Integers [temperatures]

Yesterday, I co-taught an integers activity with a colleague. It was a blast! Before I share the lesson, I'll back up and give the backstory on the context. During winter break, I ventured over to Brian Head, Utah to do some snowboarding. I knew it was going to be cold so I went into the trip with the intention to frequently check my phone's weather app and take screenshots of temperatures. I figured I might be able to make an activity out of it and/or use it with 6th graders at some point when discussing integers. (official lesson page with resources)

My fellow displayed this slide and asked, "What do you notice? What do you wonder?"
(the 3 in the lower right is the slide number)

Students noticed and wondered great things. Here are just a few:

• What's the temperature at 9am?
• Why is it warmer on the days it is supposed to snow?
• Thursday is the only day with a negative temperature.
• It's 4:13 am.
• It's zero degrees at 5am.
• It's cold!
• How cold does it get?

We established that -4 degrees Fahrenheit is cold, below zero, and the temperature at 4:13 am. Let's plot this on a vertical number line today, just like a thermometer. Does -4 degrees go above or below -5 on the vertical number line?

I told students that we're going to show them five more times and their temperatures throughout the day. Most importantly, I asked students to first predict the temperatures at those given times (tap into student intuition). Here are the times:

• 6:00 am
• 7:00 am
• 9:30 am
• 2:30 pm
• 8:00 pm

Essentially, we're tapping into student intuition, a free resource in our classrooms. I want them to predict the story of temperatures and degree change for the remainder of the day. If anyone has experienced winter weather, they know it gets cold at night and warmer during the day, possibly peaking midday. It's a small part of the activity to keep it moving along and gain student investment.

Here come the temperatures. For each time and temperature revealed, here's what were going to do:

• Plot the temperature on your vertical number line.
• Find the degree change between the last temperature given.
• At the end, we'll find the largest difference in temperature during the day.

Here's a few of our whiteboard representations:

This was a simple and fun context to work with integers and the vertical number line. I also took screenshots of the temperatures in Celsius and might be able to make a Math 8 activity out of it. Here's the desmos rough draft.

The best part for me (as a teacher) was listening to students make sense of the temperature changes and explaining their thinking. There were so many opportunities to help students with their vocabulary. For example, when asked, "what's the difference between 12 degrees and -8 degrees?" it was interesting to hear how students wanted to change -8 to a positive in order to add it to 12. There was our intro to absolute value and a number's distance from zero. Love it!

One student came up to me on his way out and showed me his paper,
"Hey Mr. Stadel, I predicted the temperature correctly for each time!"
High-five!
I asked, "Do you want to pick my Powerball lottery numbers for this week?"
He declined. Drat.

Again, official lesson page with resources here.

Brrrrrrr! it's cold,
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Make It Happen

"This could have easily been a two- or three-day activity."

I literally just met with a fellow who made this comment after we debriefed on an activity she prepared for her sixth grade students. This comment was the result of how rich the activity was. The math was rich, students were collaborating, there was student thinking, great visual representations and she did a wonderful job facilitating the activity based on 6.NS.C.6.A.

I'm sure you can relate with the sentiments expressed by this teacher. You had an activity where students would benefit from it and you wish it could go on forever. I remember experiencing this too. Here are two solutions to this [great] challenge:

Short-term:
Subtract your homework, guided practice, and housekeeping stuff from your Classroom Clock for that day (and maybe the next day).
*I'll blog more about my definition of a Classroom Clock soon.

Long-term:
Do bits and pieces of this type of activity most days of the year so students are more and more familiar with these concepts and practices.

I prefer the long-term goal! (and assigning extremely little-to-no homework EVERY day)

Example:
In this case, the teacher prepared a Pear Deck activity where students would drag colored lines to represent the placement of fractions and their opposites on a number line.

The ratio of students to devices was 4:1. After each Pear Deck slide, the teacher provided a validation slide displaying the correct placement of each line. In addition to the pear Deck activity, she also had a clothesline number line across her room where students interacted with these same fractions in a different yet still dynamic way.

