To Prove or Disprove

Today in class we were analyzing the pattern of ninths when converted to decimals.

Students had to write the decimal equivalents in the table and notice patterns.

It went according to plan.  Students write 9/9 as 1 but noticed the pattern of the numerator repeating in decimal form, which begs the question: is 9/9 or 1 equivalent to 0.9 repeating?

It's always a fun debate because the thought that it could be just blows their minds.  Thus setting the tone for 7th grade math where we take everything they think they know (rules and overgeneralization) and turn it upside down.  

Today's math debate however just blew my mind.  For those who have not been following my blog and tweets, I have been working with the ideas in Becoming the Math Teacher You Wish You'd Had by Tracy Zager.  Everything we do I put through the mathematician filter.  With that filter on, the debate was really a chance for us to question what we were seeing in this pattern and prove our conjecture (whichever side you were on).  

As usual, most kids were on the side of 9/9 is not equivalent to 0.9 repeating.  Student on both sides were giving their reasons for why they stood on that side of the room.  The patterns show they are equivalent on one side.  0.9 repeating is close to 1 but there is a number that we cannot name that would have to be added to get to 1 on the other side.  

Then one boy crossed the room, very nonchalantly while someone was talking.  The conversation went something like this:

Me: Tyler, I noticed you switched sides.  Do you want to talk to us about that?

Tyler: As I listened I realized that we didn't really have an argument that proved the other side wrong.

And that, my friends, is the difference between teaching students to explain/justify and having them prove.  In order to prove you are on the correct side of the room, you need to disprove the other side's thinking.


If you are interested in my other  Becoming the Math Teacher You Wish You'd Had blog posts you can find a complete list here.

Participating in a Twitter Chat

Twitter and Twitter chats have been transformational for me the past few years.  I want others to have the same amazing experiences that I have had.
I have many colleagues who have attempted to participate in Twitter chats and have felt overwhelmed.  I admit that depending on which chat you are joining some of them can be extremely overwhelming.  In an attempt to help my colleagues (and anyone else who may be reading this) feel less intimidated by the chats, I created some screencastify videos during one of my favorite chats, #msmathchat.

*Disclaimer: I did not script these videos (that will be clear when you watch them).  I thought about what should be included ahead of time, but I was also trying to participate in the chat which was more difficult than I thought.  Nonetheless, I think the videos will be helpful.

Preparing before the chat:
I like to have both Twitter and Tweetdeck open for chats.  This video explains more about my pre-chat rituals.

The start of the chat:
Chats typically start with introductions.  This helps you know who you are talking to.  If you find someone who teaches the same course as you it is a good person to follow on Twitter.

Participating in the chat:
Think of the chat as a social gathering.  They can range from small to extremely large gatherings.  It is not a whole group, wait your turn, type of discussion.  There are many side conversations.  You do not need to read everything that is a part of the chat.  You can jump into any side conversation at anytime.  Ask questions. Express agreement. Sometimes you will find that as you are typing someone posts exactly what you were thinking.  Don't let that stop you from posting.  It is ok to say the same thing.  That is what connects us as a group.

• Always include the hashtag so others see your tweet.
• Moderators will label questions with Q1, Q2, Q3...  You should label your responses A1, A2, A3...

Responding to Others

If you watched the video you understand the disclaimer above.  But here is the thing about about Twitter, blogging, and social media for teachers, if you wait until it's perfect it will never happen. It is an easy and informal way to share all of the great ideas and resources teachers have. No one is judging.

If you have any questions or there is something else you would like me to explain in a screencast, just use the comment section below to let me know.  I look forward to "seeing" you in a chat soon!

This site claims to have a complete list of Twitter chats for education.  I'm not sure if has them all but there is definitely something for everyone.

Purposeful Popsicle Stick Picking

I am so excited about this idea.  Before I explain it let me give you a few background links.
After reading Tracy Zager's book Becoming the Math Teacher You Wish You'd Had I created these posters and hung them in my room.

On the wall before the posters it says Mathematicians... and then these are all in a line following that.  

Can you picture it? Great!

Then during the first week of school I asked students these questions about the posters and we used them to set goals.  

I have done goal setting in the past.  Not necessarily regularly because it is one of those things I'm bad about following through with.  The goal sheets get put to the side and at best I would pull them out later in the year and ask how we did with our goals.  I don't know why it hard for me to follow through with goal activities, but it might have something to do with the fact that the goals always seem to be grade oriented.  Meaning what students will do to get a good grade.  I am not a big fan of grades.  I get why we have them, but students (and parents and sometimes teachers) lose sight of learning.  It all becomes about the grade.  

I want this year to be different.  The goals were multiple choice.  9 choices that are all focused on what mathematicians do.  The students had choice in what they wanted to work on, and I only have 9 goals to focus my instruction around - not 26 or 80 personal goals to meet with students about.  This all kind of came about this week and I was loving it, but I didn't really have a next step.  Until this morning.  

Colored popsicle sticks.  How many teachers have those popsicle sticks with all the students names on them?  Before you call on a student you pick a name out of the popsicle stick cup. Sound familiar?  

It is so brilliantly simple.  I could not be more excited.  Here is what I have done:

Each student's popsicle stick is color-coded based on their goal!!!

