Designing a curriculum: essential questions for Geometry & Algebra 2

I started working at my new school last week and it's been full-speed, non-stop all the way since day 1. I feel like I'm pedaling like mad down one of the steep slopes that are so popular around here and my brakes are broken.

Yup, I live here now

We're in year 2 of a new Upper School and the challenge is to rethink the traditional Math curriculum so that it promotes engagement, interdisciplinary connections, design thinking, and deep content learning without compromising students' ability to do well on the SATs, AP exams, and you know, their future life in a potentially traditional world that may frown upon our hippie ways.

After an amazing day working with Denise Pope, a professor at Stanford's School of Ed and writer of the book "Doing School,"who basically schooled us in the purpose of education, I felt ready to tackle some curriculum writing. She reviewed some of the ideas from backwards design from Wiggins: don't start with activities, but with the desired results. Then, figure out how you will know whether the desired results have been achieved. Then create activities that lead to the desired learning. Her guidelines for selecting enduring understanding goals were:

• Represents a big idea with enduring value beyond the classroom
• Resides at the heart of the discipline
• Geared to students' interests and developmental needs
• Open-ended, complex, provocative
• Fits with school standards/expectations

So, I'm putting together my list of "essential questions" and "enduring understandings" for the integrated Geometry/Algebra 2 class that most students at my school will be taking during 9th and 10th grade. Please, please, please give me all of your feedback and criticisms and suggestions for improvement either here or on twitter (@borschtwithanna). I hope that this can be a fruitful springboard for discussion and curriculum framing.

Geometry:

• How is a system of definitions, postulates, axioms, and theorems created and made cohesive? What does it mean for something to be true?
• How can objects be transformed? Which of their characteristics stay constant and which change and why? How can we measure and describe what changes? How does this relate to concepts of proportionality and similarity?
• How are angles on a plane related to each other? How do we know when two lines are parallel or perpendicular to each other?
• How can we do algebra on a coordinate plane?
• What is congruency? How do we know when figures are congruent to each other? Why might that be useful to know?
• How can you determine whether a triangle is isosceles or equilateral? Why might that be useful to know?
• What are some special relationships for right triangles? How do they relate to polygons? Why might this be useful to know?
• What can we measure about figures? How do we measure these qualities? How do measurements change when we change dimensions?
• What relationships are formed when lines intersect circles? 

Algebra 2:

• What are the underlying principles of solving equations? How do we solve specific kinds of equations: linear, absolute value, quadratic, exponential, radical, and rational?
• How are equations and inequalities related to each other? How are inequalities represented graphically? How are inequalities solved?
• What are functions? How can they be represented and what do they represent? How can they be combined or reversed, algebraically and graphically? Why might any of this be useful?
• What are the key characteristics of the following specific functions: linear, absolute value, quadratic, exponential, logarithmic, radical, and rational? How can we represent situations using specific functions?
• What are asymptotes? How are they related to graphs and to equations of functions?
• How are functions transformed? What is the relationship between the equation of a function and its graph?
• How do functions model data? How do you know if the model is accurate?
• How can multiple constraints be represented with systems? How can systems be solved?
• How can expressions be simplified? How can expressions be combined or operated upon? How can expressions be factored? Why might any of this be useful?
• What are different ways to categorize numbers? How are the different categories of numbers related to each other? How can different categories of numbers be represented graphically? How can we perform operations on different categories of numbers?
• What are polynomials? What operations can be performed on them and how? How can they be represented graphically? How are their roots related to their graphs and equations? How can their end behavior be described and related to their graphs and equations? Why might any of this be useful?
• What are properties of exponents and logarithms? Why are they true? How are exponents related to radicals?
• How can sequences and series be described and evaluated? How are recursive and explicit formulas different from and related to each other? How are arithmetic and geometric sequences different from each other? How do we know whether a series diverges or converges? 
• What are the measures of variation and how are they computed? Why might they be useful?
• What are the measures of central tendency and how are they computed? Why might they be useful?
• How can probabilities of events be determined? How are theoretical and experimental probabilities different from each other? 

