Hello, my name is Vaishnavi and I recently finished with my senior year. Having recently wrapped up 13 years of education at home has got me thinking about the importance of a classroom’s environment in learning. Now more than ever, I realize the value of interaction between students in an educational setting. An interaction I believe is best emphasized in a student-centered classroom because it motivates students to learn. I have struggled to find this motivation several times these past twelve years. It’s not like I was a bad student. I got my work done, got good grades, and got along with my teachers. Still, I was motivated by an outside pressure to do well. I learned whatever I needed to learn because I was told to. My focus on my grades outweighed my focus on the content of the subject. I was always passively processing what was taught to me. However, in the long run, this mentality is dangerous. Especially in a subject like math, where the intention is to take abstract concepts and see their connection to the real world. My lack of critical thinking made it hard to care about what I was “learning”.

That was until I entered high school and saw a shift in teaching styles. Some lecture-based classrooms were replaced by more activity-based ones. Only when I experienced this combination did I see the gap in productivity between these learning styles. In the more lecture-based classrooms I found that we got through the material quicker and had a predictable schedule. Although there was a sense of security in always knowing how the classroom would go, the methodical nature of it made it hard to get excited about learning. On the other hand, there were more activity-based classrooms. These were a lot less predictable and were often slower paced. The activities were there to cement understanding and for that reason, it took a long time to complete them. To be honest, I hated these classrooms at first. I thought the activities were excessive and the unpredictability was annoying. It was much easier being told what to do. It was only when I left those classrooms did I begin to appreciate them more. I realized what I learned in them stuck with me longer. Having to figure things out on my own instead of being told how to do something made it easier to recall the memory. It’s like my brain understood not only what I was doing, but also why I was doing it. After noticing this, I found myself more motivated to learn than before because I could see the fruits of my labors. It was a lot more challenging, but at the same time more exciting.

The thing that I found most interesting though was that my growth as a learner was not limited to one subject. As soon as I started thinking critically in one class, that thinking carried onto other classes. More thinking also led me to make more connections. I could now take what I learned and apply to situations around me. This was especially true when it came to the sciences. I took the knowledge out of the classroom and began looking at the world differently. This provided me a new motivation to learn. No longer was I learning to get good grades, but I was learning to understand. Grades still felt important, but they took a back seat to the content I was learning. I think this domino effect is really valuable in education. Starting to teach students how to learn enables them to carry those skills with them outside the classroom. I am very grateful to all the teachers that have given me this opportunity. Even though their classes were harder at first, I feel more ready for college and life because of them.

That was until I entered high school and saw a shift in teaching styles. Some lecture-based classrooms were replaced by more activity-based ones. Only when I experienced this combination did I see the gap in productivity between these learning styles. In the more lecture-based classrooms I found that we got through the material quicker and had a predictable schedule. Although there was a sense of security in always knowing how the classroom would go, the methodical nature of it made it hard to get excited about learning. On the other hand, there were more activity-based classrooms. These were a lot less predictable and were often slower paced. The activities were there to cement understanding and for that reason, it took a long time to complete them. To be honest, I hated these classrooms at first. I thought the activities were excessive and the unpredictability was annoying. It was much easier being told what to do. It was only when I left those classrooms did I begin to appreciate them more. I realized what I learned in them stuck with me longer. Having to figure things out on my own instead of being told how to do something made it easier to recall the memory. It’s like my brain understood not only what I was doing, but also why I was doing it. After noticing this, I found myself more motivated to learn than before because I could see the fruits of my labors. It was a lot more challenging, but at the same time more exciting.

The thing that I found most interesting though was that my growth as a learner was not limited to one subject. As soon as I started thinking critically in one class, that thinking carried onto other classes. More thinking also led me to make more connections. I could now take what I learned and apply to situations around me. This was especially true when it came to the sciences. I took the knowledge out of the classroom and began looking at the world differently. This provided me a new motivation to learn. No longer was I learning to get good grades, but I was learning to understand. Grades still felt important, but they took a back seat to the content I was learning. I think this domino effect is really valuable in education. Starting to teach students how to learn enables them to carry those skills with them outside the classroom. I am very grateful to all the teachers that have given me this opportunity. Even though their classes were harder at first, I feel more ready for college and life because of them.

My colleague, Roger Winn, and I presented at the IB Global Conference held in New Orleans from July 24th - 28th, 2019. Our session had two goals: 1) to have attendees revise a current lesson that is more teacher-centered into one that is student-centered using the Cultures of Thinking framework, and 2) to have the attendees experience what it is like to be in a student-centered classroom by running our session as we run our classrooms in Michigan.

