Numbers, Numbers, Numbers!

I've been finding it hard to write this blog lately. It's not that I don't want to - I do! But, we have been working so diligently in class on deep understandings of numbers, that I don't have many intriguing stories to tell. They are all intriguing to me, as everything that comes out of every student's mouth is a treasure trove of assessment-worthy detail, but for you, dear Reader, probably not.

What is important to share is just how much TIME has been dedicated to number sense. It is now October 20th, and we have focused on that one strand until now. I needed to know that these kids really KNEW their numbers. So often, we gloss over it, and wonder why later everything is not working quite well enough. I have cleverly layered in skill after skill, which will now be used, starting Monday, to build an arsenal of strategies in every other strand.

We have learned how to represent numbers, build numbers and know numbers. We have learned how to explain how numbers are bigger or smaller. We have learned to order them. We've learned to compare them, not just saying that they are greater than or less than, but by identifying things like their digits, their worth and value using base tens or columns, and have found incredible differences and similarities between even the strangest pairs of numbers.

Tomorrow, we will measure everyone. Then, we'll work with those numbers for the rest of the week.

Great, deep, rich understanding takes time. I am determined to give it to them this year. Not ready to move on? Then we aren't - because it would be a shame to say "Okay, we're moving on even though you don't know about the tens and hundreds, and I now expect you to add up the perimeters of things beyond 10!"

In our understanding of numbers, we've also explored patterns. They are finding them everywhere. Yet everyday, I have changed very little in the activities. I have modelled, we have shared, and they have thought.

And so, we began "At-Home-Practice." I'll be frank here - I don't really care about homework all that much. I do believe that practicing at home is a great way to cement their understanding. But, I also believe that it becomes a fight, rather than a learning opportunity, in many cases. Parents didn't learn like this, and that is okay! If I wasn't in teaching, I wouldn't have a clue what this stuff meant. I don't expect them to. Which is why I have worried about homework for so long. I chose to send home practice questions similar to - but easier than - what we've done in class. Four a week, due back on Monday instead of Friday, in case they wanted extra time. I will not be assessing this. I know what they can do based on class work. But, it is also important that their parents know what they are up to in class. In order to smooth things over and make sure that understanding was easily attained, I started the creation of a series of videos, where I have modelled what we've done in class. Have a look, and share if you're so inclined! Feel free to use them with your own parents, kids or students.

This link will take you to the At-Home-Practice Playlist! 

Balance & Stamina & Time

I had a great chat with my neighbouring teacher today, who also teaches the third grade. The culmination of our conversation was all about the necessity and importance of balance and stamina and taking the time to build a skill.

You see, she thought I wouldn't agree with her when she said that it's okay to only give one thing to do, rather than options, at the start of an exploration. (We were actually discussing open-ended art, but it led into a math chat. The big idea was that if there is no skill yet built, then we should give the kids some exposure before saying "go wild and free!") I think I'm sometimes misinterpreted because I don't have desks in my classroom, but she was happy to hear me go on about it elaborately. I explained that while I'm a little out there with my ideas, they are grounded in pedagogy and research driven data.

We eventually ended up agreeing that with the proper skills, built through a balanced and thoughtful facilitation, will eventually allow the students to have more freedom over their work. Building their stamina and allowing time to do all of these things are how we can have them approach open-ended activities. This is where we are right now, as we balance inquiry and the culture of the pros and cons of our educational system.

This post is inspired by that short conversation, which was the fruits of some ideas that have been bubbling this month as I settle into a group of students far from the independent thinkers I worked with last year.

Stamina is something that is undervalued and overlooked. I realize this now. I didn't always. In fact, I hadn't thought explicitly about it until I'd read the Daily Five. When I read it, it clicked. Stamina is so important to build, in anything, regardless of being a part of the Daily Five language program or something else. The kids need to build their stamina for math problems, through repeated activities which increase in time over a period of time. They need it for independent reading. They need it for writing. They need it for art. They need it for transitions. So we started explicitly building our stamina with all of those things. Things are calming down as the kids start to see how pleasurable working can be.

