Guided Math, Number Sense & Numeration, Number Strings

Addition of double digit numbers

There were about 50 of these on our whiteboard at different times over the last few weeks. We’ve gotten pretty good at adding the tens using mental math strategies. 20+20 -> 2+2=4, so 20+20=40…no problem! But it’s was time to move on!

I really wanted everyone to learn how to effectively use a number line. We’ve been working our way through a Context for Learning kit called “Measuring for the Art Show”. I demonstrated it about 1 billion times. First we used cubes to make a line, then we annotated this on some cash register tape, and then we moved to the whiteboard. Finally, I gave everyone some problems (from the kit) and some paper they could use for drawing the number lines.

As I walked around I could see lots of kids with lots of right answers but no number lines. “How are they doing this??” I wondered. So I asked. And I was amazed! So many of them were using the mental math strategy of splitting. They thought about how many ones there were in each number, and how many tens were in each number, then they found a total.

But the number line isn’t the go-to strategy yet. So I’m annotating the problems two ways now. The way lots of them are doing it, and the way some of them are doing it. Both are ways I hope all of them can do these problems eventually.

Our next step…my next step…is to organize into small groups (I know…back to some guided math. It keeps coming up!) I need to help the kids who are splitting learn the number line, and help the number line kids do some splitting and help the “I have no idea what to do kids” get some ideas.

I’m ALWAYS happy to reach Winter Break, but it always comes about a week before I’m ready. I don’t want to interrupt our math learning. But I’m confident the stuff we’re doing now will stick. I won’t have to start from scratch on January 7.

Here is more work from today:

math

More on Assessment

I’m nearly finished with my math interviews (minus the two who were absent today, of course!) One interview really sticks out because It wasn’t a good interview and I finished it off thinking, “WOW!  There are basically no skills here.”  Now, this isn’t the first time that has happened to me, and I know what to do about it. It’s just that this child had come to me with comments from last year’s teacher that lead me to expect some skills.  So…I went to her for some clarification.  She had saved the interview they did together last year in June.

So instead of a score from a test, or a report card mark and comment, I got to see exactly how he had answered the questions she had asked.  In our board, there is a set of interview questions that most people use, and she had used those.  I use different ones, but have been choosing 3 or 4 of my students as “marker” students, and I do this longer interview with them so I can have the same data as my colleagues for some discussions in the building.  I was so grateful that my colleague still had the exact interview sheet lying around, and wondered why I hadn’t thought to keep more of them.  Likely because I have decided not to be a hoarder in my classroom, which is an important goal, but at times like this I question my reliability as a goal setter.

This weekend, one of my school jobs is to go through my interviews and sort the data.  The board math people (I can’t ever remember all the job titles) have provided us with a tracking sheet.  I haven’t spent any time really looking at it, but I’ll probably give it a go so I can use it to participate in math conversations in our building. I also need to plot everyone on the Landscape of Learning.  Even though I feel pretty confident about my decision to start with the “Collecting and Organizing” Context for Learning unit, I have this niggling suspicion that I will maybe have to run two units at once because a few of my friends are in a very different place than the rest of us. That’s the beauty of the Landscape!  I will know who needs to start here, and who needs to start there. It’s a counting unit though, so I think it will be the only one for now.

I know that people love Teachers Pay Teachers units.  They are super easy to print and photocopy. And I know it’s much easier to mark 10-20 questions on a math test or quiz than it is to conduct individual interviews and then plot each individual on the Landscape.  But when I am considering taking some assessment shortcuts, I can’t stop thinking about a girl I taught a few years ago.

In grade 4, I gave everyone a quick multiplication sheet so they could do some practicing.  She got every single question right! But along with the quiz, she had also been working on a separate sheet of paper.  I knew she was drawing pictures to solve some problems.  But when I sat down to look at everyone’s work, I realized she had drawn a picture for every single question.  For 8×7 she drew a picture, which is fine because 8×7 can be tricky.  But for 2×2 she  also drew a picture. And for 1x 6 and for 3×3, and on and on. Her score said “Level 4”, her drawings said “Level 2…maybe actually 1”.   And back to my friend from this morning – I would have had to consider that child a “beginning” mathematician at best, but thanks to information that came out in interviews with his previous teacher, I know him to be much more.

