Joy Ride: An Introduction to Riemann Sums

This year, I tweaked the order in which I teach integration a lot. Since we begin to talk about differentials when we talk about linear approximation in the applications of differentiation unit, it seemed natural to me to flow directly into differential equations and indefinite integration. Instead of leading with the area problem, we worked on general antiderivatives and differential equations. Since my students are now familiar with moving between functions and their antiderivatives, I am now starting to move into area applications to introduce definite integrals. They are starting to ask about "going backwards" from acceleration and velocity naturally, instead of having to force the area problem somewhere that it seems unnatural. This will also give me time to re-visit u-substitution since I got to teach it without having to worry about changing bounds in our indefinite integration unit. 

For the last 3 years, I've taught in an entirely project based program. The switch back to regular ed has been an interesting one for me, as I can see the ways my pedagogy has changed but don't always have the time, resources, or freedom to implement these changes. I knew when I was given the opportunity to have a 2 hour block for a project instead of giving my AP Calculus kids a midterm that I wanted to jump on the opportunity. I adapted this Gorilla Jump activity from MAA (which is an amazing activity if you've never seen it!) into a project that would require more data collection. It was a real challenge for the kids, but they seemed to walk away with the big idea and were asking the right questions (even if they didn't have all the answers yet). 

Students worked in small groups using this Driving Simulator (made on Scratch from MIT) to generate velocity data over even intervals. 

They were free to decide what intervals to use and needed to be mindful of units as most were measuring in seconds while velocity was in miles per hour. This simulator also has weather and varies the time of day, so I had my kids react to these variables so it would affect their velocity. Some drove responsibly at 68 mph on the highway. Some just accelerated as fast as possible the whole time.  I know who to watch out for in the school parking lot now. 

Using that data, students generated estimates for total distance travelled using the lowest velocity on the interval, the highest velocity on the interval, and the average velocity on the interval. They were asked to represent these estimates graphically, which naturally leads to a rough version of a Riemann Sum. In addition, I asked them to challenge themselves to see if they could write an equation (using sigma notation if they were feeling extra fancy) to represent how to generally find the total distance travelled. This was frustrating for the kids, for sure. I used it mostly as a pre-assessment to see what they remember from the previous year on sigma notation. (Verdict: we've got some serious work to do there). They were able to discuss upper and lower bound and postulated that collecting data over smaller intervals would lead to more accurate results. All in all, it led to extremely positive conversations and I think they'll have a very solid foundation when we attach formal names and notation to these ideas next class. 

Joy Ride Project

A few notes of things that jumped out at me, having implemented it once: 

• Since units were in mph and sec, there was converting needed. The kids didn't struggle with this, but it caused the outputs to be extremely small decimals for some groups. Just something to keep in mind.
• A lot of kids discussed ways you could have changed your driving to make your estimate more accurate instead of changing your data collecting methods. While a valuable conversation, some kids got lost on a tangent here and struggled to finish in the allotted time. 
• Since there was no requirement that their velocity function be monotonic, using the lowest or highest velocity on the interval did not lead directly to a right or left hand Riemann sum. I know that's something that won't be a hard jump for my kids next class, but again it's something to keep in mind. 
• The kids got really cranky having to make 3 of the same graphs by hand. If you have a way to photocopy them quickly, you'd have a lot fewer cranky 17 year olds on your hands. We didn't at the time (I wasn't in my classroom and we had limited time). 
• The car has an odometer. It would've been interesting to copy down the actual mileage of the trip and see how far off we were, especially since we're talking about error. Missed opportunity, Gironda. 
• There's got to be a cooler final product for this. Again, I had a limited time from in which to do this with my kids, so posters were concise and manageable. I know this could get pushed to be a lot better. 

Let me know what else you might do to make this better. I liked it enough and it was worth doing, but would like to improve it for next year! I'll update with finished products from the kids when I get them! 

Blended Learning: Lessons Learned

One of my biggest regret after moving districts this year was that I never sat down and put all the lessons I'd learned as a blended learning teacher together in one place. Luckily for me, at the end of last year I got to sit down with the people at Opportunity Culture, the group that had been instrumental in helping my district move towards blended learning. They put together this vignette summarizing my experiences and my best advice to new blended learning teachers. 

