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.

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!


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.

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?


*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!

A Duh Moment

The penny finally dropped.

I blogged on March 1 of last year about the apparent inconsistencies between coordinates and tables organized in one order (x, y) and rate/slope in another, as y/x.   I played around a bit with writing slope as x/y (which feels more intuitive to students) and found that when graphing, you get the same graph as long as you keep the change in x horizontal.  This post is a lengthy update to that one.

In a predominately DI classroom where students are shown How and expected to memorize processes, they tend to not raise questions about Why.  I certainly never did.  Like others, I might’ve notice the inconsistency and accepted it because, well, math is just confusing and you have to just memorize and follow rules in order to pass a test to get an A and look/feel smart.  Many practices in education not only encourage this approach but also award those who can use it well.

I started wondering about the Why of (x, y) vs y/x  last year when I started regularly volunteering in an 8th grade classroom.  Students were being show a whole lot of How, with plenty of examples to copy, and then given HW to practice.  When I heard one student mumble mostly to himself, a little to me, about why you move horizontally first when graphing coordinates and vertically first when determining slope from a graph, I was delighted by him yet frustrated with me because I had no response for him other than I was wondering about that, too.

Which bugged me, so I kept picking at it.  I even got brave and posted my musings.  I really hoped I would figure it out and get back to that curious student.  I let him down.

Fast foward ten months to yesterday.  I woke up early, and for some reason, it popped into my head WHY rate is written y to x and not the other way around. Isn’t that weird? I suspect seeing this  “Silent Solution” video the previous day on writing rate from a table lit up that part of my brain again.

It was in front of me all along, I already knew this, so it was my Duh moment.

If there is relationship between two things that co-vary, then you can see a pattern and develop generalized equations.  The three equations you can write for a proportional relationship are:

Dependent Variable = rate • Independent Variable

IV = DV / Rate

Rate = DV / IV <<<——Whomp, there it is!   Why (algebriacly) m = y/x and not the other way around.  (I still am leaning toward convention as to the order of coordinates.  I does feel more intuitive to list the IV first.)

Students should be empowered to explore ideas like rate and slope in such a way that they do the sense-making.  I am confident that they are fully capable of understanding Why, that the How can emerge from them and be owned by them.  And it will be glorious.

What do opportunities for student-generated sense-making look like in your classrooms?  What do you do to nurture curiousity and facilitate learning?  What do you struggle with?


PS.  I am also pondering the duality of the way rate is represented.  As a ratio, it is a comparison of two values, a quantitative description of a c0-varying relationship.  A change of two in This for every change of three in That.  We even need two number lines to graphically represent This and That’s relationship.  When rate gets put into a proportional or linear equation, it suddenly behaves like a single value, locateable on a single number line:  two-thirds.  What’s with that?


There are definitely different and often conflicting schools of thought about what it means to learn, the purpose of getting an education, and how we define (and measure) success. There are plenty of polarities– instrumental vs relational understandings, concepts vs skills, performance vs learning, student-centered vs teacher-centered. Do we want our children to understand in a way that they can think and work and solve and create in a novel situation or do we want them to be able to flawlessly perform a well-rehearsed, standardized process on demand.  Or something in between, some sort of blend or balance. Growth Mindsets, Learning Targets, Multiple intelligences, NPVS, IXL, DOK, SBG, WTF. The list goes on.

I am sure you can hear my bias.  All preferences, beliefs, experiences, information and misinformation, and yes, even biases play into what ultimately informs and shapes every single teacher’s practice and become the experiences students receive at any school, in any classroom, anywhere and everywhere. These experiences are as greatly varied as the teachers and students themselves, and certainly are not equal nor equitable. Intentionally or not, they create gaps, provide or limit opportunity and potential, and ultimately perpetuate privilege and oppression. There are so many factors and factions, a unwieldy number of entangled variables. Where to begin?

Those who persist in their instance for accountability/standardization/testing are either grossly ignorant to the complexities of teaching and learning, or are aware but chose to ignore it for a cause that is self-serving, a desire to have power and control over others. They do not/will not recognize that education involves honest-to-goodness real-life people, not clones or puppets or empty minds and blank slates. Every child and every teacher arrive at school every day with life experiences, with intuitions and self-perceptions, with personal strengths and obstacles to overcome.  I cannot think of a single teacher who does not care nor of a child who does not deserve to be cared about.  Those who continue to make noise about back-to-basics and kids-these-days do not/will not consider the vast differences of human condition in our country, created and perpetuated by a society that is so eager to blame, so easily divides humanity into “We” and “Them”, a society that ignores compassion and commonalities and instead feeds voraciously on fear and falsehoods in order to justify their hatred-fueled actions.

