Third Year Thoughts Part I: My Reflections

My last day is Tuesday, but my students promoted last Thursday.  Naturally, the last couple of days have led to a lot of introspection about the past school year as I’ve packed up my room and taken care of a lot of odds and ends with data reports.

When I look at the content of my blog posts this year, I notice that I didn’t talk about content as nearly as much as I wrote about the human element of my teaching.  I also notice that I didn’t write often and I fell out of the routine of keeping my personal journal.  More often than not, I discussed lessons with colleagues or just prayed about students on drive home.  The following may seem like a stream of consciousness without unity, but I need to write these thoughts down as reminders for the future.

This year was my first teaching 8th graders, planning for subjects besides math, and teaching without a block schedule.  I walked into the year apprehensive at the thought of teaching some students for the second time, not knowing how to make a science class work, unaware of how unique teaching PE would be, and uncertain about how much more responsibility I could expect out of my students.  I was also kind of bitter with all of the changes I was experiencing.

In retrospect, I think my attitude at the beginning of the year characterized some of the struggles I had later in the year.  Objectively, I was probably just dealing with typical growing pains.  Subjectively, I was frustrated that I was spending so much time adapting instead of deepening the quality of my questioning, scaffolding, and other academic elements of my teaching.  My attitude got in the way of seeing the truth of the situation.  Ironic, when you consider that I teach 8th graders.

In terms of my management, this year revealed my weaknesses and strengths.  Routines were fine once we established them, but I found myself altering what a “typical day” looked liked as the year unfolded.  I realized the value of asking students continually about what works, what motivates them to work, and offering options.  I held class meetings, which always sounded dumb in theory, but proved to be valuable in practice.  Sometimes the meetings were collaborative (“I don’t know what to do. You tell me what to do about it.  I can be the teacher you want me to be.  Let’s use this time.”), while other times meetings felt polemical (“You know what to do. I’m reminding you to do it.  I’m being the teacher you need me to be.  Don’t waste this time.”)

I realized how much I want my classes to be relaxed, but productive.  I know learning can occur in silent classrooms or loud classrooms, but for my sanity I need something in between those extremes.  I learned that I need organized group work as much as my students need it.  I need the days where students are working through practice or projects just as much as they probably need that time.  I learned that I need the balance between me doing stuff and my students doing stuff.  I have to stop rushing and be more flexible with my plans.  At the same time, I need to help my students find the happy medium between rushing to get things done and somehow only getting 6 problems done in 20 minutes of partner work.

I discovered how much I need to create time to talk with students individually.  I was inconsistent with this practice this year, but I know my students need these conversations (both good and bad) as much as I need them.  In some cases, I regret that I didn’t really recognize how much a student changed or worked this year until the last week of school.  I also discovered that I honestly knew very little about some of my students as the year drew to a close, which I cannot change but I still regret.  I know talking to students more will help me be more aware of how much they are growing and who they are outside of the four walls of my classroom.

I need to say, “Thank you,” more often and for more reasons besides a student or class working hard.  I need to say, “I’m proud of you,” more often.  I need to say, “I’m here for you,” more often.  I need to say, “I’m sorry,” more often.  When I consider the students who worked hardest for me this year or filled me with joy to the point where I was holding back tears, I realize those phrases popped up multiple times in conversations with them or with their classes.

I need apologies and forgiveness to become part of the culture of my classroom and my teaching.  I apologized on many occasions this year when a lesson flopped because of me.  I admitted when I was too harsh on a class.  I told students to apologize to each other for rude comments.  I asked classes to show me they were sorry, which in a way was asking for an apology.  For as much as I promote responsibility through my teaching, I know that I need to include apologies as part of that expectation in the future.

I regret how often I put off calling parents or sending a quick email home.  In the past week, I called some parents to let them know how much I enjoyed having their children in my class.  While it was a great way to wrap up the year, I realize that I should’ve made more of these calls throughout the year.  These calls might have been the third or fourth parent contact for me during a normal week, but it might be the first teacher contact for the person on the other end of the line.  I also look back at the year and realize how many behaviors or attitudes could have been improved simply by sending a strongly worded message home about consequences.  I know I teach students who are reaching the age where I don’t have to call home about everything, but I need to improve the frequency I communicate with parents.

This post was really heavy on the first person metacognitive language, but I need to have something to look back to in the future.  It was the product of processing the events of the year.  In my next post, I’ll include some of the things my students and coworkers taught me throughout the year.