In conclusion, I’m busy too, but I make time for this even though “all the other kids” get a bit loud playing their games, and they spent more time on Dreambox this week than they’ll spend in a single week for the rest of the year. It’s time well spent.

 

Guided Math, math, Number Sense & Numeration

Math Interviews as Assessment

On Friday I was working on finishing math interviews with my students.  I am ¼ of the way through the class already! (growth mindset)

Me:  What is 4+3?

Grade 2 student, without hesitation: 7

Me:  How’d you do that so quickly?  

Student:  Because last year my teacher had us work on things like that on this app on the iPad. We had to do those kind of problems every day.  It was so annoying (insert eye roll) day after day! But now I’m really quick at it.

Me:  That’s great! What is 8 + 14?

Student (blank stare):  Um, yeah. We aren’t on that level yet.  I think that’s like level 17 and I was only level 16, so I don’t have that memorized yet.

Me:  Well, do you have any way of figuring it out?   (I’m trying not to look at the 20 “special stones” that were just counted out, or the Rekenrek sitting to the right, or the whiteboard and marker to the left.)

Student: Ummmmmm……no.

Me:  What if you were using some of these tools? Or your fingers?

Student:  Ummmmm…….no.

Apparently they valued memorizing the basics at his previous school.  You can draw your own conclusions about how I feel about that. But this isn’t about judging another’s teaching based on the word of one 7 year old.  The real question for me is this:  What am I going to do about it for this child?

The start of the year is always a tricky time for me.  No matter what “last year’s teacher” has shared with me about a child, I feel I never really know them until I complete a reading Running Record, a writing conference, and a math interview.  This conversation above is exactly why I think a math interview is important. Thanks to this and a few other questions, I now know that this child is subitizing numbers less than 6, can skip count, count on and count back when counting objects but not when presented with just the numbers, can draw a number line that shows where 7, 10, 20, 30 50 and 48 would be, but that these will not be drawn in iterated units, and can count from 30 to 100 but has a long pause when moving between decades (48, 49……….50!)  I also know that though this child has some basic facts memorized, there isn’t a whole bunch of understanding behind that fact, so little understanding in fact that the child had no idea what to do when the answer wasn’t already known.

Years ago, I had a math textbook to use that provided a “pretest” and a ”post test”.  They were of very little use. Sure they gave me a score I could use for a mark, but that was often all they gave me.  The interview, however, tells me so much more about where to start, where to go and how fast to try and get there. A follow up interview after a few months tells me if I am moving at the right pace, if someone needs to be pushed faster or slower, and if I need to circle back to a place I thought we’d already covered sufficiently.

The major questions that always comes up is this: What do the other 20 children do while I spend 10 or so minutes with a child one on one.  During a Running Record, they all read and look at books. During a writing conference, the others are all writing. But we can’t just have a bunch of 7 and 8 year olds “do math” independently in September.  Here is what I have them do:

  1. Dreambox.  If you do not have access to this, I am VERY sorry for your luck.  It’s great. It’s a bit expensive, but if your luck is good your board has purchased a license and your class can use it.  If not, try Prodigy.
  2. Dice Games
    1. Race to 100:  Here is one version I’m excited to try.  We actually play for counting chips.  Every child has one die.  Simultaneously they are rolling that die, then taking the same number of counting chips.  When one person gets to 100 (we were playing to 50 this week) the game ends. 
    2. Tenzies/Yahtzee
  3. Card Games
    1. War 
    2. Addition war
  4. Math Manipulatives
    1. They love to use pattern blocks and there is a lot of spatial reasoning work than can happen even if the class seems to be “playing” with these blocks.
    2. They like to “play” with 3D shape blocks for the same reason and it gives them the same learning experience.
  5. Finally, if you haven’t read “What to Look For” by Alex Lawson, you should.  At the end of this fabulous book, there are a collection of games that can be played to help move students along in their mathematical understanding.  You can look at specific skills your students need, then choose a game that helps them work  on that specific skill.

