math

OAME (part 2)

I’ve been interested lately in the whole idea of spiralling curriculum.  I attended the first of two workshops about this today and it was great!

Spiralling in elementary, I think, will look quite different to secondary.  I feel like I have naturally been looping back to topics we’ve covered.  I will say I’ve been pretty haphazard about it, and that is one of the things that I recognize as a problem that I want to solve.

In the workshop the presenter (Jennifer Thiessen) showed us how she had cut apart all the math expectations and then sorted them herself into common themes.  This was how she created her units so she was integrating strands.  I am going to do this!  She found Rich Math Tasks from a variety of sources, then used a matrix to plot which areas of the entire math curriculum were covered with the task. She found those that could be taught during other parts of the day and moved them there so she could spend more time on Number Sense.  For example, I don’t teach my class about temperature during math.  Instead, all winter long, we check the Weather Channel website to see if we are going to be able to go outside or not and that short (mostly daily) conversation covers all the expectations in the document. But again, I have been more “Oh look!  A connection!” about the whole thing and I want to have it more planned.

She also showed us how she went through with different colour highlighters and picked out expectations that were brand new material and would require a fair bit of teaching time and those that were building on knowledge that kid already had and could maybe be taught in a Number Routine or Number Talk setting. For example I don’t need to do lesson in grade 2 about counting, but we had to practice counting a lot – especially skip counting – so I can practice that 150 different ways during 4-5 minute counting routines at the end of a lesson. But I will have to spend a fair bit of time doing actual lessons about adding double-digit numbers because that is going to be challenging new learning for most of the class.

I’m really excited about this, which I know is a bit weird, but I’m looking forward to spending some time analyzing the curriculum and sorting through tasks.

math

OAME 2019 (part 1)

I am attending my first ever Ontario Association of Math Educators conference.  Yesterday was the first day.  I told my children I was going to school to be a student and learn more about teaching math, and that is exactly what happened.  I signed up for a variety of workshops, and have not been disappointed in any.

The very first session I attended was called “Where Fractions, Area and Volume Come to Play” and was presented by Kawartha Pine Ridge DSB teachers Brandi Hollinger and Laurie Moher. It was great!  They shared an action research project teachers in their board had been involved with.  They studied and learned how to support students with learning difficulties in the math classroom.  They shared a really great document, available on Edugains called “Supporting Students with Learning Disabilities in Math“.  I already spent some time with this document last night, but will need to really digest it when I don’t have a brain that already feels stuffed (with more to come today!) I was so excited about it I put it out on Twitter, and someone else recommended that I take a look at “The Waterfall” document produced by York.  HOLY COW!  It’s a document full of information that is specific to math and learning disabilities and I feel like I am going to use to be a GENIUS IEP writer/implementor.

I also attended a workshop put on by the amazing Aviva Dunsiger. She teaches math all day long in her Early Learning Kindergarten classroom.  ALL. DAY.  And it’s integrated into the children’s play.  So what does this have to do with me, a not-kindergarten teacher? I’m thinking a lot about how I could be doing little bits of math throughout the day.  My friend, also at OAME with me, did a class with ideas for math DPA. So I’m thinking a lot about that, and how I could be doing math in little tiny snippets all day.

Day 2:
I am now in my 3rd workshop of the day and haven’t had time to even finish the first post.  This conference has cost me some money (though my board is supporting me with PD funds for most of it) and has cost me some time. But it is seriously so worth it.  I’m thinking about what I might teach next year. I’m thinking about what I am going to teach in June this year.  I’m thinking about improving assessment and note keeping (Oh the note keeping!!)

 

 

 

math, Number Sense & Numeration, Number Talks

Slice of Life: Multiply the Money

I wrote yesterday about a Number Talk I had worked on with my class.

Today I extended that activity by making an array using money.  I used the Mathies money tool to display an array made of nickels, then I asked, “What do you see?”

We built this display of our thinking a bit at a time, so I am sure it made more sense to us than it might to you!  Someone saw 45 cents.  Then someone else saw 15 cents (3x5cents) and then 3 groups of 15 = 45.  Some saw the array 3 x 3 = 9 nickels all together. I pointed out 3 x 15 and 9 x 5.  Thanks to our work with fact families, a few realized 45 “shared by” 3 people means they get 15 cents each.  Seriously!  They didn’t just point out that 3×15=45 so 45 divided by 3 = 15.  They actually explained what was happening! (Full disclosure: not all of them.)