Balance is the other piece to this puzzle of not wanting to jump out the window all year. Balancing through careful releasing of responsibility is such a necessity that it should be somewhere in the curriculum. Then again, I also think that successfully transitioning should also be a curriculum expectation, and I have no desks, so maybe I'm a little weird. So what?! When I'm talking about balance, I'm referring to everything from how much they are asked to do independently and how often that happens in comparison to large and small group opportunities; I'm talking about balancing my modelling with their own exploration, determining when it is a good idea to model first or to model after (thereby flipping the lesson around, which in my opinion and experience is often the best way to introduce something new!); I'm talking about balancing activities with quiet reflections and transitions. I'm even talking about balancing our "calm" through meditative opportunities, to turn down the volume inside their busy brains, and to give them an opportunity to have peace for a minute or two.

When the right combination of balance and stamina is combined, the results are wondrous.

I recognized, almost immediately, that my students had many misconceptions about numbers. I was trying at first to give them too many activities, to bounce around too much from adding to breaking down and building numbers and so on and so forth. I wanted them to be excited about math, but in my own excitement, I lost theirs. Lost. That's what they were. I could see it in their eyes.

So I had them work at numbers up to ten for about a week. We used ten frames and other manipulatives until getting up to ten was comfortable.

Then for two solid weeks we ONLY played with the base ten blocks. This was more modelled at first, and then I shared the conversation with the kids, and then they would work for a couple of minutes with a partner to build any number that they wanted. At some point, one kid noticed that he really didn't need more than nine single blocks. Once he reached ten, he could use a ten-rod. About a week later, someone connected that to the tens and hundreds. Bingo!

 


Time was so important. I couldn't rush it. These guys needed to become independent workers with these blocks, representing the numbers over and over and over. But it didn't get boring, because they were picking the numbers themselves. They owned the numbers. And, when their partners got it wrong, it was freakin' exciting! They got to teach their partners how the number should have been built, according to our agreed-upon system of using the most efficient blocks only.

So I would begin each lesson by inviting them to sit around the perimeter of the carpet (thereby sneaking in a math vocabulary word and making a connection available for them when it is time to look at perimeters in Geometry). I would spread out some blocks and have them count with me (because we need to count everyday). They would need to pay close attention because I might switch from ones to tens at any moment. Of course, I only did this in increments that they would be instantly successful with, in order to build their efficacy.

I would build a number and ask them what it was. They'd talk about it and call it out and we'd do it again. And again. Then, I'd ask someone to give me a number to build. Back and forth with me in the forefront (although I was sitting on the floor with them), they were able to successfully participate in the activity and practice for their independent time, with a partner. They'd break off and go to work. I'd stop them after 3-5 minutes and we'd come back together, cleaning it all up. Then, I would model some more and ask about their experience. I'd send them back to work. Having them tidy set their minds at ease and they were able to focus on me. It didn't matter that they'd be heading back out soon.

This week, I introduced a new hurdle to the mix. You see, I really, really, REALLY want them to understand numbers and work with them with ease. So the base tens were a great start after the ten frames from week 1, but they needed to work more with it, but with a challenge now. This week, they needed to draw out their blocks into the proper columns for their number. AND, they needed to write the number in words, giving us a language connection and words for our word wall. AND ... they had to write down the equation for the number. For example, if the number was 27, they would write:

Twenty seven

20 + 7

If it was 124, they'd write:

One hundred twenty four

100 + 20 + 4

The chart gave them a framework to organize their thinking, an opportunity to practice addition, writing, representing numbers in pictures, and a chance to see how complex the numbers are.

At first, this was a struggle. Now, I have kids flying through their own pages, even working with thousands. I'm not saying that they are geniuses or experts. I'm saying that they are comfortable and grasp the concept.

Next week, we will start comparing and ordering these numbers. Once this is established, I would like to work in some measurement and continue on down the path. It is all about building that stamina through balanced opportunities, which will drive their efficacy upward, to infinity & beyond! Then they'll be able to do anything.

Here is a link to the organizer we have been using, in PDF:

Base Ten Organizer (Hundreds/Tens/Ones)

You can make your own in less than two minutes, to suit your own needs!

Does 8 = 14?

Welcome back!

In a flurry, summer flew by and here we are, sitting back in our desks in rows.

Wait a minute.

No, we're not. We're not, because that's not how today's students learn. At least, that's not what I think!

I've been assigned to a new school, a new set of kids, a new grade - basically, everything is new again. I've been wracking my brain trying to figure out what and how to teach this group. It's only been a few days (6 to be exact), but the picture is beginning to clear up a bit.

