## Making Sense of the Infinite in Calculus

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’.

## How Can I Communicate What I Value?

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.

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…

## Exploring Sequences

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.

## Trying to Understand what my Students Understand

Starting to think about school again and this question has been clanging around in my brain. On my last test for my AP Calculus BC kiddos I included the following question:

My BC gang absolutely nailed this question. Almost every single one cited concavity for part b noting that a function with positive slope AND positive concavity will increase at an increasing rate while the tangent line increases at a constant rate. So, moving to the right of the point of tangency means that the function has pulled away from the tangent line. They almost uniformly used the language I just used with slight tweaks and maybe a little less detail since they were operating under time constraints. I was proud of them for such detailed answers to an important principle of graph analysis. However, after the happiness faded there was a nagging concern that arose. I worry that they are SO good at citing this language that perhaps they are simply responding to a familiar prompt. I am not here claiming that these talented students do not understand this principle. I am here claiming that I am concerned that I have ‘trained’ them too well in responding to certain prompts, that I have enabled them to simply repeat a claim that I have made convincingly in their presence. I want to do some deep thinking about how I can circle back to this idea and ask this question in a form that is similar enough that it is clear what I am asking, but different enough that my students will have to say something different to betray their understanding. I would love any advice on how to continue to poke at/probe how deeply my students understand this concept. Any clever ideas out there? Drop a line into the comments section or tweet me over @mrdardy

## The Decisions We Make…

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…

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.

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.

## Assessment Is On My Mind

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

## Linear Functions

Some of my Geometry students are wrestling with being able to accurately write linear functions given information about points and slopes. I am struggling with how to help them overcome this and I have been doing quite a bit of thinking about how we teach this and what kind of sense it might make to my students. I know that I have these fundamental ideas in my head – an equation is a relationship between the variables stated in the equation (it tells me how to turn an x into a y or vice versa) AND the graph of an equation is the set of points that makes the equation true. I know that I say these things and I am fairly certain that previous teachers have said similar things. I know that my students have repeated some of these things and they can (periodically) carry out these operations. Where the mystery lies for me is why this skill can only be inconsistently displayed and I have a couple of thoughts. I am interested in any wisdom you have about this question.

Almost every single one of my students prefers the slope-intercept equation of a line to any other form. Partially because of my history as a Calculus teacher and partially because I favor more direct problem-solving approaches, I am an advocate for the point-slope equation. I consider it a minor success that most of my students answered the first question on their recent quiz in this form. Here is the question: Find an equation of the line that passes through the points (3 , 1) and (5 , 4). Now, I am careful to ask for an equation rather than the equation but I do not know how much of an impression this might leave on any of my students. Almost every Geometry student answered this correctly and most left it in point-slope form. I was pleased. The next question was this one: Find the coordinates of the following points on the line you found in problem #1.

• The x-intercept.
• The point with an x-coordinate of 1.
• The point with a y-coordinate of 1.

Here is where things started to fall apart for many of my students. I have been thinking about the mistakes I see in class and on assessments and it occurs to me that there might be a fundamental problem that I do not know how to solve. When a student wants to write the first problem in slope-intercept form I instruct them to first find the slope, then replace the x and y in y = mx + b with coordinates of either given point to find the b value. I tell them that this way is harder, but many want to hold on to that equation form. If they want to approach the first problem with the point-slope form I tell them to first calculate slope then replace x1 and y1 in the equation    y – y1=m(x-x1)  with the coordinates of one of the points they know while leaving the x and the y alone. I am embarrassed that this inconsistency has never jumped out to me before, but why is it that in one equation we leave the x and y while in the other we replace the x and the y with coordinates of a given point?!???!? I have to imagine that some of my students are absolutely baffled by this inconsistency. I wish that they could verbalize that sense of confusion, but I just now figured it out for myself, so why should they be able to lock in on this? So, dear readers I implore you – help me figure out a better way, a more logically consistent way, that I can help direct my students. This is not an intellectual task that is beyond any of them, but I have to guess that a handful of them are so tired of being asked to do this and have sort of given up on the idea that they will ever master this concept. It is way too easy to just write it off and hope it will go away. I do not want this to be their reaction and I want to see them reliably be able to answer these questions.

## Busy Days

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 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.

## My ECET^2 Experience

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.