Author: mrdardy

Experimenting with Visible Random Groupings

In some ways I think that I am intellectually adventurous, that I am willing to try something new in my classroom. In other ways I struggle with change. I try to make myself feel better about this by reminding myself that we all struggle with this in varying degrees.

This past summer – my third lucky summer at Twitter Math Camp – I finally committed to trying visible random grouping (to be referred to as VRG for the rest of the post) for this academic year. A little background here.

When I moved up north in the fall of 2007 I made a commitment to not have my students in rows and columns. I no longer felt comfortable with most of my students looking at the back of other students’ heads. So, I rearranged the seats at my old school in ‘pods’ of three or four desks. However, I always let students pick where they wanted to sit. As most of us know, even if WE don’t assign seats, the students essentially do this themselves. I comforted myself by thinking about the camaraderie I saw, by listening in on the lively conversations that did not happen when my students sat as if they were in a matrix, and by the fact that I know that I would have preferred life this way as a student. When I moved to my new school I had two long conference style tables so I had two largish groups of students working with each other. Two years ago I ditched the conference table and went back to pods.

Over the past three summers I have heard more and more conversations about the power of rearranging the students, about shaking them out of these simpler comfort zones and encouraging everyone to be comfortable sharing ideas with everyone else in class. Alex Overwijk (@AlexOverwijk) has been an especially articulate proponent. So, this summer I learned about a pretty cool website (flippity.net) where you can build a roster for a class and anytime you want this program will randomize your class. In groups of 3 or 4 or 5, by the number of ‘teams’ that you want, etc. It creates a cool visual that you can project and the kids get rearranged instead of staying in their friendly neighborhood comfort zone. I committed to trying this for a number of reasons, the primary one being my experience the past few years in Geometry. I had been teaching mostly AP and upper level honors classes and these students mostly knew each other for awhile and they were comfortable sharing ideas and debating/challenging each other at times. Not true of my Geometry class. Even last year’s class which was outgoing, chatty, and engaged. They did a great job in their pods discussing ideas but did not do a good job projecting those ideas out to the class. They always wanted to filter ideas through me and, over the course of the year, inevitably fell into some ruts about who took command when I asked them to work together.

Now, enter VRG. The strongest proponents discuss doing this every single day to continually shake things up. I got a little scared of this because I really value the sense of camaraderie that I have seen developing over the years, so I came to what seems like a nice compromise. On the first day of each week, I shuffle the class. I am now ending the fourth week of the school year and I have some observations I want to share. I am particularly motivated to do so by a twitter chat this morning.

There is sound research in the field about VRG and its effects. This research suggests that the positive effects of this practice are most clearly seen when this happens every day. I do not want to discount this and I do not want to feel like a contrarian. What I want for my classroom is for my scholars to not only know everyone else and hear the ideas of their peers, but I want them to be in a zone that feels comfortable and safe. My prejudice is that this zone is more likely to happen if I have some time to get used to my new teammates. What I have seen in four weeks can be summarized as follows (and I will make separate remarks for my AP Calculus BC group and my Geometry group)

  1. In BC Calculus I have also been incorporating whiteboards that the pods write on together. The combination of whiteboarding (and presenting the ideas of the pod) out to the class along with VRG has been pretty spectacular. Again, these are kids that know each other well, but I have been seeing active conversation across table groups to former teammates that is lively. I can step out of the way and let them bounce ideas around as I wrote about yesterday.
  2. In Geometry we have not done as much whiteboarding, I want to improve on this. What I have seen is students talking to people they did not choose. I see them making guesses to/with their neighbor. I have seen students more willing to stand up and talk. I have heard some lively discussion between students and I know it is not just with their good buddies unless they all magically happen to love each other.
  3. I have been able, in both classes, to call on a wider variety of people because even the shy/underconfident/nervous kid has someone in their group whose ideas they can paraphrase. In the past few years I felt that there was more of a posture of looking to one person in each pod to be the spokesperson. I see less of that now.

 

When I tweeted out my happiness about weekly VRG I was promptly congratulated AND reminded that this would be even better if it was done daily. I may get there, but I kind of feel that this is my 10% moment. That place where I am making a change I know is for the better but I am limiting myself in my own discomfort a bit so that I can still feel sane and effective in other arenas.

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.