After this conversation, I cleared the board and made an array using toonies. Now, you might have been expecting me to use dimes, but I thought the $2 coin was better considering that we haven’t talked about multiplying by 10 and my grade 2s would get a lot more out of the conversation if we were thinking about multiples of 2 rather than 10. I will probably give 10s a try tomorrow on their own.  After our success today it seems like a good way to get everyone to start thinking about multiples of 10. I’m looking forward to it!

 

slice-of-life_individual
Just about every Tuesday I blog for the Slice of Life challenge over at Two Writing Teachers. You can read more posts on that blog.
math, Number Sense & Numeration, Number Talks

What you see isn’t what I see

For Number Talks this past week I made some arrays on Smart Notebook and projected them on the board. We spent time each day talking about what we noticed and thought. Early on the fact families emerged. I’m glad because we’ve just finished up some multiplication and division learning and I was glad to see this being put into practice.

On Friday I displayed the picture below:

As you can see there were multiple ways to see this picture. Immediately people saw the array of 4 rows with 3 bikes in each.

One child kept insisting it was 2 groups of 6. It took a minute for him to convince his peers, and I had to help by circling the 2 groups. I’m glad we took the time to let him explain! He was clearly showing some beginning “partial products” thinking and I wouldn’t have known this if I hadn’t probed for an explanation.

Finally someone started talking about the bike tires. I don’t have a photo of that annotation, but it was interesting to see how the students went in to figure out that 12 groups of 2 = 12+12 = 24. Of course they were able to complete the fact family. At this point very few were counting by one’s. I was quite happy about that for sure!

It took me very little time to use the tools in Smart Notebook to make these arrays. I’ve definitely used Mathies for this as well, but the pictures in Smart Notebook led to a deeper conversation. This week I need to find some cars with 4 wheels to expand on our conversation.

This work also builds on the “Eyes on Math” number talks and picture-based number talks we’ve been doing all year. Tomorrow we are going on an array hunt around the school with our iPads. That activity has been preempted a few times but I think it will end up being a better activity now that we’ve discussed the picture arrays a few times.

math, Number Sense & Numeration

Math at Home

My son, who is in grade 1, has really good number sense.  He has a lot of mental math strategies that he uses efficiently and flexibly.  He adds on, he counts back, he finds landmark numbers, he even splits numbers!  And no, this is not because we spend a bunch of time every day drilling math.  It’s because we play lots of games and have math conversations that pop up throughout our day.

As I watched him play “Sorry” I was surprised that he was having some counting trouble.  He has been able to count in sequential order with one-to-one tagging for quite some time. He can count a variety of object by ones, more than 100, and when he makes a mistake he notices it on his own and fixes it.  He subitizes, and I feel like this what he is doing  while he counts and that his how he notices his own mistakes.  But that’s a tangent I won’t go on right now.

What surprised me as we were playing “Sorry” this week was the trouble he was having  moving his pawn the correct number of spaces on the board.  He recognizes every number in this game, and connects the number symbol with the amount. He’s done this with other games many times, such as when we play other games and he has to compare which of two numbers is larger. (I had a hard time writing that sentence because I kept thinking about how we haven’t played War in a long time!)  When he drew 5, for example, I know he knows that is 1, 2, 3, 4, 5.

When he would draw a number he would count to that number as he bounced his pawn around the board, but invariably any time he had a number higher than 3 he would bounce a different number of spaces.  Sometimes he would go fewer than he was allowed, and sometimes he would go farther than he was allowed.  If you draw a 4 in this game, you have to go backward, and he did OK with that but he would count slower than usual, so I built that into my intervention. I told him about the problem.  “Just like when you are counting things, your pawn has to touch each square when you count it.” I started by putting his hand in mine, and making sure that every bounce had his pawn landing in just one box without skipping any boxes.  After several rounds of this, he started doing it on his own.  He would slow down his counting and he’d land in the right spot.

The next day we played again, and the problem resurfaced.  This time I explained the problem to  him, then instead of holding his hand I put a finger on the square as he counted.  If he got ahead of me, or skipped a square, he would recognize this on his own and correct himself (and sometimes his big sister had to butt in and point out his mistake, but that’s a different post altogether!)

