To model or not to model. That is the question.

This is the problem I started with on the second Monday of 3rd grade:

Mrs. Cleveland found a bunch of pencils in her garage. She organized them into bags and boxes. Each bag could hold 10 pencils and each box could hold 10 bags. If she had 3 full boxes and 7 full bags how many pencils did she have? 

Multiplication you say?  Place value I say.  I needed to assess their place value knowledge.  What did I find?  Nearly half my students showed evidence that they knew 10 tens is 100.  Many of the other students had good counting strategies to figure it out.  I did have a few students who were unsuccessful but only three who didn't have a place to start.  I counted it as a success.  The next day's discussion revealed that even some of the students who didn't show evidence of place value understanding when solving the problem actually did have some understanding, they just got lost in the steps of solving this particular problem. 



The next problem I posed was a measurement division problem using groups of ten.  There were 270 pieces of candy being put into goodie bags.  Each bag was to hold 10 pieces of candy.  The students were asked to find out how many goodie bags could be made.  With this problem, the same group of kids that were unsuccessful the day before were unsuccessful again plus a few.  The difference was that this time, most of them didn't even have a valid strategy. Perhaps they weren't ready for this type of problem, or perhaps I didn't support them enough in making sense of the problem.  The day before we had done a little skit and acted out putting pencils in bags and bags in boxes (though we didn't do any solving together.)  On the other end of the spectrum I had 6 students who wrote some sort of multiplication number sentence such as 27 x 10 = 270 or a string of multiplication sentences to arrive at the total. 

After Classroom Discussion

With the variety of strategies my students used, I felt like I had the perfect papers to introduce multiplication in our next discussion, but I didn't.  Mostly because I was worried about those students who weren't making sense of groups yet and weren't using grouping strategies on these problems.  Instead I shared a valid strategy and represented all 27 bags but displayed an answer of 26 bags.  We focused on the standard for mathematical practice 6: attend to precision as well as grouping.  We circled groups within the strategy to double check the answer and realized it was written incorrectly.  
As a side note, I was careful not to have the students go back and count one by one to check the work, because that encourages the most inefficient strategy... counting by ones.

After our discussion about grouping, I decided to return to a multiplication problem but this time with predetermined groups of 100.  I was hoping the students wouldn't get caught up on the multiple steps aspect of the problem solving.  The problem I posed was this:

All of MMJ students are going on a field trip to the Zoo.  The buses will hold 100 students and the vans will hold 10 students.  If there are 6 full buses and 4 full vans then how many students will go on the field trip?

Direct Place Value

Success!  19 of my 23 students were completely successful.  Many of the kids were still using a counting strategy, but most of them were writing repeated addition sentences.  Almost all of the students grouped their hundreds together and then the tens when writing these sentences.  This is likely due to the structure of the problem, but it gave me a great platform to introduce the multiplication symbol.  Again I had several students who relied on direct place value understanding to solve the problem.  I even had 3 students who had not yet used place value understanding move to using place value strategies in this problem.  I was excited about the progress.  

On Thursday, I decided that I would pose one more problem before jumping into introducing the multiplication symbol.  Because of the success on the most recent problem, I decided to take problem solving one step further and introduce multiple number sets.  This, I feel, is essential to differentiation.  I have a rather large group of students who have direct place value understanding and I needed to push their thinking.  So I chose to make my third number set 18 groups of 5 and wondered if any of them would make the connection back to the groups of 10 that we had been working with. 


With the implementation of multiple number sets, I felt like I needed to model for my students.  They needed to be shown how to plug in the numbers and keep their papers organized.  So I used a basic problem with blanks and number sets and we worked together as a class to go through the process.  First we entered the numbers from the first set and solved the problem.  We discussed how they should show what they did with either a picture or a number sentence or both.  I also pointed out the need for labels and showed them how to draw a line to separate the work for each number set.  Then we moved on the next set of numbers and went through the process again.  After solving the problem 3 times, we did a brain break (www.gonoodle.com).  They needed it!

I then posed a multiplication problem with 3 number sets: 8 groups of 10, 18 groups of 10 and 18 groups of 5.  After a quick story telling session and visualization of the problem I sent my students off to their quiet spots to solve.  You will not guess what happened next.  They ADDED!  I looked at the first completed problem and said, "Would you please go draw me a picture that matches the story."  Then another student showed me the same strategy.  Student after student brought their paper to me with three addition sentences and exclaimed, "I'm done."  Nearly half my class.  These were students that had been using grouping strategies and place value understanding just the day before.  What happened?

I modeled.  I showed them how to solve the problem.  Well not really, but really.  Unintentionally, while modeling for them how to solve with multiple number sets, I showed them how to solve a part part whole problem with addition and they took what I showed them and applied it to the new problem.  I was in shock.  All week these students had been problem solving and progressing.  Several went from not having a strategy to having a viable strategy to being successful.  The minute I provided them with direct teaching, no matter what my intent, they latched on and all sense making went out the window. 

