Monday, September 14, 2009

Exercises in Zero - Nothing and Placeholding - Part 1

Long time readers of this blog may remember my earlier post on the empty tray as a placeholder. In my training, we were told to have the child take the work tray from the shelf, remove all of the items onto an oil cloth or place mat and then return the tray to the shelf. The empty tray served as a placeholder preventing children from misplacing another tray in the spot.

This misplacement of trays is common in environments where the child keeps the tray at the table with them while doing the work. The empty place on the shelf seems so in need of something that a child places a different tray there and walks away. When the first child finishes his work and attempts to return it to the spot where he found it on the shelf, he sees that there is now another tray sitting there. So this first child stands there with a tray looking for somewhere to put it and yes there is a place. It is where the second child's tray should have gone and so the first child puts his tray there instead. Soon, either the lead teacher or the assistant comes by and notices that once again trays have been misplaced. This generally continues for most of the first several months of school.

Returning the empty tray to the shelf is an easy solution that indirectly presents placeholding. Later, when the child receives their first lesson on zero as a placeholder in the math area, they will recall this experience with the empty tray.

(Note: I am aware that many AMI environments limit the use of materials other than the Montessori materials. I, living on the East Coast, have most often worked in AMI/AMS environments. It is somewhat like that cuisine referred to as East/West Fusion. In AMI/AMS schools a collaborative relationship is maintained between teachers of both trainings. The classroom reflects this collaboration. Adding additional math materials to the shelf is up to the individual lead teacher and perhaps the head of the school)

My first placeholding exercise is for the younger students in the environment. This work is presented to a child who has had a lesson on using a hole puncher, has done that work many times (found on the shelf along side cutting work) and has had a lesson on how to use a glue stick. Also, the child has worked with both the spindle boxes and the cards and counters. Therefore, the child can recognize the symbols 0 - 5 and their quantities. Lastly, they may have played the zero game reinforcing that zero is nothing.

(See my post on musical chairs and zero: )

Setting up the tray:

In a beautiful box I placed a small bowl to collect punched circles, a glue stick and a few small pieces of colored paper to be used for hole punching. This box was placed on a slightly larger tray that held several half sheets of paper. Each sheet had six large boxes marked off and inside of and at the top center of each box was a single number from 0-5.

The work itself:

The child punches out several, construction paper circles and collects them in the small bowl. Next, they glue the correct amount of circles under the specific number in an individual box. They do not glue any circles in zero marked boxes. See the series of photos below of a child doing the work:

When she repeated the work, the second sheet did not have a zero included.

I have found that young children really enjoy this work and that the careful manipulation of the small, construction paper circles adds to their ability to later build a bead tower with the colored beads - also circles but three dimensional. Too, they are similar in size in regards to their circumference - the construction paper dot and the colored bead.


I have tried using small sticker dots but they stick on the children's clothing and they eliminate the hand strengthening and manipulation work that the hole punching provides.

Saturday, September 12, 2009

Exercises in Placeholding and Place Value - Part 2

This second placeholding / place value work is for a slightly older and even the oldest child in the class. This math activity is also an arts and crafts project. It has been a favorite amongst my students and their parents. I have had more than one parent say that they loved writing out a figure and having the child place the correct beans in their Placeholder Dragon. That's right, a Placeholder Dragon. I found this lesson in an old math magazine I picked up at a garage sale. Each year, they turn out different as each year I have different supplies left over from various art projects done the year before.

This past year, I made a really fancy one for myself. I gave a workshop on placeholding for the annual Montessori Schools Of Massachusetts Conference. Yet, I have seen some pretty amazing ones made by my students. It is a fairly simple project. Have parents bring empty cardboard eggs cartons to school. Be sure to send out a note saying you have all you need when you do or they will keep coming. Prepare the egg cartons for use by removing their lids and cutting them in half the long way so that there are six egg cups on each.

(photo credit:

For the head, cut one of the egg cups from the rest leaving five cups connected.

This single egg cup is glued at an angle in the first of the remaining connected egg cups. It should be positioned so that the top, open part of the cup looks like an opened mouth. This is the head of the Dragon.

For a tail, I cut long, thin pieces of construction paper and curled them with a pair of scissors. These I stapled to the inside of the last egg cup.

I used more curled strips of paper to make a fiery tongue.

After the head and the tail were assembled, I labeled the remaining four cups - Thousands, Hundreds, Tens, Ones. The thousands cup is directly behind the head/mouth of the Dragon. The ones cup is directly in front of the tail of the Dragon. In the photo below I used a permanent marker, but I have also used pre-cut, small rectangular labels that had the words thousands, hundreds, tens and ones written on them. The students were given one of each and then they glued them in the appropriate spot.

