I’m re-listening to some of the Cathy Fosnot #notabookstudy interviews on Sound Cloud this summer.  Today I am listening to Week 4.  In this episode, the discussion is mostly about when to jump in and tell children how to do something, and how long you let them wade through the muck trying to figure it out on their own. She quoted Jean Piaget:


Further, she went on to talk about how even parrots can learn to repeat what has been told to them, but can we ever know what is understood versus what is just being repeated.

I have been thinking a lot about assessing just that.  Because whether I told children how to follow a particular strategy or they had a peer show them how to solve a problem, I’m thinking it’s pretty rare for a child to just figure everything out on their own.  Maybe an only child who is homeschooled would.  But in a busy classroom there will always be someone who knows a strategy or figures out a strategy and then shows it to everyone else.

One of the things I’m thinking a lot about is what we ask in an interview with a child that helps us get to the heart of their understanding.  It’s easy to see if they got the right answer or not. What’s harder is figuring out the thinking behind it.

I am currently reading a book called “Dynamic Teaching for Deeper Reading” by Vicki Vinton.  Even though it is a book about teaching reading, I have made  connection to assessing math understanding. On page 82, she shares a list titled “Steering the Ship: Teaching moves to support thinking and meaning making.”  It lists things a teacher says to move students along in their thinking.  Even though this month is dedicated to literacy related professional reading, I can’t help but see how some of the same moves would help in math. I have turned the suggested teaching moves into questions that could be asked during an interview, while coaching a child or group through problem, or even during a Number Talk.

Steering the shipTeaching moves to support thinking and meaning making by Vicki vinton

Isn’t it great how teaching moves in one subject can be used over and over throughout the day regardless of the subject being taught?


#notabookstudy: Constructing Fractions

OK you guys!  I am so excited about my math class on Friday.  Perhaps too excited??  IMPOSSIBLE!

