The other day a colleague of mine sent me a text with this picture of their counting circle time (4th grade class).

She followed up with this text a little while later.

She reminded me of the prodding and pushing that I do after this routine has been set in place and students start getting comfortable.

Students won’t venture out and try something new unless we push them a little (a lot in many cases). If we don’t push them they’d still be counting on their fingers by 1s! My colleague refreshed my memory of the similar things I’d say to my students:

“How can you get better at different strategies if you’re not TRYING new strategies?”

“You CAN’T use _____ strategy!”

“You must use a strategy you never used before.”

“I want to hear your least favorite strategy and why?”

This is where the setup of your counting circle is first tested. Is your environment safe enough for students to try and make mistakes but also be brave enough to share it? Is your “team” really working as a “team”? Are you being less helpful and allowing students to learn from each other?

How am I pushing students to higher expectations but continue to support them through the learning process? I force them to try new strategies by “being mean” (for lack of a better word). If my colleague wouldn’t have implemented the “no more counting on rule” do you think “E” (lower right corner) would have shared such an AMAZING strategy? Would “S” (left corner) have pushed herself to look for a pattern?

Here is another example of how counting circles allows teachers to push the standards for mathematical practice and also to push the mathematical habits of mind. We are forcing kids to find better, harder or easier ways to solve problems. To defend their answers and maybe strategies. To look for patterns in the numbers.

Now that counting circles are becoming a more solid routine, start pushing your students the same way my colleague has and see how your students rise to the continuously increasing high expectations. You might be surprised!

Ok so I didn’t come up with this idea out of nowhere. I was reading this awesome book–> Number Sense Routines by Jessice Shumway and I had this awesome class of students who were lacking in number sense. They were also a class of students who were very used to being unsuccessful in math and most of them did not enjoy interacting with mathematics. After reading through chapter 4 (Counting Routines) I thought my high schoolers would be much more receptive to sitting in a circle and counting than using visual routines (Chapter 3). So I decided to try it. If you are going to dive in and try this I HIGHLY recommend reading chapter 1 and Chapter 4 (if you can read the whole thing, please do). Chapter 1 discusses what number sense is and talks about subitizing, unitizing and lots of specific skills that most math teachers never thought of. My definition of number sense was almost verbatim described in Chapter 1 of this book! In a nutshell, its a great book for any math teacher even though the book states K-3 grades.

Moving on to WHY? I came up with this idea. You can read about it here (THE BLAME GAME) and read through my #TMC13 presentation here. In a nutshell, I am unable to live with myself if I allow students to graduate high school (pass my class) without having mental math strategies. It is equivalent to allowing a student to graduate from high school without being able to read. I refuse to be part of this movement anymore.

So I start this idea with my high school class of 12 students who’s only relationship with mathematics was very negative. To be completely honest, these students’ relationship with school was very negative and they were kind of ready to give up on school all together.

Counting circles in general seem on the surface like a routine that is just done 10 mins a day however it is much more than that. They help start a culture change in your classroom. They give students the opportunity to own math, to be successful in math, to “create” and share their strategies, to make mistakes and learn from them. They are asked to verbally describe their mathematical thinking not just write it like we have always asked kids to do. You will find that students are NOT used to these kinds of activities in math. Learning a new way to feel and be in math class (at least if you are doing these in gr 4-12) is not a FUN process for students. They are scared! School has trained them, for s0 long, to conform to learning as a feeling of success and ease at all times. For anyone that has actually LEARNED anything this is very far from the truth. The learning process is messy, ugly, scary, tiresome, filled with failure, extremely hard and you normally pick up scrapes and bruises along the way. So when you start counting circles in your classroom it is extremely important to start slow. Start with something students will be successful with: Gr 3-5 start with counting by tens starting on either a decade or non-decade number, Gr 6-8 start with counting by tens on a non-decade number, Gr 9-12 start with counting by tens on a non-decade number. (These recommendations all depend on where your students are, do not start with these if you know they will be unsuccessful)

When I started this with my class I started with counting up by 1s. Yup I know what you might be thinking…”Why 1s? That is too EASY?” YOU. ARE. EXACTLY RIGHT! My students were so hesitant to do math that I had to start with 1s. I had to build their trust with me and the trust within themselves. Everyone can count by ones so I knew that my students would have NO excuse not to be able to participate. I also wanted to change their mindset about mathematics and the relationship they have had with mathematics (extremely negative). So I needed a hook. I need to show them that they CAN DO math, that they aren’t unsuccessful ALL THE TIME. This is where you start the culture change in your classroom! You are giving them the opportunity to change the mindset have had about themselves for years!

