Block 2

SCROLL DOWN

Block 2

Expect to read the entire gamut of classroom activity, from "big picture" overviews to detailed reflections on one day's work. Posts are authored by faculty.

An introduction to our Block 2 theme, which puts social and emotional learning into practice.

In Block 2, we turn our mathematical attention to algebra.

We're writing code and making art.

We're focusing on collaboration, and our school library is the byproduct.

In this math class, we emphasize the geometry behind algebraic symbols.

At Proof School, we communicate the value of the fundamentals.

Students are learning that limits are set by one's creativity, not by "the way things are."

Students compared world views ranging from Aristotle to Hubble.

For our mid-December activities, we focused on community, kindness, and service.

"In the classroom" posts.

Our Block 2 theme

Our Block 2 theme

On Friday we kicked off our Block 2 topic of “Kindness and Service” with a Flex Friday devoted to discussions of and activities about compassion. Students began by writing out their own definitions of compassion and expressing their ideas out loud to the large group. Then we looked at someone else's definition of this concept, that of Pema Chödrön, a famous Buddhist nun and writer:

“Compassion is not a relationship between the healer and the wounded. It's a relationship between equals. Only when we know our own darkness well can we be present with the darkness of others. Compassion becomes real when we recognize our shared humanity.”

After pulling apart Chödrön's meaning here and discussing the nuances of this definition of compassion, students brainstormed how the concept of gratitude might be related to that of compassion. They suggested that being grateful for the things we receive in our lives helps us to be aware of our “shared humanity” and interdependence—that we are all needy, or “wounded,” in various ways throughout our lives, and that other people need our help just as we have needed theirs.

Reflecting on this idea of gratitude, students were then guided through a short meditation, called the Empty Bowl Meditation, the purpose of which is to open ourselves up to all the things we receive from world and from other people. We were all skeptical of how well it would work, but the Proof students pulled it off with maturity and grace: a roomful of 44 silently meditating kids!

After their meditation, students broke off into smaller groups to consider how to take compassionate action in a variety of specific scenarios. For example, “Your sister didn’t make the basketball team. You did make the basketball team. Ever since she got the news, she has been snapping at you—saying mean things. Now she is crying in her bedroom, with the door closed. How can you act compassionately in this situation?” The scenarios were purposely challenging, and the students often struggled to come up with what the most compassionate response would be.

Students certainly rose to the occasion, however. Here are a few of the ideas they jotted down during one of the moments for reflective writing:

“The difference between compassion and pity is that compassion is feeling with someone and pity is at someone.”

“I learned than compassion can be expressed by doing little things and big things. Compassion is when you can put yourself in their shoes and try to help them.”

“Compassion is being the same as someone else, giving them comfort in the fact that you are with them”

“It's good to be compassionate, but there has to be a limit. For example, you can't and you don't have to give all your money just to be compassionate.”

“We are compassionate to others even when others are 'bad.' We do it because we are all humans. It doesn't matter how bad they are, we are all equals [….] I have learned two things today: compassion starts with me, and I'm not doing enough.”

*-- Sydney CochranNovember 1, 2015*

In the classroom

In the classroom

One of the immense advantages of giving our students a chance to delve into this subject every year is that by senior year they will have a much clearer sense of how broad and rich this subject is than many college graduates. There's a lot more to algebra than the quadratic formula! (Although this can be a gem also, when viewed in the right way. My Algebra II students just learned all about it this past week.)

During block two our math courses are Algebra I, Algebra II, Number Systems, and Ideals and Varieties. The first two encompass a traditional middle/high school algebra curriculum, although taught "Proof School style." The third course introduces students to the algebraic structure within the most well-known number systems (rational, real, and complex numbers) along with a few fascinating but less common examples. Finally, ideals and varieties is a college level introduction to polynomials in several variables with a view towards algebraic geometry.

