Category Archives: Student Reflection

Who Will Work with Whom?

How do you and your students determine who will work with whom?

Elizabeth has been reflecting on teaming her speed demons with other speed demons and her katamari with other katamari. She is grouping and regrouping often, paying attention both to how students work and how students work together.

As part of SREB’s Mathematics Design Collaborative, we use the work from student pre-assessments to pair students homogenously on days when we are doing a formative assessment lesson from the Mathematics Assessment Project. Many of our teachers have worried about homogenous pairing. They wonder how two students who have little understanding of the material will learn anything if they are paired with each other. What we are finding, though, is twofold. Since we don’t have to spend as much time with pairs of students who have demonstrated understanding or some understanding, we can focus our time on the pairs of students who have little understanding. In addition, neither student can sit back and rely on the other student to do all of the work. Together, they end up doing something. The formative assessment lessons are written so that all students have entry to the content. Some items are more challenging than others, and we are slowly learning that every student doesn’t have to get to the same place in the collaborative activity. Students work for a certain amount of time and share what they have learned, even though they might not finish the entire activity.

Others (Alex Overwijk and Dylan Kane) cite Peter Liljedahl’s work on visible, random assignment of student teams.

It takes me a long time to get to know someone and feel comfortable sharing my ideas. For many years I let students choose their teams and work together for the entire year. More recently, though, my coworkers and I have used a card sort activity for teaming students on the first day of a unit. Teams work together throughout the unit unless we are enacting a Formative Assessment Lesson (FAL), in which case we team students homogenously based on their pre-assessment.

In geometry, we’ve made card sorts that introduce students to some of the terms and diagrams that we will study in the unit, often leading right in to the first lesson. It often takes a while for students to find their other team members since they don’t already know the content. Alternatively, we could use content/card matches from the previous unit to team them randomly and visibly on the first day.

For the first team sort, I emailed a preview to students the night before class so that they would have some idea of what to expect/what they might do with their card when they came to class.

Many students noted in their end of course feedback that we should keep the team sorts:

I think you should keep putting us into teams, as we can learn from others who think differently or similarly to us. I think you should also keep switching the classes some. I feel like this helped me a lot this year.

I would keep the different groups that are paired up. I feel that the groups helped me to see others point of view not just my own.

switching classes to see different teaching styles and having different groups throughout the year.

The changing of groups because it has helped me make friends and learn to work together with people who frustrate me.

All of our geometry team sorts are linked here.

I’ve heard others talk about teaming and re-teaming several times during a single lesson based on what students know and don’t know yet. I’m not there yet, but I am intrigued by the idea and would like to learn more both about the value of moving around so often and the logistics of what happens to students’ stuff.

And so the journey to figure out who will work with whom continues …


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When Your Team Is Working Well Together

Have you seen Jill Gough’s blog post Strategic Teaming: leadership, voice, our hopes and dreams? Jill reminds us that strong teams both set norms for their work together and then self assess to ensure that they are functioning within their norms.

How do you provide your students the opportunity to set norms for the work that we have to do together?

I asked my students what it looks like when your team is working well together.

Here’s a wordle of their responses.

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I see communicating, cooperating, talking, participating, strategies, but what strikes me most from their suggestions is everyone.

Some lengthier responses from the students:

We are all talking about our strategies. Everyone considers all possibilities presented by the team. Everyone is contributing and listening to what each other has to say, respecting each other. We communicate reasons the answers may be correct or wrong. We will work together to figure out multiple solutions, or the one correct solution, or if there is no solution.

We’ve agreed to these norms.

Everyone …







Since I want to be transparent about formative assessment being for students as well as teachers, I showed them Popham’s levels of formative assessment.

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We are working well together when the whole class is using formative assessment (and not just the teacher). We want all students in our class to meet the learning goals. Not just the “smartest”; not just the fastest. This isn’t survival of the fittest where some can adapt and others will grow extinct. Everyone can learn. Everyone will learn.

The start of another school year has come and gone as the journey continues …

Popham, W. James. Transformative Assessment. Alexandria, VA: Association for Supervision and Curriculum Development, 2008. Print.

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Posted by on August 18, 2016 in Geometry, Student Reflection


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For my students, on their graduation

You might have heard me say before that I believe that my students and I enter into a community of learning together at the beginning of each school year. While it is our tangible goal to study the measure of the earth (geometry) and change in motion (calculus), our intangible goal is to enter into the practice of learning.

We are not here to celebrate because you’re smart. We are here to celebrate what you’ve accomplished because you are committed to the practice of learning. We are here to celebrate the perseverance that you’ve shown, all of the hours of studying and practice that you have put in. We are even here to celebrate the synapses that have fired in your brain every time you’ve made a mistake – every time you’ve learned something new. We are here to celebrate your kindness to us and to each other. And we are here to celebrate the questions you have asked.

In The Falconer – What We Wish We Had Learned in School, Grant Lichtman suggests that “Questions are the waypoints on the path of wisdom”. We are here to celebrate your journey towards wisdom. I know your parents agree that you are well on your way to learning the art of questioning. You have been asking questions since you could talk: Why is broccoli green? Why is 2+2 equal to 4? Why do dogs bark and cats meow? What if my internet goes out on the night of the deadline? Who came up with the number e? What if it snows and we don’t have class tomorrow? What if my alarm doesn’t go off? Why does the unit circle go counter-clockwise?

