math, Number Sense & Numeration

Money, Money, Money

According to the TIPS4Math scope and sequence maps found at edugains, it was only going to take me 5 days to teach money.  I had 5 days before March Break, so it seemed a perfect match.  I mean, I thought it would take more than 5 days, but I had 5 days and wanted to believe that if I had enough faith I could finish all of the math stuff in those 5 days.

Then on Monday I found out we didn’t have to be at school on Friday.  (I pay really close attention to calendars, obviously.)  But that didn’t matter, because I had technically started doing some things with money the week before, so I was going to be OK.

Let me stop right here and say that if all I wanted to do was check the curriculum expectations off my list, then I would have been “finished” with all of the money stuff after 5 days.  I seriously would have.  But, I don’t know.  Money just seems like one of those things kids should not just do.  They need to understand money, even when they are 8 or 9, right?

Here’s what we did:

  • We dumped money out and identified coins/bills.
  • We made combinations of coins that were equal to a dollar, or $20 for grade 3 students.
  • We counted piles of coins & bills to find their total.
  • See…I’m practically done a day early!

If you haven’t ever seen The Pancake Menu, take a look here. Such a fun book and activity! We read that and talked about it, but I didn’t want to make pancakes again, so I told my grade 3’s to come up with their own cafe idea.  Then we ran out of time, so on Wednesday night after school I bought a bunch of stuff for ice cream sundae’s, plus root beer and dragged it all to school on Thursday.  I gave the grade 3s a blank menu and they figured out prices.  I told them each grade 2 would have $1, so they had to keep the prices low.  They did pretty well with this, double checking the addition with different combinations of toppings for the ice cream.

They gave each child a plastic Loonie, which I had to explain was a problem if they didn’t want to make all sorts of change.  So they counted out a dollar in different coin combinations for each of their grade 2 classmates.  Except some of the kids got more than $1 and some got less than $1.  Then the grade 2s came to the table to give their orders and everyone gave up on counting money because there was ICE CREAM within their line of sight and the grade 3s were in a panic that we’d run out before it got to them.  They were just taking all the money and the grade 2s were trying not to climb over tables to get their sundaes and thus did not care about receiving change. I was busy scooping out ice cream and training kids to use the whipped cream in a can (#VitalLifeSkill).  Then when everyone asked for seconds, I said, “Depends on how much money you have left.” and that’s when they realized they’d been ripped off by the ice cream sellers.  The entire class stared silently at me and my half-full bucket of ice cream, unsure if I was actually going to keep the rest for myself.  (I’m not going to lie – it crossed my mind. I had homemade hot fudge!)

Morals of the story:

  • You can cover all the money expectations in 5 days.
  • You will not have actually taught very many kids to use money.
  • You can’t assume children will have all had root beer floats. (#LifeChangingEvent)
  • You can assume someone will complain about not getting a third sundae, even though they have all received a second sundae for free.

When we get back from March Break, I am going to have everyone write out their orders for the grade 3 students, and they will figure out the actual totals, and then I will have them count out the right amount of money they would need to pay a cashier.  I will not be giving them more ice cream until at least June.

One day, not that long ago, my daughter, who is in grade 1, was convinced she had $21.  Can you see why?

She wanted to buy something that cost $21 and I told her she’d have to use her own allowance because it was something junky.   She names every coin, but she’s not unitizing the money yet, and doesn’t really get that each coin has a different value.  She can count by 5s and 10s, so she can count dimes and nickels if I am next to her walking her through it.  She’ll get there though!

math, Number Talks


Found this on Twitter the other day, and decided to give it a try.  We are “finished” with some learning about adding and subtracting and unitizing and counting.  We need to move on – both for my sanity and because there are other things that need to be covered in the curriculum.  We will circle back to this in the new year, and we’ll keep practicing when we do a daily Number Talk at the beginning of our day.