Now all that remains is to decorate the dragon. I give the children artistic freedom here as long as they don't cover the quantity labels. Suggested materials include feathers, beads, colored markers, gold markers, poms poms and any other decorative items you might have on hand.

My only caution is that if they are heavy items, even pom poms, they may need stronger glue. Last year, my assistant used a hot gun glue to help secure some of the items in place.

Here is a photo of the one I made for my workshop:

If you think this may be a project that looks great but you aren't sure that your students will be able to put all of these pieces together in a time frame that suits your classroom needs, then do what I did once. I cut and glued the head/mouth in place, I labeled all of the quantities and I put the tail on for each of my students so that before the project was presented I had a completed Placeholder Dragon for demonstration and I had partially assembled dragons for all of my students. What they did was decorate them and use them for math. Do what works best for you. If dragons just aren't your thing, have your students make a caterpillar or an inch worm place holder.

How to use the Placeholding Dragons -

On strips of paper write a single four digit number. Make sure to include several numbers that have at least one zero. Place these strips in a basket. Fill another basket with beans. Now, select one of the pieces of paper with a number on it. Ex. 9,520. Next, place 0 beans in the ones egg cup, 2 beans in the tens egg cup, 5 beans in the hundred cup and 9 beans in the thousand cup. Empty the cup and choose another number. This work can be done with two students. One places the beans and the other checks the work and then they switch.

I have the children take their dragons home with them along with a baggie that has several beans and 6 or so slips of paper with a four digit number written on each.

Art and Math, its a good thing....

Thursday, September 10, 2009

Metal Inset and Fraction Work - Don't Forget the Frames

I have observed children in many different classrooms doing metal inset work using only the insets having left the metal frames on the shelf. In my training, we never left the metal frame at the shelf. Both the inset and the frame were meant to be brought to the table together.

The child is suppose to first trace the inside rim of the frame without the inset.

The child then removes the frame, positions the inset over the initial outline and then draws the outside rim of the inset. When the inset is lifted there are two outlines - not one.

Regarding the empty space of the frame, this space is a visual reference for the child when they are drawing around the rim of the inset. If the child looks up for a moment from their work, they see before them the image of the shape they are tracing. Often a child's hands cover the majority of the inset's shape. I have heard many children say out loud to themselves something like, "Ellipsoid. Okay, now I remember," when they are doing metal inset work and forget for a second what shape they are tracing. They stop in the middle of the work, hold their hands still, visually check the frame's empty space (which echoes the shape they are tracing) and then complete their work.

A second work that I have observed children leave half of on the shelf is the fraction circles, specifically the metal, Montessori fraction circles. This material is both a sensorial and a math material. Four year olds should be using this work to simply manipulate the material and position/reposition the fractions into the formation of a whole or 1.

In regards to its use as a math material, the frames need to come to the table or floor with the fraction insets or a major element of the work is missing. Maria Montessori built into these materials zero. When the child lifts the whole circle representing 1 from its frame what remains is an empty circle or more specifically zero. Don't leave the zero on the shelf. Also, each time one of the fractions is removed from its frame the negative or empty space represents the absent quantity.

If the metal insets didn't need the frames to be brought to the table then why have them and their space occupying shelf? The same could be asked of the fraction materials. If the frames have no significance than why purchase them? The answer is of course that they do have a significance. They are part of the presentation/lesson.

Final Comment:

In the photo above you can see that both the inset and the frame where brought to the table by the child. But, (and I have talked to both AMS and AMI trainers about this and they agree) look at how the child's wrist is lifted up onto the rim of the tray. This lift in the wrist is maintained while the child (not seen in photo) bends their wrist down so as to lower the pencil to meet the paper. This is not a proper positioning for a child to write or draw. The metal inset work is a preliminary writing material. This lift in the wrist is actually a similar positioning to that made by adults that work at older computer keyboards. Many of these adults suffer from a wrist injury commonly known as carpal tunnel syndrome. I never write on a tray. Do you?

Also, the materials in the photograph are too crowded and the child does not have ample room to use them. Metal inset work should not be done on a tray. And yes, I too have seen the lovely metal inset work trays sold by various Montessori material companies.

But, at the AMI Centennial Conference in San Francisco, the trainers warned teachers not to be seduced by unnecessary and, at times, inappropriate materials. Believe me, I have purchased my fair share of similar materials at various conferences. Some of these materials are wonderful extensions to the Montessori work, but others do not serve the child and their development.

Next winter when the AMS National Conference is in Boston, I am sure I will be right up there at one table or another eyeballing all of the Montessori candy. If you see me there remind me of this post...

Saturday, September 5, 2009

The Sensorial Materials Are Scientific Materials: A Testimony (Revised)

I was in my first year of teaching. I had graduated from my training only a few months earlier. I had also just moved from Minneapolis to Boston. It was a period of enormous change in my life. Still, I was very excited about becoming a lead teacher and having my first classroom.

