Talking Circles in Math Class

Earlier this week I went to a professional development day where I had the pleasure of listening to Janet Milanowski present on Talking Circles. Prior to this professional development I had experiences and knowledge about counting circles but I had never heard of a talking circle before. I was intrigued with these talking circles because I could tell instantly that they would be a great way to build community within a classroom. During a break that day I pulled out my November issue of Mathematics Teacher (this month's issue is about creating classroom communities) and I thumbed through to the first feature article. To my delight it was an article written by Marcus Hung on, you guessed it, talking circles! So I figured that was a sign from the math deities to write a post about talking circles, so here we go!

What is a Talking Circle?

Well, it is actually kind of what it sounds like...a circle where people talk! More specifically though it is where students in a classroom pull their chairs (or desks) around in a physical circle and everyone takes turns responding to a prompt in a structured way. These prompts can be anything from community building questions, to conflict resolution, and even content-based questions. Talking circles can last anywhere from ten minutes to a whole class period depending on the goal of the circle. There is normally a facilitator or "keeper" or the circle, their job is to lead the meeting, uphold the integrity of the circle, create a welcoming and risk-free environment, and most importantly participate in the circle! The keeper is usually a teacher, but once the routines and expectations of the circle have been well-established a student may also be chosen to be the keeper of the circle. 

In addition to the keeper there is another important piece to a successful talking circle, and that is the talking piece. The talking piece is what is used to help keep the flow of the conversation positive and productive. Only the person with the talking piece is allowed to talk in the circle, which helps to prevent students from being interrupted. The talking piece moves in one direction around the circle until everyone has had a chance to share about the given prompt(s). (Some students may want to pass and that is okay! But they should be given the opportunity to share at a later time in the circle if they would like) Some circles have the talking piece go around twice before the circle is completed to allow students to respond to what another student said or to add a new idea or thought to the discussion. The talking piece prevents one-on-one debates within the circle, and also encourages shared responsibility for the discussion. 

So what is the Format for a Talking Circle? 

Talking circles normally begin with an opening. The purpose of the opening is to create a sacred space where everyone will come together, and also to set a positive tone to transition into the circle process. After the opening is completed the guidelines and values of the talking circle are explained. These guidelines should be something that are decided upon as a class and something that every participant is held accountable to. Once the guidelines have been reviewed then it is important to explain or review the purpose of the talking piece. Before the discussion begins it is important to check-in with the students to see how they are feeling, physically, mentally, and emotionally, at the moment. An example of this might be to have the students describe how they are feeling by explaining what weather pattern would describe their mood today. Now that all of the guidelines have been set, the talking piece explained, and you have checked-in with your students the discussion can begin!

The discussion is the "meat and potatoes" of the talking circle. Like I stated previously, these prompts can be used for community building or to talk about content. One of the prompts that we discussed in my professional development meeting what "What was your favorite Halloween candy as a child?", and then later on we had a prompt that was to explain "Why you decided you wanted to be a teacher.". The responses to the prompts should be consolidated, to prevent rambling and also to give everyone an equal chance to respond.

The talking circle is concluded with a check out. The check out is an opportunity for students to express how they are feeling at the moment, or explain what they are taking away from the circle. This gives time for reflection which helps to consolidate what was just discussed.

How can Talking Circles be Content-Based?

Our professional development meeting mainly focused on talking circles as a way to build classroom community and culture, but what interested me the most was the ability to bring math into the picture. In Marcus Hung's article in Mathematics Teacher he talks about how he uses talking circles for equitable student participation in his math class.  He observed that during whole-class discussions the majority of students who were responding were students who were mathematically confident, and they were the same three or four students every time. Even in the small group discussions in his classroom Hung observed that while there was more discussion happening it was still done mainly by students who were mathematically confident. This did not give the equitable participation that he desired in his classroom so Hung made the choice to use talking circles (which he had already implemented for community building) to discuss mathematical content. Hung used prompts that are both open-ended and direct to get his students to open up and share their thinking about the topics they are learning about in class.

I imagine using talking circle to introduce new topics and start conversations about students' prior knowledge. Talking circles would could also be a great way to review material before an assessment by going around the circle and taking questions or having students explain a concept that their are still unclear about.

(Figure taken from Marcus Hung's article "Talking Circles Promote Equitable Discourse" in Mathematics Teacher)

Pros and Cons of Talking Circles

As mentioned before the two biggest benefits of talking circles are that they create community in the classroom and also that they allow for equitable discussion and participation (and get the quiet students in class a voice). When students (and teachers) participate in talking circles they get to know each other and build relationships, which will create a welcoming environment in the class, and also potentially increase participation outside of talking circles as well. Talking circles also promote problem solving and critical thinking skills while students listen and respond to their peers' thinking. 

One of the biggest downfalls with talking circles is that there is most likely going to be a lot of repetition with responses, and students might just share what another student already shared as a way to avoid answering the prompt. Another downfall with talking circles when used for community building is that there are some questions that are very high risk. For example something as simple as "How was your night/weekend?" can cause some students to shut down if there is something they are struggling with going on in their lives. It is important to know your students when you are planning the questions to ask in a talking circle. 

I think that talking circles are a great way to add variety in a classroom. Talking circles are not meant to replace whole-class or small group discussion but to present a different form of discussion. While I have not personally planned a talking circle yet I am very excited by the idea of them! Doing number talks in a talking circle format would be a great way to increase students number fluency and get them comfortable with explaining their thought processes. I look forward to reading more about talking circles and updating this post once I have done one myself!