Granted, not every student regressed.  As a matter of fact, even many of the students who added revised their strategy when asked to draw a picture to match the problem and support their number sentence.  Still though, initially they adopted my strategy and thought nothing of it.  So hear me out, I believe that students have an inate problem solving ability.  When given the opportunity to think through a problem on their own and make sense, they can do it.  Some will make sense faster than others, but still they can.  They do not need us to show them how to solve the problem, they need us to facilitate their thinking and question them so that they make connections.  At least for a certain population of our students, modeling and mini-lessons impair their own problem solving abilities.  If you want multiple strategies, don't show them yours. 

90 Minutes of Math

Recently I've received several requests to share the schedule I plan to use in my math block.  I must first say that the schedule I am about to share has not yet been put to the test and I will likely tweak it throughout the year.  Also, keep in mind that I have a straight 90 minute block with no interruptions.  I know many don't have that luxury, so as you read this, you can think of it in parts, that could be scattered throughout your day.  Here's what I'm thinking:

15 minutes - Number Talks

A Number Talk is a short, daily routine that provides students with meaningful ongoing mental math practice.  During Number Talks students are expected to use number relationships and the properties of operations to add, subtract, multiply and divide.  Often a number talk is comprised of a string of related expressions that are intended to elicit a specific strategy or operational property.  For instance, in my classroom this past week one of my number talks featured the following string: 

100 - 89

100 - 49

250 - 24

This string was intended to elicit the "add-on" strategy and while a few of my students counted back by tens and then ones, I did have many that added up, and we were able to discuss those strategies and make comparisons between the two.  

It is important to keep Number Talks short, as they are not intended to replace current curriculum or take up the majority of your math time.  Number Talks are most effective when they are kept short and done every day.

5 minutes - Brain Break 

Because my math block begins at 8:00, the kiddos have already been trying to focus for 45 minutes to an hour at this point in the morning and I've found that they need a brain break early.  Depending on when you get started with math, you may choose to move or eliminate this break.  As I mentioned in my last post, I'm using GoNoodle, and so far the kids love it!
 

20 minutes - Discussion of Previous Day's Problem or Task

In the Purposeful Pedagogy and Discourse Instructional Model (PPD Model) a huge part of the students learning and growth happens during the discussion of student strategies that emerged while working the task or problem chosen by the teacher.  When I first saw this model in action, the teacher posed a problem and then walked around the room watching the students work, making mental notes of which student strategies she wanted to share during the whole group discussion.  As I watched the discussion that took place after problem solving I remember thinking, "Wow she's good to facilitate a discussion like that on the fly... she couldn't possibly have known what the kids were going to do with that problem."  I found out later, of course, that she had planned that discussion based on the strategies she had anticipated her students would produce.  At that point I thought, "Wow, she's good, she must have lots of experience in order to be able to anticipate what her students will do with every problem." 

What I'm trying to say is, facilitating a discussion on the fly takes practice.  Even if you plan in advance, you have to have a really good idea of the students will likely do, and that takes experience and deep understanding of student thinking.  So, in the meanwhile, until I get good at that, I plan to have my discussions the following day.  This will allow me time to look at my student work, sort it, think about it, and plan a discussion based on what they actually did, not what I think they'll do.  Look for more details and examples of this process in future posts.  

20 minutes - Introduce Today's Problem/Task & Work Independently

I'm not going to lie, choosing the just right problem or task is not easy.  Sometimes I might choose a problem from my CGI or ECM book or sometimes I might choose a problem or task from our district resource Stepping Stones, by Origo Education.  At other times I may use Contexts for Learning or I might even write my own. No matter where it comes from, I try to choose something that has the potential to produce the discussion I need, in order to push at the learning goal I have chosen for my students.   

This past week I was focused on these two common core standards as my learning goal:

3.NBT.2 - Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

3.OA.9 - Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations.

Posted with permission: Origo Education

I chose the task you see here from Thinking Caps (an add-on to our district provided resource Stepping Stones).  I posed this task to students without a mini-lesson, and without providing them an example.  The strategies that students came up with to write multiple subtraction sentences with a difference of 115, generated a fabulous discussion of patterns along with the relationship between addition and subtraction.  It was really quite exciting. 

5 minutes - Brain Break

Have I mentioned www.gonoodle.com?

25 minutes - Number Sense Routines and Fact Fluency

I don't have this time completely worked out in my head yet.  This past week I used this amazing number board which is again found in our district provided resource Stepping Stones.  We did some count around the circle and identifying patterns/missing numbers on the hundreds board during this time.  Then I used a problem solving activity from Stepping Stones that had students using what they knew about patterns on the hundreds chart to identify missing numbers on a given piece of the hundreds chart. 


In the future I intend to use this time to focus on fact fluency strategies at least a few days a week.  The first few minutes will be used to introduce/elicit strategies.  After that I will use the rest of the time to reinforce strategies and allow students to practice basic facts.  This time will likely be partner work and purposeful game play.  I can't wait to get started!

