Category: Uncategorized

One of the ideas that has been pinging around my brain recently is that the order of questions on a quiz or a test has a pretty large, but unmeasurable, effect on student performance. Not performance of the whole group, mind you. I am thinking on the granular, individual level. I am now reading Thinking Fast and Slow by Daniel Kahneman (it’s pretty fantastic so far) and he just described an experiment that got me thinking (quickly!) and I put the book down to bang out this quick post.

He describes a survey done of some students where one group of students saw these two questions in order.

• How happy are you these days?
• How many dates did you have last month?

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

 

Some of this blog post is kind of embarrassing to write, it regards an idea that I have worked with for years and I feel I should have had a better sense of it than I do. I’ll start with a conversation I had with my wizards in BC Calculus recently.

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’.

 

An email from a colleague a couple of days ago has my brain buzzing a bit. Here is the note he sent me:

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…

In Discrete math we are exploring recursive sequences and talking about how to make them explicit. When given a table and a recursive definition, my team of Discrete Math warriors has gotten pretty good at examining first differences, second differences, etc. and relating them back to the degree of an explicit formula. I recognize that some of this is rote, but sometimes skill development looks like that. It was not until I presented the following problem that I realized how rote some of the problem-solving has been. Here is the problem:

 

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:

1. How much quadratic function review do I want to do to help set up a meaningful context for these recursive functions? My gut feeling was that I did not want to go into those thickets with these kids. Many of the students in this class are realizing that they can do some mathematical thinking once they removed themselves from thinking that mathematical thought only looks like equation solving.
2. How do I balance the discrete nature of this problem with the inherently continuous point of view that students have regarding quadratic functions?
3. How do I help my students focus on building a table of data that is clear and meaningful? How to focus more clearly and quickly on the pertinent data in the problem?
4. How can I carefully structure a positive class discussion around one in-depth, challenging problem like this in a class where too many of the students have felt defeated by math one too many times? I feel great about the general atmosphere we have created together and I want to keep that while extending their thinking.

Sadly, I have to wait until next year to make this better.

Thank you thank you thank you, John Golden. John commented on my last blog post and gave me some important wisdom regarding my frustration with my own decisions and the decisions that my students had made last week. As expected, the quizzes were subpar. In the class where I had chosen not to explain the permutation notation I made the following grading decision. I graded the last problem as if it were a 10 point problem as advertised. However, when calculating their grade, I counted it as a 5 point problem. So, the students who had learned the notation earned some bonus points while those who had not were not stung quite as severely. Not a perfect solution, but it did open the door to a public conversation about my frustration and about how we might avoid their frustration AND my frustration moving forward. Don’t know yet how that will sink in, but at least it was received as a good will gesture on my part and no one complained out loud that it was unreasonable for me to have expected them to read that definition. We’ll see what happens in the next week or so as we have two more opportunities for showing some learning here.

I have two sections of Discrete Math this year, one in the morning and one right after lunch. During the fall term, each of these sections had 7 students. We all sat at a single group of desks together and had some great conversations. A number of the students have spoken to me about how much they enjoy this atmosphere. It does not work for everyone of course, some students prefer not to have the expectation of participation, they would prefer to quietly observe and have more time to think before speaking. Our school is on a trimester schedule and this Discrete course is set up as a trimester course where students can move in or out and not have the demands of previous knowledge from this course. So, I have done some thinking about how to make this course modular. One of my sections expanded from 7 students to 16 this term and we are in the process of figuring each other out and how this new group will mesh. One of the students who has been in the class the whole year commented that class seems more quiet this past week. Interesting that more than doubling the size of the class has resulted in a quieter atmosphere…

All of the above is just to sort of set up what our week together was. We started a probability unit this week and so far all of our energy has been spent on counting techniques. When does or matter? When does and matter? What is the difference between them? What is the deal with that ! function anyways? When can we tell whether replacement matters? These are the kinds of conversations we have been having and I have had in-class activities for us to work on together while I have been asking them to do some reading and some HW on their own outside of class. The great Wendy Menard (@wmukluk) shared some fantastic resources that one of her colleagues shared. She was also kind enough to spend some time on the phone last weekend to serve as a sounding board. One of the decisions I made was not to spend much time emphasizing notation together in class. For example, our text explains permutation notation pretty cleanly, points out that our calculator writes 10P4 while you might also see P(10,4). It clearly shows that this calculation is 10!/(10-4)! while also introducing this notation in more general form of P(n,r). In class we had a number of examples of drawing some subset of members from a group, so I thought that the text’s approach and our class approach would support each other. I also figured that any students flummoxed by the text notation would ask me in class what the deal was. So, the first HW question on Wednesday night was this in fact – Which is equivalent to P(10,4), 10!/4! or 10!/6! ? We had a quiz scheduled for Friday and on one of the questions I gave the students the numerical value of P(26,3) and asked for an explanation of how to get that answer. On Thursday I had a couple of review problems thrown together from the textbook author’s supplemental test bank. I planned on starting class by fielding any HW questions then turning them loose to work on the review problems. In my morning class I projected their HW from Wednesday night and had the first HW question on the board. Not one student knew what P(10,4) meant. They asked whether that was a point on the plane. I have to assume that they did not do the reading or the HW on their own. I quickly untangled the notation, pointed out how it matched some other conversations we had and then gave them their review sheets to work on. That was my morning class of 7 students. After lunch I had the book projected with the first HW question. Not one student in that class knew what P(10,4) meant. I decided to remark on the importance of doing the reading and the HW and then just gave them their review sheets and sat down.

