This past week at OAME, I was pretty focused on spiralling the math curriculum and on finding more problem solving tasks to use with my class.  I find that a lot of the tasks are a bit beyond our reach, which is frustrating.

One of the things I was introduced to was Graham Fletchy’s 3 Act Math Tasks. I so appreciate when a person is willing to create a resource like this and then share it with the wide world!  While planning my week, I picked out a few in particular that I thought would engage my students, while also spiralling us back to some Big Ideas we haven’t worked with for a while.

Today we did this task, called Snack Machine.  We have had a lot of practice working with each other.  We have had a lot of practice thinking about a strategy to use to solve a problem.  But this task, and others on the site, really allow for a lot of divergent thinking.   There are multiple entry points, and multiple paths to a solution.  It’s great!

In the Snack Machine, a video shows a girl buying something from a vending machine.  We watched, then talked about it, then watched again, then talked again.

At this point, the children didn’t know what the problem would be.  They were simply looking at the video and mathematizing it. The discussion started off with someone suggesting that the girl in the video looked at her change and was disappointed.  That definitely had people thinking about why.  I got a kick (as my grandma would say) out of one of them suggesting that the machine scammed her.

After the second viewing, we had things to add.  We heard 4 coins fall, so which coins might they have been?  That lead to a long conversation, mainly because 4 toonies would make that sound, but would be an awful lot of money for a bag of chips, but 4 nickels wouldn’t really make sense either.  In act 2, there is a picture of the vending machine showing us that the chips actually cost 60 cents. Then another video shows the machine counting up the money.  We added that to our board:

Sorry about the cropping – I have written the initials of the person who contributed the idea and don’t want to publish them. Also, SO THAT’S WHERE MY ERASER AND RED MARKERS HAVE BEEN ALL DAY!

After this, I sent them off to figure out the coins she must have used.  Amazing things happened!  After everyone had a pretty good shot at solving the problem, I showed the final video.  In that video we see that the change was 2 dimes.  They used this to confirm that 80 cents had gone in, 20 cents had come out + 60 cents worth of chips, so it all made sense. No scam!

This friend needed help putting in the + sign, and also knowing where to put the $ sign.

 

This friend needed help knowing that she’d arrived at the answer. Annotating our thinking continues to be a skill we need to practice.

The money used was American money, and of course a little bag of chips would cost more than 60 cents in a Canadian vending machine. But I told them the two coins we saw were dimes, and that was good enough for them.

Yesterday we worked on Sliced Up, which had us estimating, thinking backward from oranges cut into wedges to whole oranges, and finally multiplying (5 whole oranges, 4 wedges from each orange so how many wedges in all?) For tomorrow, I am debating between It All Adds Up which is a nice money connection to Snack Machine, and The Whopper Jar   which is a nice follow up to the estimating we did in Sliced Up.  Whichever problem doesn’t make the cut tomorrow will our Monday task.  I’m learning toward the money problem because I have a bunch of activities we could do as Number Talks to stretch that learning all week.

It’s EQAO week at our school and I like having some fun, confidence building task for my students to work on.

Which Way Do I Go?

The beginning of the year is hard for me in math. There are so many things that need to be done!  This is especially true for those of us who are teaching split grade classes.  Some things are the same: number sense, for example.  I can figure out where everyone is and take them to where they need to be.  But my grade 2s are supposed to learn about some things that the grade 3s are supposed to already know (which sometimes they do and sometimes they don’t) and the grade 3s are supposed to do things that the grade 2s are not (which sometimes they are ready for and sometimes they are not!)  And I know I can still do the things, and it won’t hurt anyone to learn about something a year early, but it all takes time. And even though it’s only the 29th of October, I feel like time is slipping away and I need to GET ON IT!

So this week, I was feeling like it was time to move on from adding the tens and the ones.  I gathered the balances so we could talk about balancing equations.  I started planning in my mind where we’d go next.  But by Friday, I realized that I might be moving on a bit to fast.

