We have two cats, one is named Olympic and one is named Titanic. Here is a picture of them.
The black one is Titanic and he is the subject of our conversation.
We have a neighbor cat across the street who closely resembles Titanic. The neighbor cat, my daughter has dubbed this one Mr. Whiskers due to his long white whiskers, was sitting on his porch Monday morning. Lil Dardy says that she likes Mr. Whiskers because he makes her think of what Titanic will look like. She said he looks like Titanic five years from now. I proposed that, perhaps, this was a Titanic from the future who traveled in a time machine to look after his younger self. Lil Dardy responds by telling me that scientists are working hard on building a time machine. It would be great, she informs me, for kids who don’t like school. They can just skate past their school days in the time machine. She then gets serious and says ‘Dad, I’m sorry but I think I like science more than math.’ I assure her that this is not an insult and I follow by asking why it is she likes science so much. Her quote was pretty great. ‘In science you think of something and then try to make it true.’
Pretty great conversation to start my day.
]]>In one of the wonderful synchronicities in life that make me so happy, I received our staff received an email about a program led by a local leadership group. They launched a class last year for local teachers and the capstone of the year long class is a school improvement project. Two of my colleagues participated last year. One of them launched an initiative related to the libraries on our two school campuses and the other launched a character awareness/character development project that is run by students. I saw this email as my opening to formalize this goal of mine and to have a reason to seek participation from a number of my colleagues. I asked for permission to apply since the program would require me to miss one day each month from September through April (our last class meeting is this Thursday) and this would put a bit of a strain on my students and my colleagues. I do not think that I have missed 8 days of school combined in the last three years, so a guarantee of 8 absences in one year felt like a huge commitment. I was approved by my school and accepted by the program and I am glad that it worked out that way.
Over the course of the year, we have had a number of pretty inspirational speakers and conversations. I have met colleagues from local schools and learned about their school cultures. I have learned a great deal of local history that I was unaware of before. Overall, it has been a worthwhile experience and I am recommending one of my department colleagues for the program for next year.
I pitched the idea of launching an #ObserveMe initiative at our school to my upper school peers at a faculty meeting late in January. We had five weeks of uninterrupted school scheduled between the end of our spring break and a long Easter weekend. I pitched this time period for the project and I solicited volunteers. I had nine colleagues volunteer to join the project and they came from everywhere in our school. An administrator who teaches history, a school counselor who does not currently have classroom duties, our lead college guidance officer who teaches a section of French, and six other volunteers who together represented every major academic department at our school. I asked them to commit to one class visit per week over the course of the five weeks I had targeted.
The project had a rough start. One day after returning from our two week spring break, a record snow storm hit us dropping over two feet of snow on our school. We were out of school for the remainder of the week. I regathered and asked folks to still try to commit to one visit per week, but it would now be four weeks. Since we had ten participants (myself included) I hoped for 40 class visits over a month. I set up a shared google spreadsheet where participants would have their schedules posted and they could make notes for days/times where visitors would not be appropriate. They also all made notes about what they wanted their visitor to focus on during the time in class. I tried to make sure that people were comfortable with visitors and that they understood that these visits were not for evaluative purposes, they were for sparking conversations. If the observer kept comments focused on the concerns raised by the classroom teacher, then (I hoped) the conversations would feel supportive and instructive.
While life got in the way of some of the participants, we got close to my goal. A total of 37 class visits occurred during the 19 day span of the project (we were off on Good Friday during the fourth week of the project window.) I sent out a questionnaire and received a number of positive responses, some of which I will share (anonymously) below.
- More to the point, I like that having visitors in my class keeps me “honest” in a way; I find that having someone new paying attention to what is happening really helped me to focus on my own words and interactions with students.
- I remember up until about a year or 2 ago we were required to be visited by one colleague and visit one or 2 colleagues each year. Then we had to fill out a sheet saying who we visited and who visited us. I always found this to be a chore, something to check off my “to do” list. Although your project was more involved (many more visits to be made and many more visitors than the old requirement), it felt more helpful and less annoying. I believe the reasons for that were that it was more of an exchange (you visit me/ I visit you) and there was a purpose – we wrote in the google doc the feedback we were looking for. This structure really helped make it worthwhile.
- It makes me want to take classes again. I appreciated seeing how everyone engaged their classes, especially the quiet students.
- Overall I really loved the opportunity to see my peers in action. It brought me a sense of respect for the energy they put out with students and pride about the quality of education the students are receiving.
