Dave Hewitt on Arbitrary vs. Necessary

 

Hewitt refers to the ‘arbitrary’ if it is something that students need to be informed by someone else and is concerned with names and conventions that students have to memorize (1999). It is the role of the teacher to assist students with retrieving this information through memory. The ‘necessary’ is something students can become aware of without being informed by someone else (1999). The teacher should provide tasks to help students educate their awareness to the arbitrary.

Thinking about this distinction between arbitrary and necessary will help me plan my lessons and develop as a teacher with regards to how I teach. Hewitt refers to ‘received wisdom’ as the fact given to students by the teacher and perceived by the student to be true as a result (1999). For example, when learning about the order of operations, students are told about BEDMAS OR PEDMAS by the teacher, which is an acronym reminding students the order they should perform arithmetic operations. If a student does not memorize this acronym, then they will incorrectly solve order of operation algebra problems. Even myself, I do not know why this order is the standard, but it is the agreed upon convention used everywhere. One of the points Hewitt makes that stands out to me is to give students time to use their own intelligence to become aware of why certain procedures must give certain answers instead of constantly giving them teacher’s wisdom. For example, when determining the area of a rectangle, students first must be told the definition of the area, and then be given the opportunity to “discover” how they can determine the area before being told the equation for finding the area. This allows for self-discovery before immediately being told the equation and eliminates the big question of “why is this the equation the area of a rectangle?”

 

Reading:

Hewitt, D. (1999). Arbitrary and Necessary Part 1: A way of viewing the mathematics curriculum. For the Learning of Mathematics, 19:3, 2-9.

Teaching Perspective Self-Test Results

This is the result I got from my TPI graph. It appears my nurturing perspective is a dominant feature, while apprenticeship is just above the mean and social reform just below the mean. Transmission and developmental are both almost recessive traits according to this test.

I am not surprised that nurturing is a beyond exceeding trait for myself as I often find myself being a very caring person and focused on social and emotional wellbeing of others as well. I was surprised at how apprenticeship was a little above the mean. Apprenticeship refers to teachers being able to know what learners are capable of and where they may need guidance. As teachers mature and become more competent, the teacher role changes and may offer less direction and give more responsibility to students (Pratt & Collins, 2002, p.3). When educating students, I like to let them ‘struggle’ a little bit when solving problems rather than giving them the answer immediately. The development was also not a surprise. Pratt & Collins (2002) refer to developmental as effective teaching planned and conducted “from the learner’s point of view” (p. 4). During my practicum, I noticed that I was sometimes explaining concepts a little more advanced than my students could understand; likely due to my additional knowledge coming from post-secondary math experience. It is important that as an educator, I teach concepts that could be understood by my students at a level to their understanding.

Given this information provided from the TPI test, I am curious to see how my results change after a few years of teaching and whether or not I will still score the same results. I hope to still score high on nurturing aspect, as I believe that one of the first avenues towards making math fun, is for the students to see that through a caring and passionate individual.

 

Source:

Pratt, D., Collins, J. (2002). Summary of Five Perspectives on ‘Good Teaching.’ TPI – Teaching Perspectives Inventory.

Thinking about math textbooks

As a student, I personally enjoy having textbooks as alternate sources of information to supplement teacher instruction. I also find that I greatly benefit from questions in textbooks to practice my understanding and refer to sample worked out questions if I need clarification. Comparing my answer with the textbook answer helped solidify my understanding of the material. From the teacher-bird view during my two-week practicum experience, all of the teacher I met referred to the textbook for practice questions, but did not know what questions were off the top of the head; when students asked, “I don’t understand question 9” or “Can you help me with question 18” teachers needed students to read out the question or refer to their teacher solution handbook before attempting to help. This changed my perception of how teachers use textbooks. When first developing lessons, they refer to textbook as sources of information and find relevant questions to assign students; then over the years, they may not refer back to the textbook unless they need clarification or a quick review.

I believe that textbooks are extremely valuable resources, however, I do not agree with having students to spend hundreds of dollars for new texts. An example of this is with booklists in post-secondary courses; students are required to purchase specific texts as described by the instructor. The Math department at UBC is doing a fantastic job towards making textbooks electronic and free to access. This would be an excellent idea for secondary schools to try and implement; to make education more accessible, especially if the school is in an area where many students have financial hardships.

