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.

Geometry, math, Math Workshop, Number Talks

Real Geoboards vs. Virtual Geoboards

This past week we’ve been doing some geometry work in class.  The grade 2 curriculum expectations for geometry are fairly simple:  name, sort and make 2D and 3D shapes.  In general, children arrive in grade 2 already knowing most of these.  The more common the shapes are in the natural environment, the more likely this is true.  Octagon and hexagon usually give everyone a hard time with their tricky names, but by the end of grade 2 few children can’t recall these names.  In grade 3, we have to do a few more things. The vocabulary is increased (quadrilateral, angles) but again other than folding and unfolding nets of 3D solids, it’s nothing too complex.  Of course, I say that from this point of view – some kids do find it a bit tricky.  In all, however, it’s about 1 week’s worth of expectations.  I like to teach them early on because there are a lot of problem solving opportunities that can involve geometry and once we have the vocabulary learned the problem solving comes more easily.

This week I had rubber bands on hand.  That’s not actually something that happens all the time.  Since we had them, I pulled out the old geoboards.  Lack of rubber bands is actually one of the main reasons I don’t always pull them out.  The virtual geoboards, available here, here and here, are so much more reliable.  And nobody can shoot a virtual geoband across the room at somebody.

In the activity shown below, our Friday lesson, students were asked to make some shapes according to a rule.  Then their classmates had to figure out the rule.  Was the rule: shapes that have 3 sides?  Shapes with 4 corners?  Shapes I enjoy making because they create cool patterns? Here are some of our results:

A few years ago I remember reading an article about how important it is for students to have a real experience with a manipulative before they move to the virtual version.  I tried to find that this morning and couldn’t.  My brain doesn’t remember the source!  So I put it out to my virtual PLN (Professional Learning Network) on Twitter, and found a lot of teachers agreeing with my thinking. Reflecting on our week today, I was so glad that I had used the real geoboards.  There was some really interesting stuff that happened.

First, students were making shapes of different sizes over and over in different ways.  On the apps, they tend to get busy playing with other tools – like changing the colour of their geobands, or colouring the shapes in with a variety of colours.  They get focused on the wrong things and come up with rules like:  Shapes that are orange.  And I’m sorry, but orange is not a geometric attribute.

Second, I noticed that some children were struggling to stretch the bands across the pegs.  Some of the rubber bands are smaller than others, so this became a problem solving challenge.  I feel like they were motivated by the task to work through the challenge and find a new rubber band or change the perimeter of their shape.  This simply doesn’t present as a challenge to be worked through in the virtual environment. I had forgotten about this part!  Developing learning skills needs to be embedded in every part of our day, and I’m glad they got this chance to work on some problem solving skills.

Finally, there were some social things that we could work on.  Sharing is always an issue for 6 and 7-year-olds. They had to cooperate and collaborate to share the rubber bands on their table, and to decide who would get which geoboard.  I tried to make sure that every board at least matched the other boards in a group, but I didn’t always make that happen.  Again, students had to talk through this because everyone wants the one that is different, and therefore special.

Now that we have spent some time with the geoboards,  they can become one of the activities students can do during a Math Workshop session.   I can put them on a table with some task cards, or the students can request them to help solve a problem.  When we move on to perimeter and area (after we spend some time working with number lines for the next 3 weeks!) I can incorporate questions about shapes and feel confident that everyone knows the shape, and can work with the shape.  And I can add some

 

math, Math Workshop, Number Sense & Numeration

Games

This past week I had a chance to think about, and talk to colleagues about, how my family did math at home when I was a kid. My answer: we played games. I don’t recall having much homework until high school. I always got off the bus (5th grade and up) by myself and had about an hour before my mom came home. I typically did my homework while watching Days of Our Lives.

After dinner, however, it wasn’t uncommon for us to play Rummy, Gin Rummy, or Uno. We also enjoyed Yahtzee and Monopoly. If nobody was into it, I’d play Solitaire by myself (yes, I was a kid before it was possible to start a game online with someone.) When I was a teenager my dad taught me to play Black Jack. (It’s much less stressful with my dad than it is in Atlantic City for sure!)

Now I like to play games, in class and at home with my own children. Some of our favourites are War, Addition (or multiplication) War, and Tens Go Fish. You can play any of these with a standard deck of cards. Remove the Face cards if your children aren’t ready for adding, or multiplying, 11-13!

Today in class we did 100 chart puzzles. I copied 100 charts on card stock and cut them apart. Since I made these I have been using a 120 chart in class, but I couldn’t let the game go to waste. I suppose I should pass it off to the grade 1 class and make a new set. It was easy, and cheap. I’ve got nothing to lose. You can see here that some of my puzzles are more challenging than others.