Unfortunately, I wasn't greeted as warmly as I had hoped by the parents of my new school. Even though I was a published author, had spent ten years as a non-violence educator in a domestic violence program serving men who had been arrested for the crime and was starting the fourth decade of my life, the parents were openly concerned about having a "new" teacher lead the classroom. I cried a lot at night and on weekends during the first several months at the school.

As I had moved far from my training center and my trainers, I had no on-going mentoring. The head of my school was also working as a lead teacher in the second class. She was very busy juggling her administrative work and her classroom work. We really did not talk much.

But because I had just graduated, the lessons resonated within me. It was almost like I could hear the words of my trainer speaking to me in the moment as I presented work for the first time. It was an enormous comfort. Too, phrases like, "Never underestimate a child's creative intelligence" repeated themselves over and over again in my mind like Montessori mantras. This too was comforting. Finally, I had my albums which I carried with me like corporate employees carry their briefcases. They were my life line.

During my training, the Sensorial materials were presented as scientific materials designed for a scientific classroom. The color tablet boxes could be used to determine if a child was color blind. The thermic bottles could determine numbness or hyper-sensitivity in a child's hands and fingers. The sound cylinders/boxes could assist in determining issues regarding a child's hearing.

Each material has a built in abstraction and a built in scientific mode of measurement. These are two of the reasons that they are only to be used (initially) for individual presentations as each child responds uniquely to them. It is with these materials that so much knowledge about an individual child is gained.

It was work with the Sensorial materials that changed my relationship with the parent community at the school. I was re-presenting the sound cylinders/boxes to a young four year old. I had first presented the work to him a few weeks earlier. At that time he was able to matched all of the cylinders/boxes correctly. What I was seeing during this follow up presentation was quite different. He matched the loudest and the softest quickly but mismatched all the others. I smiled and invited him to put the work away. While he sat having snack, I wrote two pages of notes based on my observations.

The very next morning I spoke with his mother when she dropped him off. I mentioned that I was concerned about his hearing and that I had noticed a recent change. As she was still not warmed up to the "new" teacher, she did not seem to think it was a big deal. Since he didn't have a fever, she felt confident he didn't have an ear infection. She suggested that I might have made a mistake as I might be more exhausted then I realized, having started a new job after a big move. Her final suggestion was that I do the work with him again in a couple of days to make sure my observations were correct. I respectfully agreed that repeating the work in a few days was a good idea.

After observing the same response again from the child using the sound cylinders/boxes (not being able to match most), I approached the parent a second time. She told me that he was going to the doctor's that afternoon for a routine check-up and that she would ask the doctor to check his hearing. She also stated, "Maybe he isn't really interested in that work. He likes math. You could do more math with him instead." I remember the head of my school watching me from down the hallway as this parent spoke. I knew that my observations were correct so I thanked the mother for agreeing to have his ears check.

The boy's father brought him to school the next day. He was in a hurry and didn't say anything about the doctor's visit. However, at the end of the day, his mother stood outside my door with a bouquet of flowers. When I looked at her, a tear ran down her face.

"You saved my son's hearing. The doctor said he could have gone partially deaf or even worse. I am so sorry I doubted you," she said as more tears ran down her face.

She explained to me that she had taken him to visit her grandmother two weeks ago. She said that her grandmother used a hearing aid and that her son had seen her place the hearing aid in her right ear many times.

During their last visit, he watched her change the battery in her hearing aid and then put the aid back in her ear. Apparently, the grandmother left the old battery on the table within the child's reach.

What the doctor found when he looked in his ear was the old battery. It was wedged deep inside of the little boy's ear and took several minutes to remove. After examining the battery, the doctor noticed that it was corroded and leaking. He explained to the boy's mother that it could have caused brain damage if it had remained in the ear and had continued to leak.

After she explained all of this to me and thanked me four or five more times, I told her it was the Montessori materials that revealed the change in her son's hearing abilities and that I would not have been able to recognize this loss without them. After this conversation, the entire parent population grew much warmer towards their "new" teacher.

For me, it was an amazing confirmation that the Montessori classroom is a scientific environment. After the child's mother left, I sat alone in my classroom and wept. My heart wasn't sad. It felt instead overwhelmed by the emotions of joy. I was so glad I had taken my training and become a Montessori teacher.

(*Revision note: I received an email from a reader informing me that my previous copy of this post had a few spelling and tense errors. My apologies. I was writing on the run. I was trying to get a post in before driving my son Ian back to college. He was asking "How much longer?" and "Why can't you write that later?" I just felt such a strong urge to post the story about the hearing aid, the battery and the little boy, a pivotal moment in my first year of teaching, that I wrote quickly and checked spelling later. Again, my apologies. The above copy is hopefully 99% error free - I am sure there is one more that I didn't catch...Susan Dyer)