Guided Math, Number Sense & Numeration, Number Strings

Addition of double digit numbers

There were about 50 of these on our whiteboard at different times over the last few weeks. We’ve gotten pretty good at adding the tens using mental math strategies. 20+20 -> 2+2=4, so 20+20=40…no problem! But it’s was time to move on!

I really wanted everyone to learn how to effectively use a number line. We’ve been working our way through a Context for Learning kit called “Measuring for the Art Show”. I demonstrated it about 1 billion times. First we used cubes to make a line, then we annotated this on some cash register tape, and then we moved to the whiteboard. Finally, I gave everyone some problems (from the kit) and some paper they could use for drawing the number lines.

As I walked around I could see lots of kids with lots of right answers but no number lines. “How are they doing this??” I wondered. So I asked. And I was amazed! So many of them were using the mental math strategy of splitting. They thought about how many ones there were in each number, and how many tens were in each number, then they found a total.

But the number line isn’t the go-to strategy yet. So I’m annotating the problems two ways now. The way lots of them are doing it, and the way some of them are doing it. Both are ways I hope all of them can do these problems eventually.

Our next step…my next step…is to organize into small groups (I know…back to some guided math. It keeps coming up!) I need to help the kids who are splitting learn the number line, and help the number line kids do some splitting and help the “I have no idea what to do kids” get some ideas.

I’m ALWAYS happy to reach Winter Break, but it always comes about a week before I’m ready. I don’t want to interrupt our math learning. But I’m confident the stuff we’re doing now will stick. I won’t have to start from scratch on January 7.

Here is more work from today:

math, Math Workshop, Number Strings

They heard me. They really did!

Last week, I was ending the week feeling like I may have spent a few days talking to the walls. (You can read about it here.)   This weekend, I feel much better.

We spent the week working on building an understanding of number lines. After making a measuring strips, in groups of 5’s and 10’s, and measuring some things, we needed to start thinking about how a person could skip around on that number line and use it for adding.  When I taped a 100 strip to the board and started asking kids to tell me the number of a certain cube on that number line, it was like a miracle had occurred.  Because nobody could reach the number line to touch each square, and because we’d talked a lot in our math congresses about how we could use the 5 and 10 structure of the paper number line to skip count, they started actually using the number line tool and the skip counting strategy to find the answers I was seeking.  THEY ACTUALLY DID!

Oh, and no big deal, but they were finally counting on from a known number instead of starting back at zero every time.  Seriously.  I’m not even exaggerating to make myself look/feel better.

Here’s the lesson for me:

  1. Trust Cathy Fosnot.
  2. Sometimes moving forward helps some kids who appeared to not be ready to move on.  I thought I would do a quick number string, sort out who needed some more help with skip counting and counting on, and then make up some Math Workshop groups.  But, low and behold, some of the kids who haven’t been counting on started counting on!  And many who had been fully committed to counting by ones were using the 5s and 10s.

So there you have it:  Valentine’s Day, Winter Electives, and a field trip, all in the same week, and we still moved around on the Landscape of Learning!

 

Guided Math, math, Measurement, Number Sense & Numeration, Number Strings

Use the 5’s and 10’s, PLEASE! I’m Begging You!

This week I started a new Context for Learning unit with my grade 2/3 class.  Prior to this unit, we have completed the “Collecting and Organizing” unit, which encourages the use of the 5 and 10 structure to organize and then count large groups of items.  We counted books in our classroom because that was a meaningful thing for my class.  The parent council had recently offered up money to buy more books, so I tied that all together. After that, we completed the “Double Decker Bus” unit, again using 5’s and 10’s and thinking about adding and subtracting.  Simultaneously, my grade 3’s – who were already doing well with the models and strategies taught in the bus unit – were working on “The T-Shirt Factory” unit.

Measuring for the Art show comes next on the recommended order list.  I should be starting “Grocery Stores, Stamps, and Measuring Strips” with the grade 3’s.  However, I really want to solidify this number line business, so I am not going to go forward with that unit for another week…maybe two. I am going to extend the numbers well past 100 in this unit so the grade 3’s are still challenged. Picking the numbers is my job this weekend.

So…here we are, measuring for a fictitious art show, and also thinking that we will run this year’s school art show.

I gave groups of children baskets of cubes in 2 colours and set them the task of using the blocks to measure the papers.

