The home thinking that the students had to do at the end of day 2 was the following:
The students turned in their documentation of their thinking as they walked in to class. I looked at it briefly, checked that they completed it, and then handed it back to them. I used the "Team Shake" app on my phone to generate groups of 3. I then asked them to run through the MicroLab protocol to discuss their thoughts on the values of a, h, & k (If you are not familiar with the MicroLab protocol Visible Thinking routine, see the image below). I displayed the slide to the right in class and did not have to explain too much because we had used it before. I just reminded them that as each person is talking, the other two needed to listen only. We ran through the protocol, and then I asked the groups of three to have an open discussion. After this I had the students go back to their home tables and to verify their thinking about the values a, h, & k, and then we had a class discussion about these values.
As a final check for understanding, I had the students close their laptops and answer the following questions:
I circulated around the class and looked over their shoulders at their responses. After about ten minutes I told them to open up Desmos.com to check their answers. I circulated and the students entered the 5 equations in Desmos to check the 'width' of the graphs to verify if they had them in the right order. Some of them had questions, which I addressed and overall the problem went well.
The second question, though, was a different story. I thought it was going to go quickly. I thought to myself, "If you take 3x^2 and shift it left 4 units, you obviously get 3(x+4)^2 and you go on your way." The students had real trouble with this. Many of them wanted to add an x term and a constant term to get the graph to shift. Most students shifted the graph to the left, but not by 4 units. Some of them shifted it left 4, but then it moved up or down as well. I was puzzled as I thought since they had played with the equation y=a(x-h)^2+k in desmos, and tried different values, they would understand the connection. As I questioned more students and tried to get the big picture it was clear to me that they did not see y=3x^2 as possibly in vertex form. So, I wrote the equation on the board as y = 3(x-0)^2+0. Bingo! Lights on! They caught it immediately and went on their way. One of them said to me, "But I thought y = 3x^2 was in standard form?" Another responded to her saying, "It looks like we have an example that sort of fits both forms." Still another said, "What is standard form?" I laughed out loud, not at the kids of course. I explained to him that this was one lesson that did not go as I had scripted, but it was necessary because that is what the class needed.
After we got this squared away, I had them open the Marbleslides activity (they can log in so it is saved - see the sample screen below). I walked around and checked if they had plotted the vertex on their parabolas and they had, and asked them how they found the vertex. One student said, "I just looked at the graph and knew where the vertex was at and made a best guess on what the coordinates would be." I asked how many had did it this way, I about ten of them raised their hands. I said, "How did the rest of you do it?" One student chimed in, "I just looked at the vertex form, took the h and the opposite of k, and knew that that was my vertex. Then I plotted it."
I said, "It sounds like we have two methods: one is looking at the graph, the other is to look at the equation. Which is better?" A student said, "I like looking at the graph and making a best guess. I'm visual" while another said that using the equation to get the vertex was more exact. I said that sometimes in math we have several methods to do things. However, I did mention that if they didn't have the graph in front of them to look at, that they needed to be able to use the equation exclusively to find the vertex.
At this point I projected the marble slide screen shown above for the whole class to see. I then typed in a different form of the equation that was already there, but this time I put it in standard form (see picture below). I asked the kids what was happening here, and I made sure to give proper wait time. Here are a few responses:
I asked the class how many of them had familiarity with FOIL. Most of them raised their hands but there were some that did not. I wrote the conversion on a sheet of paper and when I was finished, I projected it on the document camera. I then asked them to take a couple minutes at their tables to read what I had written and to discuss it. I purposely put a mistake just to see if they would catch it. The discussions were rich as the students that remembered FOIL were explaining it the students that either never saw it before or forgot about it. The one comment that I took away from this was from a student that said, "I knew how to do FOIL, but I never knew why I was doing it."
I then gave them their thinking for the rest of the class period and for home thinking for that evening:
Go to slides 2, 4, 5, 6, 8, and 10 in the marble slides activity and convert your equations from vertex form to standard form. Do the conversions on paper so you can get the mechanics down, and then plot your answer on the marble slides screens to make sure it completely matches the graph that is already there.