So you’ve decided what you are going to count by and what you are going to start at….Now what?! YOU COUNT IN A CIRCLE! I normally ask for volunteers (“who’s wants to start for us today?”) and we start there and I decide to go counterclockwise or clockwise. Depending on the number of students my classroom (I have done it in as big of a classroom as 30 5th graders) I normally go one time around the whole circle and sometimes more. It really depends. When I am starting the routine and want them to get used to the routine I make sure that the focus of the counting circle is not on the math but more on feeling successful (hence the choice of starting easy) and the way the routine functions. So students count around the circle and I’m paying attention to what they are saying, how much time it is taking them to answer, any fingers being used to count or any other type of non-verbal clues students are showing me as to how hard the adding is for them or strategies they are using to find their answer. In addition to this pay attention piece, I am writing their answers on the number line as we go around the circle.

This is an important visual piece to counting circles that makes it attainable to all. Also it allows me as a teacher to hear any mistakes students might be making with place value or verbal descriptions of numbers. It also gives students who don’t have a lot of mental math strategies to use the number line to look for patterns to help them with another aspect of the counting circle: Stop and guess (I will get to this shortly)

As students are counting I am writing the numbers they said on the number line and I am also pay attention and making note of any common errors I see. As a teacher you already know some hard transitions for students to make while counting. For example:

Counting up (or down) by 10s on a non-decade number common error happens when counting to the next place value unit and also the teen numbers (i.e. 97, 107, 117,..)

Counting up or down by tenths common error happens when counting to the next place value unit (9 tenths and one more tenth makes the next whole number not 11 tenths)

The basic errors that happen can all be pretty much pinpointed to place value unit changes. As you do more and more counting circles you will see common errors and will be able to anticipate errors so you have an idea of what you want to ask in the stop and guess part.

While we are talking about errors another part of the counting circle is that I (THE TEACHER) DOES NOT correct ANYONE! If a students says an answer I write it up and I DO NOT EVEN HESITATE!! I keep counting, I keep on moving along. I wait to see if any students will speak up or if there are lots of whispers happening after a couple other people have said their numbers. If a students says something I make sure that the way they say it to other students is using what a colleague of mine calls “synergizing” language. For example: “Can you please check your answer?” “I want you to check your answer” The key to this is that students are NOT being talked to negatively. You do not want to hear “that is WRONG!” This ties back into the culture change in your classroom you want students to feel safe and comfortable to make mistakes and learn from them instead of them being reprimanded and feeling stupid for making those mistakes.

After you count around the circle, see some common errors, now is the time to stop at someone. This is where my paying attention part allows me to decide where I should stop. I normally stop on a number that students made errors on or in the past I have seen students make errors on or if I want to see IF they are going to make a common error. When I stop at that student I write his/her name under the number and then I choose how many people down the circle they have to guess. For example: “What number is Ashley going to say?” <–Ashley might be 6 people down the circle from the person I stop at or 7 people or whatever you decide based on what you see students struggling with previous) In the example below, we stopped at Kauhi and I asked them “When you figure out what number Megan is going to say, put your thumb up.”

Now we be patient and wait. We give kids as much time as possible. I have done this in a class of 30 fifth graders and I will wait and expect them to also wait patiently while everyone is able to come up with an answer. If someone doesn’t put their thumb up we continue to wait until they are ready or at least pretend they are ready.

Now comes the fun part! NUMBER TALKS!

I have students share ALL their answers. EVEN the ridiculous ones! I can’t stress enough how important this is…ALL ANSWERS!! Take them all! The wording I use “What answers do we have?” “Are these all the answers? The number you were thinking about in your head is on the board?”

After I take all answers from any students I ask for volunteers to share HOW they got their answer. This piece I am not going to describe because it is basically a number talk and you can find lost of info on the internet about number talks but I have also added my video of jotting down one of my students strategies. I normally try to take around 3 DIFFERENT strategies and I make sure to put the students answer, their name and to write ONLY WHAT THEY TELL ME VERBALLY! NO ADDING TEACHER THINKING IN!! VERY IMPORTANT HERE!! We are asking kids to be verbally specific about math and I want them to learn that their words are not specific enough or maybe they are but I write exactly what they say. I often here “you know what I meant” and I tell them “I am only writing what I hear you say” This forces my students to think about the way to verbalize their mathematical thinking (addressing SMP 2, 3, 6). After they have shared sometimes I have to verify the answer so I will start where we stopped and continue to the person we were guessing and confirm what is the correct answer. This normally only has to be done in the first half of the year because after they get accustomed to the routine they are critically thinking about answers and are able to catch their own mistakes more often and don’t need the validity piece. After the number talk is over the counting circle is over.

