Monday, November 20, 2017

Mathematicians...A Follow Up

Last week I had parent teacher conferences on Tuesday and Wednesday.  On Monday I realized that I hadn't had students complete any sort of end of quarter reflection like I have in the past.  This year, after reading Tracy Zager's book Becoming the Math Teacher You Wish You'd Had, I have been helping students focus on what we do as mathematicians.  I decided to take the posters I made (blog found here) and turn them into our reflection sheet. 
Here is what I came up with:

I had students mark which areas they felt were strengths and which ones had room for growth.  There are many bullet points for each characteristic and many students wanted to just put marks down the middle for all.  I made them choose and offered room for them to explain their thinking.  I only gave them 20 minutes so many students were not able to explain all of them.  They were really being thoughtful about their choices and I appreciated that.  To be honest, all of my students have strengths and room to grow in each of these areas.  I was more concerned about their perception.  

I absolutely loved reading the students' responses.  What I loved more was sharing these reflections with parents.  My conferences had a different feel and the conversations were more meaningful.  Here are some examples of responses and conversations that surrounded them:

 Many students interpret "Ask Question" as asking when they don't understand.  I want students to tap into their natural curiosity.  Noticing, Wondering, and asking Why does that happen? are characteristics of lifelong learners.  I want this for my students.  (Insert plug for Table Talk Math Here - flyer for conferences)

 Yea! The vertical non-permanent surfaces are a success!  We often use the whiteboards to show our proofs.  That way everyone can see our work when we are explaining our reasoning and can compare strategies. 

Many students think they are bad at math because they don't get it instantly.  They want the teacher to explain it step by step so they can follow those steps.  That is not how I teach math because I don't believe that is math.  Math is what happens in the confusion.  That is where students are working to make sense of numbers and relationships and looking for patterns.  Of course we need to come out of this fog at some point but some days it is just to move to a different fog of confusion. 

 We do a lot of sense making in class.  Often times the models that parents don't understand (because we didn't learn with them) are not necessarily how we will solve the problem, but instead they show us the number relationships and deepen our understanding.  This helps engage our intuition when working so we can see if our process and answers make sense.

Some students make errors because they don't understand the concept.  The fix? Help them understand the concept.  Then there are students who make errors even though they understand the concept.  What is the fix?  Focus on precision.  A benefit to learning multiple strategies is that we can go back and solve another way to see if we get the same answer, or use a model to check our thinking and engage our intuition (see above).

Another strategy for checking over your work is to go back and tell yourself it is wrong.  It is one thing to look over work to see if there is a mistake.  We are much more analytical if we go back and look at the work as if there is a mistake. 

These two make me so happy.  Both students recognize that they need to grow in these areas, but these responses show me that understand their depth of understanding is important.  They can no longer have just surface level knowledge (memorizing rules).  We are moving to variables and more abstract math.  We need to understand these number relationships so we can apply them to problems where the numbers are not as concrete. 

Most students marked that they needed to improve their ability to prove and reason.  At conferences I often get a peek into the past.  "Last year they had to show their work in a specific way and would lose points if they didn't have it exactly as they should." "He always was marked down in his ability to explain."  I do think that different teachers have different expectations in terms of work and what constitutes as an explanation and that is a vertical discussion that districts should have.  Here is what I tried to focus on at conferences:
  • Showing work is part of what mathematicians do to prove their answer.  They need to show their thought process.  It also help us understand mistakes.  It is hard to reflect on an error if we cannot see where it happened.  
  • Explanations cannot just have the steps taken: first I multiplied then I added.  The explanation is a proof.  It must explain why those are the steps that make sense for this problem.
  • How do I help students to improve their explanations?
    • Models: Some students need models to figure out the answer.  Because models show the number relationships they can show what operations make sense for the problem.  
    • Vocabulary: A good explanation is not determined by its length.  We focus on vocabulary so we can be concise and clear in our explanations.  Many students understand the relationships in their head but struggle to put it into words because they don't have the words.  
    • Skeptic:  Your explanation is persuasive writing.  It has to convince someone who does not agree with you.  You aren't just explaining to your friend.  You said you multiply and you are convincing someone who thinks you should divide.  In class we share our reasoning and help each other find holes in our reasoning.

I did pull out some samples of student work at the conferences.  After going through the reflection sheet, we were now looking at these characteristics of mathematicians in their work.  It was a different conversation.  I was in my nerdy math zone but still able to explain things to parents in ways that they were able to connect with and understand.  

