## Imagine IT Phase 3

This academic year I will be teaching an advanced math class at Lindblom, affectionately called Kick-Ass Mathematics, officially called KAM. This is a class for seniors in high school who have either finished Calculus or are concurrently taking Calculus. Similar classes have existed in other parts of Chicago, but this is the first time such a class will be offered at my school. Thus, I have significant freedom to create a class that I believe will best embody the transition from high school mathematics to post-secondary mathematics.

Math can seem like such an imposing subject with many rules, procedures and algorithms to follow. There’s such a long history to it! It’s no wonder that it seems that way when we try to distill thousands of years of mathematical work (from Plato’s arithmetic to Newton’s Calculus) into a dozen years. Of course, a skilled mathematician will be able to solve several algebra, geometry and calculus problems. But how did those problems come to exist? Who created them? How do we discover mathematics? What does that even look like? I see these questions at the heart of my work this year, because I believe that students will be more likely to think mathematically when they see it as something that they can create. Moreover, when students can see that the math they create can be used constructively by themselves or other to create even more math….well that just seems exponentially wonderful.

In order to cultivate this in my classroom, I will introduce content from mathematical theories like number theory, graph theory and Euclidean geometry. This will be the framework that we use for mathematical growth. An essential knowledgebase for students to build and share their work will be the language of mathematical proofs. This is an essential skill for the social aspect of math, as well as creating math. To show understanding, students will present their work in a visual and written form for the class as well as outside of class to the broader public.

With the future in mind, I hope that when students leave my classroom, they will be able to create and share their mathematics. I hope that they will be able to take all the external facts, axioms, the building blocks of a theory and use those as tools to create an enduring theory that is not only mathematically sound but also aesthetically pleasing. I hope that students will be able to present their fully formed thoughts and explanations as proofs that can be published to the class, as well as perhaps a wider audience. I hope that students will be inspired to dig further into the mathematical universe when they leave my classroom.

Clearly, if my big idea for ImagineIT centers on student-created mathematics, a performance of understanding will have to include…creating mathematics!

Before we begin that process, I believe it’s important for the class to measure where they stand when it comes to mathematics. What to students think higher-level math is? What are some moments where they really felt connected to mathematics? How did that feel? And what about the inverse, with moments where they felt a real disconnect? What are some other passions that they have? Are there some mathematical connections? I think these sorts of questions are great to pose at the beginning of the year, end of the year as well as some times throughout the year, as a form of reflection.

As we continue throughout the year and explore different math theories, it will be important for students to track their progress. In fact, let’s extend this further and recognize that it’s important for students to give constructive criticism to their peers as well. Then, after feedback, they would be ready to present and we’ve taken a small step towards the culture of collaboration common in the “adult” mathematical world. Students will assemble a portfolio of work (proofs, explanations, solutions, etc). My role will be to help students use appropriate math tools in class as well as outside of class so that they can solve new problems. In fact, students might even be able to organically create extensions or generalizations of theorems from class and explore those as part of their portfolio. On a personal note, I really appreciate it now that I had a teacher who set that expectation in one of my math classes, because now I can look at all that previous work.

The development of the portfolio is a weekly ongoing project, but another level of understanding comes with a capstone project. I expect students to take a topic from class (or even something tangentially related) and expand upon it, work out some new math, solve some examples and present their work to the class not just as a lecture, but as an actual class. I want students to develop something new and I want to learn something from their work. This is something we have to scaffold over several weeks, but I think this would be a fantastic use of class time and a real culmination of math creation, collaboration and growth.

I’ve talked about how I want students to create and share mathematics and a few possible sources of evidence that they are creating and sharing mathematics. It is time to talk about how I plan to do this. Let’s TPACK.

**Identify Desired Results**This academic year I will be teaching an advanced math class at Lindblom, affectionately called Kick-Ass Mathematics, officially called KAM. This is a class for seniors in high school who have either finished Calculus or are concurrently taking Calculus. Similar classes have existed in other parts of Chicago, but this is the first time such a class will be offered at my school. Thus, I have significant freedom to create a class that I believe will best embody the transition from high school mathematics to post-secondary mathematics.

Math can seem like such an imposing subject with many rules, procedures and algorithms to follow. There’s such a long history to it! It’s no wonder that it seems that way when we try to distill thousands of years of mathematical work (from Plato’s arithmetic to Newton’s Calculus) into a dozen years. Of course, a skilled mathematician will be able to solve several algebra, geometry and calculus problems. But how did those problems come to exist? Who created them? How do we discover mathematics? What does that even look like? I see these questions at the heart of my work this year, because I believe that students will be more likely to think mathematically when they see it as something that they can create. Moreover, when students can see that the math they create can be used constructively by themselves or other to create even more math….well that just seems exponentially wonderful.

In order to cultivate this in my classroom, I will introduce content from mathematical theories like number theory, graph theory and Euclidean geometry. This will be the framework that we use for mathematical growth. An essential knowledgebase for students to build and share their work will be the language of mathematical proofs. This is an essential skill for the social aspect of math, as well as creating math. To show understanding, students will present their work in a visual and written form for the class as well as outside of class to the broader public.