I am pleased to report that our system of identifying students' backgrounds and placing them in an appropriate class was extremely effective this block. We made a (very) few adjustments to class rosters after the first day or two and now feel confident that each of our students is in a math class that is nicely suited to their level and interests.

From what I hear the math courses are all off to a fine start this week. Best of all, there was a significant amount of remixing of students between the previous block and the current one, which gives students the chance to work alongside new classmates and leads to a more cohesive community overall.

*-- Sam Vandervelde**November 1, 2015*

Introductory Python is a course for students with minimal to no previous coding experience. So, in keeping with tradition, we began the class by writing a "Hello, world!" program, which consists of one line of code in Python:

print("Hello, world!")

Much of my previous teaching experience has been in C++ and Java, and I am growing increasingly appreciative of how much easier it is to write such programs in Python!

Then students learned how to accept input from the user in Python, which is equally straightforward:

name = input("What is your name?")

print("Hello, " + name)

To practice these ideas, students either wrote a "Madlib", or a unit conversion program. I try to make the programming exercises in this class as open-ended as possible, to cater to students with a wide range of previous coding experience. One of my students wrote a Madlib with 22 different requests for input from the user, and another student wrote a conversion program with 12 different options (one of which was "miles to inches").

Moving on to loops and conditional statements, the main project for the unit was to write some sort of gambling game, which could be played over multiple rounds, and which used some sort of random process to decide whether the player wins or loses in each round. Among the games written were roulette, a slot machine simulation, and a program which plays "Rock, Paper, Scissors" against the computer.

Right now we are wrapping up a unit on Python turtle graphics. The main project for the unit was to create a "Moire pattern", which consists of many copies of the same geometric shape, drawn inside a loop, and with one or more properties of the shape changing each time it is drawn. For example, you could draw a sequence of circles of increasing diameter, or a sequence of congruent squares, each one rotated a bit from the previous one's location.

*-- Steve Gregg*

In the classroom

In the classroom

This block in Language Arts 1, we are focusing almost exclusively on learning through collaboration and peer interaction. Such an approach to the classroom purposefully shies away from individual accomplishment but achieves so much more class-wide. What’s more, this approach focuses on the key dispositions that make learning possible over the long term: listening and communicating to others, navigating differences of opinion, and discovering internal motivation to tackle large, challenging problems. Our focus this block therefore not only underscores Proof School’s mission and axioms but also our theme of community, compassion, and kindness.

Since we have accomplished so much, and have learned even more, this post for "In the Classroom" is long. But I hope you will read it to see how we are approaching social-emotional learning in the classroom--and what’s possible when we rethink education to approach learning in a 21st century context.

**Class-Wide Achievement**

Before we jump into the process of how this is coming alive in our classroom, let’s get right down to some preliminary results. In the first three weeks of the block, our Language Arts I class accomplished the following:

- We created an interactive timeline of the US Library of Congress that involved researching its history; creating multiple databases of information and media; writing journalism ledes and headlines; fact-checking information; and a rigorous revising process for both accuracy of information and creative direction. It’s still in progress.
- We sorted, categorized, and organized our growing library of hundreds of books, and developed an in-house, simplified classification system and a process by which to check out books.
- We invented and embedded a secret code into our classification system. I would tell you what it is, but that would be giving it away! Suffice it to say that the code is complex but still decipherable, and it leverages the mathematical instincts of our students. This code will become a key theme in next year’s Building Things with Books class.
- We researched and vetted nine digital library systems, including web-based and app-based programs that can help us manage our books, make the check-out system easy, and track our holdings. We are now choosing among the top three as recommended by students: LibraryThing, Libib, and Home Library.
- We built a dozen bookshelves and anchored them to the walls; set up seats and step-ladders; and outfitted our new library with task lighting.

While I think these accomplishments speak for themselves, I think they deserve some conscious recognition: we have done more in three weeks of class--as a class--than what is individually possible. In fact, we have accomplished more in three weeks as a class than what is imaginable.