Here is a the link to Marcus Hung's article in Mathematics Teacher if you are interested in reading more about how he using talking circles in his math class! 

  

The Calculus of Friendship: A Book Review

Recently I have had the pleasure of reading a great book by Steven Strogatz called The Calculus of Friendship. In his book Strogatz writes about the letters that he sends back and forth with his high school calculus teacher, Mr. Joffary. The book contains over 30 years of correspondence, however the interesting twist is that the letters are not personal in nature, they are about mathematics. Strogatz, after high school and college, goes on to teach at MIT and also Cambridge, and during this time Mr. Joffary sends Steven math problems that his students present him. Strogatz then in turn sends Mr. Joffary proofs to the questions he asks and poses new questions for his students. Throughout their years of correspondence they touch topics such as infinity and limits, chaos, bifurcation, Hero's Formula, and relativity. The book contains a mixture of letters, proofs, and explanations about the topics discussed, and as each page turns the reader is able to see the relationship between teacher and student blossom.

Now I won't lie, a lot of the proofs presented in this book were beyond my level on mathematics, but the reason I kept turning the page was for the friendship that was growing between Steven and Mr. Joffary (or Joff as he would later call him). Both Steven and Joff go through a lot of ups and downs in life, however these are never present in the letters, only math. I got the feeling though that both Steven and Mr. Joffary knew that they had a lifelong friend, even if they never discussed it.

I think that the reason that I loved this book so much is because Mr. Joffary reminded me of my grandfather. My grandfather, Brian, taught math at Muskegon Community College for over 30 years and he is just as kind, thoughtful, and smart as Mr. Joffary. Even the jokes that Mr. Joffary includes in his letters, and his mannerisms reminded me of my grandfather and I couldn't help but smile. This was a book that not only makes you think, but also warms your heart. It is about the bond that is formed between a teacher and a student. and how sometimes the student becomes the teacher. If you are looking for a story about friendship and integrals, I suggest you pick up a copy of The Calculus of Friendship. I will leave you with my favorite quote from the book, and a video of Steven Strogatz talking about his friendship and his letters to Mr. Joffary, you can clearly see how much this relationship meant to him.

"This was a different kind of infinity than Joff had ever encountered before, a higher order of infinity. While I explained it to him, the sun began to set. We sat together on the beach and solved the problem, surrounded by the waves in Long Island Sound."

How to Create Better Classroom Discussions

One of the struggles I have been dealing with this semester while I do my teacher assisting is trying to create better classroom discussions. I am doing my teacher assisting at Grandville Middle School in two Math 8 (pre-algebra) and two algebra classes. The atmosphere of the classroom is very collaborative (the students sit in groups of four and spend a lot of time doing small group work) but I want to become better at facilitating rich classroom discussions. I am currently reading NCTM's Principles to Actions and I just finished a section on facilitating meaningful mathematical discourse and in that section it outlines five practices for using student-responses in whole class discussions. I also found an article by Margaret S. Smith, Elizabeth K. Hughes, Randi A. Engle, and Mary Kay Stein called "Orchestrating Discussions" where they propose the same five practices, called The Five Practices Model. The five practices are:

1. Anticipating student responses prior to the lesson
2. Monitoring students' work on and engagement with the tasks
3. Selecting particular students to present their mathematical work
4. Sequencing students' responses in a specific order for discussion
5. Connecting different students' responses and connecting the responses to key mathematical ideas

In this post I want to take these five practices and apply them to a problem introducing the concept of the variable that my Math 8 students will be working on next week. The problem comes from Ana Stephens's article called "Developing Students' Understanding of Variable" The rest of this post is a reflection on how I would apply the five practices to Stephen's problem.

The Problem:

Ricardo has 8 pet mice. He keeps them in two cages that are connected so that the mice can go back and forth between the cages. One of the cages is blue and the other is green. Show all the ways that 8 mice can be in two cages. 

Anticipating:

In my opinion the point of anticipating is so that you are prepared for what your students are about to do. This can range from anticipating student mistakes or misconceptions, to anticipating different strategies that students will use to solve the problem. For this activity I had two groups of anticipations, one was related directly to the problem the other were anticipations about my students' understanding of variable in general. 

Anticipations about Variables:

• I anticipate that some students will view variables as labels rather than a symbol that represents a quantity. So in this problem some students might use b and g for variables just because they are both the first letters of the words blue and green. 
• I anticipate that some students will not understand that variables are changing quantities and they do not have only one specific value. In the context of this problem students might not realize that b (the number of mice in the blue cage) can range from 0-8 at any given time.   
• I anticipate that some students will believe that two different variables must have two different values. So in this problem students might think that b (the number of mice in the blue cage) and g (the number of mice in the green cage) can never have the same value.

Anticipations about Students' Strategies:

• I anticipate that students likely begin this problem by drawing a picture or creating a table. Some students may also use a graph or an equation but I don't expect either of this to be their first step. 
• I anticipate that some students might not consider having zero in the blue or green cage.

Monitoring:

During this monitoring time it is my job to be walking around and trying to understand my students' thinking, I will most likely do this in the form of questioning. In the book Principles to Actions there are four major questions types. I will list those types and they provide a couple examples of questions I might ask students during this activity.

Gathering Information: students recall facts, definitions, or procedures. 

Examples: "So if there are ___ mice in the blue cage, how many are in the green cage?", "How many mice will there always be?", "Is it possible to have zero mice in a cage?"

