The Market Scales Problem

 

Extensions:

What pattern can you find from the solutions?

What happens if we want to measure whole-number amounts from 1 to 41 grams? Do we need to have more weights? 

DRAFT Unit Plan + 3 Lesson Plans: Relations and Functions

Relations and Functions Unit Plan + 3 Lesson Plans

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.