We started out the session by utilizing the Compass points thinking routine, a routine included in the "Making Thinking Visible" book by Ron Ritchhart, Mark Church, and Karin Morrison (note: a second book on Visible Thinking, Making Thinking Visible 2.0, is due out soon!). If you are not familiar with the routine, the points on the compass are triggers for participants or students to think about different aspects of whatever it is that you are doing. In figure 1, the prompts for each direction of the compass are given (Note: these prompts are a bit different than what is described in the official Compass Points thinking routine description). Participants had about ten minutes to think individually and jot down thoughts on sticky notes. After this individual thinking time, we had the participants stand up, find a person that they did not know from the conference, and share their thoughts about what they wrote. This time served two purposes: 1) to give participants time to talk over their thoughts, and 2) to give Roger and I time to circulate around the room to listen to our participants. This is very common in our classrooms as well - active listening to our students.

We started out the session by utilizing the Compass points thinking routine, a routine included in the "Making Thinking Visible" book by Ron Ritchhart, Mark Church, and Karin Morrison (note: a second book on Visible Thinking, Making Thinking Visible 2.0, is due out soon!). If you are not familiar with the routine, the points on the compass are triggers for participants or students to think about different aspects of whatever it is that you are doing. In figure 1, the prompts for each direction of the compass are given (Note: these prompts are a bit different than what is described in the official Compass Points thinking routine description). Participants had about ten minutes to think individually and jot down thoughts on sticky notes. After this individual thinking time, we had the participants stand up, find a person that they did not know from the conference, and share their thoughts about what they wrote. This time served two purposes: 1) to give participants time to talk over their thoughts, and 2) to give Roger and I time to circulate around the room to listen to our participants. This is very common in our classrooms as well - active listening to our students.

After the participants had time to talk over their thoughts from the Compass Points activity, we showed the four pillars that we use to revise a lesson: novel application, meaningful inquiry, effective communication, and perceived worth. These pillars come from the 'Opportunities' cultural force that is part of the book "Cultures of Thinking: The 8 Forces We Must Master to Truly Transform Our Schools" developed by Ron Ritchhart, Senior Research Associate for Project Zero, at Harvard University (see Ron's website here). The framework consists of 8 cultural forces that exist in every classroom, and it is the harnessing of these forces that can create a collaborative, student-centered environment. In this part of the session, Roger and I mentioned a quick blurb about each of the pillars as well as the key questions one should ask when revising a lesson with that pillar in mind, as shown in the blue box below.

To give the participants an idea of how we approached this idea of lesson revision, we handed out an example of a lesson revision for math and one for chemistry. Each lesson showed the way that we previously taught the lesson, which was in a more teacher-centric classroom, and compared it to how we might teach it now after applying the guiding questions from each pillar shown above. We gave the participants some time to annotate the documents and generate questions surrounding their thinking (one item worth noting here is that in my math classroom, I wouldn't have my students annotate revised lesson plans, of course, but I do have them annotate other things, like student work, technical writing, or thinking routine notes). Then we used the Q & A feature in Google Slides to send a link to the audience so they could submit questions. If you have never used this feature, it is very worthwhile! Once you go into Q & A mode, google slides will put a link so participants can ask their question as shown in figure 2. A sampling of the questions from our participants is shown below.

We then took a few minutes to address some of these questions while tabling some of them, noting to the participants that in our classrooms, we do answer some questions directly, but there are also those questions that are 'left hanging' at the end of the block, which is perfectly acceptable!

The next portion of the session allowed for the participants to form a group of two or three and revise a lesson of their choosing, using blank copies of the templates discussed in the math and chemistry examples above. The participants were encouraged to find other people that they had not met to work with. This time allowed for Roger and I to circulate and join in the conversations, and to help where needed. Here are some pictures that capture this part of the session:

The next portion of the session allowed for the participants to form a group of two or three and revise a lesson of their choosing, using blank copies of the templates discussed in the math and chemistry examples above. The participants were encouraged to find other people that they had not met to work with. This time allowed for Roger and I to circulate and join in the conversations, and to help where needed. Here are some pictures that capture this part of the session:

In my opinion, listening and talking with small groups like this is the most exciting part of presenting at a conference, and is the most exciting part of a classroom! Take a look at one of the conversations I had with one of the groups:

Jeff: "I like to call it home thinking - it takes the stigma away."

Participant 2: "Home thinking - I like that. I'm going to use that."

Jeff: "So, to answer your question, the home thinking is directly related to what the thinking was in class for that particular day. It could be a desmos activity, an annotation of another student's thinking, a partner thinking activity where they have to collaborate and solve problems together before the next class, or, it could be a set of problems that they have to practice. It all depends."

