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.

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…

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.

Collaborative Classrooms

When I moved up north from sunny FLA in 2007 I made a big decision about the geography of my classroom. In my last school I had seats arranged in groups of 3 or four, I referred to them as pods and my students selected their pod mates. They usually stayed with the same group all year long and built up some real team identities centered around their pod. When I moved to my new school in 2010 I had two large conference style tables brought to my room and had students sitting in big groups of 10 each. Typically, students stayed at one of the two tables all year long, but there was a little bit of movement at times. In 2015 I moved those tables out and got moveable desks and I am back to my sets of 4 seats. However, this year I am using visibly random grouping (using Flippity and happy with it) to shake things up and encourage more of a sense of collaboration across a larger group of their classmates. I begin each week with a new seating arrangement. I have been pretty happy with most of this evolution and I am convinced that these arrangements have increased student communication over the years. However, I saw Susan Cain’s TED Talk not too long ago where she talked about introverts and the impact of all of this collaborative time on their learning and their comfort. Since seeing that talk I have had a bit of an internal struggle about how to try to compensate for the fact that not everyone wants to talk as they learn. Not everyone is comfortable thinking out loud before they have come to some sort of a clean conclusion. Not everyone thinks at the same pace. Luckily for me, Brian Miller (@TheMillerMath) wrote a blog post that has come to the rescue for me with ideas about how to address some of these fears. Brian wrote about a great idea for balancing the principles at hand that seem to be in competition with each other. His post is over at and I strongly encourage you to read it if you have been thinking about some of these conflicts.

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.