Remember when I wrote about how we were having trouble communicating our math thinking? Well, that hasn’t gone away yet.  Now that we are adding, and even subtracting those double-digit numbers, I thought, wouldn’t it make sense to stop there and do some problem solving?  Wouldn’t it make sense, I asked myself, to take this thing we are pretty good at doing and use it to practice the communication piece?

So this is what we are doing.

1. Trip over the balances that are shoved out of sight behind my desk. It was a pain to get them into the room so I’m just going to live with them for a while.
2. Monday’s problem:  (Two versions because I am differentiating!)There are 14 red apples, 15 green apples, and 8 yellow apples.  Can each child in the class have one apple? 

    There are 4 red apples, 5 green apples, and 8 yellow apples.  Can each child in the class have one apple?

    

3. Tuesday’s Problem: I bought some Halloween candy this weekend!  I have 15 suckers, 23 Smarties, and 30 Kit Kat.  Do I have enough for every child in our class to have 3 pieces of candy?   

   I bought some Halloween candy this weekend!  I have 10 suckers, 12 Smarties, and 4 Kit Kat.  Do I have enough for every child in our class to have 1 piece of candy? (The Smarties are stressing me right now because I mean 23 of those little boxes of Smarties, but there are 10 actual Smarties in each.  There’s a unitizing thing in there.  I think I’ll just have to verbally clarify with the class before moving on.  I’d just take out the Smarties all together, but I’m sort of feeling committed to them now because it’s going to give us something good to talk about.)

4. Wednesday: Give in to the evil of Hallowe’en and graph some candy.  (I try to do random survey’s and graphing instead of a data management unit.  I’m going call it spiralling, like all the cool #iteachmath teachers.)  Then they’ll work on these alone, not with their Learning Partners:Make a list of 10 ways you can add two numbers and get the answer 37 every time. 

   Make a list of 5 ways you can add two numbers and get 10 every time. 

5.  Thursday and Friday: Depends on how the other days are going.  I really want to make sure that I am not rushing through.  I want to take the time to congress the solutions properly, and to talk about what makes a good visual representation of the groups thinking.  We are starting up with November Learning Partners (a few days early because we were all just DONE with the October groupings!) I have a fun nrichmaths activity that we will do if things are going well.  And I have some 100 chart puzzles we can do, which will help reinforce the work we’ve been doing about noticing patterns in the 10’s and ones that help us take leaps of 10 and 1.  We are on to Measuring for the Art show next, and this is an important understanding for that unit.
6. Then it’s Monday again, and we can balance some equations.  Probably.  Most likely. “It is highly likely that the class will work on balancing equations next week.” to put it in data management and probability  language.  And then we should move on to some geometry because that is something I have a hard time integrating on it’s own at this particular grade level.

Even though I am feeling compelled to get moving, what I really want to do is make sure everyone understands what we are doing now.  These adding and subtracting and patterning and data management skills are so important and there’s no sense in moving on until everyone is ready, not just me.

 

Mapping

I sat down with the social studies curriculum the other day, double checking to make sure I hadn’t forgotten anything.  As I looked at all of the expectations, I realized I hadn’t done that much with mapping this year.  We did a bit, but not much.  Then I looked over the geometry things I knew I had left and realized that I really hadn’t done enough mapping.  So last week, we worked on mapping!

After talking about maps and their need to make things really clear and help people find things, I gave everyone a piece of graph paper.  They all drew a map of our school yard.  They did a pretty good job!

Then when we went on a walk for science, I asked everyone to pick up a small rock.  Back at the class, we painted them different colours.  The next day, I asked everyone to think of a place where they could hide their rock, then mark that spot with an X.   We were going outside to hide our rocks in that spot, then trade maps and see if someone could find our rock. Everyone was very excited to head outside and get started.

I hadn’t, of course, anticipated that the kindergarten would be outside at this exact time. It’s ridiculous because they go outside at the same time every day, but I was so excited about my own thing that I forgot about their thing.  Soon, as you may have guessed, there were 10 or so kindergarten completely over the moon because they’d found a gorgeous rock unexpectedly in the yard.