**Class-Wide Process**

So, how did we do all this? The short version is that we worked collaboratively using principles of project management, especially SCRUM.

This means I consciously shed the role of “teacher” to play the role of project manager. That doesn’t mean I didn’t teach, or that we didn’t learn. In other words, we adapted SCRUM for a classroom environment in which we learned by doing.

The basic principles we applied to the classroom included:

- breaking down each of our five accomplishments into component parts;
- forming teams to work on these parts simultaneously and interactively rather than sequentially;
- appointing team leaders who brought questions and answers to other teams; and,
- bringing the teams together for frequent, quick check-ins and class-wide presentations and feedback.

Breaking tasks into component parts gave students wide latitude in choosing what they did. Even in making the timeline, we formed different kinds of teams at different points of creation. At one point, students joined one of four scrums: Engineering (coding, database creation, software selection); Creative Direction (visual rhythm, typeface selection); Content (writing, researching, editing); Fact-Checking (sourcing, annotating, and documenting). Because we reformed scrums on a daily basis, students had daily opportunities to change the nature of their involvement.

Some students discovered in the process that what they thought they would like doing ended up being not as fun, while others returned to their original choices. I worked with kids each day on different kinds of dispositions. We talked about persistence and commitment in the middle of activities, but we thought about other activities they could choose in the next class. We talked about broadening their experience of the class on day three if repeated an activity on days one and two. We were working, in other words, on identifying the way choices and self-motivation played an important role in their learning. Over the next three weeks, we will pay more attention to talking and writing about the motivations behind their choices.

**Bringing It All Together: Reflection as a Mode of Learning**

Thinking of the instructor as a Project Manager rather than a teacher challenges certain assumptions we often make about education, but it only underscores the core foundations of learning itself. By that distinction I mean our activities the past three weeks have looked little like a school classroom, but students have learned about themselves as a community and more about their individual way of learning.

The question for me is the following: how do I help students become aware of their learning? Arguably, an important distinction between doing and learning, between the office and the classroom, is the degree to which we are cognizant that we are, in fact, learning.

We’ll do that in our classroom through reflective writing. After the Thanksgiving break, students will write and revise an essay in which they address their own contributions to the class and their own learning process. They will describe both challenges and accomplishments, and their respective dispositions and the experience of learning. The essay will be each student’s opportunity to reflect on what they have been doing as a way to consciously address the process of learning.

For students who have remained engaged in the process of doing, the essay will be their chance to take a step back and realize what in them was motivated by the projects, so they can tap that motivation in the future. For students who at times felt adrift or challenged by the activities, it will be their opportunity to dig into themselves to discover what they could have done, and what they need, to feel more engaged. I want all of us to realize that we have something to learn from our experience, no matter our experience.

When I wrote, some time ago, that Proof School builds social-emotional learning into the classroom, I had in mind something akin to the activity of our Language Arts 1 class these past three weeks. From my research into social-emotional learning, I knew that using activities to test and express our dispositions toward learning would materialize into an amazing achievement. I just didn’t know it would materialize so quickly, nor result in so much.

Please join me in saying “bravo” to our Language Arts 1 students!

*--Zachary Sifuentes*

In the classroom

In the classroom

Ideals and Varieties asks students to think geometrically and reason algebraically. Algebra, even at a college and graduate level, can feel like intricate symbol manipulation. In our class, we emphasize the geometry behind the symbols.

In one problem from class, we graphed the equation y^{2} - x^{3} - x^{2} =0. We avoid using computers to do our graphing for us. Instead, we think about the symmetries of a function, its similarities to well-known functions, and what solutions it has when we plug in specific numbers for x or y. This equation is the nodal cubic, which looks like:

In a calculus class, students might learn to parametrize this equation by manipulating x and y until they find a relationship between the variables. In our class, we never set out to solve something algebraically without first thinking about the geometric picture. The geometry of the nodal cubic tells us which algebraic manipulations might take us to a parametrization. This allows us to both understand more deeply why it is possible to find a parametrization, and also avoid spending an hour bashing through algebraic equations. In this case, students realize that almost any line through the origin will intersect the nodal cubic exactly one more point. By looking at lines through the origin of varying slopes, we sweep through the entire cubic, giving us a nice pair of formulas, y = t^{3} - t and x = t^{2} - 1.