Probing Thinking: students explain, elaborate, or clarify their thinking, including articulating the steps in solution methods or the completion of a task.

Examples: "When you drew this first picture, what does this situation represent?", "What made you decide to create a table for this problem?", "I noticed you wrote down b=3, what do you mean by that?"

Making the Mathematics Visible: students discuss mathematical structures and make connections among mathematical ideas and relationships.

Examples: "How does the table you made relate to how many mice are in each cage?", "Can you use this information you generated in this picture to create a graph or equation?" 

Encouraging Reflection and Justification: students reveal deeper understaning of their reasoning and actions, including making an argument for the validy of their work.

Examples: "So I see you wrote an equation b+g=8, can b ever be equal to g. Why/why not?", "How can you check your equation to make sure it is true?"

Selecting: 

What students' work you select to show all depends on what your mathematical goals, or learning targets are for the activity. Your goal is what is going to drive your selection process and decide whether you will select a misconception to show, or maybe a unique way of solving the problem. For this activity my mathematical goal is to show how variables are changing quantities, so therefore I would select students' work that represents this idea. I would want to show a student who represented this problem with a graph, a student who did it using a picture, a student who used a table, and finally a student who wrote an equation or expression. 

Sequencing:

Now that I have decided what student work I want to show, I have to decide what order I want to present it in. Again, the order that I choose should reflect my mathematical goal(s) of the lesson. Sometimes you might want to start with a common misconception and explain why it does not work that way, other times you might want to start with the strategy that the majority of your students used and then go on to the more unique strategies. For this activity I want to sequence the student responses like building blocks, so I would start with the picture, then show the table, then the graph, and finally the equation. This way I am working from more concrete thinking to abstract thinking. In the article "Orchestrating Discussion" they give a great suggestion of creating a table that can organize your sequencing. I have taken that idea and created a table that I would use while I am monitoring this activity. 

Connecting:

This is arguably the most important practice of the five practices. This is where the students consolidate what they just did and they can see the "big picture" of what it is important. This normally will take the form of a whole class discussion after presenting the sequenced answers. As I was thinking about this activity these are some of the discussion points I would want to present to the class:

"So what I am noticing from a lot of your work is that the number of mice in both cages isn't always the same. How can we represent the number of mice in each cage in a general way?"

" So if we let b equal the number of mice in the blue cage and g equal the number of mice in the green cage, how can we write  relationship between the two quantities? Is there more than one way?"

"Are b and g always, sometimes, or never equal? What makes you say that?"

"So in math we call b and g variables. Based on what we have discussed what is a definition we can come up with for variable?"

At the end of this activity I would hope that my students would be able to understand that variables are changing quantities, that we can represent with letters or symbols. Each of the five practices above reflects my goal, and every move I would make in this activity would be intentional. I think that the five practices are a great way to structure mathematical discussion in a classroom. They do require some planning in advance but it will help to direct the discussion towards the mathematical goal of the lesson. I believe that by incorporating these practices into my teacher assisting placement I will begin to see better whole-class discussions in my classes.   

Tessellations...Are They Math?

The more math I learn the less it fits into the conventional definition I once had of mathematics. This semester I am taking my capstone class, The Nature of Modern Mathematics, and this class has been pushing the boundaries of what I consider mathematics. Our discussions have ranged from talking about mathematicians such as Archimedes, Brahmagupta, and Leonardo of Pisa (also known as Fibonacci) to talking about the Pythagorean Theorem and arithmetic in Roman numerals. I have been learning such a rich history of mathematics but there is one topic in particular that has stood out to me, Islamic tessellations.

So what exactly are tessellations? Tessellations are tilings of the plane using one or more geometric shapes, called tiles. The important part though, is that those tiles cannot overlap, nor can there be any gaps between the tiles. In order to understand Islamic tessellations, it is important to understand some beliefs of the Islamic faith. Most interpretations of Islamic law discouraged the portrayal of humans or animals in art for fear that it would cause people to idolize those humans or animals. Therefore Islamic art centers around three main elements: calligraphy in Arabic script, floral and plant-like designs, and geometrical designs.

The Islamic geometric designs often included a lot of repetition and variations. Even though the designs may have only consisted of a couple shapes, the patterns those shapes created couldn't have more variety. The Islamic tessellations also included a lot of symmetry, and many designs included translations and rotations of tiles as well. Here are some famous examples of Islamic tessellations.

The Alhambra in Granada, Spain (Google Images)

The Alhambra in Granada, Spain (Google Images)

The Alhambra in Granada, Spain (Google Images)

Alcazar in Sevilla, Spain (Google Images)

Alcazar in Sevilla, Spain (Getty Images)

Alcazar in Sevilla, Spain (Getty Images)

Alcazar in Sevilla, Spain (Getty Images)

There is no denying that these tessellations are masterpieces, but are they mathematical? After some research on tessellations (check out this awesome Wikipedia page!) I have discovered that there are three types of tessellations: regular, semi-regular, and irregular.

Regular tessellations have both regular tiles and identical regular corners and vertices. There are only three shapes that can form regular tessellations, equilateral triangles, squares, and regular hexagons.

Semi-regular tessellations are tessellations that use regular tiles of more than one shape with every corner identically arranged. There are eight semi-regular tilings.

Irregular tessellations are tessellations made from shapes such as pentagons that are not regular. M.C. Escher was an artist who was famous for creating tessellations that used tiles shaped like animals, humans, or other objects.

