What Does Division Really Mean

Fraction division is a messy business.   Now that we have Common Core State Standards, students must be able to model fraction division in addition to performing the task algorithmically.   This is no simple task for many of them, because it forces them to grapple with the question of what it means to divide.   In order to do it successfully, they must really understand that division represents one of two things.   It can be dividing something into a specified group size to find the number of groups. (I have twenty four cookies and I want to make packages of 2 cookies, how many packages can I make?)  It can also be dividing something into a specified number of groups to find the group size.  (I have twenty four cookies and I want to serve 12 kids, how many cookies can they each have?)  They have to be able to read a problem and figure out which of those two types of problems it is and then form groups accordingly.     Making sense of these ideas and constructing a real understanding of fraction division is hard, even for a lot of adults.

As we were wrapping up our work with these ideas this week, I wanted to do a quick formative assessment to see where everyone was on these ideas.   I took two of the ACE  questions (this is the set of problems from which we draw homework assignments)  from the Connected Math textbook that I use.   Instead of using them as part of a homework assignment, I turned them into a Vote With Your Feet activity.

Vote With Your Feet is a Marzano high engagement strategy that incorporates movement into a lesson. In the activity, students are presented with a multiple choice item.   They move to different locations in the room based on their chosen response to the question (e.g., north wall for A, east wall for B, south wall for C, west wall for D).  The activity gives students a chance to get up and still stay focused on the task at hand.   It also is a really quick formative assessment, taking only a minute or two to see what each student thinks and where their misconceptions are (if the question is well-designed).

Here are the two questions that I posed.

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I had students vote on each of the two items without commenting on their choices.    Afterwards, we debriefed both questions by discussing what problems each model might represent.   Kids talked to partners and in table groups.   The class talked about it together.    As we talked, I asked students to give me two division problems for each model.   What is the problem if you are dividing by a specific group size?   What is the problem if you are dividing by a certain number of groups.

These are two of my favorite problems for a formative assessment on fraction division.   This year, I used them as a Vote With Your Feet activity.   Next year, I might take the four options and make them Quiz/Quiz/Trade cards.   I might take them and just use the model and present it as a “here is the answer, what is the question” formative assessment.   In the meantime, I will probably go ahead and make Quiz/Quiz/Trade cards with problems like this to use as a quick review of fraction division from time to time later in the year.

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What I Should Have Said

This week, I was presented with a powerful teachable moment and I dropped the ball.   A student said something and I was so stunned that I just stood there.   I needed words that I didn’t have.

I had commented to the student that I was surprised another teacher had commented that she wanted him to speak up more in class.  I was surprised because he didn’t show any reluctance in my class.   He was always ready to jump in with his thoughts.    He looked down at the floor for a minute and then this is what he said.  “In that class, I don’t want to make a mistake in front of everyone.   The questions have an answer and you are either right or you are wrong.   In your class,  there’s not just one answer and it’s OK to make a mistake.    We learn from our mistakes.  It feels safe.”     I looked at him and after a long pause, I said “I’m glad you feel safe in here.”

I’m glad that he feels safe in my class, but I feel like I let him down.   I didn’t have any wisdom.  I still don’t.   I want to tell him to be brave, to be fearless.    I can’t, though.  I’ve seen too many scars from the things kids say to each other.

Maybe I should have told him to believe in himself.    Maybe I should have told him that I believe in him.  Maybe I should have told him a thousand things.    I just don’t know what the right truth was for that moment.     I should have said more, but I still don’t know what it should have been.

 

One Brave Thing

Be kind and be brave.   These have been my parting words to my students each period of each day for 3 1/2 weeks.   I want these words to have power, to be words that echo in the hearts and minds of my students as they face all the big and small decisions that fill their days. I want these words to become the actions that populate their lives.   Saying the words matters, but it isn’t enough.

