A Picture is Worth…..Everything.

I spent about an hour yesterday playing with geometry tiles.  I was:

  1.  Fully engaged and enjoying myself.
  2. Experiencing OMG, WTF, and AHA moments.  (AKA actively learning.)

The inspiration for my mathematical play time was this photo from  Sara Vanderwerf’s recent post  on Stand and Talks, a bold routine she uses to increase the number of students talking about math.   Really great stuff and worth your time to read it through.

IMG_2094But back to the photo, which is her example of using a Stand and Talk to introduce a new idea, in this case, the Pythagorean Theroem.   My eyes perk up (is that a thing?) because I have been pondering about this very topic.  Specifically, wondering about what activities/tasks would place students in the active role of sense-making.

The first thing that popped into my head was….wait, trapezoids?  (Thank you, Sara!) Then I started some noticin’ and wonderin’ from my teacher-y perspective and also imagining/guessing what students might say.  The more I noticed and wondered, the more I realized how FULL OF MATH IDEAS this one photo is.

I find this exciting.  Really, really exciting.  Can you imagine starting here, with this one little image, asking students what they notice and wonder, and then letting them run with their curiosity?  Isn’t it joyful to know in your heart of hearts that this is so much BETTER than, “Hey kids, here’s a little something called the PT, copy it in your journal/onto your pre-made “doodle” notes, plus copy these three examples of how to get the right answer, and do this worksheet”? How superior this single photo and all its potential are to “I do, we do, you do” and /or anything else that keeps students in the passive role?

It didn’t take me long to drag out my tub of tiles and start messing around with them.  TOTALLY what I would want my students to ask beg to do.

The questions I decided to guide my play were:

First, can a proportional, scaled-up version of each polygon tile (that I have) be built with only that particular tile?
If so, why? Is there a pattern?
If not, why not?

Second, can any three similar polygons be used to make any type of triangular “hole”? Why or why not? Which ones make right triangles? Why? Is there a pattern??

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I started with squares.  Don’t know why, maybe because that’s what is already familiar to me.  No surprises, the areas are the square numbers, 1, 4, 9, 16…..

So I moved on to the equilateral triangles, wondering/kinda expecting that I will make triangular numbers.  Imagine my surprise IMG_2119(my OMG moment) to find SQUARE NUMBERS AGAIN!  Holy shit!  NOT what I was expecting!  Now I feel compelled to keep exploring.  COMPELLED.  (Imagine your students feeling compelled!!)

I quickly confirmed that the next in the series takes 16 triangles, and moved on to IMG_2125the rhombus.  YES, the areas are square numbers AGAIN!  Which gets me reconsidering my understanding of square numbers and forming conjectureish questions:  Can square numbers be made out of any polygon, not just squares? These are all regular; what if they are irregular?

I’ve been reading Tracy Zager’s fabulous book Becoming the Math Teacher You Wish You Had.   She starts Chapter 7, Mathematicians Ask Questions, with a quote from Peter Hilton: “Computation involves going from a question to an answer.  Mathematics involves going from an answer to a question.”  Brilliant distinction.  Looks like I’m living Chapter 7.  Know I want this for my students!

Remember, one photo.

I finally get to the trapezoids, which are more challenging to tile and require some analysis.   I managed to construct one, but the number of tiles it took (7) was a red flag for me, so I  examined it closely to verify if it was proportional or not.  It wasn’t, so I kept playing, keeping theseIMG_2126 two beliefs in my mind: a proportional trapezoid was possible, and it could be made out of 4 tiles.  And lo and behold….

TA DA!  “Square” Numbers!  (Whew!)

 

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By now, I have even more questions.  By now, I have reviewed and applied my prior understanding of proportionality and similar figures, without ever touching a worksheet.  By now, I’ve taken risks, persevered, reasoned, looked for patterns, used vocabulary, formed conjectures, and justified.  All from looking at one picture and allowing myself both time and pleasure to play and be curious….and I haven’t even begun exploring that thing with the triangular “hole”.   Yet.

Finally, the hexagons.  And….What?!?  You can’t even make a similar hex without using some non-hex friends!  WTF!

New question!  WHY NOT?

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One.

Little.

Picture.

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Competencies

Its been so long since I’ve blogged that I had to look up my password.