In Ideals and Varieties, students are developing a toolbox of concepts which are applied to many different fields of math. These concepts include basic definitions from algebraic geometry, such as fields, n-dimensional affine space, ideals, and varieties. However, they are not specialized concepts: they appear all over. During the second half of the course, each student is writing a paper on a topic related to our class. While writing the paper, students are realizing the connections between the tools they learn in class and big areas of research in math. They are also developing their expository writing skills and learning some of the conventions of writing papers in mathematics.

Over the next few weeks, we will continue our study of multivariable polynomials, and prove the Hilbert Basis Theorem, a foundational result in algebraic geometry. We will study Groebner bases and solve the ideal membership problem, wrapping up some of the major questions we have been asking all block. By the end of the course, students will have a mastery of basic algebraic geometry, as well as a related topic which they have chosen to write their paper on. Students are doing truly impressive work and engaging with hard problems every day.

*--Sachi Hashimoto*

Anyone who has taught calculus to undergraduates will tell you that the main thing these students struggle with is not calculus itself, but basic algebra. Algebra I is where many students fall behind, and without a solid foundation to build on, they never manage to recover. I therefore feel an enormous sense of responsibility teaching this course: all the mathematics that my students will ever do, at Proof School and beyond, relies on what they are learning with me this block.

If the hardest part of calculus is algebra, then in some sense the hardest part of algebra (or at least Algebra I) is arithmetic. One has to understand how numbers work at a very deep level in order to manipulate algebraic expressions. For instance, in order to multiply x² + 3 by x³ + 4x² - 7, one has to understand the distributive law and the laws of exponents. In order to compute

one has to truly understand integer factorization and fractions. Of course, one can also learn how to do these things by memorizing a bunch of rules without understanding; this works in the short term, which is why so many students graduate from high school convinced that mathematics involves moving symbols around according to a set of arbitrary procedures and formulas. But in the long term, this strategy fails: by the time they get to calculus, these students can no longer manage the keep the increasingly complex sets of arbitrary rules in their heads. In contrast, I want my students to know -- to truly believe, in their heart of hearts -- that the symbols in mathematics always stand for something real (in Algebra I, it's actual, honest-to-goodness numbers) and that the rules for manipulating them always reflect this reality. If you want to know whether a certain manipulation is valid, you don’t need to consult your notes; you just need to consult your understanding of reality!

To drive the point home, I told the class a story about a friend of my family who, while traveling in a remote region of Russia in Soviet times, had a chance to observe a math class in a small village school. He was shocked by what he heard: the teacher was telling the students that to add fractions, you just add the numerators and add the denominators. After class, our friend politely explained to the teacher that he was a mathematician from Moscow University who happened to be visiting the area, and that the correct way to add fractions was, in fact, quite different. He sat in on the next class to see what would happen. The teacher told her students: "Children, there is an important man here from Moscow. From now on, we have been instructed to add fractions in the following way..."

It took a short historical aside to help the students understand the humor of the story, but they loved it! I hope this anecdote serves to remind them that mathematics possesses a truth that is not based on the authority of a teacher, or a visiting mathematician, or even the Communist Party of the Soviet Union. You never need to memorize arbitrary rules; you just need to understand what is really going on.

