Limits at Infinity- Understanding Dominance with "Infinity War"

I recently met with one of our Pre-Calculus teachers to talk about what else they could do make sure students were better prepared for Calculus when they get to me. Unsurprisingly, this set me on a thought path about what experiences I wish my students had before my class, in general. 

I can remediate skills, I can teach more factoring, I make you do all my goofy dances to help remember the algebra rules you may have forgotten. One thing I can't do is time travel back to when you first learned a concept and ask you to think deeply about it before you are told all the "rules" you need to play the game. So many of my students get to me with "rules" and then are surprised that they can't remember them all. They are hesitant to think about the "why," even if it might save them time and mental energy in the end. 

One of first times I encounter this is in teaching limits at infinity. Predicting end behavior is such a beautiful exercise in logic, testing your understanding of algebra and concept alike. For students who've only memorized the "top heavy" and "bottom heavy" rules, anything beyond a rational functions becomes a challenge. 

To combat that this year, I created my very own "Infinity War" (cue gifs and memes of the Avengers). 

1) Students are given a deck of function cards to be dealt and divided among the group (I am doing pairs!)

2) Each round, every student flips over 1 card. They work together to figure out which would be "dominant" as x approaches infinity. 

3) Player with the dominant card wins! 

4) Play repeats until one player has all cards OR Mrs. G panics about the amount of time left in class and calls "TIME!"

Here is the file: 

Dominance "Infinity War"

I'm hoping this helps build more conceptual understanding of why end behavior occurs the way it does and makes limits at infinity a more intuitive topic. I'll report back after I use the activity....10 hours from now in 1st block... :) 

Composite Functions Task

One of my goals for this summer has been to rewrite a bit of my non-AP Calculus course. After so many years of teaching AP and focusing on making sure I was being challenging enough, I found that my tendency was to give tasks that were a bit too far out of my students' ZPD. I adjusted quickly and got better as I got to know their strengths and weaknesses, but any time it happened I knew it would hurt some of their trust in themselves...and maybe in me.

Many of these students were not enrolled in an honors level pre-calculus course last year, so algebraic concepts are fuzzier than you might expect at the very beginning of the year. They need more explicit practice with those background skills that AP students have mastered. One of those, I knew, would be dealing with composite functions- both their composition and decomposition. This is a key skills for the remainder of the course, with special importance to limit definition of the derivative, chain rule, optimization, and integration by u substitution. 

As both a pre-assessment and a task to get them thinking in groups, I drafted this sorting activity. It gives students notational representations of composite function and expressions to match. There are a few that don't quite match, and students must work to answer questions about those.

I also am going to be using Index Card Questions (as described here in my Necessary Conditions blog) with my students for the first time this day, so I'm having students reflect on the classroom norms we've used to emphasize them as we move into more content-centric lessons. 

This is very much in draft form, so if you have any feedback, please let me know! 

Composite Functions Sorting Task

Algebra 2 Trigonometry Practice Circuit

Sharing a quick practice activity from my trigonometry unit in Algebra 2 this year!

Trigonometry Circuit

This doesn't cover the whole unit, but you can plan to see:

• Coterminal Angles
• Reference Angles
• Sine and Cosine as Coordinate Points
• Finding Exact Trig Values of Sine/Cosine/Tangent
• Radians

Feel free to steal or adapt! Happy Trig-ing! 

My Favorite Last Day of School Traditions

So many people are spending #MTBoSBlaugust writing about what they'll be doing on the first day of school, but I want to talk about one of the most special days of my year- the last day of school!

I teach primarily seniors. We typically have 4-5 weeks after the AP exam and by the end of June,  class can feel a lot like this:

Actual Photo of AP Calculus Class in June

But the very last day is special- a sacred time where students can reflect on their growth as a person and be sent on their way into life after high school with love and support. The things below are some of the things that make me love it so, so much.

An Individual Card

I live for a custom card. I made my husband and dog dress up in matching pajamas last year for our christmas card. It never gets old to me- truly. 