New plan:  The next day, I drew a map.  We were talking about symmetry, and I had shown a picture of school on Google Earth.  It’s a completely symmetrical building!  My map was an ariel view of the space just outside the fence.  We’ve walked there a million times.  During recess I ran back to a less than secret spot, and hid a bag of candy.

When the bell rang, I met everyone outside.  I had them join their May Learning Partner and share a copy of my map while they searched for the treasure.  Everyone wandered around for a about 10 minutes before someone said, “This doesn’t make sense.  The X is behind the school, not in the play yard.  It doesn’t make sense!”

“Why not?” I asked.  He just kept repeating “It doesn’t make sense.”  Finally I prompted, “Well, are you trying to say we need to leave the play yard and go behind the school?”

He gave me a blank look and then said, “Yeah.  I think we do.”  He then went about trying to see if anyone agreed with him and soon we were headed to the back yard.

“Pick a landmark,” I told them. “Someplace where you want me to stand.” They picked a spot; I reminded them that if they found the treasure they were to keep it a secret and come sit beside me.  Two people actually did that before another person was overcome with excitement and gave the hiding place away with a loud, “Here it is!!”  But they were all in the general vicinity when he did that, so I’m calling this a success.

For the past few weeks I have been experimenting with using the outdoors as the classroom.  We go out to do work we could be doing indoors, but I am also trying to do things that teach about the outdoors, and use the outdoors as a resource, not just a work space.  I feel like this activity would have been less successful inside because we would have all been a bit more stressed about the noise we generated.  (Ok, mostly that would be me.) I also feel like it would have been less successful if we hadn’t been outdoors so much lately, taking the time to notice the yard and the trees, and exploring the landmarks (natural and human made) around us. I feel like putting the math and mapping skills into this context helped everyone see meaning in the activity.  (Another example of how Cathy Fosnot is right about everything!)

Technically I have now accomplished what I wanted to accomplish.  But there are only a few weeks left until the summer break and that is when the really good stuff happens if you ask me.  I am quite sure that I could do this activity a few more times.  My map drawing skills will greatly improve!  I also think I can get some members of the class to draw the maps for other members to follow.  I have 6 grade 3s and the drawing is more for them anyway.  They will love drawing a map to help their classmates get to the ingredients for an ice cream party on the last day!

 

Math Is Everywhere!

The other day we arrived at school about 5 minutes later than usual.  It isn’t much, but it means we have arrived after the first bus has dropped off some children.  My 5 year old walked up to the boot line and started counting.  “1, 2, 3, 4, 5, 6, 7 and I am 8!  I am the 8th one here today Mommy!”  He was unitizing the pairs of boots and knew that 2 boots is equal to 1 child.  I was also impressed that he knew about “8th”.  He, for a long time, only seemed to understand 1st and last.  I suppose some of that comes from being the second of two children, one of whom is keen to point out when she has beat her brother at everything. He understands first because he’s been last a lot.  But eighth?  That was interesting.

This week I started participating in an online writing challenge I’ve been participating in for 11 years!  The participants are mostly writing teachers, definitely people focused on literacy.  I’ve been surprised how many, in just the first 2 days, have posted about their gratitude for not having to teach math because they 1) don’t feel competent at math, and/or 2) don’t like math.   I honestly don’t know how anyone could possibly spend any time in an elementary school environment and avoid doing math!  The boots are just one example of the math that I see all around us daily.

My school’s electives have just wrapped up, and every single week I found myself integrating quick math lessons into our cooking.  I know you’re thinking about measurement, but there were a million opportunities to count things. Did we have enough cookies for everyone to make an ice cream sandwich?  Were there any cookies to take home?  How many brownies would each person get?  Did we have enough bowls for everyone to have a serving of apple crisp, including the office staff?  If we counted up the number of children in the group, took into account the number of kids who said they hated pineapple-upside-down cake, and divided all the pineapple up, would there be enough pineapple to eat plain?  These were important questions that had to be answered.