Besides practicing math, games are a great way to practice taking turns, and losing with grace. These are important skills for kids to learn too!

math, Math Workshop

Communication

Communicating about math is a whole skill set of its own.

Case in point: each of these pictures is supposed to show you how 3 children would share 10 granola bars.

I’ll probably write at some point about the actual math. But I was most struck by the issues we encountered with communicating their thinking in writing. They basically got correct answers, but I’d never really know that with a couple of the groups if I hadn’t talked to them, and if I hadn’t helped them through the communication piece.

We’ve done a bit of this, and it’s clearly one of the things I need to focus on. I give them paper and ask them to communicate their strategies; “Show me how you got your answer!” I say, and I get little cartoons of kids writing the correct answer on a paper. Seriously. I’ve been doing a lot of modelling of writing to record our discussions during Number Talks and Number Strings. Alas, we still find ourselves in murky waters.

To be clear: it’s not just this year that my students have struggled with this. They are 7 and 8 years old…some still 6 at this point in the year! Writing is a skill they are learning. And by writing I mean printing letters and numbers, translating a stream of consciousness into written words, pictures, and numbers, and doing all of this while remembering what it is that needs to be said. They, pretty consistently across the class, thought they could write 3 names on that 10th granola bar and be done. They kept saying, “What is this big paper for?” And I kept pointing to the front of the room where our last set of math posters was still taped to the board.

I’ve had students who use manipulatives to show how they got an answer, but then not have the words to explain. They just point and smile.

I know we’ll get there! I will keep demonstrating. I have given everyone a small math journal, and for the last question during a Number Talk I ask them to write their strategy. I think it’s helping. Actually, I know it’s helping! We’re only in the middle of October. There are a lot of days left to practice mathematical communication skills.

math, Math Workshop, Number Sense & Numeration

Counting

I never used to worry too much about teaching students to count. I mean, the year I taught kindergarten we did a lot of counting, but in grade 2…or 3…or 4? Nope.

In the last few years I’ve become aware of how important counting is, and the layers of skills that are involved.

After interviewing my people, I discovered most can start at 30 and get to 100 without difficulty. Some had trouble not starting at 1 – and they also had trouble assembling a 100 chart. This is not, I think, a coincidence.

One of the classes figured out we need 12,000 laps around our track to equal Terry Fox’s journey across Canada.  We’re keeping track of our contributions with tally marks!

Some students were very organized with their counting…

And some students were not organized with their counting.

After we congressed these photos, I sent everyone off the do more counting. I asked them to count out 17, or 24, or 52, or 65 of the math tool they wanted to work with. Everyone tried an organization strategy of some sort! Even when they sorted into rows or groups of 5, many were counting by tens.

I loved this one because this child said, “10, 11, 12, 13, 14, 15, 16, 17!” It was great to see her counting on!

I think when we look at these images and congress them today, I need to make sure I talk about how the organizing helped. One of the things I know I need to do better is point out these things that seem so obvious to me.  Some students will have already realized the advantage of using groups, but some will not have.  I need to help them with that.

I used to always start the year with addition. This made more sense I guess when I was teaching grade 3/4 classes. But starting with counting and making sure everyone has a strong foundation of number sense to build on has truly made my addition and subtraction units go more quickly. Everyone seems more prepared for the addition and subtraction work once I know they are solid with counting skills.

Data Management, math, Math Workshop, Number Sense & Numeration

Candy Math

If one buys a bag of Sour Patch Kids, will there be an equal distribution of the good colours and the gross colours?  Because if I could buy a bag of just red and avoid the gastly green, and blue, I’d be happy about that!  The children in my class disagreed and hoped that there would be an abundance of blue.  But we all agreed that it should be equal.  Time to test it out!

I put a bag of Sour Patch kids on every table, and told the students they could pick their own groups.  This doesn’t happen often for us, but I wanted to see if they would distribute themselves evenly. One group of 3 got a whole bag to themselves because nobody else wanted to work with them. That left one big group of 6 to share a bag.  I think the kids in that group will think twice before they settle on a group next time (cause you know there will be a next time!)

I was very interested in the strategies students would use.  I had predicted that there would be some organizing into groups by colour, and I was right for all but one table.  It took them a few minutes of debate before they all agreed to do this.  At first, each was starting his/her own groups, stealing from the others to try and create one pile for each colour, except they were all trying to create the pile right in front of themselves.  They had 4 red piles, 4 blue, etc. Finally they realized, with a tiny bit of prompting, that one pile for each colour would suffice.

Since September, we have been talking about how organizing into groups of 5 makes counting a lot easier.  But…still…lots of kids were counting by 1’s.  *sigh*

Over time, however, they switched to grouping, usually by 2’s, but at least it was grouping.  One group, the group I would least expect to struggle with this, organized each colour by a different number.  Then they couldn’t figure out how to make that into a graph.  We had a very interesting conversation about this, so I’m counting it as a win.