With that in mind, I started making custom cards for my kids and writing them each personal note. Staples almost always has a Groupon to get enough cards for all my kids for $20 and I use E-Bates too, so it's not particularly costly. I include my favorite quote, a math-y pun, and my email sot hey can keep in touch. Here was this year's card:

It takes a long time, but I write each senior a card. I get so many emails in October or May of the next year, when kids come across the card in their dorm room and decide to drop a line and say hi....one of my favorite day-brighteners. 

A Keepsake

One of my middle school teachers did this for me and I continue to pay it forward. I ask all my seniors and many of my colleagues to tell me their favorite trait about each student that I teach. I put them all together anonymously and give them out to the students to keep. Here are a few of my favorites from this year:

I laminate these and hand them out to wallet-sized cards.

An Explanation 

The first thing my senior gift includes is this small note of support and an explanation of the keepsake:

"Your time at SSHS is coming to an end...your next big adventure awaits! Know that life has lots of twists and turns in store for you-some amazing and exciting, some terrifying and heart-breaking.When life gets hard, it’s easy to forget how important you are to others and how much power lies within you. I hope that this card serves as a reminder of the impact you’ve made and the amazing things others see in you, even when it’s hard to see them within yourself. Tuck it away and take it out on a day when it’s hard to remember who you are and why you’re so important. You always have a home and team of cheerleaders here at SSHS. So proud of you!-Mrs Gironda"

An Inspiration

I try to either show a video or read something that will inspire my students. This year, it was this. I think it might be this forever. I cried reading it in every. single. class. 

Let's Hear It For The Average Child

The ending of the reading, just like the end of the school year, just hits me right in the heart:

"Summer beckons, a great, green, gorgeous gift. We’ve already kept you far too long, so let us send you forth with just one last reminder of a truth that somehow you already understand, though school is not the place where you learned it:

Life is not a contest, and the world is not an arena. Just by being here, unique among all others, offering contributions that no one else can give, you have already won the one prize that matters most."

Necessary Conditions: Effective Facilitation & Wrap Up

Last day of my 3 part series on Necessary Conditions! The last part of the book focussed on Effective Facilitation of tasks, the 3rd and final element of the pedagogy outlined. One of the things that makes observing other classrooms so special is when you see the magic that goes on within those walls. Effective facilitation seems like a gift to some, but the further I've gotten in to teaching, the more I realize it's half science and half art. It can be learned, it can be practiced, it can be honed. Academic safety feels like something I try to foster in my classroom already. Quality tasks can be found in so many places and Twitter and the MTBoS has been such a huge wealth of them for me. But effective facilitation, that takes time and intention and research and growth. It's where most of my goals lie in my professional growth each year and it's the part of the book that drew me in the most.

Ideas I'm Implementing/Saving for Later:

• Index Card Questions
• Oh baby, am I stealing this idea!! The teacher in the book gave out one card per week at the beginning of the year with a question that would help facilitate mathematical conversation. Each week, they'd add in another question. They included (but were not limited to):
• Can you justify that?
• What assumptions are we making?
• Does that align with our initial estimate?
• What other problems does this look like?
• Publicizing Feedback During Group Work
• I loved the suggestion of using your document cam to record a plus/delta and recording group behaviors during work time- both commendable one and ones that need to be changed. Here, you can record specific phrasing, questions, use of Standards for Mathematical Practice, etc, and use those as evidence in a whole group conversation about how the class is attending to different norms.
• Reflecting & Debriefing Norms More
• I make a huge point of establishing norms at the beginning of the school year, but I liked the ideas for revisiting them throughout the year.  Questions like "What norms did you exhibit well today?" or "Which norms would you like to focus on tomorrow" are great, quick warm up prompts. These discusses can result in reflection, public praise, and re-emphasizing what we worked so hard to establish as important on Day 1
• I want to use rubrics for more reflection here- doing more of these tasks and having students use rubrics throughout the year to have them reflect on how they are hitting those norms
• Reading Designing Groupwork (just ordered used online) and learning more about Complex Instruction
• Make Sure Tasks are Group-Worthy
• Tasks that are too rote backfire in group work and I've seen that in my own instruction. I want to be more selective and utilize group and partner world more strategically next year. 
• Establishing Group Roles
• You guys, I am the worst at this. I read it every year and I never do it. I'm going to be doing more research and trying it....anyone have any favorite resources? 
• The Know/Need-to-Know/Next Steps Process for Facilitation
• This gives a real structure to posing a task to students. I think it's something I try to do intuitively, but I loved the step-by-step goals and structure this gives. I also like that this encourages teachers to pre-empt the paths students may go down, making you more prepared to facilitate no matter where the lesson goes. 
• The Workshop Model, Including Use of Sign Up Sheets
• I see myself trying to integrate workshops first into my review days to try to get some more small group instruction to students who need it. I love all the suggestions for using it during task facilitation, but it may take me some time to get there! 
• Solution Presentations
• I want my students to be sharing their solutions more- whether that's in a small group, gallery walk, writing, videos, or actual presentations. 
• One of the things I love was "Externalizing the Enemy"- if students need to present to an external group, then you as a teacher become a resource to help them prepare. This really tied to my experience working at the pharmaceutical company....my group grilled me on my presentation, but when I had to present to the entire program, I was truly prepared. I felt more comfortable being wrong in front of my group because I knew they could help me catch things I'd missed or said wrong and they'd give me the feedback I needed to make sure I ready for the "enemy." They talked a lot about how it was better to get caught by your lawyer in private interviews than to get caught in cross-examination in court. I like this idea that you're on the students' team and the added professionalism of presenting to an outside group. 
• Think about why students are presenting, how they will present, who will do the presenting, and to whom they will present
• Lesson Planning Structures
• Using the lesson plan templates and rubrics in the book will be a great resources for building lessons myself
• Make sure launching the task includes space to assess and hone student understanding of the task
• Use overly structured questions and tasks to create "hint cards" to hand out as you simplify the tasks and allow (require?) students to use their question prompt cards
• Set up a space in my classroom for "workshop" time
• Plan the debrief
• "Make homework work for your students, not the other way around"
• Plan to identify moments of brilliance, peer appreciation and encouragement, and speaking with every student
• Limit the number of questions students can ask the teacher during a task- "the one question"
• Assessment
• Analyze student work first, look for patterns and then aggregate data
• Spend more time with students helping them understand why they fall in one column of a rubric. This act allows for much more metacognition! 
• Have students help develop exemplars at the beginning of the year
• Shorten quizzes, add tasks to tests, allow for student discourse and collaboration on parts of the test, allow retakes for full credit
• Quick Strategies/Phrases
• "Teachers need to listen to students' ideas rather than listening for a particular response"
• "Permission to be inarticulate" 
• "It's easy to forget the Important for the Urgent"
• Full Year Planning
• Choose "Anchor Problems" to assign throughout the year so they are on the calendar, even if the dates are off
• I bought this tripod off Amazon so I can start recording my lessons and analyzing them myself. I will always be more critical of myself than an administrator- I'm looking forward to using this to improve my facilitation. I also liked that this has a bluetooth remote so I can record without students getting squirrely and noticing. 
• Practice routines more- even if it feels silly!
• Plan for panels/check ins with students to see how class is working for them (Information like that is priceless and "Pizza is Cheap")

Questions I Am Pondering:

• How can I best assess my own facilitation skills? 
• How can I help students be willing to scrap a failed solution, but continue to persevere? 
• How can I work within the confines of my grading system and PLT to improve my assessment?
• How can I be more of an agent of change in my own department? 

I can't recommend this book more! Mine is highlighted, written all over, dog-earred, and well-loved already.  I feel ready to jump into the rest of MTBoSBlaugust and all my planning for the new school year! 

Necessary Conditions: Quality Tasks

Before taking my current position, I worked at a STEM magnet school. Few things put me on the course to my current pedagogy as much as my time there did and tasks were a really key part of that. I did a lot of stumbling through the dark, experimenting, and seeing what worked (and failing forward when things didn't). I often feel like the idea of implementing tasks is one of the most intimidating to teachers- how can you give students all this time when this image makes us all feel so personally attacked:

And math teachers aren't wrong- a poor quality task is not worth the time it takes to implement it. That's what I really appreciated about the treatment of tasks in Necessary Conditions. Quality is king. 