Participant 3: "It's not just drill and kill everyday then?"

Jeff: "No - far from it. Drill and kill is needed at times, but not everyday, and not just because of the old adage 'well that's how we have always done it.' I try to give home thinking that makes the students think as critically outside of class as they do inside of class."

Participant 1: "So the planning for this type of classroom really revolves around the students and where they are at, and not necessarily a lecture where you finish talking with one minute to go and you say 'okay, we've covered it.' "

Jeff: "Correct - although you do have to have guidelines and general time frames in mind. This is where the real 'art' of teaching comes in to play - how much time is spent here? how much do I talk? how much do the students talk? what are the students actually doing? are they passively receiving information, or fully engaged. This is why I think 'section teaching' in math doesn't work well in a student-centered classroom."

Participant 2: "It seems like a large shift, and one that could take considerable time."

Jeff: "It is a large shift, but just start small. Change a few small things in the classroom, and see how it goes, and then expand from there!"

| As you can see, listening in and joining in is the fun and informative part, and I find the same to be true in the classroom. Although the box above shows a dialogue between the participants and I, many times in my classroom I just circulate and listen to them, without interjecting. I learn so much just by listening to my students, it is mind blowing! I used to think that learning in a classroom meant that the teacher needed to primarily talk, while the students listened; now I think learning in a classroom means that the students primarily talk, and the teacher listens! Once the participants had time to revise their lesson with other people, we had everybody move around again and find a new person that they did not know. The idea this time was to share one excitement and one struggle about their lesson revision. Again, Roger and I listened in, and interacted as appropriate. |

Our final part of the session was to have the participants utilize the **I used to think....now I think** visible thinking routine. The participants were asked to write down what they used to think about student-centered learning, and what they think now about student-centered learning after their time with us. Here is a sample of responses:

As Roger and I reflected on our session, as we currently do with our lessons, of course there were tweaks that we would like to make, but for the most part we met the two main goals outlined earlier: to have participants revise a lesson and to experience a student-centered classroom!

If you would like Roger Winn and Jeff Watson to run a session on student-centered learning at your school or district, please contact us at rwinn@bloomfield.org or jwatson@bloomfield.org.

]]>If you would like Roger Winn and Jeff Watson to run a session on student-centered learning at your school or district, please contact us at rwinn@bloomfield.org or jwatson@bloomfield.org.

Cultures of Thinking is an educational framework developed by the Project Zero team at Harvard University. The framework brings to light the eight cultural forces that are at play in our classrooms: Expectations, Language, Time, Opportunities, Routines, Environment, Interactions, and Modeling. These forces are at play in every classroom; CoT provides tools for the teacher to properly harness these forces to produce 21st century, globally minded learners and thinkers. Teachers can use a variety of methods to create a Culture of Thinking in their classrooms, such as allowing critical time when asking and answering questions, using key language to drive deep thinking, allowing key interactions to occur among the students, to name a few. The tools are varied and many, and since this is a framework, not a program, it isn't a matter of doing a couple of things and suddenly the classroom is transformed - CoT is an iterative, exciting, and invigorating journey.

Many times over the past year I've heard people say, 'Well, I can see implementing a CoT in humanities, but it won't work in the Math classroom.' My response to this has been, 'Wait, there isn't any thinking in Math classrooms?' Hmmmmm. After spending the better part of four years on this framework, and doing an extensive study on the book, "Creating Cultures of Thinking", I can assure you that creating a Culture of Thinking in Math class is possible, re-invigorating, and your students will thank you for it!

How can you begin to transform your class and begin to implement a Culture of Thinking in your math classroom? Well, the good news is that you are probably already doing many things that address some or all of the eight cultural forces.

Here are some ideas to get started:

Many times over the past year I've heard people say, 'Well, I can see implementing a CoT in humanities, but it won't work in the Math classroom.' My response to this has been, 'Wait, there isn't any thinking in Math classrooms?' Hmmmmm. After spending the better part of four years on this framework, and doing an extensive study on the book, "Creating Cultures of Thinking", I can assure you that creating a Culture of Thinking in Math class is possible, re-invigorating, and your students will thank you for it!

How can you begin to transform your class and begin to implement a Culture of Thinking in your math classroom? Well, the good news is that you are probably already doing many things that address some or all of the eight cultural forces.

