Final reflections

*NEW* Read through all your blogs for the course and reflect on

(i) what you have learned,

The thing that I learned that stands out the most was the thinking classroom. Using vertical, erasable surfaces to explore ideas and try things without a fear of failure makes a lot of sense to me and is something that I am eager to try out. I have noticed a lot of people “hiding” during math because they don’t want to expose their inability. This approach addresses that problem. On that note, the overall theme of the course really resonated with me. It is so helpful for learning to have hands-on activities to engage students. I agree with this because I feel that the main issue in teaching math is motivating students and engaging activities increase and inspire motivation in students. Also, these hands-on activities help students see that math is not strictly a pen and paper or computer related discipline. It extends outside the classroom and can be applied to help with almost anything.

 

(ii) how your ideas may have changed throughout the course, and

My ideas in regards to hands-on, engaging activities have changed because I now see them as much more valuable and effective than I did before when I thought of them as fun but maybe not so helpful for the test. Now I see that they promote mathematical thinking more generally which is very helpful on the test but also the tests are not that important because there are many ways to assess, evaluate, and judge understanding.

 

(iii) suggestions to improve the course for next year.

I would say keep the articles and posting about them. That was good. More discussion on them every week would also be fun. Maybe reconsider the art project to make it smaller because I had a hard time with that one. For example, maybe just a standard blog post instead of a presentation.

My favorite math thing

One of my favorite math things is the movie A Beautiful Mind starring Russel Crowe as Steve Nash, an extraordinary American mathematician of the twentieth century who pioneered the mathematical field of game theory. In the movie, he battles his personal demons while working on high level mathematics similar to the recent phenom Queen's Gambit about an extraordinary American chess player of the twentieth century battling demons while working on world-class chess strategy. Game theory is the most interesting branch of mathematics. There is a scene in the movie that shows how game theory was born in a bar in an entertaining way. The main character and his friends are all at a bar trying to decide how best to succeed with a beautiful woman when Steve Nash has a eureka moment that they can all increase their odds of success by working together - and game theory is born.

When it comes to mathematical food, one thing that I learned in my years as a chef was that when presenting food, odd numbers are more aesthetically pleasing then even numbers. So when presenting food, always try to have odd numbers of things. Kitchen math is also a favorite math thing of mine and I feel a good way to teach and exemplify math similar to teaching math from a financial literacy perspective. This makes it feel more real, relevant, tangible, useful and hands-on. Next to financial literacy, teaching math from a culinary point of view is one way that I will connect with some students. Everybody eats and almost everybody wants to know what's going on in the kitchen so it's a good way to connect.

Another mathematical kitchen thing of interest is "turning." Turning is shaping vegetables so that they stand up on their own and look pretty. Personally, I have never seen a use for it but here it is if you're interested: Go to about 3.5 minutes and 5.5 minutes to see the knifework.

https://www.youtube.com/watch?v=_WyzGfu-AK0

Unit Plan - Math 8 - squares, cubes, area, volume, Pythagorean theorem

 

EDCP 342A Unit planning: Rationale and overview for planning a unit of work in secondary school mathematics

 

Your name: Ben Gregson
School, grade & course: Eric Hamber, grade 8, math 8

Topic of unit (NOTE: This should be a unit you will actually be teaching on practicum!):

 

Squares, cubes, surface area, volume, Pythagorean Theorem

 

Preplanning questions:

 

(1) Why do we teach this unit to secondary school students? Research and talk about the following: Why is this topic included in the curriculum? Why is it important that students learn it? What learning do you hope they will take with them from this? What is intrinsically interesting, useful, beautiful about this topic? (150 words)   We teach this topic to secondary school students because it is a necessary part of understanding the world around us, it is required knowledge for many tasks, jobs, and careers and it ends with these young people getting their first introduction to any kind of mathematical theorem and it is one that is beautiful in the sense that it shows an underlying geometrical, mathematical order to a world that can at times appear utterly disordered. It is important to learn about squares and how they are related to areas because problems involving area arise frequently in careers such as engineering and architecture but also painting, carpentry, and numerous other trades. Therefore, everybody can benefit from a solid understanding of area, which requires understanding squares and square roots. This also leads directly to cubes and volumes. The Pythagorean Theorem also requires understanding squares and square roots and it has a beautiful simplicity in addition to being extremely useful for understanding triangles, trigonometry, and hands-on math such as designing or building a staircase or roof as just one example. After this unit, I want students to understand squares and square roots, cubes and cube roots, surface areas, volumes and the Pythagorean Theorem by being able to exemplify their understanding numerically as well as verbally explain them.

