I was not intending to write again today, but here we are.  It’s just that it was all so exciting our math class today that I couldn’t put off telling this story until the end of the week.

As you’ll recall, we are working toward developing and strengthening the Big Ideas of hierarchical inclusion and part/whole relationships.

I wrote 8+4 = on the board before anyone arrived, and 30 minutes before math was supposed to start, kids were already coming to tell me that they knew the answer.  I’m going to start doing that more often!   I reminded them that yesterday we had discovered that within a number, say “4”, there are other numbers.  There are two 2’s,  a 3 and a 1, and even 4 four 1’s.  Then I asked for answers to 8 + 4 and most everyone agreed it was 12, except for one person who was sure it is 11.   Time for the second problem:


Almost as soon as I had it written up on the board, thumbs were going up.  I want to say that we have been using this “thumbs up” routine for a while, and I still (DAILY!) have to remind people of that routine.  *sigh*)  We agreed it was 12, and then someone said, “I know what you did.  You just took one away from that 8 and gave it to the 4, so it’s still just going to be 12.

So, I make a huge deal out of that idea.  “What?  Do you mean if I take 1 away from the first number, and give it to the second number, I will still have 12??”  They said yes.  So I said, “Will that always work?”  and we tried some other examples.  It did work!  We even noticed we could use facts we know, like 10 + 10 to help us with other facts, just by moving a number from one side to the other.  HOLY COW!  Isn’t math amazing??  I probably said that 5 times.  I wrote this on the board:  “If you take one away from the first number and give it to the second number, the answer will stay the same.” I do have a photo of that, but forgot to edit a kid’s name out.  I had written down the name of the child who first said it, and the names of a few who were quick to agree with this.

Soon they were asking for harder numbers.  “The problem is that I can’t really give you harder numbers.  See, once you know that this works, they’re all going to feel easy.”  That’s what I told them.  They didn’t believe me, so I gave them a 1000+1000 example and they were stunned.  STUNNED! Then I asked the 100+100 = 50 + 110 problem you see in this picture, and we had to add the addendum:  It only works if you take add back the same amount you took away.

I know…it says, “…others numbers…” which doesn’t make sense.  But sometimes I am so excited about what I am writing, or my mind is actually on 2 or 3 things simultaneously, and I don’t write everything 100% correct.  I fixed it when I made our semi-permanent chart for the wall. 

So it was all great and exciting and I kept thinking, “This is awesome!”  But the surprising thing I didn’t expect was how excited kids were too!  “Don’t erase it!  Leave it up forever!”  they kept saying.  And “Can you take a picture of that to show my mom?”  (We use SeeSaw and they love when photos go home to mom and dad.)  While they were doing their Writer’s Workshop writing, several kept saying, “Are you going to add that work to our math board?”  So instead of doing some writing conferences, I was doing that. I’m sure you’ll agree it had to be done right then.

One thing I said yesterday is that I wanted to make my own work more organized.  I think I was able to do that.  But I can’t prove it because I left a kid’s name on the board.  I’ll try not to tomorrow!  Another thing I am noticing now is that I should have a model up there.  I am going to, right now, e-mail my colleagues at school and see who has Cuisinaire rods we can use sometime this week, and I am going to make sure I draw some models tomorrow.

And I feel like I should keep it all real here, and confess that after all that,  one child raised a hand to say, “I still think 7+5 is 11.”



math, Number Strings, Number Talks


Today was the day that I was going to “do something different” based on all the learning I did with Cathy Fosnot last week.  I sat down carefully on Sunday night and planned out the Number Strings I wanted to do each day this week.

I started with this goal:  Guide my students to discover the Big Idea of “Part/Whole relationships”.  I feel like this is one thing we can work on in our Number Talk time that will help everyone understand numbers a little bit better, and help us understanding adding and subtracting.  I also feel that too many of my grade 2s are still having trouble counting on from a whole number, and are struggling with hierarchical inclusion.  There’s is too much “counting 3 times” and starting at 1 every time something is counted.

