Math art reflections

 Artistic Representations of Numbers and Ideas

Assignment 1: Artistic Representations of Numbers and Ideas

 David A. Reimann is an artist and a professor who combines his expertise in mathematics and computer science with art to create visually engaging pieces that embody mathematical principles. His work centers around using geometric shapes, patterns, and numbers to build often complex visual representations that reveal more details as viewers change their perspective. Reimann’s work emphasizes themes like symmetry and repetition, and it shows how simple shapes can transform into intricate designs that reflect the interconnectedness and balance inherent in mathematics. This combinations invites viewers to engage with his art on many levels, experiencing different aspects of the work depending on their distance and perspective, making each viewing a unique exploration of the mathematical concepts depicted.

In our extension, we created a fractal that represents a person's full (average length) life. Each level zooms in to show the next measurement unit of time. It goes from years to month to days to hours to minutes to seconds. The idea behind this fractal is to demonstrate the interconnectedness of everything in our lives and how there are endless ways to make connections between everyday tasks and life experiences. The full sequence of zooming in can be seen in the slides.

 

 

 

 

 

Experience of the Project

Working on the 'Artistic Representations of Numbers and Ideas' project was both challenging and deeply rewarding. The creative process demanded a lot of problem-solving, especially when it came to conceptualizing how abstract mathematical ideas could be visualized artistically. It was intriguing to explore the blend of symmetry, patterns, and shapes, and how these could convey meanings beyond their numerical value. However, I found it frustrating at times when trying to balance the artistic side with the mathematical accuracy required for the designs. Still, once the pieces started coming together, it was incredibly satisfying to see how numbers could take on new meanings through visual representation, offering fresh perspectives on the interplay between math and art.

Takeaways as a Teacher

As a teacher, this project provided invaluable insight into how interdisciplinary projects can deepen students' understanding. I believe that the approach of combining math with art—using patterns, symmetry, and visual structures—would be a powerful tool in the classroom, especially for visual and kinesthetic learners. This idea could be expanded to other subjects like history, where visual timelines or spirals could represent the progression of events or historical patterns. One area that might be challenging to implement would be the mathematical precision required for more complex concepts, but simpler, intuitive projects that connect math to other disciplines, like music or science, could work well. I look forward to incorporating this in future classes, making learning a more connected and holistic experience for students.

Textbooks

 The article helped me reflect on how math textbooks can shape students’ experiences. The examples show how textbooks often talk at students, making math seem rigid and impersonal. For example, phrases like “The graph shows you...” make it seem like the math happens on its own, without human involvement. This can discourage students from feeling connected to the subject. I also found the point about different types of instructions interesting. Some phrases like “explain” or “describe” encourage students to think deeply and work together, while others like “calculate” or “write” feel more mechanical. As a teacher, this makes me think about how I can balance the textbook with more engaging, interactive lessons.

When I was a student, math textbooks often felt boring and difficult to relate to. They used formal language and didn’t feel connected to real life. However, when teachers explained things in a more personal way or used group activities, math felt more engaging and easier to understand. This shows how important it is to go beyond what’s in the book.

Why Use Textbooks?

1. Clear Structure: Textbooks help organize lessons and ensure all important topics are covered.
2. Extra Support: Students can use them to review and practice on their own.
3. Expert Design: Textbooks include problems created by experts, often building from easy to hard.

Why Not Use Textbooks?

1. Too Rigid: Textbooks don’t always meet the needs of different students or fit their lives.
2. Boring Examples: The language and problems can feel distant and uninteresting.
3. Limits Creativity: Overusing textbooks can stop teachers from trying new, creative ways to teach.

Math textbooks are changing because of technology. Interactive tools and online resources are becoming popular alternatives. These tools:

• Let teachers adapt lessons to their students’ needs.
• Include features like videos, instant feedback, and interactive exercises.
• Are often more engaging than traditional books.

In summary, while textbooks can be useful, they shouldn’t be the only tool we use. Combining textbooks with hands-on activities, technology, and real-world examples can make math more exciting and meaningful for students.