Here are some ideas to get started:

- Perform a self-assessment of the Culture of Thinking in your classroom. Use the rubric here.
- Watch this video to see if you can pick out any of the cultural forces in action. Some of the forces are 'addressed' in this snapshot, and some are not:

- Pick one of the elements on the wheel above, buy the book 'Cultures of Thinking', and begin to read about it!
- Please look through my other blog posts! There is a lot of information and I would love feedback.
- Take the online course "Creating Cultures of Thinking" through the Project Zero team at Harvard University. I am currently an online instructional coach for that course and I can tell you it is well worth the time and effort to take it! Click here for the link to the course.

- Section 1: Graphing quadratic functions
- Section 2: Translating parabolas
- Section 3: Solving equations using square roots
- Section 4: Solving equations using factoring
- Section 5: The quadratic formula
- Section 6: The discriminant and complex numbers
- Section 7: Quadratic systems

Many times at the start of this unit I would show the students how to graph basic parabolas - I would define the key items such as the vertex, line of symmetry, and intercepts, and then graph them. The students would copy what I was doing and then do homework problems essentially replicating this behavior. The students' thinking only went as far as the rote mechanics they were employing when finding the key items.

As I approached this chapter this year, I was thinking about how to organize the lessons to best suit the students. It then led me to the question, 'How can I organize this unit so that I am able to leverage the 8 cultural forces described in 'Cultures of Thinking: The 8 Forces We Must Master to Truly Transform Our Schools' by Ron Ritchhart, in a way that helps students the most?'

I have organized this blog post in to several days. Today I will cover day 1, with other days following soon.

On this day, I wanted the students to speak the language of parabolas. Our foreign language teachers tell me that the best way for students to learn about the language is to speak it - so it seemed reasonable that the best way for students to begin to learn about parabolas is to speak about them. I decided to start the unit by using the "Polygraph Parabolas" activity on www.teacher.desmos.com. All the students need a computer and sign in to the activity after receiving a special code that Desmos generates. I decided to sign in as well so I could do the activity with them. When everybody signs in, Desmos pairs everybody up randomly.

The opening screen of the activity looks like this:

Student 1 of the pair chooses their 'favorite' parabola, and then Student 2 of the pair asks questions to try to figure out which parabola that Student 1 picked. Student 2 has to ask yes-no questions, and then Student 1 answers them. Take a look at the sample interactions below. One nice feature of desmos activities is you can anonymize the names so the students' real identities are hidden:

As you can see from the above interactions, the students must use mathematical language in order to communicate with each other. I used this in more than one block, and the activity would start out very quiet as the students communicated via the computer as they typed in and answered their questions (much like they interact when texting each other). However, in every class, as time went on, the students began to interact with each other with live conversations. It was really awesome to see that unfold. I would hear things like "what do you mean by ...", "when you say the graph is positive, what do you mean exactly", and "when you say minimum, do you mean the vertex or another point?" The best part about this computer interaction and live interaction is the students were communicating with each other with mathematical language. At the beginning of the activity, the language was imprecise but as time went on it became more detailed and mathematically correct. The best part of it was when the students used their own language to start with, such as "x-axis crossing points" and "bottom point", and then to see it slowly transform into terms such as "intercepts" and "vertex."

I let the students play this activity for about 30 minutes. After this, there is another desmos activity related to this called "Polygraph parabolas, part II" that I let the students explore. This activity takes some of the imprecise language, such as 'turning point', 'smile', and 'touches the middle line twice' into language such as 'vertex', 'concave up', and 'intercepts.' At the end of this activity, the students are asked to create parabolas with certain characteristics. Here is an example:

I let the students play this activity for about 30 minutes. After this, there is another desmos activity related to this called "Polygraph parabolas, part II" that I let the students explore. This activity takes some of the imprecise language, such as 'turning point', 'smile', and 'touches the middle line twice' into language such as 'vertex', 'concave up', and 'intercepts.' At the end of this activity, the students are asked to create parabolas with certain characteristics. Here is an example:

There are many cultural forces at play during this day in math class. Here is a look at some of them:

1.**Language**: The language of community is very evident as the students are creating the vocabulary that is used to communicate about parabolas. Often times during the course of the polygraph activity, students ask me 'is this the way to say it' or 'is this right?' My response is always 'are you able to communicate with your partner effectively with the words you are choosing? Perhaps you should talk with them and get the vocabulary ironed out. Later in the week we will see what other mathematicians have used to communicate, and we will also talk about why they might use the language they use.' This allows the students to have a sense of community and to see that math is not something that is defined outside of our classroom - rather it is something we develop as a community of mathematicians.

2.**Modeling**: By participating in the activity with the students, I tried to model that I am a learner with them and that I am not the sage of parabola vocabulary. I wanted them to see me as a role model instead of an exemplar. As they worked through questions about parabolas, I noticed the students leaning on each other to make sense of what was going on - they slowly became independent and confident in talking about parabolas. If I would have used transmission of knowledge as the mode of teaching on this day, students would have missed out on this rich activity and would have seen me as an exemplar instead of taking control of their quadratic vocabulary.

