Over the past couple of weeks I have been posting about the quadratic unit that we are covering in my freshmen math class.  Today I am going to deviate from that a bit to talk about a lesson that occurred today.  The first part of the next unit is using exponent rules to simplify expressions.  There are a lot of mechanics involved, and the practice problems can get quite tedious.  
    Lately I have been telling my freshmen that teachers are like personal trainers.  I gave them this scenario: imagine you go to the gym with your personal trainer.  For an hour you watch your trainer run 5 miles, then afterwards you watch them lift weights.  At the end, you leave the gym and say that you "worked out."  Ridiculous right?  The personal trainer should show you some exercises and guide you to best practices, but at some point you need to perform those exercises and routines yourself, and go through the sweat and tears.  Teaching and learning is much like that.  The teacher can guide and show certain things, but the students must perform the exercises and go through the sweat and tears much the same way as in the gym.
   I thought to myself: 'How can I guide my students through these exponent rules while giving them the opportunity to go through the sweat and tears necessary to deepen their learning?'

​​First, I created examples for them and handed them out.  Above is one of the examples that I gave the students.  You can download the full worksheets here.  As you can see, the worksheet isn't anything special - it is something that I've more or less handed out every year.  I then organized the class in to six groups using the Team Shake app.  You can see the organization of the teams on the right.  After the class re-organized themselves in teams, I said the following:
"Okay, all of you are in your teams.  Team 1 is responsible for Example 1 in the packet I gave you, Team 2 is responsible for Example 2, and so forth.  Your team needs to think about the problems in your example and talk together to determine how to do them.  Perhaps some of you have seen this before and you can start the group in that direction, or maybe you want to go to www.wolframalpha.com, type in your question, see the answer, and then work backwards.  As always, you can ask me well thought out questions.  In addition, each team gets one "spoon feed" - I will work out ONE example for you if your group requests it.  I won't tell you what I am doing, though, it will just be the work and process written on a small white board.  Okay - enjoy!"

​I left the team arrangements showing on the screen. The students went off in their teams and began their problem set.  Here are the things I observed:

• Some of the teams would call me over for their one spoon-feed problem, but others in the team would say, "wait, don't use it yet, I can explain it to you!"
• Many students would ask me 'is this right?'  I would turn it back on them and ask them to type it into www.wolframalpha.com.  It turns out, though, that wolfram alpha didn't always have the answers in the perfect form.  Below is one example.  On the left is the original problem; on the lower right is Wolfram alpha's simplification of the problem.  It turns out that it is correct, but not quite as simplified as one would wish.  This led us to a discussion of helpful math websites.

Before we embark on day 4, here is a quick summary of the first three days of the unit:
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.

I started the class with the following prompts:

I had the students read and think about the above prompt quietly for ten minutes.  I then had them have a discussion at their tables.  I picked up some of the things the students were discussing:

• "I think we learned how to convert from vertex to standard form, didn't we?"
• "Standard form tells you which way the graph points"
• "All of the forms tell you which way it points, not just standard form"
• "Isn't the quadratic formula another form, so there are four forms?"
• "It looks like the intercept form gives you the y-intercepts"
• "Having multiple ways to write a quadratic is sort of like having different languages to communicate with each other"

As a teacher, It's always hard not to step right in when you hear incorrect information, but I resisted, choosing to let the 'incorrect' stuff to hang out there for a bit.  After about 10 minutes, I then announced: "Ok, in a moment you are going to have Watson time - a ten minute period where you can ask me anything you want.  So, take a moment at your tables and come up with the burning questions that you just can't seem to answer amongst yourselves and jot them down.  Take a moment...."

This always sparks rich discussion as the students always want to use Watson time wisely.  The funny thing is that they can ask me any questions at any point in time during class, at lunch, after school, or by email.  However, when I set aside a block of time in class, they always use it as if it is an exclusive time to ask questions!  Here are some of the questions that they came up with:

Question 1 comes from Luke
Luke: "Does intercept form mean x-intercepts or y-intercept?"                      
Me: "What happened when you graphed it in Desmos?"
Luke: "I didn't."
Me: "Take a minute to graph it, jot down the intercepts, and then we can talk some more."
<A couple minutes go by as I answer another question from a different student>
Luke: "I graphed it.  The y-intercept is 6.  The x-intercept...I think there are two.....1 and 3?"
Lizzy: "Yeah, there are two.  Just take the opposite of the numbers in the intercept form and they are the x-intercepts."
Me: "So, what do you think about your original question?"
Luke:: "I forgot what it was.....ok, wait, I remember 'does intercept form mean x-intercepts or y-intercepts'.  It should be called x-intercept form, it means x-intercept form."
Saif: "Why do they call it that?"
Me: "Who?"
Saif: "The people that named it."
Me: "Well, I got this name from a textbook and a website that talked about quadratics.  They are mathematicians just like you - they chose a convention and it stuck."​ 
     
