Blogging on Arbitrary vs. Necessary

 According to Dave Hewitt,

"I describe something as arbitrary is someone could only come to know it to be true by being informed of it by some external means - whether by a teacher, a book, the internet, etc. if something is arbitrary, then it is arbitrary for all learners, and needs to be memorized to be known."

In other words, when things are "arbitrary", it means that something has been informed by someone or something, and there are no questions or opinions to this piece of information. Whatever is informed is meant to be true and without any doubts. 

Hewitt continued to explain that "all students need to be informed of the arbitrary. However, the necessary is dependent upon the awareness students already have."  In "necessary", students will take the "arbitrary" part and work out the properties that are in learning. "Necessary" requires students to discover their learning. 

In planning lessons and units, teachers need to effectively utilize and distribute the "arbitrary" and the "necessary" aspects of teaching. There are learning points in which students have to memorize (arbitrary) and there are learning points in which students have to develop their thinkings from what is being informed (necessary). In designing lessons, teachers need to make sure that they are not teaching students that they have to memorize everything being told. Instead, there should learning activities that show students' thinking processes, and that they can develop and derive new information from the arbitrary. 

TPI Test Results

The TPI test result page indicates that perspective totals above the score of 35 are DOMINANT for me. I received the highest scores in APPRENTICESHIP and NURTURING, so they fall under the dominant area. Perspective totals on or below 35 are RECESSIVE for me, and I have TRANSMISSION, DEVELOPMENT, and SOCIAL REFORM under this category. 

My scores for the five teaching perspectives are mostly in the 30s. I would say I hold a moderately average amongst the different perspectives. It may seem like some perspectives have dominated over the others, but through examining the summary graph, I would say that they are pretty averaged out. (Except for the highest and lowest scores)

I am a little bit surprised to see apprenticeship in the dominating category as I have not actually started teaching yet, so I haven't had any opportunities to apply apprenticeship into my teaching practices. Although I aim to include apprenticeship (linking math to real-life) in my teaching, it is often challenging to do in math; especially with students at the secondary school level. I think it is more ideal to address apprenticeship in math for college students. Therefore, I am not sure if I would focus on the area of apprenticeship at the beginning of my teaching career. 

 I scored lowest in the l social reform perspective. According to the website, social reform is defined as the following:

"From the social reform point of view, the object of teaching is collective rather than the individual."

 I would interpret this as letting the students know that we are learning because we are striving to make our lives better with knowledge and skills. I think this is important that our students need to know why we need to be educated, but again, I am not sure how much emphasis I will put on this perspective once I start teaching. 

These results make me wonder how I should teach with the goal to incorporate all five perspectives evenly. I feel like all of these are important in cultivating successful lessons. 

Again, I have not started teaching yet, and I think the scores in these teaching perspectives will change once I start teaching. I plan to come back to this website and do the test again after I have started teaching for a year.  

 

Blogging about math textbooks

  This article analyzes how textbook position students in relation to mathematics, theirs peers, teachers, other people, as well as their own experiences. I find this article interesting because the authors are putting a lot of emphasize and value on textbooks that should be used as facilitators in learning about math. In my perspective, the crucial element in learning math at school is still on how the teachers deliver the lesson. In other words, what matters the most is how teachers teach math. It's not like students are being asked come to a math class, open their textbook, and learn math on their own. Teachers and students can utilize textbooks as tools to facilitate understanding in math, but it is not the central element. 

As a former student, I actually like textbooks. I remember I would go home and do a lot of practice questions from the textbook before a test. Although the curriculum is suggesting students to learn math in a new way, I personally enjoyed learning math the traditional way by doing lots of practice questions. I get the feeling of satisfaction after solving a math question correct, and that also proves to me that I am learning something because I know how to do it. Whenever I get stuck on something, I would always refer back to the textbook. Sometimes I can find questions on the textbook that were not covered by the teacher in class, and I would self-learn it by reading the textbook. Also, doing many practice questions from the textbook actually helped me achieve high grades back in high school. 

Shifting my role to be a teacher, as much as I liked textbooks (I still do), I think I have to slowly move away from it because it actually belongs to the "old" way of teaching. With the new curriculum, teachers are being asked to shift away from the traditional-based teaching styles, and implement pedagogies that are more relatable to the real-world. During my two week practicum, students would still have their textbooks, but teachers rarely refer to the textbooks, and they would design teaching content and practice questions on online platforms such as IXL, Desmos, and CEMC. In this technology-rich world, teachers have to incorporate the aspect of digital literacy onto their lessons.  Now going back to the topic of textbooks. As a teacher, I would still use and refer to textbooks in planning my lessons. Textbooks could be rich resources for explaining concepts and finding examples. Nevertheless, I can foresee in the future, textbooks will eventually retire and be supersede by computers in mathematical education. 