Quality Tasks

Ideas I’m Implementing/Saving for Later:

• Tasks don't just allow for creativity- they should rely on it
• It's so easy for kids to hide behind rote, low level learning in many traditional classrooms. If they can regurgitate it back to you, GREAT! They can even do it with the numbers changed?! AMAZING! But those same students often cannot think around a problem posed differently or see when their knowledge might be necessary. They think math relies on memorizing processes and then implementing those "when the question looks like this" and who in their right mind would want to study THAT? Real world problem solving involves creativity that is intrinsic to true mathematics. I want to prioritize that in my classroom more. 
• I loved this quote from the book:  "If you deny students the opportunity to engage in this activity- to pose their own problems, make their own conjecture and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs- you deny them mathematics itself" (p.59)
• Two solutions from the book for this: 
• tasks that are primarily student generated 
• tasks that have a singular solution that can be arrived at through a multitude of solution methods 
• Curiosity is key
• Tasks should naturally spark curiosity- building engagement and buy in. I had 2 favorite suggestions from this section of the book:
• Solicit Predicitions! Start with one that is obviously too low and one that is obviously too high, then a best guess. These get students to think about what their results should and should not be- sense-making and allowing them to justify what they think. 
• "Steeped in Dissonance"- cognitive dissonance is such magic in the classroom. It gets kids thinking, wondering, trying to explain (or argue against!). This phrase gave me that warm, fuzzy feeling all over. You have to let kids sit in their dissonance a little to let them process when something is counterintuitive to them.
• Access is non-negotiable
• We talk a lot about "low floor, high ceiling" tasks in math ed and for good reason. If every single student in your class can't access the meaning of the task in some way, they have no way to be curious about it, much less get started. 
• One suggestion from the book for sussing out the level of access is asking students to restate the task on their own. This is so simple, but brought me back to my time at the NYC Math Lab. Kids will explain to you what THEY understand, whether that's what you want them to or not. It's my job to find where that knowledge is anchored- what they DO know, instead of what they don't. Again, let them talk. Strategies after empathy. 
• Simplifying tasks can lead you down a more complex road in the end
• I loved the discussion of simplifying what we ask- not only to extend access to more students, but also to open up the doors of creativity. It set off a lightbulb in me....I have been in a graduate cohort for the last 1.5 years that partnered with NASA and I couldn't figure out why so many of the NASA resources felt "off" to me. Here's an example about radiation in astronauts.

https://www.nasa.gov/sites/default/files/files/SMIII_Problem28.pdf

• It's got good bones- I like the real world applications and the use of data. But I've often felt like these feel "forced" to me- like a person said "HEY! There's math in this! I bet we can make this seem interdisciplinary!" Last year I started trying to open up my tasks more and it made a huge difference. This was my favorite- discovering the derivative. I asked me kids "What's the actual instantaneous velocity of a marble at this point?" and made them figure it out. It being open ended allowed them to ideate, test ideas, fail, and learn. It wasn't just a bunch of steps they were following because I asked them to do so. They were answering a big question in a way they saw fit. My goal this year is to simplify more. 
• There is task inspiration everywhere! 
• I loved the practical suggestions the book gave for adapting existing tasks. Textbook writers take a lot of time to write application problems, but they're often too scaffolded or too contrived. These tasks- which often rely on simplifying, as discussed above- help transform a meh question into a great task:
• Start with the answer
• Remove the steps and sub-problems
• Make the problem one of optimization (hellooooo, calculus)
• Choose your own problem
• Blur or withhold critical information
• Move from application problems to authentic experiences
• Keep fighting the good fight, even when kids fight you back!
• Sometimes you just need to hear this. A student that I taught 2 years ago and who graduated this year wrote me a card at the end of the year thanking me for not giving up on teaching the way I do, even though he fought me on it daily. He said it took him being years removed from it to realize the impact it had had on his ability to think. So for any of you fighting the good fight, keep fighting! 

Questions I’m Pondering:

• What are good strategies for a student who really struggles with accessing at-level tasks? Especially with open enrollment in classes?
• What are more resources for upper level tasks? Who wants to share some with me (hint, hint ;o) )? 
• What are the best ways to balance tasks with timing in an AP course, where you are already missing a month of instruction?

Last day tomorrow- effective facilitation & wrap up!