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Planning for Grade Level Work Sessions

Tomorrow is a day filled with grade level work sessions focused around starting number talks in classrooms.  We will be using Sherry Parrish’s book, Number Talks, to help frame each of the conversations as we discuss the purpose of number talks–to help students build mental math and computation strategies.

We will select an operation, a focus strategy, and the set of number talk problems that will fit the group of students in each of the classrooms where we’ll be working.  Each of the grade level conversations will then focus on making some predictions about the strategies the kiddos will use; which students will participate; how the level of participation might change during the number talk; and how kids use of strategy might change over the course of the number talk.

And, then we’ll try out each of the number talks we planned–fishbowl style.  I’ll facilitate the number talk with the kids and my colleagues will collect data, make notes about how kids’ thinking and ideas change over the course of the number talk, and record the various strategies kids use.  Afterwards, we’ll debrief the learning that occurred during the number talk:  what did we see and hear, why do we think those things happened, what did we learn, and what’s next. (Debriefing protocol based on the work of Barry Tambara.)

Tonight, I’m organizing the notes I made yesterday from the book Intentional Talk:  How to Structure and Lead Productive Mathematical Discussions by Elham Kazemi and Allison Hintz to include in tomorrow’s conversations, as well as, selecting some number talk problems that we might want to consider using.  The note taking/record sheet is all printed and ready to go.

Here’s one of the number talk problems that I predict might generate a variety of strategies as kids determine how many dots there are in total in the three ten-frame cards, by using the question structure from Sherry Parrish’s Number Talks:

How many dots do you see and how do you see them?


What strategies do you think the students will use to determine the number of dots?






Addition Strategies

Last month at NCTM 2014 in New Orleans, I picked up a copy of Intentional Talk:  How to Structure and Lead Productive Mathematical Discussions by Elham Kazemi and Allison Hintz.  Chapter 2 is focused on the idea of open strategy sharing.  The authors define an open strategy sharing talk as one that is designed around a mathematical problem that “lends itself to multiple solutions.  Students listen for and contribute different ways to solve the same problem….The goal of open strategy sharing is to bring out a range of possible ways to solve the same problem and build students’ repertoire of strategies.”(page 18)

The authors’ idea of an open strategy sharing talk connects very nicely to the work of Sherry Parrish in her book, Number Talks.  Cathy Fosnot discusses the idea of number strings in the Young Mathematicians at Work Series.  Their collective ideas offer suggestions to and encouragement for teachers as they design learning opportunities for students that are focused on building strategies, developing flexibility in thinking, and analyzing how to approach mathematical situations.

If I were to set out these three ten-frame cards for a number talk with some first graders, what strategies do you think the kids would use to answer:

“How many dots do you see and how do you see them?”

photo 1

(question structure is from Number Talks by Sherry Parrish)

Here’s my list.

1.  Counting All:  The child (typically) starts at the top left and counts each and every dot.  She may start at a different spot or with a different card, but she count by ones.

2.  Counting On:  The child knows that the top card has 9 dots.  She moves to the second card and continues counting, “10, 11, 12, 13, 14, 15” and then goes on to the third card and counts each of the five dots, stopping at 20.  As with the counting all strategy, the child may not necessarily start with 9.  She may begin with the card that she just knows, that she can subitize, and move into counting on from there.

3.  Making Tens:  The child looks for dot combinations that make 10 or ways to make a ten.  In this case, the student visually moves the dot from the second row of the 6 card into the empty space on the 9 card, decomposing the 6 into a 5 + 1.  Now, he has created a 10 and a 5.  Then, he can combine the 5 (from the 6 card-with one dot moved to the 9 card) + 5 (the third card); 5 + 5 = 10.  So, 10 + 10 is 20.

The child might start with the 5 card, and decompose the 6 into a 5 + 1, and say that 5 + 5 is 10.  Then, one dot from the 6 card is put with the 9 card, creating ten.  10 + 10 is 20.

4.  Doubles:  In this example, the doubles strategy and make 10 strategy overlap in the fact that double 5s is 10.  The change in language is subtle.  The child might start out decomposing the 6 into a 5 + 1, and move the 1 with the 9.  Double 5s in the top card is 10.  That leaves two 5s in the last two cards.  The child says that I know that double 5s is 10.  So double 10s is 20.