The question of “what next” has been hovering at the edges of my mind for weeks now.     Today, my students and I took a first step in our journey.   I had a few extra minutes in class today (which never happens), so I decided to use them in a new way.   I asked my students to each take a post it note and to write down one brave thing that they had done or that they had seen one of their 6th grade cohort do in the last week.   They then put their post-it notes anonymously on the “One Brave Thing” table.

This is what they created.

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It is only a beginning, but it is a beginning filled with being brave enough to reach out to someone else who needs a friend, being brave enough to stand up for someone else, being brave enough to ask questions when they don’t know something, being brave enough to ask for help, being brave enough to persevere.   It is a beginning that fills me with hope.

Anecdotal Records – Three Ways

Anecdotal Records Take One – What Can You You Do

When we are about to start working with a mathematical process fraught with pitfalls and errors, I make an anecdotal record form unique to that process.   The form consists of a bunch of boxes (one for each student) filled with key components of the process.

img_1819 This  is one that I use when I teach students how to make a line graph.   I put a student’s initials in each box.   Each day, I walk around and watch my students work with this form on a clipboard, making note of what they can do correctly with a check and what errors they are making by circling an item in the box.   At the end of the period, I have a snapshot of exactly where each student’s mastery is. The next day, I know exactly what conversations I need to have with each student.   Sometimes, I use the same form multiple days and track progress by using a different color ink on different days.   Sometimes, I just use a new sheet on different days.

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This is another one that I use for proportional relationships

I love using these kinds of anecdotal records because I know exactly what each student knows and exactly where I need to target instruction for each one of them.   The record keeping is easy to do during class and easy to access when I want to check progress.

While I primarily use this to target instruction, it is also very useful when I need to track progress for an IEP or SAT intervention.

 

Anecdotal Records Take Two – Knowing What Do You Do When You Get Stuck

While knowing what students know is important, knowing what they do when things get tough is equally important.   I found this form in  Mathematics Assessment – A Practical Handbook .

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I like this form because it gives me a snapshot of where the student is getting stuck and what he or she does when that happens.

Anecdotal Records Take Three – Knowing Whatever Else 

I keep a folder like this for each class period.   Each index card corresponds to a different student.   (No, these are not all the students in the class.   There are more on the other side of the folder).  I put the student name on the bottom of the index card, and then use the lines to record relevant observations.  I date the line and then write whatever I observed.   I use these to document whatever I need to remember about a particular student.   It can serve as a record regarding a particular intervention.   It can serve as a record of behaviors.   Mostly, it serves as a means for me to detect and recognize patterns.   By using this, I am able to recognize things like a particular student being more of a sequential thinker or being more of a conceptual thinker.   Recognizing the patterns in the way a student thinks helps me do a better job of meeting his or her needs.

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I keep the forms in “Anecdotal Records Take Three” in each folder.   The set of them are in a file holder on a table beside my desk so that I have easy access to them as needed during class.

Sunshine and Blues – Giving My Students a Voice

So much of school is filled with feedback.   Grades.   Comments.   Red and green marking.   Comment coding.   Rubrics.   Notebook checks.   Contracts.   Conversations.   The thing is, though, it is mostly a one-way street.   The feedback is mostly given to the students.   Only rarely do they get to give feedback to me.   Granted, I have them fill out a survey telling me about themselves and their feelings about math and math class at the beginning of the year.   That’s about it except for the occasional conversation in which a student chooses to tell me something he or she likes or hates.   This year, I decided I wanted that to change.

I want to give my students more of a voice.   I want to know what they think about math class.   I want to know what they like.   I want to know what they don’t like.   I want to know what they think is working and what they think isn’t working.    I also want to give them a sense of safety as they speak.   I don’t want them to be afraid to say something.

Today, during the last two minutes of class, I asked my students to take a post-it note and write one thing that they want me to know.   It can be something that they like about class or something that they don’t like, something that they think is really working or something that they think isn’t working so well, it can be whatever they want to tell me.   I asked them not to put their names on the post-it notes (unless they really wanted to and wanted me to follow up with them individually).   I told them to put the positive things on the “Sunshine” poster and the negative things on the “Blues” poster on their way out the door.   Then, I stepped out into the hall just before they left so that I wouldn’t see who put up any particular post-it.

