Monday, September 11, 2017

Storyboarding to Introduce Limits

I had the rare opportunity this year to start from scratch with my AP Calc students when introducing limits. Due to adjustment to anew curriculum, our Pre-Calc PLC just didn't get there last year so we are doing more thorough coverage in class than I've done since my Pre-Calc teaching days. I wanted to try something new this year to get my kids thinking about the concept of a limit. 

I was planning to put this slide on the board:

I'd then asked for 4 volunteers, 2 to be a part of Scene #1 and 2 to be a part of Scene #3. Scene #2 was a mystery that would come back later. I'd give each group a set of a secret directions only they got to see and about 30 seconds to plan, then had to pose to represent their "scene" for the class. 

Scene #1
Two friends are having a frisbee toss. Friend #1 is ready to toss frisbee to friend #2, who is wearing sunglasses.

Scene #3
Friend #2 is holding broken sunglasses, having clearly gotten hit in the face with a frisbee. Friend #1 is reacting to having just hit his/her friend. 

The class has to study the scenes and make a prediction as to what Scene #2 would look like. Most would go for the obvious choice. A few would go for something crazy (a bird flew in his eye, he got punched in the face, etc), which is great. We start to discuss what it looks like should have happened in that scenario, given the information we were given. We also discuss the fact that this predicted scenario doesn't necessarily have to be what actually happened. The kids grasp on to this pretty easily and it becomes an easy framework through which we can view a limit, especially a limit where the defined value of the function and the limit are not equal. And it's a goofy, fun way to get seniors out of their seat! 

But, of course, I forgot we had a senior assembly this morning and I would lose 45 minutes of my 2nd period.....teacher problems for sure. It remains hypothetical, but I have a lot of hope for it. Sharing if it can be a fun and goofy thinking activity for anyone else and storing it away for a rainy day for myself!  



Saturday, September 9, 2017

The First Week: Building a Triad of Responsibility

First Week 2015
First Week 2009
I have a history of very interesting first weeks of school.  There have been good ones, meh ones, and downright weird ones. My very first first day of school ended abruptly in an early dismissal and subsequent closing of school for days due to a massive flood. Two years ago, I sliced the tip of my finger off trying to make pickled radishes and went to school in full middle finger bandage....basically flipping off my classes for the entire first week. (Luckily, my husband got to see silver nitrate in action when I got my finger cauterized and had a cool story to tell his chemistry classes that year...)

NYC Math Lab Triad of Responsibility

This week was, by far, my favorite first week I've had in my 7 years. I loved all of my classes, felt like I got to know more than usual about my kids, and it's been the most perfect September weather here in Upstate NY. I tried a bunch of new things and brought out some old favorites. I celebrated with coworkers and went to bed by 8:30 pm on Friday. 


The biggest difference maker for me this year was the marriage of 2 different activities I learned about this summer. First, I had my classes participate in Sara Van Der Werf's 1-100 Task as one of our first day activities. I decided to fuse this with one of my experiences from NYC Math Lab this summer. In Math Lab, we used the Triad of Responsibility to help set norms for the students. I loved this idea and thought it was a natural fit for the first day. 

Before we started the activity, I had my students individually brainstorm their responsibilities to themselves as students. It was amazing to see the difference between answers in a Geometry with Lab class (1.5 times the class minutes of a typical Geometry class) and an AP Calculus class. 

Side Note: The Calc ones almost made me sad....so grade oriented, no examination of yourself as a whole person. It would have been how I answered in high school and it's part of the reason I love doing this particular work with these kids. I hope they all begin to see that they are valuable for more than just a number. 

Next, I had the students perform the 1-100 task as described in Sara's blog. Overall, we did it 3 times and discussed group norms or "Responsibilities to My Group" in reference to the activity. 

Lastly, we discussed "Responsibilities to our Classroom Community" in reference to our whole group discussion. Each class had common themes, but brought many unique answers. Even if lot of math teachers aren't into them, it was a warm, fuzzy day and I ate it up. 

Here are what my kids generated from our conversations:






These are nice to hang in the classroom, sure, and I've already found myself referring back to them in our first 3 days of instruction. But you cannot imagine the difference this short activity made with my students.  I have seen the type of group work I normally don't see until November or December during the 3rd day of class. I have seen students checking in with each other to make sure everyone in the group understands. I have seen communication from every student in a group. Students are taking small risks, testing the reactions of their classmates and seeing that it is okay to be wrong. And they're asking questions! I'm very interested to see how this initial sense of community carries on through the year and I feel a huge responsibility to live up to the expectations of a safe environment that I've set from the first day. Anyone have any favorite ways to keep that sense of community and safety going all year long? 