I’d decided to move on the geometry, mostly because I’m a sucker for all the Christmas tie-ins.  I love making 3D shapes into ornaments and decorations! But then this fun fraction activity popped into my life, and I thought it would make a nice transition from Number Sense into Geometry.

On day 1, a shortened math period due to irrelevant circumstances, everybody cut out their pieces.  (As an aside, I know that cutting isn’t math, but we have some fine motor issues, and it’s really good to for us to cut as often as possible.  Also, I have a life to live and it’s takes 20 kids 15 minutes to cut these out so I let them!)  I started by holding up the “whole” and one of the fourths, and then asked the recommended question:  What do you see?

They named a bunch of stuff, unrelated to math:  I see white and yellow, I see a fried egg.  Then they moved on:  I see a small square and a large square; they both have 4 sides, and 4 corners; they are the same but different.  Then I let them go play with their shapes.  We came back and talked about what we saw:  the different shapes in different sizes, teh same shape in different colours, etc.  Then we had lunch.

On the second day, I asked them if they could use the paper to show half.  I was interested in their understanding of half, and also, would they see that you could make half of all the shapes, not just the whole.  While I walked around and talked to different groups, I was so glad that I know that I should ask them to explain their thinking.  Some of them had very unexpected answers that showed partial or complete understanding in a different way than I was originally thinking.

All of these came with justifications for how it shows half. One is half yellow (front) and half white (back), one has purple lines dividing it into sections – half the sections are yellow and half are white, and one uses the purple “lines” to divide a design in half.

We talked about symmetry, which I hadn’t intended to do, but it naturally fit.  I really thought I could do this activity and move on to fourths and thirds.  It was clear we weren’t ready.  Luckily I am SO organized (haha) I was able to grab this book off my shelf:

Give Me half by Stuart J. Murphy

It’s about a brother and sister who do not want to share fairly, but an unspecified adult (mom?  dad?  babysitter??) tells them to share.  They each get half of the pizza, the dessert, the juice.  Then I sent everyone back to show half again.

We had better results, and each came with a really good justification.  Then, just to make sure we all understood that each shape could be divided in half, I sent everyone to fold each piece in  half.  Which they did.


My favourite was this one:


He explained that both show half because the one folded twice shows half of half. 🙂 That’s going to get us started on fourths!

On Monday we are going to find other things in the room and in the school that show half, and then we’ll move on to fourths.



New School Year Eve

Tomorrow is the first day of school for teachers in my board, and really most of the province.  The students don’t start for 6 more sleeps, but tomorrow morning school libraries across this vast city will be full of educators ready to discuss how we are going to make this year the best one yet.

Or, maybe, full of people hoping we’ll have a few minutes at the end of the day to make sure the classroom is ready.

From home, on rainy days, I have been making plans.  I think I know what I am doing on the first 2 days.  I’ve spent a lot of time thinking about how I am going to start math this year.  Every August I feel like I have never taught before.  I can’t remember anything I did on the first day in previous years. I have vague memories of starting off with art projects and activities related to identifying 2D and 3D shapes, or of asking students to play math games.

The awesome math facilitators in my board have created a “First X Days” of school document for students in junior and intermediate grades, and are working on a primary version.  I attended workshop in June and so have decided to arrange my first 2 weeks recommended in this document.

I am really excited about 2 of the activities! I mean, I’m excited about the whole thing, but REALLY excited about 2 specific things.

First, Amy Krause Rosenthal and Tom Lichtenheld wrote a really great book called “Duck! Rabbit”.  Throughout the book, there is one drawing and unseen people are arguing about whether it is a duck or a rabbit.  (I’m assuming they are people.  It doesn’t really say.)

From Duck! Rabbit!  by Amy Krause Rosenthal and Tom Lichtenheld.  

Each gives his/her/its well supported opinion. I think it really shows an important idea we need children to consider in mathematics:  There isn’t always one right answer. Two people can have different opinions and still be right.  The important thing is being able to justify one’s opinion.  (PS:  I know that’s not just in math.  I’m trying to stay focused here though.)