As your journey continues, we urge you to keep asking questions – to keep learning –to seek peace and defend justice – to live responsibly – but we also want to warn you away from only doing enough to get by.

In his book about ethics, Sam Wells insists over and over that you cannot know what to do and how to act without preparation. Don’t expect to be able to lead later unless you’ve done the hard work of becoming a leader. You can’t sleep now and expect to do the right thing later. So, he tells the story of a surgery that took a tragic turn in an Edinburgh hospital in the 50s resulting in the death of a young child. Later that week two friends were discussing the tragedy, and one of them expressed sympathy for the surgeon who had run into a completely unexpected complication. The other friend disagreed.

I think the man is to blame. If somebody had handed me ether instead of chloroform, I would have known from the weight it was the wrong thing. You see, I know the surgeon. We were students together at Aberdeen, and he could have become one of the finest surgeons in Europe if only he had given his mind to it. But he didn’t. He was more interested in golf. So he did just enough work to pass his exams and no more, and that is how he has lived his life – just enough to get through but no more; so he has never picked up those seemingly peripheral bits of knowledge that can one day be crucial. The other day [at that table] a bit of ‘peripheral’ knowledge was crucial and he didn’t have it. But it wasn’t the other day that he failed – it was thirty-nine years ago, when he only gave himself half-heartedly to medicine. (74)

Our hope for you is for you to be who you are called to be – to find something to which to give yourself whole-heartedly – something about which you are passionate – and for which you learn all of the peripheral knowledge crucial to doing the right thing. (We aren’t suggesting you should never play golf.)

We’ve spent the past several years convincing you that the part plus the part equals the whole. You remember, right? In geometry – the Segment Addition Postulate – If I have a piece of wire that is 4 m long and another that is 3 m long, then together, I have 7 m of wire. It works for angles, and it works for area. When it comes to learners, though, you give us evidence that maybe Aristotle knew more than Euclid: The whole is greater than the sum of its parts. You are better together than you are alone. And we are better teachers and learners because of you.

Take good care of yourselves and keep in touch with us and each other. Don’t ever wonder whether there’s someone who’s cheering for you. We are, and we look forward to hearing about the next part of your journey.


[I shared this with some of my students at a luncheon celebration last month and last year.]

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Posted by on May 18, 2016 in Student Reflection


Collaboration & Perseverance: What Do They Look Like?

I recently wrote about this year’s circumference of a cylinder lesson.

As I was looking through some pictures, I ran across these two from last year’s lesson.

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What do you see in these pictures?

I was struck by what I saw: collaboration and perseverance.

What do collaboration and perseverance look like in classrooms you’ve observed? What about in your own classroom?

How do you create a culture of collaboration in your classroom?

How do you make sure your students know that we want them to learn mathematics by making sense of problems and persevering in solving them?

Thank you to all who share your classroom stories of collaboration and perseverance, so that we might add parts of those to our own classroom stories, as the journey continues.

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Posted by on May 8, 2016 in SMP1, Student Reflection


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Our First Day Message

What message do your students leave class with on the first day? How do you craft the first day learning episodes to promote that message?

Our students walked into the room with two Which One Doesn’t Belong scenarios.

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Which led to a discussion about working on our math flexibility throughout the year.

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We wondered what one word students would use to describe their feelings about math.

Algebra 1:

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AP Calculus:

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(I wasn’t surprised at the negative feelings towards math in Algebra 1 … but I was surprised at some of the responses from geometry and AP Calculus students.)

That led to the Quick Poll that we’ve sent now for a few years from Mindset.

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We did a short number talk that I saw on Twitter:

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We watched Jo Boaler’s Day 1 Week of Inspirational Math video on mindset and mathematics.

We ran out of time to do our normal opener where students find more than one way to complete a sequence of terms.

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We thought about what it looks like when a team is working well together in math class.

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TeamWork – Calculus (using Navigator for handhelds)

cohesive         1

everyone is building on each other’s idea 1

synchronization         1

organized discussion            1

clockwork       1

coordinated   1

everyone ends up with the correct answesr and understands   1

everyone ends up understanding the problem    1

communication and listening          1

people help each other, in an oreo 1

everyone gives input            1

everyone talking/ explaining at the same time.    1

clear discussion         1

All ideas and all teammates are listened to.           1

a well-oiled machine.            1

lots of words, pointing, and ideas    1

It looks synchronized            1

Like a well-oiled machine.    1

you are productive and no one gets left out          1

A solid conclusion shared by the entire group is reached based upon the well thought out ideas contributed each individual   1

dividing and conquering      1

efficent           1

We are very focused and productive.         1

lots of high fiving and excitement   1

lots of talking and no one hates each other           1

looks like an oreo      1

Organized discussion            1

explaining thought processes to each other          1

success           1

helping each other and an oreo      1

TeamWork – Geometry (using Navigator for Networked Computers)

People are building off of other ideas and are able to make an educated descision using everyone’s input.           1

When a team is working well together they may not always agree on the answer or how to get to it; they always work to find the right one no matter how difficult or stressful it may be.            1

When a team works well together they are listening to each other and trying to trying all methods (to solve). Everyone listens as much as they talk.     1