One student came by on Friday morning during my free time to ask me a question about the notation and she remarked that it was clear I was disappointed (annoyed?) that no one had done the HW. She wanted to make sure she understood that notation. Each class on Friday began with me answering questions before the quiz and I do not recall anyone in either class directly asking me to revisit the P(n,r) notation at all. I have not graded the quizzes yet but I know that there were a number of students in my second class that either left the question about P(26,3) blank or simply wrote something to stumble into extra credit. A number called me over to ask about it and I said this was something they needed to know. I do not remember my morning class as clearly, they may have been in a similar boat.

So, as I think about this I realize that I made two very different decisions with my two groups of students and I am not happy about either of them. In one class I came to their rescue and explained something that they clearly could have come to terms with – in some way – on their own. In the other class I let my annoyance take over and I did not address the question at hand. I also realize that my students, especially those in my second class had two decisions to make. On Thursday night, after seeing my disappointment/frustration they could have gone back to their reading and either understood it themselves or they could have checked in with me during review on Friday. It is clear that a number of them did not do that. So I am faced with yet another decision when I grade what are likely to be disappointing papers. I feel that I want to get across a pretty clear message about responsibility but I also need to recognize my responsibility here. It is reasonable, I think, to see my role as someone who expands the conversation from the text, not as someone here to simply recite what the text already explains. But I also recognize that I have 9 students who are new to our class and all of them are new to me as a teacher. If they are used to teachers making sure that every question in their text is also addressed in class then my idea about my role might be a bit of a shock and I did not spend much time together on Monday explaining this about myself. However, I also have 14 students who were with me all fall and it is pretty clear that none (or very few) of them did the reading and the HW either.

I am not happy with myself that I let my annoyance get in the way of clear thinking. I am also not happy that I was not more clear with my morning class about my disappointment that none of them had done what I asked. I am not happy that so many students did not do the reading or the HW. I AM happy that I had a student come by and clarify the question for herself while also recognizing that she should have done so on Wednesday night. I feel that including the question when I compute the grade will likely have a pretty significant impact on many grades as it was one of four questions on the assignment. I also feel that it is a reasonable question to ask, but it relies on notation that I did not explicitly present.

I have been reading a number of the DITLife blog posts and there is a constant reminder about the number of decisions that we make on the fly everyday. These are complicated decisions and I know that I hope that I make them clearly. Here is a case where I think I was probably not as clear thinking as I should have been and I will likely need to make a decision about grading that will, luckily, not have to made on the fly. I have a bunch of new students who are only one week into their experience with me. I want it to be a good experience where they grow as scholars. I need to think carefully about how I respond to this disappointment – in my own behavior AND in the decisions they made.

Our school is making a pretty big change next year. We are moving from a static schedule where we have seven classes that meet in the same order every single day. Our class lengths are either 40, 45, or 50 minutes depending on the length of our assemblies and our ending time each day. Next year we will be on a rotating schedule in 7 day blocks. Each class will meet four times for 50 minutes and once for 90 minutes over that interval. Five classes will meet every day, four of them will be 50 minutes long and one will be 90 minutes long. As a result of this upcoming change we are being asked to do some deep reflection about our practice and our curricular choices. As department chair I have been encouraging my department to think deeply about trimming or eliminating items from our curriculum to make time to think and have more open ended (and open middle!) problems. This schedule change will really push us to do this and I suspect I’ll be thinking out loud on this space as we move through this process. Where my mind is tonight is on the subject of assessment. I was asked to facilitate a group this morning to talk about our assessment habits and goals. Years ago, I moved to a school that had a rotating schedule with a long block. At the time I was not the same teacher I am now (at least I don’t think I was) and I don’t remember being all that thoughtful about the impact on my practice. The fact that I had moved to a new school and just had to adapt probably diminished any sense of dramatic change that I might have felt. Around here right now we are having some pretty valuable conversations and I was part of one this morning.

