What I get out of Student Writing

I have been using journaling in math class since 1996 – which was a really important year in my teaching career for lots of reasons, but it was definitely because I was introduced to the idea of math journals.  Since then I’ve done many different iterations for what my expectations are.  Even this year I did something new where I allowed students to write about errors they made on assessments in order to attempt to compare their assessment problems to what they did on homework in the hope of reflecting on the work pre-assessment for future problem sets.

However, a lot of students still use their journal almost like a problem-solving conversation with me, especially after we have already gone over a problem and they still don’t understand a method.  Here is one I ran across just the other day in my lower-level geometry class and thought it just perfectly expressed some of the goals I am hoping to accomplish with journaling.

I’ll call this student Cindy and we had just introduced the theorems about parallel lines through a geogebra lab and this had been the first problem they looked at that took the concepts out of the context of the lines and threw it into a triangle.  For many students this might be an easy transfer of skills (including the algebra, other theorems, etc.) but for the kids I have – not necessarily.  Here is what Cindy wrote:file_001-1

The first thing that Cindy does in her journaling is make her own thinking explicit (which I love).  She is stepping me through her thinking and the questions that arose for her.  This is actually a major step for many students who are confused – are they able to even know what they are confused about?

She writes: “I know the problem probably deals with the parallel line theories that we dealt with.” and then lists the types of angles we studied and then with a big “OR” says “maybe it has something to do with the sum of the angles of parallelograms and triangles?”  Little does she know that what she is doing is practicing synthesizing different pieces of prior knowledge – is it overwhelming to her? – possibly, but she went there and that’s so great!  I wanted her to know that I was excited that that she even thought about the sum of the angles so I gave her some feedback about those ideas.

She wrote down what she knew about the sums of the angles which we had also studied.

She writes her first equation to think about: “5x-5/=180” using one of the angles in the top triangle.  I would’ve loved to know where that was coming from.  What made her write that?  She then notes that “but it wouldn’t work because if x is the measure of the angle than the equation should be set to 180”

There is so much that this tells me about her confusion.  She is not understanding what the expression 5x-5 is supposed to be representing in the diagram I think, or she isn’t connecting what x is “not representing” (the angle) and the whole expression is representing too.  She also is confused about between the sum of the whole triangle’s angles and just that one angle.

She then looks at the two expressions she is given, 5x-5 and 4x+10 and I think makes a guess that they are corresponding angles – she doesn’t give any reason why they are corresponding.  She just asks the question.  But the cool thing is she says “Let’s try it.”  I love that.  Why not – I am always encouraging them to go with their ideas and the fact that she tries it is wonderful.  The funny thing is she does end up getting the same value for the two angles so she asks: “Does this mean that this is correct?” and then “What do I do for “6y-4?” and still has not connected many of the ideas line the fact that these angles are a linear pair and that’s where the 180 comes into play, or even why the angles were corresponding in the first place.  So many questions that she still has, although I am encouraged by her thinking and risk-taking.

This journal entry allowed me to have a great follow-up conversation with Cindy and she was able to talk to me about these misconceptions.  I’m not sure I would’ve had this opportunity to clarify these with her if she had not written this journal entry and then she would not have done so well on the problem set the following week.  I just love it!  Let me know if you use journals and if you feel the same clarifying or communicative way about them too.

See my website for lots of sample entries and also other blogposts and resources about journaling if you are interested.

 

Yours, Mine and Ours

Yesterday we had a speaker in our faculty meeting who came to talk to us about decision-making process in our school.  He spoke about the way some colleges, universities, independent schools are very different from businesses, the military, and other governing bodies that have to make decisions because we are made up of “loosely-coupled systems.” These are relationships that are not well-defined and don’t necessarily have a “chain of command” or know where the top or bottom may be.  They also don’t necessarily have a “go-to” person where, when a problem arises, the solution resides in that location.  The speaker said that this actually allows for more creativity and generally more interesting solution methods.

About mid-way through his presentation he said something that just resonated with me fully as he was talking about the way these systems come to a decision cooperatively.

“The difference between mine and ours is the difference between the absence and presence of process.”

Wow, I thought, he’s talking about PBL.  Right here in faculty meeting.  I wonder if anyone else can see this.  He’s talking about the difference between ownership of knowledge in PBL and the passive acceptance of the material in a direct instruction classroom.