As I contemplated what to have the students do after their 'Polygraph: Parabolas' activity on Day 1 (see Day 1 post here), I needed to take inventory of what the students learned up to this point. I tried making a list, and then realized that it would be better if the students made the list instead. This led me to the new thinking routine, the '+1 Routine', under development for the Making Thinking Visible 2.0 book, which is under development by Ron Ritchhart and Mark Church of the Project Zero team at Harvard University.
The routine goes something like this:
I had one person from each group read their list, and then I said the word 'popcorn.' This is a signal to students that I am looking for a comment or feedback on what was just read. The student that did the reading then called on a person of their choice to share their thoughts.
The +1 Routine provided a great recap of the previous day's lesson, and sparked more of the vocabulary that I wanted the students to continue to use.
Where did we go from here? The polygraph activity on the previous day involved only graphs of parabolas - the students did not see the corresponding equations that generated those parabolas. It seemed logical to me that they should spend time associating parabolas with the equations that generate them.
This led me to the Desmos Marbleslides parabola activity. As you can see from the sample screen below, in this activity the students have to use the vertex form of a parabola in order to 'catch' stars with marbles rolling along the parabolic curve. They are given an initial curve, and then have to change the values of a, h, and k in order to get the parabola in a certain shape to catch all the stars. Once they feel confident that their shape will work, the students hit the "Launch" button which releases the marbles (the first marble is the purple dot) which 'hit' the stars. Once all the stars are hit, the student gets to move on to the next screen which is a new curve and a new set of stars.
After the students got a feel for the first couple of slides (teachers are able to pace the activity so students can only work on certain slides at one time), I paused the activity to give a new set of directions. I had them each take out a piece of paper and explained that they would be documenting their thinking as they went along. I felt it was important for them to do some additional thinking about what they were doing with desmos in this activity and to write it down. I have students record all of their desmos activities so they can go back to them later, but I also wanted to see some documentation of their thinking. I wrote the general form of the vertex form on the board, and proposed the following questions as they were going through the activity:
These prompts gave the students some things to think about as they changed the graphs of the quadratic equations. I gave the them time to complete more slides and think about the above prompts, and then explained that their home thinking due the next class was to finish the marble slides activity, and to answer the above two prompts.
In the Day 3 post, I will discuss how the students converted between vertex form and standard form, and how they used Desmos to 'check' if their conversions were correct.
This week in my freshmen integrated math class, the students needed to review and comprehend laws of exponents. I created a sheet with three different types of problems (see figure below). I gave the students 5 minutes to read and annotate the document. Many of the comments I saw were pretty typical: "Why isn't the three inside the radical?", "Does eight to the zero just disappear?", "What does 16^(1/4) mean exactly?" The students then discussed with their groups the various questions and comments that they originally wrote down.
As I listened to the conversations, I began to realize that the understanding of the concepts and rules of exponents was not going quite as I expected. One of the students asked me, "Mr. Watson, are you going to go over these?" Translation: "Mr. Watson, are you going to go up to the front and go through each one for us?" I did not think that going to the front and lecturing was the right thing to do at that point in time. Sure, I would go up and present the rules, answers, and be proud of the clarity of math that I shared with the students, but going by the old phrase 'the person doing the talking is the person doing the learning,' I realized at that point we needed a different direction as a class. In the past, I may have given in and spent the next twenty minutes spoon-feeding the information. Instead, I tried something different.
I asked the class the following question, "Which of you understand all nine problems, and no longer have questions?" Six students joined me at the front and I called them the 'group leaders.' At that point, I used the Team Shake app on my phone to generate six teams (it literally takes a few seconds to generate the teams). I then assigned one group leader per group. I explained to the group leaders that they needed to take their group to some space in the room to ask and answer questions about the different exponent rules and examples that we were learning about. All the groups went to various places around the room - some took small white boards for their group and some gathered around the large white board. The group leaders then began to ask and answer questions to their team. The conversations began to grow and soon the groups were off and running.
I was sure to listen in as the groups discussed, debated, and collaborated on the problems and ideas. It was invigorating to say the least. I was a guide for the groups - giving proper direction when the groups veered off track, answering questions at the appropriate times, and ensuring that all team members were involved in the learning.
