Intentional Flexibility - A Reflection on an Idea

Intentional is my new favourite word.  If I was running the educational world, this would be the new "differentiation," or "problem based learning," or "accountability."  Intentional is such an important word, in so many ways.

Today, I want to focus on intentional flexibility, rather than something in particular that has happened in our learning environment.  Intentional flexibility is the combination of careful, curriculum and research based planning, alongside the ability to know where you're coming from and where you would like to go, so that you can bend when it is needed.

I see this all the time: a student tackles a problem in a completely different way than I ever would have, and therefore I missed the option when I was working through the problem myself.  In doing this, the student goes off on some other tangent.  You know what?  That is OKAY!  Just because it isn't what I would have done, doesn't make it wrong.  In fact, if the links that they are seeing are logical, then I believe it is my role to facilitate, understand and encourage the connections made.  So I was trying to get them to work on fractions, but they skipped over that and worked on something else, that still gives them a logical answer?  No problem.  Tomorrow, I'll bring it back, or maybe I won't.  Maybe what they did was so genius, I will just go that way with the rest of the group, and we'll come back to fractions when the time is right.

When the time is right?  Yes.  I know, we are strictly bound by the curriculum and the school calendar. Our report card marks and comments are completed by early June.  That leaves us with a few weeks to "fill in the blanks."  I remember sitting in a staff room one year and hearing a colleague say at the end of May, "Well, I finished the text book.  I guess we just won't do math now!"  They were proud of themselves!  Where is the intentional flexibility?  There is none in text books.

My nose is in my curriculum document all of the time.  Almost every day, I look at it.  I have learned to flip back and forth and find the links that make sense between "strands" or "units."  Units are our worst enemy, by the way.  In being intentionally flexible, I can jump back and forth between strands and hammer out a load of them, while giving students meaningful, real learning experiences.  I'm not saying I have it down to a perfect art - no one ever could - but what I am saying is that I allow for logical connections between strands in order to "layer" expectations and spend more time on problems.

Why GIVE students geometry nets, when they can make their own?
In this example, the kids (Gr. 2) made their own nets by collaborating (not cooperating, there
were arguments, but that is part of it) and then we printed them to test them.  It wasn't
what I had started with the intention of, but it certainly filled the expectations, better than my
initial plans.  Follow the students.

I am intentional in my questioning.  I know what I want to see from them.  However, I am also flexible.  Math problems need to be open, as Marian Small has so successfully promoted and researched.  (Have I mentioned that I am a HUGE Marian Small fan?!?)

I am intentional in my communication with students.  I rarely say whether or not a student is right or wrong.  I don't often point out errors, but I am flexible in this: if a student is having a misconception or is completely halted, I can communicate with my students carefully in order to guide them, without leading them.

I am intentional when I assess.  And I assess all the time.  Every pencil mark, every dialogue, every consolidation - it's all assessment.  I am flexible in how I look at what I'm given.  It is difficult to explain, as it is an intuitive science rather than an exact science, but my assessment aligns with curriculum standards, while at the same time allows me to determine what I need to explicitly teach, who I need to work with beyond the whole-group setting, and what my next steps are.

Who cares about seating arrangements?  When I got over this, my teaching flourished.  They sit where they choose, when they choose, as long as it is on-task and appropriate to the activity.  My intention is that they learn.  I am flexible in how they do it.  In a recent reflection survey, 18/21 students said they did not want assigned seats. Hmmm ....

I am moving away from my long range planning, and following my student's lead.  I am able to see their connections between expectations as they work.  If they are naturally starting to use decimals effectively, and are asking me questions on their own on how round numbers, then I know that the time has come to teach the group about it, using the students who are interested in it as my catalyst - they are going to come up and show how they solved the problem, and it will lead into the next thing.  Along the way, we started rounding decimals.  Check.

She is working hard by recreating the prompt.  She is using what she needs.
She is intentional in her use of manipulatives, and is flexible in working
with her partner, who has some other ideas.  It's not just for teachers.

While it may seem like I don't know what I'm doing, I would happily tell you my exact intentions - curriculum expectations and all - at any point in the classroom.  I would dare you to come in and follow me as I interact with my students, jot notes on their work, record videos of how they work, and sidle up to strugglers without making it obvious that I am helping them work through their larger misconceptions or struggle-beyond-safe spots.

I don't have a teacher desk.  It is intentional.  It allows me to be flexible in where I am and what I am doing at every point throughout the day.

They're so happy!
It's because they enjoy working collaboratively, on hard questions,
and make lots of mistakes, but learn the whole time. 