Part of my own research had to do with how girls felt empowered by the ownership that occurred through the process of sharing ideas, becoming a community of learners and allowing themselves to see others’ vulnerability in the risk-taking that occurred in the problem solving.  The presence of the process in the learning for these students was a huge part of their enjoyment, empowerment and increase in their own agency in learning.

I think it was Tim Rowland who wrote about pronoun use in mathematics class (I think Pimm originally called it the Mathematics Register). The idea of using the inclusive “our” instead of “your” might seem like a good idea, but instead students sometimes think that “our” implies the people who wrote the textbook, or the “our” who are the people who are allowed to use mathematics – not “your” the actual kids in the room.  If the kids use “our” then they are including themselves.  If the teacher is talking, the teacher should talk about the mathematics like the are including the students with “your” or including the students and the teacher with “our”, but making sure to use “our” by making a hand gesture around the classroom.  These might seem like silly actions, but could really make a difference in the process.

Anyway,  I really liked that quote and made me feel like somehow making the process present was validated in a huge way!

End of Term Reflections

Phew…exams given…check…exams graded…check…comments written…check…kids on bus…check.  Now I can relax.  Oh wait, don’t I leave tomorrow to drive to my sister’s for Thanksgiving?

Such is the life of a teacher, no?  Just when you think you are on “vacation” there’s always something else to do.  I had an exam on Saturday then worked the rest of Saturday and Sunday finishing up that grading and writing my comments that were due this morning at 9 am.  But wait, I told some people I would write a blogpost about what my classroom is like, so I really wanted to do that too.  That’s OK though, I think it’s important for me to reflect back on this fall term – what worked and what didn’t for my classes.

I have three sections of geometry this year that I teach with PBL and a calculus class that I would say is something of a hybrid because we do have a textbook (as an AP class I needed to do what the other teachers were doing), but I do many problems throughout the lessons.

In my geometry classes, the student have iPads on which they have GeoGebra, Desmos and Notability where they have a pdf of their text (the problems we use) and where they do all of their homework digitally.  My class period for that course alternate between small group discussions in the Innovation Classroom in the library on Mondays and Thursdays and whole class discussions with student presentations of partial solutions (a la Jo Boaler or Harkness) on Tuesdays and Fridays. (We meet four times a week 3 45-minute periods and 1 70-minute period.)  Because my curriculum is a whole-curriculum PBL model, we spend most of the time discussing the attempts that the students made at the problems from the night before.  However, in class the discussion centers around seeing what the prior knowledge was that the presenter brought to the problem and making sure they understood what the question was asking.

classroom-shot1

Whole Class Discussion in regular classroom

 

geom-class-2

Small Group Discussions in Innovation Lab

If this didn’t happen we end up hearing from others that can add to the discussion by asking clarifying questions or connecting the question to another problem we have done (see Student Analysis of Contribution sheet).

One of the things that I had noticed this fall in the whole class discussion was that the students were focusing more on if the student doing the presentation was right immediately as opposed to the quality or attributes of the solution method.  There was little curiosity about how they arrived at their solution, the process of problem solving or the process of using their prior knowledge.  Unfortunately, it took me a while to figure this pattern out and I felt that it had also weeded itself into the small group discussion as well.

One day in the small group discussions, it became clear to me that the students were just looking for the one student who had the “right” answer and they thought they were “done” with the question.  This spurred a huge conversation about what they were supposed to be doing in the conversation as a whole.  I felt totally irresponsible in my teaching and that I had not done a good enough job in describing to them the types of conversations they were supposed to be having.

This raised so many questions for me:

  1. How did I fail to communicate what the objectives of discussing the problems was to the students?
  2. Why is this class so different from classes in the past (even my current period 7 class)?
  3. How can I change this now at this point in the year?
  4. How can I stress the importance of valuing the multiple perspectives again when they didn’t hear it the first time?

In my experience, sometimes when students are moving forward with the fixed mindset of getting to the right answer and moving on, it is very difficult to change that to a more inquiry-valued mindset that allows them to see how understanding a problem or method from a different view (graphical vs algebraic for example) will actually be helpful for them.

My plan right now is to start the winter term with an interesting problem next Tuesday.

“A circular table is pushed into the corner of the room so that it touches both walls. A mark is made on the table that is exactly 18 inches from one wall and 25 inches from the other.  What is the radius of the table?”

table-picture-problem

I have done this problem for many years with students and I have found the it works best when they are in groups.  I usually give them the whole period to discuss it and I also give them this Problem Solving Framework that I adapted from Robert Kaplinsky’s wonderful one from his website.  I am hoping to have a discussion before they do this problem about listening to each other’s ideas in order to maximize their productivity time in class together.  We’ll see how it goes.