3.**Interactions: **It goes without saying that the Interactions were a big part of this day. Most students interacted with over half the class because when one partnership ends, desmos automatically matches the students up with somebody else that is waiting. As I said earlier, what started out as virtual interactions slowly turned into live interactions as the students were trying to sort out their thinking.

4.**Opportunities**: The best part about providing this opportunity for the students was that it was a low threshold opportunity - all students could participate. Some of the students already knew the 'perfect' parabola vocabulary, while those that didn't improvised by using words that made the most sense to them. Some students were using phrases such as 'bowl shaped', or 'smile' to describe a certain parabola. I think that was great. If I would have just *told them* to say 'concave up' instead of 'smile' it would have been my words and not their own.

5.**Environment**: The environment on this day was one of organized chaos. There was structure for sure as the students were at their computers and playing a polygraph game, but the overall atmosphere was one of curiosity, thinking, and laughter as the students tried to communicate with each other about a topic that was new to many of them.

In upcoming posts, I will describe how to leverage Cultures of Thinking with the rest of the quadratics chapter.

Later this week, I will cover what happened on**Day 2: Marbleslides: Mapping the shape of a parabola to the equation that generates it. **

Please visit again!

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In upcoming posts, I will describe how to leverage Cultures of Thinking with the rest of the quadratics chapter.

Later this week, I will cover what happened on

Recently we started a lesson on similar triangles. I thought a lot about what to have the kids do, and a small part of me (very small) wanted to 'direct teach' the lesson to save time - we are behind, after all, and if I just told them what similar triangles were and what to do with them we could move on to the next lesson. In the past, I would typically give them scripted notes that had a set of similar triangles, such as the example in figure 1. I would explain what similar triangles were, and how to find different things about them, such as angle measures and/or side lengths. The students would typically do well with this and then we could say that we had 'learned' similar triangles.

This year I wanted to provide a richer opportunity for them. I put up the following directions:

**Use your math resources to figure out what it means for triangles to be similar.****Once you are comfortable with the notion of similar triangles, take the colored paper in the back and create a set of similar triangles. You can use any other tools as well, such as rulers and/or protractors.****After you create your set, be able to convince somebody that they are similar by using a mathematical argument**

I purposely made the direction vague. Some students asked me what similar triangles were, but I pushed it back to them to use their resources, such as their phone, computer, or textbook, to figure it out. Some students also asked what I meant by a 'set' of similar triangles. I asked them to look up that word as well and then decide what to do. I had to laugh because some students just really needed me to tell them 'how many' triangles they needed to make. I did not want to tell them because I did not want to give them a threshold at which to stop learning.

As time passed, many students were using their protractors to create two triangles that had the same angle measures. Two seemed to be the most popular number of triangles. There were a lot of struggles going on as measurements were being made and sides were being measured. Some students used a lot of paper, while others used a sheet or two.

With about ten minutes to go in the block, two of my students, Filip and Haden, asked me to come and look at their set. I walked over to their desk and this is what I saw:

As time passed, many students were using their protractors to create two triangles that had the same angle measures. Two seemed to be the most popular number of triangles. There were a lot of struggles going on as measurements were being made and sides were being measured. Some students used a lot of paper, while others used a sheet or two.

With about ten minutes to go in the block, two of my students, Filip and Haden, asked me to come and look at their set. I walked over to their desk and this is what I saw:

Needless to say, I was floored! The expectation in my head was to see two triangles that were created using straight edges and rulers. I of course inquired about what had happened and what led them to this result, and I found out what actually happened: they used one sheet of paper and created thirteen (or so) triangles in about 5 minutes. All of the triangles were similar, but the beauty of it was that they did it by folding the paper - rulers and protractors were not needed (one is shown in the picture, but it wasn't used). The best part was that I was going to demo to the whole class how to 'quickly' create two similar triangles (by overlaying them on each other), and instead I had these two students demonstrate a much more powerful demonstration.

I was really thankful afterwards that I didn't give in to the urge to instruct directly because this opportunity would have been completely lost. Instead, it was gained, and was a nice lead in for the next class, as I had Haden and Filip lead the class on how to construct so many similar triangles in such a short time.

]]>I was really thankful afterwards that I didn't give in to the urge to instruct directly because this opportunity would have been completely lost. Instead, it was gained, and was a nice lead in for the next class, as I had Haden and Filip lead the class on how to construct so many similar triangles in such a short time.