With the future in mind, I hope that when students leave my classroom, they will be able to create and share their mathematics. I hope that they will be able to take all the external facts, axioms, the building blocks of a theory and use those as tools to create an enduring theory that is not only mathematically sound but also aesthetically pleasing. I hope that students will be able to present their fully formed thoughts and explanations as proofs that can be published to the class, as well as perhaps a wider audience. I hope that students will be inspired to dig further into the mathematical universe when they leave my classroom.

**Determine Acceptable Evidence (Performance of Understanding)**Clearly, if my big idea for ImagineIT centers on student-created mathematics, a performance of understanding will have to include…creating mathematics!

Before we begin that process, I believe it’s important for the class to measure where they stand when it comes to mathematics. What to students think higher-level math is? What are some moments where they really felt connected to mathematics? How did that feel? And what about the inverse, with moments where they felt a real disconnect? What are some other passions that they have? Are there some mathematical connections? I think these sorts of questions are great to pose at the beginning of the year, end of the year as well as some times throughout the year, as a form of reflection.

As we continue throughout the year and explore different math theories, it will be important for students to track their progress. In fact, let’s extend this further and recognize that it’s important for students to give constructive criticism to their peers as well. Then, after feedback, they would be ready to present and we’ve taken a small step towards the culture of collaboration common in the “adult” mathematical world. Students will assemble a portfolio of work (proofs, explanations, solutions, etc). My role will be to help students use appropriate math tools in class as well as outside of class so that they can solve new problems. In fact, students might even be able to organically create extensions or generalizations of theorems from class and explore those as part of their portfolio. On a personal note, I really appreciate it now that I had a teacher who set that expectation in one of my math classes, because now I can look at all that previous work.

The development of the portfolio is a weekly ongoing project, but another level of understanding comes with a capstone project. I expect students to take a topic from class (or even something tangentially related) and expand upon it, work out some new math, solve some examples and present their work to the class not just as a lecture, but as an actual class. I want students to develop something new and I want to learn something from their work. This is something we have to scaffold over several weeks, but I think this would be a fantastic use of class time and a real culmination of math creation, collaboration and growth.

**Plan Learning Experience and Instruction**I’ve talked about how I want students to create and share mathematics and a few possible sources of evidence that they are creating and sharing mathematics. It is time to talk about how I plan to do this. Let’s TPACK.

__Context__: I work as a high school math teacher at Lindblom. Lindblom is a selective enrollment high school in West Englewood, which means students test into the school. The school itself has a rich history (almost 100 years) and active alumni. Students come from various neighborhoods in the south side; the student body is 70% African-American and 25% Latino. My class will be the most advanced math class Lindblom has ever offered, and I really have a blank slate to create this class. I am going to model this class off of my experiences as a high school student taking college-level math classes. As far as I can tell, nobody has done something like this on the Southside in CPS. We have occasional access to laptops, and regular access to calculators. From a technology point of view, I’m hoping that we can do things like video, surveys, etc using our phones.__Content__: I’ve talked in length about the content, so briefly I believe that by starting with the first formal mathematical theories (number theory, geometry, and graph theory), students will be able to access the content without too much technical prerequisites. One challenging aspect of my BIG IDEA really comes down to the minutia of acceptable mathematical evidence. What constitutes an airtight proof? What assumptions do we simply have to make? I once took a formal mathematical logic class where an entire lecture was spent examining the framework of truth and numbers so that we could finally add numbers together. WHAT!? Clearly we are not going down that rabbit hole, but we will also need to accept some level of formalism. I think we shall need to spend time examining proofs that have small flaws in order to gain a deeper understanding of proofs and mathematical truths.__Pedagogy__: I think the best approach here is to start with the mindset of doing as little of the mathematics as I possibly can in order for the class to run. I’m inspired by a class I took in college, using the Inquiry Based Learning approach. In this class, the teacher provided a “script” of definitions, theorems, conjectures, etc. It was scaffolded in such a way that it was possible to solve later content using previous content. The teacher didn’t proof any major theorem on the board; it had to be done by students. I hope to emulate this as much as possible, and in fact I think I can even extend it so that students can show new math on the board as well (as part of a longer project). In addition, I’m inspired by the public framework from MSUrbanStem. If the students are going to do the bulk of the mathematics, let’s broadcast it! With videos or blog posts, I will include assignments where students share their work with the public at large. I will wait to see what the class prefers, but there will be a public space. In terms of collaboration, I think students will naturally work on a problem together in class, but I also want to include a component where student are able to give constructive criticism on written work. I am thinking that I will make a copy of a student’s written work and give that to another student; with a reasonable framework (similar to the note-taking for our Amazing Stem experience), I think it can be an excellent low-stakes space for improvement.__Technology__: Of course, there should be a way to include technology in a meaningful way here. I think it arises organically when it comes to sharing work. Can we create an online space where students can share their work? It could be just the class or even the public at large. Wouldn’t it be great to create a YouTube channel where we show our proofs? Khan academy is pretty cool and all, but just watching videos is too passive. Otherwise, I’m thinking about how to incorporate technological elements with the Genius Hour aspect of my Deep Play group. Maybe there’s a way for students to create really cool artistic visuals of the work they are doing (this becomes more in focus for me when we transition to graph theory).