Goodbye, U-Substitution

 I wrote a bit about my journey of saying goodbye to U-Substitution on the One Good Thing blog (here and here). As I said in that blog, I will probably never go back to teaching U-Substitution. When I see u-sub, I see it as a way to simplify the integrand so I can easily reverse Chain Rule. But, no matter how many colored highlighters I got out or how slowly I went, my students seemed to only ever see it as a set of abstract steps that they would get tangled up in. Mathematics should be revealing. For my kids, u-sub was muddling. The whole idea of u-sub is to simplify. If it's not simplifying things for your kids, it may be time to leave the u-sub train.

Some have asked for more details, so here we go!

I spent about four days on this:

Day 1
Review of Chain Rule and begin to reverse Chain Rule only when the inside function is linear

Notes |  Problem Set

My kids worked the problem set together around the room on vertical whiteboards (absent students were asked to work questions on the PDF).

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Day 2
Anti-Chain Rule Part I: What happens when the inside function is not linear but when our integrand is in the perfect form of f'(g(x))g'(x)?

Video lesson | Blank Notes

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Day 3
Anti-Chain Rule Part II: What if there's a coefficient left over?

Video lesson | Blank Notes

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Day 4 
Anti-Chain Rule Part III: Can we apply what we've done to definite integrals? 

Video lesson | Blank Notes

I taught these lessons live (typically I flip) so these videos are my first draft, but will give an idea for what we spent time on in class.

What else do we insist on teaching in muddling ways? What do we need to reinvent? 

Prayer for 2020-2021

I ran into a former student today. “Mrs. Peterson?” she asked (it’s been years since we’ve seen each other).

“H? How did you recognize me with sunglasses and a mask…?”

“It was your voice…”

Your voice. Our voice...

************

Compassionate Heavenly Mother,

As we enter a school year unlike another, grant us grace—grace for ourselves, grace for our students, grace for their parents, grace for our coworkers. Remind us—constantly but gently—that every single person has been through so much in the last six months, including ourselves. Free us from the temptation of comparison, as we look at what used to be and what could have been. Rescue us from our own thoughts.

We are so anxious, Mama. Still, help us find creative solutions to meet every kid. We ask that we would feel a closeness and kindredness to our students that can only come from the Divine; that we would be flooded with renewed patience and understanding; that amidst all the unknown we would search for and find true JOY...the kind of joy that still makes room for hurt and anxiety and loneliness and can say, “Baby, I’ve been there. I see your ache. I know the pain. But I’ve come out the other side and I say let’s enjoy this day together.”

May our voices be a soothing salve to the hurting. May they find comfort and solace in our rooms—whether physical or virtual. Grant us the tenderness of a new mama. Open our eyes to see and honor the Light in every soul. Open our ears to hear each story. Open our hearts to let each story move us. 

Give us the words when we start to grasp. Grant us the wisdom when silence is needed. And give us the courage to sit and cry with the brokenness.

In the midst of the unknown, in the midst of the chaos, remind us that You are the God who sees us tossing and turning at night, the One who collects our tears and records them in Her book (Ps. 56:8). Remind us we are not alone. We are not unseen. 

Finally, I ask for what feels impossible right now. I ask for unity. I ask that we see both the humanity and the Divine in one another. I ask for less strife and more open doors. I ask for less talking and more listening. I ask for less judgment and more grace; fewer assumptions, more understanding. And start with me, Oh God. Start in my heart. Purify my heart so I can see the world through Your eyes. 

Amen. 

Virtual Assessment Ideas

Type #1: Explain 

Have students take a test on their own (don’t grade the tests). Once they’re done, give students the key and have them grade their own. 

Ask students to make corrections and then create a video of their explanations of the three (or however many) most difficult problems for them. They may or may not be questions they initially got incorrect. Students should submit their initial test, corrections, and video. Grade is given on accuracy of grading, accuracy and cleanliness of corrections, and appropriate video explanations. Grade is not based on initial number of questions missed as this would encourage nefarious means. 