Once this routine gets used regularly and gets built into your classroom it should take no longer than 10 mins. I normally only do one circle a day, everyday for 8-10 mins. That is basically it. I have added one more video for an example and I still working on more videos to show the various ways counting circles change based on classroom and students. There is no one set way to really do it. I am kind of outlined a good start here but there have been lots of days that I ask different questions: “What mistake do you see in today’s counting circle?” “Who is going to say (certain number?” etc. I am in the process of starting a website that will have a post a day for a counting circle progression. I am hoping to get that started by next school year. Sorry for the delay and thank you for being patient. For now I have compiled a album of pictures that I have taken for some important reason of my counting circles throughout the 3 years I have been doing them. Feel free to ask me on twitter (@wahedahbug) or via email (hhs.mathplc@gmail.com) if you would like a description about the picture or if you want to chat more about counting circles.

I am briefly going to just list some of the math “things” that counting circles address without really going into them:

Problem solving

Standards for Mathematical Practice: 2, 3, 5, 6, 7, 8

Perseverance

Linear equations

Patterns

Mental math strategies

Multiple strategies for 4 basic operations in all different number rings (whole numbers, integers, rationals, etc.)

Adding polynomials

Metacognition

Formative Instruction

Self confidence (in relation to mathematics)

Igniting math interest

Skill practice

Classroom Routines

So much more that I will be updating continually

Here is a good overall video of how counting circles works in my classroom. Please note that this video is with a class that needs a lot of support with counting circles so I don’t normally help as much but because I need to adjust to what my students need I have to give more support for this class of students.

I was cutting oranges with one of my nieces for my OTHER niece’s soccer game. She was helping me put them in the bag and she proceeds to tell me:

N: Aunty, you are cutting it in half first and then cutting again and then again

Me: Yes you are right. So when I cut it in half the first time how many pieces does that make?

N: two half pieces.

Me: Right so then you said I cut it in half again? How do you know it is half again?

N: Because it is 2 more equal pieces that you cut it into.

M: Ok, so then what do I do?

N: You cut those pieces in half again. And you do the same thing to the other half of the orange.

M: Are you sure? Watch to make sure I do it exactly like you described. [I proceed to another orange the exact same way as before]

N: Yeah see you did the same thing. So you come out with 8 pieces of oranges.

M: So what does that mean about the “size” of this smallest piece of orange?

N: **thinking** It is the same size as the others.

M: True it probably is very close to being the exact same size. What about what fraction of the orange is it?

N: Oh, it’s one eighth.

M: How do you know that?

N: Because there are eight pieces and this is one of the eight. So this is 1/8 of the orange.

M: Ic, so what is half of half? [I ask this question as I cut the half into fourths]

N: one fourth?

M: How do you know? I don’t see 4 pieces.

N: Because if you did the same thing to the other half you WOULD have 4 pieces. so one of the oranges would be 1/4.

M: Ic, so what is a half of a half of a half? [I ask this question as I cut the fourths into eighths]

N: one eighth?

M: How do you know?

N: Cuz it’s the same as I said before but with more pieces. You are just doing the same thing to both halfs of the orange. So each orange is making eight pieces.

M: Ic, so how many pieces would 2 oranges make?

N: 16.

M: how do you know?

N: 8+8=16

M: Ic, you like doubles yeah?

N: Yeah, doubles are easy.

Orange cutting was completed and we put them in our ziploc bags. I asked her to take pictures of the ziploc bags and our eighths.

If you didn’t know I attended #TMC13 and presented about my counting circles there. After my session the GREAT Kate Nowak mentions to me that there was something I said that stood out to her during my presentation. It was about that “blame game” that teachers LOVE to do when they get students that aren’t at grade level in their classrooms.

Let’s just state one VERY important piece of information about students coming in to your classroom….majority, and I mean > 50% of them WILL come in below grade level. I am generalizing this for MOST public schools. If you work at a private/independent school then I don’t really know if this pertains to you or not. So there’s that.

One of my main motives behind the counting circles in my classroom is that I am TIRED of this “blame game”. I KNOW students will not be at grade level. I personally have worked in elementary school level classrooms and know how hard it is to get students to be at grade level and I have personal relationships with most of my elementary teachers at my school (well, old school). Hence why I don’t really like to partake in the “blame game”. So I decided instead of being part of the problem and continue to feed this fire of negativity and unsuccessfulness I would be a solution to the problem. I changed and implemented something in my classroom that allows students to work on these skills. I have made it a daily routine (only 8-10 mins a day) that helps build kids strategies and the it also gives them a feeling of success with mathematics. With success comes greater willingness to learn, a feeling of ownership and the courage to try new ideas/strategies. As students start to feel the benefits of success in math class it will be hard for them to contain themselves to the 8-10 mins of success they are used to getting. Those habits (for lack of a better word) will be pushed out into other classroom activities. Eventually you might just get kids who are willing to dive into mathematics and swim through murky waters in order to learn new things.