I will admit that it gets a bit loud while we work on this. This might feel like a waste to some teachers.  I think it’s a great chance for us all to practice what we do while the teacher is busy with just one person. I also feel that the information I gather from each child saves me so much time down the road that it more than makes up for these first few days of playing at math. I am not guessing about a starting point – sometimes missing the mark by a mile and starting too far ahead or too far behind my students.  I can confidently set up my guided math instruction in a way that is truly differentiated for the class. Finally, who says math can’t or shouldn’t be fun?  

 

*I’ve had a few requests for a copy of my assessment.  I hesitate to share it, but I’m not sure why.  I can’t think of a real reason not to, so I guess I will.  It’s going to be most useful to you if you are familiar with the Landscape of Learning, created by Cathy Fosnot.  I don’t ask every child every question.  If they are having a lot of trouble with the first 1 or 2 addition or subtraction problems, I don’t ask the others.  If they are having trouble with the number line, I don’t ask all those questions.  But I don’t stop asking after the first mistake, because sometimes the child will go back and revise their number line, and that’s useful information for me too!

Here it is.

 

 

math, Patterning & Algebra

Patterning

So – Patterning.  I’m thinking a lot about this skill and how to make it meaningful for my mathematicians. I’m thinking a lot about its connection to algebra and how to set my grade 2’s & 3’s up for success and start them on the road to algebraic thinking.

I put them to work on Monday.  I put baskets of math manipulatives out and told them to go make patterns.  As predicted, they made a bunch of repeating patterns.  They were quite proud of them in fact.  On Tuesday, we talked about growing patterns.  They weren’t really showing an understanding of reading the pattern left to right, so we had a bit of a chat about that on Wednesday when we talked about shrinking patterns and about how the direction matters. As seems to happen often this year, they were amazed by this knowledge.  I think it will stick!  Here is one of the examples I built to show them that direction matters:

Today, Thursday, I asked everyone to actually put their pattern on a number line.  We have done a lot of work with number lines this year, and with the 100 chart.  I feel like it is really paying off!  I started with some guided inquiry.  What, I asked, would my pattern look like on a number line?

Together we constructed a few:

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Really glad I put this on top of the gross old tape line that children have been slowly picking off for 2 years!

Then I sent them to make some patterns of their own, and map them on number lines.  I didn’t hand them the paper until they had their patterns made and could talk to me about how the pattern was growing and shrinking (by ones, by 3’s, etc.  Actually, no “etcetera” because everyone either did ones or threes, like our example.  I’m not worried though because tomorrow I can tell them they are too good to stick with ones and threes and they need to choose something else!)

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This one led to a conversation about how they actually have 2 different patterns: add 3 tiles to each term, OR add one column to each term. They hadn’t noticed the second and were trying to figure out why I was so excited about it! I feel like there is a double number line opportunity here, but we aren’t ready for that!

I know it might not be right to have favourites, but this is my favourite conversation:

First, there was this:

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The child who made this pattern was insistent that it was a growing and shrinking pattern.  His partners were not convinced.  In fact, they were downright mad because he was so sure and they couldn’t see it. I couldn’t see it either, to be honest.  I wanted so badly to tell him that this was not going to work!  But Cathy Fosnot’s voice echoed in my head, “Productive struggle…productive struggle…”  so I handed him the strip of paper and a marker, and walked away.  A few minutes later, I returned to this:

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He’d figured out on his own that to make a number line his “special stones” needed to be laid out in a straight line.  He was also able to finally show us that the green stones aren’t actually part of the pattern.  They just mark the end/beginning of each set of clear stones.  As soon as it was straight, he could help his partners see his thinking – he could explain it so much more easily.  He’d made it through the struggle and came out successful on the other side.  (He did write in the numbers and finish the number line – I didn’t get a picture though.)

Two others made this pattern. (I’ll add the picture later!)

When we chatted about it, they told me that they knew 22 should come next, but didn’t have enough special stones. This was a huge piece of info for me! I thought they’d just been rote counting, but are perhaps ready to make a line without having to build the concrete pattern first.