When a team is working well together, there will be a lot of communication and listening to others’ ideas and meathods.          1

The team looks uniform and united, and if you fail, then your team can figure out what to do better next time.   1

People bouncing ideas off of each other and creating new ideas or improving old ones. Even if some people are wrong, people correct them in a kind way and tell them how to correctly solve a problem. All team members are putting in effort and carrying their weight, not just leaving others to do all the work. The team finds multiple ways to solve the problem and chooses the best one.    1

All team members working together and solving problems at the same time individually then comparing answers and learning from different views.  1

When a team is working well together, each individual student listens to one another and actually thinks about each teammate’s idea and sees it as a viable solution. A team is working well together when they get along and are respectful to one another.        1

When a team works well together, they work easily and doesn’t argue when someone has a different answer than the other person. And they get the answer right. They should be able to recieve and give feedback from the other.       1

When a team, in math, is working well together, I think of a deep conversation of different ideas. I see different solutions and ways a proplem could be solved coming from all sides of the table… if you know what I mean. 1

When no one argues and everyone considers others solutions to a problem.  1

It looks very good and fluent when your team is working well together. When you have a team to help bounce ideas off of each other and to help each other reach the goal they need it is very useful. Everybody is going to make mistakes, and when your team knows that and will help you to find what you did wrong, you will have a larger success rate.    1

When your team is working well together, every member is sharing their thoughts and ideas, right or wrong. The team members aren’t embarrassed about getting the wrong answers because they know that their other teammates will help them to understand their mistakes and learn from them. Each member comes up with different ideas when the team works together, so that way every person benefits wih a new way of thinking.     1

The team is able to get more done, and do it quicker.      1

When a team is working well the group closer together and they’re all listening to other team members input and are also giving their own input.  1

It looks productive and focused. We are all concentrating on the problem that the team has to solved.      1

It looks like we all know what we are doing. It looks like we have more ideas and know more ways to solve our problem. It makes us look as if we understand the problem more and in most cases we probably do, when we work together well.     1

You are makeadvancements and improve one another and also agree upon an answer.       1

When the team is working well together it helps other people of the team to increase their knowledge because each person may see things in a different way.            1

Everyone is not arguing. People are using teamwork and getting the right answer while teaching others on the team how to do it differently.            1

Everyone is learning and helping each other when they may not know the answer to something or need help figuring out how to do something. No one makes fun of anyone if they get a wrong answer because we all need to learn and grow.    1

Somebody will suggest something and people will get excited or say “”yes!””. Then another person will suggest something and everybody will enjoy making progress. People that originally disagreed will change their minds because of something another person explained. They will keep working together until the project is finished.      1

The team is working well when they all have corresponding ideas that come together to get a problem correct.   1

When a team is working well together the team everything flows and everyone is participating. Everyone is helping, ideas are exchanged, and people are learning. Everyone is set on one goal and everyone is headed to achive that goal.           1

It is when everyone in that team is listening to and coming up with ideas. The whole group is cooperating and completing the task given. That is when a group is working well together.     1

Everyone understands the objective and is comprehending well. They understand why and how the team got the answer. No one is confused and everyone feels like they are making a contribution to the work     1

We talked about Popham’s four levels of formative assessment.

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Our goal is for all students to reach the learning goals and not just for the “smartest” and “fastest” to do so. Which means that we will have to help each other. Which also gets into Wiliam’s Five Key Strategies for formative assessment, in particular, activating students as learning resources for one another.

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The bell rang before I could have my students reflect on what they learned and what they will do in geometry this year, so I sent them a Google form to complete outside of class.

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Algebra 1 students answered a Quick Poll before leaving about what they learned.

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today, I learned that anyone can learn math

This year i am going to improve in math   1

About my teachers to be better at math.   1

I have learned If I practice I can become better.

In math this year, I am going to practice more.     1

be able to learn more about math and I will try harder. 1

I have learned that I have three very cool teacher that will make math fun this year and I also learned that math isnt everybodys fav but evryone a math person

In math this year i am going to get better at it      1

You can’t be a math person everyone is good at math it you pratice, so i am going to pratice.           1

I have learned that all three of my math teachers went to NWRHS and in math class this year i will not get bored in class         1

i have learned to think outside of the box. i am going to learn to thinkoutside the box          1

that we have three teachers.

try to work out everything better. 1

that math will be exciting this year and every one is good at math       1

That everyone is good in math in some way. In math this year, I am going to have an A.       1

I have learned that my teachers are cool

in math im going to inprove.            1

I have learned how to think outter box .

I am going to try to do better than the year before in math.       1

I have learned this year to be more excited to more about math.

In math this year, I am going to try my hardest.   1

That I have 3 math teachers. Also math wont be so bad. 1

pay attention 1

nothing and this year I’m going to start fresh       1

I have learned about my classmates and teachers.

In math this year, I am going to try harder.           1

I learned my other classmate’s names and how to do Four 4’s.

In this year of Math I plan on working harder and to advance in my knowledge of Math.     1

I have learned that you need to thing hard to get some of the questions right.