One of the concepts we have been studying in my freshmen math class is solving systems of inequalities, such as the system seen below.

In the past, the lesson would go something like this:

1) I would model how to draw each curve in the system.

2) I would draw dashed or solid lines and explain why they were dashed or solid.

3) I would shade the appropriate region and explain how to determine this.

4) I would ask if there were any questions and would address them with each student directly.

5) I would give the students a different, but very similar example to do and monitor their progress.

At the end of this lesson, students could claim that they knew how to solve a system of inequalities by simply replicating the steps that I gave them in class. I really wouldn't allow much variation in the procedure - I had the math degree, after all, and I told the students that we all need to be on the same page as far as the 'right' way to do this procedure.

This approach made me think of some of the belief sets that are discussed in the Expectations chapter in "Creating Cultures of Thinking: The 8 Forces We Must Master to Truly Transform Our Schools." The belief sets, which are outlined as natural tensions, that came to mind for this lesson are**focusing on the learning vs. the work**, **teaching for understanding vs. knowledge**, and ****encouraging** deep vs. surface learning**. As I reflected on the way that I used to teach this, I imagined that the students had the **knowledge **of how to graph an inequality, that they knew how to complete the **work** required (especially given the fact that I would assign a bunch of these types of problems for homework), and they at least had developed learning on a **surface **level. Obviously, these are not bad things in and of themselves, but how could we develop a **deep** level **understanding **of **learning** that was meaningful? I attempted to do this with the following lesson.

I started class by displaying the problem above. I asked them to draw the solution on their desk (their desks are white boards) using whatever prior knowledge they could remember. There were definitely some cob webs as the students tried to remember what they had learned previously. A lot of discussion ensued: 'how do you draw this parabola?', 'I remember having to use dashed lines - do we do that here?', 'is shading a thing with these problems?', 'I think we shade everything, don't we?'

It was tempting at this point to walk around and quickly check these for the kids and tell them right or wrong. It certainly would have made the lesson go faster, but in trying to foster independence I tried something different. I had the students check in with their group members (the students are in groups of 4) to compare and contrast their graphs and to come to an agreement on what was correct. Then, they had to erase the four graphs and create one that the group could agree on and draw that one rendition. In my classroom there are eight groups of four, so at the end of this there were eight graphs total. Each group collaborated, questioned, erased, shaded, and played until they were happy with their final product. I then had the groups rotate around the classroom. Each group was instructed to carry one marker and to 'annotate' the final product at each group. If they saw something that somebody else had already written, they had to put a star next to it. After a slew of rotations around the class, everybody came back to their home group and were able to see the comments, suggestions, fixes and feedback that were left at their table. I gave them a few minutes to digest everything that they saw.

In the past, the lesson would go something like this:

1) I would model how to draw each curve in the system.

2) I would draw dashed or solid lines and explain why they were dashed or solid.

3) I would shade the appropriate region and explain how to determine this.

4) I would ask if there were any questions and would address them with each student directly.

5) I would give the students a different, but very similar example to do and monitor their progress.

At the end of this lesson, students could claim that they knew how to solve a system of inequalities by simply replicating the steps that I gave them in class. I really wouldn't allow much variation in the procedure - I had the math degree, after all, and I told the students that we all need to be on the same page as far as the 'right' way to do this procedure.

This approach made me think of some of the belief sets that are discussed in the Expectations chapter in "Creating Cultures of Thinking: The 8 Forces We Must Master to Truly Transform Our Schools." The belief sets, which are outlined as natural tensions, that came to mind for this lesson are

I started class by displaying the problem above. I asked them to draw the solution on their desk (their desks are white boards) using whatever prior knowledge they could remember. There were definitely some cob webs as the students tried to remember what they had learned previously. A lot of discussion ensued: 'how do you draw this parabola?', 'I remember having to use dashed lines - do we do that here?', 'is shading a thing with these problems?', 'I think we shade everything, don't we?'

It was tempting at this point to walk around and quickly check these for the kids and tell them right or wrong. It certainly would have made the lesson go faster, but in trying to foster independence I tried something different. I had the students check in with their group members (the students are in groups of 4) to compare and contrast their graphs and to come to an agreement on what was correct. Then, they had to erase the four graphs and create one that the group could agree on and draw that one rendition. In my classroom there are eight groups of four, so at the end of this there were eight graphs total. Each group collaborated, questioned, erased, shaded, and played until they were happy with their final product. I then had the groups rotate around the classroom. Each group was instructed to carry one marker and to 'annotate' the final product at each group. If they saw something that somebody else had already written, they had to put a star next to it. After a slew of rotations around the class, everybody came back to their home group and were able to see the comments, suggestions, fixes and feedback that were left at their table. I gave them a few minutes to digest everything that they saw.

