Final Reflection

It was interesting to see math from a different lens from what I grew up with. Going over topics like teacher bird/student bird, instrumental/relational learning, the state of "flow", arbitrary/necessary, seeing the history of math in schools in North America and how it has changed were interesting and helpful. My favorite post is probably the math question puzzles that made me think of how to solve these word problems given what information I have, what I need to obtain, and to obtain it. The Dave Hewitt fraction problems made me think the hardest. Being in this class, I am able to see a variety of information from people of diverse backgrounds giving their take on math education. This will better help me understand and work with students who are struggling in their learning, or what kind of extra challenges they want to experience. Seeing how to creatively implement art into math in a visual way helps students see the abstract concepts more vividly. Indeed, this course has brought to my attention many things I have not encountered regarding student's learning. So being exposed to such things earlier on is helpful for teachers to know what to look out for. 

EDCP 343A Unit Plan

Assignment 3 Unit Plan

Nov 27 Reflection on Textbooks

Looking at these examples from the lens of a teacher, I would look out for/be aware of the language used in the textbook and that it can influence student engagement and understanding. Some textbooks are quite old and isn't up to modern times. For instance, some textbooks have instructions on using the CD that correspond with the it on the first pages. This may make students feel like the knowledge presented in the textbook is old and cannot connect it with modern technologically advanced times. Old textbooks can be easily damaged and become dirty over the many years. I would also look out for textbooks that are very guiding, like a story that takes you on a journey exploring a topic. There is an introduction, examples, solutions, and practice problems with different levels of difficulty (practise, apply, extend). Having lots of pictures is helpful too. I think cartoon drawings are more fun for students compared to real life pictures, so there should be more creative drawings. But real life pictures are necessary because it should give a sense of skills learned being applied to the real world. Personally I was never a fan of textbooks. I found that the teacher's notes were sufficient for my studying. Textbooks were cumbersome objects taking up space in my locker. In fact I used my grade 11/12 physics textbook as my laptop stand (I coincidentally got the same physics textbook in gr 12 as in gr 11). My teachers never assigned homework questions out of textbooks. Even in university, I never used math textbooks (but still kept PDF copies for reference). In terms of textbooks being handed out in schools, I think textbooks should still be used as a handy reference for students to self study/learn off of. However they should be updated after a few years, as information presented may become obsolete, eg. some data and statistics regarding world informations. Textbooks should be an engaging/interactive way to connect students to the subject. Some teachers may assign homework questions from the textbook. It really depends on the situation, like how much emphasis a teacher uses them for the class. For the most part, textbooks should be an optional resource for students in case they feel like they want more practice. 

Nov 20 Mihalyi Czikszentmihalyi's TedTalk Reflection

Czikszentmihalyi's idea of flow is interesting since it is a nice middle ground to be in. There is a balance between the challenge difficulty level and the knowledge/skills an individual has. During a state of flow, the person is focused and feel the task is rewarding, which leads to a sense of happiness. This works through meaningful pursuits of tasks that are a bit complex that stretch our capabilities. For me, I have experienced a state of flow with the card flip game done in class. Experimenting around and understanding the purpose/rules of the game allowed me to formulate a way to arrange the cards. I also experienced a state of flow from answering the riddle regarding "brothers and sisters I have not, but that man's father is my father's son." I'm interesting in family trees and relationships between the members. I think interest prompts flow and allowed me to use my knowledge to solve this problem. For whether it is connected with math experiences, I would say family trees could be represented using nodes and edges, to be studied in graph theory. One tree for one individual is very different from another's, yet they might share one link together. I do believe a state of flow is achievable in secondary math class because that is when student inquiry beings to grow rapidly. They are exposed to many elements in life and will question many things. Giving those students an opportunity to exercise their knowledge set will improve their analysis. To create this state of flow as teachers, questioning students on their explanation is helpful. It makes them check for any loose ends. Designing tasks that are neither too easy nor too hard helps too. The activity should be challenging enough to stretch students' skills without overwhelming them. Also, most students are interested on how math is relevant to their lives. Real world applications can make abstract concepts more tangible and worth investing effort into.

