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! 

The Final Countdown: AP Week is Here!

Here's the motto I'm trying to instill in my kiddos this week:

During the middle of last year, a colleague and I were discussing the fact that our school only offered BC Calculus and we wished there was an option for students who were interested in pursuing Calculus but might not be ready for the demand of BC. Heck, my bachelor's is in pure mathematics and I did just fine having only been a product of AB Calculus. We approached our administration about this and luckily for both ourselves and our kids, they allowed us to take on the challenge of designing and implementing a full-year AB course. The course would be an "Honors Calculus" course in the fall, leading directly into an AB course in the spring.  It took some selling to get our enrollment up and I am forever grateful to the 20 kids who took the risk of a brand new Calculus course with me. The enrollment is already more than doubled for next year and that definitely wouldn't have been possible without this pilot group. 

I feel a little like the cast of one of those youth sports movies sometimes when we are in class- despite the fact that Emilio Estevez had consistently better hair than me in any number of those movies.  We are an unlikely mix.....17 boys and 3 girls (I promise our demographics are more evenly split for next year!!) with extraordinary variation in past experience with math, intrinsic motivation, and consistency in completing tasks. But they all showed up every day and tried. Cue the Mighty Duck's-esque montage where everyone learns that "Ducks Fly Together" and learn to skate without falling.  I've gotten to watch this unlikely mix of kiddos turn into a team who can function at a really high level (or more concretely, get about 5 to 9 points on the famous 2007 AB 3 problem).

Because of the extra time built into our course, I really do feel like I've given the kids all the tools I possibly can at this point in the game. I'm okay with not trying to "cram" any extra information in their brains on the last day....which is a feeling of peace as a teacher that I don't normally feel. With my unlikely group of mathematicians and the mindset that we've prepared with the most gusto we possibly could, I'm taking a very different approach on our last day in class tomorrow. Here's the plan:

1) A sincere "thank you"
Y'all.....I'm so proud of these kids. Because I teach mostly in a STEM program, this is the third straight year I've taught this particular group of 20 and they've been a huge part of my journey as an educator. Beyond all of that, they've been with me through an incredibly difficult year personally. They've accepted me on my bad days and pushed me to be better even on my best days. They've driven me a little crazy, of course. But I am so grateful for the hard work and determination I've seen from each and every one of them. They've made me laugh and learn and grow. I don't know if we ever thank our kids enough. I'm trying to remember to do it more. 

Good Luck Gift Bags (because my inner middle school teacher still hides deep below the surface)

2) A little more shop talk
Because I don't think you can emphasize "DO NOT write 'it' if you mean f'(x)!!!!" enough. I don't mind saying it for the 16,000,001st time if it means they remember. And for Heaven's sake, don't leave off the + C !

3) A real, honest talk about anxiety, nerves, and some brain chemistry
Here's why: I am someone that naturally appears incredibly collected and outgoing. I seem to thrive on being constantly interacting with others. This is a completely intentional choice I make everyday- I am much more of an introvert than anyone would suspect.  It was actually my time as an AP student that started my journey to discovering this. AP US History basically turned me into a robot; it was the highest expectations to which I'd ever been held in a subject area that wasn't my strength. Friends tried to talk some sense into me, but I assumed it was just stress and it would pass. As I got older, I started to become more cognizant of the amount of anxiety I felt on a regular basis (although I'd still fight you if you tried to tell me I was at all abnormal in what I felt). Only this year did my anxiety finally bubble over into something I needed to address and doing so has totally changed my perspective. 

I think a lot of kids feel this kind of anxiety and pressure. Our guidance counselors get crying AP students in their room regularly. We expect to see some mini-breakdowns as exams approach. And let's be real....most 16 and 17 year olds aren't equipped to deal with it. So here's what we're doing:

Activity #1:  Article Reading

I'm going to have the kids read this silently, marking a "?" next to anything they have questions about and a "!" next to anything they want to remember for the test. Then we're just going to talk and see where it goes. 

Activity #2: Journaling Activity

Since my very first standardized testing week as a 1st year teacher, this article has been in my arsenal. I love it. I love the idea that psychologically the symptoms of being "pumped" are the same as being terrified.  I try to get kids to get pumped to show what they know, not fear what they might not know. 

