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:

Guided Math, math, Number Sense & Numeration

Math Interviews as Assessment

On Friday I was working on finishing math interviews with my students.  I am ¼ of the way through the class already! (growth mindset)

Me:  What is 4+3?

Grade 2 student, without hesitation: 7

Me:  How’d you do that so quickly?  

Student:  Because last year my teacher had us work on things like that on this app on the iPad. We had to do those kind of problems every day.  It was so annoying (insert eye roll) day after day! But now I’m really quick at it.

Me:  That’s great! What is 8 + 14?

Student (blank stare):  Um, yeah. We aren’t on that level yet.  I think that’s like level 17 and I was only level 16, so I don’t have that memorized yet.

Me:  Well, do you have any way of figuring it out?   (I’m trying not to look at the 20 “special stones” that were just counted out, or the Rekenrek sitting to the right, or the whiteboard and marker to the left.)

Student: Ummmmmm……no.

Me:  What if you were using some of these tools? Or your fingers?

Student:  Ummmmm…….no.

Apparently they valued memorizing the basics at his previous school.  You can draw your own conclusions about how I feel about that. But this isn’t about judging another’s teaching based on the word of one 7 year old.  The real question for me is this:  What am I going to do about it for this child?

The start of the year is always a tricky time for me.  No matter what “last year’s teacher” has shared with me about a child, I feel I never really know them until I complete a reading Running Record, a writing conference, and a math interview.  This conversation above is exactly why I think a math interview is important. Thanks to this and a few other questions, I now know that this child is subitizing numbers less than 6, can skip count, count on and count back when counting objects but not when presented with just the numbers, can draw a number line that shows where 7, 10, 20, 30 50 and 48 would be, but that these will not be drawn in iterated units, and can count from 30 to 100 but has a long pause when moving between decades (48, 49……….50!)  I also know that though this child has some basic facts memorized, there isn’t a whole bunch of understanding behind that fact, so little understanding in fact that the child had no idea what to do when the answer wasn’t already known.

Years ago, I had a math textbook to use that provided a “pretest” and a ”post test”.  They were of very little use. Sure they gave me a score I could use for a mark, but that was often all they gave me.  The interview, however, tells me so much more about where to start, where to go and how fast to try and get there. A follow up interview after a few months tells me if I am moving at the right pace, if someone needs to be pushed faster or slower, and if I need to circle back to a place I thought we’d already covered sufficiently.

The major questions that always comes up is this: What do the other 20 children do while I spend 10 or so minutes with a child one on one.  During a Running Record, they all read and look at books. During a writing conference, the others are all writing. But we can’t just have a bunch of 7 and 8 year olds “do math” independently in September.  Here is what I have them do:

  1. Dreambox.  If you do not have access to this, I am VERY sorry for your luck.  It’s great. It’s a bit expensive, but if your luck is good your board has purchased a license and your class can use it.  If not, try Prodigy.
  2. Dice Games
    1. Race to 100:  Here is one version I’m excited to try.  We actually play for counting chips.  Every child has one die.  Simultaneously they are rolling that die, then taking the same number of counting chips.  When one person gets to 100 (we were playing to 50 this week) the game ends. 
    2. Tenzies/Yahtzee
  3. Card Games
    1. War 
    2. Addition war
  4. Math Manipulatives
    1. They love to use pattern blocks and there is a lot of spatial reasoning work than can happen even if the class seems to be “playing” with these blocks.
    2. They like to “play” with 3D shape blocks for the same reason and it gives them the same learning experience.
  5. Finally, if you haven’t read “What to Look For” by Alex Lawson, you should.  At the end of this fabulous book, there are a collection of games that can be played to help move students along in their mathematical understanding.  You can look at specific skills your students need, then choose a game that helps them work  on that specific skill.