To Hillary, With Gratitude

This morning as I woke up and found out about the results of last night’s election I was at first filled with despair and finally got myself somewhat out of that funk.  Then I thought about what Hillary Clinton must be feeling – she must be exhausted of course.  What did it take to put all of that energy into this campaign?  And those years of service to this country? And to put up with her husband? And the criticism?  This is not to say she didn’t make mistakes in the public eye of course.  I’m not saying I didn’t disagree with some of her stances, but I just want to look at it from the female perspective.  What I want to say to Hillary right now is thank you.  Thank you for being the first woman to have to go through the ordeal of running for president and dealing with all of the mess that goes with that.  I can’t imagine what that was like.

I have to say that in my career I know what it’s like to be one of two women in a meeting room and have to work extra hard to get a group to listen to your point.  Or perhaps to couch what you want to say in terms that the men will want to hear until they come over to your side in order to get them to vote your way on a certain agenda item or thinking twice about what I wore so as not to get judged.  The diplomatic skills that are acquired in just being a woman in an administrative position are invaluable because of the ways in which you know you need to listen and be heard. Being a woman in mathematics, the message is usually clear at national meetings when the majority of conference-goers are female classroom educators and the presenters are more often male speakers who are not currently classroom teachers.  In my graduate school education in mathematics I had one female professor and I was the only female in the Masters program.  You learn to “blend in” by speaking like them, working like them and going about your business like them.

I wonder if there isn’t a little part of Hillary that this morning just said “Phew, no more of that faking it.”  She was tired of being the male-culture-created part of herself that she had to be in order to run for president.  A few female heads of school that I have spoken with have said that in order to lead, many women are expected to downplay their feminine qualities – to not cry or be emotional, to be sure they are surrounded by male advisors so no one can say you made mistakes because you “are a woman.”  Spending so much time worrying about balancing speaking your mind with being nice to everyone so you are not labeled “bitchy” gets really tiresome.

What this election taught me overall is that misogyny is alive and well in the U.S. (not that Hillary needed to learn that) even more than racism.  My guess at this point is that we will elect a gay man in the future before we elect a woman but either way, I am grateful for Hillary and all she has done to pave the way for each other woman who comes next.  I read that Kamala Harris (CA) was the second black woman to be elected to the Senate, Ilhan Omar (MN) was the first Somali-American woman elected to Congress and Catherine Cortez Masto (NV) was the first Latina Senator to be elected.

I’d like to think that Hillary is waking up today really looking forward to spending some time as a grandmother, writing a book and working on the next great way to help kids, health care reform and education.  Sure that’s just me being idealistic, but as a woman, I would like to think that’s what I would do – well, after crying after losing for a little while.

Modeling with Soap Bubbles

I am so very lucky to have a guest teacher with me this year at my school.  Maria Hernandez (from the North Carolina School of Science and Math) is probably one of the most energetic and knowledgeable teachers, speakers and mathematicians you could ever find – and we got her for the whole year!  We are so excited.  I am working with her and she is so much fun to work with.  I have been teaching calculus with PBL for almost 20 years now and thought I had all the fun I could but no!  Maria is bringing modeling into my curriculum and I’m enjoying every minute of it.

As we started teaching optimization this week, Maria had this wonderful idea that she had done before where we want to find the shortest path that connects four houses.

picture-of-houses

I let the kids play with this for about 10 minutes and then did this wonderful demonstration with some liquid soap bubbles and glycerin.  We had two pieces of plastic and four screws that represented the houses.  As the kids watched, I dipped the plastic frame into the liquid and voila-file_000

Right away the students saw what they were looking for in the shortest path.  Now they had to come up with the function and do some calculus. As they talked and worked in groups, It was clear that using a variable or one that would help them create the right function was not as easy as they thought.  However,  I was requiring them to write up what they were doing and find a solution so they were working hard.

file_000-1

We have been doing a lot of writing in Calculus this fall so far and they are getting used to being deliberate about their words and articulating their ideas in mathematical ways.

Here is the outline of the work they did in class: Shortest Path Lab

and here is the rubric that I will be using to grade it.

rubric-for-lab-3-2

The engagement of students and the buzz of the classroom was enough to let me know that this type of problem was interesting enough to them – more than the traditional “fold up the sides of the box.”  The experience they had in conjecturing, viewing, writing the algebra and solving with calculus was a true modeling experience.

