Category: Uncategorized

My ECET^2 Experience

This past weekend I traveled to Ewing Township in New Jersey to attend the third annual ECET^2NJPA conference. A bit of alphabet soup, this name. Let’s dissect it – The ECET^2 is for Elevating and Celebrating Effective Teaching and Teachers. The NJPA refers to New Jersey and Pennsylvania. I found out about this, I think, through a tweet where we were asked to nominate ourselves or somebody else. Since I was not sure of what the conference would be like, I did not nominate one of my department members (but I will for sure next year!) and I nominated myself. I also pitched a conference session. I modified one that I had presented in summer 2015 at a Pennsylvania Council of Teachers of Mathematics conference. I was flattered to have my proposal accepted, especially since we were told at the conference that about 200 proposals were submitted about 46 were accepted. It turned out that the timing of the conference was terrible for my family. My wife works at a nearby college and they were celebrated their homecoming weekend. A time of much stress and many hours for my wife. We also had committed months ago to traveling to see Brian Wilson perform The Beach Boys’ album Pet Sounds about an hour and a half from our home. So, I left around 5 AM Saturday morning and we had a family friend come and watch our 7 year old all day while I was gone and my wife was working. I drove home Sunday afternoon – essentially right past the town where the concert was – to get the family and turn around to the show. Long story short – both trips were WELL worth it. I won’t use this space to expound on the wonders of Brian Wilson, but I will use it to talk about my tremendous experience at ECET^2NJPA.

First things first – I have mostly been going to math professional events the past few years. I attended an EdCamp last year but in the recent past I have gone to 3 twittermathcamps and the aforementioned PCTM. This year I am going to NCTM in Philadelphia for a day, so this event with educators from different fields – administrators, elementary school teachers, special ed folks – seemed like it would be refreshing. I certainly ended up gravitating toward some math peeps, but it was great to be immersed in wider conversations. I also made a commitment to myself to try and go to some sessions that were not particularly math-y. This commitment is tied to the fact that I recently joined a local leadership program being run for educators. One of the goals of this program is to try and develop a school improvement project and to discuss aspects of leadership ranging from department to school to district. So, I am trying to broaden my horizon a bit and immerse myself in school conversations outside of math curriculum, and pedagogical techniques which is where my heart and mind have been living for some time now.

I will start off with my only complaint about the whole weekend. I am a bit of a holdout when it comes to phone technology. I stand out like a sore thumb at twittermathcamp because I am the only one who does not have sore thumbs from texting. I do not have a smartphone and the devices I brought with me (an iPad and a MacBook Pro) both belong to my school so I am reluctant to load much in the way of applications on them unless they are school related (with a few indulgent exceptions like Spotify). So, when I arrived at the conference there was no printout of the sessions being offered or indications of where they were located. We were all encouraged to have an app called Whovo to navigate this. Submissions of session evaluations were also to be done this way. Again, I recognize that I am a holdout, but it felt odd that I had to make a special request to see to agenda for all the sessions. To the staff’s credit, they printed one up for me right away. So, that is the end of my complaining. Now, on to the praise!

The first session I attended on Saturday morning was focused on effective feedback and was presented by Dr. Stefani Hite and Dr. Christine Miles. We discussed some ideas I was already familiar with but the real takeaway for me was a phrase they used that caught my attention. They talked about what they called ‘Feed Forward’ this is feedback or information to share with students about their work that allows them to move forward to grow. I feel that I do a good job of getting work back quickly and discussing issues with problems together in class. But I realize that most of my feedback in this format is looking backwards over what went wrong, not looking ahead to help prevent future problems or prevent the same problem from arising again. I hope that I can wrap my head around this and give my students constructive feedforward to help them grow.

The second session I attended was run by Baruti Kafele a former principal (in fact he exists in our virtual worlds as PrincipalKafele.com and @principalkafele) His session was called Critical Questions for Inspiring Classroom Excellence. He challenged us to answer the following four questions:

  • What is my classroom identity? (Who am I?)
  • What is my classroom mission? (What are you about?) – He called this our what
  • What is my classroom purpose? (Why do I do this?) – He called this our why
  • What is my classroom vision? (Where am I going?)