I may stop a lesson to teach a concept.  I may bring them together to consolidate at the end of the problem solving.  I may stop a more formal assessment altogether if I know in the moment that they're just not ready.  I know my students, I know my curriculum, and because of that, I can be intentional and flexible.

I do make plans each week.  I have a list of questions, and see a flow through them as they build on the prior learning experiences.  I make notes as to where I believe lessons will need to be introduced.  However, I follow my students instead of my plans.  I will tweak the questions, revise the lesson time, and extend the class time permitted depending on what they need.  It is intentional to start, but moves flexibly.

Think about this.
IT IS THE MOST IMPORTANT THING!
IT SHOULD BE ON THE COVER OF EVERY CURRICULUM DOCUMENT EVER!
Go to edutopia.org if you are a progressive educator who wants more ideas and to hear
from other people who think like you. 

This allows for a culture of independent and collaborative (NOT cooperative) workers and learners, rather than a culture of students who need to go to the teacher when they don't get it right away.  I can't tell you how I cringe when I see that sort of thing happening, or when I hear people complain that their kids just don't get it, still don't understand, or that the curriculum just doesn't make sense for the age range.  Seriously?

Intentional flexibility - it's the next big thing!

Struggle Space, Misconceptions & Reasoning

I feel like every time that I log in to write a new post, I want to start with the line, "I've been wondering a lot about ______ lately."  Well, I have.  I am constant wonderer, and while it is infuriating to never be satisfied with my current place as an educator, I have to say that this struggle space is what drives me to improve, and I am satisfied with my progress as an educator.

Today, I want to touch on struggle space, misconceptions, and reasoning skills.  It sounds like a lot, but it is all intertwined.

We are reaching the end of our explicit look into fractions, and are starting to explore measurement expectations.  Yes, I do tie my "units" together,  because all of the components of math are so important that I don't want students to feel that they should "close the book" once they've had a test.  Hence, no tests - just constant, daily assessment.

Today I wanted to assess some of their fractioning skills (can they take a whole number like 1000 and split it into fractions, instead of looking at the "whole" in the fraction as "1"?)  I also wanted to diagnose their prior knowledge of linear measurement, namely the conversion between metres to kilometres.  So, I asked:

Mr. Patrick went on a trip this weekend. 

He travelled 1km.

If he had only travelled a third of that, how far would he have gone in metres.

I wanted to know:
- do they know how many metres are in a kilometre?
- do they know how to use fractions as pieces of a whole, thereby splitting/dividing the number into the fractional amount, in this case being a third (1/3)?

I specifically wrote "how far would he have gone in metres" because I wanted to cue them to the conversions without explicitly telling them.  I felt that if I hadn't, they would have taken 1km and split it into thirds as kilometres, which was not the point of the activity.

This was just the first part of the question.

To be honest, I was a little bit surprised that most had difficulty figuring out how many metres were in a kilometre.  Fortunately, they saw that they needed to convert it (even though they didn't have the language to use, like "convert").  This was the intended struggle space.

Most struggled for a solid ten minutes.  Most knew that it would be something like 100 metres.  This was a great space for them to discuss and debate, and to work on their reasoning skills.  Take a look at the video below, of one of my students explaining why she believed that the kilometre held one hundred metres.  It is actually fairly convincing!

So, she wasn't correct.  That's okay!!  In fact, it's great, because she will remember her reasoning, and will correct the misconceptions.  After a few more minutes of wondering, I directed some of the students who were struggling the most to grab one of the three text books that I keep on the resource shelf.  They located the index, and found kilometres.  They identified pretty quickly that 1km=1000m.  Excellent!

I decided to use this opportunity to give them a little more confidence, as they often struggle more than others.  I called the class's attention, as I felt the struggling had gone on long enough, and I wanted to see them actually working on the thick math of the question (there was a second part, which you'll see momentarily).  I told the class that I appreciated all of the thinking and work they'd done so far, but that it was time to reveal how many metres were in a kilometre.  I asked the kids with the text book to deliver the message, which they did proudly, and everyone said, "OH YEAH!"  Then, they went back to work.

Some struggled to do the actual division.  I encouraged them to struggle a bit through some long division, but allowed them to use calculators when they had shown me that they understood why they were dividing or splitting one of the numbers by another.  This is so important - like any technology, it is useless if you don't understand the reasoning behind it.  If you were to split 3, because of the thirds, into 1000 pieces, you would come out with an entirely wrong answer, and you would likely be baffled as to why it was so teeny-tiny.  It could be a great learning opportunity, as well, which is why I stuck close by any groups using calculators.