When the groups told me they were 'finished', I gave them some extra practice problems. I had the group leaders come up so we could give them a round of applause for leading the groups. After that I told everybody to go back to their home tables, and I explained to them, "In your home tables, take a few minutes and formulate any lingering questions that you may have. You are going to have a chance to ask me these questions so be sure they are good ones!" The groups discussed for a few minutes and when I could tell they were ready, I set my timer for ten minutes. I then told them "You now have 10 minutes to ask me the lingering questions. Let's do this!" There were questions, but I found that they were much deeper than I had expected. Here are some examples:
I believe that both of these questions came to light because the mechanics and rules were discussed, explored, and answered in their groups. Then they had time to ask me more of 'wonder' type questions.
Which of the eight cultural forces were leveraged during this class period? Here is one view:
Expectations: Student independence was actively cultivated by having them work in groups with group leaders and to be actively engaged with each other. The students directed most of the activity.
Language: During the lesson, I tried to give specific action-oriented feedback, such as, "I like how you explained the fractional exponent rule here and the example that you used" and "This was a clever way to explain this to your group - it seemed like they really grasped the concept."
Time: I really tried to monitor the amount of time that I talked during this lesson - it was limited to small chunks of time. The most that I was at the front talking at the class was the 10 minute session near the end.
Interactions: Groups acted independently and many times I just listened in to hear their thinking. Many students challenged ideas, not the people pitching the ideas.
Opportunities: Students got the opportunity to direct their own learning in their groups. The group leaders led the way, but all group members were allowed to questions and direct what was happening.
In a previous article, I spoke of a "lawn" math classroom vs. a "ravine" math classroom. I believe that this class period was definitely exploring the "ravine!"
At the 2017 Project Zero classroom in Cambridge, MA, David Perkins described a setting in which he was sitting on a well-manicured, perfectly green lawn, while a few feet away at the edge there existed a colorful, wild, interesting ravine, full of life and questions. He asked the participants: 'Is your classroom a lawn - well manicured, predictable, and neat, or is it a ravine - wild, messy, with organized chaos?'
This got me thinking about my own classroom, of course, and how for many years, my room primarily fit the description of a well-manicured 'lawn.' But what does a 'lawn' math classroom look like? What are its characteristics? I came up with a short list:
After thinking about this for a while, I remember an example math classroom from "Creating Cultures of Thinking: The 8 Forces We Must Master to Truly Transform Our Schools" by Dr. Ron Ritchhart, a member of the Project Zero team at Harvard University. In it, Dr. Ritchhart describes a math classroom in which he "had a hard time finding moments when students were truly engaged in any thinking" (Ritchhart, p. 39). The teacher was described as very personable, and was reliably consistent with her students, but that it seemed the students "made an internal calculation regarding how much attention needed to be paid to complete the homework successfully or prepare for the looming test" (Ritchhart, p. 40). This classroom seems to be primarily a 'lawn' classroom, with little, if any, ravine characteristics.
Now, to be clear, there is nothing inherently 'wrong' with a having a 'lawn' classroom some of the time, perhaps for part of a class period, or for an entire class period. There needs to be a good balance. The issue, in my opinion, is when a math classroom becomes a lawn day after day, and week after week, because this causes the teacher to do a majority of the critical thinking and the rich opportunities for student learning are lost.
This idea of lawn versus ravine ties nicely with the eight cultural forces that make up the Cultures of Thinking (CoT) framework: expectations, language, time, modeling, opportunities, interactions, routines, and environment. If properly leveraged, the eight cultural forces could create a classroom culture that mimics the 'ravine', with some 'lawn'. However, if left to fester and not leveraged properly, the classroom could become primary a 'lawn', with little 'ravine': predictable, well-managed, but lacking the deep thought necessary for 21st century learning.
So, how do the eight cultural forces 'look' in a math classroom that is primarily a 'lawn'? Let's take a look at each force:
Expectations: Many things are expected of students in this type of classroom. The teacher may say things like:
Language: The key language moves to create a culture of thinking that Ritchhart describes are the language of thinking, community, identity, initiative, mindfulness, praise and feedback, and listening. In a 'lawn' classroom, some of these key language moves may not exist at all, or may be counter-productive to a culture of thinking. For example, if a teacher is continuously the sage and provider of information in a classroom, then students may never hear the language of community and identity. Furthermore, the language of praise and feedback may strictly be a language of praise without a language of feedback. Generic praise comments might include 'good job', 'great', 'brilliant', instead of action-oriented feedback such as 'I like how you completed the square here - I have not seen that before' or 'your graphs here are very detailed and it took very little time for me to get a clear picture of what is happening.'