In math this year, I am going to make my math skills better and i want to work math fast.   1

I learned that everyone can do math. I math this year, I will work really hard and do the best I can.          1

I have learned how everybody can do math even if you are not a math person and your brain grow if you work it out. In math this year i am going to try to learn more and get a better grade in math.   1

I have learned in this class period to hold a more open mindset towards math and the many different answers to a mathmatical situation. Different methods can be accepted, and not every person will see math in the same light. I am going to learn to find an excitement in the subject of math this year, hopefully.           1

i have learned peoples names in my class in math this year, i am going to make good grades           1

I have learned that anybody can learn algebra and be good at it. This year I’m going to payattention and do my best at algebra.         1

I have learned that everyone is smart at math, people just need to embrace it.

In math this year, I am going to become smarter in math.          1

I have learned… that you can get differnet answers just by using 4 4s, also I learned how everyone is well in Math, it’s just kids with more experience are more better in it.

In math this year, I am going to.. learn more about math and hopefully actually start to like it.        1

i have learned everybody is good at math and i am going to try my best to make good grades         1

Everyone can learn about math very well, and no one is a math person. I am goin to learn about everything that i can and try my best at it.I am also gonna try and figure out problems the best of my ability.   1

I have learned that the more you practice and work your brain the more it grows.

In math this year, I am going to practice and work my brain so I can learn and get better.   1

Today I have learned that practicing something, even if you don’t fully understand it, still helps. This year I am going to try my hardest to achieve high grades in math.           1

I have learned today in class, everytime you have to find something out there can always be more than one way to it.

In math this year, I am going to try and learn new things that I didnt understand last year.            1

Today I learned that no matter how much i get frustrated that i will always have someone to leaN ON THIS CLASS TO HELP ME WITH MY SITUATION … In math thjis year imgoing to succeed in all things all I do my grades will also be also better.           1

i have learned that anyone can be good at math with good practice.

in math this year im going to learn different things and hopefully get better at math.           1

practice math everyday and that it’s okay to make mistakes because we learn from them.   1

that we can learn things easier by practice so in math this year, i am going to practice if i have troubles until i fully understand it.       1

I have learned that the brain can grow with what you learn.

In math this year, I am going to study more.         1

I have learned that your brain is always changing and to get better at something you have to keep practicing.

In math this year, I am going to practice what I’ve learned in order to get better at it.           1

I have learned that your brain can grow just by learning new things.

I’m going to practice more on math because, if you practice more on something you can get better at it.    1

That math doesnt always have t0 be boring, it cab be reall        1

i am going to try my best to past     1

I have learned that everyone is capable of being in the highest math class there is. There is no such thing as a “”math person””. In math this year, I am going to try my hardest to maintain the highest grade I can, and pay attention in class in order to make good grades.   1

practice and do my best to make a good grade even thooe me and math REALLY dont like eachother. but im going to try to do my best            1

IF you practice you can get better at it       1

I learned that anyone can do math well and in math this year, I am going to try.        1

that your brian can grow the more you practice something , this year i am going to pay more attention.    1

I have learned that anybody can be good at math, you just have to work for it.

In this year of math, I am going to pay more attention than I did last year so I can get better and so I can get good grades.        1

I have learned if I keep practicing and going to different levels I will become better.

This year, I am going to study and practice math on different levels so, I will become better at math.         1

We didn’t really have a lesson, but I have learned about how the brain works with how good you are at something. In math this year, I am going to have no tardys, have straight A’s, and not disrespect the teacher in any manner.    1

that i dont have to dread coming to math. i can grow and learn from it. im going to try to make all a’s        1

You can get better at something if you practice at it.

I am going to try harder. And give up if I don’t understand it.   1

I have learned that as long as I continue to practice math or algebra I will slowly get better at it.

In math this year, I am going to attept to be more optemistic about the work and try harder to get better grades.           1

be able to be good in math by the end of the year. and that i will be able to succeced in anything that i do by just practicing.    1

I learned that when you have a difficult time with a question that your brain is growing at the same time.           1

We went from math is “complicated, hard, frustrating, …” to “I can do this” in 95 minutes. I believe our students left hearing our first day message:

Everyone can learn math.

Our brains are growing when we struggle to solve a problem.

There isn’t just one way to solve a problem.

Learning more than one way to solve a problem grows our mathematical flexibility.

Working with a team is an important part of how we learn mathematics.

And so the journey continues as a new school year begins …

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Posted by on August 17, 2015 in Student Reflection


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The Last Day of Class

Another school year has finished (two months ago), and a new one is about to begin.

Our teachers did a lot to promote growth mindset this last year.

Many of us sent our students a poll with statements from Carol Dweck’s book, Mindset, on the first day of class

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and again on the last day of class.

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You can see some change in the way that students responded.

I have wondered whether talking about mindset and promoting growth mindset makes a difference in what and how students learn. I know plenty of teachers are skeptical. I’m convinced that it matters.

Many students participated in Jo Boaler’s How to Learn Math: For Students online course through Stanford University. I heard students talking about making mistakes, their brain growing, and synapses firing on many occasions throughout the year.

