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,

840

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

Counting Dots

A teacher asked me about the Counting Dots activity I did in her teacher workshop I facilitated a couple weeks ago. I did the Counting Dots activity as a follow-up to Max Ray's Ignite talk: Why 2 > 4. I believe we teachers need to experience how valuable it is to listen to each other share strategies. If we're going to do it in our classrooms with students and value student thinking by listening to them, then we need to practice ourselves. You know? Build that muscle memory.

I was inspired by Dan Meyer's 2014 NCTM talk titled Video Games & Making Math More Like Things Students Like. You can find his specific reference to Counting Dots at the 30-minute mark. This link includes Dan's NCTM session and references to Ruth Parker's work, who according to Dan popularized Counting Dots.

I also made a video of the exact slides I used with teachers, a few extensions to counting dots, and a behind-the-scenes for anyone interested in making their own.

Questions? Let me know.

Dots,
1200

Featured comments:
Graham Fletcher shares an insightful article on Subitizing.

Dan Kearney shares more goodness from Steve Wyborney.

New Glasses

I have a new set of glasses. With my new role as a Math and Digital Learning Coach in my district, I get to spend a lot of time in the classroom with teachers and students. Let me explain the set of glasses.

When my son was born, I received a new set of glasses. When my daughter was born, I received an additional new set of glasses. As a new parent, I saw the world in a completely different way than I had before being a parent. The oversimplification for me was the stuff that I used to think was important before being a parent, instantly (and gladly) received a huge shift the second I started caring for and loving another human being that I called my son and daughter. The perspective from behind my parent glasses continues to change and update with time as my children mature. It's AMAZING! If you're a parent, regardless if your child is adopted or biological, you know what I mean. If you're not a parent, you simply have no idea. No offense. Trust me, I had no idea either. And if you're a parent and don't know what I mean, then please check yourself. But chances are good that if you fall into this latter category, you probably aren't reading this blog to begin with.  I'd imagine anyone who does read this blog is in education, and why would you be in education if you don't care about children, yours or someone else's? This could possibly be the harshest thing I've ever said on this blog. However, I feel that if you're in education, it's your duty and responsibility to care about children. I digress. Back to the new set of glasses.

We obtain new sets of glasses throughout our life. By the way, if you haven't figured out by now, anytime I refer to glasses, it's in the figurative sense: how you perceive the world around you through the lens created by your position, role, and/or relation in life. My new set of glasses as a Math and Digital Learning Coach allow me to be a student again. Some might think I should see the role of a teacher differently. This is true, but I think that's putting the cart before the horse here. I get to be a student again. And by being a student again, it helps me refocus as a teacher.

With my new set of glasses, I get to feel like a student again. Which is humbling and powerful at the same time. It has allowed me to focus on the student perspective of learning, exploring, and receiving knowledge as opposed to the teacher perspective of preparing activities, assessing, and delivering knowledge. Think how one role greatly affects the other.

I remember when I first started subbing and was helping students with math questions. At the time, I hadn't done math in a few years, so I was rusty since it wasn't fresh in my mind. Therefore, I was literally working alongside the students trying to figure out how to solve these questions. I was trying to tap into anything I could remember, which usually meant I had to derive some solution or maybe even take the long route to the correct answer. It was exhausting, thrilling, honest, raw, and exhilarating all at the same time. That moment in time flooded my head tonight on my drive home. That was such an awesome feeling, because I felt like a student of math, exploring and persevering. It made me want to pursue teaching. But then I got my teacher glasses and my student glasses went out of focus at times. It wasn't the glasses' fault. It was me. And I'll be the first to apologize for some lousy teaching moments because I lost focus. I wish I had a mulligan for one year in particular: 2012-2013. I tried a lot of new instructional/grading/facilitation/pedagogical strategies and shifts that were not necessarily well received. But let me tell you, I not only learned what to do that year. I learned what NOT to do and I am eternally grateful for the students and parents who (indirectly) made me strive to improve and refocus on the student perspective.

The point of this post was for me to reflect on the power of having these new (or rediscovered) glasses. I'm seeing that students need less and less from the teacher so that they are given the time and room for more ownership of their learning. That's an awfully general statement so I will explain. I believe that students benefit more when the teacher keeps the lesson introduction, question, or objective simple and succinct. The students respond better when the teacher is actively listening to them, not for the correct answer as Max Ray would say. I noticed that teachers are able to better listen to students when they ask interested questions like, "Can you tell me more about what you did here?" or "I didn't get all of that, can you explain it to me again?" or give simple commands like, "Walk me through what you did here." or "I noticed you did ____, please explain that."

As the student, I'd want you to ask me questions that show me you care about what I think and not judge me because I'm wrong and you feel you need to fix me (or my math work). With my glasses, I see students are drawn toward visual representations of mathematical situations: rectangles, patterns, estimations, soda, water bottles, etc. instead of just pure numbers (pure math). With my glasses, I noticed that students appreciate someone interested in their work. I noticed that students like to share with each other, especially when they know how to do something correctly. I noticed that students like to hear a voice other than the teacher.