First, I’ve been thinking a lot more about the learning that takes place when students develop a model on their own.
Second, last year I worked a lot on the open number line with my grade 3/4 class.  I thought we were doing so well!  Then we started working on fractions and they didn’t know where 1/2 fit on the number line.  They could identify 1/2 of a set, or half a whole, but 1/2 of a distance was too much. I was going to say “at first”, but I never really felt like they owned that piece of information.  They just sort of borrowed it from me for a while and then silently put it back on the shelf, I think.  Though this is giving them some credit for putting things back where they found them, and that might also be misplaced. So…this year I wanted to add fraction of a distance to our study of fractions.
Today I wanted to start talking about fractions.  Yes, this is a bit last minute.  (I’m not going to tell you the whole story of why this happened.)
So how to develop the context and ask everyone to develop their own model?
Typically, I would gather the class, show them a half, and them let them go make some halves. Then we’d do fourths and thirds, and then we’d try out other things that make kids happy like 5ths, 6ths, etc.   I would do fractions of a set one day, then fractions of a whole another day, then practice them a variety of ways – sometimes alone and sometimes side by side. Then we’d make some connections to half of dollar, or half of an hour, and then we’d be done with fractions. We are a grade 2/3, so equivalent fractions would show up in there, and so would mixed numbers, but that is not our focus.
But new-me wanted them to develop more of this on their own.  All of it, I hope.
Students in my class sit at tables.  There are 4 of tables, so 4 groups.  I put colour tiles on one table, pattern blocks on another, LEGO on the third, and snap cubes on the last. I told them to go play with the blocks for 5 minutes, and then we’d start.  It’s May, and we’ve used the math tools a lot, but they seem to still need to just play with them for a few minutes every time.  If you can’t beat them, meet them where they are, I always say.  Or at least I say that sometimes.
After 5 minutes, I had them all join me on the carpet.  I wrote 1/2 on the board. (I had 1 on top of 2) “What is that?” I asked.  I got a variety of answers.  The 4th or 5th student called it “half” but others called it 1 divided by 2, 1 out of 2, a 1 and a 2, and 1 = 2. I decided to give it to them.  I wrote “half” under the fraction.  “Oh, half, right,” a few responded.  “Now, go to your table and use the tools I put out for you to show me half.”  Away they went.
Here are some responses:
I was surprised at the number of students who weren’t sure what to do.
One student had this:
I asked, “Is this half?”
She said, “I’m not sure.”
I turned to her neighbour who had this:
I asked, “Is this half?”
He said, “Yes.”
I asked, “Can you explain to (her) how you know it is half?”
This was really interesting.  He kept saying, “Well, it’s half. See?  It’s half.”  Then after a few repetitions for this, he used his hands to separate the two halves and said, “See?  Half are here and half are here.”  He’d separated them by colour.  He wasn’t sure how to explain it, even though he knew he had half.
I asked her if she could see it, and she could.  So I challenged her to make her own.
At the pattern block table, I took this picture:
“No,” the student said.  “You have to take the picture from above.  Then people can see that half is  yellow and half is red.”
“What about this one?”  I pointed to 2 red trapezoids.
“It’s cut in half by that line.”
Here was a very interesting one:
When I asked him to explain to the class, he said, “Well I have 4 blocks.  2 are over here, 2 are over here, so half are in each place.”  I was really surprised by this because I thought he was showing me two models: one half orange, half red, and another that was 2 halves making a whole. I explained this to the class and they got it both ways.
One of the people at the connector cubes table had this example:
I use the Show Me app and the Apple TV to project pictures during an activity like this so we can congress what has taken place.  Sometimes I write within the app, and sometimes, like in this picture, I write directly on the whiteboard.
He had set it up two ways. First he said, “Half are here and half are there.” Then he changed it and said, “I mean half are yellow and half are something else.”  I think he was surprised when I said that each was a different way to show half and he’d been write two times with two different answers.  They are used to me saying this kind of thing, but I think he thought he was wrong the first time.
After we congressed this, I sent them back to their tables.  “Now that you have seen some other models, go show me 5 ways to make half.”  And they did.  I mean, AND THEY DID!!!!!
We had to skip math on Thursday because of an assembly, so I decided to do double math today. I told them that after recess I’d have different blocks on their tables and they could use a new tool to show me half.  They came in and got right to work.  That’s a lie.  They got to work eventually.  But they all showed me half in a few ways with a new tool.  Photographic evidence:
One child had this, which she explained like this, “When my mom says, ‘Eat half.’ this is how much I would eat.”  I was happy to see that connection to fractions in her real life, even though she didn’t quite understand that each of her piles should have an equal amount in them.  In reality, who counts every exact pea?  But I think that might start happening, and I should probably call her mom and apologize in advance.
I then read them a great book called, “The Cookie Fiasco” by Dan Santant.  It’s part of the “Elephant and Piggie Read” series.  It’s funny!  Four friends, 3 cookies, how can this be solved? I read up to the part where the friends are completely befuddled about how to solve this problem.  Then I gave them 3 square cookies made of paper, and asked them to figure out what the friends could do.  In the book, the nervous hippo had already broken the cookies in half, but there still weren’t enough cookies for everyone.  So, all of my people cut their “cookies” in half. Most of them cut them again, and within a few minutes everyone had figure this out.
Congress:  “What did you find?”
Everyone agreed that everyone got 3 pieces of cookie.  I asked, “Ok, so how much cookie did they get.”  This took some discussion.  I had to reassemble them into wholes, then move them around until someone said, “Well, they are quarters.” and then we counted by quarters, “1/4 + 1/4 + 1/4= what?”  Finally someone said, “Well, it’s 3 quarters.”  and the lightbulbs that flashed on above all the heads nearly blinded me! It was awesome!  They got it!  I mean THEY GOT IT!!!
Then I showed them how some of our classmates had cut the “cookies” into squares and some into triangles and some into rectangles. It took a bit more discussion, but we all saw how they all ended up with 3/4 as the answer to how much cookie each friend got. I told them they could put the cookies in the trash, and about half of them put the in their backpacks instead because sometimes trash = more important to take home than our actual work or, you know, report cards.  You’re welcome, Moms!
Finally, fraction of a distance  Last week in gym I asked everyone to run halfway across the floor and stop.  It was trickier than you might think, especially given the fact that there is a giant black line halfway across the gym. But then I said, “From there run halfway to the wall.” and this time there was no line to signal halfway.  Some of them did it, and some of them didn’t.  We ended our super day of math by going outside into the glorious sunshine.  I had two hula  hoops and asked kids to stop halfway between the two.  More of them did than didn’t.  Of those that didn’t, most overshot the mark by a good amount. We’ve obviously got some work to do around fractions of a distance.  I’m thinking they need to do it in a place where they can’t really run.  I think they are going too fast and 1 meter into the full out sprint they have forgotten they are supposed to stop halfway, then one classmate stops, or they see the end goal, and something deep inside them says, “Wait? Was I supposed to stop somewhere?” and by then they’ve gone more than half way.
Tuesday we’ll do some work with writing down fractions, looking at the models and pictures and writing the number.  Then on Wednesday we start EQAO with the math section and I am super paranoid about being accused of cheating so I don’t teach math during the 2 days when we are doing the math sections (which is where we begin this year.)  That gives me some time to think about a few more ways to connect their  fraction learning to their past learning.