I then put up the following drawing using desmos. I asked them to compare what they had with what they saw from desmos and to write down what they were wondering about. Here are some samples:

- is the blue and the red region necessary?

- is the white region part of the answer?

- i still don't understand the dashed lines

- is desmos right or am I right?

I asked the class "Which region is the correct region? Is it the blue, the red, both of these, or the white? Many students chose blue, some chose red, while some chose both. I called their attention to the fact that in the original system, the word "AND" was used, which in math means intersection.

- is the blue and the red region necessary?

- is the white region part of the answer?

- i still don't understand the dashed lines

- is desmos right or am I right?

I asked the class "Which region is the correct region? Is it the blue, the red, both of these, or the white? Many students chose blue, some chose red, while some chose both. I called their attention to the fact that in the original system, the word "AND" was used, which in math means intersection.

I then showed them the same graph in desmos, except now with some points plotted in the different regions. I asked students that were comfortable to go and plot more points in the various regions, so they slowly went up and put more points on the screen. Instead of asking which color regions were the solutions, I asked which points were the solutions to this problem. Many students mentioned the point (0,4), as well as others that were in the blue, but some still called out points (4,2), and (10,-1) in the red region.

At this point, I displayed the following Venn diagram on the board. I asked them to draw this Venn Diagram on their desk and to place the coordinate points in the appropriate places in the Venn diagram. They had been exposed to Venn diagrams in the past, and for the most part knew that the AND part was the portion in the middle of the diagram. They slowly started placing points in various spots on the diagram, and were trying to figure exactly where the coordinates would fit.

However, the most fun occurred when I asked each group to create their own systems. They could include any functions that they were familiar with as long as they had general knowledge of how to draw them. This was fun to see develop as some groups created a system with a line and quadratic (just like mine!) and I told them to go back to the drawing board and make it a bit more interesting. One group brainstormed a list of the functions they were familiar with before choosing. Still another group picked three functions instead of two. Regardless, the kids came up with much better examples and insight than I could have, and it certainly was much more rich than me just handing them a system to solve!

The eight new systems served as a springboard for more practice. I gave the students a choice on which systems they wanted to tackle, and when they wanted to tackle them, whether it be at home, during class, or after school.

]]>The eight new systems served as a springboard for more practice. I gave the students a choice on which systems they wanted to tackle, and when they wanted to tackle them, whether it be at home, during class, or after school.

Day 1: The students used Desmos polygraph to discuss the vocabulary surrounding the graphs of quadratics.

Day 2: We used the +1 Routine to generate all of our thinking about parabolas and quadratics.

This led to the Desmos marble slides activity where the students wrote equations of quadratic functions in vertex form in order to generate graphs that 'catch' stars.

Day 3: The students continued the marble slides activity by plotting the vertex on each page of the activity, converting the equations from vertex form to standard form, and looking at the 'wideness' and 'narrowness' of parabolas.

For their home thinking that was due today, I had asked the students to manually find the vertex for each of the parabolas in the Desmos marble slides activity and to check their answers as they went along. Each student logged in to their Demos marble slides activity and had it showing on their computer screens, with the home thinking that they created sitting out as well. I used the Team Shake app to mix the class up in to pairs and gave them five minutes to discuss the results of their thinking. I walked around and answered various questions about finding the vertex, looking at the math work as I went. Some students were still having trouble simplifying correctly, while other students had mastered the concept. I then had a discussion with the class about doing more practice with this concept. It went something like this:

<The class and I all laugh at this comment>

At this point I present the class with the following pictures of two different calculator screens:

I then pose this question to them: "Take a look at the calculator screens that I have put up on the overhead. Take a minute and process what you are seeing here. Think about what is happening on each calculator screen and be ready to discuss." The students thought for several minutes. I realized after we did this that it would have been a perfect moment to have the students do a 'See-Think-Wonder' routine; this is essentially what we did, though, but with different words. I asked the students for some input and gave appropriate wait time. The students chimed in and realized that what they were seeing was the calculation of the y coordinate of the vertex, done two different ways. I then asked them which solution was 'better.' This part surprised me, as most of the students chose the picture on the left. Here were some of the comments:

**"I would pick the one on the left because it makes more sense to me"**

"The one on the left makes you think about what you are doing"

"The left one is easier to see what is going on"

"The one on the left gives you the steps, but the one on the right is quicker. So, if you know what you are doing, do the one on the right; if you are still learning it do the one on the left."

