Monday, November 30, 2015

Coherence & Canoe Trips: Common Language for Understanding Vectors & Polar Coordinates

First, let me begin with this:
Okay, now that we've gotten that out of the way...

One challenge that I find with teaching Pre-Calc is that it's the culmination of all the content the students have learned up to this point. Very few of the topics are completely new to the students, so often there is a "we already know this" attitude when we intro a topic. However, the course requires more than just the knowledge of each specific skill; it demands an integrated understanding of the concepts. Disconnected thinking just isn't going to cut it here....coherence among pedagogical approaches needs to be a priority if we want kids to have an integrated conceptual framework in their minds. Kate Nowak mentioned it in her talk at NCTM Nashville- you don't want to piece together a string of "short stories" in your class; you want to reveal to students the "whole novel." 

My goal years ago was to develop some sort of common experience to use with my kids to talk about the concepts of vectors and polar coordinates since they are so closely related but often taught separately. I wrote and have been tweaking these explorations for the past few semesters.

Canoe Trip Part I- Finding Vector Components

Canoe Trip Part II- Converting Between Rectangular and Polar Coordinates

Photo from
Both of these are introductory level activities and I've found that some kids latch on to the "rule" that they develop quite easily, while others cling to the "drawing a triangle will always work" approach. Both methods are 100% valid to me, so I don't make much a distinction at this point in the concept (although the "rule" method is much faster!). When I see those right triangles being drawn out, I get the "teacher stock photo" feeling. My heart feels like this picture- a student choosing to use the concept instead of just blindly memorize the formula! I have a "moment", for sure. 

I like the way these work right now, but I'd like to make them more open ended in the future. Maybe give different groups different scenarios and have them critique each other. Maybe ask them to create their own scenario to model a situation. Unfortunately, these topics always wind up getting smushed at the end of the course and we don't get to have the "play time" I'd like. Anyone have any tweaks or any ways they love to teach these topics? I've love anything to improve or enhance these! Comment away! 

Thursday, November 26, 2015


We have an amazing art teacher at my school who dedicates the day before Thanksgiving Break to gratitude each year. She has each student write notes, some anonymous and some signed, thanking someone who has influenced them at school. This year, one of her students chose to write this to ALL teachers instead of just picking one she sent it out to all of us. It was my email I received all day:
I'll be the first to admit that I love a good pity party. I think about not having to choose between paying off my student loans and using my 20's to travel the world. I think about having evenings and weekends to relax....maybe even go out! I think about being evaluated on my own merit alone, not on the amount that I'm able to overcome the host of outside influences on a  group of teenagers. It all sounds pretty appealing, especially when you're in a job that even said teenagers can see is under-appreciated.

But then I take a second and I think about those teenagers- the fun that I have with them on 8 out of 10 days, getting to witness them understand something for the first time, earning their trust enough to have tough conversations about what's going on at home or with their friends, the list goes on and on. I can't imagine a career where I don't get to share dynamic moments like that with a group of people who are excited about all that the future holds. We talk about "kids these days," but I have been lucky enough to know great kids who are going to make the world better.

So here's my yearly list of things for which I am grateful, work edition:
  • The opportunity to teach in a program that supports innovation, most notably in it's allowance for failure in the interest of growth
  • That I have spent my entire career surrounded by the most incredible colleagues
  • The people who have shaped my teaching, whether it was in person or through online collaboration (I'm looking at you, MTBOS). The whole "standing on the shoulders of giants" thing is so true- I am truly a better teacher for having worked with each and every one of them. 
  • The trust of those around me which has opened new doors to me as a teacher
  • That I married an educator who not only understands my insane work schedule, but pushes me to think differently about my own practice and my kiddos
  • nearly any form.
  • The truly amazing kids who challenge me every single day to bring my best and forgive me when I don't
Time to put on my stretchy pants and prep for my 3 favorite things: family, stuffing, and football. Happy Thanksgiving! 

Sunday, November 22, 2015

Prunes and Cowboy Boots: Tales from NCTM Nashville

I've been home from Nashville for 2 days and trying to somehow simultaneously process all the awesomeness I just experienced and entertain family that is in town for the holidays....not an easy task so far. After packing up the in-law's van this morning and eating about 15 candy canes (it's the holidays, isn't it?), I'm hoping a little writing will do the trick. 

