## The Infinite Series Sorting Hat

Looking for a way to keep all those convergence tests for infinite series straight? Looking for a cultural reference that some of your calculus II students will still find timely and relevant for a few more years? Look no further than the infinite series sorting hat.

Get the printable versions:

##### Assuming the first term of each sequence is n=1:
1. $$a_n = 4-4n$$ and $$a_{20} = -76$$
2. $$a_n = \frac23(n-1)$$ and $$a_{20} = \frac{38}{3}$$
3. $$a_n = 36-11n$$ and $$a_{20} = -184$$
4. $$a_n= -5 + \frac72 n$$
5. $$a_n=6 – \frac12 n$$
6. $$a_n = -70+9n$$
7. $$a_n = -5+9n$$
8. $$a_n = 22-4n$$
9. $$a_n = \frac{12}{5} – \frac25 n$$
10. $$2585$$
11. $$n = 5$$
12. $$a_n = 2\cdot 3^n$$
13. $$a_n= \frac{15}{2} \cdot \left( \frac25 \right)^n$$
14. $$a_n = 8 \cdot \left(\frac12\right)^n$$
15. $$a_n = 60 \cdot \left(\frac12\right)^n$$
16. $$a_n = -3072 \cdot \left( -\frac14\right)^n$$
17. $$a_n = \frac45 \cdot 5^n$$
18. $$a_n = \frac32 \cdot 2^n$$ or $$a_n = \frac32 \cdot (-2)^n$$
19. $$a_n = -160 \cdot 2^n$$ or $$a_n = -160 \cdot (-2)^n$$
20. $$a_n = 14\cdot \left(\frac12\right)^n$$ or $$a_n = -14 \cdot \left(-\frac12\right)^n$$
21. $$5115$$ (Note, problem should read “…of the geometric series.”)

There’s also a chance that the teacher uses $$(n-1)$$ more than $$n$$ in the answers to 1-9 and 12-20. If so, then the teacher’s answers can be simplified into these answers.

## Welcome to Matt Salomone’s Mathematics Home

Welcome to my newly-updated web home at matthematics.com .

I enjoy thinking, tweeting, and sometimes blogging about emerging issues in the teaching and learning of mathematics and quantitative literacy. Some of what you’ll find here includes:

Open Courses/Text Projects:

## Bringing Their Brains

The next time I teach a web hybrid course, I want it to be a great online course with the meetings as a bonus (rather than a great face to face course with web content as a bonus).

I’ve been thinking a lot about the Community of Inquiry (CoI) model of online learning during the #HumanMOOC course on humanizing online instruction. The model (Garrison, Anderson, and Archer 2000) positions student learning as a function of three interrelated forms of presence in an online environment:

Community of Inquiry (Garrison et al, 2000)

For me,

• Teaching presence means making myself an active part of the course for all students. The folklore on my campus is that many students in online courses don’t even know who their professor is. On the internet, it could be anyone. A dog. A robot. If I’m doing my job as an instructor, it shouldn’t be different from my job as a face-to-face instructor: teach. Create conditions for learning, guide students through the process of learning with feedback, and assess whether learning has happened. These are hard to do face to face, just as they’re hard to do online, but no more or less important. This helps students to know I’m engaged, which is necessary (though not sufficient) for them to engage.
• Social presence means helping students to connect with one another and cultivate a sense of belonging in the online environment. Online learning, like in-person learning, can only work so well when it’s hub-and-spoke, with each student interacting with me and not with each other. (It also centers me as the authority in the class, which I need my students to do less.) So I need to use my teaching presence to create opportunities to build social presence too: by initiating (and requiring, and (!) being active moderating) threaded discussions, but also by exploring multimedia tools such as the nifty Flipgrid to give students space to interact through short video clips. I could see these videos as both opportunities for students to reflect on their learning, and to record student-voice “how-to” tips on a topic to help their classmates in their study process.
• Cognitive presence, the topic of this week, is where the higher-order learning happens. But insofar as students struggle to be cognitively present in face-to-face classes, especially those that meet before 10:00 in the morning, I would expect it’s even more of a struggle for them to be cognitively present in an online learning environment.

I wonder whether trying to do higher-order thinking in an online class is like trying to do physics homework at a rock concert. Distractions are everywhere, and the milieu isn’t designed for “system-2” engagement of higher cognitive function. (Kahneman, 2011) The internet, like the rock concert, is fundamentally a vehicle for dopamine delivery, not deep discourse.

But there is hope, as Peter Shea reports in today’s HumanMOOC webinar on cognitive presence:

He and his team did a quantitative content analysis on all (tens of thousands) student interactions in an online course and connected them with measures of teaching, social, and cognitive presence, and was able to describe the relationships between these forms of presence using a structural equation model and CHAID (Chi-square automatic interaction detection) tests to show that students who were most cognitively present in an online course were those who:

• Felt most comfortable contributing to online class discussion (social presence), who in turn
• Most felt that the instructor helped keep those discussions focused on relevant course concepts (teaching presence).