I am, however, left with one question:  How does one put a repeating pattern on a number line?  ABABABA patterns, or ABCABCABC patterns – can they be put on a number line?

*update* Today I challenged everyone to try something besides 1 & 3.

They tried 2, 4, 5 & 10.

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…backwards…. We still have work to do!
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This one went on and on, but there are kids in the picture near the end.
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They made a “counting by 1” pattern and had to be convinced to add the second row and then to understand the second row. I had to start the number line for them, but it felt like an appropriate scaffold.
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I can’t show the “100” at the end. But I can say there was a great conversation about why 10 and 50 were so close, but 50 and 100 were so far away! They decided they needed a “refresh” and drew a new underline with perfectly iterated lengths between the numbers. Also, they realized they didn’t need to build every term in the pattern to draw the number line.
math, Number Sense & Numeration

Proportional Reasoning is so cool!

This image is from the “Grocery Store, Stamps and Measuring Strips” Context for Learning unit.  I love this unit and think it is a great way to introduce multiplication to students.  At the end of the unit, students are asked to look at this image.  (Note:  I am only including part of the image because this is not my work and I don’t want to violate copyrights!)

We had done all the proportional reasoning work before this:  figuring out how tall or long everything on a city street (trees, a bus, a few buildings) might be in relationship to 4 foot tall Antonio.  It was time for some final assessment.

“How many design elements are on the curtain?” I asked. One image shows a full extended shade, with 16 (maybe 20?  I forget)  oranges in an array.  The grade 3’s easily told me the total, and explained how they had counted.  Lots of multiplying I was very happy to note.  But they said there were 14 stars on this shade,  and 12 or maybe 18 diamonds* on the other curtains. They debated it for a while.  Then I said, “What if I stretched that shade so it covered the whole window, just like this one with the oranges?”

Blank stares.  The 5 of them looked at each other.  They looked at me.  They recounted the 14 stars and 18 diamonds*.  They  weren’t sure what I was asking.  “Well, we can see part of the window in each of these, and there is light coming through.  But what if the curtain was closed?  What if we could see the whole shade?”

They thought some more.  They used their fingers to measure.  They finally decided that if there were 12 on one curtain, there must be 12 on the other, and 2 groups of 12 = 24. . The roller shade wasn’t as easy, but once they figured out the curtain they had a strategy. “I think,” said one, “that there would be 3 more rows of stars.  So thats 7+7 doubled, plus 7 more.” OH MY GOODNESS!  Proportional reasoning AND partial products???  I could not have been happier.  Everyone agreed, then took turns explaining how they’d counted the stars and diamonds.

I’m calling this unit a success!

 

*We are calling them diamonds instead of rhombuses because they don’t have parallel sides that are straight, and the angles aren’t right for rhombuses.

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

We can make our own number lines!

I’m not going to lie:  making the jump to drawing number lines independently has taken a while!  All the grade 2s can explain what I am doing on a number line, and all of them (ok, most of them) can describe a strategy and when I draw it on a number line they confirm I have drawn what they were doing in their heads.  But to make their own?  That’s been hard.

We had completed all of the activities in “Ages and Timelines”, one of the Context for Learning Units, and people were still referring to tools (hundred chart mostly, they they tried to use math racks unsuccessfully) so I wanted to spend an extra week just talking about how to use that number line AND draw it independently.

One day last week, I created some Smart Notebook slides and we all sat down the chalkboards.  Here’s the first one,along with some notation to show that people were flexible with strategies…they used both addition and subtraction to find the answer. I will say that those who added were surprised some had used subtraction, and those who subtracted were surprised that addition could be used, so we had a great conversation about this slide! (Oh, and we are collecting paper towel tubes for science! 🙂 )

Here is another of our questions:

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I wandered around and captured some number lines.  Now, this might not be beautiful to you, but darn it!  It is gorgeous to me!  Look at the line, the iterated jumps, the acknowledgment that 4 jumps of 1 is the same as 1 jump of 4…*sigh*  I’m smiling again just thinking about it!

