Wednesday, April 6, 2016

New TYPES of puzzles

We're over halfway done with the semester and my Games and Puzzles class has been fascinating and rewarding in many ways.

As I mentioned last time, our first foray in pencil puzzles was into Battleships, after which we've explored the pencil puzzles Nurikabe, Skyscrapers, and Masyu for a few weeks each. I was starting to become wary of the structure rut that I had been finding myself in, in which we would start with a new puzzle, investigate strategies in detail, try creating a puzzle of that type, and then the final result of the unit was the creation of a puzzle of that type given some twist.

So in class on April 6, I suggested the following challenge: work in groups of 3 or 4 to create a hybrid puzzle incorporating concepts from at least two of the four puzzles we have mastered this semester. I was encouraged by how quickly they took to the task. The discussions remained focused on the challenge and the discussions included all members of the group. In my other project-based classes I have a tendency to be very hands on. But for this challenge the students didn't need me at all---which is great! The ideas are flowing and the students are pulling in all the expertise that they have accumulated throughout the semester. I stepped in a few times to diffuse some concerns such as what is allowed in this new type of puzzle (Anything!) or when a group was having a heated discussion about the type of clues that they should include (What if you created two different types of puzzles?!)

My students started the semester solving the logic puzzles haphazardly and moved to thinking critically about solving strategies. They have progressed from creating puzzles of a specified type often without regard to uniqueness, to thinking critically about creating puzzles using a puzzle solver's eye. Now they are creating new types of puzzles! These transitions have been gradual and I can see my students' confidence building.

Wednesday, February 24, 2016

Announcing the Circular Teaching Squad

Abstract: I discuss the rationale for changing the name of our lunch discussion group. Join us March 22nd!

For a couple years now I have been moderating lunch discussions about teaching and learning on campus here at Queens College.  I had been doing so under the guise of the Experiential Learning Group started back in 2012.  After our last lunch, my friend and kindred spirit striving for excellence in teaching Nathalia Holtzman came up to me and suggested that I move away from its association with the defunct Experiential Learning group.

I was especially swayed by the argument that the discussions we have been having about teaching and learning are much broader than "experiential learning", and perhaps that misnomer was keeping other faculty away who might want to participate in a more general discussion about teaching and learning but were either less interested in experiential learning, or unsure of what this buzzword entails.  This set in motion the idea to change the name of the group.

The name I chose for the group is the Circular Teaching Squad.  I like this name for multiple reasons.  First, it is unusual, and would make people pause, question, and consider it, if only for a small while.  Second, it is playful, which is a quality I very much enjoy.  Third, it can be easily shortened to "The Teaching Squad" or just "The Squad".  Fourth, isn't SQUAD an amazing word?  It even has a Q in it, just like Queens!  Fifth, it conveys the idea of a "Teaching Circle", where we share ideas as equals.

Last, what truly inspired me to choose the name is the concept of a Circular Firing Squad, in which a group of people (often political candidates) attack each other, often weakening themselves so much that they lose the bigger fight.  Instead of weakening each other, our discussions involve shooting ideas about teaching and learning into a circle of peers who teach diverse subjects, transmitting practical knowledge and strengthening our bonds as educators at this public university.

The Circular Teaching Squad will meet from 12-2 on Tuesday, March 22 in the President's Conference Room #1 in the Queens College Library.  (Stay the whole time or drop in informally!)  The topic of discussion will be about the ways in which we teach students how to learn.  Should we teach these ideas in a more explicit way?  In what ways do we teach students how to take ownership of their learning? While the is the first meeting under the new name, it will continue our tradition of invigorating the discussion of teaching and learning on campus!  If you're in the neighborhood, we'd love to see you there!

This blog post is part of the Queens College Teaching Circle blog; it is cross-posted on my personal teaching blog, Math Razzle.

Tuesday, February 23, 2016

Student Forays into Pencil Puzzles

Abstract: In the first six days of my pencil puzzles class, my students have had to think deeply about solving and creating Battleships puzzles. This post discusses the day-to-day class mechanics and assignments.