Likewise, the most cognitively present students were also those who

• Most felt as though they got to know other students and felt a sense of belonging (social presence)
• (I’m sure this doesn’t happen by accident either, and that a measure of teaching presence, such as the requirement to interact with other students and the instructor’s ability to communicate approachability, availability, and sensitivity to student needs likely explains this.)

So the moral of the story in my mind is that students will “bring their brains” to an online course that makes them feel connected to one another, positive, and engaged in focused, topical discussions of course content. I like to think I do well enough with the connectedness and positivity — encouragement of social presence — in my courses, but what I most want to work on is my creation of, and especially my presence in, discussions of course material. I’ll be thinking a lot about how to do that in ways that will work for my students as well, as next semester approaches.

Garrison, D. R., Anderson, T., & Archer, W. (2000). Critical inquiry in a text-based environment: Computer conferencing in higher education. The Internet and Higher Education, 2(2), 87-105.

Kahneman, D. (2011). Thinking, fast and slow. Macmillan.

An article came across my Twitter feed that hit upon a challenge I didn’t realize I had in thinking about how to run an effective online class:

In the article, not only does Steve Kolowich make a point that I fretted about in the beginning of this blog:

[W]hat often makes students choose the flexibility of online learning — being too busy to enroll in a classroom course — can also make it harder for them to keep up with their studies.

What’s more, he also talks about how the primary point of contrast between students and instructors — the learning management system (LMS) — is often designed around asynchronous, text-based interactions (posting course materials, dropboxes for written work, text-centered discussion boards) in ways that make it difficult to integrate richer media interaction. In short, for instance, Blackboard isn’t set up to be a transformative platform for multimedia engagement — it’s a bulletin board-cum-gradebook for trading in text.

Which is fine as far as it goes. But what I miss most when I teach web-hybrid courses is the social interaction that happens in the classroom (and hopefully also out of class, in the hallways or in office hours). And I have tried to use video to amplify my pedagogy in the past but struggled to connect it naturally with the rest of my course. So I says:

And I didn’t realize it at the time, but my frustration with getting students to engage in text-based interaction spilled over into frustration with getting them to interact with a text more generally. That is, it’s not just that I’m unsure how to invite students to meet with me via text, but also that I’m unsure how to invite students to a text at all. Partly because I teach in a discipline that, particularly at the undergraduate level, values parsimony in its written expression (“the best” proof, or “the most popular” textbook, is often the shortest or the one which requires a minimal level of textual engagement), I haven’t been forced to reckon with my students’ reticence to read as much. It may also be why I’ve found some success in video-based flipped pedagogy: for my students, I got the sense that videos weren’t supplementing their reading, they were supplanting it. Was that what I really wanted for them?

That said, as I puzzled through this in the public arena of Twitter (yay for Open Pedagogy), I was fortunate to have one of my colleagues push back from a different disciplinary perspective:

Her point, as I took it anyway, was twofold: (1) that the message is more important than the medium, and (2) that engaging with text is essential for all students, whether online or in person, whether studying Victorian literature or algebraic topology. So I’m coming full circle to the question:

How do I use the tools at my disposal in a course, particularly online tools, to invite students to engage with their text?

What I’ve tried, that mostly hasn’t worked, so far:

• Reading quizzes. These were a particularly unpopular element in a particularly difficult class the last time I taught multivariable calculus in 2013. Despite my best effort to make the quizzes short, low-stakes, and formative in nature, students still felt like they were a “gotcha” — they’d do poorly on them either because they didn’t read, or didn’t know how to understand what they read, and then neither they nor I knew how to follow up on that fact.
• Team readiness tests (RATs). I used this Larry Michaelsen technique in a business calculus class. Each test was taken first as individuals, then immediately thereafter as a team. The individual test was on paper, while the team test was given via an LMS (Moodle at the time) that permitted teams to revise each answer until they got it correct. This went over much better than my previous reading quizzes, maybe because of the self-correcting revision mechanism that made it feel less adversarial. But it was a lot of work to sustain because of the need to create both paper and LMS versions of each quiz. (Plus, at least one student, it turns out, blew off the individual portions because he incorrectly thought that only the team score counted.)
• Social reading. The newest attempt, I’m using this method along with the exciting Hypothesis web annotation platform to create a conversation among my students right in the margins of online text. It’s been limited so far since I’m not using an open, online textbook, but at least I can (1) tell that my students are, in fact, reading, and (2) see exactly how they’re building understanding around the text, where they’re getting stuck when they read, and how they’re helping each other to get unstuck. I’ll be using a free online text next semester so I’m looking forward to using this more consistently throughout the course.

So what are some other ways to use technology to transform how we invite students into a text?

## Catching Up

Learning is inherently a social activity, so an online milieu that does not foster social interaction is not one that will foster learning. How do you create social presence in online learning?

The prompt that got me thinking this week was

• Consider how students must feel when they cannot meet with their classmates and instructor in the classroom, how can you make them feel connected to their learning community?