Time: In a lawn math classroom, time may be used very well and there may not be a second to spare, but the time spent may not be on critical thinking. For example, if a lot of time is spent in lecture or sit-and-get mode, then time is allocated and used, but students are merely scribes at that point, with little or no thinking being accomplished. If this is the case, there might not be enough time for students to process ideas. As a general rule, a teacher should not talk for more than ten minutes at a time in order to give students processing time.
Modeling: The cultural force of modeling as described by Ritchhhart is the teacher as a role model of learning and thinking; in a lawn classroom the modeling may be predominantly instructional modeling which consists of the teacher showing techniques and methods to 'show the work' and 'get the right answer.' As a role model of learning, the teacher should be modeling how to take risks and reflect on the learning. It's ok for a math teacher to say, 'I'm not sure why dividing by 5 stretches the graph - I need to look into that more. It does seem like the graph should shrink. Why don't we do some research and we can talk about it next time and coordinate our ideas?'
Opportunities: What opportunities are 'lawn' math classrooms providing for students? Ritchhart states that 'opportunities that teachers create are the prime vehicles for propelling learning in classrooms.' In 'lawn' math classrooms, the opportunities for rich thinking in which the student examines, notices, observes, identifies, uncovers complexities, and captures the essence of something may be limited. If a math classroom consists of homework check followed by direct instruction, day in and day out, then it is very possible that a student does not think at all, outside of the elementary tasks of 'paying attention' and 'taking notes.' As a math teacher, do you take quality time when planning to ensure that your students have rich opportunities?
Interactions: A 'lawn' math classroom may consist of a lot of QRE interactions, which stands for Question-Respond-Evaluate - the teacher asks a question, a student responds, then the teacher evaluates that answer. This results in a 'Ping-Pong match back and forth between the teacher and a single student, leaving much of the class out of the interaction' (Ritchhart, p. 212-13). The interactions that should be fostered are ones where the students are pushed to reason and think beyond a simple answer.
Routines: A 'lawn' math classroom may be littered with procedural routines, for example:
Environment: Take a look at the image created by Thomas Murray (email@example.com) on the left. Does this look familiar? Not much has changed if you compare the class from 1916 to the one from 2016, yet the world around us has changed in so many ways. In many math classrooms, desks in rows are still a reality. Is there a place for this? Sure there is, perhaps during tests or at a time where a mathematical procedure needs to be shown. However, this should not be the primary setup; it should be changed to allow students to have richer learning opportunities, interactions, collaborations, and discussions.
So, what is a 'ravine', and what do the eight forces 'look' like in that type of classroom? Stay tuned as I will look at that in the next blog post!
David Perkins spoke on the last day of the institute about how to take these ideas back to our schools. He mentioned the 5-year effect: new ideas get implemented, get traction, then slowly lose that energy and eventually die off. So, how do we make sure the energy does not fall away, and that key ideas and frameworks can have a long life? One thing that many schools do use is what Perkins calls the "installation model" of implementing change. Some training happens, maybe posters and brochures are made, teachers implement it, but then over time things begin to change back to the way they were. How do we combat this and ensure longevity? David talks about the "ecological model" of introducing change which has 4 main "legs":
Please check out the full article on "Giving Change Legs"
Wow! What a week! Things are beginning to wind down here in Cambridge and will culminate tomorrow when Tina Blythe and David Perkins, both of the Project Zero team, give a talk on "Giving Change Legs," which I believe will help us to utilize many of the awesome things we have learned here this week. First I included a few pictures. Then I include some discussion on the course "All Learners Learning Every Day."
All Learners Learning Every Day (ALL-ED) by Rhonda Bondie
Creating a Culture of Thinking right from the start
Creating thinking opportunities from a mathematics perspective
What is a color, symbol, and image that captures the Project Zero classroom thus far? Here is my take!
Color: Yellow - I chose yellow because I think yellow represents energy, brightness, and excitement. These qualities have been on display here for two days!