What has convinced me more than student responses throughout the class, though, are the voluntary reflections that my students have offered. I received a handwritten letter at the beginning of May from a former student who no longer attends our school. An excerpt follows:

“When you taught my geometry class last year you polled us at the beginning and the end of the year to see if our opinions on innate/static intelligence vs. one’s ability to improve intelligence had changed. I just want to say that though I was doubtful at the time, this idea of an evolving and increasing intelligence through questioning and learning through wrong answers has stuck with me and served me well. I was once pretty insecure in my academic abilities: yes, I made good grades without much trouble, but there’s always someone faster or more confident or more eloquent, and so much of my identity was wrapped up in being a ‘smart’ kid that I was often afraid to speak up and make mistakes. Now, though my grades and academic integrity are still very important to me, I don’t see successes and failures quite so black and white. Rather, I try to see it all as a learning moment, and I thank you for introducing me to some of the ideas of growth mindsets and ‘GRIT’.” – CM

This thoughtful reflection a year after the class ended is coupled with a thoughtful reflection from another student who wrote as the class ended last year. You can see his reflection in this post.

We will start another school year on August 6 … our students are going to hear the message not only that they can be successful in mathematics but that we, their teachers, want them to be successful in mathematics … our students are going to be greeted with open-ended problems that are accessible to all (many of which will come from youcubed’s Week of Inspirational Math) – problems that allow them to realize from the beginning that we don’t all think the same way and that making our thinking visible to others is a good and important learning opportunity for all … our students are going to set norms for how the class will learn together throughout the year … our students are going to hear from Carol Dweck on the power of “Yet” and they might even hear from Sesame Street, too.

What message will your students hear on the first day of class? What will they say about your class when asked how they think classes are going to go this year?

I look forward to school starting again, as the journey continues …

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Posted by on July 25, 2015 in Student Reflection


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The Wilson Technique of Elusive Answering

Earlier this year, I received a precious gift from a student. Sydney emailed me an essay that she had written for a scholarship that included a reflection on what it’s like to be in my class. I asked for her permission to include it as a blog post, as her reflection gives me some hope that my students are learning practices and habits from our time together that transcend subject, grade, and location.


The Wilson Technique of Elusive Answering

In my 18 years, I have not once taken a class that was not important, but I have taken a few that made me feel not important. This, of course, is not the worst feeling to have, as it is largely true. In the grand scheme of things, I am no better than any other person or creature. If a comet was heading towards my state’s stronghold, it would not wait for me to evacuate my suburban residence before it obliterated our, not one, but two capitol buildings. However, this is not about comets; this is about 9th grade geometry. More specifically, this is about the teacher of 9th grade geometry.

Little Sydney (That is, me from the past. Although in reality I was only a few inches shorter than I am at the present.) was a little high school freshman on her way to her first class with the one, the only, Jennifer Wilson. Her name might not be spoken around the globe, and there are certainly no sonnets in her honor, but Mrs. Wilson is definitely a celebrity at home in Small-Town, Mississippi. As if I was not nervous enough walking into my first high school math class, I soon found out that my teacher’s name was listed as a reviewer in our textbook that I would be using, worshipping, and clutching to my frail body in the wee hours of the night in hopes of perhaps absorbing its knowledge in some bizarre form of literary diffusion throughout the year. She is a big deal. I did not know all of this, of course, not yet. I was too concerned with the boy with the curiously curly hair that sat across from me and the question of whether or not to attend the bowling team tryouts next Thursday to recognize that this woman, this marvelous teacher of the extraordinary, was changing the way that I would not only view mathematics, but also how I would approach challenges and obstacles for what I am certain will be the rest of my life.

There is no question that Mrs. Wilson’s geometry class was unlike any class I had ever been in before. She was the first teacher I had that responded with “I don’t know; what do you think?” when a question was asked of her. Looking back, she totally knew the answer. She knows every answer; she’s Mrs. Wilson. But she would not give it to us without a bit of work on our part. She taught me that the best conclusions are the ones that you can come to all by yourself. To the untrained eye, this technique may seem like just a lazy cop-out for being a teacher, but this is not the case. Mrs. Wilson wanted you to know the answer, but not without you first deciphering the question. For a teacher in the system of public education, Mrs. Wilson showed a remarkable ability to genuinely care about each and every one of her students getting the most out of every class period. The Wilson Technique of Elusive Answering is not idiosyncratic, I am sure. It has definitely become pattern for the rest of the teachers in the math department, as it is something that I have come in contact with every year since the first day in her class. This Technique, while somewhat annoying at times (in those days my habit of giving up at the first sign of trouble was much more engrained into my being than it is now), it proved to be the single greatest educational method that I have encountered. I would give up on myself, but Mrs. Wilson would not. She believed that every experience was a learning experience, and that, even on the test, you should discover something that you did not know. I learned much more than geometry that year, I learned how to learn. Mrs. Wilson taught me how to look at a problem, no matter how big or complicated, and think to myself “There is a way to figure this out, and I can find it.” She showed me that the obstacles of mathematics could be overcome with cooperative effort and excessive perseverance. Even greater, Mrs. Wilson showed me that any challenge, not just the ones involving the mass of the sun or the sum of interior angles, can be conquered with knowledge and persistence as opposed to disgrace and defeat. This was the greatest thing that I, as an impressionable teenager could have been taught, inside or outside of the classroom. The impact she has had on how I approach challenges and difficulties is remarkable, and, the funny thing is, she has no idea. To ask that every teacher be as caring and engaging as Mrs. Wilson was (and continues to be) would be a practically impossible feat. But, in a fashion that Mrs. Wilson would be proud of, there is a way to make sure that every student get as much out of their educational experience as possible.