After, we are going to read this awesome book:


I’m waiting for some permission to photocopy a page for students to write on.  I am going to ask the students which one doesn’t belong, and then let pairs of students talk for 10 minutes about their opinion. Then they will present to the larger group.  My goal with this day’s activity is to have the students really think about justifying their thinking.  I want to introduce that word to them early on.  I’ve read Cathy Fosnot’s book “Conferring with Young Mathematicians” and she talks a lot about asking good questions that get the students to really reason their way through and around an answer so they understand their thinking deeply, rather than just accidentally stumbling upon the right answers and moving on. That should get us through one day of the first week.

The second thing I am excited about is also related to a read-aloud (because that’s how I do things!)  This one is called “Great Estimations” by Bruce Goldstein.  Inside, there are pictures  of every day items.  A small amount is sectioned off and labeled, a larger amount is also sectioned off and labeled, and then there is a large amount of the same item and readers are asked to estimate how many there are.  Here is an example:

From Great Estimations by Bruce Goldstone. I highly recommend this book! I think it is appropriate for use in math lessons for students in grades 1-12. And I think kids will love the pictures and challenges enough to read it on their own time, no matter how old they are.

After we talk about the pictures in the book, which will be too far away from the students to count by ones using their fingers to touch each one therefore “forcing” them to estimate, I am going to send them off to their tables.  On each table I will have 2 jars of everyday items.  One will be full, the other will have 10 of that item.  I am going to ask them to work in pairs and estimate how many are in each jar, record their answers, and then move on to at least 3 more sets of jars.  I am thinking this will take 2 days.  I’d like them to have to work out a strategy, and then use it again after a good night’s sleep to make sure they are really getting it.  I’m also hoping this will reinforce the idea that we must record our thinking clearly!  The estimation jars are not my own original idea.  I know I have read about them in an Effective Guide to Mathematics Instruction, but I can’t remember which one and I am not going down the long and windy road of to try and find it for you.  You’re on your own! Or you could just take my word for it. I’ve used them many different times, sometimes even having a new set of jars every month and revisiting this idea of estimation often.  I like it, the kids like it, and they develop a pretty good sense of magnitude, comparing, and counting if you ask me. Oh, and estimating.  They get pretty good at estimating.

In conclusion, I think this really will be my best start yet.  I am loving this idea of focusing on setting up the class so that our first 9 conversations are all about problem solving as a good thing, mistakes as an important part of learning, and about being able to participate in a productive argument about your math.  I’m glad I had someone else guide me in that direction.



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


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

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

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

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

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

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


#notabookstudy: Constructing Fractions

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

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

Guided Math: #2

I started out last week with a plan to do Guided Math for one week.  I KNEW I would hate it, but that I might be on my way to hating it less.  So, really I should be proud because MISSION ACCOMPLISHED!

We made it through the week, with just one interruption. Junior track and field events on Friday left us with some extra students for the day.  They were grade 3 students from a 3/4 split. (Primary track and field is another day.)  Because of this, our routine was quite out of whack.  Friday was supposed to be the day that my class and I sat down to go over, in a congress-y kind of way, the math that we did this week.  I will have to do this on Monday.  I think that my class actually liked the Guided Math activities, but I am anxious to hear their thoughts and ideas.


  • I had 10-15 minute rotations going, with one grade staying with me for about 40 minutes while the other grade rotated through 3 activities. I think the timing was right.  Keep in mind that I have a grade 2/3 split, so though our stamina would probably allow for them to stay with the activities longer, our wiggly butts were happy to move along after 10 minutes.
  • If I were to start this in September, I think a math folder would be necessary.  I had them putting away work in their cubbies at the end of the rotation, but a folder would have kept it all more organized and easier to find the next day.
  • If I spent September training the class, as I did during our literacy block, they would know that they need to work quietly and not interrupt the group meeting with me.
  • I met this week based on skills I wanted to teach. Grade 3s needed to work on multiplication, and I wanted my grade 2s doing some more addition/subtraction work.  I had different curriculum expectations to focus on.  This week, however, I am going to be teaching fractions to everyone, and I think my groups will be divided based on abilities rather than grade assignment.
  • Yes, you read that right. I am going to give it another go this week.

