Take-away From Math & Art Project

    I appreciate impressive presentations from all other groups and I truly enjoyed making this project with my great teammates Michelle and Do Hoon. 

    The biggest take-away for me from the math & art project is that there is no perfect answer for Math. Sometimes, students and even teachers focus too much on digging the perfect solution and forget that math problems can be open-ended, fun and artistic. In my group’s project, I was fascinated that dots and lines could form patterns that secretly deliver messages. I tried to use dots, lines and curves to form more combinations in my extension. At first, I was not satisfied with it since I felt there were many limitations. However, I was surprised by the beautiful patterns my classmates made with my extension in the interactive activities. Then I realized that there was no perfect answer for the project and the most valuable parts were the process of thinking, inspiring others and getting inspired by others. 

Math & Art Presentation

Do Hoon, Michelle and Yiwei: Artwork by Owen "OAT" Rohm

Google Slides

The Dishes Problem

Solving without algebra:

The first step is to find a number that has factors 2, 3 and 4. The smallest number that meets the condition, i.e the least common multiple, is 12. Thus, the solution must be a multiple of 12. The possible solutions can be 12, 24, 36, 48, 60, 72 etc. The next step is to figure out which one meets the other condition: the total number of dishes is 65. By trials and errors, we can find 60 is the final answer as 60/2+60/3+60/4=65.

Math problems, puzzles and histories from diverse cultures foster students’ interests in math. Since these questions could have different approaches, students can practice their mathematical thinkings and have deeper understandings of the development of math. When students from other ethnic groups find that the math questions are from their cultures, they could be more excited and actively engaged in the class. During the interactions, other students could also benefit from the problem-solving process. 

Word problems, puzzle stories and imageries play important roles in math learning and teaching. Students could develop essential skills such as understanding the backgrounds of problems, collecting useful information, visualizing problems and utilizing different approaches to solve problems. Moreover, they could feel connections to the ancient time and appreciate the efforts and wisdom of ancient people. By successfully solving these kinds of problems, they could gain more fulfillment and joy than figuring out the solutions to boring equations. 

Letters from Future Students

Letter from a fan:

Hello Ms. Liu,

    I am writing to bring you great news about me. I just got an offer from the math department of MIT! Thank you so much for fostering my interest in math. I still remember the math stories you told and the interesting puzzles you introduced. My classmates and I had a lot of fun in your class!

    I hope to see you soon and talk to you in person.

Cheers,

Ms. Liu’s big fan

Letter from a hater

Hi Ms. Liu,

    I still cannot understand why we have to learn math. I struggled a lot in your class. Math is BORING — this was the only thing that I learned. Now I am doing history in university and I do not need to deal with math at all (except aftermaths).

Best,

Math hater

Reflection:

I hope my students can see the beauty of math as I do. I wish they could have fun and feel my class relaxing. This awareness guide my teaching strategies. Moreover, it reminds me that students may have different feelings when doing the same activity. I should prepare for more diverse approaches. 

Response to Lockhart's Lament

    I truly agree with the point that Lockhart made about the professionalism and passion for math teachers. He stated that math teachers should “at least understand what mathematics is, be good at it, and enjoy doing it” (p.11). I believe my least favourite teacher did not meet the standard here - giving away arithmetic worksheets was all her job. Another related experience for me was in a university geometry course. The professor went to a conference and invited his colleague to be a substitute instructor for two weeks. Nevertheless, since the colleague actuarially knew nothing about the topic, all he could do is presenting and reading content from slides. There was no doubt that he was a mathematician and must be an expert in his study area, but he could not be a good teacher for me for this course.

    I do think math is beautiful and it is an art. However, I disagree with Lockhart’s idea of equalizing math with other art subjects. He thought that if students could work on difficult topics such as Shakespeare in English, then they should also be capable of creating math by themselves. He argued that “the problem is not that the students can’t handle it, it’s that none of the teachers can” (p 17). In this case, he ignored another big problem: the amount of basic knowledge required for people to freely play with the subject. The main reason that students can interpret Shakespeare in secondary school is that they had already developed a solid language foundation: they practice reading and writing skills every day in their daily life. In contrast, math is not only an art but also a science. Similar to other science subjects, it heavily relies on assumptions or definitions and most importantly, it is developing. To memorize, understand and interpret these “prerequisites”, people need to put in all their effort for a long time. As a result, it is reasonable that students in the secondary level are not “capable of operating on such a high level” (p 17) for math. In other words, even if all math teachers are able to create their own math, it is also hard for secondary school students to achieve a high-level standard like this. 

Locker Problem

    First, I had a sample of 10 lockers and 10 students and tried to observe some patterns. I drew the line that separated the final states and changing states. I also underlined numbers where the change of states happened.

    Since there seemed to be no obvious patterns, I turned to collect information on the total number of open and closed lockers for each locker and each student. Unfortunately, I still could not see any special.

    Then, I focused on the last row and surprisingly found that the only lockers that were closed are perfect square number lockers. However, this finding should be rationally supported before I made a conclusion.

    Thus, I looked back to the sample of 10. I counted the numbers of changes in states for each locker and realized that they are the numbers of factors for each locker number.

    Finally, I double-checked my solution with a sample of 20 that I made in a spreadsheet. It matched what I had got!

    Reflection: Although I underlined numbers that change occurred, I did not get it in the first place since I forgot to include the initial states of lockers. 

My Favourite and Least Favourite Math Teacher

    My favourite math teacher is my father. He told me math stories to simulate my interest and fostered my mathematical thinking by valuing the process instead of the final answer. As I mentioned in my EDCP 442 blog, my father read interesting mathematical episodes in history as my bedtime stories. These stories stimulated my interest in math. The most impressive one is the mistranslation of rational numbers in Chinese. The original idea is a rational number can be represented as a ratio. However, since rational also can be interpreted in another way, the Chinese translator eventually named rational number as a reasonable number. This story reveals that although math is regarded as a universal language, there are still possible misunderstandings due to different backgrounds. Moreover, my father educated me to have strong mathematical thinking. When I was solving homework problems, he encouraged me to find solutions in multiple ways. Sometimes I got low marks in math quizzes since I made some trivial numerical mistakes and the teacher took off all the marks for these questions. He always accepted my mistakes and told me that the process was more important than the final answer. He is my role model of being a math teacher. 

    My least favourite math teacher is who taught me in elementary school. She forced the class to do lots of mental arithmetic practices every day. I still remember the pain and stress I felt. However, I also benefited from these exercises. I can do calculations relatively fast and produce my way of visualizing numbers in my brain. I believe developing mental math practice is important, yet in this case, quality is way more important than quantity. 

    In conclusion, I learned both from my favourite and least favourite math teacher. I hope I can motivate and inspire my students in math learning like my father. Additionally, my experience with the least favourite teacher makes me pay attention to the balance of quality and quantity of assessment in math teaching. 

Comment on Discussion of Skemp - Digital Literacy

    Thank you Mariana, Alan, Yiru, and Rebecca for bringing out the topic of digital literacy. I believe that digital technologies should be applied more frequently in math teaching. First of all, using digital technologies in teaching is a good approach to attract students’ attention. Every time I used Kahoot for quizzes, my students showed more interest and engaged actively. (Normally, they started to sigh after hearing the first syllable of the word quiz.) Also, students could get deeper impressions and quicker understandings through visualizations. When teaching topics such as the slope of functions, I found they mastered the topic easily when I used online graphing resources. Additionally, applying online tools improves students’ digital literacy and benefits their future learning. My students were interested in the digital tools I used in the teaching process, so they asked me to show more functions of it. Eventually, they were taught how to use one tool named Desmos. Later, they freely worked with it and explored more on the learning topics with it by themselves.

Response to Skemp's Article - The Balance of Relational and Instrumental Teachings

    This article was published in 1976! I am so amazed that Skemp foresaw the most critical point in the development of mathematics teaching in the current time. Nowadays, relational understanding turns to be more essential due to the rapid development of technologies and an exponential increase in knowledge databases. Students are educated for jobs that have not been created. To add the current situation to the analogy of map of the town, now our students are required to be capable of exploring out of the town all by themselves. However, instrumental teaching is not good-for-nothing. I appreciate that Skemp addressed the advantages of instrumental mathematics (and I like the section was named as devil’s advocate). 

    In my perspective, teachers should apply both relational and instrumental teachings though I did not think this way not long ago. The change happened during my experience of being a private tutor. Students came to me for improving their marks in a limited time. Additionally, their parents were worried about they did not get enough practice and thus asked for more homework. I was expected to teach instrumental mathematics. Painfully, as a believer of pure relational understanding at that time, I had to meet their requirements -- they paid me for these. My blue days ended when a student approached me and claimed of creating a new formula -- the sum of roots to a quadratic function is -b/a. He generated it based on the worksheet for the axis of symmetry of quadratic equations. Although I had to tell him it had already been introduced by Vieta hundreds of years ago, I started to reconsider the role of instrumental teaching. A good analogy of the relationship between instrumental and relational understandings is the regression model: the more the data is collected, the better the fit is found. I believe that to reach a relational understanding, students rely on instrumental practice. Thus, teachers should balance the use of relational and instrumental teachings to help students to become autonomous learners ultimately.