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. 

Thursday, August 3, 2017

WODB- Limits #1


Taking July off from blogging has been extremely restorative after a long school year and I'm ready to get back in action! I'm starting to create a series of WOBDs for warm ups in AP Calculus and am looking first at my limits unit. This one is just a start, but it's getting my blog moving back in a productive direction as August gets underway! MTBOSBlaugust was pretty powerful for me, so while I may not post daily, I'm trying to get back on the wagon this summer! 

I want my kids thinking about both continuity and limits in this one, so I've tried to vary the reason I see things "don't belong." These are just my thoughts, but we know the kids will always find more:  
A-  Continuous
B- Removable Discontinuity
C- Limit Does Not Exist at x=1
D- Does Not Have Y Intercept of 2 OR Limit DNE at x=0

More to come, but wanted to get this blog train back on track!