 

 

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“You tried so hard…”

Last week I was chatting during lunch with another teacher in his classroom.  After a couple minutes, a student walked in and joined us.  The student had me for 6th grade math two years ago and has the other teacher for 8th grade math this year.  Somehow or another, the conversation meandered from discussing random things like the taste of La Croix and tattoos to a critique of how different things are now compared to two years ago.  Along the way, the student mentioned the teaching style and attitude of the other teacher, my current demeanor, and what I was like as a teacher two years ago.

Before I write further, I must state for the record that I was a first year teacher when I had this student in class.  I was adapting to teaching 6th grade from student teaching in a high school.  There were many days where it wasn’t a question of if a lesson was mediocre.  I’ve also seen how this student has grown to be incredibly honest with teachers from being very reserved.

A few comments this student made stuck out to me as proof that I have grown as a teacher.  The same comments made me wonder if the challenges I’ve had this year would elicit similar comments from students a few years from now.  Rather than attempt to write a bunch of paragraphs, the student’s comments are in quotations below and my related thoughts are in bullet points.

“You tried so hard that year to find something that worked for us.”

  • It’s true.  I tried a bunch of different homework arrangements, class activities, exit tickets, quiz formats, management strategies, and notetaking arrangements.  By the time I finally hit upon the “typical” class that worked for that group of students (checking homework, a guided note page that would be secured into spirals, a short break, a practice activity, and five minutes to start a brief homework assignment), the year was practically done.  When I used this format with my next group of 6th graders, it worked fairly well from the start and I tweaked it along the way.

“Every week was you trying something different.  Some things worked, but a lot of things didn’t.  A lot of times it wasn’t you…  It’s just that a lot of us didn’t know what to expect.”

  • Consistency was a struggle for me that year.  Everyone likes to have some predictably about class.  Even though teaching necessitates breaking from routine from time to time, there needs to be routines to break from if I want changes to have an impact.  I recognized I needed to change things up over the course of this year, but I at least tried to stick with things for a while because I knew it would help students feel comfortable.  It also helped that I had class meetings with my students this year to figure out what was working and what needed to be changed.  During my first year, I think I underestimated the value of asking students about the structure of a class.

“You put up with a lot that year… I was disrespectful a lot, but you kept trying with me.”

  • Hopefully, I won’t lose that determination.

“I finally figured out that if I just worked, everything went a lot better for me.”

  • I talk about work a lot when I teach.  I stress its value.  I remind students that learning takes work, but work also says a lot about a person’s character.  The quality of a person’s work can communicate respect, attitude, and motivation.  I cite work ethic when I have individual conversations with students, call parents, or write positive referrals.  I sometimes doubt if I make any difference when I talk about work on an almost daily basis.  I guess I shouldn’t be so pessimistic.

This post is future reminder for myself.  I know I’ll need at some point when a lesson completely flops or I’m wondering if my persistence is in vain.

 

Genuine Enthusiasm

Last week, my science classes learned about fossils and fossilization.  The lessons were meant to be a brief introduction and support our study of superposition and relative dating of rock layers.  Out of all of the geology topics my science classes have explored since November, fossils was the topic I looked forward to teaching about most.

I was the kid with the dinosaur book in elementary school.  Although my love of math led me to specializing in that subject in college, I took paleontology and a mix of other science courses, too.  I’ve always loved Earth history and I wanted to have another endorsement on my teaching license.  Along the way, I got to examine a bunch of fossils in labs, went on a trip to the Field Museum to visit the off-exhibit areas where scientists conduct research, and I dug up fossils at a quarry.  Besides these experiences, I traveled around Arizona over the course of a few summers and paid close attention to how many of the impressive natural sites formed.

Unsurprisingly, I brought genuine enthusiasm to my lessons about fossils.  My questions started off simply asking students about prior knowledge, getting them to reason that most fossils record the densest body parts of an organism, and relating the ages of fossils to the rock layers within which they are found.  I showed a video from the Field Museum, then I followed up with questions about what students watched and I shared my personal stories.  In all of this dialogue, I reflected later that my voice was stronger, my eyes opened wider, and I moved slowly about the room with a swagger that I haven’t felt in a while when teaching.

