Category Archives: Teaching and Learning

Lesson Close

I have a hard time blogging during the school year, but I love to spend the summer reflecting on the past school year and thinking about my goals for the coming year. I have been watching #lessonclose and loving all of the ideas. And then I saw this one….

It all came together for me. I love the flow chart, I love the google sheets, I just love! Thank you!

I started using exit tickets more consistently in my seventh grade class this past school year. I saw was able to use the data to form flexible groups based on my plans for the day. I saw a lot of growth and could pinpoint which concepts students were struggling with. I sometimes had days that students weren’t ready for the exit ticket, and I had to quickly change my plans and save it for another time. I want to see some self-reflection about group work and so I created the rubric below.

I also wanted to find a better way to quickly assess different skills and I’ve been using so I created this…

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I did all this before I read @rawsonmath‘s post. Now, I’m seeing things in a new light. I think I can still use some of the tools I have created, but I’m seeing the organization of everything a little differently.

Thank you!

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Are stations the answer?

My time as a math coach taught me that stations are the bomb! I observed masterful first grade teachers regularly manage many different math levels without breaking a sweat. I experienced fourth grade teachers manage a room full of wiggly students during flex blocks and everyone was “getting what they needed.” Somewhere along the way we begin to believe that our students can sit for longer, focus for longer, and we don’t need to use stations to reach all learners.

A few years ago my district purchased the Connected Mathematics curriculum. I had used previous versions of the curriculum in the past and many of the problems were interesting and rich. The new version is clunky, disjointed, and difficult to use. Many of the problems are long, requiring several class sessions to complete with very little conceptual understanding. My students lose interest and are unmotivated by the time we finish a problem. The math they may have gleaned from the problem is lost in the euphoria they feel to be done.

As the year came to a close and I reflected on what went well and what didn’t, I was reminded of the times my students were engaged, challenged, and motivated to learn. The times this happened the most this year was when I was using stations in my classroom. They felt as though the tasks were interesting and at their level.

I began to think about my first unit in terms of stations using the math workshop model thinking about how I could use the best parts of the CMP curriculum. I came up with four stations including one where I will meet with students in small groups. These small group meetings will force both remediation and enrichment to happen within the station rotation.

Then I ran across this tweet….

After I read a little about responsive stations it seems like this is exactly what I need to add to make my stations even more productive. By first spending time to teach the necessary skills before starting a new unit I can really meet my students where they are. Looking at what students know and how to build off of that I can make all students feel successful.

So, my first unit is no longer my first unit. I need some station work before we start the first unit. The skills they really need to have solid are their multiplication and division relationships. I started planning my stations here. I think this will also be a great time to teach some of the classroom routines that need to get done at the beginning of the year so I’m only going to use three stations. More to come…

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Technology and Formative Assessment

Now that I’m back in the classroom I have access to some great technology. My district is piloting 1:1 iPads in one cluster and the rest of the clusters have an iPad cart. I’m in a cluster with the cart so we have to share our with three other teachers. Luckily, this hasn’t been too difficult. We setup a shared google calendar and whoever gets there first gets the cart.

Our students all have access to a google email and apps account so we have a lot of options. When I started out the year I tested out a few different formative assessment options. Using the cart in math class it is easy to just have students log in and practice. I love to use the iPads for formative assessment or as a combination formative assessment/connection builder.

Before we had technology for each student I used Mastery Connect. I used bubble sheets for each student and scored with an Ipevo document camera. I never gave more than 5 questions and used them to gauge how students were doing with new concepts. This gave me a snapshot of what each student understood within each standard. If my district bought into this system I think it would be amazing for assessing and keeping data. Alas, they didn’t.

When I started working with the iPads Mastery Connect was the first place I went. I knew that they bought Socrative and was excited to see how they would be integrated. Unfortunately, they didn’t really integrate the best part. I loved seeing the standards covered as we moved through the curriculum on Mastery Connect. There is no way to link Socrative questions to the standards, so I decided to keep looking. Socrative and Mastery Connect were a little clunky on the iPads and didn’t really give me an easy way to track student understanding.

Next, I tried Exit Ticket. If my students traveled with their iPads I think this would be the system for me. Students have access to their results from previous exit tickets. The teacher view can be switched to projector view so a whole group discussion can happen. Love it! Students can use the same account for multiple classes and store everything together in one place.

Then came Pear Deck

Pear Deck takes formative assessment to the next level. I was intrigued when I realized it was integrated into google. My students wouldn’t need another login. (They can barely remember one password.) Pear Deck takes power points to the next level. I have a projector view, teacher view, and student view. There are different slide types for students to interact with: free response-draw, free response-write, free response-number, draggable, and multiple choice. I have the ability to insert images into the slides for students to graph or write on. I can switch the answers on so others can see answers anonymously. We can have a conversation about incorrect answers. I can see in teacher view how each student answered each question. At the end of class I can save every student’s answer to look at later. The possibilities are endless.