One of the many reasons for my little hiatus is that there are plenty of UH-MAZING blogs out there– that people actually follow and read– making what I think and wonder and write about rather…superfluous.

For example, Ilana Horn’s insightful, intelligent, inspirational blog.  Informed and intriguing.   I recently started following, and back-read several posts she has written on status in the classroom, in part because I believe know status is deterimental to equitable learning yet is created and deeply ingrained and even actively perpetuated in the Old School system/institution of teaching and learning.  She explains it all  very clearly in a series of posts:

Status: The Social Organization of “Smartness”

Seeing Status in the Classroom

What Does it Mean to be Smart in Mathematics

Recognizing Smartness and Addressing Status in the Classroom

This morning (since I kind-of overdid it hauling bark dust yesterday), I decided to chill a bit and create a list of competencies I value in my classroom.  Not besides “fast calculations and right answers”, but instead of.  A definite and requisite shift in classroom currency if one is striving to achieve an active and equitable learning culture.

In no particular order….

Curiosity.
Perseverance.
Observation.
Students in the role of sense-makers.
Connections between mathematical ideas.
Connections between representations and models.
Clear communication of thinking (the WHY), even if incomplete or unsure.
Active and intentional listening to all peers.
Self-reflection.
Metacognition.
Multiple strategies and solution paths.
Gaining insights by making mistakes.
Willingness to revise thinking and understanding.
Great respect for the value of every person, their learning, and the strengths they already have.
Genuine Questions and Wonderings.
Collaboration in learning as a community.
Flexible thinking.
Creative thinking.
Visual/alternative representations of reasoning and ideas.
Connections between multiple representations.
Connections between different strategies.
Patience.
Perplexity.
AHA and WTF* moments.
Growth.
Active awareness and regulation of learning.
Attention to reasonableness of solutions (yours and others’).
Private time to think (and respecting it).
Critique of thinking, reasoning (not people).
Critical and deep thinking.
Understanding the thinking of others, even when it differs from your own.
Respectful disagreement.
Respect for (and celebration of) strengths and strategies that differ from one’s own.
Genuine/legitimate peer support in learning.
Growth mindset.
Engagement and involvement.
Willingness to start even if you are not sure.
Equitable collaboration.
Consideration of ideas other than your own.
Ability and willingness to adjust your reasoning/opinion and change your mind.
Learning from peers.

Wow, that list is a lot longer than I expected.  Which would you add, revise, or omit?  Why?

Here’s what I might do with such a list.  At the beginning of the school year, cut it up and have students in small groups sort them into 2-5 or so categories, their choice.  Sorting activities are a worthwhile way to get kids talking to each other, voicing opinions, making choices.  Listen in, because you’re finding out about them, too.  Notice common choices as well as different ones.  Ask groups to explain their categories to you.

Then, as a whole class, share and discuss.  Ask them to notice things.  I have NO IDEA what will happen here, but I’m wondering if anyone will notice that “right answers”, “smart”, “good grades”, “fast thinking” and those types of competencies typically over-valued (and detrimental to learning) are MISSING.  So are generic behavior-type rules, like arrive on time, do your homework, pay attention….Will they notice the focus on inclusiveness and learning instead of on first and fastest?  Will they identify with some of them?  I’m really curious about how kids will sort these and what they will say!  Finally (if there are enough common themes?), use their input to develop a SHORT list of classroom norms that recognize and support these valuable competencies.

 

 

*Probably should change this to WTH What the Heck, or HIW Hold it, What!? Or some such thing more socially appropriate, right?

WTF moments are not moments of frustration, though.  They are moments of realizing something is amiss, some reasoning, intuition, or process is not going the way you expected, or the solution make no sense.  Disequilibrium and perplexity reside here.  In a sense, these moments are insights, too, a realization that an adjustment is needed;  understanding WHY one path works and the other does not paves the way to the bigger insight (AHA!) and gains in learning and understanding.  When students people share their thinking, they tend to leave the WTF moments out and share only what worked, saving face and strengthening the currency of “right” answers.  However, in a healthy, inclusive culture of learning, WTF moments are valued as an important and natural part of the learning process, worthy of sharing, even celebrating!   “First, we thought….because….then we. saw…realized…tried….because….figured out….learned….”. Even “First we tried….because…not working……and now we wonder….not sure….have some questions…..”