(2) A mathematics project connected to this unit: Plan and describe a student mathematics project that will form part of this unit. Describe the topic, aims, process and timing, and what the students will be asked to produce, and how you will assess the project. (250 words)   Our project is to figure out how many cubes and how many tiles as well as how many and what shape of block we would need (with dimensions) to make our cubes into the shape of a longhouse with tiles for the roof. The cubes will be the basis of the house and the tiles will go on top of the house to make the roof. We will need a 3D non-cube shape to get our tiles to fit nicely on top of our house. It will be the student’s project to figure out what shape or shapes we would need to connect our flat roof tiles to our little cubes to make a solid longhouse. We will need a non-cube shape to build with our cubes and tiles to make the angled roof. I will do an example on the board of how to figure out what kind of block shape an exemplary design will require so students can figure out what kind of block shape they will need for their own design. I will also show how to find the dimensions of the required block piece. Students can find an image of a longhouse online to emulate or be creative and design with a variety of angles and a simple roof or a complex one with changes in height, skylights, or chimneys. Longhouses are part of some Indigenous cultures in BC as well as other cultures from around the world such as the Vikings.

(3) Assessment and evaluation: How will you build a fair and well-rounded assessment and evaluation plan for this unit? Include formative and summative, informal/ observational and more formal assessment modes. (100 words)   Students will be informally assessed on how they participate in their groups. For example, are they able to properly combine their tiles into squares and cubes into larger cubes properly? Also, they will get formative assessment at this time. There will be formal, summative assessment on the work that they produce individually during the individual work portion of class time as well as a unit test after the unit. There will also be completion marks for the work done during the group portion of class time. For example, predicting and recording the sizes of the squares and cubes that they construct in their pairs and groups.

Elements of your unit plan:

a)  Give a numbered list of the topics of the 10-12 lessons in this unit in the order you would teach them.

Lesson	Topic
1	Squares using little tiles (hands on)
2	Square roots
3	Cubes using little cubes (hands on)
4	Cube roots
5	Area of a rectangle
6	Area of a triangle and circle
7	Volume of rectangular prisms
8	Volume of pyramids & cylinders
9	Longhouses
10	Pythagorean Theorem part 1
(11)	Pythagorean Theorem part 2
(12)	Unit Test

b) Write a detailed lesson plan for three of the lessons which will not be in a traditional lecture/ exercise/ homework format.  These three lessons should include at least three of the following six elements related to your mathematical topic. (And of course, you could include more than three!)

 

These elements should be thoroughly integrated into the lessons (i.e. not an add-on that the teacher just tells!)

a) History of this mathematics

b) Arts and mathematics

c) Indigenous perspectives and cultures

d) Social/environmental justice

e) Open-ended problem solving in groups at vertical erasable surfaces (“thinking classroom”)

f) Telling only what is arbitrary, and having students work on what is logically ‘necessary’

 

Be sure to include your pedagogical goals, topic of the lesson, preparation and materials, approximate timings, an account of what the students and teacher will be doing throughout the lesson, and ways that you will assess students’ background knowledge, student learning and the overall effectiveness of the lesson. Please use a template that you find helpful, and that includes all these elements.

 

Lesson Plan

 

Teacher’s Name: ___Ben Gregson_______

 

Subject: ____Math 8______________

 

Lesson Topic: ____Squares____________________

 

