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.

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!

Reflection on Eisner's Three Curricula

    The first stop for me was when Eisner argued the reasons for aspiring students’ preferences of universities. Since I did not fully experience the K-12 system in North America, this example was more clear to me to understand the explicit and implicit curriculum he discussed. It reminded me how I chose which Canadian universities to apply to from another side of the earth— I checked the world rankings. Similarly, there is a school ranking website named Fraser Institute for K-12 schools in BC. However, these rankings actually mislead parents or students who have no sense of implicit curriculum for the schools. Now, the new BC curriculum explicitly explains the hidden curriculum and sets standards with the three core competencies. The introduction of these core competencies not only reveals implicit information to parents and students but guides the direction of the development of schools and teachers as well. 

    The concept of the null curriculum was my second stop. I appreciated that Eisner brought this term into the discussion of the school curriculum. Other than simply neglect some knowledge and skills, it is important to know the reasons for ignoring them. Currently, many new subjects such as art designs are rapidly added to the new BC curriculum. Moreover, more new content is expanded under the same subject area. For instance, geometry, statistics and the history of math are attached to mathematics. These subject areas and content were selected to move from the null curriculum since educators saw the necessity of them and understood why other concepts could be neglected. Also, designers of the BC new curriculum changed the approaches of describing courses. The content and skill development section was replaced by two parts: content and curricular competency development. As a result, schools and teachers have a clear image of what to teach and what to put less weight and students understand the exceptions of learning.

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. 

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. 

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.