TPI Results

The Teaching Perspectives Inventory (TPI) is like a personality test, where teachers better understand their views and perceptions about teaching. There are five perspectives which correspond to different questions from the test:

1. Transmission
2. Apprenticeship
3. Developmental
4. Nurturing
5. Social Reform

Below is a screenshot summarizing my results:

 

The higher the score, the more I foster this view in my teaching. From highest to lowest score across the five perspectives, I have the following: 

1. Nurturing
2. Developmental and Social Reform
3. Transmission and Apprenticeship

With Nurturing well at the top of my results, I will further look into what the TPI says about this.

Effective teaching assumes that long-term, hard, persistent effort to achieve success comes from the heart, not the head.

I am happy that the TPI informs this as my top perspective because this is central to my teaching philosophy.

Market Scales Puzzle

In my scribbles, the puzzle came down to the following question: What are four numbers in which the addition or subtraction of one, two, three, or four of these numbers equals 1, 2, 3, ... 40?

Rough Work

I started by selecting masses of 1, 2, 3, and 4 grams. Under the question, this permitted the scale to balance all masses 1, 2, 3, ... 10 grams. With masses of 2, 3, 4, and 5 grams, I could balance all masses from 1 to 14 grams. However, with masses of 3, 4, 5, 6, I could not reach masses of 12, 14, 16, or 17 grams. 

I then tried 1, 3, 5, 7 and this worked for all masses 1 to 16 grams. However, when I tried 3, 5, 7, and 11 I could not get 21 grams. I noticed there were redundant combinations to produce certain masses, such as 2 for instance. I went back in my development to state that I needed 1 or a difference of 1. This rule holds true for all numbers up to 40. I then approached the problem with using prime numbers. 1, 3, 7, 13 work to balance masses of 1 to 24 grams. However, when I kept increasing the fourth mass, I reached the set of 1, 3, 11, and 19 where I couldn't balance 24 grams. 

After a few days, I returned to my sheet of scribbles to realize that I could follow an more general approach.

A Solution

Consider a, b, c, d whole numbers.

For a = 1, b = 2, we have the ability to balance 1, 2, and 3 grams. Therefore, for any c, we can balance masses that equal c as well as plus or minus 1, 2, or 3 grams. 

For a = 1, b = 3, we can balance 1, 2, 3, and 4 grams. For a continuous balance (with maximum combinations for a, b, c) we must have c such that 4 grams less than c is 4 + 1. Therefore, consider c = 9. With 1, 3, and 9 we can balance masses of 1 to 13 grams. Under the same idea, in choosing d, we let d = 27. Therefore, we may balance masses of 1 to 40 grams. Thus, a solution for the four masses is 1, 3, 9, and 27 grams.

Are there other solutions? I don't think so with only four masses. However, I will have to think about it and I look forward to seeing what my classmates have done.

Herbel-Eisenmann & Wagner (2007)

As anecdotal evidence to align with some of the narrative in the reading, I reflect on my math learning experiences with and without textbooks. I have had good to bad experiences with textbooks. Throughout high school and university, almost all of my math courses had a textbook, however a large portion of those courses, the textbook was an additional resource with suggested problems rather than a key piece to the learning puzzle. I think that in the high school setting, the textbook is often seen as an enemy because there may be inconsistency between what the teacher is doing and what the textbook has written. The authors of this article inspect a the relations between the student and their textbook with five different entities. 1) Their peers. 2) People. 3) Their teacher: 4) Mathematics 5) Their own experiences. Regardless of how a textbook is written, the nature of what is included and the manner in which it is included is complex, driven by politics (as we have seen in earlier readings), from local and global discourse. Additionally, "Language indirectly indexes particular dispositions, understandings, values, and beliefs." The authors conclude, "there is room to draw awareness to the dance of agency between particular persons". I think it is most important to connect this community of relations between the student, textbook, and their environments through focusing on the who. Who found these equations? Who discovered this phenomenon? Where do they come from? This brings the math to life for the students and the textbook can be written to emphasize the diverse humans who made the discoveries we teach.

Group Microteach + Reflection

Links to our teaching materials:

• Lesson plan
• Slides
  

Reflection: The three of us worked well together as a team to deliver this microteach. However, as our peers unanimously agreed upon, we had far too much content for 15 minutes. Each of us rushed through our allocated five minutes, which resulted in us brushing over important teachable moments. For me, I missed the opportunity to answer "my students" in the concerns they had in doing the example problems before approaching the more challgening ones. For this microteach we learned our own lesson, that less is more. We could have preserved the learning outcomes but focused on one example for each of the three segments of the session.

Soup Can Puzzle

This puzzle reminds me of Fermi problems, which my biophysics professor in my undergraduate studies taught me. More than a mere order-of-magnitude approach to solving problems, he taught us a beautiful new way to think about science.

To begin this puzzle, I purchased a can of Campbell's tomato soup. 

I made the following measurements with a ruler:

• Can radius: 2.3 cm
• Can height: 10cm
  

Next, I approximated the bike's height as equivalent to the radius of the water tank. Without the bike as a reference, we cannot approximate the radius of the water tank because there would be nothing to scale it to. I consider the bike's height to be 1 m. This tells us:

• Water tank radius: 1 m
  

Since the water tank has the same proportions to the soup can, we can find the height of the tank with one unknown in two equivalent ratios where three out of the four values are known. 

• Water tank height: 100/23 m
  

Next, the volume of the water tank is given by the volume of a cylinder:

• Volume of tank: ~ 14,000 L
  

Lastly, if we consider the average bungalow home to be 1500 square feet and that 1 gallon is required for every 3 square feet of fire, we would require 500 gallons, which is 1890 L.  

Note: Unit conversions are required from the imperial to the metric system because homes and fire data are commonly represented in square footage. 

 

In summary, when I consider a Campbell's soup can with radius 2.3 cm and height 10 cm and the bike as the radius (with height of 1 m) of the water tank, I find about 14,000 L in the tank, which is enough to put out about 7 house fires of 1500 square foot bungalows at a rate of 1 gallon per 3 square feet.