Multiplication Models

I wish I could remember how I was taught to multiply.  I have absolutely no memory of who first taught me about multiplying…which is kind of weird because I remember all of my elementary school teachers. I do remember doing “mad minutes” in Mr. Goodreau’s 6th grade class. I would bounce around doing all of the 1s, 2s, 5s, and a few other random problems that I actually had memorized. And I remember being in high school algebra class and writing down my skip counting for problems with 6s and 7s. Heaven forbid Mr. Creager should ask a question that involved 8s or 9s.

Excuse me while I go deal with some residual anxiety related to multiplying.

As a teacher, I’ve often wondered what would have been different for me had someone along the way pulled out some manipulatives, or showed me an array, or, you know, explained that there’s more than one way to answer a multiplication problem.

I introduce multiplication to my grade 3 students using a “groups of” model. Using lessons from Marilyn Burns “Lessons for Introducing Multiplication” book, I let them investigate (though I’m not now 100% sure I have been allowing them to investigate enough…topic for another night!) ways to figure out how many legs there are in their group, or how many pencils we’d have if I bought 4 boxes, or any number of other questions designed to push them to use skip counting or repeated addition and then connect that to multiplication.

I also teach my students about arrays. I get out the colour tiles and ask them how we can arrange them to show 4×3, for example, and we move on from there.

Why arrays?

Well, because Marilyn Burns told me to, at first.  Then I stuck with it because it made so much sense to me.  And years later I am still doing it because I see it as a model that makes sense to kids.  There are arrays everywhere, so it is easy to find ways for them to practice using them, as well as a way to connect them to why multiplication is meaningful.

Today I wanted my grade 3s to work on arrays.  Usually, I would show them an array and then we’d practice them together and we’d all walk away pretty happy.  However, today I decided to let them discover the array on their own.

I pulled out the colour tiles and put them in the middle of the group.  “We’ve been playing “Circles and Stars” and I am wondering if that always has to be played with chalk and a chalkboard, or if we could play it with these colour tiles.”  The colour tiles were our manipulative, and the array was the tool I wanted them to work toward understanding.  I am going to admit to being a bit nervous about this, and thought it might be a disaster.  But who am I to question Cathy Fosnot, right?  So I kept going.

A few of them suggested that they could use tiles to make circles and then fill the circles with other tiles.  That would be like circles and stars.  I asked them to show me, and they started arranging the tiles into circles.  They soon decided  that it is hard to make a circle out of squares.  Then one boy said, “You could think of the sides.  It has 4 so you could do 5 x 4.”  This is what he made.


“That’s a Minecraft circle!”  someone said.

I asked, “You think we could do it like that?  For every problem?”

A students said,”That would be hard.  That would be a lot of work!”

Another student had been watching the whole thing.  She finally said, “I have 2 plans now.  You could use the tiles to make dice….wait…you could have imaginary circles.  You don’t actually need the circles, just the groups of tiles.”  HOORAY!!

We took turns modelling that:

I asked them to count what they had, and some interesting counting things happened.  I know Melissa will be sorry if I don’t explain that, but it is kind of a tangent. 🙂  Let’s just say that even though they had groups of 4, they still counted by 2s and 1s.  (sigh)  But then another child said, “Well if we put them in lines, we can count by 2s.”  and he made this:


Someone counted by colours in groups and then they started, on their own, talking about other ways to arrange the 16 tiles we had randomly started with.  Like, seriously, they started making the division connection ALL ON THEIR OWN!

(By now I had switched out all the tiles so they were the same colour.  I wanted them to think about organizing the arrays without worrying about making designs with the colours.  This is a real problem with colour tiles and can often get in the way of the math learning.  I mean, patterns are important too, but making checkerboard patterns for everything doesn’t seem to help with the counting or understanding in anyway. It just slows kids down and causes them to hate one another because someone is always hoarding the good colour that everyone else wants.)

We talked about how each of the arrays we had made matched with the dice we use in “Circles and Stars”. Then I asked, “Now do you see how to play ‘Circles and Stars’ with colour tiles?”  And they did!

So I sent them off with a partner and a pair of dice.  We are now using a 10-sided die to show the number of groups and a regular 6-sided die to show how many in the group.