Nov 13 Dave Hewitt's Secondary Teaching Video Response

The first thing that I realized was that the pupils were very attentive. Pupils back then were in my opinion more disciplined and didn't have a lot of external forces and influence pertaining to their lack of attention/self control. They were not chatting amongst themselves and followed Hewitt. Or maybe they acted more polite because they knew they were being recorded. The uniforms caught my eye as well, and those can be a whole topic on its own (on whether or not school should have them as a practice. . . not speaking about some merch clothing that the school may sell). I think it's because the pupils don't have to think about what they wear each day, and that they don't compare their outward clothing appearance with each other (so that everyone is represented on the same level). It is interesting to dive deeper into the topic of having school uniforms in schools. From what I know, independent schools have school uniforms and not public schools. There is probably some school pride associated with it as well. Secondly Hewitt has a unique teaching style. I don’t prefer his method when he bangs the wall with the stick since this creates a lot of noise pollution which irritates me. It might be a way to wake up some sleepy pupils but overall I think it is disruptive, especially if there are classrooms on the other side of the wall. He could’ve used his hand/arms and point/touch those places instead, since it is a nice prelude to the main topic. Also, this was a classroom setting in the 80s, so digital projection/smart boards didn’t exist. So to teach on the chalkboard is something rarely done nowadays since there are usually premade notes for pupils to fill out, and the teacher projects those notes onto the screen/smart board. However Hewitt used that space for pupils to work on problems together, which allowed collaboration and teamwork. I see that he asks and goes over lots of questions verbally. He puts finger by ear for a stronger response from the pupils answering in unison. This is nice since it isn't just the smart kid that shouts out the answer. He also says that he isn't a fact checker and won't really answer "yes" or "no". In fact he asks questions back to the pupils "is this correct". There is also a wait time for the pupils to respond, as the gears in their brain turn as they ponder on the questions. So being patient is crucial.

I think Hewitt created the fractions problems by taking two fractions that are close to each other in simplified form to see if there can be a fraction with particular numerators that is between those two fractions. This is an existential problem which is used in many areas of math. When we know something exists that satisfies a property, then that object may be further used to continue a problem or inquire newer problems that branch off to multiple areas. For me these problems are interesting and it makes one dump all of their math knowledge out of a box and find which tools to use. They make people think deeply and understand the underlying methods to tackling these problems. 

Task #1 Fraction that is between 5/7 and 3/4:

Using the lowest common denominator of 7 and 4, then by making equivalent fractions we have 20/28 and 21/28 respectively. Since there is no whole number between 20 and 21 we can multiply top and bottom by 2 for each fraction to get 40/56 and 42/56 respectively. Thus, we can have the fraction 41/56 s.t. 5/7 < 41/56 < 3/4.

Task #2 Fraction that is between 5/7 and 3/4 where the numerator is "11":

Consider 5/7 < 11/x < 3/4 s.t. x is a positive integer. 

5/7 < 11/x < 3/4 

⟺ 4/3 < x/11 < 7/5 

⟺ 220 < 15x < 231 

⟺ x = 15    ∵15*15 = 225

∴ 5/7 < 11/15 < 3/4

Task #3, 4 Fraction that is between 5/7 and 3/4 where the numerator is less than or greater than "11":

This part took me a long time to figure out as it's one of those problems I see in number theory classes. To deal with a sufficient condition that gurantees a fraction falling between 5/7 and 3/4, first take a look at fractions 0 < a/b < c/d. 

0 < a/b < c/d

⟺ ad < bc  

⟺ ady < bcy   ∧   adx < bcx    x,y positive integers

⟺ ady + axb = a(bx + dy) < bcy + axb = b(ax + cy)   

∧   adx + cyd = d(ax + cy) <  bcx+ cyd = c(bx + dy)

⟺ a/c < (ax + cy)/(bx + dy) < c/d

Given this result, 5/7 < (5x+3y)/(7x+4y) < 3/4. So for example numerators 8, 11, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23 work.