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It was the first time I have done this, but it won’t be the last.   I got lots of “sunshine”, but the “blues” were more important to me.   It gave someone a chance to say this.

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I don’t know what I am going to do with this yet, but I’m glad I gave this student a chance to say it.   It gave me something big to ponder.   How am I going to help this student feel that faster doesn’t always mean better.   (Usually, this is a question I have to address from the other perspective – a student who isn’t thinking deeply enough thinking fast is the goal).

 

Be Kind and Be Brave

At the start of each day, one of my friends tells her little boy, “Be kind and be brave” as he heads off to preschool.   I find it the perfect distillation of what I want for my children and what I want for my students.

Be kind.   What more is there to say here?  Just be kind.

Be brave.   Be brave enough to not be perfect so that you can learn and grow.   Be brave enough to try new things and new ways of doing things.   Be brave enough to persevere when things get tough.   Be brave enough to ask for help when you need it.   Be brave enough to offer help when someone else needs it.   Be brave enough to seek out an adult when someone needs more help than you can give.    Be brave enough to befriend someone who needs a friend.   Be brave enough to stand up for someone else when they need it.   Be brave enough to do the right thing instead of the easy thing.  Be brave enough to be different.   Be brave enough to be your very best self.   Be brave.

“Be kind and be brave.”   I think this is going to be my new way to send my students out into the rest of the world each day.

First Days of School

Is there anything scarier than the first day of middle school?   Probably not in the lives of most 11 year old kids.   Will I be able to remember where to go?   Will I get lost?    Will my teachers be nice?   Will I be able to manage 6 different classes with 6 different teachers who all want something different from me?   Will I have any friends in my classes?   Will I have anyone to eat lunch with?   Will I be able to do the work?   Will I fit in?   Will I be able to put on a brave face so that nobody knows I’m scared because I’m in middle school now and I don’t want anyone to know I’m scared?

These “not quite little kids anymore” and “not quite teenagers yet” arrive looking a little bit like deer in the headlights and leave utterly exhausted.   For most of them, it takes three or four weeks before they finally feel at home.   Given the high level of anxiety, I really want the first days to set a tone that is welcoming, accepting, and engaging.

Building a Growth Mindset  

  1. A Picture Book with A Dose of Girl Empowerment on the Side  Rosie Revere, Engineer tells the story of a second grade girl who secretly constructs great inventions out of rubbish.    She hides her inventions away because she is afraid of failure.   One day, she has a life-changing visit from her great-aunt Rose (nice reference to Rosie the Riveter included).   Her great-aunt shows Rosie that a first flop is something to celebrate because it is a first step toward success.   The idea that an initial “failure” (or mistake) is the beginning of the road to success is something that I want to impart to all of my students.   I like the idea of sharing this idea with a picture book because it’s a lot less like preaching and is a nice tie back to the normal events in an elementary classroom (maybe making this new place seem a little less foreign).  I also love that the protagonist is a female.  I want both my male and female students to see it as “normal” for women to be engineers and scientists.
  2. The First Penguin – Hooray For Mistakes  Randy Pausch, a former professor of Computer Science at Carnegie Mellon University, always gave an award to the group in his virtual reality course with the most collosal failure.   In his book The Last Lecture  , he explains that he wanted his students to realize that we learn more from our failures than we do from our successes.  He wanted his students to realize that being willing to take risks and make mistakes is an important part of the journey.    After sharing the story, I will introduce the class to our “First Penguin Award”.   When someone in class finds an error in someone’s work/thinking and can help them “fix” their thinking, both of the individuals will get a piece of candy because they both helped to move thinking forward (the “fixer” in finding/fixing the thinking and the “error in thought” in allowing everyone to examine his/her thought and think a little bit deeper).
  3. Lure of the Labyrinth – Powerful Problem Solving Hiding in a Video Game – Lure of the Labyrinth, a video game developed by MIT’s Education Arcade, engages students in rich problem-solving as they play a video game in which they go on a quest to save a pet.   The tasks are challenging and require a lot of perseverance.   My goal is to help students to grow a growth mindset by growing problem-solving skills.