Here are some highlights:

  • On their own, one group found out about ones and zeros and what happens with them in multiplication.  🙂
  • On their own, each group figured out that organizing the groups into straight lines made them easier to count, and easier to count without having to go back to ones.
  • One girl started counting by ones.  I stopped her and encouraged her to try a bigger number. (Array pictured below) “I can’t count by 2s because I get confused!” I said, “Then try something besides 2s.”  I thought she’d do 5s, but she counted by 4s.  🙂
  • Slowly 2 of the 3 groups moved to this sort of group arranging:

I feel like my goal today was accomplished, and then some! The students discovered the array.  I felt like I was scaffolding, not rescuing them.  (A personal pursuit of mine.) And I think there was a healthy bit of discovering things that can lead to other things for us.  (Division, for example.)

I have an idea for my lesson tomorrow.  Two of my grade 3s were away today, and should be back tomorrow.  They will need to be in on this discovery.  And I want to really solidify what we did today.  BUT…instead of sharing my idea, I’d love to hear ideas from others.

And by the way, the grade 2s were all doing their Guided Math rotations while the 3s and I did this.  I blogged about my experiment with GM earlier in the week. I am still on the fence about it, but will finish out the week before I talk about it again.

Finally, because I know that many people visiting this post are new to blogging, I want to mention that it’s great to leave a “drive by” comment.  But the best conversations happen when people return to read comments left after their own, and then comment again!  After you write a comment, WordPress blogs allow you to click a box that will “subscribe” you to a post.  Every time someone comments on that post you’ll get a notification that will keep you in the conversation.  I recommend it!

You’ll have to log in to comment on my blog, leaving your name and e-mail address.  Rest assured that only I will be able to see your e-mail address.  It won’t be visible to anyone else.  I have this setting on because it helps to eliminate many spam comments.


#notabookstudy: What Next?

We are still working on the “Ages and Timelines” investigation.  We worked out the difference between the ages of Carlos and his family, etc.  Next, it was time for everyone to calculate the difference between themselves and their family.

I expected, anticipated, that some would have a tricker time than others.  Given the ages of their siblings, which are not always nice landmark numbers  like 10 or 15, I knew some would struggle a bit.  I also anticipated that because this part of the investigation is more for individuals some would struggle without a partner to talk through the work.

I was right.

But what I didn’t expect was the number of children who came with the answer.  I said, “How old is your mom?” and they said, “She is 33, and she was 25 when I was born.”  I didn’t expect so many children to then have trouble putting this together as a number line.  I mean, if you know where to start (8) and where to end (33) and you know the number of jumps in-between (25) then making the number line should be easy, right?

It wasn’t.

So my next step feels like a backward step, but I have decided it is a sideways step.  We need to go around this obstacle, learn a little bit more about number lines, and then move forward.

ALSO:  I didn’t anticipate the halves.  As in, “I am 9 and 1/2.” or “My brother is 14 and 1/2.”

I feel like this week, so far, 2 days in, has been all about being responsive to student understandings.  In the example above, I felt I had two choices.  One would have been to teach that child how to deal with the halves.  We could have talked about using a decimal, or about using the fraction to figure out a more accurate answer.  My other option, the one I chose for this child at this time, was to say, “We are going to forget about the halves for now.” He has a tentative grasp of this whole “finding the difference” concept.  Letting him stick with the whole numbers will help him solidify.  Talking to him about the decimals or fractions would muddy the water.  He’s in grade 3, the curriculum doesn’t require us to talk about the decimals or adding/subtracting fractions.

For the next two days I am going to do a part of this unit that my colleague skipped with her class: creating a timeline.  She felt it was confusing her class, but I feel like it will be useful to mine.

And at the end of the day, isn’t this responsive teaching what ditching the textbook is supposed to be about?



#notabookstudy: Landscape of Learning

So, I started this blog a few years ago…a week, if I remember correctly, before The Board decided we shouldn’t be using blogs with students unless they were accessed through the something or other that I never have figured out. It’s been sitting idle ever since.  I think my last post was in 2008! Seems like the perfect place to start blogging about some math learning.

Confession:  When I first learned about the Landscape of Learning in the “Young Mathematicians at Work” books by Cathy Fosnot and Maarten Dolk, I dismissed it.  It is a lot of shapes and though it is visually appealing, I didn’t take the time to figure out what all of those ovals and triangles are for. I was too busy teaching!

Fast forward: Last year I did some work with the Student Work Success Teacher.  She suggested using it to assess where my students were.  I liked that idea!  It looked so cool with all the triangle and ovals coloured in with pretty highlighters. But…there were a lot of words I didn’t understand, and I didn’t really understand the difference between the “Big Ideas” and the “Landmark Strategies”.  I did understand the tools.  Hooray for me!

This year I have been doing some work with other teachers in my school and in the board, and I feel like I am starting to understand the Landscape.