Best of all, the posters got approval from a very harsh and deciphering academic who only wags her tail at her favorite first day strategies. 










Wednesday, August 30, 2017

Basic Algebraic Limits Circuit


Given our high number of snow days last year and changes in curriculum, our Pre-calc classes didn't get as far into the Limits unit as they have gotten in the past. With that in mind, our PLC has tried to put some extra umph into our limits unit to cover not only the advanced applications, but also the basics. This is a practice activity for the first day of algebraic limits and covers evaluating by algebraic simplification with factoring, rationalizing, and trig identities. This just requires students' basic knowledge of trig identities- none of the special trig limits needs need to be memorized for this!

This would work for the end of the year with Pre-Calc too! Let me know if you see anything that needs fixing! (And don't mind my name line....I always put some pun or saying there and I've been watching too much Game of Thrones sooooo here we are).




Monday, August 21, 2017

NYS Common Core Geometry Standards and Assessment

Happy Eclipse Day! 

As the new school year inches closer and closer, my thoughts are turning back to my classroom. After having our first PLC meeting of the year this morning, I'm at least being productive today! 

Wanted to share a resource for any other NYS Geometry teachers who might be interested. I spent a lot (....like a lot...) of time this summer going through every released Common Core Geo Regents and sorting the questions by standard (side note....I am sure there was an easier way to do this as the Regents already has an alignment document and would love to be enlightened if anyone has ideas!). I recorded next to each standard how it has historically been assessed on the Regents and 18 pages later, we have this. These are sorted according to our pacing and unit layout, but can easily be adapted to whatever your school does.


Feel free to send me ideas if there are things you would add! I made this to be used, so please steal, adapt, and keep sharing more ideas! 

Tuesday, August 8, 2017

WOBD Limits #3

I feel like this needs a tweak for some reason. Letter D seems a little...forced?....to me. Anyone have any feedback that might help? I will update if anyone has any great ideas! 

A- Taking limit at a real number x value(not negative infinity), limit is one sided, limit is taken at an infinite discontinuity
B- Function is continuous, does not have an infinite discontinuity
C- Limit exists 
D- Limit approaches positive infinity **This is the one I want to make better. Ideas?

Friday, August 4, 2017

WODB- Limits #2

Here's the 2nd in the Limits WODB series. I am open to ideas and feedback on any of these, so please feel free to suggest improvements! 

A- Can be evaluated with direct substitution
B- Limit exists at a point where function is undefined
C-  Limit does not equal defined value of function, not as x approaches 2, is a graph
D- Limit does not equal 5, Limit DNE

NYC Math Lab (Alternate Title: Elementary Teachers are Superheroes)

Real talk: one of the joys of teaching is summer....time to step back from the day to day grind and reflect on the year behind us, while looking to the year ahead. I make it my mission every summer to get outside my comfort zone for at least one major challenge. Know what's outside my comfort zone? 

10 year olds. 

Most of my best friends are elementary school teachers and I genuinely believe they are superheroes. I hug crying kids who didn't get into their first choice college or whose boyfriend broke up with them and I live through the day in and day out struggles with factoring, but they teach little humans how to read....how to share....how to exist in regular society. They work with the same group of kids all day (no bell to save the day when little Johnny is having a rough one and you're ready for him to go to history now kthanksbye) and they have to be masters of every subject. They are magicians of classroom management and engagement.  They still clean up bathroom accidents (and they do it with a smile). Seriously....superheroes. I've always known I have a lot to learn from them and NYC Math Lab was an amazing opportunity to do so. 

The Math Lab is the most unique professional development I've ever experienced. Each day, the teachers in attendance were able to observe instruction for a group of rising 5th graders in a "fish bowl" environment and then begin to work individually with a student as the week went on. Afternoons were spent analyzing the lesson and sharing ideas while looking ahead to what would be best for kids as the week progressed. It was a collection of people there to learn from each other- the students, the lab instructors, the participants. Everyone was working towards a common goal of students' individual conceptual understanding and sense-making and that made it feel like a true professional learning community. These weren't hypothetical kids....they were our kids. They had funny shirts and favorite books and bad days and amazing successes.  

