Teaching Geometric Proofs

There’s a debate about the relevance of geometric proofs and then, about the importance of writing them formally. For me, geometric proofs are more about the analysis, the support and the organization of an argument. I generally find that my strongest writers are usually the first to see how a proof ought to flow, with the rest of the class eventually catching on. After that, their written responses become more direct and they include more evidence to substantiate their answers. Even in other subjects, several of my proof-writers have begun to write more coherently. In that way, proofs are very useful.

What I don’t find useful is forcing students to memorize postulates or theorems. In years past, less invested students never had the proper names at their fingertips; without the correct property, they often wrote little more than the ‘givens’.

Last year, in a sudden burst of frustration, I decided to allow open note assessments for the unit. I even went one step further by purchasing 1-subject notebooks for the class that we would fill in together. At the front of the notebook, we made lists of theorems, formulas, angle properties, and vocabulary. Starting from the back of the notebook, students would include the proofs we practiced in class. Eventually, I learned to photocopy notes if I needed to move the class along quickly (since some of the students took forever to make their notes beautiful). Students also needed diagrams printed out for them so that a poor illustration wouldn’t throw them off as we interpreted proofs together. Towards the end of the unit, we began several classes with 5 minutes of cutting and pasting, with much more time for productive discussions then onward.

Looking back, the geometry notebook wound up being my greatest success in organizing and teaching last year. I don’t remember a class ever picking up proofs so quickly – and most of them took pride in informing me that they liked writing them. The notes lessened the stress at test-time. They also helped students focus on interpreting diagrams, gathering support, and organizing a proof – which were my key goals for the unit anyway. Better still, a majority of the class hardly used the notebooks on the final assessment, since they were so comfortable using triangle properties and the like. If I taught an advanced class, I probably would expect them to memorize theorems, organize themselves and demonstrate their understanding without any help. But for my regular level, the notebooks proved to be a great resource.

Note: This year, I will have an origami-lover in my geometry class. It’s a great opportunity to do some paper-folding proofs, but I really don’t have much experience with them. Any suggestions?

Roasted Squash Potage with Spiced Creme Fraiche

My father came to hang out with the pup, but that also gives us a good excuse to do some cooking together. Last winter, we ate a lot of this soup, which is thick and warm and not too complicated. My father had a craving and the pup is already thinking about chasing autumn leaves. So despite the hot temperatures, we returned to this soup, which came out to be as delicious as we remembered.

The recipe comes from The Flexitarian Table by Peter Berley. My brother gave me the book, and as he’s a trained cook and engaged to a vegetarian, his recommendation carried a lot of weight. I have loved all the recipes I’ve tried – but this soup is a true favorite.

The following is slightly modified from the book’s recommendations. My father likes to add sausage or bacon to every soup; he cooks it first before adding to the pureed soup. Peter Berley recommends a garnish of freshly ground cinnamon and chopped fresh parsley. We add some high-quality croutons to give the soup some crunch.

soup ingredients

3 pounds kabocha or butternut squash, peeled, seeded and cut into 2-inch chunks

2 tablespoons extra-virgin olive oil

sea salt or kosher salt

2 tablespoons unsalted butter

2 cups chopped onions

1 tablespoon minced peeled fresh ginger

3 garlic cloves, peeled

4 cups stock or water

1/2 cup apple cider or apple juice

freshly ground pepper

bouquet garni ingredients

handful of celery leaves

3-4 sage leaves

1 2-inch piece of cinnamon stick

3 whole cloves

creme fraiche ingredients

1/2 cup creme fraiche

freshly grated nutmeg

freshly ground black pepper

sea salt or kosher salt

Steps

1. Preheat oven to 400 degrees.

2. On a rimmed baking sheet, toss the squash with the oil and season lightly with salt. Spread the squash out and roast (turning a few times) until tender and lightly caramelized, 25-30 minutes.

3. In a 4-5 quart Dutch oven, melt butter over medium heat. Add onions and 1/2 teaspoon salt and cook, stirring, for a few minutes. Add the ginger and garlic, cover and reduce the heat to medium low, and cook for 15 minutes until the onions are soft. Try to keep the vegetables from browning.

4. Put the bouquet garni in a cheesecloth and tie with kitchen twine.

5. Add stock/water, apple cider/juice, roasted squash, and bouquet garni to the vegetables. Raise the heat to bring the soup to a boil, then reduce the heat to low, cover and simmer for 20 minutes. (If my father is thinking about sausage, it’s at this stage that he’ll put it in the oven.)

6. Make the spiced creme fraiche by mixing the creme fraiche with 5 or 6 gratings of nutmeg, pepper and a pinch of salt.

7. When the soup is ready, discard the bouquet garni. Puree the soup using a handheld blender or a regular blender (working in batches) – or use a food mill. Adjust seasonings. (And it’s at this stage that my father adds the cooked sausage or bacon.) Reheat the soup – and add more broth – if necessary.

8. Garnish each bowl of soup with a dollop of the creme fraiche, a sprinkling of cinnamon, parsley – or if you’re with my father croutons.

Favorite math help websites

One of my students last year began to teach himself material before we started it in class. He was clever in recognizing that he could search for a topic on the internet and find many lessons and sample problems.

My favorite website is purplemath. The explanations are rather sophisticated, but the example problems are challenging. I often use their word problems as models for my own.

In class, I use a smartboard to show students virtual manipulatives from the NLVM website. Although these may not be as useful to students at home, they can be excellent tools for understanding more complex topics. (For example, I use their algebra tile manipulative to model multiplying polynomials with 7th graders – well before they study polynomials formally.)

Salman Khan has been in the news for his video lessons that you can access through the website Khan Academy. I don’t find all the lessons to be very efficient, but they are excellent for students who either didn’t understand my presentations or who missed class.

I will continue to look for more websites – please send suggestions! – but the more important comment I’d like to make involves websites in general. Students do not need to be as dependent upon their teacher, their textbook or their tutors to build their skills. Thanks to the internet, students can find plenty of explanations and sample problems on their own. When my 8th grader figured that out, not only did he leap to the head of his class, but he also began to consider problems beyond the elementary basics. We were both very excited.