I will admit that it gets a bit loud while we work on this. This might feel like a waste to some teachers.  I think it’s a great chance for us all to practice what we do while the teacher is busy with just one person. I also feel that the information I gather from each child saves me so much time down the road that it more than makes up for these first few days of playing at math. I am not guessing about a starting point – sometimes missing the mark by a mile and starting too far ahead or too far behind my students.  I can confidently set up my guided math instruction in a way that is truly differentiated for the class. Finally, who says math can’t or shouldn’t be fun?  

 

*I’ve had a few requests for a copy of my assessment.  I hesitate to share it, but I’m not sure why.  I can’t think of a real reason not to, so I guess I will.  It’s going to be most useful to you if you are familiar with the Landscape of Learning, created by Cathy Fosnot.  I don’t ask every child every question.  If they are having a lot of trouble with the first 1 or 2 addition or subtraction problems, I don’t ask the others.  If they are having trouble with the number line, I don’t ask all those questions.  But I don’t stop asking after the first mistake, because sometimes the child will go back and revise their number line, and that’s useful information for me too!

Here it is.

 

 

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!

img_9218

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.

Guided Math, math

Sometimes I get into deep Twitter conversations that leave me thinking for days.

A few days ago, I got involved in one of those Twitter conversations that reminds me why I love having a professional learning network of strangers.  I love being pushed to think about my practice and how I might change it, or why I might keep it the same.

I have settled, I think, into a Math Workshop teaching structure rather than a Guided Math structure this year.  And by settled, I mean we do what suits us when it suits us.  When we are deep into a Context for Learning unit (a.k.a. The Fosnot Units) the workshop hums along.  When we are working on some of the other problems and activities, I feel like it still moves along, but not quite as smoothly. I’m cutting myself some slack for that.  But I am definitely not running a centre-based program with a set schedule that we follow (today is game day, tomorrow is iPad day, and blah blah -whatever else people do in centres.)

One day last week Mark Chubb posted about targeted instruction, and one method for organizing groups so those with similar ability levels, based on their answer to 1 question.  In summary, he wasn’t sure if students should be organized into ability groups.   It got me thinking about how I organize my students so I thought I’d write about that this week.  (You can read the post here.  It’s about a lot more things that what I have written today!)

I agree with what he says about students who are struggling being paired into a “low group”, and that this will only really serve to teach them basic skills and not push them into higher level thinking, thus widening the gap between them and peers who “get it” quicker. In his exact example, it would keep kids in a vocabulary learning group while their peers are moving forward to problem solving. I wholeheartedly agree that a child can learn lots of things about geometry while still, occasionally or always, calling a hexagon a “pentalellogram”.  

There are lots of ways groups and partnerships are organized in math class – or really any class.

  • Random partner assignments:  pulling sticks with names on them, or slips of paper, in order to create partnerships.
  • On the fly, sort of organized partner assignments:  the teacher calls out pairs, figuring out who to put where in the moment and trying his/her best not to leave two for the end who shouldn’t work together.
  • Planned in advance partnerships:  the teacher makes intentional learning partnerships in advance of the activity.

I have been using a method I love, and will continue to use, for a while now.  I plan out the learning partnerships I want in my class, and then I have those students work together, throughout the day, for an entire month.  Here is why I like this way of doing it:

  • I figure out who will work well together because of their learning skills, not just their subject specific skills.  I think this helps children see each other in a variety of contexts, and helps them polish their collaboration skills.  I like to pair two really quiet people together, for example, so they are “forced” to start talking and collaborating.  I like to pair two children who naturally fall into leadership roles in a partnership together so they learn to negotiate in their communication – hopefully learn to listen to each other and not just talk.  I like to pair three kids when one is perpetually absent – with a group of three it’s OK if one is away.  And on, and on, and on.
  • Working together for a long stretch of time means they get a lot of the social stuff out of the way early on, and spend the rest of the month more focused on learning. You know those kids who will spend a whole work period moving around the room trying to find the perfect spot?  Or the whole work period arguing over who is going to be the scribe (this mostly, but not only, happens when markers are involved!)  Well, when kids are working on the 2nd or 3rd task together, they don’t even talk about where to sit, or who will write.  They just go to the spot they picked the time before, check their memory to see whose turn it is to scribe, and get to work.  
  • I can make sure that students don’t have the same partner two months in a row.  Inevitably they will work with each other again and again.  This is especially true since I find myself using triads quite often and there are only 20 of us…sounds like a math problem!  10 months, 20 kids how many different combinations can I make??