Nov 8 Arbitrary and Necessary Article Response

The concepts of "arbitrary" and "necessary" hasn't explicitly occured to me until now. I like how it begins with a scenario about a child's question regarding the name of a square, which leads to a discussion about the arbitrary nature of names and labels in math. The author points out that many aspects of mathematics, such as the names of shapes or the number of degrees in a circle, are arbitrary and are social conventions that have been agreed upon and must be memorized. The "necessary" parts however are those that students can work out for themselves, which are not dependent on memorization but on understanding/awareness. For example, a half plus a half gives a whole (two halves make a whole). A lot of formulae use notation that is arbitrary. I think mathematicians make up a lot of notations for certain notions. These notations are arbitrary. In fact a symbol (whether in Latin/Greek/any other language) is used more than once to denote concepts that are very different (eg. P/p can denote perimeter, pressure, polynomial, primes, and p-value). So knowing the context is important. Richard Feynman came up with his own notation in his teens for sine, cosine, and tangent. He says "While I was doing all this trigonometry, I didn't like the symbols for sine, cosine, tangent, and so on. To me, 'sin f' looked like s times i times n times f! So I invented another symbol, like a square root sign, that was a sigma with a long arm sticking out of it, and I put the f underneath. For the tangent it was a tau with the top of the tau extended, and for the cosine I made a kind of gamma. . .so my symbols were better." 

The fact that certain elements of mathematics are arbitrary (such as the naming of shapes, the order of operations, or the way we write numbers) means acknowledging that these will require different teaching strategies focused on memorization and recall. This is more instrumental approach. The relational approach kind of delves deeper into the root of the necessary truths of a statement. It is emphasized that the "necessary" is not about memorizing facts, but rather about what can be logically deduced or inherently understood from given mathematical properties. So, outlining the necessary elements is key, and it'll be helpful if students could explore some aspects on their own by reasoning from prior knowledge. This not only makes mathematics more meaningful and engaging for students but also helps develop their critical thinking and essential problem solving skills. 

Nov 6 The Giant Soup Can of Hornby Island Problem (Updated Nov 6)

From my research a can of Campbell's Condensed Tomato Soup typically has dimensions where the radius is approximately 3.8 cm and its height approximately 10.8 cm. However there were many other dimensions so I just chose the most common one. The average road bike length is approximately 177 cm. One can measure the length of the bike in the picture with a ruler and the length of the can in the picture (which is the height as it’s sideways). Divide the length of the can by the length of the bike. Multiply that number by 177 to get the actual length of the can in cm.  I will say three bikes fit alongside the length of the can so 177 cm times 3 equals 531 cm. Since the proportions are the same as a normal sized can, the radius of the water tank can is 531 times 3.8/10.8 which equals around 187 cm. Putting it all together the volume is π[(187)^2](531) which equals around 58 334 786 cm^3 = 58.334786 kL. This is more than enough water to put out an average house fire, since only several kL of water is needed. My experience doing this question initially wasn't fun since I couldn't find consistent values for the dimensions of the soup can. There were different versions of the cans and sometimes the dimensions given to me were for the box that contains them! The image of the water tank can was also taken at an angle so measuring it from side to side wasn't even. Thus I just approximated that three bikes fit across. This question made me relate to a typical question grade 10 students would do with scales and proportions, say finding approximately the distance between two city centres in real life. On the diagram a cm could represent 50 km. Students measure two dots representing the two city centres and find their distance, say 6 cm. So that means they are 300 km away. Other times you walk into a museum and there is a big display of a lego model representing ancient Rome. There might be a scale at the bottom so one is able to figure out approximately how big ancient Rome really was back then based on how big the model was.

