Thinking and Engaging…A Week of Math!

What would school be like if every so often (or always) we didn’t race in the Daily Academic Decathlon: teaching  reading, writing, math, social studies science, art, engineering, music (oops….), PE, and technology?

dan-obrien-and-dave-johnson-before-the-1992-olympic-gamesI grew up watching Dan and Dave, but I bet they didn’t train for every event all day every day.  I bet they focused and went deeper in their training.

Yes, yes, yes…I don’t believe those are all isolated subjects and most aren’t subjects, but we do this in school all the time. I have completely gotten pulled into this and because of trying to do it all, at times we are accomplishing nothing (ok, that’s not true, but sometimes it just doesn’t seem feasible to do it all and then we are always a dollar too short.)

What would happen if we took a week and focused on becoming a scientist, an author, an engineer, a mathematician? What would happen if we just dove into a content area, topic, current event and engaged our entire being in it? What if we became mathematicians for the week? What if we, “play(ed) the whole game”  as stated by David Perkins  in Making Learning Whole? What if students came to expect that, “a large part of their learning in the subject area involved acquiring the thinking abilities and processes of the discipline, not just learning about it for the test.” as stated by Ron Ritchart in Creating Cultures of Thinking? What if parents would focus on the learning and not just the test grade or the completion of the HW assignment?


Well, we went for it. We spent the last week as Mathematicians investigating math. Where did this idea stem from?

  1. I always feel crunched for time. I have 1 hour (ok…1 hour 15 mins…I never finish in the hour) that we must get our math in.  What would it be like if we could work on a problem until we were done instead of waiting to finish it the next day? What would school be like with no time limits? What if we took a break when we needed a break instead of when the bell rang? Who lives their life on a preplanned schedule?It is a little bizarre to be honest.
  2. I had just had a lot of conferences and was reflecting on the need for parents to experience the math that we do. I kept hearing, “I’m not good at math. My husband is the math person.” Ughhhhh….Makes me want to throw up.  Everything was about the answer.  I wanted them to stop and see that math is anything but boring and their children are completely engaged. So I thought, let’s open up the class for the week. Come in and do math with us any time you can.

So what did we do? We emerged ourselves in math. We played. We modeled mathematics. We looked for patterns. We read about Mathematicians. We explored the mathematics of ancient civilizations.

As Jo Boaler wrote in Chapter 4 of Mathematical Mindsets: Unleashing Students’ Potential Through Creative Math, Inspiring Messages and Innovative teaching,

When students see math as a broad landscape of unexplored puzzles in which they can wander around, asking questions and thinking about relationships, they understand that their role is thinking, sense making, and growing. When students see mathematics as a set of ideas and relationships and their role as one of thinking about the ideas, and making sense of them, they have a mathematical mindset.


And I think that this “mathematical mindset” is really a thinking mindset.  No matter what we are doing, the fun comes when we think, question, and reflect.

As always, there were things that I loved and things I wish I had done better.

What I loved….

Each day we had a Big Idea and Details to help us focus


Screen Shot 2015-12-22 at 10.07.41 AMBig Idea:  Mathematicians create viable arguments and critique the reasoning of others.


  1. Use examples to prove a claim is true and non examples to prove a claim is false.
  2. Explain with words and evidence why a claim is true or untrue.
  3. Question others to find holes in their arguments or to better understand their argument.
  4. Use drawings, tools, and representations to support a claim.

Another day we focused on…

Screen Shot 2015-12-22 at 10.07.41 AMBig Idea: Mathematicians look for patterns and relationships to make generalizations they can use.


  1. Look for things that are repeated, things that are the same, things that are related.
  2. Question…How is what I am working on similar to….?
  3. Does what I am working on remind me of something else I have done? What is the relationship?
  4. Look for smaller parts that compose the larger problem. Are there patterns or relationships in the smaller parts?