Mrs. Wilson teaches the universal language of mathematics, but mathematics, unfortunately, is the only language that can boast this sort of ubiquity. Speech, one of the primary forms of human communication, has over 6,500 languages and variations. Even taking into account the fact that 2,000 of these languages have fewer than 1,000 speakers, there are still 4,500 spoken languages that very important people with very important things to say use in their everyday lives. How many of these 4,500 do I, a high school senior in America, comprehend enough of to understand these important peoples’ important thoughts? I can think of only one. Other than the foreign language courses required by my educational institution, two measly years of Spanish or French, I have had little or no exposure to languages of a foreign nature. If I were put into a room with 4,500 people, each with a different native language, the only person I could talk to would be myself. The blame for my lack of language, however, does not rest on my schooling or myself alone. No one had ever required me to venture into bilingualism prior to high school, thus my high school courses had no solid foundation to build upon whenever the time of learning came. Learning a second language takes constant practice and effort, so attempting to introduce students to a new language once they are already half-grown and expecting them to learn enough to communicate with native speakers of that language, though well-intentioned, is almost naïve. The only way to ensure that young students develop the skills and ability required for learning a second language is to introduce them to one as early as possible. Preliminary language courses almost always require chapters and chapters of vocabulary, vocabulary that could be easily taught alongside the usual English vocabulary of most American public schools.

However, speaking a second language is only the start. It is the key to a chest that unlocks hundreds of new possibilities and hundreds of new people to talk to and share ideas with. For example, the perfect team of people to solve age-old medical mysteries could be roaming the earth this very moment, but those in it are at a loss for words, quite literally, if they are not able to communicate with one another once they meet. Opening these lines of communication early on in students would allow them to jump the first lingual hurdle, increasing their chances of staying in the race. Rather than viewing the language barrier as a wall, students will view it as a staircase, with each step bringing them closer and closer to the world around them. Schools would provide their students with a solid foundation for lingual growth and fluency by introducing them to foreign languages on an elementary level, enabling these students to open their minds on a global scale. The educational reform that I am proposing for my country, as well as any other country not already on the bilingual bandwagon, would take years to introduce and perfect, but it would be a major step in ensuring that students of all ages live their lives with the world in mind. One must speak the languages of the world to be a true citizen of it.

Works Cited

“How Many Spoken Languages Are There in the World?” <>. Pearson Education, n.d. Web. 26 Oct. 2014.

University of Haifa. “Bilinguals Find It Easier to Learn a Third Language.” ScienceDaily. ScienceDaily, 1 February 2011. Web. 26 Oct 2014.


Posted by on February 7, 2015 in Student Reflection


Growth Mindset & GRIT & SMP1

If we want our students to be mathematically proficient, and if we want mathematically proficient students to make sense of problems and persevere in solving them, how will we help them when they don’t? or won’t? or feel like they can’t?

Jill Gough and I have been working on leveled learning progressions for the Standards for Mathematical Practice. Here is the visual for SMP1.



I wonder how much making sense of problems and persevering in solving them has to do with the work of Carol Dweck on Mindset and Angela Duckworth on GRIT. I had the opportunity to hear Angela Duckworth speak at the AP Annual Conference a few years ago.


One of the ways that our students can earn Problem Solving Points in our course is to determine how much GRIT they have:

Angela Duckworth says that the key to success is GRIT. Watch her TED Talk. Then determine how much GRIT you have. Then email your instructor a reflection with a response to at least one of the following prompts:

I like …, I wish …, I wonder …, I will …

We have enjoyed reading our student reflections on GRIT.

I like this idea and I do believe in it. I believe a lot of people don’t really understand how extremely important it is though. I think a lot of people would watch the video and think “oh cool grit whatever” and not realize that that’s more than likely is what will get you hired coming out of college and that it will probably take you farther in life than anything else. I wish more people understood that. I wonder if GRIT is something you can turn off and turn on, like we know it can change but can you just decide you want to be gritty for this one thing and be gritty.



I like that Angela Duckworth and Princeton (and you too, Mrs. Wilson) are speaking out and beginning to normalize this idea that intelligence is a fixed point, that we can’t change, is all wrong. Yes, it’s true we are not all rocket scientists- but should the people with less of an initial gift for learning have any less of an education? I’ve felt that in our school and our society there are a lot of limitations, including how high you rank in standardized tests, that influence how much you are pushed and expected to succeed. However, I don’t think that people who rank lower in testing scores should be shoved aside and given just the bare minimum. If the fear of failure was not so prevalent in the school system, maybe kids would believe that they can succeed after the initial failed attempt; that not just the ‘smart’ kids will be the ones to succeed.

I wish that someone had told me about this sooner, and that we were setting the goal at something more like GRIT, not just if you get the answer faster or easier than someone else. I’ve been in the smart track my whole life so I might sound out of line, but even I know that I won’t be a mathematician or the one to find a cure for cancer. No matter how hard I try, there is reality to remember, and though I’ve had encouraging parents and many very helpful teachers, I’ve still had the idea of my failure put into place. Can I wipe away that misconception that I was hardwired with a certain capacity for greatness? Even if I do, I feel that I just wasn’t born with a lot of determination. I’m sad to say my GRIT score was only 2.7 or so.