Breakthroughs

Today was a pretty blah day until my last period class. My first three classes all had assessments so I had no fun conversations and I watched work pile up. As I came in to my last class of the day – my Geometry class – one of my Geometry teammates was waiting in my room to share that his students had been making some great strides in GeoGebra. He told me that a number of his students were really beginning to dig into what GeoGebra could do for them, especially now that we are talking about transformations. I used Geogebra extensively when writing my text and I borrowed from resources around the web for activities. One of them was an activity called A-Maze-Ing Vectors which had been created by the amazing Jennifer Silverman (@jensilvermath) and we used that activity the past two years. My teammate who had been waiting to share his good news had asked me this past summer about modifying this activity. We had had trouble completing the activity in one day and it did not take up enough for two solid days. He also had an idea about combining vector transformations on objects more complex than points. He created a pretty wonderful adaptation of the activity (you can find it here) and my students worked through it yesterday. I opened class today by projecting the last page on my AppleTV where we had to navigate a triangle through a maze and I invited a student to come up and draw on the TV (with a dry erase marker, don’t worry!) and I cannot tell you how great the conversation was in class. I sat down – a commitment of mine based on my #TalkLessAM session at TMC16 – and just watched the fireworks unfold. Kids were challenging each other, going up to the TV to draw their ideas, debating distances, talking about slope, worrying about vertices colliding with walls and discussing the option of rotating the triangle as it moved. I was SO thrilled with the engagement and the level of conversation. I credit this to a number of factors. The original activity was terrific and my colleague’s rewriting of it is creative and concise. Kids like drawing on a TV – it feels naughty or something. I sat down and got out of the way. Kids had worked this through the day before in their table groups and were invested in both supporting their teammates and making sure that their memory and their perspective was clearly heard. They were supportive of each other and slightly defensive if someone else had a different approach. After a pretty uneventful day at the end of the week it would have been easy to just limp tot he end of the day, but these kids brought each other to the finish line for the week sprinting. I am optimistic that we can pick up with a similar level of energy on Monday.

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!

Questioning Myself

This is going to be a super quick post and I would LOVE some feedback here and/or over on the twitter (where you can find me @mrdardy)

I am one of three teachers of Geometry at my small school and I am also the chair of our department. I feel that the three of us ought to have pretty similar policies to make life feel a little more unified and fair to our students. I have opted in the past to keep most of Geometry calculator free. I feel that this is one of the last opportunities to try and help firm up some number sense and some self reliance with minor calculations. I also encourage my kiddos to leave answers like the square root of 160 as a final answer or even something like ‘the sum of the first 98 natural numbers would be the number of handshakes’ to help offset any anxiety about calculations beating my students down. I also place very little emphasis, pointwise, on arithmetic mistakes. One of my colleagues pretty vigorously disagrees and feels that having a calculator by their side eases pressure, and is simply a more realistic way to approach life for her students. I find myself questioning my decision here since I do not restrict calculator usage in general in my other classes. I do, however, worry a bit about all of this since I am hearing pretty consistently from recent alums that they head off to college and are not permitted to use calculators in their freshman classes. My recent Calculus students report this very consistently, they take freshman Calculus at college without a calculator. I know that there are all sorts of reasonable arguments that we should not make high school decisions based on college realities. I also know that I am hearing back from a small group of students.

So, I guess all of this rambling is really about one thing – Give me advice! Let me know how you approach this question. Does it depend on the level of the class? Is it a departmental decision? A school or district policy? Am I simply holding on to some quaint idea that mental arithmetic really matters? I fear that I am not being coherent or consistent in how I think about this issue. HELP!

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.

 

 

Curious Creativity

A quick post here reflecting on a great solution presented by one of my AP Calculus BC kiddos. They had their first assessment on Friday. At this point, we are doing a quick and deep review of last year’s work. In our school, BC students have already completed AP Calculus AB and we spend this year digging deep and moving into the BC only topics. So, one of the questions I posed was in two parts:

  1. For what value(s) of x is x^10< x^6
  2. For what value(s) of x is x^7 < x^3

 

This, by the way, was a non-calculator assessment. I will be writing soon about my wavering on this issue. One of my students presented the following work:

 

  1. x^10 < x^6 becomes x^4 < 1 and this is true whenever |x| < 1 (other than x = 0) so the intervals are (-1, 0) and (0, 1)
  2. x^7 < x^3 becomes x^7 – x^3 < 0 which becomes x^3 (x^4 – 1) < 0 since x^3 < 0 for all x < 0 we need x^4 – 1 to be positive. This is true when 1 < |x|. So, the overlap here is x < – 1. If x^4 – 1 is negative while x^3 is positive, then 0 < x < 1.