If you decide to do this problem or have done something like it before, please share – I’d love to do more like this.  I am very lucky to have a live-in PD person with me this year and am grateful every day for Maria!

 

Need Some Help Looking Forward

So I’m trying to figure out how to reach more people and thinking about the future of my professional development plans with PBL for all levels of teachers.  I’ve gotten some great feedback from people about the PBL Math Summit so far (from the two years we’ve had it) and I have some ideas about how to create some better online resources too.  If you have the time, and are interested in helping me out, would you please fill out this short survey about PD Needs for PBL Math Professional Development.  Also, tell others who could give me insights too.  Thanks so much for reading my blog and for also being inspired to be interested in PBL math teaching!

 

A Math Girl’s Story or the Introduction to My Dissertation

I am not really a negative blogger but I do have to say how tired I am of research reports that over and over again talk about the way we are not doing enough to support girls in math education (or other underrepresented populations of students).  There is enough evidence now from many research reports (NCTM, 2016, Why so few? AAUW, 2010, Riegle-Crumb, et al, 2012 I could go on…) that show that there is little difference in math ability by gender and that the reasons that girls and women choose to leave STEM fields are culturally related.  And yet, we still need a white male to make statements like:

“I believe that this issue of women’s confidence is cultural, not biological. It fits in with all we know about stereotype threat. When the message is that women are not expected to do as well as men in mathematics, negative signals loom very large. Calculus—as taught in most of our colleges and universities—is filled with negative signals.”

  • David Bressoud, MAA Blogpost, Launchings, October 1, 2016

Now, I don’t know Mr. Bressoud and perhaps this most recent research study really pushed him over the edge to being a believer, so no offense meant.  But I’ve just had enough of it.  My life experience had been based on all of this and it’s enough for me.

We need to do more to change the way math is taught in the U.S. so that more girls (and other underrepresented students) feel connected and desire learning, feel like they belong and their ideas and voices are valued within the context of mathematics and the community of mathematics learning – at the secondary level and the college level. Period.

Here is the introduction to my dissertation, “Dismantling the Birdcage:  Adolescent Girls’ Attitudes towards Learning Mathematics with a Relational Pedagogy in a Problem-Based Environment” (2013) (don’t feel the need to read the whole thing).

“If you look very closely at just one wire in the cage, you cannot see the other wires…You could look one wire up and down the length of it, and be unable to see why a bird would not just fly around the wire any time it wanted go somewhere…There is no physical property of any one wire…that will reveal how a bird could be inhibited or harmed by it except in the most accidental way.  It is only when you step back, stop looking at the wires one by one and take a macroscopic view of the whole cage, that you can see why the bird does not go anywhere; and then you will see it in a moment.  It is perfectly obvious that the bird is surrounded by a network of systematically related barriers, no one of which would be the least hindrance to its flight but which by their relations to each other, are as confining as the solid walls of a dungeon”. (p.5)

-Marilyn Frye, Oppression, in The Politics of Reality (1983)

I will begin with a story.  It is the story of a young girl excited and interested in learning and doing in all aspects of her elementary education.  Luckily, her parents were always encouraging and supportive of her learning goals and her initial schooling included “enrichment” class for which she was chosen to receive out-of-class group instruction in advanced topics – including mathematics and science.  The girl was confident, motivated and eager to move forward in her exploration of new topics and share these ideas with her friends and family.  As middle school approached, it became clearer to the girl that categorizing students by ability became more important and she realized that her work and grades in her classes, as opposed to her interest in mathematics, would begin to determine her path through her education.  The pressure of this realization, and possibly other determinants, affected her performance and she was placed in a pre-algebra course in the eighth grade, which she knew, even then, would set her on a trajectory that somehow indicated less success.

However, the following year, the girl’s work in algebra was so successful that her teacher that year recommended that this adolescent girl now double-up in her mathematics courses and take geometry and a second year algebra course concurrently. Reinvigorated and more confident in her abilities, she regained her momentum and faith in herself as a mathematics student, although the fun with her peers and connections with the teacher from her “enrichment” classes were now a thing of the past.  Mathematics seemed made up of a set of disjointed courses that needed to be passed sequentially in order to fulfill the requirements for graduation.