 

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The graphing was fun to watch too.  We’d talked about how we don’t always have to count by ones on a graph, but we clearly have some growth to do in this  understanding.  Though I’d given them a task that required more than the number of rows I had given them on the graph template, they still thought they could just fudge it.

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Let me interpret this for you:  this group had 21 Red, 17 Green, 15 Orange, 13 yellow, 22 Blue, and 1 split (I have no idea why, but one child was obsessed with the possibility that 2 colours might have melted together and therefore 1 SP kid fell in to 2 categories AT THE SAME TIME! This did not actually happen, but this student really wanted me to believe it was not only a possibility, but an inevitability!)    I have just one picture of this to share, but it happened all over the place.  I eventually pulled them together and reminded them about choosing a scale.  The grade 3’s got it after that, but the grade 2’s not so much.  It’s not one of their expectations anyway, so I’m not worrying about it. They could read the graphs the grade 3’s created, so we’re good!

This week, assuming the one day I have to do some actual teaching actually doesn’t get interrupted by the unexpected, we are going to find out if we get more caramel popcorn or more cheese popcorn in a bag of Chicago Mix.  We are going to compare the store brand to the Orville Redenbacher brand and see if one is more even than the other.  We are also going to compare the Humpty Dumpty brand “party mix” and the Doritos brand “party mix” to see if I am truly being ripped of, as I suspect, and getting more than half a bag of pretzels.  I firmly believe there should be a “No Pretzels!” option here, just like there is a “No peanuts!” option for a can of mixed nuts.  Am I alone in this?

I used to do food math all the time.  Froot Loops, Smarties, M & Ms – they all make great math manipulative.  But for several years I have had students with food allergies and bring any sort of food into class was so stressful for me that I avoided it at all costs.  This year I have very little of that to contend with, so I’ve been going for it.

Oh…almost forgot!  One group got two half Sour Patch Kids.  They weren’t sure what to do about it.  We had a great conversation about the 2 halves making a whole.  They were reluctant to believe me, which just shows that I didn’t do quite enough with fractions this year.  I’ll have to rectify that next year.

 

math, Math Workshop

Mea Culpa

I never give out traditional worksheets for math.  Like, seriously…never.  Maybe once time this whole year, and that was for a child who was absent. But I am starting a new Context for Learning unit with my grade 3’s and they need all of my attention to get them going.  So I need the grade 2’s to work quietly.  I came up with some good independent workshop centres for them to do, with plans to bop in and out as my grade 3’s got busier.  But they really needed me, so my time with the 2’s was limited.  The activities were really good too, did I mention that?

On Friday (I know…who does this kind of stuff on a Friday?)  I gave them their games and put them work.  Over the weekend I rethought some of the noisier activities and added two that would require basically no talking between the participants.  Then on Monday I spent the whole time redirecting grade 2’s and asking them to work quietly. It seems they were having too much fun, so even though they were doing the math I had given them, they were also loud.

I decided they could do some worksheets today. They could quietly work on some addition and subtraction practice without me having to intervene at all. I dug out my dusty grade 2 commutation worksheet black line masters and copied a few. I passed them out, sent my grade 2’s to tables, and got to work with the grade 3’s.  Things were going OK!  I knew it wasn’t the most engaging work, but hey…Mercury is in retrograde and there’s a full moon on the way. Sometimes you gotta do what you gotta do.

As I started marking the worksheets later in the day, I saw lots of problems that looked like this:

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For those who have not taught Primary students, let me interpret.  Both of the answers are correct.  The little 10 floating in space between the 8, 2 and + is the correct answer to 8+2, and the weird 114 under the 7 is there because 7+4 is 11.

I have always had a child or two do this, but never the whole class.  I’m gonna take full responsibility here.  I mean how did I not see this coming, right?  I think this is why teachers in older grades think we aren’t teaching math to the little people. They arrive in a class where the teacher expects students to complete worksheets or do work from a textbook and the kids look like they can’t do math when really what they can’t do is a worksheet. That’s a whole set of skills that they just aren’t going to be taught in my class. I don’t think I’m alone in this.  Every child answered all or most of the questions correctly, it’s just that they didn’t know how to communicate this…there were weird circles all over the place, floating numbers, and even a few pictures that seemed to come from nowhere. A bunch of them did the worksheet in marker, even though there were sharp pencils aplenty in the can. It looked, at first, pretty terrible.  But because I know my mathematicians, I could find the answers and interpret the hieroglyphics.

Next week my grade 3’s will be in a more independent place and I can get started on a unit for the 2’s so everyone will be engaged in work that is moving them forward instead of practice.  We’ll be able to do our regular Math Workshop work then, and I can spend more time conferring with groups instead of trying to get one group off the ground and the other group off the ceiling.