Rationale	This is a hands on activity meant to be a fun introduction to the idea of squares. It is leading up to learning about square roots and surface area. Students at this age have a solid understanding of the difference between adding and multiplying and multiplying is to adding what exponents are to multiplying.
Big Ideas	squares, square roots, surface area
Curricular Competencies	Reasoning, analyzing, understanding, representing, connecting,
Student Learning Objectives (SLO’s, SWBAT)	Understand, calculate and explain squares, square roots, and the surface area of a square
Required Vocabulary and/or Language Strategies	squares, square roots, surface area
Introduction   __50_ min.	For this lesson we will be playing with little tiles. Cut up pieces of paper will work if nothing else is available. Students pair up and get a small bag of approximately 100 little tiles to combine together to make big squares.   Students will be asked to each individually write down the length of the side of each square that they can make, as well as the total number of tiles needed to make each big square before moving on to a bigger size. But first, students must predict the biggest square that they will be able to make with their bag of tiles.   After approximately 20 minutes, I will ask students to join their pair with another pair to make groups of 4 and see how big of a square their bigger group can make by combining their tiles. Again, they must first write down a prediction. After another 10 minutes or so, if space allows, groups will combine again and repeat the process.   Afterwards, we will split the tiles back into their original amounts of approximately 100 and they will be available for the rest of the class if students want to use them as a tool to answer questions.
Body/Development   __40_ min.	Next, I will spend about 5 minutes doing a short presentation on the board explaining squares and the arbitrary notation involved in representing this idea in writing. Also, I will explain that the total number of tiles is the area. We can calculate the area of a square by multiplying the length and the width together.   Next, students will have about 30 minutes to work on numerical problems (each of 1 to 20 squares) and a word problem involving squares.   Q1. You are moving houses. Your new house is getting the backyard redone and a new patio built out of square-shaped stones. The new patio is going to be square-shaped as well. The patio stones cost about $15 each and another $15 each to install. You have about $2000 to spend on this. You want to spend as little as possible but you also want as big of a patio as possible. How big of a square-shaped patio can you afford to get built with $2000? How much money will you have leftover afterwards? How much more would it cost to go up to the next biggest size of square?
Closure   __10_ min.	I end the class with a quick recap of what we learned about squares and how the number of tiles that are in each big square is the surface area of that square. Therefore, we can find the surface area of a square by multiplying together the length and width of the square. Then, as part of a flipped classroom approach, I briefly go over the idea of square roots, which we will be doing next class.   To explain square roots, I start by asking, “If I had say 80 tiles total, what would be the biggest square I could make, what length? And how many tiles will I have leftover?” Then, I connect it to squares as a counter-operation to squaring a number, which also explains the name square root. Just like how subtraction is the counter operation to addition and dividing is the counter operation to multiplication, squaring has a counter operation called square rooting.
Extensions/ Modifications/ & Early Finishers	Is there a pattern for the difference between perfect squares?   I would also present the locker problem to students who are interested and ask them how that problem is connected to squares.
Assessment	Students will be informally assessed on how they participate in their groups. Also, they will get formative assessment at this time. There will be formal assessment done on the work that they do individually but they can get help with that if they need it. There will also be completion marks for predicting and recording what they construct in their pairs and groups.
Materials	100 little tiles for each student. Cut up pieces of paper can be used if nothing else is available.

 

 

 

Lesson Plan

 

Teacher’s Name: ___Ben Gregson_______

 

Subject: ____Math 8______________

 

Lesson Topic: ____Cubes____________________

 