Two Lizards (1942) M.C. Escher

Johannes Kepler was one of the first people to study regular and semi-regular tessellations in 1619. In 1819 Evgraf Stepanovich Fedorov marked the unofficial beginning of the mathematical study of tessellations when he discovered that every periodic tiling contains one of seventeen different groups of isometrics, now known as wallpaper groups. Some people claim that all seventeen of these wallpaper groups are represented in the Alhambra Palace in Granada, Spain but it is still being debated.

There is no doubt that there is a lot of sophisticated patterning behind tessellations. Not only are they visually stunning but they are also rich in mathematics. I encourage you to try creating some of your own tessellations, here are some great resources to get you started!

John Golden: Islamic Tessellation Helper
Victoria and Albert Museum: Maths and Islamic Art & Design
The Metropolitan Museum of Art: Islamic Art and Geometric Design

Also here are links to isometric and square paper.

Happy Tessellating!

First Week of Teacher Assisting

Well I have officially made it through my first week of my Teacher Assisting placement at Grandville Middle School! This semester I will be working alongside Mrs. Jamie Stuart and her 8th grade math students. I am at the school Monday-Friday from 7:20 am-12:30 pm. I start my day off with two sections of Math 8, which is a pre-algebra class. Also, the first section of Math 8 has one paraprofessional in the classroom, and the second section of Math 8 has two paraprofessionals. Then I have a planning hour before two sections of algebra. I think I am really going to benefit from the set-up of my schedule because it allows for some great reflection time in between classes. In other words, Mrs. Stuart and I will be able to discuss what we can do to improve our lessons from one hour to the next.

The classroom setup is made for a lot of group work and peer collaboration. The students are arranged in groups of four, and there are eight groups in the classroom. One thing I really love about the arrangement of the classroom are the small round tables on both sides of the room. Both of these tables have two stools at them, and they are a great place to be able to pull students aside for some one-on-one work. I imagine myself working with students individually at those tables and helping them with any content they might by struggling with. Here is a picture I created of the classroom layout!

This first week of school we have been mainly going over rules and procedures and building the classroom community, so we have not covered any content yet (except for a pre-test the students completed on Thursday). So while I am not 100% sure what the daily schedule will be like in the classroom, I think I have a general idea. The students will come into class and begin working on a warm-up problem while I go around and check their homework. Their homework is graded on effort and completion rather than for correctness. Once the students are done working we will go over the warm-up and any questions they might have had on their homework. After this is completed the students will begin working on their notes and vocabulary for the day. Mrs. Stuart has the students create interactive notebooks as their main resource for their content. (If you are interesting in learning more about these notebooks you can check out them out here!) Once the students are done with their notebooks, they will most likely begin a small group activity. This can range from anything like a card sorting activity to maybe some challenging practice problems. Mrs. Stuart has told me that a lot of the students time in class will be focusing on peer collaboration work.

Mrs. Stuart's classroom is based on inquiry-based learning. This quote is taken from her beginning of the year parent packet:

"My style of teaching is very hands-on and inquiry based.  I believe that students learn best when they can take ownership of what they are learning.  My job as a classroom educator is to facilitate the learning for the students.  This is done by giving a group of students a task to explore, providing them with any tools they need, and monitoring a student lead conversation about what they discovered.  At the end of this process the students should have a deeper, more conceptual understanding of what is being taught.  Along with this I also try to incorporate real-world examples though the use of technology and real-world tasks. "    

I am very excited to see how much I learn this semester! I think I am very lucky with the placement and the mentor teacher I have received. Mrs. Stuart seems very warm and excited about learning, which makes me excited to learn from her. I know that this semester will be a huge learning process for me, but I am ready for the challenge! I hope that I can have a growth mindset as I go through the semester and I can learn from some of the mistakes I am bound to make. Here's to a semester of new opportunities and lots of learning!

Wait...What is Math?

Have you ever tried to explain what math is to someone? If you have, chances are that you probably approached the task with confidence but quickly realized that it was not as easy as you had thought. Maybe you gave a definition, something like "math is the science of numbers" or maybe you focused on a specific area of math such as geometry or algebra. However, if you really think about it, math is so much more complex than that. 

Just to humor myself I looked up the defintion of math on dictionary.com, this is what is says:

Mathematics [math-uh-mat-iks]: the systematic treatment of magnitude, relationships between figures and forms, and realtions between quantities expressed symbolically. 

Woah. That is mouthful, and I don't think it gets me any closer to an explantion of math. So here goes my own interpration of math.

I believe that math is our human desire to give order and regularity to the world. Humans love to have an answer and reason for why things happen and I think that math is a logical way to do that. For explain, how did people discover that we can model tidal changes using trigonometry? Honestly, I don't have the answer, but my guess is that someone wanted a way to predict the height of the tide and to do that they had to find a pattern. Once they found that pattern or "rule" they could then apply that to predict tides in different places. So I guess what I am trying to get at is that I think math is about trying to find universal patterns that we can apply to infinite situations or problems. 

So in honor of the complexity of math-uh-mat-iks here are five of my favorite math milestones:

1. Euler's Identity (I swear this is the closest thing the real world has to magic.)

2. Euclid's Elements 

3. Pythagorean Theorem 

4. Blaise Pascal's invention of the mechanical calculator 

5. The Seven Bridges of Konigsberg problem

Stacking Cups Activity (Systems of Equations)

This winter I observed a lesson in an 8th grade math class that used a cup stacking activity to introduce systems of equations. I really like the activity and I thought it provided the students with some rich mathematical exploration. This activity was something that I was very interested in using in my own classroom someday. I made a few changes and decided I would share!