Out of the Woods

UPDATE 1/20/27:  Difficulties concentrating today, so I am submitting this already published post for the MTBoS 2017 Blogging Initiative, Week 3, “Read and Share”.  Looking foward to some feedback!

I am struggling to write this post, and I am not sure why. I want to offer a thorough response to a blog post as evidence of my growth, but am battling (maybe?) feeling underqualified and lacking in credibility. And strangely vulnerable.

I could describe my eight little years of teaching as a classic case of not being able to see the forest for the trees. Its like I downloaded the awesome constuctivist app I really wanted, but never thought to or knew how to update it. My current situation allows me to finally view the forest, a chance to look around and consider the bigger picture. This perspective has helped me recognize and understand my shortcomings and offers me insights for moving forward. MTBoS is helping me update my app, and essentially, I want to test it out to see if I’m understanding how to use it. Consider it a formative assessment.

The inspiration for these musings comes from Justin Aion  . Actually, I mentioned him (and This Post) the last time I blogged, so thank you, Justin, for the two-fer.

It would be best for you to read his whole post, of course.  He leads with a brief description of his school’s current math textbook:

Each section begins with an introductory activity that is frequently hands-on.

The task is this:

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Justin Aion’s Photo

Draw a pentagon with extended sides.
Label the external angles.
Cut out the external angles.
Put the external angles together and make and observation.
Repeat with a hexagon and an octagon
.

Seems straight forward enough, right?  But what happened was this:

It didn’t go as I would have hoped.

Apparently, even mostly on-task groups did not get done, in spite of the fact that they had just done the same thing with internal angles. I know EXACTLY the feeling. Been there, by golly, many, many times. Loads of empathy here. Too much precious time spent on a hands-on activity and no learning taking place. In fact, it would not surprise me that this is the #1 Reason for Avoiding These Kinds of Tasks.

He also writes,

(Students) are much more attentive to the tasks when they are working individually or when I’m giving direct instruction.

I suspect that means it’s a larger issue than just these kids in this class.

Justin works hard to see the forest while standing in the trees; in my opinion, his suspicions are spot on.

What I see is a chance for me to check my understanding.

First, I connected his final comments to my recent reflecting on teaching and learning. That was my last post, and my hypothesis is that students are passive learners because they are put in the passive role. School happens to them.

Wait, this was an “age/grade appropriate” task, though. Hands-on! Engaging! Student-centered! Everything a constructivist teacher’s heart would desire!

What I am beginning to understand is that “engagement” is more complex than providing something for students to do. That not all tasks are created equal, and implementation matters.  I’m not talking about comparing mindless worksheets to learning in groups; I’m talking about those activities and tasks the look great on the surface, when in fact they do not genuinely engage because they are not designed to.   These pseudo-engaging activities are easy to miss, and I did, many times over.  Here’s what I noticed about the lesson from Justin’s textbook:

  • From the students’ perspective, the need to explore external angles is non-existent (other than compliance), so there is no intrinsic motivation.*
  • Also missing: student-generated observations and questions. No perplexity, no curiosity, no ownership.
  • The method to explore external angles does not come from students but from a textbook (and the teacher). No exploring, no creativity, no problem-solving.
  • Student role is passive: they are merely following directions. In an attempt to keep students “on task”, teachers often model step by step directions, keeping their grip on the active role. “Engaged” means “looking busy”.
  • The low cognitive demand throughout the task actually frees them up to socialize. Hence the fun (for students) and frustration (for teachers). Some may even want to prolong this “easy” part to avoid what feels more challenging: making an observation. Coping via procrastination.
  • There’s no intrinsic reason to finish, either; experience has shown these students that whatever they are ‘supposed to learn’ from this hands-on busy work will be stated for them, anyways, by the teacher (or by the “smart” kids).

I am not yet knowledgeable/confident enough to play  “What Can You Do With This?”  although I have ideas brewing. For now, what I do have to offer is this:
Assuming you want to empower students to be productive, active learners, consider developing the habit of regularly running lessons, activities, and tasks through the role-lens. All of them– your creations, the textbook’s (especially these), something gleaned from the internet– as often as you are able. Examine closely what students will be doing and keep tweaking** until you think the active role has switched to them, where it belongs more often than not. Take a risk and trust them to rise to the challenge. Be vigilant, be intentional.