One of the debates we often have concerns dress code and why we have one (versus why we say we have one – these are rarely the same thing) and he had an interesting story to share that is causing me to think a bit about my stance on this question. He talked about people who wear uniforms. If you see a fireman without his uniform you have no idea who he is and no expectations about him. If you see a chef without her uniform you have no idea who she is and no expectations about her. However, when they are in their uniform you have a set of expectations about how they will perform, about who they are. I am wrestling with where this will go in my mind, but I know that I found it to be a striking conversation. He followed this by telling a story about how his children have expectations of him as dad, his wife has expectations of him as Baruti, we in the room have expectations of him as Principal Kafele. I do want to feel different when I am dad, husband, son, brother, friend, Mr. Doherty…

 

The next session was a wonderful one run by Manan Shah (@shahlock) and his wife Meredith Valentine. Mann is a college math professor (among other things) and Meredith is a second grade teacher. I was drawn to their session for a number of reasons. I was excited to finally meet Manan in person after interacting with him on twitter for some time. My daughter is in second grade so I wanted to hear some second grade stories, and it was time for some math. They discussed some fantastic math games and strategies for sneaking in some high level math ideas with little ones without burdening them with formal notation and imposing formulas. I am going to share at least one of their ideas with my daughter’s teacher. They talked about having two classmates skip count while walking (or even skipping!) together. Imagine this, one person is walking and counts off every second step out loud because two is her number. She goes step, TWO, step, FOUR, step, SIX, … Her partner has the number three so he goes step, step, THREE, step, step, SIX, step, step, NINE,… What a fun activity to plant ideas about common factors, about least common multiples, about a number of number patterns. What was great was that they never used this formal language, just let kids notice things and ask questions.

Session four was when I was presenting my love song to the MTBoS called Escaping the Tyranny of the Textbook. The theme of the weekend was ‘The Power in the Room’ and I felt that my message of self-sufficiency and the power of the resources of educators on the web sharing resources really fit in. I’m glad that the organizing committee felt the same way.

The last session was by Steve Weber (@curriculumblog) and it was called Building a Culture of Learning. I’m glad I was there and I was moved by Steve and the conversations in the room. I’m following him now on twitter and I expect to get some great nuggets from that.

We also had a series of speakers between sessions and had plenty of time to share ideas and stories not only at the conference, which was hosted on the campus of The College of New Jersey, but there was also a lively get together in the hotel downstairs on Saturday night.

I want to make sure that include a couple of important references and thank you notes here as I sign off. The lead organizer for the event was Barry Saide (@barrykid1) but he will be quick to pass off any credit and share it with the energetic team of coordinators and volunteers. I owe a thanks to the Gates Foundation who help underwrite these events. I arrived Saturday morning around 7:30 AM and left Sunday around 1:30 PM. In between I was treated to five nice meals, snacks in between, and a free room at a nice hotel nearby. Can’t beat that cost! Anyone interested in this organization can start out by checking out the website of the local event – http://www.ecet2njpa.org

Special thanks also to Manan Shah (@shahlock) who I have been interacting with on twitter for some time now but until this past weekend he was a virtual friend. He’s a for-real flesh and blood friend now and it was delightful chatting with him at the conference and at the hotel.

 

More Creative Problem Solving

screen-shot-2016-09-22-at-4-25-26-pm

 

The problem above came across my twitter feed this morning courtesy of John Joy (@johnjoy1966) along with the suggestion that this was a problem from a trig unit. John also questioned who this problem would be appropriate for. I told him I would feed it to my wizards in AP Calculus BC. I also had a class coming in right after John posted it so I did not see any of the conversation – this way I could present it to my class with no prejudice about what to say. When they came in the next period I had the problem from the tweet up on the screen with no other support. I simply said that this seemed like an interesting problem and I had not had time to try it myself. I handed out the whiteboards to each desk group – this was their suggestion! – and I got out of the way. I heard them talk about the function being odd so that they knew f (-3) right away. One group found f (6) by imagining it as f (3 + 3). This meant, of course that they also knew f (-6). Progress, right? But nothing about knowing the values of f (1), f (2), or f(3). I asked if they wanted to hear a hint and three students quickly waived off that notion. After another couple of minutes I went to the board and started writing what we seemed to know about the function.   I wrote f (3) = f(1 + 2) and wrote out what the definition of the function suggested. We got an ugly expression for comparing f(1), f(2), and f(3) to each other but it was not promising. I was itching to give them a hint but they were holding me off. One of my students – thinking out loud – wondered if this might be a periodic function based on the values we knew on the board. Another group suggested that it might be a sine function. I hopped at this – another example of how I need to work on developing a poker face of some sort. The group backed up a bit and suggested that they were kind of joking, but I buoyed them up by reminding them of the periodicity suggestion. I finally gave them a vague clue – one too vague to have helped them at all. One of my students during a class warm up a few days before had the back of his book open to a series of formulas and review facts from their study of trig. I reminded the class that I complemented him on that and I pointed out the similarity of the given function to a trig identity involving the tangent function. The kids were a bit flustered claiming that no one remembers these formulas but they sealed the deal right away once they had this fact in hand.