Pear Deck has changed the way I teach. I can quickly see who understands and who is still confused.

I’m sharing the Pear Decks I’ve made so far. Feel free to try them out.

Pear Deck Folder

 I did pay for a premium subscription which does give me more access, but try it out anyway. You’ll love it!

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My colleagues and I decided to enrich our CMP 3 curriculum with one of Cathy Fosnot’s Math in Context Units, Best Buys, Ratios and Rates. It has been a rewarding experience and changed the way that I think about teaching.

One thing I have regained is the benefit of “struggle.” Students that are willing to struggle are able to gain more from a problem than students that insist on having an answer first. It is sometimes tempting to teach what students need to know before they have had this experience.

I have always enjoyed teaching this way, but time constraints and the need to finish the standards in a timely way often scare me into a different teaching style. I gave my students some extra time today. I wanted to finish the problem and share out by the end of the class period today, but students were engaged and learning. I let them go on with the problem for the whole period. One student looked at me so excited and said, “I’m so excited, it feels so good to struggle and then get it.” She had just discovered what I was trying to teach. I didn’t do any of the talking, she did all of it and she learned!

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Goal Setting

I’m really excited about mathematics education. I think the Common Core Standards came at the right time and the perfect storm is upon us. Jo Boaler is creating exciting content for teachers, parents and students at, Carol Dweck is putting in to words what we know is necessary to succeed, and at the same time there was enough of a shift in standards that teachers are now focusing on what students need to understand, not what they need to learn.

There are two goals I have for the upcoming school year based on all of this excitement. The first is to change my classroom culture to make mistakes more valuable. I am constantly telling students that mistakes are valuable, and they should be learning from them, but this is not evident in my classroom. I don’t show students how to learn from mistakes, and I don’t value mistakes in my grading practices.

The second goal is to come up with a more consistent classroom routine for number sense routines. There is so much good content right now that I think I could fill 180 days with number sense routines. I love Building Powerful Numeracy for Middle and High School Students by Pamela Weber Harris, Jessica Shumway’s Number Sense Routines, Number Talks by Sherry Parrish and Classroom Discussions by Chapin, O’Connor & Anderson. In addition there is so much good web content coming out of the math twitterblogosphere it’s hard to know where to start. I want to use Fawn’s math talks and visual patterns, Andrew’s Estimation 180, and Sadie’s Counting Circles.

A few months ago I attended a conference where all three authors of Classroom Discussions presented in separate workshops. Something I tool away from Nancy Anderson’s presentation really stuck with me. She said, “Math class is like a cooking show.” The more I think about this the more I love it. We should constantly be looking for the mistakes that students are making we should be bringing those to the surface and presenting them in an organized way. Don’t just hope that students choose to participate the way you want them to, have it all ready to go. While students are discussing a problem listen to the discussion and ask that student to share once you bring the class back together. Whether it is a common mistake, or a great idea the point is to make every minute in class count.

I’m not sure how to attain these goals, but I know I need to focus in on the important points. I have used Building Powerful Numeracy for Middle and High School Students and Classroom Discussions in my classroom before. I want to add in some of the other great content out there. I think I may have to rotate the number sense routines that I’m using in my classroom either weekly or daily based on what we’re doing? I’m open to suggestions.

I’m hoping that I can show the value of mistakes through some of these number sense routines and my new “cooking show” technique. One thing I’m still struggling with is how do you value mistakes while grading. If I’m encouraging students to make mistakes I need to have some way to record what mistakes they are making and what they are learning from them. Ideas?

I’m excited about a new school year. I think I’m going to grow a lot as a teacher.

Egyptian Fractions

I discovered Egyptian Fractions while studying number theory at PROMYS. It was one of the long term projects that teachers could research, but didn’t really interest me at the time. When I saw a group of teachers present their research at the end of the summer I saw the implications for teaching fraction operations.

Egyptians used only unit fractions. To represent \frac{3}{8} the Egyptians would have used the fraction \frac{1}{4} and \frac{1}{8}. Together these fractions are equivalent to the original total, but use only unit fractions to express the sum. Once students have some knowledge of fraction addition they can begin to attack these problems. My students had also covered multiplication of fractions which made the process easier. Here is the activity they did:

Before introducing this activity in class students watched a five minute video about Egyptian Fractions. If you have access to United Streaming from Discovery Education, this video was a great way to start students thinking about how the Egyptians work. Our students study Ancient Egypt this time of year in history as well so this is a great activity.

One process introduced in the video is the “loaf of bread method.” Many of my students understood this method and stuck with it through the activity. If an Egyptian wants to share 5 loaves of bread with eight people he will start by splitting 4 loaves into halves. Then the leftover loaf will get split into eighths. Each person will then get half a loaf and an eighth of a loaf.  \frac{5}{8}=\frac{1}{2}+\frac{1}{8}

In this activity students were forced to work together to discuss methods. One single method will not work for all of the fractions.  This is one of those problems that students cannot divide and conquer.