What can you let go of and turn over to students?

Will they be asking and answering their own questions?

Will they notice patterns and make conjectures without you prompting them?

Will they be curious and driven to make sense of something, even if that something is math?

Will they own the learning and all the work it took to get there, together?

Anyway, that’s what I would try to do, were I feeling a bit lost in the woods.

*Which also perpetuates the perception/myth that math is a Random Bunch of Useless Stuff No One Really Cares About.

**Here are some practical ideas and resources for tweaking from a couple of Experts that are not overwhelming. The ideas, that is, although Dan Meyer and Kate Nowak are probably also not overwhelming. They are two of many that have been instrumental to updating my app me.

From Dan Meyer:

Makeover Monday .  His MM summary is here.

His delightful  video on relevance ,  also located here.

From Kate Nowak :

Make Them Figure Something Out

Plan a Killer Lesson Today

Roleplay

Yesterday, being Tuesday, I volunteered all day in Jackie’s 7th grade math classes. Last night, being me, I started reflecting on what took place– what we did or did not do, what students did or did not do– in terms of the conversation she and I had at the end of the school day. Today, being unusually snowy (for NW Oregon) and stay-at-home-y, I’m gathering my thoughts here.

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My sister’s backyard.

Let me first say that Jackie, like many other teachers including me, wants to teach in a way that student learning moves forward. Neither one of us know exactly how to pull that off, so in some many ways, its the blind leading the blind.

She was willing to spend weeks (weeks!) on making sense of integer addition, but was, well, flabbergasted when she asked them to write in their journals (in their own words) what it means to add and, with the exception of one student (out of about 100), students wrote basically this: adding means to put numbers together to go up to a bigger number. The use of “bigger” notwithstanding, what prevented these students from writing something like this?? Adding means to combine. Sometimes when you add, you increase to a higher value, sometimes you decrease to a lower value. It leaves us wondering what in the world it takes for students to internalize concepts well enough to build on them, to move along.

Now, I don’t think at all that zero learning has taken placement but, c’mon. Nor am I going to say these kids are lazy or don’t care or don’t try hard enough. Quite the contrary; these kids are normal, but they are for the most part passive learners. It makes me wonder what’s going on here (and it went on in my classes, too) that needs addressing. Plenty, I’m sure.

Allow me to digress a bit.

(I may never get over how amazing it is that you can be pondering a particular problem or question and *ding*, the MTBoS sends you a pertinent post from a total stranger.  Its cosmic.)

This morning my inbox contained  a post from Justin Aion. One of the reasons I follow his blog is because he is so candid in his daily reflections and I can easily relate. If I understand him correctly, there’s a conflict between teaching the way he wants to teach (for deep and lasting conceptual understanding) and teaching in a way students expect him to teach (direct instruction), and feels a more than a little guilty when he gives in.

Which brings be back to some questions I have percolating*.

  • Are students ‘passive learners’ because that is the role given to them, over and over, the active role belonging to the teacher?
  • If a teacher strives to develop a thriving, student-centered learning community and struggles to make it a reality, is it (in part) because these roles have not sufficiently switched?
  • What are some obstacles to switching roles and how do you think they can be overcome?
  • If so, what can one do, alter, and even not do to make the switch and make it last?

What do you think? What do the roles look like in your classroom?  What recommendations do you have for switching the active role to the students?

 
*My fairly confident answer for 1) is YES and 2) is It’s worth considering.   My answers for 3) and 4)  are a bit more tenuous and lengthy, so I’d like to make them  another post.

Week 2, With Time to Spare!

My solution for missing the deadline for Week 1 of the MTBoS 2017 Blogging Initiative was to write a belated post (see below).  Not wanting miss this opportunity again, I am already posting for Week 2! The focus: soft skills.  That is, the part of teaching that is more about raising children, the crucial part you don’t realize about teaching until after you are standing in front of a room full of students.

Unfortunately, the more I reflect on my set of soft skills, the more I realize that, in SBG terms, they are in the “getting there” stage, and I don’t have really much more to offer other than it is primarily about building relationships. Those more proficient, experienced, and successful than me in the relationship-building arena will have oodles to share, I am sure. In fact, a lot has been written about soft skills already, as evidenced by the 2010  Soft Skills Virtual Conference recommended by Sam Shah.