(?? not really a blog post, but a copy from our class blog...)

Wednesday, September 4, 2024

Skemp on two approaches to teaching and learning mathematics

Here is our first article, an old but influential one:


Richard Skemp on instrumental and relational ways of understanding mathematics

Please read this article and write a response on your blog by 9 AM Monday September 9 (note correction!).

Your response should be brief (1-2 paragraphs), but full of interesting ideas. You should NOT summarize the article in this response, but you should talk about:

• Three things that made you "stop" as you read this piece, and why
• Where you stand on the issue Skemp raises, and why



Oct 7

Go to the pdf of Thinking Mathematically. Read chapter 3 (Responses to Being Stuck, pp. 45-57) and write your concise response to it, including two things that were "stops" for you.

Then go to Chapter 11 (Thinking Mathematically in Curriculum Topics, pp. 181-230). Choose one of these puzzles and post your choice in the comments on this post -- first come, first dibs (so that everyone chooses a different puzzle). Work on your puzzle in the form of a two-column solution -- one column for your mathematical work, and the other for your process, emotional reactions, dead ends, productive approaches, etc. The process is as important as the solution here

  

Arbitrary and Necessary

  

Arbitrary and Necessary

 

 

In Dave Hewitt's piece "Arbitrary and Necessary," he distinguishes between two aspects of the mathematics curriculum: those that are "arbitrary" and those that are "necessary." The "arbitrary" refers to conventions and labels, such as naming shapes or defining the order of coordinates, which have been socially agreed upon but could have been different. These are facts that students need to memorize, as they cannot be derived through experience. On the other hand, "necessary" knowledge refers to concepts that can be deduced through reasoning, such as the relationships between angles in geometric figures, which students can discover independently with the right tasks or guidance.

As a mathematics teacher, this perspective encourages me to differentiate between content that students must simply memorize versus content that they can actively explore and discover for themselves. For "arbitrary" elements, I need to ensure that my students understand the importance of these conventions for communication and problem-solving within the mathematical community, but also help them see these conventions as choices that could have been different. For "necessary" content, my focus should be on guiding students to build awareness and understanding through exploration, rather than simply providing them with facts to memorize. This means creating learning experiences that help students connect with the underlying principles and relationships in mathematics, fostering their ability to reason and understand rather than rely solely on rote learning.

Assignment 2

  

Basic Lesson Information
Subject / Topic	Workplace Math 10 / Creating and Interpreting Bar Graphs
Created by	Carson Hoang, Krystal Kong, Nizar Slimani
Allotted Time	20 minutes

 

Stage 1 - Curricular Elements and Pre-Lessons
Prerequisites any content required for grasping this lesson, such as homework or past instruction	- Homework from last class: watch tutorial lecture video on bar graphs and do basic starter exercises     - How to make bar graphs: assigning labels, making a scale, title and axes, bars     - How to read graphs and interpret them
Big Idea(s) / Essential Question(s) what students will understand and extract from the text that they’re reading (at a conceptual level, see connections to and between ideas, goes beyond the classroom learning)	- Representing and analyzing data allows us to notice and wonder about relationships.     - How do we choose the most appropriate graph to represent a set of data?     - How do graphs help summarize and analyze data?
Student Outcomes what students will do (activities to deepen understanding /important skills or processes)	- Explore, analyze, and apply mathematical ideas using graphs  - Model with mathematics in situational contexts. - Be able to connect mathematical concepts with each other, other areas, and personal interests. - Critically think about what graphs represent.
Content / Scope basic subject-specific knowledge, definitions, etc.	- Create, interpret, and critique graphs     - Horizontal and vertical bar graphs, comparing effectiveness of the two orientations     - Creating and reading bar graphs     - Interpreting the implications of data on provided graphs in context

 