The Scale Problem

 

Thinking process:

We want the scale to balance on both sides.

- To measure 1g of herb, we have 1g weight to balance the scale. We know the one of the four weights is 1g.

- To measure 2g of herbs, given that we already have 1g weight, we need can have another 1g of weight to balance both sides. However, the question states that the four weights are at different amounts, so the second weight cannot be 1g. Now if we put the 2g of herb to the 1g weight on the left side, we would need 3 g of weight on the right side. So we can say the second weight is 3g. 

- To measure 3g of herbs, we can use the 3g weight to balance the scale

- To measure 4g of herbs, we can use the 1g + 3g weight. 

- To measure 5g of herbs, we have 1g + 3g weight, we need another 1g to make the scale balance. Once again, we already have 1g, so it doesn’t work. Analogic to measuring 2g of herbs, we can move 5g of herbs to one side with 1g+3g weight, in total we have 9g. So the other side might be weigh 9g. We can say the third weight is 9g. Now, we have the weights to be 1g, 3g, and 9g. 

- To measure 6g, we have 1g,3g,and 9g. We can balance the scale by having 9g on one side, and 6g herbs + 3 g weight on the other side

- To measure 7g, we can have 7g herbs + 3g weight on one side, and 1g+9g on the other side

Similarly thinking process can be applied for the rest of the numbers until we get to 14g.

- To measure 14g, we will end up with the same situation in measuring 2g and 5g of herbs. From there, we can find our last weight to be 27g. 

I have provided a trial-and-error chart to support my thinking. Using the four weights at 1g, 3g, 9g, and 27g. We can balance the scale in weighing out whole-number amount of herbs from 1 to 40g. 

I believe there is only one correct solution to this problem. 

To extend on this puzzle, I would ask the students what they notice about the numbers, and if there is a quicker way of doing it more mathematically. In which number will they need to add a fifth weight? Sixth weight? What will be the values of the fifth and sixth weight

Group Micro Teaching Reflection

 Our group presented the this mini-lesson a little bit differently from the other groups. We wanted to mimic the lesson as if we were in an actual classroom setting with a group of grade 8 students. As we were teaching, we encountered some technical issues with using the whiteboard on zoom (we would write on the board if this was in class). It was also hard to monitor the students on zoom while we were focused on delivering the content. Our group should have spent more time in preparing and accommodating the issues we would have on zoom. While we were planning our lesson, we actually talked about if we should create a Powerpoint slide in delivering the content, but we decided not to as a group because we wanted to present it as if we were in the classroom.

 Our lesson today was skill based, and it was suppose to be one of those lessons where students would copy notes down. Again, it was not ideal with our audience being other math TCs. We also had some great questions from our TCs that were beyond the level that would be aksed by the grade 8 students. Overall, I think the lesson went like we expected, but we could have improved on the content and presentation. 

Group Micro Teaching lesson plan - Ivan, Zoe, Sukie

 

Title	Square Roots	Grade	8	Date	Nov 16,2020
TC	Zoe Zhang, Sukie Liu, Ivan Li	Subject	Math	Time	15 mins

Learning Intentions	Understand (big idea or SOI)	Know (content)	Do (skills)
	Computational fluency and flexibility with numbers extend to operations with rational numbers.	Square roots	-Finding the square root of a number  -Understanding the three rules for square roots (Pg 193)
	Learner Profile and/or ATLs (if not noted above but will be explicitly taught)
	COMMUNICATION - Public speaking and presenting  THINKING - Critical thinking skills
	Essential Questions
	Factual	Conceptual
	WHAT are perfect squares  HOW to utilize perfect saucers and square roots	Introduce 4 rules for square roots  Operating the rules to solve problems

Assessment	At the start (formative)	At the end (formative or summative)
	Task/Activity:	Task/Activity:
	What I am looking for: Have a basic understanding of what is radical and radical number Understand and use the 4 rules for square roots in operations	What I am looking for: Have a great understanding of what is a radical number  Be able to operate multiplication/division or a combination of both for radical numbers

Preparation	Materials/Resources	Organization (setup, pre-made things, classroom management, etc.)
	Textbook (Haese Mathematics for international students Grade 8) Whiteboard on Zoom	Open Zoom  Open textbook PDF on my laptop