So, what did I learn from them today?

  1. I need to work on my poker face.
  2. I need to stop giving clues, they are too good to need them.
  3. This group of students is super persistent and creative.
  4. The small desk groupings AND the randomization every Monday seems to be working.
  5. The whiteboards give them space to ‘think out loud’ and effectively share ideas.

 

Man, a terrific day in Calculus thanks to my wizards and my virtual friends who prod my brain with their great problems.

Scouring for Resources

In a couple of weeks I will be presenting a session at the ECET2NJPA conference. I’m pretty thrilled about this opportunity to network with other educators and to sing my love song to the MTBoS community. The presentation I pitched is called Escaping the Tyranny of the Textbook. I presented a version of this previously at a summer conference for the Pennsylvania Council of Teachers of Mathematics. The focus of my presentation is to help build a network of people to support each other in the creation (and curation) of meaningful classroom experiences without having to rely on textbook publishers to always be the resource of activities, worksheets, practice problem sets, etc. Those of us who are engaged in the MTBoS know that there are an ocean of resources available to us – blogs, twitter feeds, the MTBoS Search Engine , various virtual file cabinets, etc. I incorporated a bunch of these links into my presentation to PCTM because I had a pretty targeted audience. I expect a wider range of classroom interests in my upcoming presentation, so I want to broaden the scope of the links and resources I highlight. This is where you, my dear readers, come in. Please share with me here in the comments or over on twitter (remember – there I am @mrdardy) any rich resources for social studies teachers, or English teachers, or World Language teachers, or elementary ed folks, etc etc etc

 

Thanks in advance for any wisdom!

Some Class Silliness

One of my pet peeves is when students use acronyms to mask knowledge that they already have, thinking that this somehow makes their life easier. I cannot count the number of times I have heard Precalculus students chanting about ‘All Students Take Calculus’ as a way to remember what trig functions are positive in different quadrants. This, despite the fact that they have known for years that above the x-axis is positive for y and to the right of the y-axis is positive for x. Clearly (to me,at least) this does not add to their knowledge, it just adds a filter that obscures relationships in their knowledge. So, I tell them a story to express my disdain for these kind of meaningless mnemonic devices.

Years ago I had a hilarious student named Jeanine in one of my classes. One day, when walking to lunch, Jeanine calls out to me and says ‘Hey, Mr. Dardy I have a mnemonic to remember how to spell my name!’ She tells me that she just needs to remember

Just

Eating

Aardvark

Noses

In

New

England

I love this story for a number of reasons. More than 20 years after this happened this still makes me giggle and makes me think of a wonderful former student. It is a true story and my students instantly see how silly it is to have this sort of memory device. I do not know if this has resulted in enough of them abandoning mnemonics but it is a fun story to tell.

This year, I told the story to a class with a student named Genevieve. The class took on the challenge of trying to develop a mnemonic for her name. Here is their result:

Great

Eaters

Never

Eat

Vegetables

In

England

Very

Energetically.

 

This makes me all kinds of happy.

Beginning of the Year Updates

A brief post here as our labor day winds down. As with most boarding schools, we actually have classes on labor day.