Stage 2 - Learning Plan
Potential Barriers to Success: (Might include: engagement, motivation, organization, language ability, exceptionalities, reading level etc.)	1. They may not have watched the tutorial videos or done the basic exercises
What will you do?  (differentiation/adaptations)	1. Have a short section at the start to catch students up if they did not fully grasp the tutorial, or have time before class to answer questions
Infusing Aboriginal Education / First People’s Principles of Learning	- Learning is holistic, reflexive, reflective, experiential, and relational (focused on connectedness, on reciprocal relationships, and a sense of place).
Resources / Materials / Technology	Graph paper (cut into half sheets and provide scaffolding for exit slip responses), Slides: click link, Whiteboard and markers

 

Stage 3 - Post-Lesson
Homework	- Finish exit slip if not completed   - 1 sentence: what piece of information you can get from a bar graph   - 1 sentence: which orientation and why   - make your own bar graph exit slip activity at the end - Watch tutorial video on circle graphs and do a few circle graph exercises
Reflections

 

 

 

 

 

 

 

 

 

 

Agenda
Start - End Time	Topic/Activity	Teacher is doing…	Students are doing…	What’s the Point? (why?)
0:00 - 1:00 (1 minute)   (Krystal)	Average male height [Hook] + Outline   (slides 2-3)	- Present the graph of Average male height - Outlining the class roadmap	- Share quick thoughts	- Get students interested in the topic - Set expectations for lesson
1:00 - 4:00 (3 minutes)   (Krystal)	Tutorial   (slides 4-6)	- Explain the parts of a bar graph - Explain multi bar graphs - Explain the uses of bar graphs	- Answer quick questions such as “why bar graphs?” and “parts of a bar graph?”	- Make sure students have an understanding of how bar graphs work - Prepare students for the upcoming discussions and activities
4:00 - 7:00 (3 minutes)   (Nizar)	Demonstration   (slide 7)	- Demonstrate how to turn data into a bar graph: “How are we all feeling? How should we categorize ourselves here?” - Write on whiteboard	- Give suggestions on data to use for the demo bar graph  - Gather around whiteboard during demonstration	- Initial activity intended to connect content to students
7:00 - 9:30 (2 minutes 30 seconds)   (Nizar)	Discussion - Interpreting graphs activity   (slides 8-10)	- Display 2-3 “interesting” graphs with social commentary and ask students to interpret graphs’ implications - Also include graph from demonstration	- Answer questions by turn and on graph paper - Basic questions about data - Their meanings/inferences in short sentences	- Instill mindfulness and critical thinking into the content we are teaching in an interactive way - Embedded formative assessment of analytic skills
9:30 - 11:30 (2 minutes)   (Carson)	Discussion - Horizontal v.s. Vertical bar graphs   (slides 11-12)	- Display 1-2 pairs of graphs where one is horizontal, one is vertical (but all else is same) - Ask students to raise hands and pick the better orientation and why	- Upon seeing graphs, point to each one and ask if it’s horizontal/vertical (shout out together) - Pick the better orientation and explain why (hand raising) - Fill answers in graph paper	- To formatively assess students’ abilities to choose the better orientation of graph as a data representation and explain why - Add interactivity to lesson
11:30 - 13:30 (2 minutes)   (Carson)	Discussion - Axis breaks   (slide 13)	- Explain what axis breaks are in graphs - Show examples, including the hook graphs on male height and recontextualize	- Listening - If time permits, discuss pros/cons	- Tie in the hook with context
13:30 - 18:30 (5 minutes)   (all of us)	Exit slip - Creating your own graph!   (slide 14)	- Display the prompt - Walk around and check student progress in activity - Give suggestions based on what has been learned (horizontal vs vertical, axis breaks or not, basics)	- Using the class as data subjects, make up data about a topic (ex. phone usage, homework done during weekend) and make a bar graph on graph paper - Submit graph paper at end	- Formative assessment again - Give students chance to apply the lesson contents with their own input in a fun way
18:30 - 20:00 (1 minute 30 seconds)	Questions/Buffer   (slide 15)	- Open up for questions after “Demonstration” - Open up for questions at the end	- Ask questions if needed	- Buffer time