To a certain extent, such understanding has to come from experience, so a lot of Algebra I consists of practice, practice, practice. I tell the students that it's like learning to play a musical instrument: to get anywhere with it, you have to practice your scales. However, you can't learn to love music with scales alone, so I’ve made sure to intersperse fun activities in our class as well. We’ve had an "underwater scavenger hunt" (our class meets in the Fish Tank, which got decorated with origami sea creatures during the last Build Week) and sent each other secret messages encoded with linear equations. I often tell anecdotes from the history of mathematics to help make the subject come to life. But what the students seem to love most of all is when I give them a peek at more advanced mathematical concepts. (It turns out that Proof School students are always eager for more cool math!) For instance, while talking about Euclidean distance, we also discussed the taxicab metric, in which circles are squares. And as an application of the rules of exponents, we learned about the Ackerman hierarchy and got acquainted with some numbers that are so big that they make your brain want to explode.

Many of my students hadn't finished prealgebra in their old schools. This class has therefore been extremely challenging for them, as it amounted to leaping over a year or more of the middle-school math curriculum. However, being Proofniks, the students all rose to the challenge and never lost their good humor. Now that they've started their journey toward abstract mathematical thinking, there is no turning back. Of course, the journey does not end here: they will be practicing the skills they learned in this class for months and years to come. Still, I am honored to be the one to have given these kids their first glimpse of the great art of al-ğabr wa’l-muqābala.

--Mira Bernstein

in the classroom

in the classroom

To me as a mathematician, one of the most amazing things about my field is its continual openness to new ideas. Math isn’t just about following a set of logical rules to their conclusions; it’s about making up new rules and seeing if anything interesting happens. What if 1+1 were 0? What if negative numbers had square roots? What if we changed the definitions of addition and multiplication? All of these childlike what-ifs turn out to lead to beautiful and useful areas of mathematics—different from everyday math, but consistent within themselves. One might even say that math is more art than science: the limits are set by one’s creativity, rather than by “the way things are."

When I set out to create a post-Algebra II course for Proof School kids, my goal was for students to experience some of this creative freedom—and learn how to follow through from the what-if phase to the investigation phase.

In part, we’ve been doing this by retracing the history of the numbers, trying to see familiar things as they looked when they were new.

There is a diagram which many students have seen in their algebra classes: a circle inside a circle inside a circle inside a circle, showing how the set of whole numbers fits inside the set of integers, which fits inside the set of rational numbers, which sits inside the set of real numbers, and so forth. This diagram doesn’t just convey a simple fact about sets—it tells a story about the history of mathematics! Moving out from the center, each circle represents a newer and richer concept of number, a hard-won expansion of what math is able to speak about. The very names of the newer types of numbers—negative, irrational, imaginary—convey the “outsider” status they once held relative to established number systems.

We’ve worked our way outward through this diagram, figuring out what flaws needed to be corrected in the old number systems and how the new systems accomplish this. We’ve also looked at the axioms, or “ground rules”, for each number system, and learned how to use them to prove what we thought we knew: that 1 > 0, or that the product of two nonzero numbers can’t be zero, or that every integer has a unique prime factorization. It was quite a challenge for many students to cast off assumptions and avoid circular reasoning! Yet this is a vital task if one is going to explore the more exotic realms of mathematics, where all of the above “facts” can fail to be true.

Along the way, we learned that some “familiar” number systems can be much weirder than they appear: for example, students were shocked to figure out that most real numbers cannot be printed out in any form by a computer program, or even described in English.

We’ve also created some of our own unusual number systems—like a system in which numbers are geometric distances, and arithmetic is done not by writing columns of figures, but by drawing with a compass and straightedge. One student is on a mission to create his own number system where 0 = 1!

Each week has had its own disparate theme, but as the course has progressed, we’ve started to see some unexpected links between different areas of math. For example, a tool we originally used to divide complex numbers turned out to also help us find fractions approximating √2, draw right triangles with integer sides, and figure out which integers are sums of two squares! The class is learning that when they explore their own what-if questions, the discoveries they make may help answer other questions as well.