Yesterday I wrote about the clever solution that one of my Calculus students presented and, as I guessed, he did make a clear decision about when a power of x can be negative versus when powers of x cannot be. I had him present his solution to the class to start things off today. In my Calculus and my Geometry classes I am using flippity.net to generate random groups at the beginning of each week. I am not brave enough yet to randomize groups every day, I feel like it is important to me to have some comfort (even if it is just a few days at a time) within my small groups. I have also (finally) bought some whiteboards and have one at each table group. It has worked fantastically well in Calculus. The kids have been talking vigorously, they have been enthusiastic about sharing their work out to the whole class. One of my goals coming in to the year was to increase student voice – especially in whole class conversations – and so far I have accomplished that in Calculus. In Geometry I am also shuffling groups at the beginning of each week. They have been better at talking in their groups rather than projecting out. This is not surprising to me. They are younger students and generally not quite as confident as the Calc BC kiddos. However, last Friday I was pretty insistent about having students stand and say what was on their mind and to say what their questions were. I made a big show of sitting down and asking students to stand so that everyone could focus their attention. We had multiple solutions to a problem offered and a great question from one of the students about a particular solution. I think that as long as I can be consistent, and insistent, about stepping aside and having students take the lead in the conversations then I think I can make some progress with these students and help set the table down the line for our department in having a student body that sees participation as a central part of their job.

My other two classes are each super small right now at 5 students each. They will each grow a bit but the whole idea of random grouping does not work with groups this small. We all sit together at a conference table. These are the sections of our senior-level elective called Discrete Math. We are having some good conversations about voting and ballot strategies. I was delighted to have one of my students tell me how excited she is that a number of her classes are all touching on the same ideas. She is in an AP Government class and I love the idea that students see that ideas can work across course departments.

 

 

The Language That We Use

I recently engaged in a spirited discussion prompted by Patrick Honner (@mrhonner) on twitter and on his blog. The original post that started this whole discussion can be found here and it is well worth your time. Engaging comments there an on the twitters and a friendly suggestion by Patrick himself has me writing here, thinking out loud.  To set the stage for this post, an image from Patrick’s post is important.

Screen Shot 2016-07-31 at 8.34.15 PM

A quick glance at this certainly suggests that these are congruent figures until you look more carefully at how the question is worded. This is a pretty classic example of the kind of question that makes students think that test writers are gaming the system to catch them in a mistake. We are looking at two figures that are equivalent to each other. A rigid transformation maps one onto the other. However, that mapping does not map them in the order suggested. A classic mistake that I lost points for as a student and one that, sadly, I admit that I have probably deducted points for when grading. The debate on the blog and on twitter raised some really challenging questions about our goals with this type of specificity. Yes, mathematics is a precise language and precision is a powerful habit to try to help develop. However, I keep thinking about my fun Geometry class from last year. When we were discussing how to determine whether  a triangle with given side lengths was acute, right, or obtuse we worked out a strategy where we assumed that the Pythagorean Theorem would hold and we decided what the consequence was when it did not. This led to my students saying things like this; “If the hypotenuse is bigger than we thought it would be, then the triangle is obtuse.” Now, I know that the largest side of an obtuse triangle is not called the hypotenuse. When pressed on the issue I suspect that almost all of my students knew this as well. Optimistically, I want to say that they know this as well, but is is early August… My concern here is that I was letting them down by letting them be a bit lazy with their language. What I did at the time was to gently remind them that hypotenuse was not the best word to use there but I understood what they meant when they said it. Should I have made a bigger deal about this at the time? Was I being understanding and flexible? Was I being undisciplined and imprecise? I suspect that there is a decent amount of both of these in my actions and I have to admit that I did not think too deeply about it at the time. In the wake of the conversation that Patrick moderated, I am thinking deeply about it. It is also early August (again, I note this) and it is the time of year that my brain reflexively starts dwelling on teaching again. I am also thinking about a distinction that I got dinged for as a student but this time it is one that I do not ding my students for. I remember losing points in proofs if I jumped from saying that if two segments were each the same length then they are congruent. This is, obviously, true but I was expected to take a pit stop by making two statements along the way instead of jumping straight to congruence. I know that equivalence of measure and congruence of segments (or the same argument with angles) are slightly different meanings. A nice explanation is here at the Math Forum. But I feel pretty strongly that my 9th and 10th grade Geometry students are not tuned in to the subtle differences and I think I am prepared to defend my point of view that they do not need to be. I want my students to be able to think out loud and I DO want them to be careful and precise in their use of language but I do not want them to think that this is some sort of ‘gotcha’ game where I am looking for mistakes and looking for reasons to penalize them.

I am thankful to Patrick for getting this conversation started and for gently nudging me to try and work out my thoughts more thoroughly on this issue. I am interested in hearing from other teachers – particularly Geometry teachers – on how they try to navigate these conversations. How precise should our high school students, especially freshmen and sophomores, be when discussing these issues?

As always, feel free to jump in on the comments section or reach out to me through twitter where I am @mrdardy

I’m Moving!