I started with 4+3 = on the board.  Through conversations, we ended up with this:


So far, so good, right?  One of my grade 3 students said he’d found the double then added in the “extra” one.  Right there I thought this is the difference between Number Talks and Number Strings.  I could have written more problems that let everyone practice the “find the double” strategy.  But I was really looking for the “part/whole relationship”.   We started talking about other ways we could make four.  I made a list.


I was so determined to stay focused on my goal!!  But then someone mentioned 3 + 1, even though we had 1+3 and we had to talk about that.  HAD TO!  I mean, really, I thought, “What would Cathy Fosnot do??”  and I decided she’d explore that AMAZING discovery that someone had just made.  I had to ask, “Wait…you mean I can just reverse the two numbers I’m adding and still get the same answer?!”  I added that thinking to the board, complete with a name label, and then had to ask the next question:  “Will that work every time??”  We tried out a few examples and decided that yes, it would for addition, but no, it would not for subtraction.

Here’s something I have decided:  The goal doesn’t have to be completed in just one day. I have the whole week to explore part/whole relationships.  And maybe it’s stretching it out over a few days that really helps the kids to see some ideas.  I could have easily turned this into a full 1 hour conversation.  But after 10 minutes people start to get restless and stop really listening, so I ended it.  We’ll pick it up again tomorrow.

One of the things that really struck me in Cathy’s lesson on Saturday was that she displays all three parts of her conversations (the question, the model and the conjectures) at once.  One is not behind the other.  She said that this way kids can see all three and each helps to strengthen the other.  This is one of the main things I want to really do this week (and beyond.)  I am going to start to do these talks on the large white board instead of the small one so I have room for everything.  I also, as you may have already thought when you looked at my pictures, need to make sure I am organizing this better as I go.  But I’m okay with admitting this is going to take some practice.



PD Saturday


A few years ago, I read a book by Burkins and Yaris called “Who’s Doing the Work?”  It’s a fabulous book.  I now follow a learning community they have created on Facebook.  The other day, this photo was posted.  It was captioned:  We had such a wonderful time filming next generation shared and guided reading lessons this week! Stenhouse Publishers is accelerating the production process so that we can debut the videos at our full-day workshop in Hurst, TX on November 3rd! We are so excited! Come see!  (I pulled that directly from their Facebook page.)

This photo has been alternately bothering me and making me feel vindicated.  You know when you watch a demonstration video and the whole time, instead of thinking about how great the teaching is, you’re really thinking, “There’s no way that could happen quite like that when there are 23…25…37….kids in the class.”  Well, here’s proof that the videos aren’t actually set up in a “real” classroom setting.  I mean, I already knew that, but it feels good to have it validated.

I spent a few days this week learning from Cathy Fosnot. What I mean is Cathy and I hung out and talked about math.  There were about 25 other people there on Tuesday, and maybe 125 there today, but I still feel like it was me and Cathy getting together to talk about math.

Cathy had a variety of videos to show that showcase her work, and to help me (Ok, “us”) learn from her methods. One of the very first things I noticed was that there was so much background noise.  There were actual other kids in the classroom also having math discussions.  Cathy was talking to a small group of kids while the rest of the class was also working on their investigation.  This is how it sounds in my class too!  I am chatting away with a few kids, and brilliant things are happening, but I’m also using my peripheral vision to make sure naughty things aren’t happening, and once in a while I have to sort of tune out of the conversation, or say, “Hold that thought…X!  SIT DOWN AND DO YOUR MATH!!…ok..go on.” That’s reality.  What this showed me is that it’s okay that sometimes I am giving a group less than my full attention during a conferral.  That doesn’t mean great things can’t happen in my room.  It does mean I need to remember to turn on the voice or video recorder when I sit down for an important conferral because I just might miss a bit of it. It also showed that brilliant math conversations happen spontaneously in lots of math classrooms when the situation has been properly set up.  It doesn’t just happen with a hand-picked group of children in a quiet one on one setting, which is then edited to highlight the moments of brilliance. I feel like I am probably going to be able to pull together some examples of how this has happened in my own classroom in the past few weeks, and I’ll be able to do it easily.  (But I will probably have to wait until next weekend because I was away all day and my house shows it. But I might also decide to just throw out all of our dishes and clothes and just blog all day tomorrow.) My point is, it’s happening in my real life!

