342 Unit Plan

Linear Relations Unit Plan

textbooks and how they position their readers

How do you respond to the examples given here -- as a teacher and as a former student?
As a teacher, I value the need to understand the applications used in textbooks and how they shape students' perspectives towards their peers, the subject matter, and the world around them. The concerns raised in the reading about textbooks, like depersonalization and overlooking individual contexts, push me to be more deliberate and strategic in using them. That being said, I aim to complement textbook content with personalized materials and applications that actually matter to cater to diverse student needs.

Looking back at my student days, I remember relying on textbooks for specific aids like bolded terms and examples, which were a great help for homework and clarifying concepts. Yet, a lot of content often went unnoticed. This reminds me of the need for better readability—visuals and clearer emphasis on key points can make a big difference.

In my opinion, I believe that balancing textbook use with other resources is key. Textbooks are valuable, but it is important to go through them and check for their compatibility with students in the classroom. Perhaps you have ELL students who prefer visuals over text, or maybe there are students with dyslexia so they may prefer concise text with bolded and highlighted key terms and definitions. Mixing teaching methods and encouraging deeper engagement with textbook content can truly enhance the learning journey for students.

What are your thoughts about the reasons for using or not using textbooks, and the changing role of math textbooks in schools?
This article goes deep into the realm of math textbooks. It really delves into how these books shape students' perceptions in relation to others and the world around them. Textbooks have their strong points and provide structured content, exercises, and explanations. But there's also a discussion about their downsides, especially in terms of depersonalization as the reading mentions.
I believe textbooks, when used effectively, are a valuable tool for both teachers and students. 

Personally, as someone who looks back on their student days, I often relied on textbooks to quickly find key terms, definitions, and example problems marked in bold. It's a great fallback when class examples slip from memory and helps kick start homework. Yet, it's common to see most of the textbook content overlooked by both teachers and students. This really highlights the importance of making textbooks more readable by including graphics, images, and clear emphasis on key terms. In my view, textbooks could benefit from being more concise and positioned as an optional or additional resource for classes

Flow

Have you experienced a state of flow through certain experiences? What prompts it? Is it sometimes connected with mathematical experiences?

I experience a state of flow when I am interested in what I am doing. For example, jamming with friends to a piece that may pose as challenging can be fun and can get me into a state of flow. In terms of math, I have experienced a state of flow while doing extremely difficult problem sets in my areas of interest, like number theory and graph theory. They pose a challenge but are extremely satisfying when I find a clean way to do the proof.

Is it possible to achieve a state of flow in secondary math classes? Can we, as teachers, help create the conditions for a flow state for our students as they learn math? If so, how -- and if not, why not?

As math teachers, we can achieve a state of flow by engaging our students and catching their interest. As kids, we enjoy puzzles because they pose a challenge and catch our interest. How much different is math? Math has had a bad reputation for so many students, so as a math teacher, engaging students involves weaving the subject into their world. We can start by relating math concepts to their interests and daily experiences, showing its relevance in gaming, art, sports, statistics etc. To spark their curiosity, I like to introduce puzzles, intriguing problems, and perhaps some competition to encourage them to collaborate and explore solutions together. Making math tangible with hands-on activities or technology keeps them intrigued, while celebrating their progress and encouraging questions fosters a growth mindset, allowing for the state of flow where students are both challenged, interested, confident, and focused. Challenging them with problems slightly beyond their current grasp ignites their desire to learn more, keeping the thrill of discovery alive in every lesson.

Dave Hewitt & mathematical awareness

As for the dvd clip we watched in class, it was really interesting how the teacher taught polynomials to grade 8 students without doing much talking. Instead, the teacher used actions and let the students think and answer each other's questions. When a student made a mistake, the teacher will let other students correct them instead of he himself doing it. I found it fascinating because students were able to think and self-learn the material while remaining engaged in class. Connecting it to the previous reading, I felt that this was a good way to learn necessary information because contrary to arbitrary content, necessary content requires student understanding.

Dave Hewitt Video:

I thought that the way he explained being unable to add two things together unless they have the same name was a really profound method to teaching grade 8 polynomials. I have never thought of it that way and would otherwise just use the textbook to teach the material. It is often these interactive activities that help students both understand the practicality of and visualize their learning.

While thinking about what he said about being unable to add things without a "common name," I thought that we could even use the idea of a common name to teach substitution methods. For example, if we have (x+y)^2+(x+y)+1 and we call z=x+y, z is now a common name tand we can substitute in z for x and y. In his example, he had 2 pencils and 4 brushes. He could not add them together because they are different objects. However, if he called the pencils and brushes "things" or even more specific, "stationary," he can now add them together. Dave mentioned that this promotes awareness that in order to add things together, you have to find the same name. I found this idea really cool and applicable in my classroom. 

Key awareness allows people to understand an idea without memorizing a method or a mathematical rule. This was an interesting idea. It is kind of like the idea of using common sense to teach mathematics. For example, it is obvious that 3 apples plus two apples is 5 apples, but 3 apples plus two oranges is not 5 apples. On the other hand, 3 apples plus 2 oranges is 5 pieces of fruit. For students who struggle with mathematics or have math anxiety, we can often try to reach them by allowing them to see its practicality in the real world and using common sense to teach the subject (at least at the lower grades when math is not as rigorous). 

These examples were interesting, in the sense that if you find the least common denominator between7 and 4 and compute the two corresponding fractions, there is no integer between the numerators. Hence, we need to change the denominator in order to find a nice fraction in between those two numbers. Such a question would make students think outside of the box and help them understand the relationship between numerators and denominators and how they work.

Overall, Hewitt's ideas are very useful for teachers to think and reflect on how the material can be best communicated towards students. Maybe teaching them that apples and oranges cant add up is a better way than teaching them that x and y cant add. This speaks to the face that awareness is really important for both teachers and students. 

Arbitrary and necessary

This article has shed some light on the crucial role that teachers play in differentiating necessary and arbitrary concepts. This reminded me of teaching and proving the quadratic formula to Pre-Calculus 11 students. Naming the constants a, b, and c in the general form of the quadratic equation is an arbitrary concept, where mathematicians decided to name them a, b, and c in that order. On the other hand, understanding how it is derived is a necessary concept.

The text was discussed the teacher's role in the classroom, being to differentiate arbitrary and necessary concepts. The author says that students need to be informed when a concept is arbitrary. On the other hand, teachers need to create tasks and activities that help students comprehend and develop a better understanding of necessary concepts and mathematical truths. The biggest takeaway from the article for me is that arbitrary concepts have to be memorized, while necessary concepts have to be understood by students. As a teacher, I think that it is important for students to be aware of this, as it will help them know how to study for tests and exams.

So how would this affect the way I teach my math classes and plan my lessons? I think that the type of activity provided to the class really depends on what type of concept they are learning. For example, arbitrary concepts can be studied and memorized through flash cards and definition activities. On the other hand, necessary concepts can be studied through homework problems, proofs, group problem solving, multi-step critical thinking puzzles, etc. This shows the importance of teachers strategizing activities around the types of concepts students are learning.