I started planning this lesson with one thought: what do I want my students to be able to do once this activity is done. With that in mind I went to search for the CCSS and Mathematical Practices that aligned with this activity.

Common Core State Standards:
8.EE.C.8a Understand that solutions to a system of two linear equations in two variables correspond to the points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.

8.EE.C.8b Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.

8.EE.C.8c Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.

Mathematical Practices:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
8. Look for and express regularity in repeated reasoning

I also had some questions that motivated me while working on this activity. Here they are:

• What does understanding look like for this?
• How does this activity help with understanding?
• What will students do?
• How will I check for understanding?
• How will I have students consolidate?

This activity is meant to be an introduction to systems of equations, or an activity to do right after the topic is introduced. So I decided that if a student had understanding of these topics they would be able to write equations that represent real world scenarios, they would be able to graph these equations and find their solution, and they would be able to explain to me what this solution represents. This activity will help with their understanding because it will walk them through writing equations, graphing them, and interpreting their solutions. As far as checking for understanding, I have created a worksheet for the students to do after this activity that I might collect and check. However, this would all depend on my classroom structure at the time. To have my students consolidate I have given room at the end of the activity for a reflection.

Now for the lesson. I have broken it up into four parts: the warm-up, the introduction, the investigation, and the reflection. The warm-up consists of two questions that will brush students up on some skills that will be needed for the investigation. The introduction includes a 101qs video and a group discussion. The investigation is the Cup Stacking Activity. Finally the reflection includes another group discussion and a written reflection.

Warm-Up

For this warm-up I would either put these problems on a projector and have students write their solutions down or else I would pass out this warm-up on a paper. After it seems like most students are done I will pull the class together and we will discuss the warm-up.

Graphing and Writing Equations Warm-Up

Introduction

The purpose of this introduction to to get the students ready to complete the activity. First I would have the students watch Andrew Stadel's 101 questions video called Stacking Cups. The purpose of this is to get the students interested in finding a solution to this activity. 

I would have the students watch this video two times and then I would have a group discussion. I would ask my students what they Notice and what they Wonder about the video. I would record these answers on the board. Then I would have the students watch the video one more time. I would tell them that their task is "How many cups does it take for the stacks to be equal in height?". At this point I would introduce our activity, distribute materials (yardstick, two stacks of different cups, and the activity). 

Before I started I also wanted to anticipate some student strategies. The first thing that I anticipated was that students would assume that the height of two cups is twice the height of only one cup. If I saw any students using this strategy I would remind them to remeasure their stacks of cups and look and see if the height is really doubling. I also anticipated students not knowing where this change in height was coming from, that is why in the first problem of the activity I have the students label the different measurements of the cups. Hopefully by doing this the students can see that the height is changing each time by the height of the lip of the cup. These were some things specifically that I would be looking for throughout the activity. 

Investigation

The students should complete this activity in groups of 2-4 students. I will be walking around the room while the students are working so I can help them if they are struggling. I want a healthy bit of frustration to happen while the students are working. I think that this frustration is where awesome learning happens. 

Cup Stacking Activity 

Reflection

The reflection is the most important part of this lesson. With 15 minutes left in the hour (or once everyone is done with the activity) I would bring the whole class back together and we would discuss the activity. I would reference the Notice and Wonder that we did earlier, and see how many of the "wonders" we answered. I would also share some of the students work and we would share our answers. At this point I would show the video Stacking Cups - Act 3. Finally I would have my students complete the reflection at the bottom of their activity if they haven't done so already. I may or may not collect these depending on how well I think the activity went. 

I also created a follow up worksheet for this activity. This activity uses a lot of the same concepts used in the Cup Stacking Activity. I might have the students do this activity after they have been working on systems for a day or two. 

Follow Up Worksheet 

Variations of the Activity 

Noelani Davis- This is a variation on the cup stacking activity that uses the same 101 questions video. Noelani has a cool idea using Hint Cards with her students too! If her students are struggling she gives them a Hint Card to help them. I also adapted my follow up worksheet from Noelani's. 

Tara Maynard- Tara's class was where I first saw this lesson, her student teacher did this activity with her 8th grade students. Tara's activity has a really cool Part 2 that you should check out!

Dan Meyer- Dan has a variation that has students estimate their teachers height in cups!

Andrew Stadel- Here is the link again for the Stacking Cups video this activity is based on. 

Interview Reflection

This semester I have been observing in an 8th grade math class at Creekside Middle School in Zeeland Michigan. During my time observing I also conducted three student interviews. During these interviews I gave a student a math problem to work on, and my goal was to try and understand their thought process. While the students were working I would ask them questions to try and get them to explain their thinking. The purpose of this post is to reflect on the interview process and talk about what I would do different next time.

Before this assignment I had never done a student interview, so my first goal was to find a couple resources that would help me. I used NRICH to search for challenging problems to give the students I would be interviewing. One of my favorite problems that I found was called What's it Worth. Even though I did not end up using this problem for my interviews it was one that I seriously considered. Then I found a Student Interview article that gave a lot of great guidance on interviews including questions to ask the students, what to look for, and math problems to use.

