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.
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:
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:
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:
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:
From Kate Nowak :
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.
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*.
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.
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.
UPDATE: Serendipity: Liz Mastalio’s Week 2 post “Honestly, the Math is Secondary”.
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?
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:
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 learning–culture 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?
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.
Can you teach an intuition?
We can do this better
Thoughts about Teaching
Better through reflection
The Writings Of Alfie Kohn
Inspiration and resources for mathematics teachers
Musings on math and teaching
I research mathematics teaching and learning in secondary schools
Math Change Agents
Lighting the world with math, one student at a time.
Not everything that can be counted counts, and not everything that counts can be counted. -Einstein
In Math, the Journey IS the Destination.
Not everything that can be counted counts, and not everything that counts can be counted. -Einstein
Reflections on teaching and parenting young mathematicians
A Math Teacher Trying to Have More Wags and Less Bark
The mathematics I encounter in classrooms