As always please feel free to continue the conversation in the comment section below. 
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.

Saturday, October 7, 2017

Making Claims - Play to Student Strengths

Any of my former students can tell you, I do not like it when students quote math rules to me.  If we are discussing strategies and a student simply tells me the steps to solve, there will be follow up questions.  I do not accept "because that's how you do this type of problem" or "because a teacher told me to" as an explanation of a strategy.

I still feel this way about math rules, however, reading Tracy Zager's book, Becoming the Math Teacher You Wish You'd Had, gave me a new perspective on how I handle these situations in my classroom.  Mathematicians look for patterns.  Mathematicians make generalizations to extend what they know to other problems.  The problem isn't the math rules that my students know, it is that they overgeneralize (use them when it is not appropriate) or have lost the mathematical reasoning that led to the generalization.

Tracy suggests making a "Claims Wall".  As students start to make these generalizations add them to the claims wall and continually revisit them to be sure they continue to work in other contexts and with other number sets.  A colleague recently told me, we need to play to their strengths.  If rules are what students know, then that is our entry point into the conversation.

With this in mind, I started my review of fraction multiplication and division by having students make claims.  I gave each team a post it to write their claims and made it clear they did not need to explain them at this time.  Then we looked at everyone's claims and made a combined list.  Here is what we had:

Most of these claims came from the post its.  If you look at the multiplying fractions there is a claim in blue.  This one came up after we wrote about fractions less than one.  One boy started making a claim and I knew it was going to be good so I started writing as he was talking.  "You can't multiply with mixed numbers." He paused, thinking he was done, then added the word "easily".  Then he backtracked and added "because I guess you could".  At this point he realized that I was writing what he was saying.  He adjusted his final claim by saying "You should use fractions greater than 1".  I love that the entire thought process is written on our claims poster because now we can analyze not only that we can multiply different ways but we are going to need to have conversations about efficiency.
The other claim that was not on a post it from students is the last one on the division poster.  It is one I added.  After we had the claim that you can use multiplication, I put forward the claim that you can use division.  I asked the students if they thought that was true.  They all thought that seemed like a reasonable claim, but with only a minute or so to think about it none of them felt they knew how to use division to divide fractions.  I can't wait for that conversation!

Starting the unit this way was a big risk for me.  I started class knowing we would do claims and had no idea of where it would go from there.  I did not know what I was going to need to have prepared for our exploration of these claims (because I didn't know what claims we would have).  I had a few things on hand  based on claims I thought would come up.  It was a bit of an unsettling feeling.  I decided to analyze the claims in an order I was comfortable with, not the order they are on the sheet. That allowed me to plan our approach and prepare more for certain claims.
In case you don't read beyond this I want you to know this risk was definitely worth the pay off for me.  Because we were starting with what students knew they all had access to the activities.  Then as we explored, they were able to self assess and realize what they didn't know about the topic.  The learning about fractions became authentic (without made up "real-world" problems to explore).

Here is the breakdown of what we did and what I plan to do:

  • Multiply across: I had the students go to the white boards (I have some new non-permanent vertical surfaces in my room).  They needed to come up with mathematical reasoning why this claim works. Many groups drew unit rectangles and used an area model. One group used a coordinate grid (still an area model).  Even though they knew the visual representation, through discussion we realized they didn't know how that proved their claim.  They just knew that they had used it before in math class.  (This is an issue I have with visual representations.  We cannot just teach them to kids or it is just more math to memorize.  Students need to be coming up with authentic representations and making sense of the math.) 
  • Cross canceling: We used this worksheet to analyze why this strategy works.  I have used it in the past and had it prepared in hopes that there would be a claim about this strategy.  This strategy helps as we work with proportions throughout the year and it's a great discussion about the commutative property.  
  • Multiplying mixed numbers: I sent students back to the whiteboards to show how to multiply without converting to fractions greater than 1.  I want to pull some different problems to focus on efficiency with fractions greater than one compared to keeping them mixed numbers. We didn't get a chance to discuss yet, but here are some pics of what they created.  I am very excited for these conversations (Spoiler Alert: when we talk distributive property we will also discuss their love for the order of operations and how it doesn't need to be followed if you have mathematical reasoning.)

  • Multiplying by numbers less than 1: For this we will play the role of skeptic and see if we can come up with counterexamples.  Then try to prove it will work for every case without having to list every problem that exists.  
Division of Fractions:
I will try to add to this blog next week.  Last year I took a course through the US Math Recovery Council that deepened my understanding of fraction division.  We used this book:
I want to revisit the chapter on fraction division to prepare for analyzing our division claims. It is a fabulous book and I highly recommend it for all ages of fraction work.