After sharing about the effort required to find and collect fossils, I directed students to work on a summary page as I showed smaller groups my fossils.  Even though it meant a lot of repetition on my part, I talked to group after group of 4 students about my Silurian and Ordovician fossils.  I reminded students of the effort it took me to get the fossils as they carefully held the fossils and examined the specimens.  I used the opportunity to reiterate the concepts we discussed earlier.  I questioned students about what they noticed in the rocks, explained what organisms living today are similar to the fossils, and answered student questions.  I had students asking me questions about the fossils that I haven’t seen raise a hand for weeks of class instruction.  I was brimming with delight as students asked me why everything looked like it came from the ocean, where I got the fossils, why I studied paleontology, and how come some fossils looked broken.

I owned that lesson.  I knew it, but students knew it, too.  Looking back at that day, I definitely point to enthusiasm as the determining factor of the lesson’s success.  The same activities (short Q&A, lecture, video, worksheet, and small group hands-on activity) would have flopped if they were devoid of the personal stories, confident delivery, insistent questioning, and opportunity for students to wonder.

I share this reflection as a reminder for myself.  Regardless of the subject matter, I need to find more of the content I teach that provokes genuine enthusiasm on my part.  Even if it’s just a short personal story about how I’ve used a concept, why I value something without applications to everyday life, or my experience learning something for the first time, I need to bring that human element into my teaching.  Enthusiasm will make my delivery stronger, my questioning more inviting, push me to make lessons more interactive, and invite students to question more.  While adolescents can mask their enthusiasm and attempt to derail even the strongest of lessons, it’s hard to ignore (or disrespect) a person who is obviously passionate at the moment.  Even if a student doesn’t love math or science as much as I do, hopefully my enthusiasm will at least get students to respect the subject matter, respect people who are passionate, and feel a little better about being passionate about their interests.

I can’t be the person asking all the questions.

From the start of the semester until last Thursday, my students were working with systems in some form.

Graphing.

Substitution.

Elimination.

Word Problems.

A month and a half of teaching, practicing, assessing, reteaching, practicing, assessing, teaching, practicing, assessing, reteaching… you get the idea.

Linear systems was probably one of the most difficult concepts I’ve taught this year.  The ideas and methods involved literally pull together all of the algebra and rational number skills students learned in 6th and 7th grade.  There’s so many ways a minor mistake can derail a train of thought.  There’s also the concepts of variable, maintaining equality, substitution, and representing situations with algebra.  Even after so much time working with systems, I’m still not certain some students truly understand that substitution is just swapping out a variable for something of equal value or that we need to create an equivalent equation with one type of variable to be able to solve the system.

Without surprise, I found myself using a variety of questions throughout my lessons to help students make sense of a system.  As we got deeper and deeper into the content, I found practice activities devolved from productive group work to people attempting to do the same thing without regard for the problem or throwing up hands and declaring, “I need help.”

While recognizing that help is needed is a good starting point, later in the unit the process became exhausting.  A hand goes up from a student looking for help.  I ask the questions. Students answer correctly or incorrectly.  I tweak my approach if students were incorrect or move onto another question if the student was correct.  Rinse and repeat.  Later in the unit, my questions were pretty much the same and were able to guide students:

  • Can you read the system/problem aloud for me?
  • What are the variables in the situation?
  • How could we represent the word problem with equations? Why do we need two equations?
  • Does either equation have a variable that’s isolated?
  • Do you see any opposite coefficients between the equations?
  • Looking at the system, which method (substitution or elimination) would require less work/algebra?  How do you know?
  • Have we created an equation with only one type of variable?
  • We ended up with a nonsense equation? What does that mean?
  • We ended up with an equation that’s always true? What does that mean?
  • Do we have the whole solution to the system?
  • How can we find the value of x/y that we know the value of y/x?
  • Have we answered the question?

The interesting part about my questions is that most students were able to correctly answer them and got used to the routine. Nevertheless, students did not adopt these questions for themselves even with repeated reminders from me.

In some respects, I felt like I was getting trapped in a cycle I experienced during my first year of teaching.  I was somehow becoming the sole questioner in the class.  Naturally, I started to get down on myself about this development.  What was I doing or not doing that was causing students to avoid questioning themselves?  Was it the difficulty of the content that was driving students to only seek help from me?  Was it my explanation of concepts and examples?  Was it my progression of concepts?  Was it my pace?  Did the behaviors I was seeing reflecting a deeper problem that never got resolved earlier in the year?  Was I not explicit enough in directing students to self-question?  In a year that continues to make me doubt my abilities as a teacher, you can imagine that I was rapidly falling down the rabbit hole as I asked myself these questions about students not asking themselves questions.