Finally, the ultimate course in mathematics came during her senior year of high school where she would be able to truly show that she had made it to the top – Advanced Placement Calculus.  However, difficulties arose when little interaction occurred between the teacher and the students surrounding mathematics in the classroom.  Utilizing a textbook that was published almost 25 years earlier, the now young woman felt isolated and alone in a class where asking questions seemed to signify weakness and demanding an explanation also showed that a student was incompetent. Students who could easily and quickly replicate the mathematical exercises performed by the teacher were praised and favored whereas those with difficulties were dismissed and even asked not to take the Advanced Placement exam at the end of the year.  Sadly, our young lady was among those disinvited to be part of the elite exam takers.  This was a turning point in her desire to continue with mathematics as an intellectual endeavor.  She vowed to never take a math class again and moved on to college to pursue music as a major field of study.

On arriving at her chosen college in the fall, the young woman was required to take a mathematics placement exam in order to fulfill her natural science portfolio requirement.  Begrudgingly, she took the short test and a few days later she was told she could register for Calculus III.  How was this possible?  She did not even take the AP exam in May and barely passed the course in high school.  Would this roller coaster ride of messages of encouragement and discouragement ever end?  Who did they think were, telling her to move into Calculus III?  She would show them and just retake Calculus I and be done with it – get that natural science requirement out of the way and move onto much more interesting and meaningful courses so that she could leave mathematics in the dust.

However, something surprising happened in that basic Calculus I course that fall.  The young woman had an interested professor that saw her potential and talents.  The professor engaged her in conversation about mathematical justification and questioning. Citing the young woman’s exceptional ability in Calculus, the professor questioned why she was even in the class.  At the end of the term, the professor had convinced the young woman to continue on and even elect to take a computer programming course to see if she liked it. “Why not?” the professor said, “and it’s required just in case you decide to be a math major someday.”  The young woman laughed out loud.

One by one mathematics courses came and went.  The smaller seminar style of the upper level mathematics electives worked extremely well for her learning style.  Although she was often one of two, or the only female in the course, the girl believed that she was supported and encouraged by the professors she met.  There was a community of mathematicians who allowed her to grow and develop her skills, as opposed to suppress and discourage them.  Abstract courses like Linear Algebra, Number Theory and Topology connected much of the mathematics that for too long seemed discrete and disconnected. After serving as a teaching assistant for much of the department and receiving honors on her senior thesis, the girl was encouraged to apply to graduate programs.

In graduate school, once again the girl found isolation among a male-dominated community of academics and senior mathematics professors seemed to look differently at her, wondering why she was not in the Master of Arts in Teaching program with the women.  After two years of struggling with the environment, but quite enjoying and thriving in the teaching classroom, the young woman realized her gift and decided to find a way to make her journey complete.  Combining her love of mathematics and her talents for teaching was the way to make a life worth living while also bringing the consistent support and encouragement to students that she so greatly needed all those years.  Although it took her 20 years to realize this direction, ultimately it became a passion and lifelong commitment.

At this point in stories like these it is generally tradition to state “The End.”  However, at this point, I would change the phrase and say “The Beginning.”  Yes, it was just this story that has led me to this place and passion in my research for gender equity in mathematics education.  Now, 22 years into my teaching career I can look back and see how it began with this personal experience, but when I started my teaching and my doctoral program, I am not sure I was as aware of the implications my own story had for my research and teaching interests.  As my career brought me in and out of single-sex schools, my research interests led me towards a relational pedagogy.  As individual students that I crossed paths with shared their own hopes and fears about their relationship with mathematics, it began to be clear to me that it was more than a coincidence that my dissertation research, and perhaps my life’s work, would be centered around finding ways to improve the education of marginalized students in mathematics education, if possible.

And so it was the beginning – the beginning of a long journey with this question about how it might happen – how to improve the learning of students who feel marginalized in the world of mathematics as I once did.  But I would begin with one group of marginalized students in the mathematics classroom to whom I could relate most readily; adolescent girls.

What is “Low Ability” Anyway? Comparing a Point to “Room”

One of my big “beefs” at my school is the fact that we have three levels of tracking – count ’em, three.  There’s the honors track, that of course at a college prep school, most kids think they belong in.  There’s the regular track, that which is still pretty quick and difficult, and there’s the track that the kids who are sometimes, I would say, just not very motivated to learn math, or have less interest in math, or maybe come from a school with a less rigorous math program, are placed in.  These are the kids who probably all their lives have been told they are not “math people” and have been pigeon-holed as an “artist” or “writer” so they won’t actually need math when they get older.  This really, really irks me.  But I do it – I go along with a system that has been in place way before I got here.  I’m only one person – even though I cite Jo Boaler’s list of research showing why “tracking” in general is just a bad idea and hurtful – I know I can’t win.