- I really liked the opportunity to visit classes and talk about teaching with colleagues, and I think it would be a good thing for visitations and discussions to become part of the school culture. But, I am skeptical about it happening without teachers being made to do it.
Overall, I have to say that I am pleased with this experience. I chose not to hang a note on my door as I know many others have done because I chose not to be that public about this at this time. Since I had a small, dedicated group of volunteers and I did not want to insist that they hang such notes, I chose not to do so. I am seriously considering starting next year with such a sign outside my door. I came into this project with the idea/belief that visiting each other more regularly and more intentionally would lead to important conversations about our craft. The feedback I received, and my experience in so many different classes during this time, have reinforced and deepened that belief. I worry about the skepticism that a number of the participants expressed regarding whether this can become a part of the regular fabric of the school. I believe that this would be a much greater benefit to our students AND to my colleagues if this became a regular and widespread practice, but I suppose I should concentrate my energy on planting these seeds in my little corner of the world first.
Many thanks to Robert Kaplinsky for sparking this fire and to my colleagues who jumped in and gave their time and energy in addition to their normally busy days.
]]>I was reminded of the importance (and joy) of this recently by two events. My blog post got a belated reply from a fellow named Josh. He linked to a post he had written on this idea. This was an important reminder not to take it for granted that my students know how much I appreciate being able to leave and knowing that some good math might still occur. The second reason is linked to my leadership program. The group in our area works with business leaders, teachers, college students, and high school students. I was asked to host one of the college students recently. The young woman who was my guest is a math major in the education program at her college. She sat in on three of my classes that day. Unfortunately, she had to leave for her class before my Geometry group met. The feedback I got from her was wonderful. She remarked on the conversations that my students were having and on the level of ideas that they were willing to wrestle with. I was SO pleased not only to hear kind words, but specifically to hear her compliment the discourse in my classroom as this is a big focus of mine. The best part though, was being able to share the remarks with my students the next day. I think that they just shrug it off a bit when my colleagues say nice things, like, maybe, they are just supposed to be nice. However, there was a more tangible reaction when the words of kindness came from a stranger, especially one who is studying math in college.
The fact that these two events happened in the same week was pretty awesome for my flagging energy level and it was a reminder of just how fortunate I am.
I’ll also be posting this reflection over at onegoodthingteach.wordpress.com
If you are not a regular over there, you should think about subscribing.
]]>He describes a survey done of some students where one group of students saw these two questions in order.
When presented in this order, researchers saw essentially zero correlation between the answers to the questions. Kahneman concludes that dating is not the measurement by which students assess their own happiness. However, when the questions were reversed there was a remarkable correlation. He concludes that the first question gets students focused on that particular aspect of their lives and colors how they view the question about their own happiness.
Does something similar happen to many of our students? If an early question or two seems comfortable and familiar, does this build a sense of confidence and help lead to better results? I often find myself moving at least one of the questions that I anticipate to be a challenge toward the front of the test. My reasoning is that I want them to engage these richer questions while they still have more energy and more time to wrestle with the ideas built-in. After reading this brief description by Kahneman I am now doubting myself. I wonder if I would see better performance from some of my students if I intentionally arrange the test so as to build confidence and, perhaps, give some more built-in clues for later on in the assessment. The real problem with this type of thinking, and the type of thinking I have been more traditionally doing, is that my sense of which problems might pose a challenge do not always correlate to my students’ point of view. I think I want to have a conversation about this with my classes. My three subjects this year have such different sets of students that I suspect I will get pretty different feedback on this issue. That might serve me, and my students, well.
Any thoughts? Drop me a line here or over on twitter where I am @mrdardy
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I am convinced that the process is a meaningful one. Letting our students know that their voices and opinions matter is a powerful thing. Letting them know that the adults who have been evaluating them need to hear back on how we are doing levels the playing field (at least a little bit) and this tells our students that they can be part of an honest feedback loop where we can grow based on their experiences/opinions/successes/failures. But I also know that the process can be better than the one we have.
I know I have read a few posts along these lines and I am off to the MTBoS search engine this morning to see what I can find. I would love to hear from anyone reading this about their successes, and failures, in elucidating meaningful feedback from their students.
As always, feel free to drop a comment here or pick up the conversation over on twitter where I remain @mrdardy
Thanks in advance for any wisdom.