Here are my thoughts after this class activity.  
In class we give students tasks to allow them to explore a concept and build their mathematical reasoning behind it.  Then we pull together everything that has been discovered and come to these generalizations.  Some students get to these math rules because they work through the mathematical reasoning and it leads them there.  Others understand the mathematical reasoning but do not connect it with the rule that the class has determined.  (I have many students who feel they struggle with math but actually have great math reasoning.)  Others never understood either piece but continue to work to memorize the rules because the class has moved on and they need to be able to solve the problems.  In the end most students know the rules. They do not care about the mathematical reasoning because they can answer the problems.  How do we make them care about the mathematical reasoning when they can successfully solve problems?  Does it matter?  (I would argue yes.)  Everything we do in my class is about mathematical reasoning.  Explain.  Justify.  Prove.  Always, Sometimes, Never. Notice/Wonder. Question.  The focus is never on the answer.  But if the answer is what the student can do, I do not want to minimize that.  For them to want to know the mathematical reasoning I need to ask them to do things that require more than just calculation. If I can get students to continue to question, then they will not be satisfied with simply knowing an answer or a rule.  

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.

Wednesday, September 13, 2017

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.

Monday, September 11, 2017

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.

Sunday, September 10, 2017

A Line Up of Warm Ups

It seems like there are quite a few of you who are coming along with me on my journey to help students find their inner mathematician.
If you are just joining, you will want to check out Tracy Zager's book Becoming the Math Teacher You Wish You'd Had.

I am really just lesson planning, mapping out my week, and trying to pull all the thoughts in my head into an actual plan.  Since there people who seem interested, I decided to use this blog post to do it so others have access to it.  This is not the full lesson plan, just the highlights and mostly the warm-ups I will use to get us in that mathematician state of mind.


Tuesday: Estimation

  • Let's start talking about our intuition.  Since we are working on class norms I think this is a great activity to have some fun with math and launch into a few expectations about respecting others.  I have been saving this tweet from Chrissy Newell since July because I loved it so much.  

 If you check the responses you can see Chrissy's response with actual number and background info.
  • I always do estimating on a number line and have a clothesline at the front of my room.  This first one will take longer as I set up those routines. will explain strategies for teaching estimating and clothesline in case you are not familiar with either of these.  


  • I noticed last week that when student did not get to my room before the bell that others (probably even their friends) started giving them a hard time.  "You're late." As if it were a competition and they just lost.  I want to make sure I address this because it was only the first week of school.  These students don't know each other well.  They are late because they are trying to figure out where everything is.  If students are being competitive about arrival times it does not bode well for a collaborative environment.  
Thursday: Notice/Wonder
  • In the first day of class we did an activity where students had to define and describe math.  So many said numbers.  (I created word clouds like in Tracy's book.  If you want me to blog about this activity just let me know.)  Also, when they were creating goals I realized that many of my students interpret questioning as asking for help (which it is, but it's so much more).  I am hoping to address both of these things with this notice/wonder.  What do you notice?  What do you wonder? 

Friday: Rise to the Challenge, Take Risks, and Check up on class norms
  • I just think this one is fun.  Math is about noticing patterns.  Someone did that and create the graph in this tweet.  I will be removing numbers and having students try to figure out what information is being graphed.  I doubt that anyone will get it (I will update if I am wrong).  This will be a great opportunity to practice our classroom norms.  We will have to explain our reasoning and try to prove ourselves to others who may have a different opinion.  We can make sure we are able to do so in a respectful way.  

I am hoping that these first warm ups will set the tone for my class and help students see math as so much more than numbers and calculations, and hopefully have some fun too.  

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.

Thursday, September 7, 2017

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.

  • 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.

Thursday, August 31, 2017


Have you read Tracy Zager's book Becoming the Math Teacher You Wish You'd Had?
If not you definitely should.  After reading it this summer I created these posters for my room.  It is the key points from each chapter written in student friendly language.  My plan at this point is to hang them up.  That's really all I have at this point.  It's good to keep the bar low so I can actually follow through with this goal. 😜  I hope to refer to them throughout the year as it fits with conversations we have in class.  I will continually add to this blog as I do so that I can reflect on how to use these posters effectively in the future.  If you use them I would love to hear about it in the comment section.

UPDATE: I asked the students a few questions about what was on the posters.  I wrote about it here

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.