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Ordering Fractions

I know. How dull! If I find it boring students must find it even more boring. As I mentioned before I’m kind of a number theory nut. After learning about the Farey sequence during my summer in PROMYS I knew it would be a great addition to my fraction bag of tricks. I tried it and students did it, but they weren’t getting where I wanted them. I don’t think I was asking the right questions.

I recently found this activity and tried the Farey sequence again. Students loved it. They wanted to try to get the next sequence. I shared it with a colleague and she couldn’t believe one of her students asked if it would be all right to do F9.

Here is F4:

Fourth Farey Sequence


I love lessons like this one. There is so much mathematics packed into it. We have mixed ability classes so students that are struggling with multiplication are sitting next to students that can solve equations in there heads. This problem allows students to move through at their own pace. I want students to practice comparing fractions. The fact that they are discovering patterns and symmetry along the way is a bonus.

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I was really looking forward to this problem. I couldn’t wait to see what my classes would do with it and it did not disappoint. I mixed up my classes (assigned seats) and gave them very little direction. I wanted to see what kind of mathematical thinking they would do on their own.

The Counting Cogs is very specific and Nrich has even included step-by-step group directions so students can easily manage the problem in a group. Students need to discover which pairs of cogs will allow a colored tooth on one cog to go into every gap on the other cog. Students cut out the cogs, colored one tooth, and started spinning cogs. Nrich even has an interactive that helped me give some classes an idea of what they should be doing. Thank you Nrich for a great problem.

I was thrilled with the thinking. Some students quickly realized they needed some way to keep track of pairs and created great tables and charts. Some students wanted to immediately begin making conjectures. We talked about proving what we believe to be true, and they began to see that one example is not enough to be a proof. Some students even started creating other cogs than the ones supplied to see if their conjecture really worked. I heard a lot of conjectures! Tomorrow when we begin discussing this problem students will quickly discover the cogs that are relatively prime to each other work and the others do not. I did hear this conjecture today, but students just didn’t have enough time to really prove it.

This problem let students extend their thinking about factors, primes, and relatively prime numbers. They were actually applying these concepts rather than completing another worksheet. I really like the way multiple concepts are intertwined in this problem.

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I have resisted this acronym in the past. I just don’t think it is as useful as everyone else. I am constantly frustrated by students multiplying before dividing regardless of the problem. I gave in this year. I went for it. I circled the M and D. I put arrows between the M and D as well as the A and S.

I hoped… They have all done this before I told myself.

It didn’t matter. The same mistakes happened, they are still happening. Each time I find a student doing all multiplication before the division they look at me like I’m the one that’s confused when I try to correct them. When I show them the circled letters with the arrows between them they say, “Oh.”


Regardless, it was very clear today that we need some more practice with the order of operations. We have listened to the PEMDAS song, done a PEMDAS relay (scroll down), and today we did a Treasure Hunt (found this great idea here at mr barton maths).

To do the treasure hunt I cut out and laminated (not necessary, but now I have them for next year) each card. Students can start with any card do the problem, then they have to find their answer on another card and complete the problem on that card. This continues until they get back to where they started.

The treasure hunt definitely got everyone up out of their seats doing math. Everyone was focused doing twenty-four math problems on a Friday! Students discovered mistakes right away because they couldn’t find their answer on another card. This is an activity I’ll modify for other concepts throughout the year. I really liked the conversations that students were having. No one was giving answers, but helpful hints were happening all over the room.

Here’s the Treasure Hunt Template if anyone wants to change it for their own class.

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Long Division

Long division is always a struggle at the beginning of the year. Many students do not know the process and even more students do not understand what they are doing. I’m afraid that long division will join algebra soon and mathematics education will only require students to learn how to multiply.

Long division is a multi-step process that many students struggle with to the point of making up rote memorization mnemonics. Students that have been shown how to repeatedly subtract multiples of the divisor don’t understand how this connects to the more efficient method.

My sixth grade classes today looked at me stunned when I asked them to divide 32 into a ten digit dividend. Does it matter? Have they never been asked to do something that is not printed on a worksheet? Something that might be challenging?

Several students asked if they could just do short division. This is something that really bugs me! Am I the only one?

I think there is so much math in long division that short division loses. Remainders are so important. Which remainders are repeating? Can we have a remainder larger than our divisor? What are the possible remainders when we are dividing by 32? These are questions that sixth graders don’t know the answers to. These are questions that are important to their understanding of division.

When I divided 132 by 11 today students couldn’t tell me what it meant to say 11 goes into 13 once. Why didn’t I put the 1 over the two? Why don’t I just subtract the 11 from the 32? These students have all seen their fourth grade teacher repeatedly subtracting multiples of 11’s. What is missing? Is it repetition? How do we help students “really” understand what they are doing?

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