Another thing that stands out this week is how it feels when your teacher is enthusiastic about your response.  Cathy talked about celebrating what our students say.  She did this over and over too, and not in a way that seemed put on or like acting.  She really was enthusiastic about responses people gave, even though those people were adults.  On the second day, I was part of a large group that was sitting in the fishbowl.  She was giving us Number Strings to work on and talk about during a demonstration lesson, while others in the room observed.  Having seen this on Tuesday, I knew when she told me to sit in the  horseshoe that I was going to have to do math in front of a group of my peers.  First reaction:  PANIC!  I couldn’t figure out which chair to take with me, I couldn’t figure out where to put my chair or my phone, I sat down only to realize I’d not brought paper with me incase I needed to write something down because I can’t keep it all in my head like other people can.  My heart was beating fast!  But it took no time at all for me to feel at ease in Cathy’s class.  I knew she wouldn’t call on me if I wasn’t ready.  I could see that when she called on people, it was after she had heard that person talk to a partner and knew there was something good to say.  In a way, having her call on my partner and I to share our conversation was a confidence boost because I knew she’d already heard us say something good and important.  Her enthusiasm was confidence-building, and contagious. I want to make sure I am passing that on to my class.  If you’re near my class this week, and hopefully often after that, I hope you’ll hear that in my voice, as “Oh my goodness!  Did you all hear that great thing that X said?  Isn’t that amazing?!” echoes down the hall.

The other thing, which is really an overall shift in my thinking in general, is that subjects shouldn’t be (maybe aren’t) compartmentalized.  I’ve gotten so used to thinking about covering this idea from this subject or math strand, and then moving on to another and not letting them all weave together.  In one Number String today, Cathy had us learning about division, fractions, ratio tables, spatial representation and reasoning, and general number sense.  And I could see how with careful note taking I could have comments for a report card for 3 different math strands, without teaching 3 separate units.  See?  Master teacher!  Because math has not been my strong suit in the past, I have not always easily seen the connections between ideas.  I mean, I can see how Number Sense connects to lots of places, but mostly that was me saying, “Oh yeah!  Number Sense is everywhere!!” while secretly hoping nobody would ask me to prove it.  But if you asked me to prove it today, I could give you multiple examples.  That might not seem like a big deal to everyone, but it was a big deal to me.  Is a big deal to me.  Sometime I might show you the long range plans I have created for this year and last.  They aren’t a list of things I will cover in each term.  Instead, I made a….circle?  Maybe a mind map?  Not sure how to describe it.  But anyway, it’s a visual representation of how I am now seeing the subjects all work together -as if there aren’t 12 separate curriculum manuals, but just 1 called “Mrs. Corbett’s Class”.   I like it.

Ok.  Once again I am at the end and my post is getting too long but I don’t really know how to sum it up.  It’s all still jumbly.  But the parts are coming together and I can’t wait to get back to work on Monday!


The Math Pod: Week 1 Reflection

This weeks Math Pod podcast with Cathy Fosnot was about teaching math in context – making it relevant to students and teaching it as something they will need to use, rather than a bunch of skills they might find helpful in certain jobs or when they have to balance a chequebook or figure out their discount at the clothing store sale, and thank goodness for fractions if we want to double our cookie recipe!