There was another excellent opportunity built into this question.  1000 divided unequally, at 333.33333.... and so on, forever and ever, amen.  We've been slowly looking at how to round the decimals, which is an expectation in Gr. 5, and this question allowed the kids a real opportunity to see it in action.  Some struggled with the idea that 333x3=999 ... they wondered and were frustrated with the fact that it wasn't 1000, but that if they used 334, it would be way too many.

With the skills developed from part 1 of the question (fractional splitting and measurement conversions), they moved on to part 2:

On the way home, Mr. Patrick took the scenic route.
He travelled 12 kilometres.
What are some other ways to describe the distance?
How long would he have travelled if he only went a third of the route?

In this part of the question, I wanted to see:
- can they transfer the skills from part 1?
- can they convert 12km to 12,000m?
- will anyone be able to/think of converting it into cm?

With part 1 complete, most were able to struggle a bit and identify the conversion of 12 to 12,000, reasoning aloud to me that if 1 is equal to 1000, then 12 would obviously be 12000.  Some of them knew it right away.

One group was able to find out how many centimetres it would be, and solved the fractional piece using both metres and centimetres.  Then, they went on to determine how far I'd travelled in total, using every different measurement unit.  In one lesson, those two students demonstrated that they had already achieved the curriculum expectation.  For them, I can move on.

As we brought it all together at the end, I really had a sense that everyone was on the same page.  I knew that I would be able to present tomorrow's explicit lesson on conversion and have them all grasp what I would be saying, because they had struggled, reasoned, and had many misconceptions repaired today.

Based on all of this, I feel like it is a strong explanation of the power of struggling and having a misconception; where explicit instruction and redirection can fit in; the importance of caution in the delivery of a correction; and the beauty in strong reasoning, whether or not the actual facts are correct.

This allows me to reflect on my teaching - am I allowing the kids to grow of their own accord, while still travelling in the right general direction?  Are my questions allowing them to explore the intended curricular areas?  Am I creating the right opportunities for struggle space that is reasonable and not beyond their zone of proximal development?  Am I allowing the students with opportunities to discuss, explore and reason, for real and in depth?

Struggle space, misconceptions & reasoning - in my eyes, the cornerstones of a solid math program.

The Day I Taught A Lesson (Gasp)

My students have become so comfortable with exploring problems in math that they think having me teach them is absurd.  In all honesty, it sort of is.  I really do believe that for the most part, having the students work through problems and then consolidating the major problems is the best way for them to learn.  Now that it is their culture, it is the norm and they are on-task.

Last week, I gave them a paper with some questions about fractions.  We've done a couple of weeks worth of problems surrounding fractions, but I couldn't really tell yet where their understanding was.  They have been comfortable the whole way through, but the idea of equivalent fractions seemed to evade them.  Even when the question suggested that they use equivalent fractions, they would still be able to find loop holes and solve the problems without even touching them.  That is wonderful, and I embrace this kind of open ended problem solving ... but at the end of the day, I need to have them be able to find equivalent fractions and understand how it works.  I asked them the following questions:

1) What is a fraction?
2) How can you change a fraction?
3) How could you turn a fraction into a decimal or percentage?

#2 blew my mind.  They drew the fraction in different ways.  They discussed the fraction in different ways.  Some even wrote the rest of the whole as a fraction (1/3, 2/3).  FANTASTIC!  But wait.  They hardly touched on equivalent fractions.

So today, I did a "lesson."  (Gasp! Shock!)

As I was getting into some demonstrations (and do note, I approach my lessons in a discover sort of way - there were a lot of aha! moments for the kids, especially those who had pieces of understanding yet weren't yet bringing it all together), one of the kids piped up (with attitude):

"Uh, Mr. Patrick?  When are going to get to work?"

I was flabberghasted.  THIS IS WORK! DUH!

But I realized as I was about to react ... this isn't work as they know it anymore.  This is weird.  This is Mr. Patrick blabbing away.  Yes, many of them had misconceptions repaired and some loose ends tied up in their understanding, but it was still me delivering data.

If I had more time with them, I would have happily continued to let them explore the fraction business and work more one-on-one with them.  But, with only a half day to deliver math, language and science, and time closing in on me as we reach the final 2 months, I can't justify (at this point) another two weeks of problem discovery.  There is still a lot left to cover, and a lot of it won't layer together.