Nashville or Bust! 
This year has been my toughest yet teaching, without a doubt. I have been facing that typical year 4/5 burnout- taking on a huge number of responsibilities while trying to juggle being a newlywed and have a personal life. The balance between doing what's good for my students every minute of every day and doing what's good for my own life and's been a tough one to try to strike. In addition to all of that, I've had a heck of a fall personally. I sliced the top part of my finger off about 4 days before the school year started trying to be an iron chef and not use a hand guard on my mandolin slicer. We've been battling puppy cancer with our (way too spoiled) rescue mutt since September- and seem to be on the other side of it right now, thankfully.  We've had other personal trials we never saw coming and all of this has such a huge impact on life in the classroom. You don't get an "off" day in teaching and when you have one, it just makes you feel guilty that you're letting the kids down. I needed some time away from the classroom to get some perspective back, to reflect, and to immersed in a group of people who understand all too well the challenges I'm facing. I'd been looking forward to NCTM Regionals as that opportunity and it was that, for sure. 

Side Note: If you've felt this way, I highly recommend this blog post by Vicki Davis. We definitely all walk wounded in teaching....gotta stay in the game. 

Prunes? Why Would You Put That in a Blog Title?
Andrew Stadel (Divisible by 3) told a wonderful story in the opening session about how advice is taken best when it is timely. Or more precisely, how another exhausted parent at a playground solved his toddler's constipation issues by telling him how they'd gone through the same issue and the answer was PRUNES! If someone had told them this advice at their baby shower or months before, it wouldn't have carried or the same weight or perhaps might have been forgotten completely.  I think about it like this: In high school, I read the book The Unbearable Lightness of Being. Or should I say I skimmed "it"....."it" being the Sparknotes. It didn't resonate with me; I didn't have the life experience to be able to connect with it. In my senior year of college, I was assigned it again and was dreading it. I was also right in the middle of a breakup of the cliche big college relationship....cue the timely part. I didn't even know if I think the book was all that "good", but it was the right book at the right time for me and it helped shape my experience at that point in my life. It was my "prunes" when I was holding the constipated toddler in the middle of the playground. 

I felt that familiar feeling of resonance in Nashville....a recognition that this was the right advice at the right time for me. I am feeling re-energized not only by the presentations I heard (most of which were great), but by the people I met. I was ready to hear these things and meet these people and I'm so thankful that I had the opportunity. 

What Did I Walk Away from Nashville With?

  • Formative Assessment I got to attend the pre-conference workshop on formative assessment and left with tons of research-based ideas on how to involve my kids more in peer and self assessment and also with a new lens on the amount of grading I'm doing. I'm going to start incorporating way more self-graded or ungraded assignments which I can leave feedback on without muddling it with a grade at the top. I teach honors kids. If it's an A, they file it away. If it's not an A, they hide it in hopes no one else sees it. Either way, I spend hours grading and wind up finding papers on the floor or in the recycling bin.....frustrating to say the least. Get rid of the grade and they need to fish for the comments and the feedback more. So simple, but filled with so much power.
  • Modeling I love modeling's my bread and butter as a STEM teacher. While It might not be a new idea to me, I definitely left with a host of new ideas for every course (including Calculus! Yay!). A huge thank you to Brian Shay (@mrbrianshay), Brett Doudican (Coordinated Achievement), and Robert Kaplinsky ( for inspiring modeling sessions. 
  • Blended & Flipped Learning I've been doing blended learning for two years in my reach-model classroom, piloting it for our district. I've proven that the reach model works for the right kiddo- one who is self-motivated and takes the time to do their work outside of class. I learned that I really am towards the forefront of thinking on much of this stuff (which is nice validation), but I had the chance to really push my thinking further with great colleagues. One session in particular that I attended examined how people are using flipped models, examining the difference between videos that are purely instructive vs. videos that still drive student thinking using the ideas from Principles to Action. Yes, the videos replace the lecture. But in particular, we discussed using the video as a driving factor the conversation the next day. The video does not need to answer questions- it needs to generate them. So small, but so powerful. Thank you to Jeremy Strayer from Middle Tennessee State University for having such a productive conversation in his session. I also attended the packed session on flipping AP Calculus with Joel Evans and left feeling like my dive into flipped AP Calc (which I promised my class we'd start after Thanksgiving) won't be as scary as I thought. 
  • Networking I finally got to meet a lot of the people whom I've been following for a long time and who have inspired a lot of my to have my little math teacher celebrity moments. Getting to chat with so many of MTBoS colleagues who have supported me without ever meeting me was amazing.  And my final math teacher celebrity moment- finding out Kate Nowak (Function of Time) and I had the same graduate advisor. My graduate experience shaped so much of who I am as a wonder I find inspiration from someone who worked closely with the same people I did. 
  • FREE STUFF!! I won a copy of Principles to Action and have already highlighted the crap out of it. I ate about 37 tootsie rolls and 113 mints from vendor booths. I signed up for copious amounts of free resources from NASA. I got a brain stress ball from Gizmos when I told them I was part of a grant. I got a student sized whiteboard (because I don't already have 30 of these in my classroom....). And I heard lots of great country music in our time outside the conference. 
My Favorite "Overheard at NCTM Nashville" Moments
Pretty sure this was Robert Kaplinsky right before his session started to a participant talking about fear of things not working
"If everything always works in your classroom, you're not innovating enough" 
An awesome lens for viewing flipped instruction, when people were asking questions too specific about "videos"
"It's not about the videos, it's about the pedagogy." 
God Bless Puckett's Nashville
"Mac and cheese is the vegetable of the day"
A participant asked Kate Nowak how to do a particular problem she'd offered to us as a challenge. Her response was the perfect teacher answer. 
"It's pretty satisfying and magical when you figure it out, so I don't want to steal that from you by telling you."