Symbol: No spoon feeding - I believe that to use many of the great ideas presented here at project zero effectively, spoon feeding of students needs to be kept to a minimum. In math class, spoon feeding can take many forms, such as: showing them the recipe to do a math problem as opposed to the students creating the recipe; giving them the answers to a set of questions as opposed to them using other tools to gather the answers, such as Wolfram Alpha, Desmos, or each other; handing them a review sheet of problems that will be on the test as opposed to them coming up with the ideas and concepts that they will be tested on, to name a few.
Image: An awesome buffet - A great buffet, like the one shown at the right, presents a lot of appealing options. We have the freedom of choice to pick the items that we like the most, and also to leave what we won't eat. We also can go back to the buffet later, and pick items that we want again, or new items.
This institute reminds me of that because we have been presented with a plethora of appealing ideas, many of which we want to take, and some that we may not be ready for. We have the freedom of choice to pick the ideas that we want to try to implement, and ones that we may want to come back to later.
Definition of Slow-looking: Taking time to notice more than meets the eye at first glance
The themes of slow looking
I was thinking about the applications to math class, and at first was a bit puzzled as to what that would look like. After discussing it with my study group (STUDY GROUP O ROCKS!), the slow-looking does not have to involve an image, it could be a writing, a poem, an image, or work from a math problem. Many times in class, students will use a solution guide for a math problem, and after a minute or to proclaim, 'I have no idea what they are doing here!' It is in this instance that I think I will walk them through 'slow-looking' at the solution, and remind them of the quote to the left!
The Teaching for Understanding (TfU) framework
We assume we teach for understanding because we equate knowledge and understanding; knowledge is vital and critical, but just having a lot of knowledge doesn't make for understanding. What is it that a child will be able to do better if they understand?
The goals of the Teaching for Understanding framework:
What does it mean to understand?
Identify something that you understand really well. For me, it was finding solutions of a quadratic equation. My evidence is that I can find the solutions in multiple ways; I know what the solutions look like; all methods of solving result in the same solutions.
What does understanding look like?
1. successful at something
2. know the pattern of doing things
3. you can teach it well
4. can analyze into component part and synthesize into whole
6. big picture in relation to details
8. you can make connections
9. have a greater metacognition
10. can problem solve
11. can do it in a variety of ways
What helps understanding develop?
- doing multiple times
- errors, mistakes, mishaps
- mentor, guide, structure
- explain it to someone
- exploring various ways to do something
- different perspectives
- applying, implementing
- try it again and again
What does teaching for understanding look like in the classroom?
Some of the key questions from our study group today:
To encapsulate day 1 of the Project Zero classroom, I am going to utilize the See-Think-Wonder thinking routine from "Making Thinking Visible" by Ron Ritchhart, Mark Church, and Karin Morrison. Below I highlight the steps of the routine and apply it to the first day...
What do you SEE? Today I saw over 360+ educators from 30 countries. I saw an enthusiasm that would motivate any teacher. I saw people that I have read about: David Perkins, Howard Gardener, Mark Church, and Daniel Wilson. I saw a study group of people that I did not know at 9:00 this morning, but by 4:00 was working with them to create a visual arts representation of a poem that we were studying. I saw a campus with incredible architecture. I saw other educators that motivated me to want to be the best that I could be. I saw a thoughtful place that I want to be a part of for the next four days. I saw a place that I can't wait to tell my colleagues about.
What did you HEAR (this technically is not part of the routine, but it was too important to pass up)?
I heard many incredible things:
What do you THINK is going on here?
I think this is an institute that is a must for any educator at some point in their lives. I think I made a great choice in taking the time out of my summer to come here. I think this institute will help my students understand their content, think critically about it, and reflect thoughtfully about what they have learned.
What does this make you WONDER? This makes me wonder why I didn't come to this institute sooner. It makes me wonder how my students learned anything in my early years of teaching. It makes me wonder why I thought I had teaching perfected when I was 25 but how at age 45 I realize I have so much more to learn.
Some other pics from the day...
Many times I have heard fellow teachers say things like, "if I could just teach and not grade", "my whole weekend was spent grading papers", or "I wish I could hire somebody to do all my grading for me!" It is true that stacks of papers are an inevitable part of teaching at any grade and any level, but as Ron Ritchhart states in his book Creating Cultures of Thinking: The Eight Forces We Must Master to Truly Transform Our Schools, "Taking a stack of papers home to read through and mark is rarely an energizing task for any teacher" (p. 94). Amen to that!