Now I'll talk about the first few days of my pencil puzzles class.  (Previous post setting the stage.)  I started the first day of class by qualifying that this class was new to me and would be an adventure for them and for me.  I explained that there wouldn't be any lectures, there wouldn't be any exams, and that it was unclear exactly what the assessments would be---the initial ideas I included on the syllabus were:

  • One-page-long essays about the history of certain puzzles and related solving techniques.
  • Curation of content for a wiki about puzzles, puzzle solving, and puzzle creation.
  • New puzzles with accompanying discussion of the methods used to create them.
  • The uploading of puzzles to web repositories.
  • The creation of a puzzle trove.
I conveyed that I wanted the class to be a student-driven discussion about the solving and creation of pencil puzzles.  The rest of the first day was filled with a worksheet with four types of pencil puzzles and the students worked on two of the types.  The first homework assignment was to work on 12 Battleships puzzles I had copied from puzzle magazines I had laying around and for the students to address the following prompts:
  • What strategies are you using? (Give them names. Give a simplified example of what the strategy says. Only including the relevant parts of the puzzle board.)
  • What makes one puzzle harder than another? Try to find a way to explain it in words.
  • How does knowing that there is one unique solution help you?
  • Highlight areas where you get "stuck". If you are able to get past being stuck, how were you able to do so? Take a picture with phone/camera. What happened at that moment in time? Why are you able to make a mark? What logical reasoning are you using? What was the "AHA" moment?

On the second day of class, students discussed their strategies and I served as scribe.  In our brainstorming session before the start of the semester, Sarah Mason had drawn on her own classroom experience to suggest that we give the solving strategies names so that they would be easier to discuss, similar to how names are given to strategies on Sudoku solving websites.  I thought that this was a great idea, since it would solidify a strategy with a clever name and as mathematicians, it is always good to define commonly occurring concepts.  However, the naming of techniques didn't take hold.  Either there was a lack of creativity for punny names, or just no appetite for giving the techniques names, so I abandoned this idea after the second day of class.

I found it especially worthwhile when a student projected onto the screen a before-and-after pair of pictures.  He showed a situation where he was stuck, and then detailed the logical reasoning that it took to get to write the next mark on the paper.  This sort of play-by-play analysis is crucial in conveying one's logic and strategies to the class.  The detailed discussion of strategies we had in class was universally considered to be helpful, and led to the next homework assignment of completing more complicated battleships puzzles before the next class.

On the third day we spent the whole day working through the following battleships puzzle from start to end because every single student was stuck:

Screen Shot 2016-02-23 at 9.55.22 AM

This led to an in-depth discussion about the concept of a decision tree, and that battleships puzzles that are hard often have a wide and deep decision tree.  In other words, in order to figure out the next mark to place, there are many possibilities for the placement of the next ship (Either here or here or here or ...) and for each of those choices, the sequence of logical steps leading to a contradiction is complicated.  Our discussion about this puzzle was especially interesting because of the appearance of the 21112 clues on the bottom left.  We discussed how the uniqueness of a solution implies that there must be some symmetry in the placement of ships, restricting the possible next moves.  The homework for the next day was as follows:
  • Create two or more Battleships puzzles with some intrigue, at least one of which is a Junior Battleships puzzle. As you do so, answer the following questions.
  • What is hard about creating a puzzle?
  • How do you know your puzzle has a unique solution?
  • How does creating puzzles influence how you solve them?

So on the fourth day, the students brought in their hand-crafted battleships puzzles, we arranged the chairs in a circle and had a deep discussion about the difficulties involved in their creation.  As I moderated this lively discussion about puzzle creation, I transcribed a number of themes that kept reappearing, from which the homework assignment for the following day emerged organically:

  • Create one or more Battleships puzzles that push the boundaries imposed by the normal structure of a Battleships puzzle.
  • Write up a page of discussion about the making of your puzzle.
    1. What conscious decisions went into its construction? What boundaries did you push? Which decisions make the puzzle easier or harder to create? Easier or harder to solve?
    2. How does puzzle uniqueness play a role in its construction?
    3. How does the tension between "uniqueness of solution" and "puzzle difficulty" come into play?
    4. What direction were you using in its construction? Forward? Backward?
The last half-hour of this fourth day was spent working on peer-created puzzles that had collected copied in our office.  One issue that arose is that some of the puzzles did not have a unique solution.  This served as a warning that everyone needs to be very careful when creating the puzzles.