On his advice, I read (and in several cases, re-read) most of the contributions to the conference. Fabulous, all. What I want to share here are two related excerpts that stood out rather significantly for me. As in, holy shit!

From  Shawn Cornally, whose writing I could read all day long:

He would sit with me for 15 minutes stints, explaining things that I should have learned in high school, because he realized something that every teacher should: teach them where they’re at, not where you wish they were. You can only do that if you manage to somehow care more about the kids than your list of standards.

(Emphasis is mine.)

From Riley Lark, organizer and curator of the SSVC:

These roles [facilitator, resource manager, task manager, recorder/reporter] make me more comfortable with my guilty admission: I don’t care very much if the kids learn math. I mean, I’ll teach them some math, and when they leave they’re going to see more of its beauty and be equipped to use it in society. But which is more important, vector addition or working in a team? Factoring or formulating questions? Integrating or leading peers? Obviously, obviously, the math comes second. It’s just lucky that learning math provides so many opportunities for learning the more important things.

(FYI, the emphasis on that second ‘obviously’ is not mine.)

Talking Points!  For each, decide if you agree, disagree, or are sitting on the fence, and include WHY. It’s OK to change your mind after listening to another’s points of view, or to restate your mind and strengthen your argument.

  1.  Obviously, obviously, the content comes second.
  2. Standards are required, I must be sure to get through them all.
  3. Virtual Conferences are a fabulous idea.
  4. The MTBoS would benefit from more input from elementary teachers.
  5. To meet each student ‘where they are at’, we need leveled and remedial classes.
  6. (Insert a related Talking Point of your choice here.)

Go.

UPDATE: Serendipity: Liz Mastalio’s Week 2 post “Honestly, the Math is Secondary”.

Week 1, Late.

I was going to jump on board the MTBoS 2017 Blogging Initiative .  I was planning to take a deep breath, close my eyes, and plunge in.  But for some unfathomable reason, I read “submit by midnight Sat., Jan. 7” as “submit by midnight Sun., Jan. 8th”.   Oops, I missed that first boat; fortunately, no one is going to dock points!  I’m determined to not let Week 2 drift by, but a part of my brain is still on Week 1’s theme, “Favorites”.

My brainstorm for Week 1 included a task from Visual Mathematics Course II and the folks at The Math Learning Center.

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Original Version, Submarine Task

Its annual use qualified it as a “favorite”, although over the years I used it in various ways.  An assignment, a formative assessment, a group task, a summative assessment.   I always required diagrams and equations that supported each other, and eventually figured out to omit the question to allow students to focus first on understanding the situation.  That would be Version 2 (with all verbs in agreement as well.)

In the spirit of You Can Always Add here’s Version 3:

The Submarine Task, Version 3

A submarine cruises in an ocean. First it dives down, then climbs up, dives again, and finally climbs up.

Before you read my ideas, what would you do with this version?

What I think I would do, feel free to poke holes:

First, do some Noticing and Wondering  Ask students to make a sketch of what is going on.  Have students suggest not only what questions they could ask, but also what information they would need to answer their questions.  (BTW, typically the question in V2 is, “Where does the sub end?” I am curious what, if anything, will be different for V3.)   Then, allowing private think-time before working in small groups, give them this:

Which of these sets values fit this situation, which do not, and why?   Use visuals and equations to explore each list and be prepared to justify your reasoning.  Use values exactly as they appear, and in the order given.


-200, 150, 115, 180, 100

                                     -200 -150 -115, -180, -100

                                      -200, -150, 115, -180, 100

                                     -200, 150, -115, 180, -100 

                                      -200, 150, 115, -180, -100 

Compare what this task is asking of students, and what the others versions ask.  What do you notice?  What are your thoughts on the lists?

Continuing with my ideas:  Perhaps give the lists to groups on strips of paper so they can move them around a sort them.   I am dying to know which lists students accept and which (if any) they reject; should be an interesting discussion!  In my mind, V3 would be appropriately placed after some reasoning  and conjecturing about adding and subtracting integers, during a time when there is still room for questioning and sense-making, and before students practice fluency.