The moral of the story?
1) The kids are comfortable in a problem based setting.  The culture is strong and they LIKE to work.
2) Explicit teaching is still okay, but only when it comes after some discovery.  Otherwise they'd have had nothing to tether it to, and my hot air would have been far less effective.
3) Perhaps I need a better balance - it always seems to come back to balance - and should "talk" just 1% more.  They seemed extremely uncomfortable.  Or maybe I'm just being paranoid - maybe it is a good thing.  But they will be moving on to someone else in September, and that person or those people (who knows who they will be?!) likely will not be quite as comfortable as I am being quiet.  I know it's taken me 3 years of hard, HARD work to learn how to shut up, and I still struggle with it!  So what's the right balance?  I don't know.

Fractions Aren't Scary!

I thought I'd try something a little different today.

I've been out a lot lately for collaborative inquiries (professional development), both for math and blended e-learning.  We are gearing up to launch a blend of e-learning with live-action teaching, and I have been so overwhelmed with ways to bring tech into the classroom that I thought it was time to try something a little different.

Since I haven't been in my classroom for a full week since before the March break (that's insane - I was at a hub, a conference, a meeting, and a hub!), I knew that just giving them the question would be okay, but that I really needed to WOW! them.  From what I've been looking at, they are "getting" fractions to varying degrees.  They are extremely comfortable drawing fractions, and many can talk about equivalent fractions when they are "easy to manipulate" numbers, like 10/100 being the equivalent to 1/10.  Some can even tell me that if it is 10/100, then it is 10%, which is also 0.1, or 0.10.  PERFECT!  But, I need to make sure that this understanding is concrete and not flukey, so I thought I'd probe a bit.

The question today was wide open.  I was looking to see if they could do what I mentioned above - could they represent the fraction as an equivalent, in a decimal, or as a percentage.  While most didn't really give me that information, I realize now that the question may have led them to drawing pictures.  Here it is:

Nick is going on a trip.  His mom bought him a gigantic sub, and he needs to figure out how to make it last over the full day trip, with no other stops!  That means it will be his snacks and meals.

What fractions could represent how much he will eat each time?  Explain how you know.  Are there other ways to express those fractions?

Many students became caught up in how big the sub needed to be.  They were concerned with the measuring aspect.  Others were trying to figure out how often he would eat (one group was successful in finding this out, actually).  I would say that 99% of the class (that's a guess, and just my way of saying that pretty much all of them, so don't nail me later on doing some poor math!) didn't extend their thinking beyond showing me a circle split into however many snacks and meals he would have.  I did see some interesting things though.  I'll show you those in a minute.

You are likely wondering, "What did he do that was so different?  He just asked a question - nothing new!"

You would be correct!  Except, before I showed them the question, I interrupted their independent reading with this trailer:

It was a great way to grab their attention and whet their appetites for the upcoming question.  It made them giggle and engaged them in the math.  It made an "authentic situation" a little more authentic.

I created it in about 4 minutes using my iPhone.  I grabbed the kids from the classroom during my prep time and snapped some photos - they had no idea what it was for.  I used my question to input the text, and then that was it!  Using the "Trailer" option instead of "New Project," I was able to throw everything together with the fun template graphics and music.  The only downfall was that I couldn't change the number of photos, the length of them, or anything else.  I highly recommend using this from time to time!  It is easy, and the kids can even use it to showcase their thinking (in fact, one group was making a grand effort to use their own iPods to record their work, but we ran out of time and they didn't get to finish).

Here is some of their work.  It is not, by any stretch of the imagination, mind-blowing.  It is, however, a good dose of reality in what they know, can do, and how well they were able to read my mind.  Next step for me?  Be a little more explicit in what I ask, since they didn't go down my road this time.

This student showed his thinking using percentage and the original fraction!

This student's work is well organized, and includes a mini legend.  There is a lot of effort going into figuring out another way to show the fraction, but the student requires some "hard teaching time" now. 

These students re-interpretted the question.  They knew that the snacks would be smaller than the meals, so they split the sub equally and then took each half, splitting one half into thirds, and one into halves, to create more fractions.  Interesting approach!  It is well laid-out.  However, they still haven't shown the fractions in more than one way.

With some prompting, this student was able to find equivalent fractions using multiplication.  We used this example in our class consolidation to see how to grow the fractions.  Then, we moved into simplifying them, working backwards by dividing.  I think they are starting to really, truly see it.  Fractions don't need to be scary!

It's not about changing everything you do as a math teacher - it's about small additions and adjustments.    The trailer didn't revolutionize how they saw the problem.  It did help excite them about math again, and break up another Thursday.  A little shift is important - it's the technological equivalent to a body break!