]]>We are discussing convergence tests for series and one of the standard tests is the integral test. As Stewart presents it, at least how mrdardy presents Stewart’s presentation, we discuss an infinite series and if the integral of the function has a finite answer, then the series itself converges. Formally, what my students were told is this:
Suppose f is a continuous, positive, decreasing function on the interval [1,∞) and let the sequence a(n) = f(n). Then the infinite series ∑a(n) is convergent if and only if the improper infinite integral ∫f(x)dx is convergent. If the integral is convergent, then the series is convergent. If the integral is divergent, then the series is divergent.
Pretty clean and clear for a calculus statement, right? We had a nice discussion about the difference between looking at the integral which is a continuous summation versus the value of the series which only sums the natural number inputs. A question arose about how to compare the value of the integral with the value of the series. We decided, I really leaned them in this direction by thinking out loud, that the integral sum would be greater than the sum of the series since the integral also added “all that stuff between natural number inputs”. We all seemed happy with that conclusion until yesterday. My LaTex skills are lacking, so I will try this as best I can without that tool. Apologies in advance.
The series we were looking at was the infinite series of 4/(n^2+n). They quickly recognized that this could be rewritten using the method of partial fractions. We rewrote it and, by the magic of the telescoping series, we saw that the sum was 4. All good, right? Well, someone asked me to remind them how we could use the integral test to confirm convergence. We integrated the function using partial fractions and arrived at a sum of 4 ln2. Nice, the integral converged as well. Not so nice, the integral is less than 4. This violated what felt like a nice conceptualization that we had talked through a few days earlier. I sent out a quick call to twitter about why the integral test, that I had convinced myself of providing a ceiling for the sum approximation, would do this. I got a gentle reply from @dandersod reminding me that the integral is the floor for the approximation. I am pretty sure that I have known this at some point, but had constructed such a compelling Riemann approximation argument with my students that I was stumped. I dismissed them for lunch.
One of my advisees dropped by for a quick chat. He is one of our three Differential Equations students this year. He took a look at the problem and I tried to convince him of my mistaken impression that the integral gathers up all this other area and should be an overestimate. He paused, thinking about the problem and then had a great ‘A-ha’ moment. He said, isn’t the series just the left hand Riemann approximation? Since the function is decreasing (otherwise, it wouldn’t converge!) this left-hand approximation will necessarily be an overestimate. Nice, clear reasoning. The kind of reasoning I should have had at my command when talking with my class a week or so ago. Sigh.
My Calculus kiddos are taking a test today. I get to apologize AND praise one of their colleagues on Wednesday when we revisit this ‘mystery’.
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Hey Jim, I just made a connection from our conversation this evening with an earlier conversation. It manifests mostly as a question/challenge.
You have said, and I agree, that we ought to value what we assess and assess what we value. So, if we value collegiality and collaboration amongst students, what is a fair and appropriate way to assess that? I feel like your group quizzes are part of the answer, but I also feel like there is more to it.
The night he sent me this email I went to twitter to see if I could get some feedback/pushback and I did in a lively conversation with Michael Pershan (@mpershan) where he questioned whether grades were the proper avenue for communicating what it is I value in class. He closed with a couple of important points:
I’ve never been happy while using grades to motivate (it flops) but your experiences might be different than mine here. +
One further thought: an assessment is a promise to a kid that we can help them improve on what we’re assessing.
Some background here might help me clear my thoughts and might help you, dear reader, understand the origins of this whole train of thought.
Years ago, I read a powerful piece through NCTM written by Dan Kennedy of the Baylor School. Dan had been an AP admin for some time and was around when some of the sea changes were happening with the AP math curriculum. He wrote a terrific piece called Assessing True Academic Success: The Next Frontier of Reform. You can find it here, it is worth a read. The quote my colleague referred to is from that article and it is a point I raise with my department when we talk about the role of assessment. I have written before about my commitment to trying to create a culture of communication and collaboration in my classroom and the colleague who wrote to me had just spent a week in my Geometry class observing my team in action. He references a habit I started just a year or two ago where I have one group quiz each term here. I know that this is a baby step, but I am still trying to figure out how to find a balance between personal responsibility for showing knowledge, the ability to work productively within a group, the meaning of a grade on a report card, the reality of what lies ahead for them, etc. etc. etc. I feel that a group quiz each term is a beginning, I hope it is not the end. So, my colleague is asking/urging/challenging me (and himself, I suspect) to really dig in and think about how to effectively communicate to my students that I value communication and collaboration. I have always reflexively felt that the clearest way to communicate to students what I value is to make sure that those stated values are transparent in the assessment process. If I tell my students that I value problem solving and process in mathematics but I then give them multiple choice tests, then they will suss out that I probably do not really believe what I say I believe. In comes Michael Pershan with his measured challenges. I have a few thoughts that are coherent enough to air out here about Michael’s comments. The first comment about grades is a real challenge. I work in a college prep day and boarding school. For better or worse (probably aspects of both) grades are a HUGE force in our school. As long as I feel I am being intellectually honest about what I communicate to my students through my grading practices, I am willing to accept Michael’s statement as a deeper, long-term truth while also accepting that, in the now, my students’ behaviors (and, hopefully) beliefs can be steered a bit by how I evaluate them. Given that, the struggle is to figure out a way that feels honest, transparent, and not too terribly subjective to incorporate the values of collaboration and communication into my grading system. I have been in too many classes where students ‘participate’ in conversation by simply echoing the opinions of others so that they can get their names on the participation roll. I do not want to clutter my class this way. I also want to recognize that there are students who do not want to, who do not feel comfortable, talking out loud in whole class conversations. Most of these students are willing and able participants in smaller group conversations.