Pioneering Blended-Learning Teachers Reach More Students Vignette Series
A vignette written about my experiences with blended learning in Pre-Calculus. If you want my lessons learned (often the hard way), read this! 

I'm so thankful to have had this experience and to have had it documented! 

Hey! I said that! 

Blended Learning & Design Thinking

Being a STEM teacher makes you think about things you've never considered...a lot. I've spent more time considering rations you'd need to live on Mars and the location of the focus of our parabolic planetarium at school than I'd like to admit and generally these mental aerobics have made me a stronger thinker and teacher. This has been a direct result of my time spent in STEM and life was not always so in my previous job. One thing that was usually about as far from my mathematical mind was the life of people like my friend Jon, who is a graphic designer. I've always loved looking at his latest projects for clients and he would always explain the client's reaction to it. He even designed the layout of the cornhole boards I made for my wedding, offering me 3 different options and allowing me to give him feedback on what I liked best before he sent me the final design. Seems like a different world than mine own (or at least it did at the time). 

Cue dramatic music and study the picture here:

STEM Design Cycle

This is the design cycle by which we live and die in my STEM program. It is a process that as adults we move through almost without thinking- figure out the problem, come up with a idea, see how it works, get some feedback and reflect, and then modify to make better. It wasn't until I met a very wise art-teacher-turned-STEM-guru that I started to see the world of design and my own world start to converge. This whole process is the key to everything I do as a teacher, as a mathematician, as a team member, and in my own personal relationships. This is how we grow and improve. 

Recently I attended a PD that asked us to view our blended courses from a design perspective, something I wish I'd considered more before this whole thing started 2 years ago. When I think about the creation and implementation of the course, I realize that I'm immersed in this process almost constantly. The class is by no means perfect and I regularly try to tweak and get feedback on things. Feedback can be a scary thing when you're sticking your neck out and trying something new....the more I teach blended, the more confident I feel in asking for help in it and the more critical I find myself being of my own design. We discussed and considered research from the Schlechty Center and it's some powerful stuff. I am by no means an expert in this, but here are some highlights that jumped out at me when viewing through the blended learning lens:

Design vs. Planning

All too often we choose to embrace structure and order over chaos and divergent thinking. It's easier. It's comforting. And frankly, it's how we were taught. Because design is trying to meet the needs of it's "client" (the student, in our case) and not trying fit activities into an organized algorithm, it is messy. It involves experimentation, student choice, and being willing to admit something didn't work at all.  As a teacher, it's often frustrating when students come to you without the background knowledge they need and rather than planning for the content I'm trying to teach I should be designing for the needs of my students where they stand in order to build them up to where I need them to be. Blended is the perfect opportunity to do this since it offers so many options for personalization. However, that type of personalization takes some seriously intentional work.  

Design Qualities for Creating Engaging Work

To read these, check out this document from the Schlechty Center: 

Schlechty Center on Engagement

There's so much substance in these 10 principles. They address the big issues I wrestle with daily in blended learning: motivation, failure, cooperation, feedback, choice. 

Some more interesting reading on the subject:

Working on the Work

Focus Within Realm of Teacher Control

I am learning to remember that I have no power over many of the choices the students make, especially in a model that allows for students to complete their work in an unsupervised environment. It's a large amount of control with which I've had to part. What I came away from the PD thinking was that I need to look at the factors I can control and use design thinking to improve those. Less self-blame, more self-reflection. 

One and a half semesters into teaching using this model and yesterday I still had a "real talk" with my students about why work doesn't get turned in consistently and what I can do to help them with that.  In a short conversation, my students and I discovered that the LMS we are using doesn't notify them of a certain type of assignment and we made plans to change that for the future. Such a productive conversation, but it involves being vulnerable and admitting to others that something isn't working well- that's scary. The kids are definitely seeing that I'm human and willing to improve and learn from them. I'm incredibly grateful for everything I've learned so far and the honest and constructive feedback of almost 100 kids who've been willing to fail forward with me.