Since they have done most of the heavy lifting already, end with this:

1.  Chose one set of numbers to answer the questions you asked.  Be sure to include ALL of your work.

2.  Use a second set, including your work.  

3.  Compare the strategies:  How are they the same?  Different?  Which one is “better” for you and why?  (Do not describe what you DID; your work should already clearly show your steps!)  

A possible sequal to  V3, although probably not immediatly:

Which of these lists of values fit the Submarine Situation?  Explore with diagrams and equations and be prepared with viable arguments. 

                        -142.5, -157.8, 315.25 , -273.0, 198.75

                         -3,127, -1098, 4105, -3627, 2503

                          218.5, 105.6, 162.4, 298.3, 57.7

                          -410, 119.5, -338.26, 937.01, -705.635

                           -5/6, -3/2, 1/4, -2/3, 5/12

For each situation that DOES fit, answer the questions you previously asked. For each situation that does NOT fit, you may change ONE NUMBER so that it does work. Justify your choice.

And a couple of Reflections, if you’re into that sort of thing:

  1.  Understanding why subtracting a negative value results in an increase to a higher value is often perplexing.  Why do you think this kind of calculation exists if it feels so awkward?

        2.  Consider these two questions:

                       What is the distance between the highest and lowest elevations?

                      What is the difference between the highest and lowest elevations?

Would your answers to these two questions be the same?  Why or why not?  Would your work to find these answers look the same or not?  Explain.

If you made it all the way through this post, THANK YOU!  I would appreciate feedback on any or all of these areas in the comment section:

  • The value (or lack of value) of this post, with specific, non-judgemental suggestions for improving it and/or my blog.
  • Strengths you see or improvements needed in the task and lesson suggestions. What would you do differently and why?
  • Actually use Version 3 and/or some of the additional materials and let me know how it went!

Maybe

Mark Chubb  is wondering, in the thoughtful way he does, WHY he blogs. I wrestle with this from time to time as well, and his post has once again inspired me to consolidate my thoughts.

I read blogs (mostly math ed) because I find them to be educational, inspirational, and insightful; I love and appreciate having access to a wealth of progressive ideas and thoughtful opinions, a chance to consider perspectives that mirror or challenge my own. Its a bonus if I laugh out loud. I tend to process my ideas (and over think) slowly, so it is not unusual to have what I read suddenly catapult my half-baked thoughts into clarity. Reading blogs helps me feel less isolated in my pedagogical beliefs and my struggles. Writers’ thoughts and questions around teaching and learning keep me reflecting on my practice and keep me growing.*

I started writing because I craved community, a place to have a voice. I periodically formalize an opinion, concern, or insight and hit “post” with some satisfaction. I feel as if I have accomplished a personally significant task–to summarize my most current musings into a post I hope is readable and not too boring. My community need is being met passively because for the most part, I am talking to myself. Although blogging thus far has provided a forum for thought-collection, it is not yet for me a conversation. To that end, I am attempting to be more brave proactive and comment on other people’s posts. It’s just not quite the same as a conversation though, is it? Maybe what I simply need is feedback.

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A pretty picture for you.

At times, I feel a tad foolish. If (if) I measure the success of my endeavors by readership, then I have failed. I supposed I could put a growth mindset spin on that and say I have not been successful yet. Were my goal to inspire others, then by default, that goal cannot be met due to the fact that virtually no one reads my blog. I have written about this before, questioning why I bother blogging.  As of now I am OK with my lack of followship** because writing for me is sufficient justification. I know it is helping me and it’s a valuable counterpart to and natural extension of reading.

Me Me Me.

It seems I read and write for self-serving reasons. Not very noble or altruistic, and maybe there’s the source of my inner struggle. While I can’t imagine my words inspiring others or provoking a lively exchange in the comment section, while I realize that my insights are original to me but not exactly fresh breakthroughs for others, and while I do not aspire to become a sought-after speaker or an outspoken leader and catalyst for change, I wonder…..do I hope, deep down, that somewhere, someone, benefits?  That someone, somewhere, values what I say?  Or at the very least, is listening?

Maybe.

*I also read and write because I have time, being casually retired, so I also struggle with wondering why I feel so compelled to continue to grow when I am not teaching. Currently.

**Is that even a word?  It is now!