5.  Other:  Kiddos will come up with interesting ways because they want to share and they want to have a contribution when the group is asked by their teacher, “Does anyone have a different way?”  Kids can get really creative with their strategies and provide you with a great opening to talk about efficiency and that is a good problem to have.

FINAL THOUGHTS:  If you have been using number talks regularly in your class, at this time of the year most of your students will be able to move the dots around, visually, and hang onto and organize their moves into logical and easily-followed explanations.

You may also have some students who will benefit from working with you on this question (or a similar question) in a small group setting.  They will be more comfortable using counters and blank ten-frames to help them describe their thinking about the number string.  The counters allows kids (and you) to physically move the dots from one card to another; to directly model the thinking being shared in each student’s explanation.  If you use two-color counters as your dots or different color counters (such as UNIFIX™ cubes or base ten ones’  blocks) for each number in the number string, kids can actually ‘see’ the decomposing and composing of the numbers.

Another option you might want to consider is to use white board markers and write and draw arrows and other notations right on the cards.  The marker easily comes off the cards with an eraser or a wipe.

So go ahead, organize a number talk for your kids and see what happens.  This is a great time of year to try some new things–you know your kids and they know you–everyone is comfortable.

Auto Correct won

Just realized that auto correct does not like the word subitize and keeps changing it to subtilize. I think that subtilize is a Home Depot word, but when I looked it up, Google said that it means to make more subtle.  Your email version has 4 (count ’em) typos.  So, if you go to the on-line version–I won and subitize is spelled correctly.

Can I count this as my blog post for the #MTBoS30–’cause I am already behind a few days?

A conversation some kindergarteners had about 4

A really awesome teacher I have the privilege of working with and learning from shared this really nice opening conversation she had one day last week with her kindergarteners.  Keri used a visual routine* called “look quick” to start off their math block.


CARD #1:  She set out this card, and asked the students, “How many dots do you see and how do you see them?”

photo - Version 4

She gave the kiddos a bit of time to look at the ten-frame card and think about it.  When it seemed that they had enough time to think about her questions, she asked them to share their ideas.  Many of her students told her that there were 4 dots, and the reason that was given was, “I just know it.”

At this time of the year, they just look at the pattern and know that there are four.  And that is what we want–we want them to just know.  “Young children are able to perceptually subitize, or visualize and recognize amounts (usually five or fewer), at an early age. (page 34 in Jessica Shumway’s book, Number Sense Routines)


CARD #2:  Then, Keri set out the second card she had selected for the day’s look quick sequence, and asked the students, “How many dots do you see and how do you see them?”  They acknowledged that there were 4 dots, but their explanations were a bit different than I just know it.


Lots of the kiddos said that the card had 4 dots because they could see two and two and that makes 4. (Two dots on the top row and two dots on the bottom row).  There were a few who said that there are 4 dots “because I counted them and got 4.”


CARD #3:  This is the last card Keri had selected for the look quick routine.  She, again, asked the students, “How many dots do you see and how do you see them?”

photo - Version 2

The students had different ideas as to how they found the 4 dots.  Some kids started in the top left corner and counted by ones.  Some saw three on the top and one more on the bottom and that made 4. Others started on the bottom and counted by ones.  A few made two groups of 2.  However, they all agreed that there were 4 dots.

The students who used a strategy other than counting by ones are “conceptually subitizing; which means that they recognize small amounts and combine them to see them as a unit.  Douglas Clements explains that ‘subtilizing is foundational to children’s number sense.  He states that children use counting and patterning abilities to develop conceptual subitizing.’  This more advanced ability to group and quantify sets quickly in turn supports their development of number sense and arithmetic abilities.” (Shumway, page 34)  Those students who saw two groups of two (2 + 2) and those who saw a group of 3 and a group of 1 combined the two groups to get one group of 4 are conceptually subitizing.

This is all happening because the kids have had lots of opportunities every day, all year to think about numbers, explore numbers, and talk about what the see and what they know and what they think. Keri has made it an essential and integral part of the learning in her class.


LAST QUESTION:  Keri set out all three of ten-frame cards she used in the look quick portion of her warm-up and asked her students to talk about what they noticed.  She used a slightly amended version of the I Notice, I Wonder™ strategy from The Math Forum at Drexel University found in the book Powerful Problem Solving by Max Rey.

photo - Version 3

This is what they said:

“They’re the same. They make 4.”  

“They’re the same because it’s 4 but they are different because that card is 2 and 2, and that card is 3 and 1, and that card is 4 and 0.”