I was able to notice many of these while maturing as a teacher, but for some reason they are more profound now. I hope these are some examples of how I see the teacher doing less and the students doing more. When I had my teacher glasses on and lost focus of the students' glasses, there were times when I (mistakenly) thought that I had to put on a show for my students or needed to run around like a crazy guy checking every paper or student whiteboard. Have you ever caught yourself saying something like, "My hair could be on fire, and they wouldn't care." or "I'm up there doing a song and dance, and it doesn't matter."? I know I've said something like that. That's exactly my point. I was doing too much, and the students were doing too little. That's exhausting.

If you've made it this far, thanks. I don't plan on rambling/ranting like this in the future. I'm simply happy to have rediscovered those student glasses. Throughout the year, I'm sure I'll continue to sharpen the focus on both my student and teacher glasses. My takeaway is this: by regaining focus on the student perspective of education, I'm able to improve my focus on the teacher perspective so the teacher is doing less in order to create more opportunities for the students in the classroom.

Glasses,
1149

Explain that, please.

Recently, I've given a few teacher workshops/conferences and have had the luxury of reflecting on teacher moves as I facilitate a lesson with the attendees. One of the many things we talk about are teacher responses to students.

Me: Did anyone hear me say, "No. That's wrong. You're wrong. I don't like your answer."

Attendees: No.

Me: Right. Instead, you'll hear me say things like, "Can you explain what you did here? Explain that, please. I noticed you did [this] here, please share how you got [that]. I'm curious how you came up with that. Walk me through what you did."  

I tell teachers that I'm taking the emphasis away from right versus wrong answers and placing an interest on the student's thought process and problem-solving. I continue with teachers:

Me: By telling a student they're wrong, a student can have the tendency to shutdown [I make the sound effect of a machine shutting down, "BOOOOvvvvvvvv"]. By asking a student to explain things, it shows that I'm more interested in how they arrived at their answer. 

As teachers, we know a student can be told they're wrong and it's easy for them to give up. On the flip side, when we validate a kid by telling them they're right, the student can also shut down and never reach the higher levels of Depth of Knowledge.

Recently, a workshop attendee asked me how I respond to students who have nailed the answer to a 3 Act task. First, I have them explain their problem-solving plan to me. Second, I question any details that were unclear, encourage them to be more precise, or have them explain their units of measurement. Third, I ask them if they feel confident in their answer after explaining it to me. Fourth, I validate them by simply saying, "That makes sense to me."

I don't tell them they're correct. I treat them just like as if they got the answer wrong. If that doesn't satisfy them, I respond with, "We'll find out soon if you're correct, but that (their explanation and work) makes sense to me." At this point, I offer them an extension to the task. I'd like to talk more about this later, but usually the extension revolves around the students creating something with the new knowledge or skills they have just recently gained.

After all that, please add your favorite lines when questioning students to this Google doc. I think it's also helpful we create a list of lines we avoid using with students as they explore math.

Here are a few people with other stellar teacher moves/lines to support students.
Max Ray: 26 Questions You Can Ask Instead
Dan Meyer: You Don't Have To Be The Answer Key
David Cox: Creating A Culture Of Questions
Steve Leinwand: Accessible Mathematics

BOOOOvvvvvvvv,
645

CMC North 2013

Here's a recap of my CMC North experience from last weekend. 

Friday:

Fawn Nguyen and I flew up to Monterey Bay, checked in, and found the Fishwife restaurant for lunch. It was within walking distance of the conference grounds. Thank goodness because it was a little chilly outside. We enjoyed lunch so much we made a trip back there for the 4-6 p.m. Happy Hour so we could fine-tune our Saturday presentation. We checked in with the CMC people, and if you ever make it up to Pacific Grove for the conference, don't forget the badge they mail you. It'll cost you $5 if you forget it, right Fawn?

Later that night, we attended the opening keynote by Dr. David Dockterman where he discussed the importance of a growth mindset.  Afterward, it was great to see and meet up with Fawn, Dan Meyer, Avery Pickford, and Breedeen Murray for some pizza. Good times. Good company.

Saturday:
Session 1:
The first session I attended was titled Using Formative Assessment to Create Equitable Practices by Karen Mayfield-Ingram. We diagnosed some student work and left with the following reminders:
1) Make a concerted effort to give students detailed feedback on any assessment.
2) Try and ask students questions that will encourage them to analyze their work better.