As a more specific example, below are pictures I took on New Year's Eve 2018 at the Edmonton City Hall. It is a lego model of people (veterans, RCMP, politicians, citizens) at the Vimy Ridge Memorial. I was curious to know if this was to scale as I meticulously observed this work of art. I haven't personally been to the memorial site, but I hope to one day, remembering the victory for freedom the allies fought hard for. It's especially important now as we near Remembrance Day. To see if this is approximately to scale, you can measure the height of one lego human, which is 4 cm. The average human height is 170 cm. The actual twin pylons are 27 m = 2700 cm tall. The scale is 170/4 = 42.5, so when the the height of the lego memorial's twin pylons are measured, it should be around 2700/42.5 = 63.5 cm if this model is to scale. From what I recall, it was around that height, so the model is decently accruate. This problem is doable if we are actually allowed to carefully measure this lego structure with permission. Otherwise we will have to resort to using a photo and use its measurements of the lego human figures/twin pylons to find if the scale for the structure is consistent.   

Oct 20 Professional Development Reflection

I attended the session on Supporting Students with Exceptionalities. The first part mentioned the language usage for someone who does have physical disabilities. For example, exceptionalities, physical impairment, disabled person. We should ask which term is appropriate to use. Another example is that a student may say "I am autistic" or "I have autism". So the tone is important. The stats showed that around 86 thousand students in BC do have special needs and they are entitled to equitable access to learning, achievement, and the pursuit of excellence (however this isn't happening in many schools). An inclusive classroom is one that is supportive with students and teachers doing the best with what they got. We give 100% to the students and they would probably give 100% back. Students learn together, support each other of all abilities, participate and contribute to the life of the school. So including students into the classroom isn't just providing a desk for them. All BC school districts must work to identity, prevent, and remove barriers for people with disabilities and meet the requirements of the Accessible BC Act (2021). What it mandates are to create accessibility committees, create accessibility plans, and establish a process for receiving public feedback. I really liked the visual pictures with diagrams that showed kids being cheerful with each other. A continuum of education practice was also shown from exclusion to segregation to integration to inclusion to teaching to diversity. Learning from students with special needs is helpful since they know what they are comfortable with. There was a video that was shown where a group of young kids with special needs said what they feel inside when a teacher tells them to do something. For example, one kid said that when the teacher tells him to sit up straight, he would have to focus entirely on sitting up straight and not on the lesson. This is quite surprising and it demonstrates the complex nature of people with special needs. Another kid said that he has to move around or rock in his chair to pay attention better. This makes teachers think that we may tell students to do something that may be helpful but it does the opposite. Another surprising thing was that even with an official diagnosis document, the student won't necessarily get the funding handed to them by the school. The district handles the process. I also liked three pictures showing which parts of the brain are responsible for engagement (middle of brain), representation (back of brain), and action/expression (frontal lobe). EPSE 317 addresses more on this topic and content includes types of disabilities, roles of general classroom teachers in creating inclusive classrooms, universal design for learning/lesson planning, trauma-informed inclusive classroom practices, ableism within schools, assessment and students with disabilities, IEP's and the roles of teachers. Currently I am a bit scared because I don't want to mess up and do something that may not be great for those with special needs. Not sure how much extra work/effort I have to put in to help, so I'm worried if I can even get through the material. I have no experience so I hope to have some eye opening findings and interactions during practicum to better support these students.

Oct 16 Group Curricular Microteaching Lesson Plan Post Reflection

What I gleaned from these reflections was that my group graded ourselves relatively more harshly than how our peers graded us. This shows that my group mates were thinking along the same lines as me regarding how we performed. We all agreed that the content was too much for a 20 minute lesson. The examples given out for our peers to do were much harder than the example we went over on the board. It also contained a lot of vocabulary and steps to take that weren't mentioned. It would be wise to give out questions that are on the same level of difficulty as the ones done on the board. There were times where my group spoke softly (maybe it is because another group was much louder), so we should project our voices. Going over a new concept at first requires a lot of focus from the audience, and we were able to capture some of it by asking some simple warm up questions. We should definitely work on our time and pacing because asking questions regarding buying and leasing took longer than expected. It is also the case that the questions do take more time to do, so cutting down the difficulty and giving fewer parts should suffice. On the practical side of things the audience really enjoyed how important and applicable this topic was. Thinking financially about a vehicle is a good budgeting practice to consider. The factors that contribute to whether a car should be purchased or leased as well as the pros and cons of each option were discussed. 

