referencing as she built the shapes.

In the end, she put them all together.

And finally, just when she was about to clean up, the 100 Chain was returned to the shelf, so she was able to add a decagon.

It was a morning full of math work. One 5 year old child took out quite an array of math materials and made some interesting things. I was tempted to ask him to put some of the materials away, as he was using nearly half of the math materials available at once, but he didn't seem to be disturbing anyone, so I opted to stay out of it. He first took out the 100 chain and lined up ten bead bars alongside it. Then, he took the ribbon that we use on the "operations table" to separate addends from sums, or multiplicands from products, etc. and stretched that out beneath the chain. Next, he took all of the unit cards from the large number cards tray and spread them out in order beneath the chains. Then, he took out the number cards that we use for the number rods and cards and the cards and counters and matched those to the large number cards. It was very interesting to observe. I wish I could have heard his thoughts throughout the whole process.

A 4 year old student who has spent all of this week's work time on Beads Practice and Cards Practice continued. I have never seen a child so enjoy the process of composing a numeral with the cards and going to the bank to retrieve the corresponding quantity. His favorite part seemed to be writing the quantity afterward. He has spent a full 8 hours doing this in the past 3 days.

I'm eager to see what these math enthusiasts come up with tomorrow!

Monday, May 18, 2009

Multiplication with the Golden Beads

This is the first year that I have had the freedom to present the math curriculum just as I was trained to teach it. In the past, I have co-taught and had to follow along with the way senior teachers felt things should be presented. That was no problem, of course, but it has been lovely to be able to follow the child - and follow my albums this year.

Today was officially the first time, in 6 years in the classroom, that I was actually able to do multiplication with the golden beads. I think it was even more exciting for me than for the children. Naturally, I took some photos.

Above is the "Operations Table," where each child has brought the quantity 1,321. We always cover it with purple velvet, as that is obviously the only type of material appropriate for use with the golden beads. Unfortunately, I do not have multiple sets of small number cards (or even one set) at my school. So, I used paper to write the multiplicand for each child. Of course, it is paper that the children are familiar with and have used for writing quantities with the golden beads, and it is color-coded. You can't see it terribly clearly here.

Above, the children have put all of their quantities together to find the product, 3,963, and they have brought the appropriate large number cards.

Above, you can scarcely see, but one of the children has written the final problem of the morning and the language "Taking the same number multiple times is called multiplication."

Thursday, April 30, 2009

The Color Wheel

After reading Susanne's post over at Beautiful Sun Montessori and filling out all of my conference reports, I was inspired to do some color work. My conference reports include a place to indicate whether a child is working independently with materials that teach awareness of color, and whether they have mastered them for their age and/or level. I really got to wondering what, beyond color boxes 1, 2, and, 3, really helped the children to understand color.

I toyed with some paints and worked a bit with the color boxes and a color mixing work I had available and came up with some extension ideas.

I sat down with one of my second year students today and thought I would give her a simple terminology lesson. She already knows how to mix the primary colors together to produce the secondary colors. So, I thought I would simply teach her the names for these types of colors, and show her how to build them into the shape of the center of a color wheel. This, I thought, would lay the groundwork for a later lesson on how to produce tertiary colors.

So, we sat down at a rug where she had already matched the Colors in Color Box 1. I said, "The Colors in Color Box 1 are..." and before I could continue, she interjected, "Primary Colors!" Well, so much for my imparting of knowledge, huh?

Anyway, we built them into something like a triangle, then added in the secondary colors. Then, we got out the color mixing work and started exploring the concept of tertiary colors. After mixing a primary and a secondary color together, she chose the tablet from Color Box 3 that she thought most closely resembled the color she had produced. The result was the wheel you see below.

I think I will make a paper extension tomorrow if there's time, where children can paint the color wheel with colors they've mixed themselves.

In other news, some traveling musician friends of mine stopped in and played a couple of songs for the group. We recently began a formal music class, so there has been so much interest in music - a perfect opportunity to invite some visitors! Thanks, Leftmore!