Session 2:
I showed up early at Larry Armstrong's session titled Flip Instruction to Transform Learning. I'm interested to know more about the flipped classroom. Not because I'm sold on the idea, but because my district is getting iPads the second semester and I have a feeling we will be encouraged to implement some type of flipped classroom model. Who knows? Again, I'm not sold on the idea, but I want more information just in case. Well, it's 5 minutes before the start of the session and the front of the room is empty: no computer plugged into the projector, no one is setting up, no Larry Armstrong. Nothing!
A CMC helper comes to the front to inform us Armstrong won't be presenting and we have time to catch another session if we want. I saw Brad Fulton come in at the back of the room and I got up to ask him if he'd present, but he said he was only coming to drop his stuff off for presenting at the following session. I turned to the CMC helper and offered to present my Number Sense session I gave at CMC South. She took me up on my offer and we had ourselves an impromptu Number Sense session. It was fun! For the forty to fifty attendees that stuck around, we had a blast doing estimation challenges and talking number sense for about 50 minutes. I will blog more about my presentation over the next few weeks. Stay tuned! I was honored when Rebecca Lewis, the Program Chair, gave me this token of appreciation for stepping in at this session. This proudly sits on my desk.

Session 3:
Shelly Lawson gave a great hands-on session titled Modeling Lessons Can Work for All Students - Yes, Even Yours.  I was excited about this one because it was geared toward 7th grade curriculum. You know you're in for a good session when you see a bag full of PVC pipes, a stopwatch, a meter stick, a steel ball, and some string. There were 5 different length PVC pipes, and four connecters; two right angles and two at about 135 degrees.

We had to construct a pipe so that the ball, when dropped, would make its way through the pipe at the fastest rate possible. Here's our contraption. It was the fastest because of my teammate's design. Great job Bob!

Shelly also introduced us to the Incredible Egg, Float That Boat, and the Penny Lab. All of these activities allow students to make measurements, mistakes, and formulate conclusions based on observations and data collected. I left with a packet full of hands-on activities that I can incorporate into my curriculum. The structure of the packet gives students pretty detailed steps and instructions. Personally, I will probably revise the activities to leave them a little more open-ended and less structured. Overall, some great activities. Email Shelly for a copy. See the link attached to her name above. By this time, I think I ran into Dittmer about 50 times. Really cool guy!

Session 4:
Fawn and I presented Hotel Snap to the CMC attendees. Fawn is awesome! Tell you something you don't know, right? Okay, chances are good you've already read her blog post on Hotel Snap. If not, go now and read it here. Fawn did an amazing job coming up with this task. You might be surprised, but I have nothing but great things to say about Fawn. Her task is challenging and can be used at numerous grade levels. CHECK IT OUT!

Photo by Dan Meyer

Thanks to Brian and Dan for helping us with the calculations. Thanks John for helping us clean up!

Session 5:
My man, Max Ray, presented Becoming Better Reasoners: Supporting Students to Develop as Problem-Solvers. I was excited since this was my first time seeing Max present.  See how calm he is in this picture?

I love how Max is so patient and allows his students (us CMC attendees) to formulate noticings, wonderings, and other thoughts as we worked through some Math Forum tasks. I enjoyed this last session of the conference because we worked in groups to problem-solve a task, we analyzed student work, and Max charged us with asking students questions that would help them further their problem-solving approach by becoming better reasoners. Well done Max!

Saturday night:
Ignite talks!!! I could write a whole other blog post on these. I was honored and privileged to present with the following mathletes!

The Math Forum organized, hosted, and took over the Ignite talks this year. Thanks Suzanne for all you did! Watch them in January or February 2014 when they post the 10 videos online. It was a lot of work preparing my 5 minute talk, but was great fun! Fawn stole the night! The energy and inspiration from these talks was a great way to cap Saturday night!

Sunday:
Dan Meyer gave the first keynote of the morning titled Fake-World Math. Dan is the man! His presentation was fantastic. I don't want to post any spoilers here, but he talked about the importance of modeling and how CCSS defines modeling. If I could summarize some important points, here's what I want to reflect on for future reference.
Although important to the math classroom at certain times, the following is NOT modeling:

1. I do, we do, you do
2. Using concrete manipulatives
3. Graphs, equations, functions

Furthermore, Dan stressed that the real world is not always greater than a math problem. Vice versa, a math problem is not always greater than a real world situation. He emphasized that strong modeling starts with a simple question and allows students to identify variables, formulate the necessary model to organize and solve the task, and to validate conclusions a la something along the lines of a 3-Act task. He reminded everyone that modeling tasks are out there: Dan, Robert, and here. My summary can't do justice to his presentation, content delivery, and charisma. If you get a chance to see Dan live, DO IT!

*For the record, my initial estimate for Dan's Super Stairs task was over by fifty seconds. Bam!

Dr. Timothy Kanold gave the second keynote titled The Art of Teaching Mathematics: Inspiring Students to Learn. CMC North 2013 ended with Bill Withers' Lean on Me.

Before Fawn and I flew back to Los Angeles, we had brunch with some great math amigos (pictured below).

Left to Right: Stadel, Nguyen, Meyer, Murray, Pickford.

We also said hi to some fish at the Monterey Bay Aquarium.

Thanks Dan, Avery, and Breedeen for driving us around too!

North,
819