We truly engaged in the Standards for Mathematical Practices and made time to think. As I look at this list from Ron Ritchart, I do think we have a classroom culture of thinking.  The biggest gift that this week brought us was the gift of time.Screen Shot 2015-12-22 at 9.46.22 AM.png


My other love was having parents join us.

father-son By the way, why is it that I could not find a single positive clip art of a parent and child doing math together. Reading? It’s all over the place. Math? It is always an image of frustration. Ughhh…

“Wow, I am having fun. I wish math was like this when I was a child.” Pretty much, what this mom was saying, was that she wished she was allowed to think and make sense of math.

After doing Andrew Stadel’s Post-It Three Act Task–act-1, a parent responded,”That was great. I went first to thinking I need to find the number of post-its and what is the easiest way for me to do this. Then I looked over at my son who was finding the entire surface area and then dividing by the post-its surface area. At first I thought, that is so inefficient. We had a great conversation at our table. Then I realized. Actually, he would have a better model. If the size of the post-it changed, he would easily be able to adapt for that. I on the other hand would have to recalculate everything.”

How awesome for students to learn with their parents?

After doing a visual pattern (from Mathematical Mindset), a Physicist (parent) said, “Wow, this is what I do all day long. We look for patterns and make generalizations.” Ahhh…thinking. Then he continued, “I need more time to wrap my head around this one though.”

Screen Shot 2015-12-22 at 10.26.04 AM

Another love…allowing our next learning be driven by student inquiry. 

During the first week of school, we worked on the Four Fours: Use four 4s to make the numbers 1-20.  They loved this, but were having trouble finding expressions equivalent to some numbers so I introduced to them factorials. Minds blown!

A couple of weeks ago, we were working on a visual pattern Screen Shot 2015-12-21 at 7.06.26 PM from (amazing amazing blog of patterns that Fawn Nguyen has compiled) and one student said,

“I wonder if we could use a factorial. I see 3, 2, 1 and then 1, 2, 3 with the 1 used twice.”

Another student responded, “No, it is 3+2+1 not 3 x 2 x 1 so that won’t work.”

“Oh, right”

“Ms. S, is there something like a factorial but for adding numbers in a row?”

Oh boy, I thought. “Yes, there is. We will definitely explore that. But, we need to do that another time.”

Well, our other time came during the week of math.  And, NRich came to the rescue with two great tasks to explore (Mystic Circles and Picturing Triangle Numbers ) and then an article the read.

We noticed and wondered for Mystic Circles, noticing the adding of sequential numbers. Screen Shot 2015-12-22 at 10.28.12 AM.pngand then we looked for relationships in Picturing Triangle Numbers

Screen Shot 2015-12-22 at 10.34.25 AM

After, we went back to look at the two strategies for solving the Mystic Rose Screen Shot 2015-12-22 at 10.47.53 AM

and students developed arguments for each strategy.

What I wish I had done better at….

Preparing students for having their parents in the class.

While we as a class have developed a classroom culture where mistakes are important and our goal is to learn and not just perform, that does not mean they have that culture with their parents. I had one child breakdown due to wanting to “perform” for his parent. He has never broken down in class, but I see this desire to “perform” and the pressure (self imposed?) really got to him.  This was a good wake up for me, as while we might be on the way to creating this culture, it is going to take a lot more work to bring it into every home.

I also wish more parents could have joined us. I know there was some fear, as one parent shared, “6th grade math is intimidating”. I just wish more could have come to see how open it really is. I need to figure out how to share this with more parents.