I wonder if this idea will die away or flourish in the new minds of the next generation. Before I came to your class, I’d always had teachers who would seem to forgive our wrong answers, but never one who said that, if used in the right way, it could actually help improve our overall smartness. I wonder how I could improve my measly 2.7 GRIT to something stronger. I wonder if I’ll ever find a motive to push me through, something to fuel my resilience.


I will work on not giving up; what better time to muddle through than high school? Opportunity to dump homework and just watch netflix abounds, but I will make a conscious effort to improve my GRIT and become a more responsible, diligent person.



I will definitely try harder in school and in other commitments after watching this video. The grit survey site gave me a grit score of 3.88. It also stated that I have more grit than 70% of the US population. Wow! I am shocked that 70% of the US has a grit score lower than 3.88. I am not fully satisfied with that score, so I will try harder to increase my grit score.



I took the grit survey and my result was 3.25. That makes me grittier than more than 40% of the United States of America. I will work hard to persevere on any project I begin. When I do projects, it always feels like I work so hard when I start, but as I get closer to being finished with it, I don’t work as hard as I could. I need to work on having patience to see something completed. I will also work to not get so discouraged when I get something wrong or when I don’t understand something. Once I start to do some of these things, I will become more successful and grittier.



I like how Angela Duckworth developed a grit questionnaire and how she admitted that she didn’t know how to instill grit in kids. I also liked how she ended with “In other words, we need to be gritty about getting our kids grittier.”

I also took the grit survey, and got a 3.5 out of 5, which is apparently better than 50% of the US population. I don’t if I should be happy that my 70% is better than nearly 160 million people or sad for the same reason.



Does it help for us to make our students and children aware of growth vs. fixed mindsets? Does it help for us to purposefully use growth mindset and GRIT language with our students? And whether or not research shows that it helps, can’t it not hurt if we want all of the learners in our care to make sense of problems and persevere in solving them?


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Why I Still Believe in CCSS

I’ve written before about why I believe in CCSS, so I won’t repeat those stories here.

I want to point you towards a few resources that might be helpful to understand why many educators are convinced that we must do something on a large scale about making mathematics less about following procedures and getting answers than it is about making sense of concepts and developing sound reasoning as to why something works.

How many of you have ever said or heard someone say, “I can’t read.”

How many of you have ever said or heard someone say, “I can’t do math.”

How many of you put fractions on your “top 5 things learned in 1st grade” list and as the “best thing about 1st grade”?


As a parent, the first recommendation I have for you is to read Mindset by Carol Dweck. Reading Mindset has changed the way I talk with my daughters about what they are learning, what mistakes they make, and what successes they have.

Christopher Danielson has written 5 reasons not to share that Common Core worksheet on Facebook. If you want to know more about Talking Math with Your Kids, read his blog by that same name. Better yet, subscribe to it. In particular, you might be interested in his post on Dots!

Look at the Standards for Mathematical Practice. I’ve written lots of posts about using them with my students to learn math.

If you want to know more about research evidence for needing these standards, Jo Boaler succinctly discusses Why Students in the US Need Common Core Math.

Sign up for How to Learn Math: For Students from Stanford University. It is a free online course that is currently open and runs through September 15, 2015. You work at your own pace through the six sessions. My daughters and I have been taking it together this semester. The conversations that we’ve had about mindset and learning math have been helpful in how they now react to learning something that isn’t easy from the beginning. We’ve also had several of our students take the course. Their reflections provide evidence that their attitude about learning math has been impacted positively. I can’t imagine that you wouldn’t want the same for your own children.

“I will no longer base my intelligence on the fact that I get an answer right because when I make a mistake my brain is growing which means I’m actually learning from my mistake.“

“I like that the course points out that everyone has the ability to be good at math and that people do not require speed to be a ‘math person.’ However, I wonder if that the only reason people think that they are bad at math is because they can’t process the information as fast as other people. In the future, I will try to push myself to leave my comfort zone and feel that it is ok to make mistakes.”

“I actually enjoyed some of the material I learned throughout this course. It gave me a new perspective on learning math. One of the best things I have learned is that everyone has the potential to learn math. No one is just born as a math genius. This boosts my overall confidence about my math ability. It makes me want to work harder so that I can get better at math. It also gives me hope towards all of my classes knowing that you can do anything if you just put in the effort. Another thing I learned is that not succeeding can be a good thing. Your brain will learn from its mistakes. If you always win at what you’re doing there is no point in doing it anymore right? You will not want to push yourself to do better because you will think you have already succeeded, when you are actually nowhere close. I liked learning all of these new things. I actually wish I would have known them earlier in my life. I could have put them towards my attitude on learning and even more specifically on learning math.“

Finally, read the recently released paper Fluency without Fear and be sure that the administrators and teachers at your children’s schools read it as well.

We are better together than we are alone, and we can make a difference in how our students learn and understand math.


I have learned …

Students aren’t kidding when they ask, “When will we ever need to know this?” In How the Brain Learns Mathematics, David Sousa suggests that students need a reason to move information from short-term memory to long-term memory. What opportunity do we give our students to reflect on what they are learning and why during class?