Type #2: Create

Have students create their own test, key, and solutions. Give detailed directions on number and type of questions. 

For multiple choice questions, have students explain why the distractors are good. For free response questions, give the students the questions but have them create the stem and the answer key. 

Zoom with iPad or iPhone (so your kids still see your face!)

I wanted a way for my students to still see my face when I worked problems in front of them on a Zoom call.  I think it’s important for them to see my facial expressions and it’s important for me to be able to “look” at them.  You already lose so much via a Zoom call--I wasn’t willing to lose what little interaction I had.  I have an iPad and Apple Pencil, so I used the first method below to write on my iPad and share my screen (while also keeping the meeting open on my computer so my kids could still see me).  See photo below:

If you don’t have a tablet/stylus, you can use your phone’s camera feature to share (for example) a paper that you’re writing examples on.  Think of it as a document camera.

With iPad

1. On your iPad, first add Screen Recording to your Control Center if you haven’t already (Settings>Control Center>Customize Controls. Push the green plus sign next to “Screen Recording”).
2. Open the Zoom app and start the meeting from your iPad (this is important so that you’re the host on your iPad and not your computer so you can share your screen).
3. Turn your video off (so you don’t broadcast an unflattering view of yourself…).
4. Mute yourself.
5. Click the “More” button on top right and select “Disable Audio.” 
   This will ensure you don’t have feedback/echo from having Zoom open on two devices open.
6. Now switch to your computer.  Open Zoom and start the same meeting from your computer.
7. On your iPad, let yourself into the meeting.
8. Run the meeting as normal now.  Keep in mind, you are currently hosting from the iPad, so you’ll need to let people in through the iPad.  To share your iPad screen, hit the green “Share” button, then select “Screen.” The first time you will need to select “Zoom,” and then push “Start Broadcast.”  You are now sharing your screen and can toggle through windows as normal.  A nice feature is opening your camera app so you can show whatever documents, books, etc are in front of you.

With iPhone

1. On your iPhone, first add Screen Recording to your Control Center if you haven’t already (Settings>Control Center>Customize Controls. Push the green plus sign next to “Screen Recording”).
2. Open the Zoom app on your phone and start the meeting from your phone (this is important so that you’re the host on your phone and not your computer so you can share your screen).
3. Tap anywhere on the screen to see menus.  Turn your video off (so you don’t broadcast an unflattering view of yourself…).
4. Mute yourself.
5. Click the “More” button bottom right and select “Disconnect Audio.” This will ensure you don’t have feedback/echo from having Zoom open on two devices open.
6. Now switch to your computer.  Open Zoom and start the same meeting from your computer.
7. On your phone, let yourself into the meeting.
8. Run the meeting as normal now.  Keep in mind, you are currently hosting from your phone, so you’ll need to let people in through your phone.  To share your phone’s screen, hit the green “Share” button, then select “Screen.” The first time you will need to select “Zoom,” and then push “Start Broadcast.”  You are now sharing your screen and can toggle through windows as normal.  A nice feature is opening your camera app so you can show whatever documents, books, etc are in front of you.

Happy Zooming! I hope this doesn't have to last too much longer...

Mindfulness

I started practicing mindfulness with my students last spring.  I've gotten a lot of positive feedback on it from both students and peers.  My precalc kids from last year who lopped with me into calc this year asked (told) me to continue this tradition in calculus as well. I thought I would post my script here in case it's something you want to try with your kids.  I currently teach pre-AP and AP students but I have a colleague who does this with her on-level students as well, so I'm convinced it works for all learners.