Rationale	This is a hands on activity meant to be a fun introduction to the idea of cubes. It is leading up to learning about cube roots, volume, and then exponents. Students at this age have a solid understanding of the difference between adding and multiplying and multiplying is to adding what exponents are to multiplying.
Big Ideas	Cubes, cube roots, volume
Curricular Competencies	Reasoning, analyzing, understanding, representing, connecting,
Student Learning Objectives (SLO’s, SWBAT)	Understand, calculate and explain cubes, cube roots, and the volume of a cube
Required Vocabulary and/or Language Strategies	Cubes, cube roots, volume
Introduction   __50_ min.	For this lesson we will be playing with little cubes. Sugar cubes will work if nothing else is available. Similar to what we did with tiles, we are now going to do with cubes. Students again pair up and get a small bag of approximately 150 little cubes to combine together to make big cubes.   Students will be asked to each write down the length of the side of each cube that they can make, as well as the total number of cubes needed to make each big cube before moving on to a bigger size. But first, students must predict the biggest cube that they will be able to make with their bag of cubes.   After approximately 20 minutes, I will ask students to join their pair with another pair to make groups of 4 and see how big of a cube their bigger group can make by combining their cubes but first write down a prediction again. After another 10 minutes or so, if space allows, groups will combine again and repeat the process.   Afterwards, we will split the cubes back into their original amounts of approximately 150 and they will be available for the rest of the class if students want to use them as a tool to answer questions.
Body/Development   __40_ min.	Next, I will spend about 5 minutes doing a short presentation on the board explaining cubes and the arbitrary notation involved in representing this idea in writing.   Next, students will have about 30 minutes to work on numerical problems (each of 1 to 10 cubed) and word problems involving cubes.   Q1. Do the following question about a 3x3x3 cube and then repeat for a 4x4x4 cube and a 5x5x5 cube. See image below.
Closure   __10_ min.	I end the class with a quick recap of what we learned about cubes and how the number of cubes that are in each big cube is the volume of that cube. Therefore, we can find the volume of a cube by multiplying together the length, width, and height of the cube. Then, as part of a flipped classroom approach, I briefly go over the idea of cube roots, which we will be doing next class.   To explain cube roots, I start by asking, “If I had say 100 cubes total, what would be the biggest cube I could make, what length? And how many cubes will I have leftover?” Then, I connect it to cubes as a counter-operation to cubing a number, which also explains the name cube root. Just like how subtraction is the counter operation to addition and dividing is the counter operation to multiplication, cubing has a counter operation called cube rooting.
Extensions/ Modifications/ & Early Finishers	We have seen a pattern with the difference between perfect square numbers. Is there a similar pattern for perfect cube numbers? What if we extend our word problem to bigger sizes of cubes? Is there a pattern there?
Assessment	Students will be informally assessed on how they participate in their groups. Also, they will get formative assessment at this time. There will be formal assessment done on the work that they do individually but they can get help with that if they need it. There will also be completion marks for predicting and recording what they construct in their pairs and groups.
Materials	150 little cubes for each student. Sugar cubes can be used if nothing else is available.

 

Lesson Plan

 

Teacher’s Name: ___Ben Gregson_______

 

Subject: ____Math 8______________

 

Lesson Topic: ____Longhouses____________________

 

Rationale	This is a hands-on activity meant to be a fun application of geometry, cubes and squares. Students at this age have a solid understanding of the difference between adding and multiplying and multiplying is to adding what exponents are to multiplying.
Big Ideas	Squares, cubes, geometry
Curricular Competencies	Reasoning, analyzing, understanding, representing, connecting,
Student Learning Objectives (SLO’s, SWBAT)	Understand, calculate and explain squares, cubes, and the customizable quadrilaterals needed to make them fit together properly
Required Vocabulary and/or Language Strategies	Squares, cubes, irregular, quadrilateral
Introduction   __10_ min.	I’ll start with showing students some pictures of longhouses from BC and around the world as well as modern ones and ancient ones. I’ll explain to students that these buildings were often the first form of permanent structure used by different cultures and the vast amount of work and materials that go into building one.
Body/Development   __70_ min.	After showing students what a longhouse is, I will give a brief 5-10 minute explanation on the board of how we can figure out what shape of block we will need to make our tiles fit on top of our cubes properly. Also, how to calculate the dimensions of the required block.   Students will have access to the tiles and cubes from previous lessons in order to help them build or just visualize how they can design and construct their own longhouse out of tiles, cubes, and an irregular quadrilateral to make them fit together properly. If we use sugar cubes for our little cubes, it will be possible for students to reshape the sugar cube into the desired shape. This will make it possible for students to finish construction of their tiny longhouse in this class and present it to the rest of the class during the last few minutes of class. Without reshaping them, we can still present our ideas on paper or digitally, if preferred.
Closure   __20_ min.	Before the end of class, students can display their finished longhouses for the rest of the class to see. We can also do peer evaluations and judge each other’s creations based on creativity, aesthetics, and complexity. After peer evaluations, I will do a quick summary of what we learned today and review how we can figure out the needed shape and the dimensions.
Extensions/ Modifications/ & Early Finishers	Describe the advantages and disadvantages of steeper vs. flatter roofs.   What sorts of things would longhouse builders have to consider when deciding on a design?   Are there some roof designs for which one type of irregularly-shaped block would not be enough? Why?
Assessment	Students will be informally assessed on how they participate in their groups. Also, they will get formative assessment at this time.   There will be formal assessment done on the project work that they do individually but they can get help with that if they need it.   Peer evaluations will also be a type of formal, summative assessment used in this class.
Materials	50 little tiles for each student. Cut up pieces of paper can be used if nothing else is available. 150 little cubes for each student. Sugar cubes can be used if nothing else is available.