In Herbel-Eisenmann & Wagner (2007), the discussion on how the textbook authors use pronouns to address the reader is something I resonate with. I enjoy reading a book where I feel a more personalized connection with the author. Hence, when a math textbook uses first person pronouns such as “I” and “we”, I feel more drawn in and connected to the learning experience. When an author uses third person or phrases like, “one can see that…” I feel more disconnected and almost feel talked down upon. Therefore, although a textbook can be helpful to students to learn by providing supplementary content and practice problems, it may not effectively achieve this if the language within is not welcoming.

 

Reading: Herbel-Eisenmann, B., Wagner, D. (2007). A Framework for Uncovering the Way a Textbook May Position the Mathematics Learner, For the Learning of Mathematics, 27:2, 8-14.

Reflection from group microteaching Optimization

When presenting the economic example of optimization, I found myself rushing through working the example because of how behind timing was. One of the most important parts to these kinds of problems is the formation of the equation we want to optimize. If I could spend a few extra minutes discussing how that solution was derived and interpreting the maximum solution from Desmos, it may help students in the future understand these ideas.

It is difficult to teach an advanced math topic that requires 10+minutes for each example in a timeframe of 15 minutes. Future considerations could be to limit the group microteaching exercise to a single example and spending the entirety working through.

Group microteaching - Optimization problems

Lesson Plan: https://docs.google.com/document/d/1IXNpOCY_sDZmOao_Id_BpyAROaxSJ9mNXes5ucN4X4g/edit 

Powerpoint: https://docs.google.com/presentation/d/1DK2AGiEvZzxQZS_0X4mRnUEykQXDi5tX0PZsoPfb5QU/edit#slide=id.p 

Subject: Foundations of Mathematics	Grade: 11	Date: Nov 16/20	Duration:  15 minutes
Lesson Overview (What this lesson is about)	This lesson will introduce students to optimization of functions without the use of calculus and utilize technologies such as graphing calculators to interpret optimal solutions graphically.

Big Idea(s) (Select one or two big ideas from the new BC curriculum): https://curriculum.gov.bc.ca/curriculum

Optimization informs the decision-making process in situations involving extreme values.

 

Curricular Competencies (What the students will do)   (Select appropriate curricular competencies from the new BC curriculum):   https://curriculum.gov.bc.ca/curriculum	Explore, analyze, and apply mathematical ideas using reason, technology, and other tools. Develop, demonstrate, and apply mathematical understanding through play, story, inquiry, and problem solving. Visualize to explore and illustrate mathematical concepts and relationships. Engage in problem-solving experiences connected with place, story, cultural practices, and perspectives relevant to local First Peoples communities, the local community, and other cultures.
Content Objectives (What the students will know)	Characteristics of graphs, including end behaviour, maximum/minimum, vertex, symmetry, intercepts Maximizing area or volume while minimizing perimeter
Language Objectives   (What new language the students will learn)	Optimization

Materials and Equipment Needed for this Lesson

Students need access to Desmos or a similar graphing calculator Slides presenting the problems to be optimized

 

	Lesson Stages	Learning Activities	Time Allotted
1.	Warm-up   Get students’ attention, connect to previous knowledge and explain why the topic is important to learn.	Story relating to gillnet fishing, needing to construct a set surface area from as little rope as possible (surplus of netting itself) Before we begin: What context should we consider? What depth do the fish swim at? Is there a difference between rope along the top/bottom and rope along the sides? What dimensions is the netting in?	4 minutes
2.	Presentation   Teach the new content and language.	Using a different example, demonstrate the process of optimization  Example 3, p. 129 from Mathematics 11 Addison-Wesley	5 minutes
3.	Practice and Production   Practice, reinforcement, and extension of the new content and language.	Ask students to apply the same technique to our fishing problem, with some guidance as needed.  Once students have an equation, have students graph it.  Share screen to show what our version looks like in case they made a mistake. Explore any issues that may have arisen for students in this process	5 minutes
4.	Closure	Return to the original problem and discuss our answer. Did we consider the context? Could we adjust for the restrictions?	1 minute

The scale problem

When working through this problem, I came across the solution of the weights being 1g, 3g, 9g, 27g. To demonstrate how I solved this problem, we need to think of an equal-arm weight scale like Figure 1. I will also denote the left side of the scale with a red x. When we have x+(some amount), then the items are on the same side of the equal-arm weight scale and when we have x-(some amount), then we have items on opposite sides of the equal-arm weight scale.

