Findings

After a semester of introducing puzzles as warm-ups, incorporating them into the homework, and treating math more as a discovery of ideas rather than tasks that need to be accomplished, I have found a few underlying themes.

What I found

I want to preface that what I have learned through doing puzzles has only been possible through spending time setting up a safe environment, holding high expectations of the students, and deeply caring for every individual student in my class. These are the ingredients of any great classroom. Students need to feel cared about, know they have a voice, and trust who is teaching them.

The following chapters describe my journey from the introduction of puzzles to the insights I have found along the way. I will describe how I used puzzles in the classroom, what the effects were, and how the introduction of puzzles can have a positive effect in any math classroom. The following chapters are split up into: puzzles used as warm-ups, puzzles in the homework, and students creating puzzles. Finally there is a brief discussion about some of the puzzles that I used. I include examples of these puzzles, and discuss how I used them, in the appendix.  

During my research I interviewed six students who I picked to represent the diversity of the class in terms of race, gender, mindset and mathematical understanding. I picked the students based on my previous 2 months with them. I looked at overall scores, attitude in class, and participation. I also picked students that would be more likely to share their views and opinions with me. I also looked at their responses and their feeling toward math on the first exit card. All six students are in the same first period class and the interviews conducted were done in groups. I also include students’ voices from surveys and exit cards. In the following chapters, I also include the stories of a couple students who were not members of my focus class but nonetheless impacted my teaching practice in ways I couldn’t have known before starting my research.  For a more thorough discussion of my research methods, see Chapter 6.

Puzzles as part of the homework

"We learned that some of the math that starts out as the most complicated can be pretty simple and easy, you just have to kind of work through them. I used to look at math as an impossible challenge, and now think of it as just an incomplete puzzle that can be pretty fun once you learned to conquer it" -Kelly

An introduction puzzle that any teacher can employ in their class goes like this: There are four matchsticks arranged as the picture above with a cherry on the inside. The goal is to remove the cherry from the wine class by only moving two matchsticks. The end result will have the same wine glass with the cherry outside of it (and gravity does not remove the cherry).

The problem above, like many of the other warm-up problems is how I started my class most days. Students would work in groups on puzzles and by themselves on others. What I found was that students quickly became engaged in thinking. It seemed that right off the bat they would give the puzzle an attempt without fear of “getting the wrong answer.” They were able to express their creativity and find solutions that I the teacher would often overlook. The puzzles also had the effect of creating a positive atmosphere in the classroom and changed my perception of the students’ perseverance or willingness to sit and work on a problem.

Before I began to use puzzles in the classroom I was often frustrated by the amount of time that the average student would spend trying to figure out a math problem. It seemed that many students would throw in the towel on a challenging problem within a few minutes. They would state that “the problem is too hard” or that “I don’t know how to do it because I never learned it” instead of just trying to figure out the problem. I remember thinking that there must be a way to encourage them to think deeper, to focus for longer, and to try to approach problems from multiple perspectives. Putting math aside, even if it was a problem that they were faced with in everyday life, I wanted the students to be able to think of solutions and if possible use rational judgment. Ultimately, I found that puzzles not only increased the time students would spend on a math problem but it was an active work out for their brain in the same way that mathematics should be.

At the beginning of the year, I started by giving a puzzle to the class to work on in groups of four. I had no idea how the class would respond to taking time out of the day to throw ideas around and try to solve a puzzle. So I unleashed the first puzzle, split them up to work with their tablemates and challenged them to solve the “cherry in the wine glass puzzle”. What I experienced from the class during this time was uninterrupted concentration, teamwork, and students collaborating with one another. I found this quite remarkable, as I had never seen the entire group interact so much with one another. In fact, every time I did this I found that people would engage in trying to figure out the puzzle given that day. The students were sharing ideas of how to solve problems and challenging each other to see who could solve the puzzles the fastest.

What was interesting about puzzles was that it seemed to have leveled the playing field for all students, as higher academic standing students performed no better on the puzzles than the students that typically struggled with math. It was as if students had the opportunity to start from scratch and, depending on the puzzle, there were no students who had a leg up on the others from knowing any particular math theorem. Over time I would discover that the solving of puzzles would build confidence within many of the students that would later be applied to new math concepts.