Here are a few notes on the making of the sticks before I share my plan for them:

You can get colored popsicle sticks but I could only find 6 colors and I needed 9.  It is possible to dye them but I wanted to make sure the colors were different enough to tell them apart and was worried dye would not do that.  I simply took washable markers and colored the tips.  I tested colors until I found 9 that were different enough.  If you line up the sticks you can run the marker over multiple sticks at a time.  It did not take that long.  

For the most part the color on the popsicle stick matches the background paper of that poster.  Because I didn't know I was doing this there were some repeat background colors so the sticks don't match exactly.  If you decide to do this and don't have the posters yet I suggest a different background color for each so the sticks can match (ur maybe that is only important to me).  

Here is my plan for using them:

Just a heads up, this section is still just a big ol' brainstorm. I don't have my book with me. (Can you believe it?  How did that happen?) I plan to look through all of my highlighted goodness in the book, and I will update and add things as ideas come to me. I just want to jot down some starting ideas.

Whenever I pull a stick I can be purposeful in the language I choose to invite the student into the conversation:

• Ted, why don't you take a risk and tell us about the strategy you used or where you got confused.
• Jane, since you are working on proving (or reasoning) why don't you share your reasoning and we will be skeptics and help you see if we need to fine tune anything.
• Seamus, where did your intuition tell you where to start and how did you verify if it was correct?
• Livy, is there anything about this work that is troubling you? (I remember this question even without the book.  I loved it so much. It's a great question to get students questioning, especially after I make incorrect statements with confidence to push them to question me.)
• Milo, let's put your work under the document camera.  We will give you feedback as to whether it is clear and makes sense to see how you are coming with your precision.

What I really like about some of these questions is that they not only help the student remember the goal they are working on, but now the class becomes supporters of that student and their goal.  We are all helping each other.  We aren't critiquing your reasoning because I understand it and you don't and now you feel like I am better than you.  We are a team and we want you to help you reach your goal just like you help us.  It feels different, even as I write the questions.

Of course, there are certain activities in my classroom that lend themselves nicely to focus on certain goals.  Not that I would only call on those that have that goal, but for those who tend to not participate or the sticks that are left in the cup, it is a good way to draw them in.  I was going to list some examples for you but I don't really have anything that isn't already in Tracy's book.  You can go back and look or buy it and read it for the first time.

Also if it is an activity I am using specifically to focus on one area, like connecting ideas or making mistakes (reflecting on what we can learn from them or why doesn't that work type of questions) I could use the sticks to create groups.  I would pull the names of the students who are working on that goal and they would each be in a separate group.  The the rest of the sticks I could pull randomly.  The people who are focusing on that goal could lead the discussion.  

Like I said, this is a brilliantly simple idea.  I am sure that your brain is already coming up with questions you would ask students or ways you would incorporate these sticks.   Go back to the chapter you are reading or your favorite chapter and think about these sticks as you read.  I want to know all your thoughts and ideas.  Please use the comment section below to share.  I am so excited to see what we create together.  

If you are interested in my other  Becoming the Math Teacher You Wish You'd Had blog posts you can find a complete list here.

Setting Goals as Mathematicians

Before school started I created posters based on Tracy Zager's book Becoming the Math Teacher You Wish You'd Had.  Here is a link to the posters.  I decided I wanted to draw my students attention to these posters.  So on day 2 of class asked my students to read the posters and answer these questions.

Link to document

• What stands out to you the most? Why?
• I was curious as to what they would say.  I love everything about these posters, but maybe something in particular would resonate with a student.  
• Give an example of something on the posters that you have done.
• I wanted them to start to realize that they are already mathematicians and they already do many if not all of these things.  
• Pick one of the things on the posters that you would like to work on this year
• Typically when students are asked to create goals they say they want to get As, work hard, or study more. Having these characteristics defined should make it easier for us to monitor and track this goal.

This activity took about 5 minutes and here is what I learned:

As much as we talk growth mindset and learning through mistakes this is not the message these students are hearing. (I blogged about this a little while ago.) I need to work to create a classroom where value is placed on the process and not the answers. And in 7th grade I think it is going to have to be extreme in order to offset the effect that letter grades (their first time getting them) will have on their perspective.

Showing work and explaining thinking is seen as busy work.  "If I have the right answer why do I have explain it?" If the students are just working independently to find a solution that will not be discussed or have talked something out with a group sharing their explanations verbally, then it possible that showing work is busy work.  I am going to have to go back to Tracy's book again.  The chapter on reasoning and proving really helped me to see the mathematician view of these and I want to be sure to share that perspective with my students.  

How true is this one? My daughter starting talking to me about text to self connections in first grade.  Then I learned about text to text and text to world (she is a little teacher and likes to make sure I know my reading skills).  Seriously, let's start bringing this to math. I think an exit slip might be where I start with this one.

Yea! I am excited to have discussions about these this year.  This is why I loved Tracy's book so much.  There are key parts of being a mathematician that we need more of in the math classroom.  We need to trust and doubt ourselves.  Don't assume your answer is right (or for those lacking confidence, wrong).  Finding a solution is not an end to the problem.  There is still so much work to be done.  I absolutely love that students noticed and questioned these aspects of being a mathematician.

As for our goal setting, the top 2 responses for what students want to work on were taking risks and asking questions.  I think this example shows that for many those two go hand in hand.  I plan to use this as a launching point next week for creating some class norms that will help ensure we have an environment where students feel safe asking questions and taking risks.  I'll keep you posted.


If you are interested in my other  Becoming the Math Teacher You Wish You'd Had blog posts you can find a complete list here.