This post is not going the direction I thought it was. I was going to write about a need to examine certain outmoded yet deeply ingrained practices in education, practices that contribute to and perpetuate unequal opportunities for our children. Practices that impede learning, practices that sort, practices that raise up some while holding others back.  I was wondering if by arming ALL children with the ability to think critically and creatively, to question the reasoning of others, to nourish in each of them empathy, compassion, and kindness, and a healthy sense of self-worth, they would be better prepared to handle and even repair the damage they are going to inherit.

I know there are many who rise above the popular call to fear and hate, who are not part of the appalling racism, sexism, and xenophobia that apparently is thriving in the US. I know there are those who are already speaking up and speaking out, already taking action in the face of horror and shame. I know that pinning a safety pin to my lapel is a simplistic and virtually empty act, that I am going to have to push myself outside of my comfort zone. I have to hope and believe that my actions and my voice will matter. I have to.

After all, rebellions are built on hope.

AHA #3. If it’s broken, throw it out.

I’ve been reading Alfie Kohn’s book The Schools Our Children Deserve; it is part of the required reading for a course I’m taking. Just before summer got crazy busy with Other Stuff, I had read through Part 1: “Tougher Standards versus Better Education”. I admit I do not feel intrinsically motivated to write thoughtful any reflections on Chapter 5 (Getting School Reform Wrong) or Chapter 6 (Getting Improvement Wrong); while they contain additional compelling evidence for what is currently amiss in eduction and with ed reform, I am already long on board. In fact, even though I took a un-planned break from the course, I’ve continued to think/read/journal/blog/reflect in order to keep moving myself towards insights about teaching and learning.

Recently, I decided to take up Kohn’s book again, looking forward to “Part 2”, in which Pooh Kohn offers solutions. Chapter 7: Starting From Scratch.

Well, beam me up Scottie if this chapter isn’t fully in sync with my current thinking and wondering.  Suddenly, I feel motivated. Huh.

What I’ve been wondering about in recent weeks stems from a conversation I had with Jackie, a middle school math teacher friend. I volunteer regularly in her room and we meet to talk almost as regularly about pedagogy and moving student learning forward and other Important Things. My role in part is to support her implementation of several strategies that are intended to create an active learning culture in which students make sense of math.

She noticed that in spite of the on-going, long term math PD that has been invested in K-12 teachers in her district and in spite of her efforts to increase student talk and focus on student reasoning, learning remains for the most part passive and feels glacially slow. Mindsets cling to answer-getting, compliance, and performance, about speed and status. In many ways, what she wants happening in her classroom differs from what is actually happening.

Been there, done that.

Neither she nor I are implying that their previous teachers are at fault! Nor are we blaming the students! (I just heard again today that The Reason students don’t learn is that they “just don’t try hard enough”.  I disagree.  I may need to blog about that.) At this point, this is merely an observation that her desired change seems too minimal. Its frustrating. We know there is more work to do, more than we anticipated.  Persevere.


Every worthwhile observation should be followed by a genuine question. It’s how we learn, how we find solutions.  What I wonder is, exactly what is going on in schools and classrooms (including mine) that impedes or prevents moving from a performance culture to a learning culture?

If we only make changes- even great ones– to some of the things we do while maintaining others, the “old-school” culture prevails. This is a big AHA for me.  Most teachers seeking to improve practice consider what to do differently or what to add in, but it is equally important is to identify what needs to be (gasp) thrown out.  In October 19’s post, I listed several widely accepted, rarely scrutinized practices that I believe undermine honest efforts to transform teaching and learning. Alfie Kohn provides a similar list and includes justification.  For both of us, grades top the list. Teachers, administrators, parents, even students, should make their own lists. This is where conversations need to begin; this is where priorities get identified, appropriate long-term goals get made, resources located, and plans developed . This is where meaningful change- change that moves education forward–stands a chance.