I ended up giving all of my students the same interview question. Initially I was going to give them a choice of a couple problems but then I realized that I really wanted to be able to compare the thinking of the students I interviewed. I thought the best way to do this was by giving them the same problem. One of the problems that I found in the Student Interview article aligned with what the students had been doing in class. This is the problem that I ended up choosing:

A restaurant has square tables. Four people can sit at 1 table. Six people can sit at 2 tables. Eight people can sit at 3 tables. How many people can sit at 37 tables?

Before I started my interview I wanted to decide what I wanted to look for while the students were working on their problems. I made a list of things I would be happy to see in the students work.

My List of Awesome Mathematical Modeling:

-Table of Values

-Equation

-Picture/Diagram

-Recognizing a Pattern

I am not going to lie, I was very nervous to do these interviews at first. I was worried that I would not know what to ask the students while they were working on the problem. So again I went back to the Student Interview article and I found an AWESOME list of non-leading questions that I could ask during my interviews. 

My List of Awesome Non-Leading Interview Questions:

What do you predict will happen?

Can you solve it in a different way?

How did you figure it out?

Why did you______?

You wrote ______. How did that help you?

I noticed that you stopped what you were doing just now. What were you thinking?

I don't know what you mean by that, will you show me?

Will you draw a picture of that?

You started with _____ and then went to ____. Tell me your thinking.

Can you tell me what ______ means?

Are you right? How do you know?

Is there another way to show me? What is it?

Are there things that you already know that would help you solve this problem?

I was overall very pleased with the way that my interviews went. Two of the three students I interviewed used a table, a picture, and an equation to represent their work. All three of the students identified the pattern (increases by two) in the tables. While only one student had a correct final answer, one other student just made a mental math error that resulted in an incorrect solution. 

The most interesting interview was with my second student. This student did not use tables or an equation to model their thinking. They thought about the problem as adding "sets" of tables. During the interview I was confused why they took this approach, because I knew they had done problems similar to this in class. So then once the interview was almost finished and the student had given me their final solution, I asked them to draw what they thought the tables in this problem looked like. More specifically I asked what they thought "8 people can sit at 3 tables" meant. The student then drew 3 separate tables all with 8 people sitting around the table. Suddenly I realized why I was so confused! I had not stated in the problem that the tables were end to end, so this student thought that 3 tables meant 3 separate tables. If I would have asked this question sooner, I would have explained to the student that I meant for the tables to be end to end. Maybe by doing this the student would have solved the problem using a different method.  Here is the work of the three students that I interviewed.  

I learned a lot from conducting these interviews. I now have a couple resources that have great problems that I can use in my classroom someday, and a list of great  questions to ask students while they are working on these problems. The biggest take away I have from this assignment is learning the signs of when a student is struggling and when a student in confident in their work. I noticed that when a student did not feel comfortable with what they were doing they normally slowed down their work, looked to me for validation, or talked quietly to themselves. When one of the students was showing me one of these signs I tried to find a question on my list that could guide the student. Interviews like this could be very valuable to the teacher because by the time the interview is over the teacher will know exactly what their student knows and what they don't. The biggest drawback? Finding time in your plans to do interviews like this. 

Technology and Formative Assessment

Formative Assessment is arguably the most important aspect of successful teaching because it lets us know exactly what our students know and more importantly what they don't. I want to take a look at two online formative assessment tools, thatquiz.org and Socrative.

Both of these programs use quizzes as a way to assess student's understanding. These programs are meant to be used with technology but both also have a print option if a paper copy of a quiz is needed.  There are features of both programs that I like and also some that I find frustrating. 

Socrative:

Is there an app? Yes there is both a student and teacher app. Also quizzes can be accessed online at socrative.com

Do you need an account? Teachers do need an account but students do not. Students will need to enter room number and their name in order to access their quizzes. 

On Socrative you can create quizzes, quick questions, Space Races (a type of group quiz), and exit tickets. Here is what the teachers homepage looks like.

Best Features:

- Can create true/false, multiple choice, and short answer questions

- Can add images to questions

- Can randomize question and answer order

- Can import quizzes from an excel file or if you know the SOC number of an already made quiz you can import a copy of that quiz 

- There are different quiz preferences (Student-Paced with Immediate Feedback, Student-Paced with Student Navigation, and Teacher-Paced) 

- Can get detailed reports of quizzes for feedback 

Some things that I don't care for are that there are not many options are far as math characters (there is a sub/super script button though). I also don't like how there is not a built in gradebook or graphs and shapes cannot be created within the program. 

My favorite part of Socrative are the reports the teacher has access to after the quizzes are completed. Teachers get live results after they send out the quiz, these results give the students' names, question numbers, and whether the question was answered correctly or incorrectly. This is what a teacher would see during the live results. 

Once the quiz is completed teachers have a few opitions of how they want to recieve their reports. 

thatquiz.org

Is there an app? Yes (cost is $3.99) however, you can only use premade quizzes. On the app quizzes can not be created. The online version of the program however allows teachers to create and share their own quizzes. 

Do you need an account? No, but without an account (its free!) you cannot create and browse quizzes you can only use the premade quizzes. 

On thatquiz.org you can create quizzes that are matching, multiple choice, or short answer and multiple choice (called "slides"). Tara Maynard presented about thatquiz.org at this year's Math in Action conference and you can find her presentation here. 

Best Features:

- Manage multiple classes all from the same account. This makes it easier to have quizzes across different subjects for different classes. 

- Browse/Share quizzes (quizzes can be seached by code, subject, or even teacher's name)

- Teachers can set a number questions for each quiz. For example, you can set the quiz up so that it randomly chooses 5 questions out of a possible 20. This allows students to have different quizzes and also students can retake quizzes without getting the some questions. 