 

Oct 16 Group Curricular Microteaching Lesson Plan

 342 Curricular Microteaching Lesson Plan

You

Allyssa

Michael

Lesson Plan Workplace Mathematics 11

Unit: Financial Literacy

Topic: To purchase, own, or lease and to operate and maintain a vehicle

Big Ideas: Mathematics informs financial decision making

Curricular Competencies: 

- Model with mathematics in situational contexts

- Explain and justify mathematical ideas and decisions in many ways

- Reflect on mathematical thinking

- Connect mathematical concepts with each other, other areas, and personal interests

Prescribed Learning Outcomes (PLO’s): 

- Describe the difference between buying and leasing a vehicle

- To understand and know how to apply the factors that involve buying or leasing a vehicle

- Defining the positives and negatives of both buying and leasing

- Use rational decision-making by comparing buying vs. leasing a car

Materials and Equipment: White board, marker, activity handout

Warm up questions: How many people do you know have cars? Do they buy or lease them? Do you know what factors to consider when debating between buying and leasing?

Lesson components + Activities

Beginning (~5 minutes):

- Warm up questions

- Definitions of buy / lease 

    - Buying: outright buying vs financing 

- What factors do we consider 

- Go over formulae

Middle (Activity) (~10 minutes):

- How to calculate the approximate total cost of either buying (outright or financing) or leasing including operating and maintaining the vehicle

- Students are formed in groups, each to work on one of the scenarios as assessment. It's learner inquiry based. We will help around and go over some examples. As this is a workplace course,  it's more relational learning for the students. 

End (Wrap up) (~5 minutes):

- Summary

(Think - Pair - Share):

- Advantages / disadvantages of buying and leasing

- How different circumstances affect this decision

Oct 11 Three Curricula All Schools Teach Reflection

For me the term curriculum initially meant a set of knowledge competencies/content to learn about. It encompasses the structured series of learning experiences or activities designed to enable students to achieve specific information and skills. Each grade has its own curriculum and it gets progressively more complex for the student as they traverse through the years. I like how the article addresses a lifelong lesson that should be understood as a young kid in school: that gratification and success is to be shared amongst each other and that it takes time to get them. It’s very difficult for an eager student to release all of their thoughts and answer all questions. If they take all of the opportunities, there won’t be any left for others. I remember back in math class in grade 6, a keen student sitting in front of me would raise his hand up and want to answer every question. He had that “oh pick me, please pick me” thought running through his head. At one point he got so excited he started standing on his chair with his hand raised! Of course the teacher told him to sit back down and that he was a bright student, but should let other students speak too (I still don’t know if he was showing off, or if he was like Hermione Granger who’s so eager to share). This way people like me also had opportunities to share my answers and it made me feel like “yes! I did/get it!”. Another interesting part that popped up was using some sort of a rewards/punishment system to reinforce/control behavior is present in all cultures. For the most part, students are habituated to satisfactions that are not a part of the learning process per se because a lot of emphasis is on the rewards/punishment structure. Grades is a common factor that gets tossed into the basket when it comes to student’s attitudes. Getting bonus marks for being nice or getting lots of points deducted from cheating are all responding to behaviors. Assessments, like tests, should reflect what the student understands about content material and apply competencies to different questions. So getting marks off for external factors isn’t a compatible way to remind students to stop doing what they're doing wrong. I believe speaking with the students on some issues and trying to see where they’re getting at is more efficient, than a stick hitting a bull. It is just the case that grades are so prevalent in students' school lives that changing them would make the students react more firmly, though often in a more panicking state. Now I can see that the curriculum is not solely based on knowledge/book smart skills, but also street smart skills, ways to communicate/interact/deal with difficult situations. It may also allow students to integrate teachings of ethics, values, and character development. We see clearly that the three core competencies for the BC curriculum are thinking, communicating, and personal/social. So it doesn't just focus on knowledge.