Last year, I had four sections of one particular course. Every time I gave a test, I had about 120 tests to grade. If you figure about 15 minutes per test, that is 1800 minutes, which is 30 hours of grading, and that was per chapter. Furthermore, I teach six sections and this only accounted for four of the sections. What about the other two? I knew, and accepted, that grading would have to be done at home, but I simply didn't have time to do all the necessary grading and have the time to plan student-centered, collaborative, thought provoking lessons that I knew were so necessary for an effective math classroom.
So what was it going to be, grading or planning? Could it be both? One answer may lie in a re-allocation of time during the school day. Ritchhart mentions a teacher named Nathan that meets weekly with each of his grade 12 students to give feedback on their writing. Nathan sets up a meeting schedule and each student sits with him and they have a discussion about their paper. He gets to know his students better, is able to establish a rapport with them during these sessions, and is sure that the students receive the feedback (as compared to written feedback where Nathan wasn't sure if students read the feedback or not). When I read this, I thought 'Why couldn't I do this for math tests? Nathan did this for literature, why not Math?'
I put this into action for the third unit test of the year. I built a schedule of 10-minute time slots using signupgenius.com, based on the days and times that I was available. For some of the days, I set up three time slots for the morning, two during lunch time, and three after school. Other days, I didn't have any in the morning, but scheduled three during lunch, two during my prep, and three after school. The time slot allotment really depended on my schedule. Then, I sent a link to the students for the signupgenius.com page. After the students had a few days to sign up, I pulled the tests according to the schedule and had them ready to go. I also made sure that my key was written and that I had two copies of it, one for me and one for the student.
Once the students started showing up for their assigned times, I would give them one of the copies of the key to look at and I would go through the test and score it while they looked at the key. We would talk as I graded and I would give specific feedback on each problem and they would also ask questions about their test and about the material. The session really gave us a chance to talk about what their strengths and weaknesses were on the material, mistakes they made on the test, best approaches to fixing them, how they did on the home thinking that was assigned during the unit, as well as anything else that helped them in their journey through the chapter.
Here is one session that I happened to jot down:
Me: 'Hi. Are you ready?!?!?!'
Student: 'I'm a bit nervous'
Me: 'Well, I can understand, but everything's going to be ok!'
<I start grading the first couple problems>
Me: 'Oh, I like your method here for solving this. I haven't seen this used much.'
Student: 'Thanks. It made more sense to me to do it that way. My friends don't like it, but I do.'
<I grade another problem and the student loses all but 1 pt.>
Student: 'I didn't really have time to study that one. I know you said to look at those types, but I
just didn't have time'
Me: 'That's understandable. It happens. No worries!'
<We finish the grading of the test>
Me: 'What do you think?'
Student: 'I think I did really well. I'm a bit disappointed in myself on a couple of problems, but I
like what I did on the one you mentioned.'
Me: 'Well, don't be too hard on yourself - you have eight classes to juggle and mine is just one of
eight! Here, let's look at your class grade when I enter your test score and home thinking score.'
Student: 'That looks good. Thanks for taking the time to do this.'
Me: 'You are welcome.'
This is just one piece of dialogue, but you can see the power of sitting one on one with them - this is a conversation that I would never have had with them otherwise. When all students left the grading session, they would know their score on the exam, their home thinking score for the chapter, and their current grade in the class.
First, the pros of the new system:
Second, the cons of the new system:
I used this method for the rest of the school year, except for the last test (because I had to turn the test around in only a few days so the scheduling method would not work). I will use it again this year as much as I can. The best part about it is the immediate feedback that it gives to the students, as well as the rapport that can be built between the student and the teacher. Next year, I think I will have the students do a reflection on the grading session, and I will also have a list of the chapter objectives at the grading session so that the students and I can talk about which objectives were met during the exam and which objectives still needed some effort.
Many teachers on our staff began to use this system as well when time allowed and the feedback from them was mostly positive. Hopefully this will help you too!
Jeff Watson is a Math teacher at the International Academy East in Troy, MI. His work as a software engineer made him realize the need for problem solvers and critical thinkers in the workplace today. Jeff believes that the secondary math classroom should be a place of critical thinking, collaborative learning, and exploration which will cultivate the problem solvers and thinkers needed today.