Both in real life and in my virtual life, I will be living in new spaces. Sometime in the next month I will move out of the dorm where I have been for the past six years. I’ll still be at the same school and still on campus, but with more privacy and less responsibility.

On the virtual front, I am moving from this comfortable place over to a new address

mrdardy.mtbos.org

 

I’ll be tweeting out links for new posts from there. Apologies in advance for any inconvenience in managing blog readers or stored websites to visit.

 

 

Hello (again) world!

Thanks to the remarkably generous David Griswold (@DavidGriswoldHH) for helping me to set up a new space on the inter webs for my writing. Many of you have read me over at mrdardy.wordpress.com, but I am migrating over to this more personalized space – mrdardy.mtbos.org in large part to advertise my proud membership of the MTBoS community.

 

Later this weekend I will complete my multi-part reflections on the TMC16 experience and I will tweet out that link so that people can find me again. Apologies for any inconveniences regarding blog reader setups and I hope that you will migrate over here with me when I write.

 

TMC Reflections, Part Three

In this post I want to concentrate on a couple of the afternoon sessions I attended. The TMC program (you can find it here) was filled with so many interesting opportunities that I kind of agonized over some of the choices. One that I knew I would attend was the session run by Danielle Racer (@0mod3) discussing her experiences in implementing an Exeter-style problem based approach to Geometry this past year. Danielle and one of her colleagues (Miriam Singer who is @MSinger216) came back from the Exeter summer math program (it is called the Ajna Greer Conference and if you have never been, I suggest that you try to change that!) all fired up and ready to reinvent their Honors Geometry course. Danielle spoke eloquently about their experiences and shared out some important resources. We had a great conversation in the session about the benefits and struggles of problem based curriculum. This conversation tied in to another session I saw as well as some thoughts and conversations I have been having for years. First, the afternoon session that I think linked in here. Chris Robinson (@Isomorphic2CRob) and Jonathan Osters (@callmejosters) are colleagues from the Blake School in Minneapolis.  Chris and Jonathan spoke about a shift in their assessment policy that centered around skills based quizzes using and SBG model and tests that were more open to novel problem solving. I am simplifying a bit here for the sake of making sense of my own thoughts. I thought that their presentation was thoughtful and it generated great conversation in the room. Perhaps we (especially I) spoke out more than Chris and Jonathan anticipated and we ran out of time. Another sign of a good presentation, I would say. When there is more enthusiasm and participation than you thought you’d get, it probably means that you are tapping in to important conversations AND you have created a space that feels safe and open.

These two sessions had me thinking about some important conversations we have been having at our school and I am totally interested in hearing any feedback. The first conversation I remembered was with a student who had transferred to our school as a senior and was in my AP Calculus AB class. She was complaining about my homework assignments which were a mix of some text problems and some problem sets I wrote. She said in class, ‘You seem to think that AP means All Problems.’ A little probing revealed that she saw a difference between exercises and problems. A brief, but meaningful, description I remember reading is that when you know what to do when you read the assignment then it is an exercise. If you read it and you don’t know what to do, then it is a problem (in more meanings than one, I’d say). The next conversation I recalled was with a colleague who has now retired from math teaching. We were talking about homework and the struggles with having students persevere through challenging assignments. He also used this language making distinctions between exercises and problems and he suggested that HW assignments should have exercises and problems should be discussed in class when everyone was working together. He felt that the struggle and frustration of problems when you are on your own would be discouraging to too many students and would likely lead to less effort toward completion on HW. A similar conversation came up with another former colleague who was frustrated with some of the problem sets I had written for our Geometry course. She did not want to send her kids home with HW that they would not be able to complete successfully. I recognized that this was coming from a fundamentally good place. She did not want her students to feel frustrated and unsuccessful. However, I firmly believe that real growth, real learning, and real satisfaction are all related to overcoming obstacles. I have witnessed this recently with my Lil’ Dardy who just became a full fledged bike rider this summer. I heard it from my boy, my not so Lil’ Dardy, who made the following observation recently, ‘You know, I find that I like video games much better if they are hard at first. Why do you think that is, dad?’