Connections, Connections, Connections…

We didn’t do a bad job on this. The connections we made throughout the week were incredible. We were connecting ratios to graphical representations and then questioning/wondering and analyzing if the visual patterns we have been working on are proportional. Do all graphs of equal ratios create a line through the origin?  We were connecting our work with visual patterns, to the work we did with triangle numbers.  We were connecting and comparing the many different strategies students used when solving Sugar Packets (which I adapted and probably made a bit too hard. Though, the students persevered and definitely engaged in some productive struggle.) and the Leaky Faucet. We were connecting the representations to equations to words. We connected and compared our base ten system and place value system to the Babylonian’s base 60 system and the Egyptians base ten, but lack of place value.We were extending ideas and connecting to the math practices.  We were deepening our learning by creating arguments for a claim being Sometimes, Always or Never.

Screen Shot 2015-12-22 at 11.01.58 AM

But, it was mostly oral.  I wish I had recorded the connections so that they were more explicit, and had the children record the connections they were making. I wonder if when we come back to class, we can look back on the various problems and explorations we did and make those connections explicit. They were explicit in my mind when I chose each problem, but I wonder how explicit they are in every students’ mind. I think this will make the week even more powerful than it was.


 How can I create this type of week in my class every week?  Can I extend math on some days to allow for more processing/thinking time? Can I extend our time to write, so we aren’t constantly starting over? Can I really help focus our learning more on one topic, so we can go deeper and allow connections to be made? What do I need to add to make all the connections and thinking even more visible so that students can transfer to other parts of their lives?

It sure is a good thing I have 2 weeks to process this and develop a plan. Time to reflect is good. I will  make it happen.



My Love for Students’ Understanding of Math Grows Deeper and Deeper

I am so grateful that I get to begin each day with math! Every day starts with playing! It is incredible.

We explore and talk every morning while making connections, looking for patterns and engaging in mathematical arguments in the form of a Number Talk; a Mathematical Conversation from Intentional Talk; a Sometimes, Always, Never discussion; a True or False Reasoning activity; choral counting; or estimation activity. My biggest problem is keeping track of time.

This past week, it was time to explore ratios. I looked through our CPM Book, the 6th grade Standards, Jason Zimba’s coherence map, and didn’t feel like the Text Book was going to meet our needs exactly the way it was set up.  So, I started to think about what my goals were and what my students need to understand.

Throughout parent conferences, I kept hearing myself speak to parents about the importance of connecting mathematical concepts. Too many students were seeing math concepts as isolated and I really want them to see mathematical relationships and see all the connections.   I realized that because some hadn’t been making these connections for years, my most important job is to create opportunities and engage them in conversation so they can make these connections.

I really wanted students to understand the language of ratios, understand the different types of ratios: part to part and part to whole, and see ratios as a relationship while connecting what they knew about fractions.

I remembered a Math Assessment Project Lesson that I thought might be perfect.

  1. It allowed for multiple entry points, allowing all students to have access to it.
  2. It would allow us to make connections to their understanding of fractions, but also allow students that need to develop more understanding with fractions that opportunity as well.
  3. It would allow us to investigate what a ratio is and come up with various notations and the language of ratios without being directly told them.
  4. From this task, I would learn what my students know and where we would go next.

As I mentioned earlier, we start each day my playing with numbers. That morning we had started with an Always, Sometimes, Never for these three statements: 1/4 = 25%, 7/12>8/14, 3<4x. We had recently been comparing fractions to percents, I wanted them to make note of when comparing fractions to use a benchmark number such as 1/2, and since we have also been working with variables, I wanted them to think about what would happen if you used a fraction versus a whole number in place of the variable.  I loved that we got on the topic of the importance of the whole. “Ms. S, 1/4 = 25% only if they are both out of the same whole. 1/4 of 8 is not equal to 25% of 4.” Love!

I wanted to give them something that they might be able to connect to  and the comparison of the two fractions was perfect for that.

For the task of Fizzy Orange Drink, Screen Shot 2015-12-06 at 2.08.00 PM.pngwe first started out with some Noticing and Wondering. This I have come to believe is one of the best habits my students can develop. It creates engagement and as Jo Boaler states in Mathematical Mindsets, “the desire to understand it and to think about it.”