So I think this all fits into my ongoing thinking about Guided Math vs. Math Workshop.  I think I am grouping based a bit on ability (I don’t want a strong grade 3 with a grade 2 who is still trying to make sense of it all…for both their sakes!)  but mostly I am thinking about how a group of 2 or 3 kids can push each other across the day – in math, but also in their communication skills (written and oral).  I get to shuttled around the room offering guiding questions and moving kids along.  I also know that if I don’t get to someone on Monday, there’s always another day or another time during the day when we can connect and talk about what is happening with the group.  I like that kids work with everyone in the class eventually.  And I like that I can spent about 20 minutes at the beginning of a month organizing groups, and then not worry about it in the end.

I do ask students for their input, and ask them to do some self reflecting.  Who do they want to work with and why? Do they think they are being a good partner?  What could they do better?  That sort of thing.  They like having the input, and they do get quite good at thinking reflectively about themselves as learners.

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. 

 

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. 🙂

 

Guided Math, math

Estimation

Like I already told you I was going to (here), I started talking about estimation in my class last week.  We read a book called Great Estimations by Bruce Goldstone, tried out a few of the problems he posed, and then tried some of our own.

I had prepared some bags of stuff for us to estimate in advance. For each item, there were two bags: one had 10 of the thing in it, and the other had an unspecified number of the thing in it. Here you can see I used Mike ‘N Ike candy and mini marshmallows. I also had popcorn kernels, elbow macaroni, and Cheerios.

I gave a set to each table, and asked them to estimate.  They also had a piece of paper they could record their thinking on.  At the end, we shared our estimates.  We did not get an actual count of the items.  This was on purpose.  I wanted them to feel like their estimate was good enough.  Mostly their estimates were in close proximity of each other, and I complimented them on that.

The following day, I have them 2 bags and an item.  They were to put 10 in one, and count as many as they wanted for the next. I gave them Lego, glass beads, counting chips and colour tiles.  Most groups put around 30 in the bag. Just we had the day before, they traded bags with each other until they had an estimate for everything.  Then, we shared our thinking, and confirmed the count for everyone.

One group thought it would be funny to put over 100 counting chips in their bag.  They were each counting out 100, rather than working as a team, so we had a good conversation about that. When another group got that bag, they were sure it was impossible to estimate. Imagine their surprise when their estimate was within 5 of the actual number!  That group was composed of grade 3 children who are still adjusting to the idea that they are the older students in the class now, so I think this was a good confidence booster for them.

This is what I discovered:  they are mostly pretty good with recording their thinking so they can share it later.  We do need to do some work on labelling.  We also need to work on each person contributing to a group assignment or task.  In each group there was a clear leader who railroaded, or attempted to railroad, the rest of the group.

This is what they discovered: they really were training their eyes (as it says in the book) and their estimates were closer to the actual counts as we went on. In some groups, they discovered the need to label.  They had written a number, say 34, but didn’t label it as “colour tiles = 34”.  When it was time to share it was tricky to share! I like that they discovered this on their own.  I know I will need to talk about this again, but I’d say about half of them were able to identify this as an important thing to do going forward.

Using the website www.estimation180.com as inspiration, I am going to create some more provocations for my students to explore.  I want to add this as an activity for them to complete during Guided Math.  (Yes, I am still trying to figure that out!)

My hope is that these estimations activities, revisited throughout the year, will help my students develop a stronger sense of numbers.  I think they will develop a better understanding of magnitude, and that the numerical reasoning skills will improve.

Finally, here is an article from Math Solutions that has had me thinking about all of these things.