I know that we can anecdote each other to death on these issues and I also know that there is not ONE RIGHT WAY to do this. But I am in the process of trying to make coherent sense out of my inherent biases toward problem based learning. I want to have deep and meaningful conversations with students, with their parents, with my colleagues, and with my administration about how to approach this balance and about what a math class should look like and feel like in our school. While I have been writing this I was also engaging in a meaningful twitter chat about some of this with the incomparable Lisa Henry (@lmhenry9) and with one of my new favorite people Joel Bezaire (@joelbezaire) so I know I am not the only one struggling with these questions. Please hit me up on twitter (@mrdardy) or start a raging conversation in my comments section sharing your successes/failures/theories about how to strike a balance between exercises and problems between challenging students while making them feel safe and successful and between running your own classroom with your own standard and fitting in with a team at your school. These are all big questions and I wrestle with them all the time. I want to thank Danielle, Chris, and Jonathan for sparking them up in my mind again and for creating lovely spaces for conversations in their afternoon sessions.

 

Coming soon will be my last entry in this series where I think out loud about the amazing keynote delivered by Tracy Zager (@TracyZager)

TMC Reflections, Part Two

This was my third year in a row attending TMC and for the past two years I was co-moderating a morning session. (Thanks to both Tina Cardon (@crstn85) and Lisa Bejarano (@lisabej_manitou) for working with me the past two years!) While I enjoyed each of those experiences immensely, I must say that this TMC felt a little less stressful for me. There were a number of appealing sessions and two in particular jumped out to me. I was torn between Henri Picciotto’s (@hpicciotto) morning session called Advanced Transformations and the session run by Matt Baker (@stoodle) and Chris Luzniak (@pispeak) called Talk Less, Smile More: Getting Students to Discuss and Debate Math. I chose the latter and it was a pretty terrific way to spend six hours over the three days of the conference.

A little background into why I made this choice. Nine years ago when I moved north I made a commitment to blowing up the traditional rows/columns seating arrangement in my classroom. I had three years of small, moveable ‘pods’ of desks at my last school. Here, I had four years with two large conference style tables before asking for desks and now I am back to smaller pods. I have been explicit with my students about my expectation that they be active participants in the classroom thinking process. I think, for the most part, that I have managed this reasonably well and have generated interesting conversations in class. I believe that my students gain some important skills in being able to think out loud and I am certain that they all benefit from hearing so many voices. What I know that I do not do well enough is to decentralize myself in the classroom. Too often fantastic conversations from small pods gets directed to me instead of to the rest of the class. The students use me and their sounding board and as their speaker and I want to learn how to get out of the way ore often and figure out how to elevate small group conversations to the space of the entire classroom. The course description seemed to match this goal.

It was an extremely popular session and we were kind of crammed on top of each other in our classroom, but it helped to develop an easy, comfortable rapport in the room right away. So, my big takeaways are as follows:

  • I have to figure out some strategy for randomizing groupings somehow. I want to balance what the research says with the norms of my school. I also have to contend with my weakness in bookkeeping. Not ever having a seating chart works well with my lack of attention to this sort of detail. Conversations in this morning session and vigorous twitter conversations have me convinced I need to do something. The big debate in my mind now is how often to shuffle the pod memberships.
  • One remark on twitter today really has me thinking. In debating randomizing seeing every day versus once per week, Anna Blinstein (@borschtwithanna) observes that daily regrouping seems to focus attention on mathematics conversation while weekly regrouping seems to focus attention on classroom discussion norms. I am inclined to think that weekly regrouping will work best with my student body and with their previous experiences. I want to foster some familiarity and comfort in small group conversations and I think that daily switching might make that challenging. I am open to being convinced otherwise.
  • I am inclined to ask my boss to have my desk removed from class so that there is no longer any centralized seat of power of any sort. I think that it would go a long way to creating the classroom culture I want if students came into class and everyone had the same desk.
  • I need to get in the habit of sitting down while a student is talking and have that student stand to make sure that attention is directed to the person sharing their ideas/questions rather than being directed at me to see my terrible poker face in action.
  • I have three large walls of chalkboards. I need my students up and at them regularly. I think that this might look different in my three very different classes that I teach, but this needs to happen.
  • I need to be careful and consistent about the use of language from me and from my students. Chris strongly advocated formal language from the world of debate where students make claims and support them with warrants. This feels like it would work particularly well in Geometry this year.

 

I need to be clear that some of these remarks/reactions are directly prompted by the helpful session that Matt and Chris ran but some of these are older ideas that have been clanging around in my brain. My reactions were given shape by the meaningful conversations we had together in this morning session.