I then gave them the information that they had requested and we were off.Screen Shot 2015-12-06 at 2.08.27 PM

We came up with these three possibilities of most orangey to least orangey. Then students justified and explained, caught mistakes, and connected to the second problem in the warm up.”Remember in the warm up we said 7/12 was 1/12 away from 1/2 and 8/14 was 1/14 away from 1/2 and because there were the same number of parts we can compare them easily. We can do the same thing here.” Yes! The student’s noticing helped other students who naturally weren’t making the connections. We agreed on the Middle row, comparing the fractions or part to whole ratios using a common numerator or unit rate.Screen Shot 2015-12-06 at 2.20.38 PM

It was then time to stop, so for HW they were asked to respond on our classroom forum and reflect about the day’s work.Screen Shot 2015-12-06 at 2.24.18 PM.png

The next day they got into the big task of sorting and matching. But first they made some Noticings and Wonderings. They noticed that some cards were related, asked about the blank cards and decided themselves what they thought the task should be. It was awesome!Screen Shot 2015-12-06 at 2.08.50 PM

I told them, as they sort and order, to be thinking about what they think a ratio might be.IMG_1778


Then it was time to Troubleshoot and Revise. So for HW, the students had to analyze two strategies that had mistakes. Then the next day in class  we held a Trouble Shoot and Revise discussion.  This was so important because we were able to surface some misconceptions and then students had an opportunity to revise their sorts.

Finally, it was time to share and compare strategies. One or two people from each group was to stay with their poster and explain their process and reasoning. The other members walked around and were to ask questions and critique their peers’ reasoning.  We then discussed mistakes and corrections, analyzed the dictionary definition of a ratio and related it to what we had just done and created a list of notations and language of ratios.

IMG_1800They then had a follow up task of mixing drinks but with 3 ingredients. My job now is to analyze their strategies and depending on how they did, we will look at my 3 favorite mistakes and Trouble Shoot and Revise or Compare and Connect Strategies.  So what comes after that? Good question.

That is what I am trying to figure out now.  In our CPM book, they are enlarging and reducing pictures and shapes. That is really a 7th grade standard, but focusing on it as scaling up or down it could be a great connection to what they were doing in 5th grade. So, I think I will have them engage in that, but with the focus of connecting scaling up or down to the work we just did. I like that it is a completely different context and think that could lead into some great conversations and big understandings.

The other thing I want to do is introduce some representations: double number lines, ratio tables, and graphs to help them see the multiplicative reasoning. I’m thinking of possibly showing them a problem solved with the three representations to see if they can connect each representation and determine how they work instead of me directly teaching. This will help place more of the cognitive load on them and require them to really engage actively.

Some are also still struggling with the notion that a fraction is division. So, pulling in some equal sharing problems with proportional reasoning is also going to be helpful. I just need to determine which I do first. Ideas?

Their minds were firing all week and they are starting to make some good sense of the math. I think we have a good start. There is a curiosity and a desire to want to make sense of it all now.



Goal Setting, Thinking, and Understanding

It has been some time since I last posted, but that does not mean there has not been a shortage of things to think about. While preparing for conferences, my inner conflict about grades versus feedback was in full force. The tension was great.

rope-tensionI had to assign a numeric score to report cards, but it often felt conflicting to our classroom focus on learning. Then came conferences and instead of always just focusing on students’ thinking, learning, how they were improving, and focusing on the new goals they created, I could feel our conversation more focused on the evaluation of skills.  It is really fascinating how difficult it is to change our culture of school from performance to true learning.

One of the things that helped us stay on track though, was that each child reflected on their learning over the trimester and we then created learning contracts. We focused our goals on reading, writing, math, and Listening/Speaking and our Norms. Each student filled out a Learning Contract Template.

Screen Shot 2015-12-06 at 10.20.13 AM.png

This was definitely not easy for all students to fill out, but the process of doing it was extremely valuable.