AWESOME kids.  AMESOME teacher.


*FOOTNOTE:  Jessica Shumway has a really nice piece about Visual Routines:  Seeing and Conceptualizing Quantities in her book, Number Sense Routines.  She talks about the use of routines in life and in math class, and how we can easily expand what we are already doing with math routines to include a number sense component.  On page 22, Shumway talks about the different routines, what they each help with, how they work, and ways to use the routine and questioning strategies.

Keri took the idea of using routines to build number sense that we had explored during some grade level work sessions, and adjusted it to fit her students and their needs, and has had great results.  She has lots and lots of engaged confident math kids.



Number Sense Routines:  Building Numerical Literacy Every Day in Grades K-3 by Jessica Shumway

Powerful Problem Solving by Max Rey

Decks of ten frame cards:

Rows of Carts

When I was out running errands the other morning, one of my stops was Target.  We love TARGET.  I had finished shopping and was heading out of the store, and came around the corner to the exit doors, and encountered … CARTS, lots of red CARTS.  I stopped short, as the guy who was in charge of wrangling the carts was moving a long line of carts into rows in their designated space between the automatic doors of the entrance and the exit.

Did I say…lots of carts?

Carts at Target


Since I was stopped next to the guy, I said, “Lots of carts.”  (Not the most original conversation opener–but it seemed to work–he acknowledged my statement.)  He was super polite, and just smiled.  I then asked, “How many carts are there?”  He shrugged, “Maybe 50?”

I started counting the row closest to us–and got 14.  He said, “Well, maybe there’s 60.” (So wanted to ask him a string of questions like: why that was his second guess, what information did he consider when he revised his guess, how many carts does he usually put in a row–you know, the usual questions you ask in a conversation about carts in Target.)

But, what I did was that usual thing that happens with estimation, once an idea is put out there, most of us model our guess after the estimate that is offered first.  I moved my guess into alignment with his and figured about 75–about 5 rows of 15ish.  This is exactly why in a class you use the estimation 180 method–a guess that is too high, a guess that is too low, and then your guess.  And then, everyone shares out–after there has been enough time for everyone to consider all that needs to be considered.

photo 1

So, I counted the next row, and got 23 carts, and because I could see that each of the next two rows were the same length, and there is a nice constant with the cart length, I knew there were 46 carts combined for rows #2 and #3:  2 x 23.  Now we were up to 46 + 14.  That left the last 2 rows.  My co-cart conspirator counted the 4th row and found that there were 20–and then he stepped away, added more carts to the last row to make it the same length as the row he just counted (row #4)–doubles strategy.  Voila!  We had our final count:  14 (from row #1) + 46 (rows #2 and #3) + 2 rows of 20.

We smiled and nodded–I snapped a couple of quick photos–and we went on our separate ways.



1.  Drat.  Didn’t want there to be 100 carts if I was going to use this for an estimation task.  Too many kids guess 100, not because of number sense–but because they like the number 100.  It feels big to them–and that could be anything from a set of 50ish on up.  However, not sure that it is just kids who like the number 100…

2.  Cool guy at Target–super fabulous of him to play along and help me out.

3.  Doubles strategy.  We used it a few times:  once to count rows #2 and #3, and again, to even out the last two rows so as to make that nice round set of 100.

4.  And now we have a fun context for a great number exploration or a number talk and an estimation 180 all rolled into 1 to use to answer the question:  How many carts are there?

46 + 14 + 2 rows of 20

Can’t you imagine the big white boards or big pieces of sticky back chart paper and markers out on the tables for kids to use and off they go to find a bunch of different way to figure out how many carts were at Target the other morning?

What strategies might we see the kids using?

My predictions:  doubles, make ten, break apart by place value, adding up in chunks, open number lines, adding on…so many options and this makes for great conversations and discourse.

5.  A warm up for a percent lesson I saw in a class I was working in last week–need to still get permission to use the teacher’s idea.  The work was focused on percents, and since we have 100 carts, we should be able to do construct a problem sequence that offers great opportunities for interesting conversations, conjectures, and discussion.

6.  Finally,…a place to fit in the post about arrays that I have been thinking about so that it wouldn’t be so random.

Stay tuned…may actually make a go at the 30 blogs in 30 days challenge that is floating around out in the mathtwitterblogosphere.  Dare I say 1 out of 30 (or 1 down 29 to go)?