* I should add that I haven't forgotten about mathematical practices, habits of mind, etc. We are planning to assess on a common set of these that will be the same across all of the math courses. Just breaking that up into a separate post.

Reflections on Design Thinking Institute

The is my second blog post reflecting on my experience with the Design Thinking Institute I just attended at The Nueva School. My first blog post on the various icebreakers they used to get us in the right frame of mind is here.

So I won't go into too much detail of what the entire design thinking (DT) process entails, cause it would be impossible to summarize a four day conference like this. There are overviews out there: here and here, and even a TED talk about it. The basic idea is that the designers identify users, research their needs, generate many ideas to address an identified underlying need, prototype one or more of these ideas, get feedback and refine the prototypes, reflect on the feedback and prototypes and make improvements.

Short Version

Long version

(from http://designthinking.nuevaschool.org/dt-diagram)

You know it's design thinking when there are a lot of post-it notes involved.

Or:

Post-its danced through my dreams by the end of the conference.

The idea for how this applies to education is that this process can be taught to students as a way of framing long-term projects that have application to the real world. For example, students can be asked to design a new playground for their school, design a utopian society, design a catapult, design a way to conserve water, design a way to promote peace in the world, etc. Ideally, projects would connect several disciplines and also tie into something to which the students are connected or care about, allowing them to develop creativity and ownership of the solution process. Other key aspects of the design thinking process emphasize empathy (both for the product's users and for teammates when working with others on a project), thinking outside the box, learning by doing, persevering past obstacles, valuing intuition and informal approaches to learning, and communication skills.

Design thinking in schools looks kind of like this. 

I found it to be a really exciting way to frame projects that is flexible enough to accommodate many different curricula, grade levels, and students. It's easy to use aspects of the process, such as researching or prototyping or reflecting on your progress or brainstorming ideas, and have students focus on learning how to do one or more of them as part of one's curriculum. Ideally, studens would eventually go through the full process to get the full benefits of this type of approach. I also feel like I learned a lot about motivation and how to structure projects while still giving kids independence and ownership of the process. Seeing kids present their ideas with confidence and passion really sold me on the benefits of design thinking as a model. I also see a lot of overlap with the Maker Movement.

I was personally challenged during this conference by having to actually get my hands dirty and make things, which I am not used to doing. There are definitely a lot of connections to math teaching in terms of how we want kids to approach open-ended problems, trying things out and reflecting on how they're working, convincing others that their approach works, and the focus on perseverance and ownership. I talked with a few math teachers about the word "prototype" - it can just refer to "hypothesis" or "idea about an approach or answer" and that in math, we're constantly having students prototype, test, get feedback, revise, and improve.

I do have some hesitation about the fact that part of the design thinking process is that neither the teacher nor the students should have a predetermined solution (or set of solutions) that they are trying to reach. Authentic design thinking is supposed to be about the process and can't be directed towards a known goal. This is a bit tricky in mathematics, where my curriculum IS driven by specific content and I write problems and tasks with particular mathematics content in mind that I want to explore or teach. Not to say that I never give very open problems, but that can't be the case very often or we will not accomplish the learning objectives of the course. In addition, requiring the existence of actual users and a product of some sort constrains the topics to ones that have real-world connections. While I value real-world connections, my favorite math is the kind that is purely a construct in our minds and beautiful because it has absolutely nothing real about it. I don't want to lose sight of mathematics that is abstract and removed from real-life considerations. Abstract math can still lend itself to deep investigations and some aspects of design thinking, but I'm not sure that it can be fully incorporated into the design thinking framework.



Resources for Design Thinking:

• General design thinking resources
• K12 wiki page from the d. school
• More information about design thinking methods from the d. school

The Nueva School is also working on creating a website that will organize some of the resources out there and create a space for teachers to upload their design thinking projects for feedback and use by other teachers. I'll update here when their site becomes available.