I first set the tone of what mindfulness is and why it's so important, what focusing on the present can do for our brains.  I highly recommend showing both these short video clips (or something similar) to your students before you start this work:

• What is Mindfulness by Scientific American
• How Boredom can Lead to Brilliance by Manoush Zomorodi

I like to play quiet music or nature sounds in the background and turn off the lights.  I let my kids stay in their seats or get on the ground, but that's of course entirely your call.  Some background videos I've liked are the following:

• Nature clips with relaxing music
• Rain
• Winter calm

Alright!  Here's my script.  It's not all my own words.  When I was looking for a mindfulness script, I couldn't find quite what I wanted, so I edited others' scripts and pieced them with  my own.  This at least gives you a starting point if you're like I was six months ago and just wanted something to go off.  I will post two other scripts at the end that I've used with minor modifications.

Start by settling into a comfortable position and allow your eyes to close or keep them open with a softened gaze. Begin by taking several long slow deep breaths breathing in fully and exhaling fully. Breathe in through your nose and out through your nose or mouth. Allow your breath to find its own natural rhythm. Bring your full attention to noticing each in-breath as it enters your nostrils, travels down to your lungs and causes your belly to expand. And notice each out-breath as your belly contracts and air moves up through the lungs back up through the nostrils or mouth. Invite your full attention to flow with your breath. 

Take a deep breath in for 1, 2, 3, 4; now exhale out for 5, 6, 7, 8.  Inhale 1, 2, 3, 4; exhale 5, 6, 7, 8. Keep inhaling...and exhaling. Try to focus on your breath or the sound of my voice if you feel your mind start to wander.  Inhale...exhale. If you’re comfortable, place a hand on your heart and feel your chest rise as you inhale...and contract as you exhale. Inhale...exhale. Inhale...exhale.  Now take the deepest breathe in you’ve taken...and the longest breath out.

As you continue to breathe in and out, try to keep focusing on your breath.  

Inhale love; exhale resentment.

Inhale courage; exhale fear.

Inhale strength; exhale weakness.

Inhale joy; exhale comparison.

Inhale kindness; exhale resentment.

Inhale confidence; exhale doubt.

Inhale intentions; exhale expectations.
Inhale hope; exhale fear.
Inhale inclusion; exhale judgement.

Inhale forgiveness; exhale blame.

Inhale passion; exhale indifference.

Inhale grace; exhale the need for perfection.

And as you take a deep breath, bring in more oxygen to the body. And as you exhale, have a sense of relaxing more deeply.

You can notice your feet on the floor.

You can notice your legs against the chair, pressure, pulsing, heaviness, lightness.

Notice your back against the chair.

Bring your attention into your stomach area. If your stomach is tense or tight, let it soften. Take a breath.

Notice your hands. Are your hands tense or tight. See if you can allow them to soften.

Notice your arms. Feel any sensation in your arms. Let your shoulders be soft.

Notice your neck and throat. Let them be soft. Relax.

Soften your jaw. Let your face and facial muscles be soft.

Then notice your whole body present. 

Take a couple minutes to focus on your breath, remembering that just being in the moment--not worried about the past or stressed about the future--helps your neural resources to grow.
After about 30 seconds: If you find your mind wandering, just notice that wandering, without judgment.  Then, bring your mind back to your breath.
Another 1-1.5 minutes of breathing.

May today be a day where you know you are loved and valued.

May you accept that love and give love.

May you know you are safe, cherished, and wanted.

May today be a day where you experience grace: both to give and to take.

May today be a day where you are connected with yourself and those around you: aware of one another’s needs and willing to both give and receive help.

May today be a day of joy and thanksgiving.

Take one more deep breath, and begin to bring your awareness back to the room.  Wiggle your toes and fingers. Gently open your eyes. Lights are coming back on.

Other scripts you may want to consider/modify for your classroom:

• Gratitude meditation
• Loving-kindness meditation

I would really love to hear from you if mindfulness is something you use in your classroom. I'm a newbie, but I'm learning! Also, Calm is an amazing app if you want to start practicing mindfulness personally...PLUS educators can get a subscription for free!