How students experienced warm ups

“I think it is more of a mind thing because if I could solve one of those (puzzles) my mind does the same thing while I’m trying to solve a math problem, it gives you more options to think of it, I think that if you do more puzzles it will open your mind to different possibilities in math”

The above quote is from Spence, one of my honors math students. Spence hasn’t always enjoyed math, but he heard good things about my math class and decided to take the challenge and sign up for honors. In our classes all the students of different ability levels attend the same class and it is an option to sign up for honors.  This is a bit unconventional when compared to most high school classrooms, but at High Tech High we do not believe in tracking students. Spence, like the typical teenager, has a mild obsession with video games and enjoys being social with his peers. To his credit he is not afraid to ask questions when he is stumped or use his resources to try to complete a problem. He also has developed a knack for thinking about things from a different standpoint than many of his classmates. He struggles at times with the foundations of algebra, but shows surprising creativity and insight regarding puzzles and problems with intuitive physical applications. When I interviewed Spence regarding the warm-up problems he quickly showed enthusiasm toward them saying: “They’re fun, they definitely get your mind to think about how to solve different problems different ways and they get your mind running especially during the morning when most of us are still half asleep.” In fact, this was the sentiment shared by much of the class. The quotes below are from a conversation I had with my focus students who were very candid about their responses.

 “It is a fun way to get to start the class or end the class because after doing all this stuff it is a way to get your mind working but also to cool it down…you can just do a little fun puzzle and it kind of helps in a sense”

“It’s like a good transition especially in the morning, it makes us think that we are doing math now, it’s no longer the feeling that I just woke up and oh crap what am I going to do?”

“I liked the idea of having puzzles to do in the beginning of class, it gets our brain moving and it definitely lights up the mood”

“I always enjoy doing puzzles and the puzzles help me adjust to a logical problem solving environment”
“YES! I LOVE THEM! It gets me excited for the math period”

Students seemed to enjoy doing puzzles in class but their reasons weren’t always the same. Many commented that they liked that it lightened the mood at the beginning of every class, others liked the challenge of thinking outside, some liked working together in teams, and there were those who liked the competition of it all. Even the few students who expressed a bit of hesitation toward the puzzles admitted that it really made them think, and well, sometimes that was difficult and uncomfortable. As one student said, “They are fun just because it is unconventional math (which I love), but personally, I have always hated puzzles, but I guess with practice I could get better at doing them.” Moreover the puzzles served as a transition from the early morning tiredness into a mathematical mindset ready to tackle the problems or ideas of the day. This transition not only helped students engage in mathematical thought, but it also got them up out of their seats, moving around, and at times interacting with one another.

Thinking Creatively

“Yes. They (puzzles) challenge me to think more creatively. Its a great exercise/warm-up activity!”

I found something interesting when I began to collect student feedback about the class and puzzles. Without prompting, the students’ replies often incorporated the word “creativity.”  In my years of teaching I have often collected feedback about the class and until this point I hadn’t had many remarks about creativity. This was always interesting because as I have said before math is the pinnacle of human creativity and yet the students weren’t seeing this. Finally, I found a way in which students were thinking creatively. Shana, who has struggled in math all throughout high school wrote:
“The puzzles are great. They are a way to warm up the mind through math and have fun doing it. I think they bring groups together and make us more creative and help our ability to think outside of the box.”
Shana is a natural leader. She is funny and a bit on the louder side. She has had a hard life. Her mother passed away between her junior and senior year, her family doesn’t have a lot of money, and she at times struggles emotionally. Yet, she is powerful, proud, and can light up a room in a matter of seconds. Often when things are getting crazy in class she is the one who quiets them down and gets us back on track. I sometimes refer to her as “bad cop” as we play a “good cop/bad cop” role together to bring the attention back to the matter at hand. She has also been one of the biggest advocates of doing puzzles in the warm-ups and in the homework, and not because she thinks they are easy. She told me,  “I actually think the puzzles are harder sometimes then the homework problems, but I understand what the puzzle is asking me to do, sometimes with the homework I don’t.”
Shana’s comment highlights that there is a directness and simplicity to puzzles that is appealing to students who tend to struggle with math. In talking with Shana it seems she knows what she is working toward while solving a puzzle; the end goal is clear but the path is not. Compare this to the typical high school math problem where the problem’s path is prescriptive and the end goal isn’t entirely obvious. I have always said that the beauty in mathematics is in the steps, perfect pieces of logic used to assemble a deeper understanding of the world. In reality it seems that mathematicians on the forefront of knowledge use mathematics and the logic therein to solve their own puzzles about the world. True mathematicians are puzzle solvers, as they have no prescribed path to follow considering that they are traversing uncharted territory. In general, students found the puzzles both challenging and a way to get the “gears turning” before we started the daily lesson.