Over all, the thing that I continued to reflect on is Guided Reading.  I did my student teaching in a school that really strongly believed in Guided Reading. The district where I worked, however, believed in basal readers.  There were no guided reading supports in place.  So the teachers used the basal readers in Guided Reading groups.  Instead of using leveled readers, they used the basals to do small group instruction. They used the quizzes in the readers to decide which group each child would be in.  It was not strategy based instruction at all; it was really focused on fluency and comprehension (retelling mostly.) There were lots of worksheets for the students who were not with the teacher.  And spelling from a spelling book. Guided Reading has come a long way in the last (almost) 20 years, but teaching reading in small groups while the rest of the class is engaged in meaningful, independent literacy activities is where I started as a reading teacher, and where I will end my career.  I would never base my entire literacy program on whole class instruction.  NEVER.  I am a zealot when it comes to Guided Reading.

When my current school board started pushing people to use Guided Reading (that’s about the year I started here), there were many people who resisted this style of teaching.  They didn’t know what to do with the rest of the class.  They didn’t really know what to do in meetings with small groups.  They weren’t sure they had the resources they needed to really teach the program.   I spent a lot of time trying to use my zeal in my role as a literacy coach, hoping to convert as many people as I could to Guided Reading.

Now I am faced with some zealots who believe in Guided Math.  I have concerns:  What does the rest of the class do?  Do I have the right resources at my disposal, or will I be up until midnight every Sunday night inventing the wheel?  What am I supposed to do, exactly, with my small groups?  (Sound familiar?)

And all of those are legitimate concerns, I think.  BUT – I don’t want to be the teacher who is still teaching a “Mrs. Frisby and the Rats of Nimh” novel study every October to my 5th graders because when I was hired in 1999 that’s what they told me to do with 5th graders in October.  So, do I want to keep excusing myself from trying Guided Math because I am satisfied with the way I do math now?  Well….kind of I do. Change is hard!  But I can do hard things, right?  (I’m asking…please reassure me!)

This is how the math block looked last week, basically, and I am going to try it for another week.

M/W:  Grade 2 students at activities.  Grade 3 with me working on multiplication.

T/TH: Grade 3 students at activities.  Grade 2 with me working on adding and subtraction (skip counting on the 100 chart and number lines by 10s.  I think they all have it this time around!)

Friday:  The circus came to town.

Activities:  Students had 3 –  Take a survey and make a graph about favourite holidays, Dreambox on the class iPads, play a game to practice adding up amounts of money.

Number Talks:  Getting ready for EQAO!!!  (I know, so exciting!!!)  so our Number Talks this week were really reviews of math skills.  I pulled out the EQAO from a few years ago and picked out a question on 3 days.  We reviewed patterning, telling time, and I forget what else.  A few times I did this at the end of the math block for about 10 minutes.  Typically I start math with Number Talks, so this was different.  It was a good way to settle the class at the end I think.  Usually I use a Number Talk as a warm-up. However, last week I was using the first few minutes of class to explain the centres, and to remind everyone to just please, if they had any compassion for me at all, to work quietly at the activities, and solve their own blasted pencil problems on their own!

This week:

Activities:  Reflections/Rotations/slides, graphing, and Dreambox.  Monday well will talk about last week, and I will read them a book and do a quick activity that sets them up to do the reflection/rotation/slide work.

Lessons:  Fractions.  I did a quick assessment in gym the other day when I asked them to walk/run a fraction of the distance between walls.  Interesting results! So I know a few of them, not necessarily all in the same grade, are going to need more support at the beginning level, while others are ready to try working through an investigation of some sort.  We need to work on naming fractions of a set, and fractions of a whole.

Which I better go plan right now. 🙂