Building Interdependence – Broken Squares with a Twist 

I want students to recognize their interdependence.     To accomplish this,  I give them a somewhat challenging task to accomplish as a group and then I give each group member a limitation that makes the task even more challenging.   The limitations ensure that the task can only be accomplished if they work together.

The Task

A group of four students are given a set of five envelopes.  Each student gets one envelope and the fifth envelop belongs to the “table”.   Working together, the group must build five equal-sized squares using the pieces found in the envelopes.  No group member may ask another group member for a piece.   (Members may give a piece to another member).

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The task is taken from A Handbook of Structured Experiences for Human Relations Training.    You can download the files for the square pieces here 

The Broken Circles task in Designing Groupwork is very similar and is based on this Broken Squares task.

The Limitations (The Twist)

I want students to experience the need for interdependence and to recognize that each person has strengths and weaknesses.   Each member of the group is given a role that he or she must play as they complete the task.

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The group member with the “See No Evil” card, must complete the task while blindfolded.   The group member with the “Speak No Evil” card must complete the task without speaking.  The group member with the “Look Ma, No Hands” card must complete the task without using his or her hands.   The group member with the “Mean Girls” card does not appear to have a limitation.   However, he or she must take on a “mean girls” persona, using only put downs in his or her communication.

The Ground Rules

Groups may begin working on the task when I say “go”.   Oh, one more thing.   The task is a race.   The group to complete the task first is the winner.

The Debrief

After a team has won, everyone looks to see how they did it.   Then we talk about what happened.   I usually begin by talking to the winning team about their process.   What did they do that helped?   Then I ask other groups what they found helpful.   It becomes really clear that they had to work together, to fill in the gaps for each other, in order to succeed.  From there, we talk about what didn’t work.   At this point, the impact of the negative talk always comes out.  We wrap up the activity by talking about the implications for our work as a community.    We document this in a circle map on “Good group work”.

Building Relationships

  1. Circle Maps – Getting To Know You and Learning a ToolAt my school, we teach students to use Thinking Maps across all content areas.   Each week, one of the content areas is assigned a different Thinking Map to introduce.   Math is assigned to teach the Circle Map (which defines something in context) during the first two weeks of school.   I teach this tool as a “getting to know you” activity.  A Circle Map consists of two concentric circles inside a rectangular frame.   The center circle is the topic being defined and the outer circle contains the things that define the concept.   The rectangular frame is a frame of reference.   For this activity, students will put their name/photo in the center circle and then define themselves in the outer circle.  The circle map will tell me a little bit about each student and will be a way for students to get know each other a little bit.  It will serve as the cover for the student’s portfolio.
  2. Math Survey – Each student completes a survey telling me how he or she feels about math and what makes a good math class.   Because affect is such a big piece of math success, I want to know how my students feel about it.
  3. One Thing –  I take a few  minutes to have everyone tell one thing about themselves.   I try to start with something a little silly like “my secret super power is that I can wiggle my ears and convince small children that I am an elf”.    The next day might be something like “‘Oh Lonely Peas’ is my theme song.   It is all about peas left on a little kids plate because she hates them.   That is exactly how I feel about slimy squishy peas.”    Asking an 11 year old for a super power or a theme song seems to open up windows that a language arts teacher might see in a student’s writing but that are a little less obvious in math.

Assessment – An Elephant Never Forgets

While this is not something I would choose to do with my first days, it is something that I am required to do.   To lighten this up a little bit, each student is going to pick a stuffed elephant to help them out, because “elephants never forget”.     I stole this idea from one of my friends.   The students would put the elephants on their heads (or their desks if they prefer – but most chose their head) to help them remember.   This will be my first year trying this, so we’ll see if I can carry this off.