After a week of what felt like nonstop questioning on my part and very little productive practice, I finally reached my breaking point.  I declared to my classes, “I can’t be the person asking all of the questions.  Time after time, people stare at a problem and call me over claiming they need help.  Every time I do help, I always ask the same handful of questions every single time. Every single time. Every single time.  Curiously, people are usually able to answer these questions every single time. Every single time. Every single time. At what point do people stop asking me for help and start asking themselves the questions they already know I’m going to ask?  Since day one, I’ve told you to ask yourself questions.  What’s been happening here is exactly why I’ve told you to engage in this process. You do know. You can make sense of the problem. Even if you’re uncertain, you know something you can always do to help you reason with a problem.”

At this point, the class was dead silent and I continued on for another couple minutes acknowledging that the content might not directly to the lives of my students, but the thinking they engage in as they work with systems builds the reasoning skills they will use later in life.  I ended my speech by reiterating the need for students to ask themselves questions.  In my later class, I wrote my questions on the board and stated I expected to hear and see students using these questions during the next practice activity.

Whether it was my delivery of my message or its content, students worked incredibly hard during the remainder of class.  When hands went up and I assisted students, the nature of the exchanges were different.  Everyone had something on their paper that they referred to as they spoke to me. In some cases, students just wanted to check if they were on the right track.  In other cases, students were uncertain, but they at least used phrases like, “I tried…  I noticed… Here’s the part that continues to confuse me…”  Students were purposeful in the help they were seeking.  I could tell people were questioning themselves and my help was to confirm their questioning.  Basically, it was the type of interactions I had been missing out on for a long time.

I’m sharing this post because I’m wondering how I can make students continue to self-question.  Does anyone have any activities, strategies, or routines besides constant verbal reminders that they use to promote self-questioning?  How do I teach and structure my classroom so that students are intentional with how they seek help?  How do I get students to move away from saying, “I need help. I don’t get this,” and start having specific questions planned when they seek help?  Once again, what can I do besides verbal reminders and modeling?

 

#MTBoSBlogsplosion: Positive Praise

This week’s #MTBoSBlogsplosion is about the soft skills of teaching.  When I look at the content of my blog, I feel like my posts swing back and forth between talking about content or questioning and the more personal side of teaching.  Part of me wonders if this oscillation is a product of being an early career teacher, but I think it’s more related to my experiences teaching at the middle school level.  It took me a while to realize that I do not teach mathematics; I teach students who are learning mathematics.  I teach human beings who are in the midst of significant mental, emotional, physical, and spiritual development.  This reality makes teaching a personal endeavor as much as it is academic.

A soft skill I’m working on this year is positive praise.

When I think about the words I use and how I interact with students on a daily basis, I’ve noticed that all too often I use reactive positive praise.  If a class goes well and students worked hard, I end class by saying how well it went.  If a student does an excellent job articulating his/her thoughts, I commend that student with brief praise during or after class (depending on the student).  If a student demonstrates leadership, I’ll say something to the student individually or call a parent to let them know.  In each of these situations, the positive praise is in reaction to behavior that was obvious to everyone in the room.  I’m trying to improve the frequency I use this type of praise.  Reflecting on this school year, I recognize that I could have praised students and classes more early in the year.  In some cases, I think being more liberal with my praise could have prevent some of the less productive days I experienced and encouraged more participation.

In conjunction with reactive positive praise, I’ve been working to include more proactive positive praise into my teaching.  For instance, I’ve dropped phrases like, “I’m looking forward to seeing how well you guys work today,” “I know you’ll be able to approach this topic with maturity,” or, “I know everyone in here is going to learn from each other and ask questions during this activity,” into my directions to premptively praise students.  On days when I’ve started classes in this way, I’ve noticed more productivity and a more relaxed environment.  One of my goals is start more classes with these statements.  I’ve also found myself proactiving praising students even when addressing misbehavior.  When I was talking with a student about her group’s off-task behavior earlier this week, I said, “I’ve seen you help other students and focus your group in the past.  I know I’ll see you do it in the future.”  The next day, I saw that student redirect a group member, help out a student who was absent the previous day, and offer to post work on the front board.  While I cannot be certain about the causes of this student’s positive choices, I think some of her motivation may have been the expectation I set with my proactive praise.  I was communicating confidence in that student’s ability to make good choice.  The student was validating my positive praise with her actions.  I want to include more proactive praise into my conversations with students this semester.