Figure 1: An equal-arm weight scale

As the vendor has four different weights to measure herbs from 1 to 40g, the first weight must be 1g, to measure 1g herbs.

For the second weight, we need to be able to measure 2g of herbs. If the vendor has a 2g weight, then we can measure 2g of herbs, 3g of herbs, or also 1g of herbs by setting 2g on the left and 1g weight + 1g herbs on the right. If we had a 2g weight, we need a 4g weight to measure 4g of herbs. If we had a 4g weight, we need to satisfy combinations of x-1, x-2, x+1, x+2, x-1-2, x-1+2, x+1-2, x+1+2. Therefore, we can measure 3g, 2g, 5g, 6g, 1g, 5g, 3g, 7g herbs. As we can measure the same herb amounts in several ways, this may not be the most effective weight combination and we would need our last weight to be 40g-1g-2g-4g = 33g.

If our second weight was x=3g, then we can measure x-1=2g or x+1=4g herbs. Hence now we can measure 1, 2, 3, 4 grams of herbs with our two weights. As noted before, to find our third weight, we need to satisfy x-1, x-3, x+1, x+3, x-1+3, x+1-3, x-1-3, x+1+3. 

We conclude that the third and fourth weights must be 9g and 27g. When we have weights 1g, and 3g, we cannot have a third weight larger than 9g as we would be unable to measure 5g herbs at the minimum. 

I believe this is the only solution. I noticed that 1, 3, 9, 27 is part of the geometric sequence for 3^y, where y is the set of whole numbers. An extension for this activity could be to examine geometric sequences backwards in the form of half-life, which is the time it takes for a quantity to reduce to half of its initial value. Half-life is commonly observed in biology or nuclear physics where substances or atoms decay. 

The giant soup can of Hornby Island

Figure 1: The giant soup can of Hornby Island

When trying to answer the question on the actual size of the painted water tank in the photo, it is important to keep in mind that the water tank has the exact same proportions of a Campbell’s Soup can. We can use the dimensions of a can of Campbell’s soup and scale up to estimate the size of the water tank using the bicycle as a reference. 

Figure 2: A can of Campbell’s Vegetable Soup

From Figure 2, the Campbell vegetable soup I will be measuring from has 284mL. For this exercise, I will assume the volume of the can is as the equation for the volume of a cylinder: pi*r^2*h. Taking measurements of our can, I find that it has a diameter of 6.6cm (radius is 3.3cm) and height of 10.2cm. Entering these in our equation for the volume of a cylinder, I get 348.96cm³.

I googled ‘dimensions of a bicycle’ for diagrams and found that most bikes had a length of around 990mm (99cm) to 1080mm (108cm) as the distance between the centre of the rear wheel to hub of front wheel. Bicycle wheels have a diameter ranging between 419mm (41.9cm) to 633mm (63.3cm) according to PandaEbikes. If this bike had a 520mm diameter (52cm) wheel, then the radius would be 260mm (26cm). For this estimate, I will take the length of the bike to be 108cm + 52cm = 160cm (adding 26cm to rear wheel radius and 26cm to front wheel radius). 

Figure 3: Example of bike dimension found on google images

I then imagined how many bikes would fit length-ways across our water tank. Figure 4 shows that approximately 3.5 bikes fit lengthways, giving our height of the water tank; h = (3.5)(160) = 560cm. Comparing this to the height of the Campbells soup can of 10.2cm, this is 54.9 times larger. I can then estimate the radius of the water tank to also be 54.9 times larger, which equates to 181.17cm. Therefore the volume of water that the water tank can hold is pi*(181.17)^2(560) = 57744479.14cm³ or in terms of liquids to be 57744479.14mL or 57744.48 litres of water.  

Figure 4: Estimating height of water tank

An extension I have for an activity similar to this for secondary students, is to estimate the volume of places such as how much water could fill up the classroom, or ask students to scale up objects such as their phones and estimate the surface area it would cover. 

 

Source:

Pandaebikes. Ebike-motor-wheel-rim-size-guide. Accessed November 9, 2020 from https://www.pandaebikes.com/what-is-my-wheel-size/ebike-motor-wheel-rim-size-guide/.