The third time we played the game, he needed a verbal reminder, but that was it.  And the fourth time he needed the verbal reminder.  And if we have time to play it again tomorrow, which I hope we will, I expect he’ll need the reminder again, but I’ll wait and see.

This whole thing has surprised me some, mainly because as I said before he knows how to count with one-to-one tagging and has for a while.  So why was he having trouble? This is what I think: there was a little pressure on him this time that isn’t normally there. First, he loves to win and he knew that winning in this game requires getting around the board quickly.  That was a distraction and a stressor when he was trying to count. Second, besides just counting, there was some other thinking that had to happen.  If you land on a square with a triangle you get to slide, and if you land on a square that already has a pawn on it then you say “Sorry!” and bump that pawn back to start, and sometimes I could see that he was making a move with one pawn while also thinking about how maybe he should actually be moving a different pawn to get a better outcome. He’d be in the middle of a move, suddenly stop, put the pawn back where it was and move a different one instead.  Third, …I don’t actually have a third.  I think those two things are enough to explain why he was having some trouble. I did double check to make sure he was wearing his glasses the first time I noticed it, and he was, so we can’t blame the vision.  And his coordination is such that moving a pawn around the board is not a physical difficulty for him.

Counting is such an interesting thing, isn’t it? I feel like I have some new insight into him as a mathematician.  I have since noticed that he also needs reminders to slow down when he is doing calculations.  He also does a better job when it is just me and him and he doesn’t have to worry about his sister butting in with answers. (Are you noticing a theme here?  It’s hard to be the little brother!) Finally, he does a much better job and enjoys the whole thing more when he can do single step problems. I feel like that last part is developmental and will work itself out over time.

My diagnosis is that there is an executive functioning thing going on.  He is using his working memory to do multiple tasks each time he takes a turn, not the least of which is to manage his emotions around the fact that his big sister is always butting in.

I am, of course, thinking about how to help my son with this particular thing.  But what does this look like in a classroom?  I’m thinking it would be useful to sit down with a few of my students and play a round of “Sorry” or “Trouble” or even “Snakes and Ladders” and really play with them.  They do these sort of things sometimes during indoor recess, but if I were to set this as an activity during class it would be so a group of children would be busy while I work on the real math with other kids.

Time to rethink that practice.

math, Number Sense & Numeration, Number Talks, Patterning & Algebra

Counting

Years ago I bought this treasure at a yard sale for $1:

There are well over 100 beaded necklaces in that bin!  I use them exclusively for math, though I definitely have had some children in the past 10 years who would have loved to wear them, or just run their hands through them over and over.  (It does feel nice!)

I bought them to use for a specific counting game.  I didn’t know about this game until I came to Canada.  Seemed every Core French teacher I ever worked with loved this game, though now that I am in an Immersion/English dual track school it isn’t as popular.  In French, this game is called “Dix” or Ten. The class sits in a circle and counts to 10, each saying one number.  Whoever says ten gets to sit down, and the game is played until there is just one person left.  I bought these necklaces when I was teaching kindergarten.  I didn’t want anyone to get out because the “out” people aren’t getting any practice.  I feel like I may have read about this in the Effective Guide to Instruction in Mathematics, but I can’t be sure.

Over the years, this game has evolved. I now use it for skip counting by all sorts of numbers: count by 10s and whoever says 100 gets a necklace, count by 5s and whoever says 50 gets a necklace, and so on.  I am getting ready to start some multiplication with my class after the March Break, so last week I pulled out the necklaces and we started using them every day for a few minutes before the mini-lesson.

On Friday, I asked everyone to count by 10s, and whoever said 30 got a necklace.  After we’d made it around the circle once, I asked them to talk about the pattern they could see.  Several realized there was a pattern.  It was identified as a “no, no, yes” pattern an “ABBABB” pattern, and a “skip, skip, yes” pattern.  Finally someone said, “It goes, 1, 2, 3! 1, 2, 3!” (emphasis on the 3!) I asked what would happen if we counted by ones.  Sure enough, every time someone said 3 s/he was wearing a necklace.  Then we counted past 3 to see if the pattern would continue.  I scribed on the board for them so everyone could see the numbers while we counted, and then I circled the numbers that corresponded with a person wearing a necklace.

Sure enough!  The pattern continued.