Overall, I think my ultimate goal is the have a cycle of compliments and praise in my teaching.  I want to start classes with proactive praise, drop more compliments and praise throughout a lesson, then end with reactive praise and a proactive comment to encourage future growth.  I’m getting better, but I’m far from perfect when it comes to implementing praise in my classroom.  Tweeting #onegoodthing has helped me be on the look out for positive elements of my day.  Reminding myself about what has been getting better has been helpful, too.  If you have any suggestions of ways to incorporate more praise into teaching, please let me know in the comments.  I know you have something to offer!

 

#MtbosBlogsplosion My Favorite: Meditation?

It’s a new year, which is a great time to start (or resume) blogging about teaching! If you’re uncertain about where to start, check out the handy-dandy post on Explore the MTBoS site!  Make sure to include hashtags!

The current schedule at my school has the 8th graders (my students) have PE and exploratory classes during the first two periods of the day.  As you can imagine, this arrangement makes the start of 3rd period seem a bit hectic.  I walk back to my room from teaching PE, set up my technology, and attempt to take attendance in a timely manner.  Simultaneously, students are rushing back from the other end of the building, asking to use the bathroom or get a drink of water, and generally seem a bit wound up after running around in the gym. Early in the year, I felt like the first 10 minutes of my 3rd period was a battle to get kids focused and ready to participate in the activities for the day.

Around mid-November, I got tired of how crazy the start of my 3rd period felt.  I decided to try something on a whim based on what I know works for me when I’m feeling stressed.  When the bell rang at the start of class, I calmly walked to the front of the room and waited for the class to end conversations.  It took a minute, then I directed everyone to close their eyes.  My students are used to blind surveys, so everyone complied thinking it would be another brief round of questions about the previous class.

Instead of questions, I began to slowly describe a map of the world.  I described the outline of the continents and their locations on a typical map.  I calmly told students to highlight the east coast of South America.  I waited a few seconds, then I asked students to highlight the west coast of Africa.  After letting silence fill the room, I directed students to visualize the continents slowly moving towards each other as I walked around the room taking attendance.  Slowly getting closer.  Slowly getting closer.  A minute or so later, I asked students to visualize South America tipping slightly to match up its eastern coast with the west coast of Africa.  I could sense the buzz and typical frenetic fidgeting of students melting away as I gave my description.  Expressions changed from annoyance or uncertainty to calm.  Shoulders were relaxing.  I directed students to visualize the continents slowly drifting away from each other.  Slowly drift.  Slowly drift.  Slowly drift.  A minute later, I asked students to take a deep breath, hold it, and slowly exhale.  When the exhales were finished, I told students to open their eyes.

Everyone looked much more relaxed.  I was more relaxed.  When I began describing the plan for the day for this science class (continental drift), everyone was making eye-contact with me and students typically more concerned with their snacks or their hair were following my movement around the room.  The rest of the period felt more laid back, too.  A partner reading activity seemed quieter, but more focused than the previous day.  When I described continental drift with the aid of an animation, I calmly connected the visualization we did at the start of class to the evidence Alfred Wegener collected for the theory.

It was amazing, so I had to see if this visualization exercise was a fluke or something of substance.  I decided to give it a try with my other science class.

At the start of 4th period, the results were the same.  In fact, one student mentioned it felt like she had meditated or listened to a yoga instructor.  Without getting religious, I think that’s exactly what happened in my classes.  Heartbeats slowed, minds cleared, breathing deepened, and bodies relaxed.  Everyone in the room (including myself) felt less tense.  The level of attention was stronger and sustained longer than in a normal class.

Naturally, two isolated incidents were not enough to convince me that I had found a new go-to strategy.  Throughout the month leading up to Winter Break, I tried out visualization exercises at least once a week with my science classes.  When we learned about the layers of the Earth, I had students visualize cutting an apple in half and seeing the layers.   One day, I had students visualize a dog attempting to squeeze between two people on a couch.  The visual connected to the rise of magma during seafloor spreading.  When we learned about plate boundaries, I had students visualize shoveling a driveway with a friend.  Each time, I spent about 5 minutes calmly describing the scenes and allowed silence to fill the room between my statements.  After each visualization, the classes seemed more relaxed and attentive than normal.

Call it meditation.  Call it visualization.  Call it a breathing exercise.  Whatever it is, it’s helping my students relax and focus.  For the benefits I’m seeing in my students, I’m calling it my new favorite and I hope to use this practice with all of my classes this semester.

 

Consistency for Flexibility?