As the semester progressed I began to think that puzzles were of equal value as the problem solving techniques the students were learning in class. The puzzles allowed students to think more creatively and “outside the box” when compared with the problems I cooked up to test their ability of the tools learned in class. In order to be a good mathematician or math student for that matter, it is of equal if not more value to be able to think creatively and to visualize the problem then to have a large toolbox of mathematical tricks.  The creativity is in the connections made and the usage of the tools that mathematics provides. Being able to see from a new viewpoint empowering one’s self with big ideas is what mathematics is all about. From the book Creativity: The Essence of Mathematics, Eric Mann argues:
“Traditional teaching methods involving demonstration and practice using closed problem with predetermined answers insufficiently prepare students in mathematics. Students leave school with adequate computational skills but lack the ability to apply these skills in meaningful ways. Teaching mathematics without providing for creativity denies all students the opportunity to appreciate the beauty of mathematics” (2006, p.3)

Fredirick Fröbel spent much of his life advocating the idea of education being a guided discovery. He states that the teachers can steer students towards certain activities that will guide the student through questions and prompts. He advocated for less lecture time and more hands-on learning, and that young children should learn through play and to instill in them the scientific method at an early age. In his book The Education of Men, Frobel writes, “Play is the highest expression of human development in childhood for it alone is the free expression of what is a child’s soul” (1885, p. 8). Fröbel was a promoter of starting the education of students through puzzles. They should experience math through creativity and solving hands-on problems and going outside often. This seems directly opposed to the closed problems and closed doors of our modern day classrooms. 

The cherry in the wine glass puzzle is an interesting example of this as it requires no mathematical skill to solve. It involves trial and error, play, and thinking a little bit creatively. I find the creative side of the cherry puzzle quite similar to mathematical creativity. For instance, if you want to find the area of a triangle you can start out by wondering how much of a box the triangle takes up?

The nice part of a problem or a puzzle like this is that it leads to a deeper understanding but it is also just fun to try to figure out. Does it depend on the size of the rectangle? What if I made it a square? What would happen if I changed the orientation of the triangle, would I get the same result? Similar to a puzzle, here is where creativity comes into play. What if I were to take the creative leap and draw a line inside the triangle?

Suddenly the answer becomes obvious! The triangle takes up exactly half of the rectangle. So if I knew the area of the rectangle Area = (base)(height) then the triangle must be exactly half of the rectangle, Area= 1/2(base)(height) which is the formula for finding the area of a triangle.
There is no memorization of a formula for the area of a triangle needed, only an understanding of how it works. The creative step was creating the vertical line separating the triangle into two other triangles and seeing that they are the same size as the portion of the rectangle left over. Similarly, the cherry in the wine glass problem has a simple solution that involves moving the matchstick that makes up the bottom of the glass half a space to either the left or the right, them moving the side of the glass no longer connected to the bottom completing the wine glass.

Although this step doesn’t lead to any mathematical insight, like the triangle in the box problem, it does exercise the student’s ability to think creatively. The cherry problem is about exploring the process of finding things out and of course, playing. I have heard many times that the deepest learning that takes place is through play. The influence it had on individual math problems I’m not entirely sure about, but the students were thinking.

It was my hope that some of this same eagerness and confidence to figure puzzles out would transfer to figuring out other math problems. Along the way, I learned an important lesson about the power of puzzles. Puzzles taught me that students were more able to use mathematical tools to solve a problem if they had a sense of what the end goal should be. This was something I hadn’t expected and that I began to realize halfway through the class. With more traditional math problems, it was difficult for students to figure out what the solutions should look like or even if they had solved the problem. Another curse of the way mathematics is typically taught is that students are encouraged to simplify their solutions, which hides the logical patterns underlying the answers. Unlike traditional problems, where the goal was unclear and the logic was obscured, puzzles - especially the ones that were the right difficulty level for the students – helped students to persevere and work until the puzzle was solved.

Perseverance

“At first I honestly didn’t understand the point of the puzzles and thought they were pointless, however after I finally learned how to solve them I really appreciated the puzzles and I think they are a great way to expand the way you perceive a problem and the way you solve it” -Stacey

One of the main questions I had while doing my research was how to get students to spend more time thinking about math problems and to truly push their own understanding, specifically when it was new material and the questions were difficult.

My initial thought based on previous experience was that students would enjoy the puzzles and spend a decent amount of time trying to solve them. I had no idea how much time they were willing to spend in class and out trying to crack some of these puzzles. The results varied from student to student, but from my observation if the students didn’t solve a problem in the warm-up time allotted I would still find many of them trying to work it out during class, during lunch, and sometimes after school. When I asked students if they found the warm-up puzzles challenging I received overwhelmingly positive responses. Here is a cross section of the many responses I received from an interview with focus students and a survey.

“That nail one made me so frustrated that all I wanted to do was stare at it and figure that thing out. It would be funny if we started solving the puzzle we created after exhibition!” -Robert
“I like that part of class because it really makes me think and try to come about the problem in multiple ways” -Shana

What fascinated me about puzzles as I experimented with different types was the effect it had on certain students who would typically struggle with content. My focus student Shana had a lot to say about the time she put into the puzzles. I asked Shana about her process of trying to figure something out, and like most teenagers with homework she said that she would “try to figure out a problem for a couple minutes, then begin to use her resources such as peers, or Internet to look it up.” Although there are math books readily available in the classroom for the students to use, she admits, “Yeah, I don’t use them (textbooks) to try to figure things out.” When asked why she stated “I never really learned from text books, I have always found them difficult, and well, a little boring.”

When asked how she approached working on puzzles, the story was different. She said, “oh you know the word ones, the cryptoquips, I spend a lot of time on those.” I asked her if she worked on the puzzles with other people and she replied, “no I don’t, I know I can figure them out on my own, it may take a lot of time but I think the reward is figuring it out by yourself.” I thought that this was not only interesting but also telling.

Here we have a student who will quickly gave up on a homework problem yet will not be stumped by a puzzle and would be willing to spend time trying to figure it out on her own. As a math teacher I want the students to be able to form their own ideas and hone their critical thinking skills. It is an added bonus to have them understand how to use a new set of tools such as logarithms or derivatives. It is always great when they can use mathematical tools to understand the world in a different context, but I really just want them to think for themselves.

Not all students felt as positively about the puzzles as Shana. In a preliminarily survey, some studentsexpressed difficulty solving puzzles and seeing a connection with mathematics.

“I think it’s interesting and frustrating. I've never really been very interested in puzzles but I think it’s a good way to learn” -Alex

“Well...I am terrible at puzzles. I always feel super behind and frustrated because I can't even do those simple word puzzles where (for example) A=E and then you have a sentence that says GHSK EIG AIGJE QEUCMO or something like that. I am basically the only person who can't do them, but I work really hard and try” –Meg 

These sentiments changed over time as students began to get the hang of the puzzles. It was as if they gained confidence with practice thinking differently. As one student told me, “I think I learned to like puzzles this year. At first I hated them, then something clicked and I got a few of them. I feel like I learned something.

Connection with students

“The puzzles that should be kept for sure are the Sudoku based ones, those were simple yet challenging, and it was fun to try and race your friends to see who can solve it first” -Jana

The power of the puzzle became clear to me when I noticed one of my students, who typically had a difficult time engaging in discussions and problem solving activities, become engaged in trying to solve puzzles. Garrett was not one of my focus students because he prefers not to talk. Yet Garrett is the student that I focused a lot of my time and energy analyzing as I found out I could connect with him through puzzles and play. Throughout the year it seemed like he needed constant one-on-one attention for the entire period to stay on task. This is not to say that he wasn’t capable, quite the contrary. In fact it was my belief that he was a rather intelligent young man. He did struggle with the homework often but that was due to his aversion to asking for help or even using the resources provided to him.

Then the breakthrough came. As I introduced the very first puzzle and stood back to give them time to ponder how to do it, I glanced in the direction of Garrett. He was participating and working with the people at the table (although silently) to try to figure it out! This was incredible! I began to wonder if I had stumbled upon the Holy Grail in terms of finally engaging Garrett? So I nonchalantly took notice everyday of what was happening with Garrett during the warm-up puzzles.  I noticed that he had a preference. He enjoyed the puzzles that were physical and showed less interest in those that were thought problems or that involved writing on paper. He was willing to work on the puzzles along with people at the table seemingly as long as he didn’t have to talk or come in contact with any of his tablemates. He would spend entire class periods trying to figure out the puzzles he liked, completely engaged and deep in thought. I remember thinking to myself that I had finally found a way to connect with Garrett. Even if it was only for five to ten minutes it was a start. At this point I began to wonder if there was be a way to use puzzles to learn math concepts? I thought: “How could I teach the derivative through a puzzle?”

In part, because of these experiences with Garrett and several other students, I began to make time on Fridays, typically after project or quiz time, where I would set up chess boards, physical puzzles, and other puzzle like board games such as Go, Quarto, and Connect Four. This created a relaxed atmosphere of fun and challenge. A number of students became involved in the game of chess. Allen, a well-dressed student with a passion for the arts became the kingpin in regards to chess. I would describe Allen as very intellectual, in fact I could very easily see him as a college professor. Given his ability to rapidly process new information I became quite intrigued by the disconnect that was happening in my class. There would be times when I would find Allen lecturing in front of the class explaining concepts, ideas, and working through problems with students. But then the homework would arrive and it would be partially completed and the quiz grades would be subpar. What was I as the teacher doing wrong to not engage this student and help him succeed in math? I then began to play him in chess. I immediately saw that he had a deep understanding of the game and was fluid with thinking a few moves ahead. He made mistakes but his overall understanding was beyond almost all of the students I had played before. These chess games would open up a conversation between the two of us, and while we were locked in thought, insights to how I could motivate students began to emerge and most of the insights had to do with play and pursuing their own interests.

Puzzles as part of the homework

“I have always loved puzzles and having them on the homework made it more enjoyable and fulfilling when completed. The sentences with the mixed up letters were definitely very hard for me and stood out because they were so frustrating”  -Peter

When I began this research in the fall I wanted to find or make problems that would engage all types of learners. I remember searching through books on logic, puzzles, brainteasers, open-ended math problems, and paradoxes. I wanted problems that would be simple but lead to complex understandings. Problems that were easy to understand, that had a clear goal and that required minimal mathematical techniques, and yet made the students think a lot. I wanted the students to think in different ways and not rely on a predetermined algorithm to arrive at the solution. I remember being inspired by Paul Lockhart’s A Mathematicians Lament (2002) in that he encouraged thinking of problems as puzzles:

“A good problem is something that you don’t know how to solve. That’s what makes it a good puzzle and a good opportunity. A good problem does not just sit there in isolation, but serves as a springboard to other interesting questions” (pg 9)

So I looked at problems that would be fun, involve thinking outside of the box, and that would get students working together sharing ideas and actively thinking. Since students seemed to respond positively to puzzles as warm-ups, I began to integrate them into the homework too.

For the record I want to state that not all the puzzles I used in my classroom gave the students a deeper understanding of the world they live in.  Then again most problems given in math class do not. Most problems are practice of concepts that eventually could be applied to the real world. Factoring, which is generally learned in high school is a great example. Problems like trying to factoring degree two polynomials gives little to no deeper understanding of the world. It is merely practice for what comes later.  This seems so limited. What about factoring degree three, four, or how about degree five?  Toward the end of the class, once students had ample practice with different types of puzzles and thinking I started to introduce concepts as puzzles.  I wondered if I could teach them the basics of probability through puzzles? Would doing so help to make ordinary math problem fun and challenging, rather than just something to do to appease the teacher?

The advantage to incorporating puzzles into the homework were that students had more time than the warm-ups to ponder them, it encouraged them to get started on the homework right away, it made it more collaborative, and overall more fun to complete.  Warm-ups had the purpose of getting the students excited about learning that day whereas the homework encouraged something a little deeper. It allowed them to time to think about challenging logic puzzles, play with Kakuros, and find checkmate on chess puzzles. I began to incorporate more open-ended style problems and pose them as puzzles. Students weren’t thinking about which math algorithm to use. They were just trying to figure it on their own. This allowed for the students to use their own understanding of the world and problems solving skills to try to tackle problems. Students were thinking harder about the problems. I could tell this by simply monitoring an individual’s process, watching them struggle, try different things, and by the amount of time they would spend on a problem. The shift to using puzzles was a conscious move against skill building and toward building creativity and curiosity. I viewed the change similar to this quote by Antoine-Marie-Roger de Saint-Exupery

“If you want to build a ship don't herd people together to collect wood and don't assign them tasks and work, but rather teach them to long for the endless immensity of the sea”

This is exactly where we need to change education, we need for them to “long for the endless immensity of the sea” in terms of what they are learning in school. I hoped that puzzles would trigger something deeper and help students develop a curiosity and appreciation for solving problems. The purpose of the puzzles in the context of homework was to encourage students to think for themselves. I wanted them to build their confidence and have fun while learning. I wanted them to “long” for more challenges and find which ones they were good at, and understand why that was.

Learning through Stuggle

“With a lot of the puzzles, at the beginning of the year I didn’t want to do them and wouldn’t really try, but over the year, I tried harder on the puzzles and worked hard to finish them. Also, over the year, I tried harder to understand the math instead of just copying straight from the notes and not really get what I was doing” -Casey

At the beginning I had a few students who complained that “they had never been good at puzzles and found many of them quite challenging.” A student named Shannon expressed the opinion that the puzzles were a waste of time when we could be spending the time learning math. When asked the question “Do you see a connection between puzzles and doing math problems?” she stated: “No. I think it (puzzles) is irrelevant and doesn’t help me with math at all. I think puzzles should be done on people’s own time. We only have an hour to learn math so we should spend that hour learning math, not doing puzzles.”

She had a point, in that we were in this class for an hour and five minutes everyday to learn math. This came from a student who does really well solving problems on the homework but had a difficult time trying to come up with solutions for some of the puzzles. The same student also struggled a bit during her senior project as it was hard for her to pick one topic of math to research and to build a multimedia piece that explained the idea of that particular topic.

I began to wonder if it was the algorithmic approach to solving problems that she liked and the openness and creativity involved in solving puzzles that she found difficult. This pushed me to try to develop my math curriculum to make the problems in class involve a more open-ended approach with puzzle-like characteristics.  Thinking about what Shannon said made me wonder why students were in math class in the first place. Was it really to solely acquire techniques for solving problems or is it something bigger? What about creativity, learning perseverance, play, and learning something new about the structure of nature? And what do we really mean when we say someone is “good” at math? It would be as if an English student studied nothing but sentence structure, conjugation, pronouns and adjectives, yet was never given the opportunity to write a beautiful sentence that conveyed meaning. Would we say this person is “good” at English?

Much of the resistance to puzzles seemed to come from the students who were accustomed to finding a solution by following the well-trampled path of algebra, and geometry. But by asking a question like “Is there a way using only multiplication, division, subtraction and addition to assemble four 9’s together to equal 100?” these students were forced to use trial an error, to try to guess at the solution (common in mathematics) and to think differently from the traditional problem. It made many of them uncomfortable, which I believe was not necessarily a bad thing. What I mean by that is that I wanted to shake things up. I wanted students to grow as learners. The traditional style of lecture-based learning wasn’t necessarily going to push them to think creatively. I wanted there to be more discovery involved and exploration of passions and interests. My goal with teaching mathematics has been to the expose students to bigger ideas and have them develop a subset of skills for doing their own analysis. Problems in everyday life more often resemble puzzles than algorithms.

Making the homework part puzzle

“I think math is difficult, but it’s all a puzzle, once you solve it, it feels so good.”
-Autum

To accomplish this goal, I began placing a series of puzzles at the end of each homework assignment. Each week I would pick 3-5 puzzles of varying types and difficulty levels. I found that the puzzles took a small bit of time away the tool-building portion of the homework but the trade off was well worth it. At the beginning, the puzzles were mostly cryptoquips, kakuro, chess, logic, and sudoku like multiplication problems.  As the semester progressed the puzzles changed into more active puzzles, where students were learning mathematical concepts through discovery.  Instead of being given the algorithms, they were discovering them. If something became too difficult to figure out we would have a discussion about it. When we found that we were getting the hang of a certain type of problem we would try to generalize it. Students were reinventing the rules rather than just following them. This was done through treating the problems as puzzles to figure out rather than following a prescription.

For example, in “The Monty Hall Problem” contestants on a game show are asked to pick from three doors, two of which have a donkey behind them and one with a shiny new car. The contestant is asked to pick a door that he/she thinks the car is behind. Once the contestant picks a door, the game host opens one of the other doors revealing one of the donkeys. The contestant is then given a choice knowing this new information, if he or she wants to switch doors or stick with the original door he/she picked. The question is: what should the contestant do? Stay with the original door, switch, does it make a difference?

I have always thought that good math problems like the above are like puzzles. If the answer to the math problem was easy to get and/or one understood exactly how to get to the solution immediately, then it didn’t make a good problem. A good math problem, similar to a good puzzle, should make you scratch your head, encourage deep concentration, and push the thought process into overdrive.

During the warm-ups I felt that the puzzles were scratching at the surface of what many of the students were afraid to do in regular math class. In the past, I have found that students are often afraid to give problems a shot, to take risks, and most of all, they were afraid of being wrong. Something about the puzzles removed this hesitation of being right and wrong. It was okay not to know how to do it, to struggle, and to not succeed right away. The more puzzles became a part of our class – and just part of “doing” math – the more I saw this to be the case.

Cultivating a Growth Mindset

“I really like the crack the code puzzle. Sometimes it was frustrating when I got stuck but when I did figure it out it usually felt rewarding” – Amanda

The students’ response to puzzles as part of the homework was unanimously positive. It seemed that they couldn’t wait to try the new puzzles and hone their skills. They were challenging in ways that the more traditional math problems were not. I can’t emphasize enough the importance of sitting down and just trying to figure something out, the act of thinking and problem solving is an essential tool that can be honed. According to Carol Dweck in her book Mindset, what really matters most is students’ belief that they have a growth mindset – the belief that the brain is a muscle that is malleable and able to become stronger (or more intelligent).  In an interview in 2006 she said:

“In a growth mindset, people believe that their talents and abilities can be developed through passion, education, and persistence. For them it’s not about looking smart or grooming their image. It is about a commitment to learning-taking informed risks and learning from the results.” (pg. 57)

During this time a boost in confidence happened for Garrett and I began to see a shift in his mindset. There were five “Tavern” puzzles in my classroom that remained unsolved by any of the students. One day I handed Garrett one during advisory period and I left him alone. About twenty minutes latter I heard a clang as the piece that ring that seemed impossible to remove had fallen onto the desk disconnected from the rest of the puzzle. Everyone in the room stopped what they were doing and turned to his direction almost in disbelief. The rumor mill spread the word that Garrett had solved one of the puzzles and soon students asked him how he did it. Of course Garrett didn’t respond to their requests but I could tell that it changed something inside of him.

Throughout the year Garrett remained a struggle to communicate with and challenging to engage. I would like to report that his social interactions improved and that his overall math ability increased, but aside from puzzles he remained standoffish. The time spent on puzzles may have been the only time during class where he was actively participating with what was going on in class. The homework was a bit more challenging. However, through puzzles, I had gained enough of his trust to sit next to him and go over the problems one-by-one until he got the hang of it. He would talk with me but usually through a yes or no response and at most a sentence about the homework. He would never ask for help on his own. I had to make it a point to sit next to him and ask him how his work was coming, otherwise he would be found staring down at the table avoiding work. But give him a puzzle and he was happy and concentrating and willing to spend the whole hour trying to figure it out. This is one of the powerful lessons I learned from doing something different: that there is a multitude of different ways to reach out to students, and what works for some might not work for others.

Another student that I was able to connect with through puzzles was Monica. Monica had struggled in math classes over the years and didn’t think she was good at it. For many students, Monica included, the rigor and difficulty of my class is quite a step up from their previous classes. At the beginning of the year I worried that she was already slipping through the cracks and not putting forth her full effort. I remember her not paying attention and I called on her to answer a question that we had just talked about. All the information including the answer was on the board. All she had to do was pay attention, but she appeared to be lost. I remember putting her on the spot for quite some time to get the point across that I wouldn’t let her get by with not engaging in the conversation or participating with problem solving.

Unfortunately, I felt that putting her on the spot did more harm than good, as she was quite frightened at the possibility of being called on again. It took some time to gain trust and create that safe environment again. The road back to her success came with the puzzles that I put on the homework. I remember one day she came to me ecstatic because she had solved a puzzle before anyone else in the class. She seemed proud that she had conquered the problem by herself. I congratulated her and told her “don’t tell anyone the solution” to which she smiled and replied “okay!” This was a big deal as the conversations before this involved me trying to encourage her with little response from her.

For many students, the puzzles opened up a line of communication between teacher and student. We now had something to talk about other then math. From this point on I would always be able to come over to her and silently whisper “did you crack the code yet?” and she would smile and nod yes. This translated into a new positive attitude in math class. She saw me as not someone who was out to get her, but rather a person who was trying to push her understanding and encourage her to think. Consequently, she began to stay for office hours, finish her homework on time, and study for quizzes. Her grade shot up, but more importantly she was “getting it!” I could even call on her at times during class without feeling that she would recoil and resort back to her previous fearful stance regarding math class. This is a reminder that attitude is everything. In this case it was puzzles that led to greater confidence and ultimately, learning in math.

Just as puzzles had served as a warm-up to get them thinking at the beginning of class, including puzzles in the homework served as an entry point for students. The vast majority of the students when given a weekly homework assignment would start doing the puzzles attached to the homework first. As the week progressed toward the homework deadline the majority of students then began to complete the tool-building section that was covered in class. I actually had students come up to me several times and request certain types of puzzles be put on the homework. Students had fun trying to crack the secret message of the cryptoquips and enjoyed the challenges of finding checkmate on the chess puzzles. Learning in math class had become fun. They also developed an attitude that while a problem may be difficult, with enough time they could probably figure the puzzle out, which was greatly different from my initial survey of how they thought about problems and puzzles.

Crafting our own puzzles

During our time exploring calculus I tried to think of ways that students could come up with the concepts on their own without needing instruction from me. I found this quite difficult as their understanding of math was insufficient to become the inventors of calculus. Then I came across a website called Peanutty which enabled students to learn programming through solving puzzles. In fact, after a couple levels it becomes impossible to solve the puzzle without first modifying the code. There was the added benefit that students could become the designers of their own puzzles, and then have other students try to solve the ones they created. Now learning was directly connected with play and puzzles.