Oct 11 Microteaching Post Reflection

What I gleaned from these reflections was that I graded myself relatively more harshly than how my peers graded me. This shows that the lesson generally went well and was enjoyable. However I did give myself a 3 for organization, whereas most of my peers gave me a 2. I think it is because my pacing wasn't well prepared. I had a good structure to my slides and things I wanted to do. The problem was that I added some extra things to the slides because I thought they were interesting. Those slides contained more information that were harder to differentiate from each other. I brushed by them quickly since time was so limited. A thing I learned was that topics that are interesting to me may not be as interesting to my students, at least when they haven't had the time to taste samples of richness. Giving more time for the listeners to refresh and go over questions/earlier concepts would help reinforce learning. Again, 10 minutes wasn't a lot of time, but it was great practice. I say this because if I can keep my mind focused and my lessons short/succinct in a short period of time (not giving out too many details at first), then I am sure to be able to get through my lessons during the allotted time for a class period. My topic chosen on semantics was interesting. My peers were fascinated with how words and sentences relate to each other. They liked how they had to come up with some examples amongst themselves. The quiz at the end helped put what they learned into practice. For my next micro teaching I would definitely cut down a slide or two and spend some time reviewing and adding fewer examples (since I had too many initially). Providing a sheet with terms and definitions would help too.

Oct 10 Microteaching Topic/Lesson Plan

Topic: Introduction to semantics (a section from linguistics), on relations among words and involving sentences.

Lesson Plan

Big Ideas: Semantic relations, word and sentence relations.

Prescribed Learning Outcomes (PLOs) and Achievement Indicators: 

• describe the meaning of semantics in a linguistics sense

• explain different semantic relations among words (synonymy, antonymy, polysemy, homophony)

and involving sentences (paraphrasing, entailment, contradiction)

• identify which meaning to associate a word with based on the context 

Objectives: 

• understand semantic relations and apply them in regular readings

• analyze notions used in evaluating the meanings of words and sentences

Materials and Equipment Needed:

• tablet for presenting and to write on 

• paper to practice questions and for testing 

Lesson Components/Activities: 

Beginning (1.5 min)

• introduction to semantics topic and its applications is areas in philosophy/psychology

• check in on prior knowledge (where it be from grade school or postsecondary) 

Middle (6.5 min)

• go through definitions of relations and provide examples and discuss other examples with each other

• go through any caveats (eg. what constitutes to being a polysemy/homophony)

• address issues that may arise when using certain words in any form of communication

End (2 min)

• go over any questions then test/evaluate learning by asking questions on which relation is used

Oct 4 Battleground Schools Reflection

I was surprised that the math curriculum went under many radical changes in the 20th century alone. The layout between the two (conservative vs. progressive) approaches to view/teach math['s various areas of interests] is also enlightening as I am able to better understand them. The move from methods to deduce algorithms to a more non-standard problem solving/inquiry where students invent their own methods to solutions opens up more pathways to accessibility and reaching goals. It was unfortunate to know that many teachers who teach math (because they were assigned to teach it) are not comfortable with the topic themselves. The article mentions that they will use their own methods to avoid going too deep and maneuvering around hard lessons due to the troubles they once had. Resorting to memorization is a common practice for those who just want to get things over with, and the real crux/meat of the matter of asking “why” is ignored. Math stereotypes arise to math phobia where only a selected elite group is capable of navigating through these hard waters. For me, I want to break that belief because I wasn't a strong math student growing up. It was one of my better subjects but I had troubles with understanding at times too. Eddie Woo, who was last place in math when he was a student, is now a mathtuber and teacher. His Ted Talk video on "Mathematics is the sense you never knew you had" opens up more room for the general population to be fluent/literate in math. This demonstrates that a student probably hasn't encountered an idea/situation/scenario that kindled some sort of interest that made them see why what they’re learning works/makes sense/is useful/is valuable. It never occurred to me that the high levels of competition in the American education system was probably due to global tensions during escalating times (such as the Cold War). The anxieties of a lack of talented intellectuals forced school systems to get more involved in K-12 students learning. I think this quickly filtered out the strong and the weak students at the time since the focus was more on flushing information down student’s heads rather than aiding/nurturing students. This again created a barrier for students who needed more time with understanding math from fully enjoying the richness it may provide. So timing is what I should work on as well, allowing each student the time to see where they may have troubles and go through fuzzy concepts. Creating an environment that isn’t so competitive will also be helpful, as students are able to focus on helping each other rather than competing against each other.