 

What knocked me out here was that he divided in one case (when it was safe with even powers) while he subtracted in the other case with odd powers. Now, I have not had the opportunity to ask him about this yet, but I have to imagine that this was not just luck. I think he had some instinct, and I want to gauge how conscious this instinct is, that there is a problem with dividing by x^3 which can, of course be negative. I get to labor on labor day, we have classes here on this holiday, so I will quiz him a bit about this. I’ll report back.

 

Thanks to Sam Shah for catching a mistake in my earlier version of this post!

Long Overdue Thanks

I will be cross-posting this over at the One Good Thing site as well.

 

I graduated high school in 1982 and I just started my 30th year as a high school math teacher. During my time in school there were three particular teachers that had a huge positive impact on me. My junior and senior English teacher, Mrs. Myra Schwerdt; my junior Honors Introduction to Analysis teacher, Mrs. Sally Giles; my senior AP Calculus BC teacher, Mr. Barry Felps.

I was fortunate enough to have run into Mr. Felps about 8 or 10 years into my career at a workshop. I graduated high school in Jacksonville, FL and I went to college and started my teaching career in nearby Gainesville, FL. My graduate advisor at the University of Florida was also Mr. Felps’ advisor when he had been in school. This made me feel that there was some sort of deep connection there and, in an odd way, reaffirmed my decision to think about teaching. Anyway, I was able to see Mr. Felps in person and thank him for his influence. I hope that this meant something to him.

I never saw either Mrs. Schwerdt or Mrs. Giles again after graduating. I fear that Mrs. Schwerdt may no longer be around (this was a long time ago that I graduated!) but recently a friend and classmate sent me a link to a profile of Mrs. Giles. She had changed careers at some point and was working for a local agency in Jacksonville called Cathedral Arts Project. A local paper wrote a feature on her career there as she was preparing to retire. I reached out to the communications director at that program and she shared Mrs. Giles’ contact information with me. I just finished writing an email to Mrs. Giles thanking her for what she did for me and letting her know that she is a major reason that I chose to do what I do.

I know that it means a great deal to me when former students reach out to thank me or simply to share some story as a way to keep in touch. I have no real way of knowing whether Mrs. Giles will remember much about me as it has been over 30 years now since I was in her class. I also know that she no longer teaches, so this note will not serve as a pick me up on a tough teaching day. But I also know that I fell MUCH better having written this note and I hope that in some way it brightens her day.

Countdown Mode – Ideas I Want to Commit to This Year

Tomorrow morning I have my last committee meeting before classes. Saturday we have a series of orientation activities and Monday we finally meet our new classes. I know that I am more likely to stick to a resolution if I make it public, so here goes a brief post to hold myself responsible.

As I have written before, I attended a morning session at TMC16 this year that focused on creating a classroom environment that encourages discussion and debate. I think that I have done a good job in the past of creating an environment where small groups have meaningful conversations. What I have not done well is to shake up those group dynamics or to help my students take ownership of their own ideas in presenting them to the class at large. I will be making a couple of changes this year to address each of these issues.

  1. Visible Random Grouping – At the encouragement of a couple of Lisas (Lisa Winer (@Lisaqt314) and Lisa Bejarano (@lisabej_manitou)) I will be using flip pity.net this year. I just entered the class lists for a couple of my classes and started playing with it. Pretty pleased so far, I must say. Since I was the type of student who liked to just settle in and speak with the same people all the time, I have given in to that tendency as a teacher. I was convinced by a number of conversations – both in person and through twitter – that I should try something different. I am committed to randomizing my groups at least on the first day of each week. If there is some special activity that needs different sized groups, I will change them on the fly. I am interested in seeing how this play out and I will be writing about this as the year goes on. Two of my classes are currently small enough that we will all sit at one committee sized grouping of tables. The other two will be split into pods.
  2. I am asking the maintenance folks at my school to remove my teacher desk and chair. I want to decentralize myself. Too often students look to filter their ideas through me before they are presented to the entire class. I have a couple of ideas about how to change this. First, by not having a desk there is no logical place to look for approval. I often move around anyway, but I hope that removing my desk means that I need to mingle among groups even more and become less of a central figure int he classroom. I am also committed to an idea I picked up at TMC. When a student has something to say, either a question or a statement, I will sit and that student will stand. We will all attend (hopefully) to the person standing and talking.

 

I am excited about the upcoming year and about these commitments to creating more space for my students’ ideas to take central stage in my classroom. I look forward to reporting back to everyone.