The fifth day was a day of presentations.  Each student came to the front of the class and projected their puzzle on the screen, and talked for a few minutes about their puzzle and the boundaries they pushed when creating it.  Some students changed the shape of the board.  Some students changed the instructions.  Some students changed the number and types of ships.  One student even created a battleships puzzle on a torus!  To me the most amazing part of the day was that the nine students who presented had each pushed the boundaries in distinct ways!  We ended the day by introducing our next type of pencil puzzle: Nurikabe (ni, cp, pp), which the students would have to practice solving for the next class.  The following screenshot about Nurikabe is from Conceptis Puzzles:

Screen Shot 2016-02-23 at 11.49.15 AM

Monday 2.22 was the sixth day of class, and students came in to discuss their trials and tribulations in solving Nurikabe puzzles.  We once again arranged the chairs in a circle, and I moderated a discussion about the difference between Battleships and Nurikabe.  What was the most fascinating to me was the palpable interest in and anxiety about creating Nurikabe puzzles, not the strategies involved in solving the puzzles.  One student explained that this feeling was based on the fact that at some level she was comfortable solving puzzles, but because this class was the first time that she had to create puzzles, that is where her mind went first.

So far, I am very pleased by how well this class is going.  Every student is participating in class in productive discussions.  The students are excited about and engaged with the material.  I am a bit concerned about the class becoming tedious for the students; I don't especially want the semester to fall into a rut of "introduce a new type of puzzle---learn how to solve it---learn how to create it".  If you have any suggestions about what else might break up the monotony and let the students get more out of the class, I'd love to hear your thoughts!

This blog post is part of the Queens College Teaching Circle blog; it is cross-posted on my personal teaching blog, Math Razzle.

Monday, February 22, 2016

Teaching Pencil Puzzles

This semester I am teaching Math 555: Games and Puzzles for the first time.  The class does not serve as a prerequisite for any other class and it does not fulfill any degree requirements more than miscellaneous math credits, which means that I have a free hand to teach the content I want to teach.  This class is normally offered during the summer and recently the class has been taught with a focus on game theory.  Instead, I decided to focus on one of my favorite past times, pencil puzzles.  These are small puzzles that can be solved using pure logic.  Many people will have tried to solve the most ubiquitous of pencil puzzles---Sudoku!

(A quick tangent to provide good pencil puzzle links!  Three high-quality websites for downloads and online play of pencil puzzles are Nikoli, Conceptis Puzzles, and Puzzle Picnic (user-generated puzzles).  Two excellent puzzle magazines (which also include word puzzles) are Games World of Puzzles and Dell Math & Logic Problems.)

I wanted to get a bit off the beaten path and away from Sudoku, and not even focus on the two types of pencil puzzles that I suggest to friends (and enemies) as being "after Sudoku": Hashi (ni, cp) and Slitherlink (ni, cp, pp).  In fact, a couple students had already seen these puzzles.  Instead, I decided to first start with Battleships (cp, pp).  This is a solitaire version of the two-player game you may have played pictured here:
In both games, you are searching for a fleet of ships that are placed in a grid.  In the two-player game, your opponent places ships in a way that they think will be difficult to find, and you guess the positions.  In the pencil puzzle solitaire version, a fleet of ships is hidden and your only clues are the number of grid cells in each column and in each row that are covered by some ship.  With these numbers, there is one unique solution that can be found purely by logic.  Here is a moderate-to-difficult puzzle created by user anurag.sahay on Puzzle Picnic:

There is the additional rule that no two ships can touch, even diagonally, which reduces the amount of possible placements.  Try out a first few sample puzzles at Conceptis Puzzles.

This was the first class I have taught in which I have no idea how the semester is going to play out---I am taking the class day by day with the ultimate goal of compiling a collection of student-created puzzles of many types.  In my next post, I will discuss the first days of the semester and how things have gone.

A quick thank you to Sarah Mason, who acted as a sounding board for the course structure before the beginning of the semester, to my students for being enthusiastic participants on this adventure, and to my department for their support in allowing me to teach such an amazing class!

This blog post is part of the Queens College Teaching Circle blog; it is cross-posted on my personal teaching blog, Math Razzle.

Monday, November 2, 2015

Creative Writing is Experiential Learning!

Abstract: Creative writing gives students a chance to experience a topic in a new way and lets students try on an alternate persona. Also, I muse about how to include creative writing in a math class.

At the Experiential Learning Lunch on October 21, 2015, we had a lively discussion about creative writing and why it is such a powerful tool in experiential learning.

Keren Dali (Library and Information Science) explained that creative writing is a great way to inject energy into the classroom. She has students choose from writing a screenplay, a letter to the editor, a blog post, a love story, etc. It allows the students to change their voice by playing a role, which allows students to take risks that they wouldn't take from their usual position. For example, a student might learn about the background of an author and do some reflective writing about them. This would help the student to connect better with the author's writing because the students place themselves in the author's shoes.