As you can see, there was some great measuring going on!  We even agreed on the measurements!

Despite all the work we have done with counting things in groups of 5’s and 10’s, some of my little friends really can’t stop counting by ones.  I asked myself, “WWCFD?” (What Would Cathy Fosnot Do?) I finally had a serious talk with them about it.  “WHY?????”  I screamed. But out-loud I said, “I know you guys can count by 5’s and 10’s, but you keep counting by 1’s even when we have a lot of things to count.  What’s up with that?”  They gave me the blank stare.  “Here’s what I think,”  I continued.  “I think you know how to count by 2’s, 5’s and 10’s, but you’re not sure you are getting the right answer so you always count by 1’s because you are sure that will give you the right answer. Am I right?”  There was a lot of vigorous nodding.  “What I want you to do is keep counting by 1’s.  But do it after you count by 5’s or 10’s. Do it to double check your work.  But challenge yourself to grow your brain and do it the harder way.  I know this is going to help you feel more confident!” So now we are doing that, except a lot of them quickly realized they were getting the right answers the first time, and it was a lot more efficient to skip count.

After 2 days of this, including a congress when we had the above conversation, I asked them to help me make a number line, organizing the cubes into groups of 5.  Believe it or not, there was magic!  As soon as I had a long string of cubes up on the board, out of everyone’s reach, 15 out of 18 immediately saw the value of using the 5s and 10s.  We worked on related Number Strings for 2 days, and then I asked them to make a number line like I had been making using their own cubes and a piece of adding machine tape.

The group pictured on the left kept counting  by 5s, but when they got to the mis-matched groups of 5, they realized that maybe I am a genius after-all and they should have listened when I said, “Make all 5 the same colour!”

So everyone make beautiful number lines, with mostly iterated units.  We put the cubes away and I didn’t get them back out. When I asked them, the following day, to figure out where numbers like 13, 23, and 33, should go, they did a great job of reasoning their way through the problem.  I can look at these and see some immediate needs I need to address on Monday or Tuesday.  But I feel like we are on our way!

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In a VoicEd.ca radio broadcast (You can listen here!) , Cathy Fosnot said she hoped that teachers who were listening would stay curious and keep wondering about the things their students are doing.   For me this is some of her most valuable advice.  Being curious about why my students are doing something, especially if it is something that makes no sense to me, has paid off so many times.

So…there you go, Cathy Fosnot.  You were right again.

math, Number Strings

WWCFD: 3

I already blogged earlier today, (here) but I want to talk about another amazing math moment, so here I am again.

A few weeks ago when I went to a Cathy Fosnot learning session, someone new to her Number Strings called her work magic.  Specifically, seeing how one problem helped solve the next problem and the next seemed like magic.  He said something like,  “Now that I see what your doing with your magic, I can figure out this problem.”  She countered by explaining that it isn’t magic, it’s math!

Problems don’t exist in isolation.  The connections we find in math  help us solve problems. We can use familiar and known problems to solve unknown problems.

Because I am just like Cathy Fosnot, I had a similar moment this week.

We (me and the grade 3’s) started with 3+6 = 9.  You can see where we went from there!

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See that note on the side?  After 56+43, One of the students said, “That’s (pointing to 3+6=9) spoiling this answer. It has 6+3 in it!”  I got to say, in my best Cathy Fosnot impression, “I’m not spoiling it!  That’s math!  Math is all connected and knowing how to do one problem helps you with so many other problems!”  And then we talked about a bunch of other answers, and made some connections and found out that math is actually kind of magical.

A few years ago, long story short, I figured out that my students were not making connections in math.  They were thinking about each unit, each skill, each day in isolation.  I started to explicitly talk about connections between big ideas, strategies, models, numbers, etc.  I feel like it really pays off and helps to build understanding.

Guided Math, math, Number Strings, Number Talks

Guided Math: part X (I’ve lost track)

This past week, predictably, was crazy.  Halloween in the middle of the week?  Seriously.  Why even bother having school that day?  I know people think it’s important for kids to have good memories from their childhood associated with fun things, like a costume parade on Halloween at school.  But I think we can all agree it’s gone too far.

It’s also been a weird time for our math class.  As you may recall, 4 of my grade 3 students were my students last year.  Two of my grade three students were in a 1/2 split, and the rest of my class are grade 2 students who are all new to me.