Thursday, April 9, 2009

Love the Spring!!

Spring is my favorite time of the year for more reasons than I could list, but among them is the beauty of the social life in a Montessori classroom and the exciting work that's going on!

Today was one of those lovely Spring days that completely blew my mind.

All of you teachers (and parents) are familiar, I'm sure, with the little moments that make you realize how far a child has come. Today, two children who can be rather volatile at any invasion of their personal space, or at the slightest conflict of interests, were sitting down in a very tight spot for group time. If I were able to freeze time at the moment that I saw them both head for the spot they chose, I would have guessed that some serious elbow-ing, face-making, and "I was there first!!!" shouting was on its way.

I would have been so wrong!

Oh, me of little faith. Here's how it played out:

G - "Could you please scoot back a little bit?"
J - "Sure." Scoots back. Still doesn't have quite enough space. "Could you please scoot over a little bit?"
G - "Sure."

...and all this with smiles and soft voices.

The children do, indeed, perfect themselves.

All of the excitement of the day inspired me to post some photos I've taken this Spring. It's a rather mixed bag, but here you are.

The Geometric Solids with Bases

A five year old girl is working with the geometric solids and their bases. She has stacked all of the solids with square bases on top of one another and all of the solids with circle bases on top of another.

The first time she worked with the bases, some weeks before this, having lined up the three square-based solids, she remarked, "It's like a square club!"

Birds Eye View aka The 45 Layout.

This is one of my absolute favorite works. I have to take a photo every time it's completed. This is the most recent one.

In my training (MCI), we use a purple mat for all Golden Bead work (though for this work I do eventually hope to make something with a bit of control).

Before giving the first lesson on Golden Bead material, the teacher tells the child that "Gold is what kings and queens used to barter with ..." and therefore we only work with the golden beads on the most luxurious of purple velvet, of course.

Pink Box 1
Objects with the Large Moveable Alphabet

This is a four year old's first completed work with the LMA. He was so very ready!

ai box

This is a six year old's work with one of the green phonogram boxes. She ran out of a and i, so she made some with construction paper. It saddens me that my LMA box does not have red vowels. Oh, what I could do if I had a budget with no limits...

My lovely assistant made a small sandpaper period, question mark, and exclamation point that I use to give the first lesson on these symbols.

This was an extension of that lesson where a five year old child wrote a sentence for each symbol. This was the first time I had seen this child write full sentences fluently. She was so excited! She didn't stop or hesitate once - she just wrote and wrote. She wrote so much, so quickly, that we had to bring out a second chalkboard for her to complete her sentences on. They were used side by side so she could complete one of her sentences horizontally. When she got to the end, she had to erase what had been on the top line to complete her last sentence. So, it's a bit jumbled, but if you look closely, you can read her sentence, "I laic being a big sister." Next is her exclamation, "My baib bruthr cen screm so lad!" (My baby brother can scream so loud!) And finally, her question, "R we gn do teh tasitg bodls?"(Are we gonna do the tasting bottles?") It was a thrill to behold.

Below is a cultural activity we have been working on for some time now called the Sun Game. I don't know that this lesson is included in any training other than MCI's, but it is absolutely one of my favorite things in the whole of the universe. It is prefaced by a group discussion or two on the sun. These preferably take place outside on a sunny day. You can talk about what the sun is, and what is gives us, etc.

Eventually you involve two identical plants, one of which you place in an area where it gets plenty of sunlight, while the other is placed in a cabinet or some other place with no exposure to the sun. Both are watered and cared for equally. The second, of course, eventually dies. This gives a concrete sense of the importance of the sun to plants.

Later, the actual material is presented, usually in stages. A large felt circle, representative of the sun, is laid out on the floor and cards picturing a variety of different plants are placed around it. These are coded dark orange. Around these are placed dark yellow cards featuring photos of herbivores, and around those light yellow cards with photos of carnivores. Omnivore cards can be added in as well. Half of the result is below - click on the image to see the whole thing.

: )

Now, off to enjoy the long weekend!