Yves Cloarec (English) voiced the concern that standard writing assignments often have the students writing what they think the instructor wants to hear. An example was given in which students did background work or research that supports one conclusion and they write that it supports an opposing conclusion because that is what they think is what the research was supposed to say!

Yves also talked about the process he uses to have the students get the most out of their creative writing. His assignments often have two parts. There is the creative writing part, but there is also an academic writing part. For every two pages of creative writing that the students submit, they are also expected to produce one page of academic writing explaining the thought process behind their creative writing. The students would discuss the research that went into their work, giving citations for theoretical resources or showing analogous constructions in other works they have read. This left a strong impression on me because I often think of "creative writing" as something that is difficult to grade objectively and fuzzy in some sense, but by requiring students to theoretically justify their creativity, you can see the thought process behind the project, and see that the students actually put thought into their work.

There was a discussion about having students learn about the writing process by having the students first think about all the steps involved in making toast---it sounds easy until you go into all the steps. The students are asked not to write the steps but instead use symbols to draw the steps of the process. Once the students have tripped over themselves they are supposed to learn that writing is a process as well. That writing entails first thinking about what they want to write, organizing their thoughts, starting to write, and rewriting (multiple times).

This exercise made me think about a recent blog post that I read about teaching students how to write mathematical proofs, where every step from start to finish must be explained with no gaps in logic. The activity in question involved having the students write down the sequence of events that are necessary to make a peanut butter and jelly sandwich. (Watch a Fun music video non sequitur.) Then an outside actor "PB&J guy" comes in and does exactly what the students have written, which is not at all what the students meant. This is an exercise in getting students to think carefully and write precisely. And then in class it is possible to refer to gaps in logic in a very handy way by referencing "PB&J guy" throughout the semester. Unfortunately I can't find the original post; if anyone finds it, I will update this post. (Here is a nice handout about such an activity.)

I'm trying to figure out how to incorporate more creative writing into my own mathematics classroom. Would that mean writing a paper from the point of a view of a mathematician so that the theorem they just proved is understood more in context? I have already had students research mathematicians and write papers or edit Wikipedia on them, but this would take that to another level. Would that mean writing a play or story that had as its basis some mathematical theory or computer science algorithm? (In fact, our class acts out the Gale-Shapley algorithm in Graph Theory.) Would this involve having the students explain their thought processes when they are writing their proofs or having students submit not just their proofs but also their proof drafts and including a discussion about why they had to change their original reasoning? I will be able in incorporate one of these ideas next semester in my Math with Mathematica class. I will make sure to have my students write the thought process they went through to create their mathematical art, so student's aren't just "making art"—they are being thoughtful in the choices they make along the way.

This blog post is part of the Queens College Teaching Circle blog; it is cross-posted on my personal teaching blog, Math Razzle.

Monday, October 26, 2015

Experiential Learning Lunch on 21 Oct 2015

Abstract: We highlight a few of the innovative teaching techniques that were presented in October's Experiential Learning Lunch.

On Wednesday, October 21, 2015, twenty-eight members of the Queens College community assembled for a lunch discussion devoted to Experiential Learning. (Learn more about our group.) I had asked participants to prepare a 2-3 minute discourse about something that they tried recently and a self-evaluation of what went well and what could be improved in the future.

In this post I would like to highlight some of the discussions we had.

Zahra Zakeri (Biology) shared the idea of the "experimental essays" that her students are required to write on various topics throughout the semester. The students must develop an experiment that would test certain hypotheses, such as "How would we know whether two twins are monozygotic?", and justify why their experiment is a valid way to test their hypotheses. By having to design this experiment the students have to gain a deeper understanding of the material in her classes instead of simply memorizing facts.

Elena Mancini (German Language and Literature) highlighted how she is taking ideas she is learning in the theater techniques workshop and integrating them into her class. She asks groups of students to analyze a text passage for tone, emotions, believability, and perspective. The gropu takes this information to create a play about the passage using props. This live reinactment of the passage infuses energy into the classroom and is widely regarded as a fun and engaging assignment. This discussion initiated a more general conversation about creative writing, which I feel merits its own blog post. So keep tuned for that!