So there you have it.  My 90 minute math block.  As it stands right now at least.  I can promise you this will not be the last you hear about my schedule.  I'm already considering employing some sort of workshop model a couple of days a week.  I'm not sure how that will work yet, but I know I need to figure out how to get in some small group time with my kiddos.  So I'll be playing around more with that idea soon.  I'm so excited!

Can't wait to start math instruction? Me too!

As a part of my job as elementary math specialist this year, I am teaching a demonstration classroom full of third graders.  I am so excited about what I am going to learn this year!  Yes, I that's right, I said it... I am teaching a demonstration classroom and I AM GOING TO LEARN!  I have spent 4 years studying theory and preaching to teachers about implementing Cognitively Guided Instruction, the Common Core State Standards for Mathematical Practice, and the Purposeful Pedagogy and Discourse Instructional Model.  Now it's time to practice what I'm preaching.  I plan to demonstrate how to make it all work in a school district that uses standards based reporting.  But first, I have to figure it out.

Today was the second day of school.  I have been granted a 90 minute block for math instruction.  Today my plan was to focus on procedures... How do we treat our manipulatives?  How do we use them as tools?  How can we organize them for easy use?  That was my plan.  What transpired, though, was a different story.  I did math... on the second day of school.  Am I crazy?

Here's how it started.  First, I posed a problem to the students: We need to figure out how to move our book boxes off our table and onto the shelves at the same time as taking our math toolkits off the shelves and putting them on our table.  We want to make sure and do this without putting anything on the floor and without having all 24 students up at once.  Innocent enough right? It was.  The kiddos came up with a few possible solutions right away, and we voted on which team's procedure would be the most efficient.  Done 25 minutes down. 

Now that we had the math tools on the tables, I told the kids a story.  "This morning I got here really early... at 6:45a.m.!  I worked and worked and worked to organize your math tools, but I ran out of time.  At 7:15, you all were here and ready to come into the classroom.  So I had to hurry to finish what I was doing."  At this point, I couldn't help it, I let the first math question sneak into our day, "How long did I work on the math tools before you got here?"  Really though, I was working on procedures, the students learned how to give a thumbs up when they had the answer instead of raising their hand.  :)  Carlton was the first to share his strategy, "6:45 is just a quarter from 7:00 and a quarter is 15 minutes.  Then from 7:00 to 7:15 is another 15 minutes, and 15 plus 15 is 30 minutes."  Now the kiddos got to learn the "me too" signal. 

 



So, you see, I really was focused on procedures.  :)  After we had our short elapsed time discussion, I explained to the students that because I had run out of time, I needed their help to open up our fraction squares and circles and remove the stickers so they could be used more easily. 

The kids enjoyed this activity and I did too. They got to "play" with the tools, and I got to facilitate a discussion about how we treat our tools.

After we finished breaking apart the fraction manipulatives, we had to squeeze in a brain break from www.gonoodle.com!  This was definitely a highlight of the day.  But even as I introduced brain breaks, we had to vote on our classroom champ, and I created a tally chart.  Which turned into a quick number talk.  How many more votes did Flappy Tuckler get than McPufferson? See I really just can't help myself. 

After a quick "Happy" dance and a calming stretch we moved on to the real problem of the day.  I had passed out place value blocks, but after I had counted out all the hundreds of unit cubes, I didn't have enough time or energy to count out the tens rods and make sure each team had the same amount so I just grabbed a pile and put some in each tray.  Now, in my classroom there are four table teams, and two math toolkits per table.  I could smell a number talk/collection counting activity coming and like a preditor hunting it's prey, I pounced on the opportunity to dig into some student thinking. 

 

The kids were in complete agreement that things must be fair, and therefore we must count the tens rods, and determine how many, if any, needed to be redistributed so that each table team would have the same amount.  So it began, they counted and I observed. They added and I listened.  Then we "fishbowled" and the class counted together as Marissa moved the rods on her desk.  Then again, the class counted as Henry moved the rods on his desk.  This is what I observed. 

 

Henry's blocks

Marissa's blocks



As the class counted, Marissa moved the blocks and grouped them into tens. Two groups of ten and 6 more. 

Henry, on the other hand, grouped off the first set of ten, and then grouped all the rest together. 

What does it mean?  I don't know.  Is it significant?  I think maybe.  Am I overanalyzing?  Likely.  

Oh well, moving on.  After counting the number of tens rods in each tray I recorded that information on the board, and what do do you think I did next? A number talk of course! 

A little bit of mental math, some strategy sharing, and some answer defending and in no time at all we had totals per team... 51, 50, 48 and 62 tens rods. 

I love strategy sharing.  I am always amazed at how they think about numbers.   

So that was it.  I ran out of time.  I wanted to keep going, but the kiddos were obviously over it.  It's the second day of school and I got in as much as I could without going overboard.  Crazy? Maybe.  But do you blame me?  I hope not.  

*Children's names have been changed.