In August of this year I was at a Summer Institute session put on by leaders in my board. Someone (I want to say it was the Director, but I might be wrong) said, “If kids want to know why and the only answer you have for this is ‘Because I said so.’, then you are not prepared enough.”  Or something like that anyway – that’s not necessarily the exact quote.  I’ve thought a lot about this since and was thinking about it again while I was listening to this podcast.  I’m noticing more and more how important it is to have a “why” attached to every single thing we teach.  In my first real classroom teaching assignment (in 2000), I taught grade 5.  I was the math teacher on a rotary team, which meant I taught math 3 times each day.  It was a great experience.  I can picture a boy in my first class so clearly saying to me, “But why do I need to learn to ‘Guess and Check’ to solve problems?  I hate that strategy!  I don’t like writing down guesses that are wrong and having to start again.  I want to figure it out in my brain and then write down the correct answer.” That is a direct quote.  He inspired me to go looking for the “why” of Guess and Check, which was a favourite problem solving strategy for the math textbook publisher whose books we used. Because I needed to give him a why (and thank goodness I realized I needed to!) I spent time figuring out this strategy, and realized it was more of an estimation strategy than a real guess.  That changed my teaching of this strategy because I realized what it meant, and how and when to use it.  Can you believe I was a teacher before I ever learned this? Well, I was.  (I’m putting the blame for that on my math teachers, who are all probably enjoying a lovely pension cheque right about now, and congratulating themselves on a fabulous career.) Later in the year, I was teaching multiplication of fractions.  Before multiplying, we would “cross reduce” the fractions so we didn’t have to reduce big numbers in the end.  The same boy wanted to know why this works.  I couldn’t tell him. It was a big school and there were 13 other math teachers in the building, including one working on her doctorate in math education, and not one of them could tell me why this works. There we all were, university educated teachers, teaching this “strategy” (or is it a trick?) to hundreds of kids year after year, and nobody could actually explain it. Weird, right?  But those two experiences, and a few others that year, and the year before when I was on a short term assignment in grade 7 and grade 8 math classes, showed me how important it is to know the math deeply before teaching it, or at least be willing to learn it along the way.

Early on I found that, especially when teaching math, I really had to consider what my students already knew, or already needed to know, before we could launch into an activity. I taught grade 5 for a while, with a really nice textbook to follow, and I taught kindergarten for one super long year.  But most of my career has been in grade 3/4 split classes.  Every year I’d find myself thinking, “Why don’t they know this?  Why do I feel like I am starting from scratch?” Then I taught a grade 2/3 split for the first time and really looked carefully at the grade 2 curriculum for the first time.  I realized my grade 3 students didn’t know certain things because I was the first person teaching it to them. I knew that was true about multiplication and division.   However, I didn’t realize how true this was when I was teaching fractions, or making graphs. The amount depth of work in these two subjects in grade 3 compared to grade 2 is huge.  This seems so obvious now, and maybe a bit embarrassing to admit, but I spent so much time looking at my own curriculum requirements that I didn’t have time to look at a different grade. That’s what learning the Landscape of Learning has done for me.  I’ve stopped thinking only about what the curriculum is asking, and looking more at what the students are doing, what skills they have and which strategies they use, and then going from there.  I know that I need to pay attention to curriculum.  I know that!  And I do that. But I don’t feel constrained by that.  Are they meeting those standards?  This is a question I have to ask as I prepare report cards.  Have I covered that material?  Of course I have to look at that. But as with reading and writing, I feel like I am moving students toward their personal next steps more and more, and we are doing this along a nice trajectory instead of trying to jump to a level that we are not ready for, and then floundering.

My biggest take-away from the last round of VoiceEd podcasts with Cathy Fosnot is that I don’t need to pre-teach skills before starting a unit. I will admit to having thought that before.  But I am not doing that at all this year, and I find it so interesting to see students really build the strategy for themselves.  I think this gives them a purpose for using the strategy.  They have seen it work, they have used it to solve a big problem so it makes sense to them.  This shift has really changed how I am approaching the Math in Context units.  I love how Cathy says that our math teaching shouldn’t be about some rich tasks, but rather a series of tasks that build one upon the other to help students progress in their understanding of math concepts.

I also love how Cathy Fosnot keeps talking about opening students up to the aesthetic of math, rather than just teaching them some useful skills. The useful skills are important, but we have to go a bit further than that.

I’m reaching the end here and want to have a really good summary paragraph that pulls all of my points together.  But this is more of a rambly kind of brain-purge. Hopefully not in an “Oh, dear.  It’s Sunday night and this woman has to be in charge of 22 little people tomorrow.  Someone do something!!” kind of way, but more of a “She’s experiencing some cognitive dissonance. Excellent!”  kind of way.