Keren Dali (Library and Information Science) was happy to share how most of the courses in their department are experiential. For example, if the students are learning cataloging, they need hands-on experience in learning how to catalog. Keren was especially proud of how she helps to take experiential education to the next level—into the community. Her class partners with libraries in the community; each student group spends time with a librarian to find challenging areas for which the libraries don't have the staff or resources to address. The students work out a solution to the problem in a group and present it to the librarian by the end of the semester. Many of these solutions are then implemented by the community libraries. The students get real-world experience and the libraries have made strides toward improving their services. It sounds like a great program!

Lightning round! One-sentence summary of some other presenters:

Joe Pastore (Mathematics) uses the software GeoGebra to exhibit mathematical concepts dynamically in class, which allows his high-school-teaching students to integrate this technology directly into their own classes.

Adam Kapelner (Mathematics) has programmed software called Gradesly which integrates directly into Google Sheets to provide students and instructors with up-to-date information about course grades.

Eva Fernandez (Provost's Office and CTL) talked about the Queens Memory Project, which collects an audio and visual history similar to Story Corps.

Jack Zevin (Education) talked about this simulation game in which students play city council members and gain first-hand experience with the legislative process.

It is always great to hear the ideas others are trying on campus. In many of these situations, students are initially reluctant to participate in these "non-standard" assignments but by the end of the semester these assignments are what the students rave about on their teaching evaluations.

The topic that generated the most amount of buzz was about technology in the classroom. Should students be allowed to use their phones in the classroom? How can we integrate technology into the classroom in a thoughtful and educationally beneficial way? What works well in different size classes? I think we have found the topic of December's Experiential Learning Lunch!

I was especially pleased and proud that three of my math department colleagues came and participated. It's nice to be building a local network supporting innovation in mathematics instruction!

This blog post is part of the Queens College Teaching Circle blog; it is cross-posted on my personal teaching blog, Math Razzle.

Wednesday, October 21, 2015

The Experiential Learning Group at Queens College

Abstract: This post is a quick introduction to our college's experiential learning group. What is experiential learning? What happens in a group meeting?

I've been participating in a teaching and learning group at Queens College focused on Experiential Learning since it was founded in 2012 when Grace Davie (QC History) was taking the lead. This group of like-minded educators spans many disciplines and provides a refreshing sense of community on a commuter campus where people in different departments rarely interact. While the faculty members participating in the various events changes each time, the value of attending the group is always very high since the self-selecting members inevitably care about teaching.

What is experiential learning? "Excellent question!" We spent many of our first meetings trying to determine a good working definition for our group. Should the focus be on social justice work of our students that happens off campus? Should the focus be on college-sponsored internships where students learn skills ``on the job''? Should the focus be on active learning activities that happen in the classroom where students engage with and experience the material instead of passive non-participation? We ended up deciding to not decide. We wanted our group to be an umbrella group that allowed people with disparate interests to come together and share their innovative teaching practices.

At the beginning our group met in multi-day workshops to which were invited outside speakers to discuss teaching pedagogy and mindfulness. More recently I have been carrying the torch of our Experiential Learning group by organizing and serving as the moderator of lunchtime discussion sessions, sponsored by the Queens College Center for Teaching and Learning. I tend to favor round-table discussions where we each get to bring in ideas from our teaching, and that is the focus of today's lunch, where I have asked participants to prepare a 2-3 minute discourse about something that they tried recently and a self-evaluation of what went well and what could be improved in the future.

The most positive aspect about this format is that we are exposed to many different ideas and we get to see the diversity of techniques that are available and in practice at Queens College. This helps to give a sense that we are a college where good teaching by our peers is happening and is valued. The most negative part about this format is that for everyone to talk, people get at most five minutes, so it is impossible to get into the nitty gritty details about the module's implementation or give constructive feedback to the presenter, or have a deeper philosophical discussion about the module. These points were better addressed under the multi-day workshop format, which takes much more preparation by the organizers.

Today's session has an additional undercurrent of supporting the administration's need to collect information about experiential learning that happens through Queens College to comply with the City University of New York's task force, required by a new New York State law to increase experiential or applied learning opportunities for undergraduates. One personal goal of mine will be to have people contribute their experiential learning modules to the collection effort.

I'm sure that today will be as invigorating as always! We have another lunch scheduled for December 2, 2015. If you are around and would like to participate, make sure to join us! You can email me at to be added to my email distribution list about Experiential Learning at Queens College.

This is the third blog post that I am writing as part of the Queens College Teaching Circle blog; it is cross-posted on my personal teaching blog, Math Razzle.