Learning Engagements	Opening (provocation, APK/S, mental set)		Time
	Self-introduction  My classroom management philosophy and rules  I don’t want to see laptops and phones on  When I am talking, no one should talk, and I want you all to listen  Please put your hand up when you want to speak (zoom: tell your name and speak) Quick lesson plan rundown  Review of square roots 4 Rules for square roots  Practice questions		3
	Strategy		Time
		LEADING QUESTION/REVIEW  Understanding the relationship & similarities of following numbers  4,9,16,25 —2,3,4,5 Write the above numbers on the board, ask student what connections they see between these numbers  BASIC MATH LANGUAGE OF SQUARE ROOTS  Square root of number a  Radical and surd (show some examples)  INTRODUCE THE 4 RULES  Introduce the multiplication and division of radical numbers while introducing the three rules. Work thought 2 multiplication/division problems with students together.       ASSIGN PRACTICE QUESTION (STUDENT WORK TIME)	2 5 5
	Closure / Resolution		Time
	Where do you think students will struggle in the lesson?  What will you do to counter this?
	I think students will struggle to pick the right/more efficient way to solve division problems in fraction decimal forms. The concept of surd radicals might confuse them when they start learning how to simplify a radical number. Encourage them to use “trial and error” method to solve questions. Always double check your answer using a calculator if it’s allowed.

Homework: The Giant Soup Can

 

Question::given the size of the actual Campbell's Soup can (of normal size) and the height of the bike in the photo, what are the dimensions of the volunteer fire department's water tank? What is its volume? Does it hold enough water to put out an average house fire?

So let's look at the information we have

Given: 

- size of the actual Campbell soup can (7 x 7 x 10 cm)

- height of the bike in the photo (105 cm)

Note: I had to search up for the meaures in both given information. Measurements could be different depending on search results. 

Need to find:

- Dimension of the water tank 

- Volume of the water tank

- Whether the water tank hold enough water to put out an average house fire

we know that the volume of a cylinder is V= pi * r^2 * h 

To determine the length of the bike, we can use the proportion of the height 105 cm : 3.1 cm as a scale. Then the length of the bike be 5 cm : 169 cm. 

Similarily, the height of the water thank is 12.2 cm : 413 cm

Since the height of the actual can is 10 cm : 413 cm (height of the water tank), we can say that the water tank is 41.3 times greater in proportion. 

The radius of the can is 3.5cm, so we can find the radius of the water tank to be 144.55cm. 

Using the formula for volume of a cylinder, we have

v = pi * (144.55)^2 * (413) = 27100000 cm ^3

Now we need to find out whether the tank can hold enough water to put out an average house fire.

By doing some research online, I find that  1 gallon of water can put out 3 square feet of fire. 

27100000 cm ^3 = 7159 gallon of water 

7179 / 3 = 2386.3 square feet

The water tank can hold enough water to put out a house of 2386.3 square feet (Average 4 bedroom house). Therefore, it is sufficient to hold water to put fire in an average house. 

Extension: If the only the measurements of the soup can are given, can you still proceed with the original question?

Blogging on "Flow" - engagement and the thinking classroom

 

In the flow state, people are in high skill and high challenge mode. They are focused, determined, and self-conscious. 

The question Mihaly Csikszentmihalyi left for us is " how to put more and more of everyday life in that flow channel?" 

In order to reach "flow", we first need to figure out the goals. What is our goal? What are looking for as the outcomes? Why do we do this? What are we trying to achieve? It is important that we set goals as we do things. Goals are the motivations that keep us going and moving forward. 

In a math classroom, teachers can set goals for students, or students can set their own goals. Ask them questions like, "What isyour goal in math class? What are you trying to achieve in this class?" I think helping students find their goals would help them transition into the flow state easier.

 To promote flow and happiness in our mathematical classes, it is important to keep our students focused and on task. One good strategy to implement flow in a math classroom is to have warm up activities in the beginning of class. Warm up activities can be as simple as math puzzles, or they can be something relating to what will be taught that day. As soon as students see the warm up activities, they get into the "math mode", and they start thinking about these problems. Some students might find the problems challenging, and they can always discuss with their partners on the solutions. It is also good way for students to practice their communication skills while they are explaining their mathematical thinkings to their peers. Having warm up activites is a good way to start the lesson and prepare students to get ready to learn math.  Students get the sense of accomplishment from solving these math problems. When students are satisfied with what they are doing, they can reach the state of "flow" smoother and sooner.