Now with this “blame game” comes that ultimate word that just makes my skin TINGLE! I feel like the Hulk when hearing these terms and I might have sat and shaken my hands in order to try to calm myself from reacting.

RE-TEACH

(and any other form of this verb)

Below you will find my reaction the last time I discussed this with some colleagues at #TMC13….

Let’s be realistic, why are we calling it “re-teaching”. When I looked up the definition of the prefix re on dictionary.com I get a definition that clearly states that something is being done AGAIN, repetition, a backward motion.

That means that you actually must have LEARNED something in order to “re-teach” it. I am not sure memorizing is considered learning but if they can’t use it a year later after “learning” it THEY.DIDN’T.LEARN.IT!

So I pose a challenge to you as a teacher….STOP complaining and START doing! Ask yourself “what routines or daily activities can I build into the culture of my classroom in order to keep myself out of the blame game?” Counting Circles? Estimation180? Visual Patterns? Math Exchanges? Number Talks?

Stop being part of the problem and instead BE a problem solver! Cuz we all know:

That is our Big Idea for Algebra 1 class this quarter.

In the past couple of weeks we have been focusing our counting circle on time (minutes). Counting up/down by 15 mins and starting whatever I chose (most likely multiples of 60).

After we practiced counting a few times (couple days) I started asking questions to have students convert total minutes into hours and minutes. I then asked the question:

“Do you know what else people say for 2:15pm instead of just ‘two, fifteen’?”

Students: “Quarter after?” (as anticipated)

Me: “Why do they say that? QUARTER AFTER? what does that mean? I thought a quarter was $0.25?”

Students: “Well but its different…” (somewhat confused faces)

Couple of students: “Well yeah but $0.25 is with money, this is time.”

Me: “Can anyone tell me why its different? What is the base number for time?”

S: “60, that is why it is different because 60 mins is 1 hr”

Me: “What is the base number for $0.25?”

S: **crickets** (many students don’t know our number system is base 10)

S: “$1.00?”

Me: “nope not $1.00, the numbers we use everyday (0, 1, 2, 3…) what are they all based on?”

Students: “100, 5, 2, …” (randomly yelling out answers so of course 10 comes out)

Me: “YES! 10 base 10 that is what our number system is based on”

Now continues a little history lesson (~3 mins long) about sexagesimal (my favorite word to teach teenagers) and decimal number systems. Including a little mention about Babylonians and why I love to teach them this word. We closed up that discussion with 1/4 not necessarily equal to $0.25 but means different things based on the number we are working with. “A quarter of what?”

Following this “Math Exchange” we start this week’s succession of counting circles (we only spend ~15 mins each period on these):

Monday: counting up by 15 mins, starting at 65 mins (not divisible by 60 but 60+5).

Tuesday: No school.

Wednesday: counting up by $0.25, starting at $1.05 (I hope you see where I am going with this). After our counting circle on this day, I asked them if they think there is a relationship between Monday’s counting circle and today’s counting circle? They were very familiar with counting by $0.25 because we did that for the most part of quarter 1.

So we discussed the “quarter” idea that we have been mulling over since last week. We also discussed how $1.05 is kind of like 1 hr and 5 minutes (65 mins). Before they went to their seats to start our math lesson for the day I told them “You need to think about these relationships because tomorrow I am going to give you an assessment that asks you to describe the relationship between two situations like these. They probably won’t be the same situations but they will be mins and $ so be prepared for tomorrow”

THAT SAID………

Thursday: Assessment (see picture below) I gave them 2 different counting circle scenarios with a number line like we would normally come up with if we did the counting circle. Then I asked them those 3 questions. Notice how the questions successively get harder as you go down. I really love how only Q#1 has one answer but the other 2 questions have multiple answers and ways to prove their answers. I love how most students would be able to answer Q#1 (getting them started) and then move their way down the ladder. I also love how Q#3 basically wraps Q#1 &Q#2 up in a beautifully wrapped math present for them.

I will admit this assessment wasn’t a piece of cake. But we have been working on these connections and counting circle numbers for quite some time now. I discussed it with them yesterday!! I EVEN TOLD THEM WHAT WOULD BE ON THEIR ASSESSMENT!!

Did they struggle with it? ABSOLUTELY!

Would I change anything? NO WAYS!

Do I think these discussions are happening too late in their math career? WHOLEHEARTEDLY YES!!

When do I think they should be discussing these relationships between number systems and fractions?? Somewhere around Grades 3-5 and continue them throughout their years in math class.

Are they happening in my school’s Grade 3-5 classrooms? HECK NO!

Will I stop asking my students to look for and point out these relationships? HECK NO!