We’ve not given up on work this week. The uppers have been solidifying their knowledge of the Pythagorean theorem as expressed through triangles. This is like the final exam in this area of math. It tells me how much they have absorbed in the lessons and what will need to be discussed again.

LR and AV worked on theirs today. They discussed all the different things they noticed. It is always interesting to see the different ways people’s minds work.

DW is working through Rosetta Stone’s French language work.

JV worked on maps a bit. This is a map of Israel.

BW is working on understanding igneous rocks. He is writing what he has found important.

Yesterday we drew our cartones (rough sketch) based off of Giotto’s frescos in the Arena Chapel.

Today we braved the very cold temperatures to apply the arriccio (initial plaster coat). We applied plaster of paris to drywall.

After a VERY brief time! It was time to apply the cartone to the giornata (the area that can be done in one sitting). The cartone is lain on top of the nearly dry plaster, and using a stylus, we traced the drawing.

Then we laid a very thin layer of plaster over top of the giornata and began painting using tempera paint.

BW began with his background.

LR began with his main characters - Mary and Simeon.

LR found translating his sketch's precision difficult.

AR has worked her way through the nomenclature of polygons and spent yesterday creating lines of symmetry in four sided objects. Today she got to have some fun with it. This extension was shown to me by one of my trainers: Karen Simon.

AR created a square from a sheet of paper and then folded it all along its diagonal lines of symmetry. She then refolded it along its diagonal lines of symmetry to create an isosceles triangle. She wrote her name beginning ON the fold making one of the legs of the triangle taking care to touch the other fold making the other leg of the triangle. We worked together to create an outline version of her name and trace it in sharpie. She shaded it. We folded the paper so we could see her now backwards name through the paper and traced it; we refolded and traced two more times – each on the line of symmetry.

Her paper now looked like:

We went in and began to add geometric shapes in one quarter of the project. The tracing process would begin again after AR shaded the geometrical designs.

It is rather tedious work (especially with fine tipped Sharpies), so she completed half and set it aside until tomorrow.

In the afternoon, we worked on inscribing an equilateral triangle into a circle to create a symmetrical three dimensional object. She created a prototype out of an old Christmas card. She was very careful with this prototype and made a lovely ball.

She was not nearly as careful with the large version and it came out very un-symmetrical. Sigh.

The Upper Elementary guys have been working through the beginnings of proof making for a couple of weeks now. All of the layers upon tedious layers have come together this week in the creation of a formula for a ratio for a particular type of triangle.

We have discovered that an equilateral triangle may be divided several ways. One of these ways is to create four small equilateral triangles and another way is through three identical isosceles triangles. We then compared one of the four small triangles and the large triangle and figured that this is a 4:1 ration.

At that point, I introduced a new equilateral triangle – one that was slightly smaller than the larger one from before. Over several days, the children established that this new triangle was equivalent to three of the small triangles from before. It has a 3:1 ratio.

We then decided that the smaller triangle (1) plus the middle triangle (3) would equal the large triangle (4). I laid down the largest triangle and placed the other two triangles to form the other two legs of a right angled scalene triangle. I was able to do this because the children and I knew through a proof that the altitude largest triangle (T3) was equal to the side of the middle sized triangle (T2) and that the side of the smallest triangle (T1) is equal to half the side of T3. (Still confused? Refer to the photos of the Upper’s project proofs.)

Ahhhhh. Bells went off in LR’s mind. “Wait! Isn’t that a binomial equation.” I could have hugged him. Angles sang; birds danced; cows jumped over the moon.

Today we discussed the ratios of T1 plus T2 equaling T3 (1+3=4)

This is very basic when we were looking at only one triangle each. What would happen if we combined each triangle with another of its same size and formed a rhombus. Ahhhh. (2×1)+(2×3)=(2×4) – Two because there are two of each triangle in our new equation.

What happens if we add another triangle to each making a trapizoid. (3×1)+(3×3)=(3×4).

We did this pattern work all the way until we made a visual hexagon with each triangle and had this equation in our list (6×1)+(3×6)=(4×6).

Off they went to make their own versions. Making T2 is difficult; depending on age and understanding, each child found their own way to create these triangles.

There was some discussion of Mandelbrot sets. However that was quickly overshadowed by a statement that the three hexagons looked like gears. JV set out to explain that you can’t put three gears touching (sequence yes) which led us to the

Lego Antikythera (see below for a really cool video) which led us to Charles Babbage and his steam or hand powered computer in the mid-1800’s (and he had a female programer who was the child of Lord Byron), which led us to Lord Byron and the Ottoman Wars. Boy do we travel far in our random association moments.

My Advent season can not feel complete without the studied reading of the Glorious Impossible by Madeleine L’Engle with the illustrations by Giotto. I have love it for years.

I was excited when I discovered that this wonderful book was part of the Catechesis of the Good Shepherd’s work. There are several reasons I love using this book with children: 1. the writing does not condescend when read by children; 2. it makes adults in the story very real – fear, dread, awe; and 3.evil is real and part of the story to be overcome. Number three is worthy of a real post in and of itself. When Jesus comes into the world to overcome evil and death, it will become messy. Protecting children from the mess of the world does not provide them with the ability to overcome evil in their lives. It is part of the history and can’t be avoided. (That’s the short answer.)

Anywhoooo. Ginger over at the Cathedral of St. Luke and St. Paul lent us the Catechesis of the Good Shepherd work to accompany this part of the story. I love it because it uses real theological titles and real adult version Bible verses. It teaches the younger children sequencing and story progression. The older children not only discuss the narrative but also discuss the style, time period, painter, and geography which informed the art.

We have stepped in with both feet to understand the art of painting frescos and the pivotal artist, Giotto. To understand the process, Ms. JW presented a review history of Byzantine and early Medieval Art and a discussion of what was happening in Europe and the Near East during the 12th and 13th centuries – plague, returning crusaders, loss of trade routes, death of generational knowledge, etc. It is such a pleasure to hear her insights into culture and history. We then discussed the peasant, Giotto, and his new style of painting.

Today we discussed, the history surrounding the chapel and its patron and exactly how frescos have been created since the Greeks – there are great examples in Pomeii. The children were given the instruction to create a fresco in the manner of Giotto. They took off to work it out on paper – the cartone. LR cut down pieces of the dry wall we are using as our foundation. Tomorrow the layers of plaster go down and the painting is done.

One of the beauties of Montessori geometry is the introduction to the proof creation process. The children are lead through a hands-on example of the proof. Today we looked at equilateral triangles in a 3:4 ratio. The steps of logic have taken four days to build and review. Today we began to force the issue forward recalling geometry from the definitions of diagonals to bisecting angles to flat out logic even when our eyes don’t want to believe. The older uppers merrily went off to create their proof sheets while the first year uppers asked to visually work through it with me again. Finally all the pieces fell into place and they too left the lesson to create the proof.

One of the most wonderful things about Montessori is that it allows for the child’s own processes to work things out. It does not place my exactitude on any interpretation. This does not mean that the child may interpret fact incorrectly, rather that their interpretation may be what makes best sense to their processing style.