My grade 3 students are solidly moving along as a group.  They make a beautiful cohort – teaching important things to their younger, less experienced classmates. Up until now, I was satisfied with how they were helping to scaffold the class through Number Talks and Number Strings.  I was happy with what they were teaching the grade 2’s about communicating their mathematical thinking.  About mid-week last week, the tide shifted.  I started to feel that the grade 3’s were dominating the conversation too often.  They were figuring everything out way before the grade 2’s. If I plotted the 2’s and 3’s on a Landscape of Learning, they were in two very different spots.  So different in fact, that I felt I had to do something about it.  That something, I decided, would be to split the class into two entirely different Context for Learning units.

Now, I have taught split grade classes for most of my career. I have, many times, had the kids in one grade working on something different from the kids in the other grade.  In math, this usually looks like one grade continuing on in a Context unit that we started together, while the kids who are not ready to go on work on something else to help them solidify the part they are A) ready for, and B) required to learn thanks to the curriculum. You’d see this, for example, when it comes to multiplication and division.  3’s and 4’s have a similar starting point, typically, based on their needs.  But 4’s need exposure and practice with dividing that 3’s don’t.  To be clear, if I have some 3’s who are developmentally ready to move forward, and keen to move forward, I would take them along on the trip.  But if they need to hang out at “multiplication up to 7×7” for a while, I let them.  This would probably include them repeating some games we had played, or something like that.

But this time, I was feeling strongly that I needed to be pushing both groups, not just letting one group sit in one place for a while.  Here are two of the Number Sense and Numeration Big Ideas for Grade 2:

  • demonstrate an understanding of magnitude by counting forward to 200 and backwards from 50, using multiples of various numbers as starting points;
  • solve problems involving the addition and subtraction of one- and two-digit whole num- bers, using a variety of strategies, and investigate multiplication and division.

And for Grade 3:

  • demonstrate an understanding of magnitude by counting forward and backwards by various numbers and from various starting points;
  • solve problems involving the addition and subtraction of single- and multi-digit whole numbers, using a variety of strategies, and demonstrate an understanding of multiplication and division.

My grade 3s have mastered the grade 2 concepts.  (As they should have last year by June.)  My grade 2’s, however, are still needing a bunch of work on this. (As they should until this year in June.)

So what was I going to do??

Well, I started two Context units at the same time. Seriously.  In the same week when Halloween was on a Tuesday.

As part of my evolving thinking about Guided Math, I thought I could have one group working independently each day, while another group was mainly spending time with me and getting my attention.  So far, it’s been chaotic.  But I feel like we are accomplishing what I want to accomplish.  I’m really wishing I could do more observing and conferring, so I will be addressing that in my planning this week.  I don’t want to be occupied with whole-group teaching and missing out on the conversations kids are having while they do their work. I did throw in the towel and have a “Fun Friday” during math because I felt that instead of moving on I needed to regroup.

Grade two’s are still working on unitizing single digit numbers and using the 5 structure to help them add.  They are working on the “Double Decker Bus” unit.  We ran into some problems because I am a paper-saver and had given them one day’s work on one side of a page, and the next day’s work on the other side.  I know: rookie mistake.  Even though I clearly told them and showed them were to start, 3 out 5 groups tried to do the wrong side.  Learning experience for all of us!

Grade three’s are working on the same things, but using the “T-Shirt Factory” unit to move into hundreds, using the 10-structure, and building a deeper understanding of place value into the hundreds.  They need more help with the use of a T-Chart to organize information.

I congressed with both groups on “Not-So-Fun Thursday!” as I am now calling it.  One group was working  on something while I congressed with the other.  I think I’ll keep this.

Moving forward, I am going to continue in both of these units.  After Fun Friday, I discovered that there are still some counting issues for grade 2s. I think they can have a “Count-Everything-in-Sight Monday” or maybe a “”Put-All-These-Numbers-in-Order Monday” while I get the 3’s started on the next part of their unit.  Then on Tuesday Grade 3’s will be able to work independently while I get the two’s started on their unit, and then I can wander and confer.

And it is going to take me the rest of the day to figure out how I can work Number Strings and Talks into it all.  Cause we have different needs there, as you probably guessed.

Thankfully I have lots of Halloween candy to get me through!

 

It’s such a big week in math that I’m blogging twice!! See the other post, about a Number String, here. 

 

math, Number Strings, Number Talks

WWCFD

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:

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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.

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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.