While the content was far below the level I teach daily, I learned so much and have so many new routines I want to try in my classroom this year. The thoughtful way you must present complex ideas to young minds to help them understand is a lesson for all of us and it explained so much of why some of my students struggle to build on a shaky foundation. My biggest takeaways were:

A Math Community is Built Intentionally
I have learned more and more each year how much I believe that I teach students, not math. I love the rapport I have with my kids, but I have been trying to find new ways to build student confidence, voice, and engagement in my room. One of the key focuses of the math lab is building a mathematical community. Much of this community centered around the Triad of Responsibility, which had 3 key components:

  • Responsibility to Self
  • Responsibility to Partner
  • Responsibility to Community
This frame of reference for students to participate in the community gave all students power- not just those who understood the fastest, but also those who needed to most intervention. Praise was never given for being right; it was given for being an active participant in the community. 

One of my favorite moments came when a student, who was much younger than others in the room, felt pushed beyond what he was ready to think about that morning. I saw him getting teary as he spoke to the other instructor and I asked if he wanted to take a break in the hallway. Once we got outside, he told me how he felt overwhelmed and we talked about how proud I was of his perseverance and hard work. Just then, another instructor walked over behind me and pointed how what a huge contributor he'd be to the math community's discussions and that the community needed him and needed his ideas. He agreed to head back in and within 15 minutes was up at the board explaining his idea to the group. His exit slip that day explained how he had loved "being brave" and would be even braver tomorrow. He felt important, included, and valuable and he saw growth because of that throughout the week. 


It is the students job to make sense for themselves
As teachers, we put a lot of the responsibility for students' learning on ourselves. I think it's in our nature...we love our kids, care about their success, and are judged by the world around us by a test score. I know students need to understand conceptually to have a truly sound foundation on which they can build their knowledge and I try to achieve this in my planning, but so often in the craziness of the school year I wind up thinking "What else can I do for them?" instead of "How else can I give them an opportunity to make sense of this?" Too often it comes down to time. But I'm realizing the further I get into my career, the more I need to slow down and dedicate time to that sense-making for THEM. I can't make sense of it for them, I can only create favorable conditions for their success. What we spend time doing in class is a good indication to our kids of what we value, so I am working on learning to slow down and show kids that I truly value their understanding. 

Students were encouraged to talk to each other in a very methodical way. Instructors never said anything a student could say, often just reframing or encouraging points and debates among students. If it was clear a student has a question, the instructor would instruct the student to ask "What do you mean by that?" or "Could you explain ____ more?" Since it was up to the student to understand, it was up to the student to ask. They were learning responsibility for their own level of understanding. This is cliff notes from another great resource that we discussed, which is high on my reading list now. 

Sometimes the Best Thing You Can Say is Nothing
Before the students walked in the first day, we were specifically asked to not interact with the students on the first day. Observing in a "fish bowl" environment means you can't interrupt...you're a fly on the wall.  I challenged myself throughout the week to really listen to what the students were saying, something I know I don't always take the one on one time necessary to do well. It's hard to listen to students talk on and on at you about a misconception, but we worked the whole week from a place of "You know SOMETHING. Let's build on that." Often this meant inching backwards in the progression of understanding until you found something that you and the student could both meaningfully agree upon and using that as a building block for your conversation. I don't think in the week I was there we came to the right answer of how to implement this given the time constraints you have when covering a whole curriculum in a classroom of 25 kids and 1 teacher, especially given the confines a state test places upon the teacher. However, it made me deeply reflect on how much I assume when I talk to my students daily. They may say something that isn't quite right, but it sounds close and I let it go for the sake of time.....building a misconception that we might have been able to nip in the bud then and there. Below is the document we were given on "Nudges" and how to consult meaningfully with students one on one while not giving them too much (with some of my notes added for extra seasoning). 


Since this wasn't my curriculum, these weren't students in classroom, and there wasn't the pressure of a test, I was truly able to treat this experience as a "lab"....a scenario where I could experiment with my own practice and analyze the impact on a student's understanding. I saw a difference. 

I am taking so much from the experience into my planning for the upcoming school year and I am so grateful to have had the opportunity to reflect, observe, and learn with these other teachers and students. Elementary teaching is, much like all teaching, both an art and a science and there is so much to be learned from seeing where my kids start before they ever step foot in my room. My gaze has shifted and I know I'll be doing more elementary PD as the years progress. So to all the participants from that high school teacher from upstate that seems a little out of place in the intense 3rd/4th/5th grade conversation- thank you. Being out of place is my favorite way to grow.