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We talked about how we could use what we had learned to count by threes, just like when we count by 5s or 10s or 2s.  Everyone was amazed, and several were happy to share their strategy: say the numbers you are skipping quietly to yourself then say the third number loud and proud.

I’ve been reading the book “Number Routines” by Jessica Shumway, and this activity shows up in that book too.  She recommends that the class start with one of her many number routines, then Number Talk, and then the mini lesson.  I’ve been giving that a try this week and I like the way the counting routine lead into the lesson, which is going to lead into our next unit of study.

Well, not exactly “next”.  We’re going to spend a bit of time on time and temperature.  But then it’s off to multiplication we go!

math, Math Workshop, Number Sense & Numeration, Number Strings, Number Talks

Early Algebra

We started the “Trades, Jumps and Stops” Context for Learning unit quite some time ago. A variety of inclement weather days has interrupted us a lot, so we are behind where I thought we’d be at this point. That might be part of the issue I’m having with this unit.  I’ve not taught it before, so that has to take a bit of the blame as well.  And finally, I’m thinking I might have misjudged our general readiness for this unit.  But I talked to my down-the-hall neighbour who is also doing the unit and she concurred:  It seems that some kids are easily getting it, and some are really, really having to work hard to get it.  There’s not a lot of in-between here.

Day 4 of the unit begins with a mini-lesson that we needed quite a bit of time with.  I was to fill 2 separate bags with a certain amount of coins in each, plus a “mystery” coin.  We have had a bit of trouble adding up coins, so I decided to stretch this out a bit instead of trying to get through it in 15 minutes.  Instead of putting the bags out of reach, I gave the coins to kids.  I started by explaining, “X and Y have some coins.  They have an equal amount of money, but they have different coins.  I want to see if you can figure out which coins they have.” On the first day only one child had a “mystery” coin (a poker chip!), but on the second day both kids had one.  The children holding the coins were very excited to be given this job. It gave them practice identifying the coin by name and value.  Have you ever thought about how we interchangeable use “dime” & “10 cents” when talking about coins? Some of the kids are still calling this “The Boat”.

We unpacked the bags a bit at a time.  I don’t have pictures of the whole process, but this is how it looked in the end.  Obviously our mystery coin was worth 10 cents this time.  This is the beginning of the children learning about a variable, and I think they did OK!

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On the second day, we talked more about the signs < > and =.  We added up the coins in chunks, as suggested in the lesson, and then decided if we needed to have a < or a > or if we finally needed the =.  This time, instead of adding them up as a group, I had the students work with partners.  We knew that X had 40 cents, and Y had 5o cents, but how much would they have once they each got another quarter?  This partnership was trying to make a number line across the bottom end of this photo, without actually making the number line.  It’s a step in the right direction for them!

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They wanted to have the numbers in a long line, but couldn’t hold all those totals in their head. Writing them above helped them work on the math and compensated for the stress load on their working memory.

We had lots of people able to do this:

Finally we made it to the mystery coin.  We knew how much money everyone had in actual coins, but what was that mystery coin worth? This led to one of those really cool moments when I felt like (most) everyone was excited about the math.  You can see here where I recorded some of the different responses to the value of the mystery coin.

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I was even able to use a double number line to show two of the answers, and since that is the goal of this unit (developing an understanding of the double number line is in fact the very next lesson we will do!) I felt very good about that.

The next day I needed a Number Talk to reinforce the understanding of the variables.  I didn’t really NEED to do it, since this doesn’t come into the curriculum until a later grade, but they were excited about it, and it certainly can’t hurt them so we did it anyway. I found some images on Math Before Bed and use them for our Number Talk.  I feel like they really help to reinforce the student’s number sense because there is more than one way to make 10, or 12, or any number really.

The weather here is terrible today (Sunday) but I’m hopeful there will be plenty of actual school days this week when we can move forward with this unit.  I don’t want to lose our momentum!  Next time I do this unit, however, I am going to maybe wait a bit.  Of course, the thing that keeps getting me through is that I have to trust Cathy Fosnot! She says this will work, and she has seen it work with many, many children, so I am going to go ahead and finish the unit.  The students are mostly getting the ideas behind the math. Some of them are actually in need of more practice with adding up money.  I am going to make sure we have a day this week when we have some activities that require counting money and we’ll rotate through those to give everyone lots of practice. Often these blogs help me think through what has happened and what needs to happen next!  Time to stop writing and plan my money counting activities.