"In a study published last year in the journal Emotion, Dr. Beilock and four co-authors found that with students anxious about math, the more stress hormone they produced, the worse they did on a test; students with low math anxiety did better the more cortisol they produced. “The first group,” she said, “felt the rising anxiety in their bodies and reacted by thinking, ‘I’m really nervous about this test. I’m afraid I’ll fail.’ ” They choked. “The second group told themselves something like, ‘I’m really psyched up for this test! I’m ready to go!’ ” Dr. Beilock recommends consciously adopting positive self-talk. Remind yourself that damp palms and a pounding heart accompany all kinds of enjoyable experiences: riding a roller coaster, winning a sports match, talking to someone you have a crush on."

We are going to end the class by actually doing the journaling activity as an exit ticket. I am encouraging each of them to leave their fears in my classroom by writing them down and putting them into a box. With those left behind, I am going to encourage them to take only their knowledge, skill, and confidence with them into the test. My hope is after the exam we look at some of their fears and see if they were worth "carrying" with us or if in retrospect it was okay to leave them behind. 

I'm incredibly proud of them and am hoping to leave them feeling excited to show all they've learned. Any I will definitely be reflecting with them on how to make the course better. 

Good luck to everyone's AP classes these next 2 weeks! Cheers to all your work to prepare an incredible group of kids for an incredible challenge! 

Learning to Trust the Core: My Weekend with the #CoreAdvocates

I feel very fortunate to have spent the weekend with some amazing math educators from all over my state, digging into the Common Core State Standards. Even though the standards are in their sixth year and we've certainly seen a shift in many classrooms towards a new way of teaching, I think we've all seen that the kinks definitely aren't worked out and there are lots of people who are still reluctant. In all honesty, who could blame them? When you've been doing something for a long time in a particular way (not to mention being a student who learned a particular way), it's difficult to feel like you're being asked to reinvent the wheel. When a huge part of your evaluation is based on a multiple choice test, hard choices sometimes get made. You hope you're doing what's best for kids and try to trust your own professional judgement. 

[Brief aside: I feel so lucky to have taken the path I did to education. My graduate advisor, Dr. Jean Schmittau, was someone who was truly ahead of her time. She advocated for conceptual, student-centered teaching for decades before the words "Common Core" were ever utter together and truly pushed her students to think deeply about mathematics and pedagogy. Let's put it this way....one of our assignments was to make a concept map that showed the concept of multiplication and it's relation to the body of mathematics as a whole. I had to go to Kinko's to get mine printed because it was roughly 5 feet wide. We spent almost a month of 3 hour classes trying to answer the question "What is multiplication?" and often we were given another week to think about it because a room full of people with pure mathematics degrees didn't answer her well enough.  She and the wonderful members of my cohort shaped me into a teacher who is always seeking to improve and challenge myself and my students. I was reminded of her in so many ways this weekend. ]

We opened the weekend with this big question:

Why are we here? 

Pretty existential, right? We were challenged to examine that question from 3 different perspectives and I couldn't think of a better way to share all that's swirling around my brain right now.

Why are we HERE? 
There is something so valuable for a teacher's soul when they are able to congregate with like-minded people in a neutral site. You are able to immerse yourself in a world that centers around what you're studying and have discussions that might be difficult in a place you are too "comfortable." It may seem small, but I think it's a huge part of what makes conferences so powerful.

Why are WE here? 
 A topic that came up over and over again this weekend was the idea that teachers need to "own the standards." It's a picture that I think holds a lot of power in education. If we truly "own" what we are teaching and examine it as practitioners, we will be cognizant of how these standards connect with the conceptual structures our students have created in their schooling. A huge part of this is creating teacher leaders and I was happy to be in a room full of them. I honestly believe that these people are the key to any real change- teacher leaders need to buy in to it and model that these changes can be effective. Ideas that teacher leaders can demonstrate are good for kids (and good for teachers) are the ones that spread like wild fire. 

One aspect that we discussed was the idea of thinking critically about our own instruction and that of our colleagues. My biggest concern about doing this as a department is always that people will feel they are begin "evaluated" if the proper level of trust isn't established beforehand. Teaching is so deeply personal and it takes trust in your peers to allow them into your room to reflect on both your teaching and their own. But what if you do? The trust, collaborative spirit, and sharing of ideas that could happen is almost unfathomable and contrary to the isolationism of teaching in the past. It's using the 21st century skills we ask our students to have every day.