 

 

 

 

Hewitt Article on arbitrary and necessary

One of the things that I have always loved about math is that you don’t have to remember much. Most people would probably find this sentence peculiar, odd or strange for a couple of different reasons. First, most people don’t love math. They hate it with a passion. Second, most people think math is all about remembering formulas or theorems or procedures, but it is not. It’s about figuring them out. I’ve always loved this about math because as a student, if I couldn’t remember something, say what to do when multiplying two logs together, I would try different things, different approaches, different rules, and see what worked. I would just put some numbers in for my variables and verify. These math rules are the “necessary.” Knowing them qualifies as or constitutes mathematical understanding.

The “arbitrary” have a key role in mathematics as well and this is where most of the small amount of remembering required in mathematics lives, in my opinion. In my own teaching, when it comes to the arbitrary, I usually frame it in a linguistic context. For example, talking about degrees and radians, I describe these not as mathematical ideas to be understood but as mere English vocabulary – definitions to be learned. And if you go do math in another language in another part of the world, they likely will have different arbitraries than we do. Arbitraries need not be worried about because they offer little in the way of mathematical challenge. They are the definitions that precede the theorems and can be written on the board and left up there during the test. In my own teaching, to keep the arbitrary from getting too boring or cumbersome to remember, I like to give it a bit of historical context. For example, when talking about degrees, I tell students about the Babylonians and their love for the number 60. I find this just makes it easier to remember that there are 360 degrees in a circle. Before reading this article I had never heard that they multiplied it by 6 because of the ratio of the perimeter of a hexagon to the radius of a circumscribed circle.

Teaching Perspective Inventory

Here are the results of my TPI self-quiz:

Date of survey:11/24/2020

Name of respondent:Ben Gregson

Association group:No, I am not a member of any of these groups

Area of learning:Secondary

Subject of teaching:math

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Transmission Total: (Tr):35 (B = 12; I = 10; A = 13)

Apprenticeship Total: (Ap):38 (B = 11; I = 15; A = 12)

Developmental Total: (Dv):39 (B = 12; I = 15; A = 12)

Nurturing Total: (Nu):41 (B = 14; I = 15; A = 12)

Social Reform Total: (SR):24 (B = 7; I = 9; A = 8)

Beliefs total: (B)56

Intentions total: (I)64

Action total: (A)57

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Mean: (M)35.4

Standard Deviation: (SD)6.02

Dominant Threshold: (HIT)41.42

Recessive Threshold: (LOT)29.38

Overall Total: (T)177

 

My TPI test said that my teaching perspective is dominated by apprenticeship, development, and nurturing. However, social reform is recessive in my teaching perspective. Transmission is almost but just below the boundary for being a dominant perspective of my teaching.

I interpret these results to mean that I care about my student’s well-being more than any other aspect of my teaching. I am encouraged by this result because I believe in holistic teaching and putting what we are learning in perspective in terms of making our overall lives better. Also, students will not be able to learn if they have some problem in their life preoccupying them, so a nurturing perspective is the only perspective that can help these students learn. Furthermore, students learn better when they know that their teacher cares about them and their life outside of school. Next, I value apprenticeship, development and transmission of knowledge. Lastly, I value social reform the least of these 5 categories.

I was not surprised and I agree with most of these results because I feel like that they are a true representation of how I feel about teaching. I feel that social reform is more of a priority for me than this result shows. Moving forward, because I see that social reform is not a priority in my teaching, I will look for ways to try to include more of that in my teaching. To do this, I will include examples related to the environment, sustainability and climate change issues.

One question that this TPI result raises for me is about how to connect the subcategories within each category. Beliefs, intentions and actions are all related but how can I increase the relationship between them? In other words, how can I turn my beliefs into intentions and then into actions?