The last unit in my Pre-Algebra classes involved many related topics, but a large portion of the errors and misconceptions arose as we learned slope.  Students who rocked out rate of change early in the unit kept flip-flipping things when we dropped context and looked at ordered pairs or a line on the coordinate plane.  Students who were easily able to catch their mistake when they wrote hours/dollar now were oblivious when they had run/rise instead of rise/run.

In some respects, I think some of the errors were related to my consistency.  I kept switching my wording for slope in hopes students would get flexible thinking about slope in multiple ways.  Rise/Run.  Change in y-values/Change in x-values.  Change in Dependent Variable/Change in Independent Variable.  Steepness of a line.  All of these definitions are helpful, but switching the wording of my questioning time after time is not helpful for students working with slope extensively for the first time.

As what tends to be the case in reflections, I noticed that my questioning was probably the largest contributing factor in all of confusion I witnessed in my students.  After rethinking how I would approach slope in the next couple days, I decided one of the best ways to address student concerns would be to use a set of questions consistently.

The next day, students walked in the room and I shared my observations.  We watched Slope Dude to revisit the more basic elements of slope (+, -, 0, Undefined), completed a couple examples, then I set students loose to practice.  Throughout the lesson (and the remainder of the unit), I kept using the same battery of questions when I encountered a student exhibiting misconceptions:

  • How are you reading this graph/table/set of points?
  • For the graph/pair of points/table, how would you describe the slope?
  • How are the y-values changing?
  • How are the x-values changing?
  • Did we write y-values over x-values?
  • Are we forgetting anything?

I chose these questions based on the success I saw students have with rate of change (change in y-values/change in x-values is the working definition we developed), but I also wanted to stress the importance of reading before calculating and self-questioning.  Thinking about the majority of mistakes students made with slope, most of the hiccups were due to students skipping the basic step of reading a graph/table/set of ordered pairs and deciding if the slope was positive, negative, etc.  After making a decision about the slope, it’s really just a matter of asking how the y-values are changing as the x-values are changing.  Not surprisingly, students began to make errors less frequently when I began using these questions consistently and encouraging students to adopt these questions to guide their thinking.

Deciding upon a uniform set of questions also made the remainder of the unit easier to plan.  When I began planning to teach linear functions and all of the elements associated with this concept (rate of change/slope, initial value/y-intercept, tables/graphs/equations/descriptions, increasing/positive slope, decreasing/negative slope, constant function/slope of zero, linear relation/undefined slope), I was overwhelmed by the various ways mathematicians talk about singular ideas.  Considering the way I kept switching the wording of my questions with slope, I recognized that I was probably causing the same amount of stress in my students.  Not good.  By streamlining my questions and sticking with the same wording, I was giving students the consistency they needed to get fluent with the topics we were exploring.  Beyond this comfort, I was providing students with more incentive to adopt and use questions I was offering to prevent mistakes and guide thinking.  Once students recognized that I was thinking about a problem in a similar fashion regardless if we were looking at a graph, set of points, table, or description,  students were able to see the meaning of my exhortation that, “It’s the same idea, just presented a different way.” Basically, students were able to build the flexibility that was my initial goal.

My experience with teaching slope and my use of questions leaves me with some unresolved thoughts.  People crave consistency, whether or not they are vocal about this desire.  As teachers, we also know that a hallmark of problem-solving is being flexible in one’s thinking.  Based on the struggles and success of my students with slope, is consistency is needed to build flexibility?  What elements of my teaching need to be consistent to encourage and build flexibility?  Is it a matter of consistent questions?  Is it a matter of routine experiences like asking students to develop questions to ask themselves or providing questions to adopt for personal use?  Do my students and I need a problem-solving framework?  Since I have more experience, I often don’t consider how I am solving a problem.  Am I consistent in the way I approach problems?  Do I need to be consistent?  Is my use of flexibility consistent?  How flexible is my thinking?  How does the connect to the way my students are thinking about problems?  Are my students consistent in their approach to problems?  Do my students ask themselves questions about problems?  What questions are the most consistent for my students?  Do my students recognize when they need to flex their thinking when they hit a cognitive obstacle?  How consistently are my students changing their approach to a problem?

Out of all of the questions I just wrote, the question that interests me most is whether or not consistency is needed to build flexibility.  I’m inclined to say yes, but I’m still uncertain about how that process looks in the classroom in relation to my teaching and student activity.  I’m guessing that there’s information out there about this idea and for all I know I might have just missed the session(s) at Twitter Math Camp that were focused around this idea.  If you have any suggestions, links, articles, or knowledge about using consistency to build flexibility, please leave this information in the comments!