So to NCTM, all the conference attendees, my coworkers who tolerated my crankiness on the traffic-filled drive home, and the city of Nashville, I say: Thank you, thank you, thank you!!!  

Friday, November 6, 2015

What Did I Miss?

This may be it. Every teacher's favorite question. 

Absences are a chronic issue with particular students it has a serious effect on the classroom. Math is so cumulative....everything we do builds on what we've done the day before. I had a student return back to Pre-calculus today after a week out and in that time, here's what we've covered:
  • Right Triangle Trig Review
  • Special Right Triangles
  • Angles in Standard Position
  • Coterminal Angles
  • Reference Angles
  • Radians and Degress
  • Deriving the Coordinates of the Unit Circle from Special Right Triangles
  • Finding Exact Values of Sine/Cosine/Tangent
  • Finding Exact Values of the Reciprocal Functions

Any hope of completing the scavenger hunt we started with as a warm up was futile. She was just too far behind. 

Over the summer, I tested out a theory that I could create a sync-able folder that I could share with students so they could access all of my documents from every class and I wouldn't have to think about uploading them at the end of each day. God bless any of you that do this, but I do not have the presence of mind after a day of chaos to save all changes and upload them to my website (especially for 3 different preps). 

Here's the solution I have tried this semester:

I teach with Notability, so this wasn't a big leap. However, I am sure there are other apps that have an "auto-backup" feature. My documents automatically sync to my Google Drive and I share the folder via a link so anyone can view it. This has solved a lot of problems for absent students, but it's had some benefits I never considered. Many students benefit from being able to go back and re-read the notes (including students who need filled in notes as part of the IEP) that I've seen it as a huge benefit for everyone. And talk about a tremendous resource at finals time, especially since we don't use textbooks at my school. Tech is here to make our lives easier, right? 

When am I EVER going to use this?!?!

I don't think I'll get through a semester without showing this now that I've seen it! 
Happy Friday! 

Wednesday, November 4, 2015

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: 

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:

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. 

Monday, November 2, 2015

Sharing is Caring- Technology Edition

Technology is something I consider myself relatively fearless with in the classroom, as I am definitely a digital native who knows there's few things a quick control-z or restart won't fix. I attended a week long institute this summer on integrating technology into math instruction and I've also been teaching in a one to one STEM magnet program for 3 years and piloting blended learning for the past 2 in Pre-Calculus. This picture has been big for me....definitely taken somewhere from the Twitter world! I find it to be a good lens when I am trying to integrate tech for the sake a tech- a way to check myself before I wreck myself, if you will. 

I love the quote from Goos, Galbraith, Renshaw and Geiger (2003) about using technology as a partner in education, saying that it should be “used creatively to increase the power students exercise over their learning; for example, by providing access to new kinds of tasks or new ways of approaching existing tasks." We have so much power to do that these days and it's something I'm still trying hard to learn myself. It should be transforming the tasks and challenges we give our students. 

So tomorrow is our annual PD day in my district and there are few things I love more than sitting in a room of math teachers, nerding out about our content. It is also the first time I've been asked to plan and implement my own PD for other secondary math teachers about integrating technology into secondary math and designing it has been one of my favorite tasks all year.  Trying to narrow down exactly what to share has been challenging, but here's what I've come up with: 

  • Desmos Polygraph Activity (because if you haven't seen it before, be ready to fall in love)
  • Discussion about cognitive demand of tasks and metaphors for technology-mediated learning
  • An activity (admittedly adapted from an amazing activity we did at the institute) where students use dynamic software to find the characteristics of quadrilaterals and create a hierarchical classification system of them
  • Some quick "tips and tricks" I've learned from teaching blended and STEM
  • An hour of self-guided "challenges" that I've written to allow people to move at their own pace through using Desmos, Geogebra, and Google Docs in different ways
Here are all of my files for the PD if anyone is interested in checking them out!

Now, if I can somehow find the time to simultaneously be in 3 places at once, tomorrow will be a big success!