--Austin Shapiro

In the classroom

In the classroom

This represents a very large span in time and intellect. We explored how our current understanding of the model of the universe can be understood in hindsight, and only by standing on the shoulders of Copernicus, Galileo and Newton. The students experienced planetary motions by enacting Ptolemy’s models, and consolidated their understanding of motion with figuring out why the ‘sweet spot’ on a baseball bat really is so sweet, at least for the batter’s hands, not the pitcher!

We described motion in many forms. Noticing that parabolas popped up everywhere, we decided to create some parabolas of our own. We launched LED projectiles across the room and captured their trajectories with long-exposure photography. We cut out carpet sections in bizarre geometric ways, placed LED lights at the center of mass, and marveled at how they traced parabolas too!

A lot of the ideas of Big History and what we have been tinkering with in experiments came together. Aristotle was not to be bashed, we learned, just because his ideas were at odds with Newton’s. We learned by dropping paper cones and observing that two stacked cones actually do fall faster than a single cone – heavier bodies do fall faster! Within Aristotle's observational powers, his ideas were indeed plausible at the time--shedding light on how scientific theories develop and evolve.

*--Kaushik Basu*

In the classroom

In the classroom

We began the week by reading from a philosophical text about the pleasures that kindness brings, and we talked about whether they thought kindness might be opposed to success in some way. Students also decided on specific kind acts they would do at home and at school during the week, to test their hypotheses about kindness. Over the course of the week each grade group focused on a different aspect of kindness and performed a related service project: kindness in our school, kindness to the earth, and kindness within our larger community. We all came back together at the end of the week to discuss our week practicing kindness, as well as to reflect on our attitude at the school more broadly.

**6th Graders**

The sixth graders focused on a theme of everyday kindness. We discussed ways of being kind in our everyday lives, at home and at school. We thought about the many small kindnesses and luxuries that we benefit from each day. For one afternoon, we became authors of advice columns—putting ourselves in others' shoes, practicing empathy, and articulating advice for others facing tough situations.

Tasked with developing projects that would demonstrate kindness to the rest of the school, the sixth graders created

- a Kindness Wall, where students and staff can publicly acknowledge the kindness of others
- an elaborate snack stand that received many enthusiastically customers during Thursday lunch
- a scavenger hunt for the rest of the school (which was unfortunately cleaned up before we could do the hunt!)

**7th/8th Graders**

Our 7th and 8th graders focused their community service project on the environment, with a field trip to the Presidio to clear an area of fallen bushes and trees after a recent wind storm. At one point, we were trying to move a 30-foot tree that wouldn't budge; the park rangers suggested we go to lunch but our crew was relentless. They simply declared there would be no lunch until we moved that fallen tree—and we did.

The best moment of the week, however, was on the way back on a MUNI bus. It kept filling and filling, and about halfway back to Proof School, a class of first graders boarded the bus. Our Proof Schoolers spontaneously got up and gave the little kids their seats. I was quite proud of their service, but I was even more proud of their awareness of others and their kindness. It was truly a moment of leadership.

**High Schoolers**

The high school students focused on kindness within our larger community, learning about issues of poverty and homelessness. They read an excerpt from Barbara Ehrenreich's Nickel and Dimed and discussed the challenges faced by minimum wage workers in the U.S. and the extreme difficulty of pulling oneself out of poverty.

This led to a hands-on activity in which students simulated the unequal distribution of wealth and power in the world and had the chance to brainstorm and then vote on proposals to re-distribute these resources in some way. Students also learned about what some activist and altruist organizations are doing to help. For their service project, students volunteered at the Glide Foundation, handing out bags of groceries to the needy and impoverished individuals who live right in our school's neighborhood.

Lastly, one student took it upon himself to plan, develop, and make this gift for the entire Proof School community: a work of anamorphic art. He asked that his gift be anonymous, and for everyone to enjoy it!

*--Sydney Cochran, Kathy Lin, and Zachary Sifuentes*