Question 2 comes from Angela
Angela: "I like the vertex form because you don't have to show work to get the vertex.  It's just right there.  Do we have to show work on the test if we get vertex form?"
Me: "Can you ask the question again but leave out 'on the test'?
Angela: "Do we have to show work if we get vertex form?"
Me: "Did everybody hear that question?"
<a few students shake their heads no>
Me: I reiterate something that I have mentioned time and time again.  "Remember, this isn't just a discussion between me and the person asking the question.  It's a discussion between all of us.  Please repeat the question."
Angela: "Do we have to show work if we get vertex form?"
Me: "What do all of you think about that question?"
Will: "Wait, show work when we are doing what with vertex form?"
Angela: "If we have to find the vertex, and we get a quadratic in vertex form, do we have to show work to get it?"
Me: "Can we have some opinions on this?"
<Hands start to go up, and I give wait time>
Will: <This student looks at me, but I redirect him to talk to the person that asked the question, so he does and they make eye contact with each other> "I don't think you need to show math work, the vertex is right there and that is the cool thing about that form - you can look right at it and get the vertex."
Me: "How does that answer sound?"
Angela: "I think it makes sense."
Me: "Can we get some other thoughts on this?"
Haden: "I would do the work anyway.  FOIL it out and find it using the formula."
Richard: "What formula?"
Danielle: "Yeah, I remember a formula too.  It's -b/(2a)."
Me: "Okay, Danielle recommends a formula.  Let's try it out.  Open up your Desmos marble slides activity.  You already plotted the vertex by looking at the graph and by looking at vertex form.  Haden said we should use the standard form and then use the formula that Danielle is proposing.  Haden, is the formula that Danielle remembers the same one that you remember?"
Haden: "Yeah, I think that's it."
Me: Okay, on slides 2, 4, 5, 6, 8, & 10, try the formula x = -b/(2a) and see if you get the x-coordinate of the vertex.  After you try a few, go over to Haden and Danielle and see if what you are doing matches what they are doing."

At this point, the class is headed in a new direction.  I realize I broke my promise of "ten minutes" to ask me questions, but I think this is ok because the discussion led us this way and important points and concepts were brought up.  The students work on finding the x-coordinate of the vertex for a while and many of them do not get the same value they currently see on their marble slides screen.  Look at the sample marble slides screen below. 

Many students plugged in 3.2 instead of -3.2, which caused their value to be incorrect.  They were able to self check, though, because the vertex is plotted for them on the screen.  I surveyed the class and asked them how many were able to get the correct x value of the vertex using the formula.  A group raised their hands, so I asked them to go to other students to check in with them to see why their values were not matching. 

After a while, I ask how many of them had seen the x=-b/(2a) formula before that Haden and Danielle proposed.  All but two students raised their hands.  Twenty-six out of twenty eight had done this before, but there were a good number that were not successful in getting the correct x-coordinate.  I then asked if any students knew how to find the y coordinate of the vertex.  Several students raised their hands, and I ask them to form a group and talk over their methods, and to put any and all methods on the side board.  

What is interesting is that some students just plugged in the x value to the original equation to get the y-coordinate of the vertex, while other students used the formula y=-b^2/(4a) + c.  At this point we have two 'competing' methods on the side board (unfortunately I didn't take pictures of this math work).  

I ask the class for their attention at this point:
"Okay, please put your pencils down and close the laptops just so we can focus on something else.  Everybody look at the side board.....can we put our pencils down and close the laptops?  Let's look at the side board.  Now read through the methods you see and first seek to understand what is going on."

At this point the students read the side board for several minutes.  I then hear chatter as they try to understand what is happening.  As students start asking questions, I direct them to the students that wrote the work on the board.  Those students head to the board and explain their processes.
I then say, "I think it might be a good idea if we all go to slide 5 and try to get the x and y coordinate of the vertex using the methods we discussed.  Why don't you try both methods to get the y coordinate and see which method you like better?"  This takes a good amount of time as some students struggle through the mechanics of finding the vertex.  Several students finish quickly, so I ask them to wander and help others.  

At this point in time, class time is about over.  I realize this is not at all how I planned for things to go (I planned on having them convert between the different forms of quadratic functions), but the quality discussions and learning they were doing made it too valuable to get upset about keeping to my schedule.  This is material that we were going to cover anyway, it was just moved forward based on the needs of the class.