Guided Math

Guided Math: Where am I now

I think the biggest barrier to me having guided math up and running has been this notion I had that guided math = centres.  I kept picturing in my mind all the students rotating through activities: a Dreambox station, a game centre, ….and that’s where it all started to fall apart.  See, I’m more of a workshop teacher than a centre’s teacher. I want my class engaged in meaningful work not busy work.  I want everyone moving forward, not just doing stuff.  Guided math felt to me like stuff the kids would do while a few were doing meaningful work with me.

I don’t know why I thought this.  I don’t teach guided reading that way, at all!  I have never been a “Centre” person, or a “station” person.  First, I’m not so good at making photocopies, or games for kids to play, or doing all the background work to create independence at a set of activities.  Second, I want to be everywhere while the kids are doing the activities.  I don’t want to miss out on the conversations that are happening during a game.  I want to be available, for example, to scaffold during a Dreambox activity.  I don’t want those things happening while I am otherwise engaged and unavailable for assistance.

We are just past Day 7 of the “Organizing and Collecting” Young Mathematicians at Work unit. I kept thinking, “Oh, I should be doing some guided math with this little group.”  or “I should do a guided math group with that group.” Then one day, I thought, “Wait…I am doing guided math with my groups!”  The other kids were all engaged in their work, and I was sitting at a table with students I had intentionally partnered with each other.  I wanted to scaffold them through the activity.  All of them were counting by 1s, never anything else, and I wanted to push them to at least try counting by 2s, if not tens (which was the focus of the activity.) I was conferring with them, and I was noticing their strategies and which stops on the Landscape of Learning were evident, and all the other groups were engaged in the activity too.

So, to recap, I had not organized 50 activities for the class do to while I worked with those 4 students.  I had not set up a “rotate every 15 minutes” routine in the class.  I had not even designated a specific day on my schedule as “Guided Math” day.  But there I was doing guided math anyway. And then I did it the next day.  And then the next. And then today we didn’t do it because I was standing out of the way and doing some kid watching while they worked on the activity.

It feels like a weight off my shoulders.  I was feeling bad about not having figured out how guided math would look and run in my class over the Summer so I could have it up and running by the end of September. In a June meeting, a colleague said, “But does guided math have to include centres?” and that sparked my thinking.  I thought it did.  It took me 3 months to realize it actually doesn’t have to include centres at all.

So, I guess I have started guided math after all. 🙂



Math interviews

Everyone in my school board was asked to do a math interview with students as an assessment. We’ve known about it since August, and today, October 2, we had a PD day where these were discussed (among other things.)

I was sitting with other Primary teachers, and we talked for a long time about how students had placed a few numbers on a number line, and represented some numbers on a math rack. We really looked at their responses and strategies and noticed so many things about our learners based on these simple questions alone.

When I first started teaching, I taught math to grade 5 students on a rotary schedule, so I had 3 classes with about 30 students in each. I marked things “right” or “wrong”, and I looked for common errors in algorithms to plan some instruction. Mostly, I taught the next lesson in the textbook day after day. If re-taught a skill and people did a better job on the quiz the second time around, I felt pretty good about myself.

Now, however, I feel like I’ve learned so much about all the ways problems can be solved, and how to analyze the strategies and skills a child utilized and demonstrates. That’s been 15 years in the making. When I was first hired in Ontario and went to set up my new kindergarten classroom, I was stunned to discover there wasn’t a single math resource in the room. I had to figure it all out. It was hard! But I think my whole career has been shaped by that experience, as well as all the years after that when I had to make due without a textbook, and eventually chose to set aside the textbooks that came my way. 

I’ve made some instructions decisions recently in math that are based on so much more than a quiz score. It’s very exciting to feel like I know exactly how to move my people along in their math understanding. We’re working on the Fosnot “Collecting and Organizing” unit, and I’m seeing them grow day by day as they puzzle their way through the tasks. It’s so exciting!