It's funny, when you are faced with a new group of strangers, how little you feel you know. I was so confident in my Gr. 5 math class last year, but now thrown into a group of Gr. 2 and 3 students, I feel a bit lost! I've taught the grade before - but it's new every time. I know the curriculum expectations, and I know my philosophies. What I don't know include: the kids, their needs, their skills, their interests, their behaviour blips, their attention spans, and their pasts.

But that's okay.  In fact, I prefer to not know much about them. It stops me from forming ideas about them that may not be true, in my room or in general.  A mom told me today at our open house that her son was excited to be at school because he liked me. Had I known about his past, I'd have assumed that he didn't enjoy school and then we may be in a whole other mess!  Fortunately, him and I are in a good mess - a mess called learning!

I always start off more or less the same, regardless of the grade, with the same questions.  Ones like:

"How many numbers are there, and how do you know?" and "How many ways can you build the number 14?"

The kids rotate through these centres, 10-15 minutes at a time depending on their age and "vibe" (if you're a teacher, you know what I mean ... we are always reading the kids ... when it is time to move on, we move on).

From this, I get a sense of:
- what sort of learners are they?
- are they comfortable with manipulatives?
- do they naturally go to manipulates?
- how do they organize (or do they organize) their work/thinking?
- are they comfortable talking about it?
- can they stay on task at this point in time?
- do they "get" this stuff or is it too abstract?

I found out by day 2 that this was all too abstract for most.

One girl asked me if 8 was 14.

Yes, you read that properly.  "Is 8 fourteen, Mr. Patrick?"

"Sorry?" I thought I had heard wrong.

"Does 8 equal 14?"

2 years ago I would have said "No. What does 8 equal?"  And that may have been okay ... at the time, for me.  I know better now.

I said: "Does 8 equal 14?"

She just stared at me.  So I said, "Let's get some counters and see."


The trick here is to explore fully WITH them.  EVEN THOUGH I know it's not right.  EVEN THOUGH it logically to me makes no sense at all.  EVEN THOUGH!!!!!! I need her to experience the answer.  She needs to be in control.

We got the counters (she picked them, because there are bread tags and blocks and animals and so on).

I said, "Let's get eight."  So she did. We counted eight together.  Good! She can count.  This is giving me information.

I said, "So is eight fourteen?"

She stared at me.  "Yes?"

I said, "Okay ... how many are here?"

"Fourteen?"

"Let's count them again."  So we did.

"How many are here?"

She said, "Eight."

I said, excitedly of course, "Yes! There are eight! High five man way to go!! WAHOOOOEY!!!!" (Okay maybe I didn't get that excited ... or maybe I did.  I don't remember.)  I circled the blocks on our chalk table (see, it comes in handy!) and wrote the number 8 over it. Then I said, "If there are eight here, I wonder how many more we need to get to fourteen?"

I knew at this point that she needed a push in the right direction.  We probably could have gone back and forth all day about eight being fourteen, yes, no, yes, no.

So we counted, with my direction, the original eight, and seamlessly continued counting new blocks until we reached 14.

These blocks went in a different pile.  We circled it.  I asked her how many were in it.

She counted them and found it was 6.

So I asked her, "So, now that we can see this, is 8 fourteen?"

She said, "No."

I asked her, "How many more do we need to go from eight, to fourteen?"

She double checked by counting the new blocks and told me it was six.  We wrote 6 over that pile, and I showed her the number sentence: 8 + 6 = 14.

She smiled, and I could tell she felt better.

The big idea here is that she didn't have the concept of numbers being their own values.  It may have been some sort of weird memory lapse, but the understanding wasn't there.  I needed to take the time, even if it seems silly to spend so much time of 8 not being 14.  I needed to give her an opportunity to EXPERIENCE the numbers.  It would have been unfair to just tell her the answer.  It would have been unfair to shut down her thinking right away by saying "No, go find out how many more you need." She was brave enough to ask me, a relative stranger (this was day 2).  How dare I shut her down!  So I didn't. We built up an understanding together.



Throughout this first week, I've found some number sense skills all over the place.

We've started working with ten frames and number lines, and are focusing on building a strong understanding of how numbers work together, so that we can start thinking about mental math and everything else.

If you don't know how numbers work, how can you do anything else?


The first weeks are messy.  They're muddy and unclear.  It is a scary place to be.  But, it's also important.  Every word I say or don't say - every extra minute I give or don't give - every nod I nod, every smile I smile, every example I choose to give or select not to ... they ALL  matter.  I'm forming their love or hate for math; I'm determining how they will approach, organize and eventually think.  I need to be sure that I'm doing it well.