- Can create graphs and shapes and upload images directly in the program

- Built in gradebook that allows the teacher to see what their students answered and also how many times they attempted the quiz. 

- Teacher can choose to always save grades or else they can save once a students gets higher than a certain percentage (In Tara's class she only records her students' grades if they are an 80% or above)

Here is what a teacher would see on thatquiz.org while they are editing their quiz.

I think the best part of thatquiz.org is the ability to share and browse quizzes. My biggest compliant about this program is how times consuming it can be to create quizzes. That's why it is so great to be able to share and collobrate with other teachers. You also have the option to not share your quiz as well. 

I have seen both of these programs used for formative assessment in the classroom, and I think both have a lot of benefits. I love the reports that Socrative provides but wish they have a built in gradebook like thatquiz.org. I also wish Socrative made it easier to share quizzes with other teachers. As far as looks, my vote goes to Socrative. 

I have also heard somethings about the program Nearpod which allows teachers to be able to create presentations as well as quizzes for formative assessments. I am currently tyring to learn more before I give this program a review. Are there any other formative assessment programs that you use and would reccomend? 

Number Talk

A Brief Background on Number Talks:

Kathy Richardson and Ruth Parker began giving teachers mental math exercises at professional development settings during the 1990s. As these teachers completed more mental math Kathy saw the teachers becoming more proficient. Kathy and Ruth realized that they wanted to develop students' understanding, but they did this best by using concrete models to figure out computation. This idea lead to number talks. 

So...what is a number talk? A number talk is a short daily routine that gives students meaningful practice with computation. Number talks help students develop fluency while solving mental math problems. One great way to introduce a number talk is after a counting circle. In case you haven't heard of a counting circle here are some blogs to check out:

 Sadie Estrella - Counting Circle

 Malke Rosenfeld - First Counting Circle

 John Golden - Sadie's Circles 

I would love to explain in detail what exactly a number talk is but Jo Boaler does a beautiful job (way better than I ever could) in this video, it is worth the watch!

Here are my Top 5 Benefits of Number Talk:

1. Teaches Number Fluency 

This is in my mind the greatest benefit of number talks. The more students are involved in number talks the deeper conceptual understanding they have about numbers. Students will begin to see different fact strategies for addition, subtraction, multiplication, and division the more they practice. They will then use these known facts be able to see things like 6+7=6+6+1 or 6+7=7+7-1. Students show their fluency when they are able to explain why they are doing what they are doing, and also when they know how to solve a problem using different methods. 

2. Number Talks Build Confidence in Students

Once a student has a deeper conceptual understanding of numbers and how they work they will start to solve mental math problems at a faster pace. It might not happen right away, but the more practice a student has the faster they will get during number talks. This gives our students confidence. They can feel that they are becoming more proficient. I know that this confidence will carry over into their mental attitude while solving other math problems. 

3. Students Learn Different Strategies 

Sometimes a student will solve a problem in a way that other students would have never thought of. By sharing these idea in the form of a number talk our students gets to collaborate and may learn a more effective way of solving mental math problems. You may hear students say things like "Oh I never thought of that!" That is one of my favorite sentences! This means that our students are recognizing that other students think differently than them and that they have valuable ideas to share as well. Even the teachers have valuable things to learn from their students. 

4. How to Make a Problem Easier to Solve

Which is the easiest to solve? 

13-7?

12-6?

11-5?

10-4?

Chances are that students would struggle more when presented with 13-7 versus 10-4 or 12-6, but the awesome part is they all equal 6! During number talks students begin to see that they can change the problem they are working on to make it easier on themselves. If a student gets uncomfortable solving a certain problem they can change that problem into something easier using their fact strategies! One student might struggle trying to solve 46-29 mentally, but once they see that 46-29=50-33 they might have an easier time solving this problem mentally. Its important for students to be able to make mental math easier for themselves. 

5. Teaches Students Communication

How do we expect our students to be able to communicate about mathematics if they never practice? At first students will struggle trying to explain their thinking during a number talk but the more practice they get the more articulate they are and the more thorough they are in their explanations. One of the 8 Mathematical Practices of the Common Core is for students to be able to "Construct Viable Arguments and Critique the Reasoning of Others". This means that students need to be able to communicate about math, using mathematical language. Number talks are a great way to get our students on the track of mathematical communication, which leads to deeper conceptual understand and fluency, look at that its like a cycle!

Last semester I wrote an article about math anxiety and counting circles. I talked about how nervous I felt during counting circles and how I basically dreaded doing them in class. I can honestly say now that I enjoy counting circles. I enjoy them because of how much the number talks have helped me. Over the last two semesters I have deeper understanding of numbers and better number fluency. Counting circles no longer make me so anxious because I have different strategies to solve mental math problems. I will 100% do regular number talks in my future classroom because they will do nothing but help my students in the long run and make them better mathematicians. 

To Flip or Not to Flip

Flipped classrooms are a hot topic in education today. Many teachers believe that they can do more meaningful mathematics when they use technology to support their students' learning. The idea of a flipped classroom is that the students will learn the content, that normally would be taught with a lecture style, outside of the classroom with the use of technology. This then allows the teacher to be able to use class time doing activites that apply the students' knowlege. Normally students will learn the material by using a combination of videos and coinciding worksheets. These worksheets are normally checked and counted as a "homework" grade. Other teachers may use quizzes at the beginning of class to keep their students accountable for watching the videos the night before.