I then ended the class.  "Wow, we definitely accomplished a lot today.  This is not exactly what I had planned today, but I think this is what we needed at this point in time.  So, for home thinking for next time, find the vertex of the equations on the rest of the slides by using the formulas proposed today.  Let me know how it goes."

Please check back soon for Day 5!  I will be discussing the results of finding the vertex, and then move on to converting between the different forms of a quadratic.

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​Happy reading!

The home thinking that the students had to do at the end of day 2 was the following:

• What effect do the values of a, h, & k have on the graph of y = a(x-h)^2+k?
• Identify the vertex on slides 2,4,5,6,8,10 in the Marbleslides desmos activity.  Plot the vertex in desmos.  How does the vertex relate to the vertex form of the equation on each slide?

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:

• ​"There isn't any difference, they overlap each other."
• "Is desmos working right, I don't see anything different?"
• "It's right, you just FOIL'ed to get the new equation"
• "What's FOIL - did we do that last year?"
• "FOILs easy - we don't have to do it this year"

​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:  

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:

1. ​Recall: In 2-3 minutes and working individually, each learner generates a list of key ideas that they remember about a topic.
2. Add (+) 1: Learners pass their papers to the right.  Taking 1-2 minutes, each student reads through the list in front of him/her and adds one new thing to the list.  The addition might be an elaboration (adding a detail), a new point (adding something that was missing), or a connection (adding a relationship between ideas).  REPEAT this process at least two times.
3. Review: Return the papers back to the original owner.  Learners read through and review all the additions that have been made on their sheets.  At the same time, they may add any ideas they have picked up from reading other's sheets that they thought were worthwhile.

The reason I decided to use this routine is because I wanted to see what key ideas the students remembered from the Polygraph activity, and how well they committed them to memory.  I had the students close their laptops and put away their source material.  I gathered the students in groups of five for the routine and I handed each student in the group a different colored pencil.  The routine went something like this:

• Me: 'Okay so each of you take out a piece of paper.  I know you get to work collaboratively a lot in this class, but for right now I want to know only what is in your mind.  So, when I say 'go', I want you to write down any key ideas about parabolas that you can remember.  Remember, this is your own work.  You will have exactly two minutes.  Go...'
• The students wrote their lists for two minutes.  Each student in the group used a different color so they could easily see who was writing each part.  
• After the 2 minutes passed, I said 'Now, take your list and pass it clockwise.  Read the list.'  I paused here and gave them time to read.  Then I said 'please add one new thing to the list.'  After a minute, a student said to me, 'Mr. Watson, I added a new item, but what if I saw something wrong with what is currently there?'  This caught me by surprise because I didn't think about this.  So, I had them pass the papers again in the clockwise direction, and gave a slightly different direction this time.  I said, 'This time, read the list, and then either correct something that is there, or add something new to the list.
• This continued for five iterations before the papers went back to the original owner.  I then had the students read their original paper.  I told the students to chat with their group about what they had read.  Since the writing was in different colors I told them to chat with the person with that color if they had questions or comments.  Here is some of what I overhead the students saying to each other:
  • ​"Wait, this isn't incorrect?  Why did you correct this?"
  • "I'm not sure what you meant by 'middle number.'  Can you explain this?"
  • "Completing the square.  What is that?  Did we do that the other day?"
  • "I still don't understand 'line of symmetry.'  Somebody wrote x equals negative b over 2a.  That didn't make sense the other day either."
• This sparked rich interactions between the students.  I'm glad that we spent the time to do this as it brought their current thoughts about parabolas alive on the paper.  Some students wrote items that they had learned in middle school, but were not yet discussed this year.  In the past, I would tell students 'not' to discuss those things because we hadn't 'covered' them yet.  Now, I encourage all prior knowledge and want the students to share it.

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:

• What effect do the values of a, h, & k have on the graph of y = a(x-h)^2+k?
• Identify the vertex on slides 2,4,5,6,8,10 in the Marbleslides desmos activity.  Plot the vertex in desmos.  How does the vertex relate to the vertex form of the equation on each slide?

​
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.

​Stay tuned!

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:

1. "Mr. Watson, why is it understood that there is a two outside the radical even if it is written there."  Another student answered it, but started to look at me.  I told him to 'talk to her about it' to direct his attention to the student asking the question.  He said, "Since it is called a square root, it makes sense that the symbol for a square, a two, is understood to be written there."
2. "Why was this symbol chosen for a radical?  Why not something else?"  I said, "Well, mathematicians such as yourselves choose symbols all the time to represent things.  You could pick a better symbol, but then what would happen?"  Another student chimed in, "People wouldn't know what we were talking about if we changed symbols - they wouldn't know what we meant."  Another student said, "Well, how do we get things noticed like this?  Is there a record somewhere?"  I then replied, "There are professional math journals that mathematicians use to publish items of interest."