Volume & Capacity - A Vlog

Hello!

I decided to wrap my ideas today into a video.  Why?

1) It's easier than reading (though, there is SOME reading to be done).
2) It will show you what I've been talking about all year.
3) I got a lot of great footage today and I wanted to package it together.  I've decided to share.

It's important to note the following:

a) The students were working on a NEW concept, and were only asked the question and given access to resources.  I did not prompt them to do things - the only prompting is my questioning, to pull out some of the information from their brains.
b) This is a typical day in my classroom.  The set up is dependent on the task, and where they are in their independence with a concept.
c) This is one way that I document who does what and how they do it.  This is handy for assessment purposes.  Today's assessment was all diagnostic.

Watch, share, enjoy!  Please let me know if you have questions, and I will try to answer them in a blog (or a Vlog! What a fun new format for me!)

Thanks for stopping by!



Link to the video on Youtube

Planning vs. Prepping ... and Everything In-Between!

I was at a hub today, where we were looking at how to blend e-learning with face-to-face learning.  Over the past few meetings, I've made reference to my math program, and today someone asked me:

"So it sounds like you're right into the problem based learning.  Do you spend a lot more time planning for that?"

My short answer was, "No, I spend less."

Here is my long answer, and why.

BEFORE: 
When I was teaching from the text book, I spent my planning time reviewing the lessons, copying black line masters, digging up manipulatives, and then finding the correlating lesson in the different grade level texts, according to the IEP's in the room.  I would then spend my class in the following way:

First 20 Minutes: Introduce the lesson and have them follow along in the text.  Do some modelling and show them EXACTLY what to do and how to answer.

Thirty Minute Work Period: Answer the questions.  Not finished?  Homework!
I would spend MY time wandering the room, peering over shoulders, and chasing down hands as the room was silent.  If it wasn't, I was shushing.

The next day, I would look at some of the work, or the work that I had them hand in.  They would hand in the one "big problem" in the text book, either on a cue card or by putting a star next to the answer in the text.

I would move on to the next lesson and forget about it.  At the end of the unit, or the lessons I chose to do (because some of them I skipped), I would give them a review test for homework, take it up as a class, answer some questions, and then give them the test the next day.  I'd spend hours marking it, record the results in my neat and tidy mark book, and that was that.  Movin' on.

Fortunately, that only lasted for about 6 months before I realized it didn't work.  (6 months into my 2nd year of teaching, as my first year I was bounced around and I don't even remember it!)

NOW:
I tend to plan on Friday for the upcoming week.  I have a 40 minute prep and can typically cover off most of my math planning at that time.  This is what I do:

1) Identify what expectations we covered this past week.  I will either build on those, extend from those, or if we are wrapping things up, move on to some new expectations.

2) Once I have identified one or two "main" expectations (for example, this week we are looking at conversions in measurement), I look for ways that I can layer in expectations from other strands.  This week, I chose to look at mean again, and targeted one of my open problems around the mean.  In order to find the mean, they would have to convert the measurements to one cohesive unit.  The unit was up to them, though I hoped they would select the most appropriate unit for the length.

3) Then, I build my questions.  I aim to have 2 open questions and 2 parallel tasks.  I rarely actually use them all, or at least use them all as they are, or within a week.  Often, this becomes a 2 week plan instead of one.

4) I have to ensure that there is some stepping-stone opportunities as I build the questions.  I don't want to just reiterate the same skill over and over - that is sometimes useful, but often a waste of time.  If students are showing solid understanding, then they need to move and increase complexity.  My questions start more simply and layer in more indepth questioning, skills, and so on.  It may not be HUGELY different, but over the week, my goal is to have them moving from collaborating to independently solving, because at some point, they must be able to do it on their own.  They know that the accountability is part of the culture, because we have built it that way and regularly discuss it.

Many of my questions, prompts and basic ideas come right from the math curriculum.  When an expectation doesn't actually have one, I will check Marion Small's Big Ideas book to see what she has and often tweak it to be more appropriate for my group.  I also check the Guides for Effective Instruction, right from the ministry, and steal questions from there.  Sometimes, I even check the (gasp!) text book, because with some work, many of the questions CAN work.