One of the things I love most in life is the sense of synchronicity I have when I realize that what is on my mind is also on the minds of others. When I suddenly see references over and over to something that I think I just discovered. Well, I took a few minutes break from this post and saw a link to a lovely post crawl through on my twitter feed thanks to the MTBoS Blogbot. The post is called Why Do You Have Us Do Things That Aren’t For a Grade?. You can find it here. It was written by @viemath. Maybe this article will spark some important insights.
I have long told my students that their numerical average in my class simply represents the worst grade that they can earn. I tell them that a student with an 88% can be an A- student if they are good citizens, if they contribute to class, if they are largely consistent or on an upward arc. I also tell them that a student with a 90% is an A- even if that student is not such a great citizen. I kind of feel good about that stance. My thought at this point is that I will simply continue to emphasize this strongly and make a distinct point that one of the major mitigating factors in figuring out whether i need to lean on turning averages into grades is to attend to class engagement as the primary point of emphasis.
I am hoping for some bolt of wisdom…
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Suppose that Hamilton is playing at the Civic Auditorium. The auditorium has only one section for seating. The seats are arranged so that there are 60 seats in the first row, 64 seats in the second row, 68 seats in the third row, etc. So, in each successive row there are four seats more than in the previous row. There are a total of 30 rows in the auditorium.
How many seats are in the last row?
How many seats are there in the auditorium?
The seats are numbered consecutively from left to right, so row two starts with seat 61, row three with seat 125, etc. You purchased a ticket to the play and your seat number is 1500. What row are you in? Where in that row is your seat located?
So, the first part of the problem went reasonably well. They were able to recall that there are 29 steps of equal size to be taken in accounting for row size, but even this was harder than it should have been due to the reflex to create a table. I began to realize that what seems automatic to me, that we are concerned with row number and with accumulated seats, was not automatic to most of my students. They set up a column of row numbers followed by a column of row size. They then arrived at a first difference of 4 for each entry in their third column and they were off on finding a linear function. The linear function is correct for the number of seats in each row, but the rest of the problem depended on them finding an accumulated number of seats. When I set up a table and had row number followed by #of seats in that row and then the total number of seats, the inconsistence with previous visuals was a real problem. Getting my students to focus on the first and second differences in this new third column was a challenge. I know that I did not answer their concerns as clearly as I need to and I have to figure out how to better answer this. Once we established that this is a quadratic relationship we were able to find the coefficients and answer the second question. It took some convincing and looking at some smaller sums along the way, but I think we came to a genuine consensus. Switching over to the third part of the question was a giant hurdle. I did not intend for them to solve a quadratic since they would get an irrational solution. Instead, I hoped for some reasonable guess and check but it became clear that, for too many of them, the ladders of abstraction leading to this part of the problem completely clouded the problem for them. I have faith that this is an interesting question. In the spirit of full disclosure, it is important to note that this is simply a (slightly) modified form of a problem from our publisher’s test bank. What I need to think deeply about are the following questions:
Sadly, I have to wait until next year to make this better.
]]>Our school is moving to a new schedule next year where we will have five class meetings in each seven school day stretch with one class at 90 minutes and the other four meeting for 50 minutes. I think that these longer spans of time together (currently, the majority of our class times are 45 with a handful of 40 and a handful of 50 based on assembly schedules, etc.) will work beautifully with the ideas that Brian laid out. I am on the fence about the nature of how I want to deal with the earbud/headphone question. I like the idea that Brian has but I am not sure about endorsing any type of paid music services explicitly with my students. I know that almost all of my students come equipped with earbuds and mobile devices everyday (probably a higher percentage than those that come equipped with a writing device to class each day!) so that is not a hurdle at all. I have announced a group quiz for next Wednesday and I know that I want to have at least one day before then where I explicitly have ‘alone time’ for thoughts before reconvening our pod conversations. I am debating whether I need to physically separate seats for this alone time or whether that is too much time and interruption. I’ll be writing about how it goes.
]]>Our school uses an LMS through FinalSite, a company that manages our school home pages. It is a pretty typical looking LMS. I populate each class with my students so that when they log on to their student portal they see each class they are in (as long as their teacher uses the LMS) and they can see calendars, they can download assignments, they see their HW and upcoming responsibilities, etc. My hope is that students check in pretty regularly (daily is a pipe dream, I fear) at least on Sunday night to scope out their upcoming week. In addition to populating this calendar – usually about a week in advance, but I sometimes lag a touch, I keep a spot on one of my side chalkboards where i highlight upcoming highlights. I lagged on that recently as well. As I hinted earlier, January has been a bit of a blur for reasons I cannot pinpoint. Anyway, last Monday I had a rare Monday test scheduled for my AP Calculus BC gang. Their class met right after lunch and a couple of students came in during lunch and asked if the rumor that they heard was true. The rumor they were referring to was a rumor that they had a test that day. I mentioned that this had been on their calendar for well over a week and confirmed that, yes, this ‘rumor’ was true. Kids got on their phones to notify classmates and they frantically started flipping through their text book.
I kept my calm and I assured them that they would be fine. they had been doing their work (I said optimistically) and that last minute cramming rarely has much positive impact. Most of the class performed reasonably well – one perfect score and one student who only missed one point out of a group of fourteen – but the average was lower than usual and one student in particular was way off of his usual mark. The frustration got me thinking about a number of things and I want to use this space to think out loud about these issues.
First, I worry about communication in this increasingly digital environment. I used to print off weekly calendars and hand them out at the beginning of each week. Some kids would lose them, some would carefully put them in their folders, some would cram them in their backpack never to be seen again. Mostly, kids seemed to know what was coming up or at least kept it a bit of a secret when they were surprised by an assignment or an assessment. Now, I print almost nothing. I post on the LMS. I keep the reminder chalkboard. I send out email reminders through the LMS. I send some occasional emails from my school account with attachments for notes or suggested extra work. I hear repeatedly from students who did not know I had sent an email or that I had posted to the LMS. This makes me wonder how much of the blame lies on me for moving away from printed reminders. I mean, if 14 out of 14 students did not know that there was a test on Monday then part of the blame falls on my shoulders. But, and this is important, something is odd in the student culture around my class if 14 out of 14 students failed to register this fact in a planner or take a look ahead at their upcoming week before reporting to school on Monday. I do want to take a moment here to compliment my students in their reaction to this event. Not a single one complained about unfairness, not a single one said to me that this was my fault, and when they received their grades back not a single student voiced their unhappiness about the situation. It would have been so so easy to point the finger at me and none of them did. This is a credit to their character and willingness to take responsibility. A number of them did ask to take advantage of my policy for reassessing, but no more than usual really.
So, I am questioning my role in communication and the avenues I choose to take advantage of. The other question this raises for me is my attitude about announcements for assessment. I know that many of my colleagues, both in my building and out in the world, have as part of their practice unannounced assessments. I have never done this and it is mostly because I find myself overly sensitive to charges of increasing/causing student stress. I always make sure that there are at least three school days between announcing and administering an assessment. In the case of major unit tests, I want to have at least one weekend between announcing and administering the test. But this incident has me questioning this commitment. Am I seeing a more true reading of student mastery of material if I check in periodically when they do not know that I will do so? Am I bypassing the stress of test and quiz preparation if I just drop a quiz or test in their lap when they show up for class? How do I setup a situation where an unannounced assessment is not such a big deal for the student?
As always, I am seeking wisdom here. If you have made a practice of unannounced assessments, how do you handle that? How do the students respond? If not, is your reasoning similar to mine? How do you communicate calendars to your students? Teachers here use either our school LMS, Google Classroom, Facebook, or old fashioned paper. What are the habits at your school? What really works? Drop me a line here or over one twitter where I remain @mrdardy
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