Icebreakers that teach group norms - accept failure, learn from others, work through frustration

I'm attending a four day conference at The Nueva School on design thinking this week. I'll blog in a bit about the main content of the conference (done!), but I first wanted to get some thoughts down about the activities we did on the first day to build comfort with each other and establish the group norms that would be really vital for the design thinking process. These are centered around empathy for others' points of view, celebration of process over product, and learning through repeated cycles of trying/failing/changing. The activities that we did on the first day of the conference did a great job of both helping participants get to know each other and create a safe space, but also to set up these key aspects of learning without which design thinking isn't possible.

Activity #1:
We started with fun improv games to get people over the fear of doing something silly because hey, we're all doing something silly.

Come get your imaginary points:

We reflected throughout and were explicitly asked to make space for people who hadn't participated as much yet, which I thought was a good thing to do when doing this with students. Basic scenario was: 3 people in the middle of the circle acting out a scene, someone from the audience taps one out and changes the scene. For the last go-round, instead of tapping out, each new person added to the scene, which was a nice way to get everyone included and participating.

Activity #2:
This was perhaps my favorite thing of the day. I really wish I'd gotten a better picture of this, but the basic idea is that each person gets a sheet of paper with two concentric stars, a mirror in a stand, and a piece of cardboard with a cutout for their hand to fit through. Their job is to draw a third star in between the two concentric stars while only looking at the reflection of their hand in the mirror, not at their hand directly.

It looks kind of like this, but if you cut a hole in the cardboard, it's a little easier to draw without looking. But seriously. This activity is amazing for dealing with frustration and failure and getting people on a level playing field. Everyone is just awful at it and it's just about trying over and over again and letting go of perfection. I've also been told that this is a great activity for people who want to know what it's like to be a dyslexic since the entire premise of the activity is fighting the image you see and reinterpreting the visual cues from how you think they look. A great activity to have kids reflect on frustration.

Activity #3:
Two people face each other where only one of them is facing the screen. The one who can't see the screen has paper and a pencil and the one who can see the screen directs the drawer to replicate the image on the screen using only words, no hands (we were told to sit on our hands). I've done activities like this with students because it's really helpful for reflecting on what makes for good communication and how important it is to take the other person's point of view into account (for example, when using directions, like "up" or "right side"). This also deals with frustration, but frustration directed at another person who either isn't understanding you or isn't communicating clearly enough for you to understand and how to work through that.

Activity #4:
This was the first building activity. We were instructed to build the strongest bridge we could out of a single sheet of paper. It had to go across a bin that was about 9 inches across so that meant we had to use pretty much the entire length of the paper. We were given metal nuts to test the strength of the bridge. And go!

People's first attempts looked a lot like this:

It was really cool for me to work on a task that was so open. As a math teacher, I am used to having a lot of constraints that I need to first analyze and understand, lots of equations and thinking done in the abstract and on paper before I'm ready to actually test it with any sort of real-world application. I am very comfortable with tinkering and trying ideas in writing, but I haven't really ever done projects where you make things first. I felt very unmoored and outside of my comfort zone, but I think that it was really good for me because that's how students often feel. It was scary to do something where I had no idea how to start. This is a great activity for helping kids be comfortable with repeated failure and with just jumping in and trying things, as well as for how to look at what others are doing and integrate different ideas or build on each other's approaches. Like all of the other activities, throughout this exercise, we were also told explicitly to reflect on our progress as well as our feelings, which I think is important to remind students to do as well.

This activity reminded me of a bridge-building homework project we had when I was a student in high school. We were given a large number of toothpicks and told to build a bridge that had to withstand a certain amount of weight. But we had only those materials and no more so there was no chance to build prototypes and test, no room for failure. I spent a ton of time on my bridge, but it fell apart immediately on the day of testing. I still remember the feeling of frustration and self-criticism... I had no idea what went wrong and there was no chance to learn from my mistakes and go forward. I decided that I just sucked at making things, which has taken me years to even start to undo. For kids who are perfectionists and fear failure, it is so important to give tasks with low entry points and where failure is accepted and even celebrated, where it's the only way to make something worthwhile. I know that we say this a lot in education, but so many of our tasks and assessments are still one-shot deals. You did it wrong this time = you don't know how to do this. The more opportunities that we can create that are iterative and incorporate feedback and reflection, the more we teach to a growth mindset rather than just giving it lip service.