My takeaways from the meeting are:

• I want to have more frequent, but shorter, opportunities to check in on my students’ learning. Ideally, some of these would not have any letter or grade attached at all but simply serve as formative checks of their developing understanding of the material at hand.
• I think that the old model of a 45 – 50 minute timed sit down assessment where everybody takes the same (or VERY similar) tests needs to shrink, at least in my discipline. Our time together will be too precious to take up too much of it silently hunched over a piece of paper with problems, no matter how creative the problems are. We have 154 class days this year that are not devoted to term exams. Meeting 5 out of every 7 days changes that to 110 days. Granted, these 110 days will be a minimum of 50 minutes each (our current longest class) so contact time will feel different. But we are still only going to see our students for 110 days and we need to make those days count.
• I want to expand my palette. My students’ grades are almost entirely dependent on times tests and quizzes. I know that there are other smart ways to do this, I want to learn more and I want to grow my toolbox.
• I have been sneaking in group quizzes lately given that my class is set up on pods. I want more of this. I experimented with my Discrete Math elective in the fall. We have a stretch of days leading up to our term exams called test priority days. On these days we are limited by department for who can have assessments so that our students do not have these pile up on them. Each department has two of these days. I chose to have a group test on the first day where everyone chipped in ideas (obviously some students were more vocal than others) and they all got copies of my feedback on their test. Four class days later, after one new topic was introduced, each student had an individual test. My best students performed about the way that they always had, I wasn’t too worried about them. What pleased me was that some of my students who had struggled during the term clearly benefited from the group work not only in improving their grade by the addition of this group test score, but they performed at a much higher level on the individual test. The combination of the work with their peers and the ability to study from that work and from my feedback all seemed to work well together. I want to capitalize on this and try more ideas like this moving forward.

 

So, the reason I am thinking out loud here about this is the reason I always cite. I would love to hear from you, dear readers. Please share successes and failures you have experienced as you expand your assessment toolbox. I want to hear how you wisely spend time with your students on extended classes like the 90 minutes we will have periodically next year. I want to hear what pitfalls to avoid that we might not even be anticipating. In general, I just want to continue learning from all of you!

As always, feel free to comment here or to engage me on twitter where I am @mrdardy

 

Thanks in advance

My last post was about a professional development conference I attended and presented at. Last week I went to another and presented again! In between, we had grandparents’ day at our school, so there are a couple of things I want to share today.

Last Friday I attended the Biennial Conference of the Pennsylvania Association of Independent Schools. It was hosted at the lovely campus of The Episcopal Academy in a Philadelphia suburb. I attended two sessions and presented my MTBoS love song at a third session of the day. When I presented at ECET^2NJPA I had a small, but engaged, crowd. At PAIS I was fortunate enough to have a full room with some people sitting on the counters. We had a lively conversation and one person in particular had some great questions. Recently, NCTM issued an editorial statement about the importance of curricular coherence. It can be read, in part, as a warning against using open source curriculum without deep and careful thought about how it all fits with what you are trying to accomplish in your classroom. My presentation focuses on my journey through the MTBoS and how the resources shared by our community helped inspire me to take on the task of writing a text for our school. It is a text that I hope represents some important values in our department. The text challenge I tackled was for our Geometry course and I have to admit that I felt a certain amount of freedom that I might not have if I was writing for Algebra I, Algebra II, or Precalculus. Those courses that are more in a direct vertical relationship with each other feel like they bear more weight in terms of coherence with each other. There is also more of a feeling that these courses depend directly on each other. I mention this because of a great question that came my way about this. One of the members of the conversation directly asked me about the decisions I made regarding the course content and I had to admit that I probably would have been more intimidated if I had tackled one of these more ‘core’ courses in the high school curricular stream. I felt that we had a really good conversation in the room about incorporating different activities into the classroom. Since the audience were all members of independent schools they probably have a little more flexibility than many of our public school friends have in terms of deciding what resources to incorporate into their classrooms. I have now made this presentation three times for three different organizations and I will probably put it to sleep now, but I am glad that I have had the opportunity to engage in this conversation and to spread the word about the deep well of resources that is the MTBoS.

 

Last Wednesday, two days before the conference, our school hosted our annual grandparents’ day. I have often been teaching Calculus in the afternoon and this rarely brings many grandparents into the room. This year I end my day with my Geometry class and we had about a dozen guests in class. I found a lovely activity at the Nrich site and my students and our guests had a terrific conversation tying together Cartesian coordinate plane ideas, transformations by vectors, and the idea of being able to project ahead in a sequence. I have been including problems from the Visual Patterns website and I think that Wednesday’s activity might have been a bit of a breakthrough. The conversation we had, and the inclination to want to show off for our guests, was more lively and engaging than I anticipated and on last Thursday’s test I saw better performance on the pattern recognition problem than I had previously seen. I cannot recommend Nrich too strongly. There is a wealth of great problems there and I am using another one for this Friday’s parents’ day visit when I expect to have another crowded room. One of the grandparents was here from California to visit her grandson and she stayed after class to chat and ask for a photo with me and her grandson joining her. I was flattered by her words and by the fact that she wanted to have this memory.