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?'

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:

• 'Turn in your homework from last night so you can get the points'
• 'Pay attention while we go over the answers to the homework'
• 'Listen while I answer homework questions'
• 'Sit quietly while I show you the procedure on how to do this math problem'
• 'Practice the procedure I just showed you - you need to know how to do this'
• 'I expect you to do every problem on the homework that was assigned'

In this classroom, the expectation of the completion of math problems may trump thinking and learning.  Furthermore, student dependence on the teacher may develop where the students depend on the teacher to "answer all questions and direct all activity" (Ritchhart, p. 323).

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:

• 'Homework will be checked every class period and if every problem is attempted, you get full points.'
• 'During lecture, take notes and be sure to raise your hand if you have any questions - don't ask your neighbor because that may interrupt the lecture.'
• 'When the lecture is finished, let's have quiet time so you can work on homework.'

Again, there isn't anything wrong with these types of routines, but if these are the only ones, and visible thinking routines are non-existent, then this may be an area that could be changed.  Thinking routines, on the other hand, can be used to deepen understanding and for a "platform for discussion, rather than work to be done" (Ritchhart, p. 325).

Environment: Take a look at the image created by Thomas Murray (thomascmurrayllc@gmail.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": 

1. Frameworks - pick a few strong frameworks; don't overload with frameworks because that will cause chaos
2. Leadership - have two leaders: the political visionary and the practical visionary.  The political visionary approves budgets for the change, sells the change to parents and the staff, is a champion for the cause, and most likely is the principal and/or assistant principal; the practical visionary has a deeper role - they manage the program "on the ground", talk about it with the teaching staff, and is usually a teacher or group of teachers.
3. Community - this is the community of educators and leaders in the building; start with a small group of people invested and let the group slowly grow; ensure that all staff members are welcome to join at every step along the way; you don't want to alienate staff that isn't ready or is resistant - make sure that they have options to hop on board at all times; you don't want it to turn into an 'in-group' and 'out-group' situation
4. Institution - this does not have to be looked at until you have made it to about year 3 or year 4; once you have made it this far, the innovation has to get written into the DNA of the school;  what has to happen is an explicit effort to ensure that if principals or teachers leave that have been part of the innovation, the new teachers and leaders are brought up to speed

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
Key takeaways:

• ALL-ED defines differentiated instruction in terms of teacher decision making.
• "Differentiated instruction is a continuous teacher decision making process where teachers search for and respond to academic diversity that will either impede or strengthen effective learning within a community.
• Structures for classroom routines (see figure 1 below):
  • Note the effect that explicit instruction has on student autonomy and student engagement
  • As the structures change, students become more engaged, which isn't surprising.  However, the wow moment for me was the white dashed line that represents teacher thinking time - as structures in the classroom drift away from explicit instruction, teachers have more time to think about how the lesson is proceeding so they can make informed decisions about what to do next.

• A visual describing the learning that happens in a classroom.  If a classroom is primarily set up for 'independent' or 'direct instruction (DI)' learning,  student engagement and autonomy is low, and the teacher does not have as much time to think about the path of the learning.  There certainly is a place for those types of learning, but they cannot be the driving force in a classroom.

• There are 3 forms of differentiation, based on the goal, materials, & assessment.  When most of us hear 'differentation', we automatically think of the third option of different goals, different assessments, and materials, when in fact there are two other types. 

• One interesting method to group kids that I learned in this session was called a triad method of grouping.  In the figure below, you can see a number surrounded by the letters A, B, & C.  The number is the group number.  The letters are where each group member puts their foot and it is their spot in the triad.  One way to use this is to put these sheets around the room (there were ten of them in this session), and then have the class get up and move to a triad sheet; the students would pick a sheet at some place around the room, and put their foot at their letter.  This is one quick way to group the students while giving them a choice in doing so.  Our instructor said that she left her triad sheets around the classroom all year long - it just became part of the routine.

• One of the best parts about this session was about student choice.  Rhonda believed that giving choice to students makes the students part of the learning instead of being just an observer.  She gave a nice example from a math class.  The students were learning about fitting a line to a set of points.  One option would have been for the teacher to use direction instruction to explain the concept.  Instead the teacher gave the students 15 minutes and during that time they had the choice to do one or more of the following: 1) look at the steps to fitting a line to a set of points (the steps were written on chart paper and posted in the classroom, 2) brainstorm with a partner on how to do this process using the graphing calculator 3) watch a video playing at the front of the room that shows somebody doing the steps for line fitting.  The students knew that at the end of the fifteen minutes that they would be held accountable for that learning, but they had a choice in how they attained that learning.  Please watch this video which shows the teacher giving the students this choice in their learning: