Drawing 3D Shapes Activity

    From a student’s perspective, I love this activity! It was amazing to see how we could draw from a simple shape, cube, to a complex objective, truncated Icosahedron. By counting the numbers of vertices, edges and faces of the shapes we had in the process, my 3D sense was developed. Also, the numbers were consistent with Euler’s polyhedron formula, and thus I was more convinced of this formula.

    As a teacher, I believe this activity can engage students and make them feel the fun of math. However, I think it could be hard to follow for some students. For instance, students who do not like drawing may suffer in the process. I am thinking that it would be great to have a visualizing tool, similar to Geogebra, that could help students to draw these complex shapes and view them from different angles. 

The final product - Math Disco Ball!

Proof?

    I believe Geogebra is a good visualizing tool that students can explore to convince themselves of geometric theorems. However, I still do not think that it provides valid proofs. Similarly, students can use sine law and cosine law etc to explore trigonometric identities, yet it is the process of justifying, not presenting proofs. 

    In secondary school class today, the main goal of teaching traditional proofs is to promote deductive thinking. The introduction of proofs in math is in a scaffolding form. Students start from conjectures and inductive reasoning first and then explore counterexamples to disprove some conjectures. When discussing the topic of valid proof, here is one question I used to show to my students:

        Choose a number. Add 3. Multiply by 2. Add 4. Divide by 2. Subtract the number you started with. What is the result? Prove why it works.

    Students were always amazed by the result when choosing different numbers. After several tries, they turned to generalize and tried to do the proof by using algebra. Although this practice of proving is kind of easy, it is actually in a high degree of rigour. I believe in this way, students develop a stronger sense of proof and thus, develop more on deductive thinking. 

Mathematics Textbooks

    The book, in any form and on any subject, is a bridge between the writer and the reader. In the article, the authors mention that the language (e.g. pronouns) and examples used in mathematics textbooks reveal the distance and connection between authors and the student. As a former student, I think that in general, mathematics textbooks give me a sense of authority. When I read these textbooks, I always felt that the authors of the textbooks were lecturing me and that what they were saying was unquestionable. In addition, the language was sometimes difficult to understand and vague. The inconsistent use of command terms and terminology gave me a headache all the time. In my opinion, as a teacher, I see the teacher as acting as an agent between the student and the textbook author. Textbooks are not flawless and thus, teachers should make decisions on selecting good sections/examples to use in class or even combine different resources.

    I think textbooks are good resources for teachers but not students. I personally like to use guided notes to teach. I can gain inspiration from different textbooks and combine the best parts into my guided notes. Also, teachers are the ones who know their students best and the examples included in the guided notes could make students feel a closer connection to themselves, mathematics and the world.

Reflection on Dave Hewitt’s Teaching

1. I was a little bit scared by the sound Dave Hewitt made with the ruler at first and my second thought was that the sound effectively attracted students’ attention. Later on, I realized that the sound was not the key to engaging students — the key was the well-designed activity. In my short practicum, I learned from a language teacher to use a ring bell to get students’ attention. However, I found that it did not work as I expected. Now, I understand the reason behind it.

2. I noticed that he wrote everything he needed on whiteboards/windows in advance so he would not make students wait and have nothing to do. I think well-preparation is essential in math teaching. Due to the requirement from the curriculum, sometimes, the contents of two sections in the workbook have to be covered in one 80-minute class. 

3. I like the idea of having mental calculations. It is a good approach to practice mathematical thinkings. He also addressed the problem of solving an algebra equation as a game of guessing the number I was thinking of. Thus, students could be more willing to share what they got as it was a game and there was no such thing called mistakes in games.

I will try to hold a similar activity when introducing the number line to grade 8’s. I would like to watch the whole video and learn more from Dave Hewitt’s teaching strategies. 

Arbitrary and Necessary

    This article reminded me of the first assigned reading on relational and instrumental teaching written by Skemp.  I believe that arbitrary mathematical content requires memorization on the part of the student and is primarily instrumental teachings on the part of the teacher. Necessary mathematical content, on the other hand, is about student awareness and is related to relational teachings.

    Since I did not study in English-based education before 18, remembering and understanding math conventions have been a challenge for me at times. Therefore, when planning my lessons, I always try to see the content from the students' perspective first, especially when introducing arbitrary knowledge. As mentioned in the article, covering the history of the content is a good strategy; however, some terms are chosen at random when they were first introduced into an English context. In this case, I think it is helpful to introduce history from other cultural backgrounds. Stories like the mistranslation of rational numbers that I shared in the presentation could help students memorize the mathematical inventions or provide at least something fun to listen to in a math classroom or to share with others later on. 

    As for teaching necessary mathematical content, holding activities to assist students in freely exploring or playing around with math is critical. When planning unit plans, I always try to include at least two activities for a topic. During my short practicum, my SA also inspired me a lot. I am fascinated by Grade 8’s explorations of shapes in 3D in the Halloween cubes activity. Here is a picture of their awesome works!

Promoting "Flow"

    I got into the state of “flow” when solving differential equations for 400 level math. I did not realize that the answer was almost 5 pages long before I finished.

    To promote “flow” and happiness in math classes, I believe the top priority is to provide students with enough “weapons” such as mathematical thinking skills, calculation tricks and history of mathematics etc. My SA always teaches students how to solve the question in more than one way. When dealing with slopes, she not only showed the basic idea of “rise over run” to grade 9s but also “short-cut” for linear equations in standard form. She told students that this new method could be learned in Math 10 and complimented them that they could handle difficult problems for higher-level math. In this way, students build their confidence in math and reach the “flow” as they are equipped with skills and are able to solve challenging questions. 

    I think introducing puzzle-solving is a good activity for students to reach the “flow” of math. I was lucky to observe another math teacher’s class in the school. The topic was introducing a puzzle named nonograms to grade 8s. It was amazing that how well students follow the teacher and they even asked for more practice. Also, every time they finished a graph together, students would use their imagination and express what the graph looked like. I am more than willing to try this kind of activity in my own teaching later in the long practicum. 

Pro-D Day Reflection

    I learned a lot from the BCAMT conference. It was my first time attending a Pro-D event and meeting so many math educators. This year the theme is connecting. Thanks to the awesome guest speakers, I explored connections between different points of view, teachers and students, and math and cultural context etc. Also, I appreciate the insights and resources the participants shared in the chatbox. 

    My biggest takeaway is the recall of six areas of human activity embedded in mathematical thinking: counting, measuring, locating, designing, playing and explaining. Although the saying that math is a universal language is not true all the time, people could find universal commons of math thinking in these six areas. By being aware of the similarities and differences of math learning and teaching in different cultural contexts, teachers could be more helpful to students with different backgrounds. 

Reflection on Group Curriculum Micro-teaching

 

    It was great to work with Dion and Mike! 

    In general, we tried to cover too much in a 15-min micro-teaching. It would be better if we provide more basic knowledge on the topic and have time for students to try different methods instead of one. Also, giving clear pictures and rationale for learning the topic should be included at the beginning of the class, so students could know where they are and where to go. The good aspect is that the small group activity was very effective. In this way, students not only learned from teachers but their peers as well. If time allowed, we could hold a larger group discussion and perhaps a debate on which method is the best to use. Therefore, students can have a chance to explore the pros and cons of different approaches to solving quadratic equations. 

    Personally, I believe I should practice more on my teacher volume and use more body language in the teaching process. 

Reflection on the Non-curricular Micro-teaching

I appreciate the feedback from my peers. Their comments encourage me a lot and provide valuable suggestions.

According to the feedback, I need to work more on clarity and timing in teaching. I chose my topic to be Mahjong since I am familiar with it. However, it could be a difficult game for people who are new to it, so it would be hard to follow for them. Also, it seemed that I tried to cover too much in a 10 mins lesson. Although I went through the lesson by myself several times before, the reality was very different from the plan. People need time to digest the new knowledge and reflect on them. 

For my future teaching practice, I will focus more on timing and pacing and plan more strategies for responding to different situations in advance. 

Reflections on Math Education Conflicts in North America

    First of all, I was surprised that the close relationship between global issues and change in math education in North America. Although the progressive reform seems to start with a spontaneous debate, it was driven by global factors such as WWI and international immigrants as well. Moreover, other countries’ achievements played an important part in the rest two math education movements. The new math is mainly caused by the competitions from the Soviet Union and the big concern and anxiety in the math wars came from the better math outcomes in other countries regarding the TIMSS ranking. 

    However, I think this kind of ranking is not accurate and all it does is creating stereotypes. This leads to my second stop which is my stand in the ongoing math wars — I believe that the assessment approach is the key. Different methods of assessment can describe the same student with very different performance results. In the current math teaching process, most teachers still mainly focus on submissive assessments, yet their teaching beliefs are actually progressive. It is impossible for a teacher to evaluate students’ development in mathematical thinkings based on the selection of multiple choices. Thus, I think applying more formative assessments could be more important than changing math curricula and teaching methods. 

    In addition, I noticed that parents’ role in math learning was mentioned several times in the article. I feel strongly related to this idea. As I talked about in my previous blog, my father is my favourite math teacher. However, when I got into higher grades, sometimes he could not assist me since the content was different from what he learned. At that time, I gained a lot of help from my peers. According to this personal experience, I think to encourage peer interactions in learning could be a good strategy in math teaching. 

My TPI result

 

    I was surprised that my score for social reform was low. I do value collaboration as the learning process, yet I have to admit that I show less care for social actions. Also, the apprenticeship factor was higher than my expectation. Many students dislike math since they do not see where math could be applied in their daily lives and future studies/careers. I believe that math teachers should address math in a more practical approach so that these students can rebuild their interests in math.

    One question raised regarding my TPI results is that why there are gaps between teaching beliefs and actual intentions. For instance, I have a low score in the teaching belief for the developmental factor, but the result also shows I try hard to accomplish it. 

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. 

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. 

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.