So how do I differentiate?  It is built right in.  An open question allows for a variety of strategies, and may also allow for self-selected numbers to work with - comfortable, "just right" numbers.  A parallel task allows me to scaffold the questions, either by changing numbers, opening a question for some and closing a question for others, and by allowing students to select the one they would like to answer.  When it comes to the elementary curriculum in math, many of the differences simply come into how large the numbers become, and when there is a solid understanding of numbers, then any number is workable.

If a student has a modified math IEP, I can ensure that their parallel task is appropriate to them by cueing them, but allow them opportunities to work with other learners at other levels.  Often, they learn from each other, and the outcome is surprising when the children are permitted to just work through, without teacher talk and interference.  I've said it before and I'll say it again ... often, we just need to shut up.

Most of my real "planning" or preparing comes WITHIN the classroom.  I am constantly making decisions based on what I hear and see.  I may choose to extend the problem into tomorrow, or consolidate today rather than tomorrow, or tomorrow rather than today.  I follow the flow of the students.  If they need to be brought back in for a tutorial, I will do that.  I don't need a fancy lesson to do it, I just need their voices and our collective skill base.  I'm no expert in math - hey, I got a 56 in Gr. 11 math.  I never LIKED math, until I found problem based math.

After class, I've already looked at their work, because I no longer have a desk and spend my time wandering.  I am recording their working process, asking questions about what and why they are doing things, and taking pictures of their work.  I will sometimes look at their work to make decisions, but most often I don't need to, because I already heard their thought processes, and that is more important than what they write down.  Of course, what they write down has value, and I'm not suggesting it doesn't.  But, I know how they are recording their work and what they're writing, because I watched it happen.  I probed as it happened.  And, I'm ready for tomorrow.

So yes, my planning takes me less than an hour most weeks.  It involves creating questions and mentally working through different way I might solve it.  It requires that I have a list of questions ready to go.  But it does include making lessons for the Smart Board, or designing fancy videos.  It doesn't require that I photocopy a bunch of stuff and dig up dusty manipulatives - whatever they might need is readily available at all times.

Do I spend a lot more time planning for problem based learning?  Not even close.  And yet somehow, I am closer to my students, deeper into the curriculum, and can comment on specific strategies and processes that they have used because my energy is in the learning, not the planning.

As a final thought, here are the questions I've used this week, and how I organized them.  I've noted my Look-For's:

MONDAY (open partner choice): Harley has been very busy – she’s been taking secret weekend trips!  In the past month, she has travelled: -       12 km -       1200 m -       120 m -       1.2 cm -       120 km -       12,000,000 cm -       __________ cm -       __________ m -       __________ km What is the mean of her trips? (LOOK FOR: Can they convert? Do they use the math wall if they don't remember "mean"? Can they find the mean and label with a unit of measure?)	THURSDAY (partial independence): Use manipulatives to prove whether or not the following statements are true:           3.2 m is 320 cm           4.1 cm is 410 km           321 cm is ____ cm OR Use manipulatives or an equation to prove whether or not the following statements are true:            320 cm is 32 m           410 km is 41 cm          ____ m is ____ cm      (LOOK FOR: Are they using Tuesday's equations?  How are they understanding conversions?)
TUESDAY (teacher selected partner choice): Investigate, test and confirm an equation for converting cm to m. Do the same for m to km, km to m, m to cm, and cm to km! (LOOK FOR: Are they understanding conversions?  Can they use variables and constants?  Are they testing and confirming their equations?)	FRIDAY (independent): Describe the multiplicative relationship between the number of centimetres and the number of metres that represent a length. Use this relationship to convert __.__ cm to metres. OR Describe the division relationship between the number of metres and the number of centimetres that represent a length. Use this relationship to convert __.__ m to centimetres.  (NOTE: From the curriculum, but tweaked to require an example of converting. LOOK FOR: Can they convert? Can they express the relationships of multiplication or division?)
WEDNESDAY: I’m away – work on practice package.

This is not an end-all-be-all explanation of my planning.  It flows and grows and changes flexibly, with intentional thought, as the year progresses.  But, I certainly spend more time thinking about what they do, instead of planning and preparing lessons before hand.  The learning is deeper, their relationship with math (and their peers) is stronger and more positive, and my assessments are deeper, based on process work, discussions, consolidations, and through looking at their "Friday" (not always a Friday, but by this I mean their independent) work.

So that is how I plan.  For now.

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.