Some examples…Screen Shot 2015-12-06 at 10.31.20 AM.png

Student BScreen Shot 2015-12-06 at 10.28.12 AM.png

Student CScreen Shot 2015-12-06 at 10.27.15 AM.png

Student DScreen Shot 2015-12-06 at 10.29.03 AM.png It was clear who were independent learners and had internalized how to learn and which students still need support in thinking routines that can help them to learn and understand.

I am trying to figure out how to help them understand what actions they can take to meet their goals. That was really hard for a lot of students, which is really the most important component. What can I do to understand and meet my goal? It will be interesting when we revisit them, to see if they can develop any more specific actions to help them. I think I need to really make their thinking explicit and visible as they are doing it. I need to take the time to allow them to reflect and process.

Then came Thanksgiving break, I read Jo Boaler’s new book, Mathematical Mindsets: Unleashing Students’ Potential through Creative Math, Inspiring Messages and Innovative Teaching, and it was such a great time for me to read it.


It was very similar to her previous books, her articles and website, but no matter what, the message is so important and great to keep repeating.  This quote really hit me.

“Successful math users have an approach to math, as well as mathematical understanding, that sets them apart from less successful users. They approach math with the desire to understand it and to think about it, and with the confidence that they can make sense of it. Successful math users search for patterns and relationships and think about connections”

Excerpt From: Boaler, Jo. “Mathematical Mindsets.” iBooks.

As I mentioned, during my conferences I was getting pulled into talking with parents about math as discrete skills students needed, yet my goal is really the quote above. My goal is for all students to approach math and all subjects with the “desire to understand it”.  This is not an easy task. For some, the change is happening, yet for others they have spent 6 years in school passively doing school and to change that mindset has proven to be much more difficult than I was hoping. But I am not giving up.  See the next post.

This past week, I finally made time for the students to reflect on their learning  3 days.. I ran out of time not the others. I am so bad with time!!!! My goal is to get to 5, but 3 was a good start.

Each day, they were to reflect on our Norms by posting to our class LMS: Schoology. Screen Shot 2015-12-06 at 10.48.53 AM.png(adapted from Jo Boaler’s Math Norms…I added in what each might look like in different subjects as well since the students are in a multi-subject classroom.)


What is great, is that students are able to read each other’s reflections and their thinking is visible for all to learn from. Below are some of their reflections.

“I liked how ______ explained himself in math and that made me think a lot harder and able to figure out the problem.” -Student A

“Today I questioned my teammates work in math and I overlooked it, got into a conversation about what was right, and then figured out the right answer. I appreciate _____ for really trying hard in math and helping me and _____ find the right answer.” -Student B

“Today I thought about if it was always, sometimes, or never. On one question I said it was sometimes and it was always, so I learned from that and got the next one right. ______ made it make more sense when he proved his answer and that helped my.” -Student C

“Today during read aloud, I asked questions about words that I wasn’t familiar with in our story and learned lots of new things about Egypt.” -Student D

“I made a connection from the example paragraph to my Mesopotamia paragraph that helped me understand how to transition to different details.” -Student E

“The trait I engaged in was being perseverant. I was being perseverant because I kept going on finding a way to put a key word in my questions.”- Student F

“…But now I learned from my mistake and my brain just grew. : )” -Student G

I really think being consistent (consistency is so hard for me) with this reflection time might help some of my more dependent learners become more independent.

And then I started to read Ron Ritchart’s book, Creating Cultures of Thinking: The 8 Forces We Must Master to Truly Transform Our Schools.

ritchhart-cultures-239x300Wow!, I have so much to process and discuss with others about this one. Where do I start?

Ahhh, thinking…the culture of thinking. It really is the only way that students are going to succeed in life and I feel like it is my most important job to facilitate learning opportunities for them to engage in thinking.

More to come… reflecting on engaging students in the whole subject, thinking like a disciplinarian, and making connections.