I recommend reading Kohn’s book. In the meantime, here’s some of my additional take-aways from Chapter 7:

  • You can’t make people want to do something, like learn. You might offer rewards and threaten punishments (grades, points, stickers, perks, honor roll, a different track, detention, praise, admonishments, etc.) but you get compliance or defiance, not motivation. Being extrinsically motivated to get the reward or avoid the punishment is not the same as being intrinsically motivated to learn. Intrinsic motivation comes naturally and heightens engagement, but can be extinguished or nourished, depending on what takes place in the learning environment.
  • The purpose of school, the reasons for educating people, tend to be debated in rather polarized terms. This or that.  Kohn writes, “It’s more a continuum than an either-or, but the point on that continuum we identify as ideal makes all the difference.”  A lot of this and a little that.  For individual educators, teams, schools, and communities, identifying where that point is located will help clarify which practices support or impede genuine learning.
  • In “traditional” school cultures, there is a disconnect between long term goals (those ideals shared by most educators and parents) and short term goals (those skills, mandated and tested) and practices within classrooms. We say we want one thing, and we do and/or allow something else. (Is that the problem Jackie and I are facing??) If our long term goals are for students to be curious, to be intrinsically motivated, active learners, to listen empathetically, to construct viable arguments and critique the reasoning of others, to persevere, to make sense of complicated ideas, and so on (all valuable life skills, by the way), but we put more energy into short term goals like being sure they can correctly add and subtract rational numbers, then it is possible– even probable— that the long term goals will never be met.
  • If, on the other hand, we first identify that ideal place on the continuum, clarify our long-term goals, and then teach in such a way that always supports these goals, then both long and short term goals will be met. Frequently meeting and reflecting on our long term goals and evaluating what is happening in our learning communities is a must.  Remember, some practices may need to be updated, included, or even discarded to be sure long term goals are supported.
  • Achievement should not ever be a long term goal. Low achievement is not the real issue (although it is popular to say so), disengagement is. Its not that students are not “trying” hard enough; they just don’t see the point.  We should be wondering why.  The performance/compliance/accountability culture that is touted as the “solution” is actually a massive obstacle to achievement; focusing on grades and points and data and test scores in order to improve learning simply backfires.  John Nichols said, “Teaching requires the consent of the students, and discontent will not be chased away by the exercise of power.”  (That quote is in Kohn’s book.)
  •  In contrast, high achievement results naturally when both long and short term goals are met by capitalizing on intrinsic motivation and student interest. Common sense tells us that all people learn best when they are interested in what they are learning about; schools can and should be putting their efforts into creating and sustaining learning environments for students and their teachers.

If you have not already done so, please check out  MARK CHUBB’S informative post about how his district set and went about meeting long term learningculture goals and what happened to student achievement.

Where is the “ideal” place on the continuum?  For you, for your school, for your students.  Why?

What practices and policies do you use or see used in your school or district that are intended to create and sustain an active learning culture? How effective are they?

What else is going on that undermines these efforts and perpetuates fixed mindsets, passive learning, and disengagement?

What’s on your list? Why?  Who will you share it with?

…equals (write answer here).

When students are shown how to “do” math and are asked to perform copious amounts of computational answer-getting, they quickly and unfortunately develop the idea that math (and learning math) is primarily about right and wrong answers, void of context and steeped in mystery.  And the correct location for the answer is right after the equal sign. In such a setting, students who say 5×6 “is the same as” 6×5 get their statement affirmed and handed some new vocabulary to memorize: commutative property.

What does that statement even mean?  In context, 5 groups of 6 objects is different than 6 groups of 5 objects. That is, the pictures look different, although the total is the same. Is that it, end of lesson?  In some classrooms, yes.  But I think there is risk here by glossing over why and caring only about what or how, and that risk is losing the kids who are trying (desperately) to make sense out of math, especially when we assume they easily move between concrete and abstract. The pictures look different.  I don’t get it.  

You could use area models/congruent rectangles to try to convince them: See? 5×6 is the same as 6×5! Same dimensions! Same area! Same rectangle! Ta-da!

Wouldn’t it be amazing if school encouraged kids to be dubious and curious?  Wouln’t it be glorious if they asked, What about 21+ 9? Or 60 ÷ 2?, Or any of the infinite number of expressions that also equal 30? Are some expressions “more” equal than others? What exactly is going on here?

When people like Kristin  ask 3rd graders to sketch pictures to match two similar stories, then discusses with them whether or not order matters in symbolic notation–while remaining genuinely interested in and openly curious about their thinking– they are helping young students develop significantly different ideas about math and learning. In this setting, students know that the teacher respects and values their reasoning; together, they are all making sense of an idea of equality that reaches beyond “is” or “write the answer here” in a way that addresses misconceptions and helps them build a bridge between abstract and concrete.