Of course we do a lot of intentional math, but incidentally there is math around every corner, in every classroom, and definitely in every line of items or children!  Maybe the trick is to get the grown-ups to stop treating it like a room to be avoided.

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!

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.

I’m making a plan!

This is the point in the Winter Break when I have started to think about sitting down to do some lesson plans. Instead, I just spent 30 minutes on social media reading garbage, and am now writing.  My lesson plans will wait until Sunday night, right?

I have started to work on report cards this week though.  This always gets me in a reflective mode.  Mainly I am asking this:  Have I covered something from at least 4 strands of the math curriculum in a way that will allow me to write a good report card?  I still have 4 weeks to go, so if the answer to this is “no” then I have time to make up for that.  I know we have done plenty of Number Sense and Numeration, and lots of Patterning and Algebra.  I think we have done enough Geometry, if I spend another day or 5 (or 6) on that, and I am going to comment on Measurement this term too, but will need to spend a few days doing some of those activities.  We did a lot of measurement in science, but I haven’t asked them do anything lately.  I want to make sure I have done something recently that I can comment on.

That leaves Data Management.  We have done quite a few things that are part of Data Management, but I don’t feel like I have done enough to comment on this strand yet. One of the things I have been working on this year is integrating math into other parts of our day.  We did lots of measurement in science, for example.  I wanted to do more data management in science as well, but we got side tracked.  I have not taught a single measurement lesson during math though, so I feel good about that.

Number Sense and Numeration, as well as Patterning and Algebra, are the areas I have always felt I needed to spend a lot of time on during math.  As a result, I have often rushed through Measurement and Data Management/Probability.  It’s not that I don’t think these are important.  It’s just that I was prioritizing one over the other.  By thinking about how to teach these outside of my regular 60 minute math block, I think I am seeing connections that will help my students build connections and we can all use math in more meaningful contexts.  For science, we were growing plants on the window ledge. For 3 weeks, every couple of days we pulled out the rulers, measured the height of our plants, and recorded that in our journals.  That’s meaningful.  I also recorded the results of a mould growing experiment on a chart as part of our science learning.  But we haven’t taken the next step and graphed any of this, and that is why I’m feeling like I am not ready to report on this yet. I could get there by the first of February if I really wanted to, but I have other plans for this so I’m not going to rush it.

In the coming term, we are going to be learning about Movement, and Strong & Stable Structures.  February and March are really interesting months to track weather in Ontario.  These are things that will give us a context in which to use some data management and probability related math.  I’m not worried about making sure we get enough practice with these concepts.

To get ready to finish first term reports, I guess my math month long plan will look something like this:

Week 1:  Measure things, like temperature & time (January is an interesting time for this, I think.  We’ve talked about time on a clock a fair bit, but need to talk about this human way of measuring the passing of our lives.) (This will also lead us into a social studies connection since we will be learning about Canadian Communities 1780-1850 in Social Studies during the second term.)

Week 2:  Use pattern blocks to measure length, width, area, etc. Talk about why we get different answers when we use different pattern blocks to measure the same thing. (geometry connection…this will give me a chance to check in with a few kids who were having trouble naming attributes of some 2D shapes and see if they’ve met that goal.)

Week 3:  We’ll do this part during our science time: Build 3D shapes using stuff (cardboard, spaghetti & marshmallows, etc.) and start talking about strong, stable structures (science connection for 2nd term)  In math we will start our next Context For Learning math unit (“Measuring for the Art Show”).

Week 4:  By this time I need to be finished with all of my math recording, and should be able to write everyone’s math report card comment.  Should.  🙂  I really want to sit down with each child and ask them some of the questions from our first math assessment in September, but realistically I’m setting an “end of February” deadline for that.  If the Polar Vortex (is that what were are calling it this year?) continues to churn over North America, we’re likely to have some bus cancellation days. This will help me meet that goal since I’ll only have a few students each of those days, but will also hinder me in meeting that goal because I tend to have the same few students each of those days.

The Math Pod: Week 1 Reflection

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

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

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

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

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

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