The teacher's learning objectives were met with this activity, but let's move toward retention. We talked about doing this more each day. (long-term solution). Here are some specific ways:

• Make these activities the two-three minute warm-up at the beginning of class each day. 
• Do them as a transition in your class. For example, "I need you to put your notebook away, get your whiteboards out, and when you're ready, place 3/10 and 7/5 on a number line." 
• Make them an exit activity before student leave for the day.

If you agree to provide students with access to rich math, find ways to make it happen more often.

Long-term,
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Productive Struggle [part 2]

Recently, I started thinking more about productive struggle:

• Blogged part 1 here
• Desmos activity here
• Comments and questions contributed here

Thanks to all (approximately 100) who participated. Here's an overlay of the Desmos activity:

I noticed (of those who participated):

• there isn't necessarily a one-to-one relationship of struggle to frustration.
• most participants will struggle for a bit before getting frustrated.
• some participants thought their frustration was logarithmic or exponential

Here are a few of the graphs.

I love that there's a story behind each graph. I love that the graphs are practically all different. Looking at the responses of math educators, it convinces me that we are just like a classroom of math students. Not surprisingly, we're all different and the relationship between struggle and frustration is different for everyone, much like the students in our classes. So what's next?

As teachers, we need actionable steps when working with math students who might be at various levels of struggle and frustration during an activity or task. From my experience, I believe the best actionable step is communication. Communication is what's next.

This idea of communication was shared by many math educators in the Desmos activity. At the conclusion of the activity, I asked for some final thoughts. If students are allowed to reasonably struggle with a math task, what would be a teacher prescription of actionable steps to:

• move the learning forward and 
• avoid a meltdown level of frustration?

Here are a few responses:

• using well-timed and small hints as needed
• humor helps
• constructive feedback that promotes reflection on the students thought processes
• have a set of scaffolded supports easily available and in multiple mediums
• hints and scaffolds are appropriate, preferably after significant struggle has already been felt
• reminders that everything worthwhile takes work 
• look at mistakes together and see what we can learn collectively
• let students explain their thinking, then discuss as a class
• turn and talk to your neighbor about this problem.
• develop a series of hints/question schemes to use with students
• determine the "breaking point" where the teacher should use whole group/small group instruction to address misconceptions to alleviate frustration
• remind yourself to ask questions before dispensing information
• remind your students that taking a break and coming back to the problem is sometimes a great idea
• pair them with someone else, ask them to break the problem down
• chunk the task or you can take a break and come back to it later
• learn to judge frustration level, have some strategies ready for each level
• I like using VNPSs so that they can look at other groups for ideas.

In my teacher mind, these all include some form of communication. Communication is happening either visually, orally, conversationally, or with questions and hints. THANK YOU for sharing these ideas so we all can make our classrooms a better environment for communicating. Many of these resonate with me, especially the scaffolded hints and questions. I'd like to share a few from my experience with students (and adults in workshops):

• be sincere in your communication and questioning
• anticipate student mistakes/misconceptions before students do the task
• compliment a strategy/idea/mistake students make that might nudge them forward
• ask another student to explain the task or sticking point to a struggling student
• explore a list of 26 questions to ask students (from Max Ray-Riek)
• listen to student thinking, not for the right explanation (see Max's Ignite)
• reassure students that you're confident they can solve it (or progress before the end of class)
• give hints that might refocus their energy on a small part of the task that's solvable or might gain them some momentum
• don't game students
• don't ask, "what do you think you did wrong?"
• don't ask, "do you think that's correct?"
• don't ask, "does that make sense to you?" (about the student's work)
• don't let students work too long on an incorrect strategy

These are two LOOONG lists and I could add even more to them. These lists are not for every student or teacher in every situation. The length of the lists and their variety convince me that it's best we know our students best when communicating with them during a math task in which they most likely will struggle. Some of these steps might appear as though I'm bailing a kid out or robbing them from some learning experience. Maybe. However, I believe we all have students that need strong encouraging hints that will give them momentum instead of me letting them struggle for too long and take the chance of losing them for good. Likewise, I don't see any point in letting students work too long on an incorrect strategy if I spot it while they're working. I'd rather use that valuable time to ask them questions and redirect them. 