One of the struggles of teaching is that not all students learn at the same speed or in the same way. While some students may succeed in a traditonal classroom setting, other students may fall behind. The flipped classroom model allows students to rewatch the videos multiple times if they are not grasping a concept on the first try. Also by watching the videos before the next class students can come to class with questions, some teachers even put a spot on their guided worksheets for such questions. 

Having students learn the material independently also allows teachers to give students more individual support. Since class time in a flipped classroom is used to do activites, the teacher is then able to walk around the room and get immediate feedback about where their students are as far as understanding the content. Therefore, if a particular student is struggling then their teacher can spend one-on-one time addressing their questions.

In the Februrary 2014 issue of Mathematics Teacher there was an article about flipped classrooms. Two teachers, Amanda Moore a 7th grade math teacher from Battle Creek, MI and Matthew Gillett an 8th grade math teacher from Armada, MI, observed the effects of flipping their classrooms. Both teachers saw homework (watching instructional videos and completing guided notes) completion rate rise, with one class even increasing by 19%. After doing a unit of flipped instruction both teachers agreeded that by flipping their classrooms student engagement increased in class and they also said they had more time to do meaningful mathematics. The homework completion rate indicated that more students were doing mathematical work outside of school, and students were also engaging in more problem solving in class that previously was provided through homework. 

The last two semesters I have volunteered in Tara Maynard's 8th grade math classes in Zeeland Public Schools. Mrs. Maynard uses a flipped instruction model in her classroom, so I have been able to see the benefits of flipping first hand. Mrs. Maynard uses videos that she makes (with her own voice using an app called ShowMe) for her students' instruction. The students in Mrs. Maynard's class can access these videos without the use of internet at home through their iBooks app on their iPads (Zeeland Public Schools are 1:1). Mrs. Maynard also provides the students with a note packet to complete while they watch the videos. 

Here is what Mrs. Maynard says are the benefits of flipped classrooms:

- She gets to have conversations with everyone of her students everyday

-Truly know what the students understand and what they haven't yet grasped

-There is more time to do explorations, activites, demonstrations, labs, and deeper questions

-Students are able to move at their own pace and learn with their own style

-Supports anywhere anytime learning

-Parents can't give the excuse of "I can't help"

As with everything in life there are also some negatives to flipped classrooms as well. Here are some of the things Mrs. Maynard says she struggles with:

-Time (creating videos, creating activities etc.)

-Still giving paper and pencil assessments

-If its just a video and a worksheet it is not flipped instruction

-Videos can be done wrong (bad lecture, too long, boring, etc.)

-Students still come to class unprepared and waste time

Here is a video of a typical day in Mrs. Maynards flipped classroom.

Mrs. Maynard also asked her students what they thought of learning in a flipped classroom, here are their thoughts.

Almost all of the teachers who have flipped classrooms can agree that students are able to engage more in the mathematics they are learning. While flipped classrooms might have some downfalls I believe the positives outweigh the negatives. I would love to some day use a flipped model for my own classroom. I think that flipped instruction can provide so many learning opportunities for students and isn't that why we are teaching to begin with? 

Fears of a Future Teacher

Well I am going to be honest, I have had a very hard time writing this blog post. Each idea I came up with just didn't feel right, I would start writing and then lose interest in what I was writing about. I think my writing block came from some stress I have been experiencing lately. This last Thursday I turned in my application for the College of Education. All of a sudden everything felt so real to me. I realized soon I will be responsible for teaching students. This terrifies me. I worry whether I will be prepared for the challanges that are ahead of me. Then I realized I wasn't alone.

I, along with over 11 million other people, read the the blog Humans of New York. (In case you haven't read Brandon's blog I would suggest you take a look https://www.facebook.com/humansofnewyork) The other day I read a post that stopped me in my tracks. Over the past week Brandon has been focusing on a school called Mott Hall Bridges Academy. He has talked to the principle Ms. Lopez, students in the school, and also the Mott Hall Bridges teachers. Below is the post about a converstaion a teacher, Ms. Powell, had with Brandon that really resionated with me. 

All of a sudden I realized that Ms. Powell had summed up all of my fears about teaching. Recently I have heard all about the gaps in the education system, how students can be multiple grade levels behind by the time they reach your classroom. My biggest fear is that I will not be able to help my students be successful. How can I teach my students to factor polynomials if they have limited multiplication skills? As a teacher I know I will always have more material to cover than I have time to cover it in. These students who come to me grade levels behind need so desperatly to be successful, but where will I find the time to fill in the gaps? I am sure this is one of the biggest stuggles amoung teachers, and it cannot be fixed overnight.

I think the work of a prepared teacher starts long before they ever step foot in a classroom. My work as a teacher begins now. I am learning the skills and practices of successful teachers, but in my opinion the most important tool I can have as a teacher is collaboration. There are thousands of other teachers in this country who have the same fears I do, and they face them on a daily basis. They are my most valuable resource. Its important that I learn to ask them questions and learn from them. 

We teachers are a team, that needs to work together. We all have the same goal, we want our students to be successful. At times my fears about becoming a teacher can be overwhelming, and I think that I am not cut out for this. Then I realize all of all of teachers I look up to, that inspire me, have all had the same fears. They have had sleepless nights worrying about how they are going to fit in all of their material, and they have experienced the joy of watching a student finally understand something they were stuggling with. I am terrified to become a teacher, but I know as long as I have thousands of other teachers ready to help me in my journey, I will be alright.