For years, I didn't give the students a choice in the matter. I made them calculate the y-coordinate using the screen on the right because I thought it was easier and cleaner; I didn't consider their viewpoint in that the steps made more sense to them if they entered it like the screen on the left. I left this activity pleasantly surprised because their response to this was not at all what I had expected. A student then asked, "Well, which one do you want us to do on the test?" My reply was, "Well, can we ask that question but leave off the phrase 'on the test'?" The student replied, "Well which one do you want us to do?" I said, "Which one makes better sense to you?" She said, "The one on the left for now, but maybe the one on the right." My response was "well, do the one that make the most sense to you. As mathematicians, there are many ways to do the same thing, and we sometimes need to do what we are comfortable with."

"The one on the left makes you think about what you are doing"

"The left one is easier to see what is going on"

"The one on the left gives you the steps, but the one on the right is quicker. So, if you know what you are doing, do the one on the right; if you are still learning it do the one on the left."

For years, I didn't give the students a choice in the matter. I made them calculate the y-coordinate using the screen on the right because I thought it was easier and cleaner; I didn't consider their viewpoint in that the steps made more sense to them if they entered it like the screen on the left. I left this activity pleasantly surprised because their response to this was not at all what I had expected. A student then asked, "Well, which one do you want us to do on the test?" My reply was, "Well, can we ask that question but leave off the phrase 'on the test'?" The student replied, "Well which one do you want us to do?" I said, "Which one makes better sense to you?" She said, "The one on the left for now, but maybe the one on the right." My response was "well, do the one that make the most sense to you. As mathematicians, there are many ways to do the same thing, and we sometimes need to do what we are comfortable with."

I then showed the following slide to the class. I gave them a minute to read it and then asked if they had questions on what they were expected to do. One question that was asked was: "What do we put in each box in the chart?" I said, "Each box is a process in and of itself. Each box will most likely take up a good portion of a sheet of paper or a white board. Feel free to use chart paper, loose leaf paper, a small white board, or one of the large white boards on the wall. It is time to dig deep! One recommendation: start with the vertex form to standard form cell, and then go from there."

The next block of time, over 30 minutes, was one of organized chaos! Students were all over the place in the classroom: some chose to work with pencil and paper, some used small white boards, some stood at the classroom white board, others were using the computer to look up items on the internet. At one point, a group of students was standing around a laptop and when I went over they were watching a Khan academy video about completing the square. I asked them what they were watching and they said that they had read that to convert from standard form to vertex form, they needed to complete the square, and they wanted to see what it was. Now, I knew that completing the square was not a focus of this unit and it would be something that they would do in later years, but for now I did not want to extinguish their energy and excitement.

As I kept listening to the class, I heard things like 'foil', 'simplify', 'completing the square', and 'factoring.' I was excited to hear these key words and phrases. One pair of students were 'arguing' over one of the conversions because one of them entered it in wolframalpha.com and the other on symbolab.com and they got different answers.

One student came up to me and our conversations went something like this:

**Student: ** "Mr. Watson, I think I know how to convert from vertex form to standard form, but I'm not sure what to call it."

**Me: **"well, what is it that you did to do this?"

**Student: **"I used foil and then made it simpler."

**Me: **"well, that sounds like a good way to describe it - let's go with that and we can see what the class thinks later."

**Student: **"We haven't spent any time on this, have we?"

**Me: **"Remember when you did the conversions in desmos last week? And you checked them?"

**Student:** "OHHH!!!! That's what we were doing when we were doing that. It makes sense now!"

This made me laugh as this particular student finally made the connection of something we were doing a while back. This type of thing used to irritate me before, but now I accept that learning is a process and different students process and master things at different rates.

Class was nearing the end so I asked the students to pick two conversions that they felt the most comfortable with at this point in time. I asked them to find two quadratic equations and convert them accordingly for our next class.

Please stay tuned for Quadratics Day 6, where we will fill in the conversion chart, which will naturally lead us to foil'ing as well as factoring!

]]>As I kept listening to the class, I heard things like 'foil', 'simplify', 'completing the square', and 'factoring.' I was excited to hear these key words and phrases. One pair of students were 'arguing' over one of the conversions because one of them entered it in wolframalpha.com and the other on symbolab.com and they got different answers.

One student came up to me and our conversations went something like this:

This made me laugh as this particular student finally made the connection of something we were doing a while back. This type of thing used to irritate me before, but now I accept that learning is a process and different students process and master things at different rates.

Class was nearing the end so I asked the students to pick two conversions that they felt the most comfortable with at this point in time. I asked them to find two quadratic equations and convert them accordingly for our next class.

Please stay tuned for Quadratics Day 6, where we will fill in the conversion chart, which will naturally lead us to foil'ing as well as factoring!