I thought about writing a few conversations I've had with students, but every conversation is different because every student is different and their mathematical thinking can be different. The result is that conversations with students during a math task are many times like fingerprints; they're identifiable and we can learn a lot from them, but no two are the same.  Therefore, our communication with students should be customized to that moment. Does it help to have some "go-to" responses? Absolutely! Here are mine.

When walking up to a student for the first time, you can find me initiating the conversation with:

• Show me what you've tried so far.
• Tell me about what you've done.
• Do you know where you're stuck? If so, show and tell me about it.
• Where are you confused?
• I noticed you did this [something in their work]. I'm curious why you did that.
• I noticed you have something circled here. Explain that to me or tell me why that makes sense to you.

I try to ask questions or give commands that allow the student to communicate to me what they've tried, what they're frustrated with, or what sense they've made of the task. More often than not, I will try and ask the most-efficient and revealing question. In other words, the question that tells me the most information about their thinking in the shortest amount of time. The goal is to use this ice-breaker question/command as a quick formative assessment while I look at their work (or lack thereof). Okay, so what's next?

Depending on their response, this is where it gets tricky. Once a student has shared their work, thinking, mistake, or blank paper, we need to be in tune to what the student gives us or doesn't give us. We need to allow the communication thrive. I love the post AND comments in this blog post by Annie Fetter, titled One Example of a "Bad Hint." Bookmark this post and revisit it when you need it. There's some great insight from Annie and everyone in the comments. Since many math educators mentioned hints, it reminded me of her post and how it has helped me get better at questioning/assisting students during math tasks. What if hints don't always work? What's next?

Offer some assistance. There's nothing wrong with this. Yes, I agree with Being Less Helpful. However, we're working with students. Their brains are still developing. They (and some parents) see us as adults in positions to offer assistance in the learning process. I'm not advocating we bail students out. Yes, it's a fine line we have to walk between productive struggle and meltdown frustration with students (and parents) at times. Trust me, I've learned the hard way. I've walked away from students after making some naive comments like, "You'll figure it out." or "Keep working on it." or "Does that make sense to you?" when I knew all too well that they wouldn't figure it out, or working on it more will cause them frustration, or that of course it makes sense to them because they're the one who came up with the answer. DUH! I'm good with offering some assistance to students who really need it because it will save both of us some frustration and I'm not about gaming students. I've tried the following inexpensive response and it's gotten me some mileage so far:

How about we work on this together for about a minute? We'll get started together and then I need to go check in with other students. 

We're knocking at the door of perseverance with everything mentioned above. I should address it a bit. Again, communication is our ally here. I've found I get a lot more from my students when I communicate to them something like, 

I know today's task might be challenging. I'm confident you will work hard individually and together. Most of you will make mistakes, but I've seen how well you learn from mistakes in the past. You know I'll check in with you throughout the task, and might offer some hints if you're stuck. However, remember that I want you to first explain your strategy to me first.

Whatever you tell your students, be sincere and honest. Admit when you know a task will be challenging. Admit when you've given them a task too challenging. Likewise, admit when you've given them a task too easy. We're getting better at this teaching stuff, just like they're getting better at that "student" stuff. Honor the fact that one of your roles as a teacher is to both model and encourage perseverance. A lot easier said than done, right. But that's why we're here (#MTBoS) for each other. We share ideas with each other to make our math classrooms a better place for learning and teaching.

I'm not sitting down to write this post because I'm claiming I know all the answers. I had a few goals when initiating these two posts on productive struggle:

• learn how others perceive the relationship of struggle to frustration
• share actionable steps from other math educators
• share actionable steps that have and have not worked for me
• stress the importance of communication
• encourage others to offer timely assistance without bailing students out or being too helpful
• learn from other math educators how to best communicate with students so that they persevere through struggle

SO there it is. Your turn, please. What is your experience?

What are your go-to responses or actionable steps in moving the learning forward while encouraging students to persevere?

Steps,

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*Here's the link to the Desmos Activity if you want to adapt it and use with your students.