Many teachers give Exit Tickets, which can give teachers good information about what students have learned. However, I’ve observed many exit tickets that are more useful for teachers than they are for students. If the exit ticket requires a calculation, when do students find out whether what they’ve submitted is correct? Immediately? Or the next time class meets? How many students then complete homework using wrong ideas?

Exit Tickets can be good formative assessment. In fact, Sousa also notes that closure in a lesson shouldn’t be students packing up their backpacks and walking out of the door. Closure needs to be a cognitive process – students need to think about what they have learned and what questions they have, connecting what they have learned in class today with what they have previously learned and maybe even to what they will learn. Exit Tickets can provide students an opportunity to cognitively think about what they are learning.

My question is what types of formative assessment are we using throughout the class period, instead of just at the end of class?

Are you familiar with Dr. Sousa’s brain research on the Primacy/Recency Effect? In essence, it shows that we remember best what we learn first in a learning episode; we remember second best what we learn last in a learning episode; and we remember least what’s in the middle of the learning episode. Think about how the typical math class has been set up. Students come in, and teachers go over homework (prime learning time). At the end of class, students practice (second prime learning time). In the middle of class, teachers teach the new material for the lesson (least prime learning time).

His research shows that 20 minutes is the ideal length for a learning episode. I teach on a block schedule, and so I find that I must be deliberate about planning shorter (20 minute, when possible) learning episodes within the block.

We were finishing up a unit on Angles & Triangles in geometry earlier this week. We begin each class with an opener of questions that students work through with their teams. I collect their responses, show them the solutions, they try to correct misconceptions with their teams, and then we talk all together about any remaining misconceptions. After the opener (first learning episode) each day, students glance through our learning goals for the unit so that they can think about what they know and what they still need to know.

photo 1

Learning goals:

I can use inductive and deductive reasoning to make conclusions about statements, converses, inverses, and contrapositives.

I can use and prove theorems about special pairs of angles. G-CO 9

I can solve problems using triangles. G-CO 10

I can prove theorems about angles in triangles. G-CO 10

I can solve problems using parallel lines. G-CO 9

I can prove theorems about parallel lines. G-CO 9

I can solve problems using congruent triangles. G-CO 8

I can explain criteria for triangle congruence. G-CO 8

Because it was the last day of the unit, I asked students to answer a Quick Poll letting me know what they have learned and what they still need to know. The more I use “I can” statements for learning goals, the more I notice that they give us a common language for talking about what we can already do and what we can’t do yet.

photo 2Screen Shot 2014-10-20 at 10.00.30 AM

i have learned that i can solve problems using triangles, i still need to know how to prove theorems          1

i still need to touch up on the statemfnts and postulates            1

A.) I have learned to make conclusions, find the measures of angles, and etc..

B.) I still need to know the process in constructing parallel lines on the calc. in greater detail           1

about vertical angles            1

i have learned parrell line

i still need to know alot        1

i still have trouble ex triangle congruences           1

1.i learned prove theorems about special angles

2.i need to work on inductive and deductive statements, theorems about angles in triangles           1

i have learned conditional statements. i still need to go back over them                       1

learned how to prove why things are what they are

still need to know how to correctly prove anything from a given           1

i have learned the conditional statements

CO9 G-CO10 G-CO9,8,8

i need work on the true false charts           1

I have learned how to construct parallel lines using a point.      1

L symbolic logic

NTK proofs    1

how to find exterior angles of triangles           how to form theorems   1

i learned how to do ratios in a triangle.

need to know how to prove theroms          1

i have learned more about parallel lines cut by a transversal i still need to know more about constructing my own proofs         1

i learned about the types of hypothesis. i still need to know the different angle terms.          1

i have learned how to work with ratios.

i stll need to know how to form theorms on my own.       1

I have learned converse, inverse, conditional, and contrapositive statements. I need to learn when to use certain postulates in order to complete proofs.            1

I have learned how to prove statements using postulates. I still need to know how to explain criteria for triangle congruence.  1

i can prove theorems about angles in triangles. explain criteria for triangle.   1

how differemt types of angles are equal anb the different type of statements

i need to know the difference between converse inverse and contrapositive statements      1

learned conditional, converse, invese, and contrapositives.         1

I have learned how to solve problems using triangles. I still need to know how to do well on tables.           1

how to construct parrelel lines

how to write a hypothesis and conclusion in its different forms and determine their truth value    1

how to identify logical statements

how to do proofs       1

i need to work on converse inverse conditional contrapositive   1

i have learned how to construct parallel lines.

i still need to know how to prove the truth value of a statement.           1

learned-how to construct parallel lines

need to know-idk      1

i have learned to solve proplems using parallel lines. i still need to learn how to prove problems.   1

what aternate interior angles are;

how to figure out truth talbes.        1

i have learned if p then        1

I took the information about what students still need to know and used it to structure the rest of the class period, instead of just going through review problems in the order I happened to put them together.

A few years ago, Jill Gough and her colleagues experimented with students and faculty taking a brain break every 20 minutes to tweet what they are learning … you can read more about it here.

What will you do to ensure that you are maximizing the learning episodes in your classes?

And so the journey continues, with thanks to @jgough for making me reflect on how often I do formative assessment throughout a class period.


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