Peanutty involves learning physics and writing code to solve basic puzzles. I thought this was a perfect way for my students to explore the math of motion and change through puzzles. As I played with the program, I breezed through the first couple puzzles, but I found the third problem was impossible without modifying the code. The fourth puzzle was to build a tower by stacking blocks on a balance beam that teeters on a ball. The program states that the tower must be stable and that it is only possible to complete this level by modifying and writing in your own piece of the code. Then as the tower puzzle is solved, I realized that there are no other puzzles as the site is open source and still under development.

At first I was a bit disappointed as I felt that I had found a great learning tool but it was in its infancy. Then I realized that when the students finished the first four levels there was an opportunity for them to create their own level, in essence by creating their own puzzles. I wondered what the students would be able to create on their own given the tutorial of trying the first four levels? This seemed to be a great learning opportunity for students to create their own levels and then challenge one another to solve the levels that they had created. Below is what the learning platform looks like:

 After the students completed all of the levels I asked them to take a week and design their own level. The results were mixed. A number of students took this seriously and tried to create one of their own based off of the already existing levels. Others just modified one of the existing puzzles and were not all that creative with it. I do think there was a little bit of a stigma similar to what one would encounter in a math class when it came to writing the code. With the first iteration I felt that they really could have put more time in and made their own puzzles creative. As my research comes to a conclusion, I am still finding ways to use games like Peanutty to challenge the students, and have the students challenge one another.
 
At the end of the semester I gave a final survey to all of my classes, for a total of 76 students. One of the survey questions was: What are your impressions of the puzzles we did this year? Which ones stood out and why? The sentiments from the students were overwhelmingly pro-puzzle, which was quite different from the quotes from the beginning of the semester when students really didn’t see the point of puzzles, or they hadn’t yet found the pleasure in figuring them out. 

Below is a pi chart of student responses toward puzzles, which I coded in terms of positive, negative and neutral comments. What was interesting is that many of the “negative” responses were in regards to students being challenged, which was also cited as one of the primary positives of puzzles. For instance, Diego wrote, “The Kenken's and the Kakuro puzzles that we did this year really made me think. They were a bit frustrating and overwhelming at times, but they really gave me a challenge.”

While several wrote about puzzles as “frustrating,” many wrote about puzzles as challenging ultimately in a good way. From frustrating to challenging, it shows that students were engaged and that their brains were stimulated. In terms of exercise their brains were no longer sitting on the couch but instead were doing jumping jacks. The struggle is what makes it rewarding. And I believe this is the whole point of math class: to have students think. A number of students said the puzzles took more time and deeper concentration than the math homework. I’m not sure that the math homework had the same feel for the students? Some of the students talked about the struggle they had with certain puzzles, and that overcoming these struggles left them with a feeling of accomplishment. One student reflected:

“At first, I thought the puzzles were pretty difficult. They required some concentration and problem-solving skills. The ones I found especially difficult were the crytogram messages, because of all the guess-work. It wasn't until we worked on the sudoku-like puzzles that I actually started to enjoy solving them and felt accomplished with each puzzle I completed.”

Another student wrote,  “I feel like they were a test of our mental endurance and would force us to be creative.” This highlights another theme, which was that students felt the puzzles pushed them to think more creatively.

The students’ comments reiterated that the puzzles were a great way to get “the mind warmed-up.” Not all students come into math class raring and ready to go, so the puzzle warm-ups were a good transition into thinking. Students also liked different kinds of puzzles. Not all puzzles agreed with everyone, so they appreciated having a range of choices.

At the end of the survey, I asked students if they felt that they had grown as math-minded thinkers. I asked them to rate themselves on a scale from zero to three, where zero represented no growth, one was little growth, two was significant growth, and three was incredible growth. Out of all 76 students, here are the results:

Not a single student picked that they didn’t grow at least a little over the past year. I was happy to see that 97% of my students felt that they had experienced significant or incredible growth as a math-minded thinker. Much of this improvement I attribute directly to play and puzzles. Through this process, I have found that students enjoy challenge and feel rewarded when they accomplish a difficult task. Students like to have a choice in what they do, whether it be a puzzle or a math project. The next steps for me in the evolution of puzzles, play, and mathematics, is to figure out how to teach not only creative problem solving skills, but to also include tool building into the play. How do I teach calculus through a puzzle? Now that would be a fun and challenging puzzle to solve.

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