While we’re on the subject of equality, what about these?

4/5 = 0.8
5 – 8 = 5 + (-8)
-12 – (-7) = -12 + 7 = 7 + (-12) = 7 – 12
a÷ x/y = a • y/x
-3/4 = 3/-4 = -(3/4)
-4(9) = 4(-9)
3(n + 7) = 3n + 21
-(a + b) = -a – b
x^0 = 1
Etc., including any formula or rule “given” to students.

If I want my students to understand equality deeply so they can use it purposefully, then I’ve got to do more than tell them that these expressions are equal, show how to change from one to the other, and then test that they can. And I’m beginning to understand that it is also a disservice to impose my reasons why. The sense-making needs to be theirs, not mine; they are completely capable of figuring out why these equations in particular are worthy of their attention.

“Same value” is just not convincing enough for me any more.

Aha #2: Learning Communities

When I started teaching, I had an vision for what I considered an ideal culture in my classroom.  Every year, I tried to make one professional goal support this ideal.  My vision was of a Culture of Learning.  Not the kind of learning that is mostly about memorizing and skill performance, but the kind of learning that is mostly about exploring ideas and constructing understanding; the kind of learning that helps develop critical thinkers.   I envisioned students working closely on interesting and worthwhile math tasks, asking questions, listening empathetically, taking risks, justifying their reasoning, learning from and with each other and having glorious aha moments that moved their learning forward.  I consciously chose to use a proficiency grading policy from the get-go (and stuck to it in spite of being the only teacher using it) because I wanted my students to grow as learners and know that they were.  Ideally, they would monitor their own learning, be willing to revise their thinking and their work, and learn from mistakes.  Ideally.

I felt it was my responsibility to make my vision happen.  I still do.  The reality was, many of these things actually did happen to some degree, but never to my satisfaction.  The various moves I tried did not ever seem to make a big enough difference.  At times, it was difficult to not feel like a failure.  Each year, I would try again, because I know beyond a shadow of doubt that the culture in a classroom matters.

This post is not about woulda-coulda-shoulda regrets.  Or about blame.  It is about trying to make sense of a culture in which learning flourishes, and part of that process involves figuring out and examining what hinders, undermines, or flat-out prevents it.    Grades.  Worksheets.  Right and wrong answers.  Performance culture.  Testing.    Compliance.  Grades.  (I said that already?  Oops.)  Government mandates.  Time.  Status Quo.  “Ability” leveling.  Homework.  Tradition.  Myths and misconceptions.  Fixed mindsets.  Data overload.  Just to name a few, of course.

In an attempt to be succinct, here’s what reflecting on my experiences and efforts has revealed to me so far about culture:

AHA #2a:  In order for a teacher to improve her daily practice, in order for her to develop and sustain a classroom learning culture,  she needs to be working in a learning community.  That is, the thriving learning culture we desire for our students needs to begin with a thriving learning culture for their teachers.  In order for teachers to learn, they need a safe and supportive learning community that is willing to talk about and examine practices honestly and critically, to make time to find and use excellent resources, to implement ideas, ask questions, collaborate, make mistakes, revise, and reflect, reflect, reflect.   While fabulous online communities can and do support the learning of individuals, I am not aware of any real impact on a school’s culture.

Please read Mark Chubb’s post * on how his district made this happen.  Notice the non-performance goal, the commitment to time, and investment in people.  If we expect/want our students to be active learners, then we’d better desire and demand it for ourselves.

AHA #2b:  You can’t simply “add” in a few pedagogical moves or latest research-based ideas and expect significant change to happen, even over time.  Even if you get training and/or implement excellent ideas well, there simply is no magic bullet.  Figuring out what you personally and you as a community need to STOP altogether or significantly ALTER — and understanding why and figuring out how–are equally important as adding in the good stuff.  Less feeling like your’re making shit up and trying to survive and more intentional learning and professional growth.  See Aha #2a.

What kind of community do you work in?  What kind of community do your students work in?  Who is surviving and who is thriving?  Why?

* Mark Chubb is my latest blog crush.  Go.  Totally worth reading everything he has to say.  Not to mention that delightful photo gracing the top.  (Other crushes I’ve had are Dan Meyer, Christopher Danielson, Fawn Nguyen, and Bree Pickford-Murray.  There are many amazing other bloggers I follow and am thrilled each time a new post shows up in my reader, but these five I have gone back and read every one of their posts.)