## Subtraction

What does “-” mean in math? As in 5-2=

We had an excellent conversation about this during a Number String this week. I went off script after discovering that some of my friends didn’t know how (as in “no idea” how) to subtract 23 T-Shirts from their inventory in our math lesson.

This is what we decided:

The next day we talked more about “take away” using the math rack. Then we talked about how we can count back on a number line. Then we talked about how we can actually count up on a number line in order to find the space, or difference, between 2 numbers. One friend was very keen to keep explaining how adding and subtracting are opposites, and during one explanation solidified his own understanding of how he uses what he knows about adding to subtract.

I had my strings planned out for the week, but on Wednesday I realized we actually needed to do something different than planned. Literally everyone in grade 3 is doing well adding double digit numbers. They need more practice for sure, but the Strings were not moving us forward. At the same time a weaker skill (from way back in grade 1) popped up and I felt it was a good time to address it.

I’m thinking more and more and more about how math learning happens on a developmental continuum. Everyone travels along the continuum at a different pace. Hopefully nobody dawdles in one place for too long, and hopefully they remember what they’ve learned. I’m confident everyone has past experience with subtraction. However, for some reason, it didn’t stick. It’s for this very reason I had my grade 2’s dawdling in their own spot on the Landscape of Learning all week. They played games that had them practicing addition facts, and started creating their own flash cards to take home and practice. Next week they’ll be doing the same with subtraction facts.

## 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
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.

## 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!

## Is it a good deal?

Perhaps the only thing worse than being a teacher’s kid, is being a teacher’s spouse. My husband is both.  His mom and dad were both teachers when he was growing up – and  his dad was teaching at his high schools!  The worst possible fate, right?  His dad was also a principal, but luckily after my husband graduated, and in another town.  So, yes, it could have been worse for him.

He doesn’t remember this happening to him, but I am always experimenting on my children.  I try out new games, and practice my mini-lesson and conferring language, and I read them the books I am previewing for lessons planning.  This week, I decided to give them a break and experiment on my husband.

The next conversation with Cathy Fosnot and Stephen Hurley on VoiceEd.ca takes place tonight.  We were challenged to solve a problem in preparation for the learning we are doing.  I teach grades 2 and 3, and thought this was a problem that would not be appropriate for them, but I wanted to try it out myself.  Here is the problem:

A neighbourhood store is selling 12 cans of cat food for \$15.  Another store is selling 20 cans for \$23.  Which is the better deal?

You can watch a teacher introducing this to some students here.

My first thought was, “Division algorithm”, but I am pushing myself to think in new ways, so I did it like this:

I found halves of each side, until halves no longer made sense.  Then I found 3rds for the first store, and 5ths for the second store, all in order to find out the price per can of food at each.  I feel confident I am right. (Which I only mention because I don’t always feel confident when I don’t check my division with the standard algorithm.  I’m getting better! I teach primary students, and haven’t taught anyone older than grade 4 for a really long time, so division isn’t something I’ve had to think about as a teacher in a really long time.  I have taught basic, beginning division of course, but mostly I’m focused on adding and subtracting – the Number Sense, Addition and Subtraction Landscape skills.)

On a whim, I decided to ask my husband to do the same. He is not a teacher, so hasn’t done any learning about math since he was a kid, unless you count the stats class he barely survived when he was working on his master’s degree.  Here is what he did:

I conferred with him after, and you can see how well that went.  🙂

Here is the interesting part for me:  He has always been good at math.  He plays cribbage too quickly for me to learn from him, and has done so forever.  He calculates in his head really quickly, and can seriously look at numbers and just know (as he indicated in our conference!) He is a social worker and has to do math all the time, but all of it is calculating, and all of it the same week after week.  He calculates mileage, works with money, figures out time sheets for the people he supervises.  He’s had this job for 15+ years though, so he doesn’t ever have to problem solve with math.  Or at least he rarely has to. Mostly, he figured out how to do this 14.75 years ago, and now he just does the same thing week to week.  And he does it every day/week/month – so lots of practice.

The problem really came for him when I asked him to explain.  He was so irritated with me for pushing him to talk about his strategy, how he knew what he needed to figure out, why he needed to divide. He had trouble putting his thoughts into words, and that is not typically a problem for him at all.

This is one of my favourite things about teaching math now:  I’m not just training kids to do math tricks.  I’m teaching them to be math thinkers, and math communicators, and math users.

(Update:  Click here and you can listen to Cathy Fosnot and Stephen Hurley talk about the math in this problem, including my solution.)

## 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.

• 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

## #notabookstudy

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.

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?

math

## Reporting and Assessment

Back in April, I wrote about using the Landscape of Learning to assess some students, hoping to do two things. 1) assess where my students were on the Landcape and identify some next steps, and 2) assess were I was in my ability to use the Landscape for assessment of and for learning. I feel that I accomplished both.  (You can read about it here.)  I have continued to work on this, and will probably write some more in a few days about how these students have progressed, as well as how I will use this next year.

In the past few weeks, however, I discovered just how much all of my Landscape learning has effected my report card writing.  I think I have always done a pretty good job of reporting.  I personalize the comments with anecdotal quotes and events, I use up all or most of the 13 lines of allotted space, and I make them easy for parents to understand.  But this time around, I started write things like, “XXX counts forward and backward…” and I realized I could say so much more about the counting than just direction.  I found myself feeling that I had to comment on all the nuances of the child’s counting skills.  Is he counting forward by 1’s, 2’s, 5’s, 10’s, etc. and backward by the same?  Is she using this counting to add and subtract double digit numbers on a number line?  Does the counting happen mentally, or only with a math tool such as counting chips or a Rekenrek?

Adding and subtraction skills were no easier.  Instead of “XXX adds and subtracts double digit numbers”, I had to write, “XXX adds and subtracts double digit numbers using several strategies, such as splitting, counting on, or finding the doubles and near doubles.  He demonstrates an understanding of the commutative property when he starts at the largest addend and counts on.  He draws his own number line to communicate his strategy.  During Number Talks, he clearly explains why he chose to use a certain strategy to find the answer.”

And that’s just for Number Sense, with a little bit of Patterning and Algebra thrown in.  If you were counting, you’ve probably realized that I have used up nearly half of the 13 lines without even mentioning measurement, geometry or data management.

It was hard, you guys! I’ve always felt like I had plenty of room to report on math.  Now I feel the opposite.  In a meeting with colleagues on Friday (you know who you are, Blue Sandals!) I found out that several of my colleagues have felt the same way while writing the second term report.

I also don’t have enough space to report on literacy skills, but that’s another blog post for another day.  If anyone at the Ministry wants to know how I think this should be solved, I’m going to tell them that we should take away all that space in the phys ed section, make a few things like phys ed, health and The Arts pass/not pass subjects where I can write, “Hey!  Your kid tried his heart out and good for him for doing it!”  Because, at the end of the day, I think knowing your child needs assistance counting money because he isn’t able to unitize is way more important than declaring him an A student in throwing and catching.

And I know some people are thinking, “But throwing and catching are so important!!”  And they are.  But do we really need an equal amount of space to report on their PE skills as we get to report on their math and language skills?

math

## #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!