For the past week, my grade 2/3 class has been engaged in Cathy Fosnot’s unit titled “Ages and Timelines”.  I thought we were pretty good at subtraction, and number lines.  I thought the timeline study would be a good tie in to a timeline we made in Social Studies to go along with the Traditions and Heritage unit. (It has been, but that’s a topic for another day.)  I thought we’d spend about 2 weeks on this unit, and then my 2s would stick with it a bit longer while my 3s jumped back into multiplication and division.  So far, so good.  It is taking a bit longer than I expected, but I am happy with where we are so far.  We’ll probably spend one week more than I had originally planned. (That’s the end of me putting the rest of this post into context for you.)

After the first week of our work, we had made it past Day 1 – figuring out how old everyone in the family was when 8-year-old Carlos and 10-year-old Maria were born.  We made it past Day 2 – figuring out how many years until Carlos and Maria will reach the ages of their kin, and are in various stages of figuring out how old the folks will be when Marie and Carlos are 33, 35, 55, 57 and 87-years-old.   I spent some time on Thursday talking with a few of my groups, focusing on grade 2 students.  Then, I came home and filled out some Landscapes while thinking about what I saw in our discussions over the week.  Here are some photos:

I started by completely filling in shapes that I saw the students were clearly possessing. The two in yellow have been working as partners for this unit.  As I interviewed them, I could clearly see that one was leading the work, but the other wasn’t far behind.  One was helping the other by showing him how to get started, choosing strategies for them to use and filling in some gaps.  The other was helping too because he was challenging the thinking.  I caught him a few times saying, “No…..oh wait.  You’re right.”  Or, “NO!  That’s not right!” and then getting the other to explain his thinking. They both asked for help a few times when things weren’t going smoothly.  I found it easy to scaffold them through their thinking because they really had already been scaffolding for each other. They both had their work very organized.  They are counting backwards, making 10s, and counting on. But some skills are sort of there, but not 100% concretely. They were using open number lines, but in a way that showed me they are still sort of experimenting with using it on their own, and they often needed me to talk them through drawing it before they got started.  I was drawing the end and beginning points, for example, and even though they were telling me what to write it was always with a question mark at the end.

Me:  What should we put at this end?

Them:  57?

Me:  Yes!  

(Repeat for every thing I put down, and with everything they then started putting down for themselves, even after I walked away and they only had each other to confirm the choice.)

They were right every time, but not confident about being right. I could also see that one was “keeping one addend whole and moving to a landmark number” but the other wasn’t as competent with this skill.  He could do it, but didn’t think of it on his own or use it independently.

Then I chose a student who is struggling through the unit.  I was able to pinpoint some gaps.  The yellow students didn’t even think about a hundred chart.  They are beyond that.  But the person in pink needed one.  It was used competently to find an answer, except that it was used for counting by ones, not skip counting, and there were some one-to-one correspondence problems that led to the wrong answer quite a few times.

So…what have I learned?  I feel like I know where to go with all three of these children.  Today in a PD session, my principal compared the Landscape with running records.  It was a lightbulb moment for me. Yes!  This is the precise sort of information I have about my students as readers and writers.

So…what’s next?  Well, now I have to figure out how I react to having this new information.  I am comfortable having all of my students working on ability-level appropriate reading and writing work.  I am comfortable meeting with groups of children based on the strategies they need to develop. I clearly need to get comfortable doing this in math.

I know that I am engaged in really good professional development when I feel my world starting to shift.  I have resisted the idea of Guided Math for a several years.  I don’t want to make it sound like I am lazy, but honestly Guided Math sounds like a lot of work.  I mean, I am doing a lot of work now, and sometimes revamping my way of doing something feels like a daunting task.  But I think I need to stop resisting and just do it.  It will be work, but I am excited about the possibility of seeing large leaps in math like I have become accustomed to seeing in literacy.

Today in our PD session we went through the work of a whole group, and mapped it on the Landscape.  We could see where they class was clustering with their skills.  We could see some outliers who were both above and below the groups.  (I mean, they were doing things close to the top or close to the bottom that others weren’t showing during the activity we observed.)  This helped to paint a picture of the whole class and what goals it might be good for them to work on next as a whole.  I know that would help me to do some planning for my class.  The curriculum is certainly a map of where we need to go each year, but I feel like the Landscape might be a more precise map of how to get from point to point on our journey.  I’m thinking of it as paper CAA maps my father-in-law gets for free from the travel agent compared to Google Earth.  They’ll both get me where I need to be, but one  is going to help me avoid the sandwiches at the Ignace gas station because I’ll know there is a Subway just down the highway!