As usual, my frame of reference is face-to-face teaching, so I thought about this from the perspective of students who miss my in-person class meetings. What do they do to compensate for the (social) experience they missed? Typically, one of three things, from rarest to most common in my recent experience:

1. Catch up with their classmates. Getting notes, announcements, and assignments from peers in the class is my favorite – not only because it requires no work on my part, but also because it’s an indicator of a healthy social ecosystem in the course (or at least subsystem). But I often wonder whether students who have these social connections either (a) brought them into class on day one — taking a class with a friend group that already existed — or (b) were already sufficiently self-regulated learners that they wouldn’t need to rely on peers beyond catching up on administrative details they missed, such as assignment due dates. In other words, are these students already positioned for success regardless of whether any features of my class foster that kind of social interaction?
2. Contact me, usually via email. This happens somewhat more often, or at least I am aware of it more often, than students connecting with one another. While I know some of my colleagues have been thoroughly jaded by students’ requests for post-hoc assistance after missing a class:

I still appreciate it when my students reach out to me because at least they feel comfortable doing so (provided it’s not too late as in the tweet above). At least these students know how to grab hold of one lifeline before their ship sinks, which contrasts them with the final group:

3. Do nothing. I coordinate developmental education for my department, and every semester get to watch the faculty teaching developmental coursework for the first time have their spirits crushed by their students’ gaps in study behavior and engagement. “They show up to class without even a pencil to take notes!” “They miss three weeks without a word and then show up again out of the blue!” “They get an atrocious exam score and then blow off my invitation to talk about it in office hours!” While these behaviors are not exclusive to developmental education students, they are perhaps reflective of a weaker social fabric. We know that students in developmental education are less connected — with their campus, with their peers, even with the world-wide Web since their access to technology is often reduced — so it may be no surprise that this is reflected in a disengaged approach to struggles in their courses. And yes, it’s at the end of this list because it is my most commonly encountered response to missing class, particularly this semester when my developmental math students have ghosted at alarming rates.

So given that creating social presence is a less obvious practice for me to do online than in person, and doubly so for my students, I’m very interested in learning more about how good social connections when they have existed in my teaching — for example, in team-based learning — can be, as the frame suggests in Chapter 8 (Holton 2012) replaced, amplified, or best yet, transformed by the technology and new possibilities that online learning offers.

Holton, D. (2012, May 5). What’s the “problem” with MOOCs [Web log post]. Retrieved from https://edtechdev.wordpress.com/2012/05/04/whats-the-problem-with-moocs/

## Tools for Video Presence

Video – in other words, video of talking heads – has always been an important part of instructor presence for me, even as an accompaniment to face-to-face courses as it typically is for me. But I haven’t found a way, or at least a simple way, to replicate the “chalk-and-talk” that comes so naturally to me as a classroom instructor.

Right now my workflow is: create slides in LaTeX including green screen where desired –> capture and annotate with Doceri app –> capture video in QuickTime player and Doceri simultaneously –> use iMovie to combine –> upload to YouTube. The final results look really good and I have a lot of flexibility, but the time involved for producing a ten-minute video this way is easily a couple hours long.

Does anyone have any more “realtime” ways of replicating the chalk-and-talk in a video setting? Or do you replace this in your online teaching in other ways?

## The Expectations Game of Online Interaction

One of the reasons I haven’t been more eager to teach online is my own views about what both faculty and students expect online learning to look like. My Twitter feed today had a couple good examples. Tweeted from a big online learning conference, the faculty (or at least faculty-developer) perspective:

I’m participating in the #HumanMOOC course on human presence in online education precisely because I do think this way. (Although I am in fact ready to learn.) But worse, I’m daunted by the perspective that exists among students as well, many of whom opt for online courses precisely for reasons that make human interaction difficult:

Clearly some students expect online learning to be impersonal, even anonymous. They expect little interaction with their professor and less with their classmates. They expect an easier course in some cases. They may use anonymity for nefarious purposes (harassment, academic dishonesty). Some of this is stereotype, but I expect not all.

So when I think about a class-wide video chat such as the one shown in the first tweet above, I think about how challenging that would be to make happen. The technology – both hardware and software – however seamless on my end, I envision it being a potential nightmare on the student end. If a student “can’t get their microphone to work” for several weeks, will they drop the class? Will my campus IT support be able to help, especially if the student is not physically present on campus? What if the student can’t afford the technology necessary for this kind of interaction?

And worse, my previous attempts at synchronous interactions with large groups of my students online have flopped spectacularly due to students’ inability to schedule the time around their other obligations. It’d be a dream to have all 25 of my students in an online course meet at one place and one time virtually for a class discussion, but I know that many of them will have opted to take the course online precisely because they don’t want to abide by a regularly scheduled meeting time. So these are big hurdles to overcome, but I’m open to having my thinking reoriented. There are, I’m sure, a lot of ways to human interaction in online learning beyond the synchronous video chat, and I’m looking forward to learning more about them.