Puzzling Jugs

I was introduced to a virtual manipulative recently called the “Jugs Puzzle Game.” It outlines a riddle that can be manipulated virtually in order to solve it. Basically, there are 2 jugs of a given size that has a certain goal amount of water to put into one jug. You have an unlimited amount of water, and you can fill up jugs, pour them out, and transfer water between the jugs. It seems simple enough, right? Except you can’t fill up a jug part of the way with the water faucet because at the faucet, you can only fill the jug to its full capacity. This makes it difficult to achieve the desired amount of water all in one jug because it requires a certain order of filling, draining, and transferring water between jugs to achieve the desired output.

I struggled for a long time to complete the first level the first time I tried completing the task. I felt helpless because I thought it was impossible, but I knew that it was a solvable problem, and that others had done it before. This boosted my confidence in completing the task, so I kept clicking around until I eventually got the hang of the process. With each jug, there are only two options with what I could do next: I could drain the water, or transfer it over, perhaps leaving a partial amount of water in the jug. As I tried each level again and again, I would gradually improve my skills in deciding which move would be a progressive move to help me get closer to my desired output rather than something that wouldn’t get me anywhere. After a while, I was addicted.

I was thinking about the mathematics behind the task and trying to figure out what exactly I was doing in order to achieve my desired amount of water in one jug. I thought about other combinations of jug capacities and thought the solution to each problem would have something to do with the size of the given jugs—I couldn’t have just any two jugs with arbitrary capacities to receive a certain desired amount of water in one of them. I realized the capacities of each of the two jugs must be relatively prime to each other, making each of the two jugs lack the same amounts of water that might factor into their capacities, as this would make it impossible to fill a jug with a desired amount of water.

When a “solvable” problem is presented, I first outline where I am, what my goal or destination is, and what means I have available to me in order to get there. In this case, I had 2 jugs of different sizes, an unlimited amount of water, a faucet, and a drain. I also knew that I had to fill the jugs all the way to the top instead of partially. I knew I needed a given amount of water in a jug (either one that could hold that given amount or more). The means I had to get there included 3 options for each of the jugs. I could fill them completely, drain them completely, or transfer water between the jugs to leave partial amounts of water in one of them or both. In my understanding of the simple terms of completing the task, I was able to push past any uncertainties and fill, drain, or transfer water over and over again. In doing this, I found new ways of filling the jugs to have the exact amount of gallons of water I needed. In teaching this concept, I would likely not use this virtual manipulative, as it was difficult and frustrating to figure out. After a long time of thinking about it, I finally thought about the jug capacities being relatively prime, and I felt that this fact was not particularly enlightening. Overall, it was a frustrating way to attempt to teach a concept that could have been taught in an easier manner without the struggle of this virtual manipulative.

When I do not know whether or not the given problem even has a solution, I can tell by the number of possibilities and probable amounts of water each jug could have. If I could add and subtract the given jugs’ full capacities, zero them out, or fill them completly, I could tell if a given amount of water needed in one jug was possible to achieve before trying it.

This virtual manipulative helps students to better understand critical addition and subtraction skills and improves mental math abilities. Numerical fluency increases as students will understand what “moves” they should make in the virtual manipulative to obtain their desired answer. As students’ skills improve and they move onto higher levels, they will be better able to keep track of more anticipated addition and subtraction answers ahead of time. These abilities make me think of accomplished chess players, who need to keep track of their pieces on the board, their opponents pieces on the board, what moves each piece can make, and anticipate their opponent’s moves in order to attack the king.

Overall, as a future teacher, I would not use this virtual manipulative to teach such an important concept such as prime factorization. I felt the concept was not fully realized in my experience using the virtual manipulative, and it could have been improved using perhaps another mathematical problem or riddle.

There are a few examples of lesson plans focused on teaching prime factorization that I find much more useful pedagogically. Click here or here to find them.

Teaching Youth with Technology

Summary:

Brian K. Ashton, in his article entitled “Teaching with Tech: Engaging Youth in a Digital World,” speaks to a general audience comprised mainly of members of the Church of Jesus Christ of Latter-day Saints who often find themselves teaching religious subjects to youth in Sunday School classes. While his audience does not exactly include those aspiring to be secondary mathematics teachers, many principles he discusses throughout his article are relevant and applicable to any classroom seeking to learn using technology. Ashton gives advice on how to best utilize technology in the classroom. First, we must learn about the technology. “As... teachers [become] more skilled in using technology, they also became more excited about using digital devices to study... and the issue of digital devices being a distraction in the classroom largely [goes] away.” By learning more about the technology as teachers, many positive effects are likely to ensue, including a renewed excitement to learn as well as less technological distractions during class. In addition, Ashton says, “I rarely find students using their cell phones inappropriately in classes where teachers [help] the students [become] involved in the lesson.” Typically students start to zone out or try to multi-task (i.e. pulling out their smart phones to scroll through social media) when they are not thoroughly engaged enough in the subject matter. Overall, as Ashton asserts, there must be a balance in the use of technology in the classroom. If technology starts to be the focus of the lesson instead of the important mathematical concepts to be learned, the students are missing out on precious learning opportunities.

After reading and studying Ashton’s article, a group of my classmates gathered together and discussed some of his most striking assertions. I was reminded that my students can teach me about the different ways to use technology if I do not already know. I should embrace the technological advances as they come and always seek ways of incorporating them in my classroom. In our discussion, we heavily emphasized the need to help students engage with the material in productive ways, such as looking up a definition of a word they might not know, message a friend to ask for notes or due dates, or explore alternative manipulatives to describe the same mathematical concept in a different way. One last point we discussed was the importance of incorporating technology in our own homes to ensure our future children are better prepared in their classrooms at school to use technology in productive ways. This will help facilitate and support their learning both at school and in the home.

Critique:

One student pointed out that establishing trust with students is one of the most powerful ways we can help students use technology in productive ways. She suggested we explain principles conceptually first, then teach them a faster and easier way to do it on a calculator. In this way, she said, the students can complete the mathematical task both ways. I disagree with this idea based on personal experience as a student. If a teacher spent thirty minutes explaining something conceptually and then follow up the explanation with a simple technique on the calculator, I would relax and stop trying to learn how to complete the problem conceptually and rely on the calculator the entire time. I would then give myself permission to forget the “why” behind the concept from my mind, knowing I can instead “arrive at the correct answer” in a faster and easier way, which hinders my learning in the long run. However, along the same lines, this student also suggested something I do agree with—that teachers should build trust with students by being real. For example, a student might complain, “Why do we have to learn this the long way, when I can just punch these numbers into a calculator?” If we respond with, “Well, you’re not always going to carry around a calculator in the real world!” then we lose all trust with the students because today, most of us carry around our smartphones with a built-in calculator on it. The students might then consider us out-of-date, or irrelevant, and stop trusting us with accurate information on which they can rely. On the other hand, if we explain that it is important to know how to compute by hand because it further develops mental math capabilities, estimation accuracy, and more, then the students are more likely to trust that you are teaching them to assist in their lifelong learning rather than punishing them with busy work.

Connections:

In considering even more ways I can apply what I have learned from this article and class discussion, something stuck out to me that I want to always remember when I am applying for jobs as a mathematics teacher is to ask whomever is interviewing me, “What kinds of technology is available to me in math classrooms?” Based on their response, I can gage not only how much technology I would have available to me to use in my class, but I would know how much the school cares about providing their teachers with appropriate resources to teach the students. I know as I make a sincere effort to incorporate technology in the best ways in my classroom, learning processes will develop instead of being hindered.

Online Professionalism

Summary:

Royce Kimmons, in his article about online professionalism for educators throughout the United States, describes the specific difficulties teachers (and other school faculty) face in their personal and professional lives online. He expounds his thesis through his list of teachers who have faced severe repercussions from their inappropriate use of technology, his list of ways educators can improve their virtual footprint, his five scenarios and analyses of educators making severe mistakes, and his commentary throughout the article. Most notably, Kimmons asserts that relationships between teachers, students, and parents must be kept professional at all times. Even the slightest question about how a teacher acts in what they say or do might make the student talk negatively about the teacher to others in an exaggerated way. Simple and seemingly harmless activities such as friending a student on facebook may have serious negative consequences, and teachers often lose their jobs and/or teaching licenses as a result.

In one of my math education classes at BYU, we had an insightful class discussion on the use of technology in both the personal and professional lives of educators. Most of the students in the class are preparing to be educators in the future, but all of us have different backgrounds and experiences with the use of social media. One student voiced a concern about students approaching her in inappropriate ways rather than the other way around. This concern sparked a fascinating conversation about different ways to handle it. As a class, we concluded the best way is to pretend to be oblivious, ignore it altogether if possible, or just brush it off in the moment. However, later it is best to contact the principal directly to see if he has any alternate accommodations for those particular students. We all agreed it would not be the best idea to react in the moment to the student(s), get angry or emotional in any way, or even contact their parents regarding the situation, but that contacting the principal would be in everyone’s best interest. In addition to discussing how to deal with specific cases when students might start making moves on the teacher, as a class we discussed how to behave online in general. This included making sure privacy settings were set according to how much we would want future employers, students, and/or their parents to see if they were to search our name. Many of my classmates decided to test out what kind of information they would find on themselves if they were to log off all of social media outlets and simply search their name on google. They shared the astounding results of finding links to silly videos they recorded or embarrassing pictures or statuses they wrote years ago. The discussion opened my eyes to the types of things I can be on the lookout for in my own personal virtual space.

Critique:

As a result of reading the article and participating in the discussion in my math education class, I was convinced that as a future teacher myself, I will be keenly aware of the posts, comments, and texts I send to everyone. Cyberspace is a vast, virtual space that carries a lot of my personal information. As a result, I will thoroughly analyze my privacy settings on all social media outlets to ensure the information available to the public is professional enough to where any future employers, students, and their parents would be able to find it and I would not be embarrassed or feel the need to explain myself. This will help me keep my job.

Connections:

When I am an educator, I will establish the kind of classroom environment that encourages respect for everyone in hopes that students will not approach me in a disrespectful manner. In high school, my math teacher and I established a very good relationship with the perfect balance between professional and personal conversation. As I approached her and asked her advice about my future education plans, she gave me great advice that I still hold dear to my heart. I have never been able to find her on facebook, and we only correspond if I reach out first. I hope to someday follow her example of the perfect balance she helped us both establish in our student-teacher relationship.