 

I caught one person who was struggling, and handed that child a hundred chart, with a 25 chart on the back.  For this problem, this child was able to use the 25 chart, but for later problems, had to use the 100 chart, and did! img_0677.jpg

Another beautiful number line… Another great demonstration of the iterated units drawn evenly and there even arrows on the end!

 

One of the most amazing things that happened is shown here.  One child had a 100 chart and was using it well.  Another had no strategy and was looking around the room to see if she was alone in this dilemma.  As soon as she spotted the 100 chart, she scooted over to it.  However, upon arrival, she realized she wasn’t sure what to do.  The other child showed her!!!  (There is a sock on one child’s hand because we use them to erase the chalkboards!)

So, it took us an extra week, and I am quite sure the number of grey hairs on my head has doubled since March Break. Next week, mixed in with some probability to math workshop centres, I am going to be sitting at a table interviewing these lovely grade 2’s to see what they can really do all on their own.  Can’t wait!

math, Number Sense & Numeration

Finding Connections

A few weeks ago, I sort of made my husband famous when I wrote about how he and I had each solved a problem about a good deal.  I had used an elegant solution, to quote Cathy Fosnot, and he had used the long division algorithm, which was just fine, also to quote Cathy Fosnot.  (You can hear the whole thing here.)

I had occasion to ask my husband to solve some math problems again this week, and I thought I should make sure that everyone knows he is my go-to double-checker. His methods may be old-fashioned, but he gets the job done.

I have applied for a Teacher Learning Leadership Project grant (TLLP) and am pretty close to getting it approved (I HOPE!!)  At the beginning of the month, I received a request for some clarifications about the project, which is apparently what they do.  Makes sense.  I am asking for just shy of the maximum allowed amount of money, so I’m actually glad to know they are making sure people are being fiscally responsible with this money.

One thing  I was asked to do was to make sure my budget aligns with the project goals.  I went over the entire thing with a fine-toothed comb, making sure I had the right number of days, and had figured it all out properly.  I have to account for the number of days each member of the project will be out of the classroom, and how much it will cost to provide coverage for the class.  My principal is joining, so I have also had to account for the extra money paid to a “teacher in charge”.  One member of our group doesn’t actually teach at my school, so I have factored in some mileage for her.  Even though I’ve been over it a few times, I needed someone to double-check it all for me.

I suppose you could say we worked as a team here.  I gathered the information we needed, and organized it into problems.  My husband, new to using an iPhone took one look at the work I needed him to do, pointed at his phone and asked, “Does this thing have a calculator on it?”  I showed him, and he proceeded to answer each question  on it’s own.

Again, the interesting thing here is that he didn’t see that he could solve one and use it to solve the others.  The 4 day option is double the 2 day option, but he didn’t use this. I think I am paying extra close attention to this right now because in my class we have been talking about splitting.  If we know that 40 + 50 = 90, then we don’t have to start over to solve 44+50.  We know it is 4 more than 90!  For some people, this might not be a huge revelation.  But for me, when I first learned to do math without using algorithms, these important connections between problems were completely missing. The only time I used anything like this was when I figured out 3 x 7 = 21 (for example) and then found all the 3 x 7 or 7 x 3 on a Mad Minute and wrote 21.  But I wouldn’t, for example, notice that if I knew 3 x 7 = 21 I could use that to help me with 3 x 8.

Connecting is one of the 7 mathematical practices in the Ontario Mathematics Curriculum: Grades 1-8 (2005).   On page 16, it says:

Experiences that allow students to make connections – to see, for example, how concepts and skills from one strand of mathematics are related to those from another – will help them to grasp general mathematical principles. As they continue to make such connections, students begin to see that mathematics is more than a series of isolated skills and concepts and that they can use their learning in one area of mathematics to understand another. Seeing the relationships among procedures and concepts also helps develop mathematical understanding. The more connections students make, the deeper their understanding. In addition, making connections between the mathematics they learn at school and its applications in their everyday lives not only helps students understand mathematics but also allows them to see how useful and relevant it is in the world beyond the classroom.

Since I have started to focus on teaching students to see the connections in math, I have noticed an increase in their over all number sense.  For many, as soon as they see a connection, it’s like a switch was flipped and they “get it”.