One of the things I'm taking away from the weekend that I felt most passionate about was a really powerful tool to help start these conversations among mathematics coaches, administrators, and peers. 

Achieve the Core Instructional Coaching Tool

This tool provides feedback on what I would consider the 3 most important aspects of any lesson: alignment to standards, appropriate selection of instructional strategies, and student opportunities to engage with the mathematics. Digestible language, jumping off points for conversations, and no wording that rings of "judgement" of the teacher....love. We discussed using it as a goal setting tool in our PLC's and as a tool to start to generate trust among a department. It's something I'm carrying with me as I continue to pursue teacher leadership in mathematics.

WHY are we here? 

I consider myself someone who is pretty well versed in the Common Core, as my master's

centered around conceptual teaching and I completed it right before the Common Core was implemented. We used the standards as our textbook. I've got to be on the savvy-iest end of things, right?
 
WRONG! But let's be honest, I didn't know what I didn't know. 

http://www.360results.com.au/wp-content/
uploads/2013/06/tumblr_mh6eljSrzR1rx06nvo1_500.jpg

The picture here rings so true to me....we examined the standards as a whole, not just our own grade level. And more importantly, we examined them from a perspective of 2 of the 3 major shifts- Focus and Coherence.  So often as teachers we are focused on the "checklist" of topics we need to teach, the day by day "calendar" we've created, and not looking at what's happening outside our own course. For this weekend, I hit "pause" on my high school teacher mentality and tried to examine the standards from a mathematics educator perspective. 

Here's my biggest takeaway from this weekend:

We need to learn to trust the standards. 

The key word I've started to imagine with the regards to the standards is this: design.  These were written, revised, and critiqued by people who have spent their careers immersed in mathematics education. These were designed intentionally to create a cohesive understanding of mathematics that balances conceptual understanding, procedural fluency, and applications.  They are saying this: "Trust me. There is a method to my madness." These were my "ah-ha" moments: 

•  Situation: A 3rd grade teacher who is introducing multiplication teaches how to "carry" because it makes it so much easier for students to multiply big numbers!
• Explanation: This teacher is trying to do right by their students. They are trying to give them a strategy that will always work. However, in doing so, they're neglecting the coherence that is built in the standards to really develop students' conceptual understanding of multiplication so they can apply it later. Fluency with the standard algorithm for multiplication isn't expected for 2 more years (5.NBT) and if we introduce a "trick" like this too early it can actually be a detriment to the understanding our students are building. 
• Situation: A teacher is so used to teaching surface area in 7th grade that they continue to teach it. It's in the textbook, right?
• Explanation: The core of the major work in elementary school is focused on developing students' understanding of number and operation. Geometry can be a tool to do that, but it is regularly listed as a supporting standard. This doesn't mean it should be ignored; it just means it can be used as a lens through which you can refer back to the major topics. It isn't until 8th grade that a geometric topic is considered part of the "major work" and with good reason- the lower grades are trying to go deeper into content. As we advance in the grades, we can have a wider focus. Surface area formulas are not specifically memorized in any grade because it lacks the conceptual basis that the Common Core demands. Just because it's in our textbook doesn't mean it needs to be our focus.
• Situation: High school teachers have always heard that there is supposed to be "less content" in the Common Core, but our courses seem to be even more jam packed.
•  Explanation: The high school standards were not written to decrease the amount students learn each year. This was never their intention. By focusing on fluency in the younger grades, high school is able to become a time where concepts are connected. Mathematics is a story and high school is the time when we weave together all of the plot points.

These standards were not written arbitrarily. They have a plan and every teacher has a role to fill to make sure students are getting the most from the instructional shifts. We need to see ourselves as part a whole; we are a K-12 team that is created to support the success of each individual student. Synergy, synergy, synergy.

This is by no means comprehensive of all the things I'm leaving this weekend having learned, but it's certainly where my mind is focused right now. I feel so inspired to keep learning and leading and I'm grateful I was able to be explore so much this weekend!