Anyway, I really shouldn’t complain because the department has let me do my thing with the geometry curriculum and I have written a PBL curriculum for the three levels.  In my 201 book, I have created scaffolded problems that I think work really well with these “low-abiity” kids and often challenge them enough to make them realize how much ability they actually have.  We just started talking about dimension and we watch these clips from Flatland: The Movie, where Arthur T. Square meets the King of Pointland and then meets the King of Lineland.

We had a great conversation about why the King of Pointland keeps saying “Hello Me, Hello Me” and can’t really understand why there’s anyone else there.  We talked about why the King of Lineland doesn’t understand where the Square is because he only understands the directions of left and right.  One of the kids goes, “Is this kind of like what happened in the movie Interstellar? I think he went through a black hole and just appeared in the future or something.” Now, I hadn’t seen that movie but then another kid said, “Well, I’m not sure it was like that.” But then one of the other students says, “No, the King of Pointland is kind of like the kid in the movie Room.  Did you see that movie?”  I nodded in understanding and so did many of the others in class. The student went on, “In Room, the little boy grew up thinking that “Room” was his whole universe so that was all he understood, and that’s why the King of Pointland seems so nuts. That point with no dimensions is all he can understand – there’s no one else in the world.”  I was so blown away by that analogy.  She really had an understanding of the idea of the limitations of being alone in the universe of a point. I had never had a kid in a “regular” or “honors” class make a connection like that – but then again, Room just came out!

Repost: Always Striving for the Perfect Pose

Back in 2010, I wrote an blogpost comparing teaching with PBL to doing yoga. Since I have been doing Bikram Yoga for almost a year now and still can’t do “standing head to knee pose” *at all* – I thought I would repost this one just to give myself some perspective, and possibly many of you out there who might need a little encouragement at this beginning of the year time. I know that every year when students begin a year in a PBL math class the obstacles return. Parents are questioning “why isn’t the teacher teaching?” Students are questioning “why is my homework taking so long?” Teachers new to the practice are questioning “When is this going to get easier?” and “why aren’t they seeing why this is good for them like I do?” The best thing to remember is that it is a process and to understand how truly different and hard it is for students who are used to a very traditional way of learning mathematics. Give them time, have patience for them and yourself and most of all reiterate all of what you value in their work – making mistakes, taking risks, their ideas (good and bad) and be true to the pedagogy.

Here’s the original blogpost I wrote:

I don’t think my professor, Carol Rodgers, would mind me borrowing her yoga metaphor and adapting it to PBL. I use it often when talking to teachers who are nervous about falling short of their ideal classroom situation or teaching behaviors. I think this can happen often, especially when learning best practices for a new technique like facilitating PBL. There are so many things to remember to try to practice at your best. Be cognizant of how much time you are talking, try to scaffold instead of tell, encourage student to student interaction, turn the questions back onto the students, etc. It really can be a bit overwhelming to expect yourself to live up to the ideal PBL facilitator.

However, it is at these times that I turn to Carol’s yoga metaphor. She says that in the practice of yoga there are all of these ideal poses that you are supposed to be able to attain. You strive to get your arms, legs and back in just the right position, just the right breathing rhythm, just the right posture. But in reality, that’s what you’re really doing – just trying. The ideal is this goal that you’re aiming for. Just like our ideal classroom. I go in everyday with the picture in my head of what I would want to happen – have the students construct the knowledge as a social community without hierarchy in the authority where everyone’s voice is heard. Does that happen for me every day? Heck no. I move the conversation in that direction, I do everything in my power for that to happen, but sometimes those poses just don’t come. Maybe I just wasn’t flexible enough that day, or maybe the students weren’t flexible enough, maybe we didn’t warm up enough, or the breathing wasn’t right. It just wasn’t meant to be. I have exercises to help me attain the goal and I get closer with experience. That’s all I can hope for.

So I tell my colleagues who are just starting out – give yourself a break, be happy for the days you do a nearly perfect downward facing dog, but be kind to yourself on the days when you just fall on your butt from tree pose. We are all just trying to reach that ideal, and we keep it in mind all the time.