Oct 4 My Teaching Perspectives Inventory Results/Reflection

I wasn't surprised that my highest result was transmission. I value that the teacher of a particular subject should be knowledgeable in the field. Not only that, they should also be able to help students in any stage of learning to reach proficiency and maybe mastery. In my opinion, a specialist is most suitable to teach a specific subject area, albeit it doesn't hurt for a teacher new to the topic to learn ahead of time to get familiar with it. Having knowledge in the subject will allow the students to be comfortable asking tough questions. I know one case where when I was in grade 12, a grade 10 student asked my math teacher about Taylor series/expansion and he wasn't able to answer. It is not the case that all students will be this aspired, but it is always good preparation to be able to give a response. The TPI results also reflect my belief that attitude is important in a student's learning. Believing that you're not good at something will hinder your learning at the very beginning. If no effort or efficacy is present, then there is no assurance that those who do struggle will stop struggling. I am surprised that my apprenticeship total is lower, but it is my responsibility to know each students' zone of proximal development, so I may aid them in closing any learning gaps. Something I may want to work on is to address social issues that can be modeled with math and its realistic applications. This perspective is my lowest one and is recessive. I notice that some of my peers are also in the same boat as me. I wonder in what ways can teachers make math more interactive in real world experiences/topics? All in all I am glad to be able to come across several readings and hands on activities in these courses that allows me to better engage with these notions.

Oct 4 Math Art Project Follow-up Reflection

 

    My group decided to analyze and extend upon the artwork "Between the Devil and the Deep Blue Sea" by Gauthier Cerf, a series of hexagons embedded within each other, where their side lengths correspond to the Fibonacci numbers. Different tasks were broken down and we discussed ways in which such an artwork may be used as an active way to engage students in understanding how the Fibonacci numbers are related to each other through its recursive definition. We extended the Fibonacci pattern using the square, that generated the golden spiral (which is seen in many natural objects) and created a colourful aesthetically pleasing butterfly (credits to Esther). Linking patterns to other mathematical ideas visually allows students to see how some fomulae came to be/makes more sense, such as explaining an infinite geometric series with equilateral triangles.

    This project was initially very entertaining to do because I am intrigued by how Fibonacci numbers are implemented in beautiful artworks. The work distribution was also fair and figuring out who does what wasn't difficult. I loved the analysis by the artist on their artwork and I had to think deep about how it was constructed. Taking the time to reconstruct the artwork was also a fun process, since I love building big things from tiny things ground up. In this case it was the construction of many hexagons with two hexagon blocks. However in hindsight, one hexagon block was sufficient. Tracing objections, joining line segments with a straight edge, and cutting pieces are all activities that I don't do regularly. I also spent a lot of time preparing for my math extension part with the Golden ratio. I thought it was a good way for, say, older high school students to review/utilize some math topics, to help them use what they learned like fractions/ratios, pattern recognition, roots of quadratics, system of linear equations, and limits. Unfortunately, I got to talk about none of that due to time constraints which was frustrating as I put a fair amount of work into it.

    There was a lot of advice I took in from presenting this project. For example, preparing more materials in advance just in case some papers weren't cut to be big enough. It was fortunate to have the art supply room right next to our door, but that won't always be the case for all schools. Giving clear instructions on creating an artwork is crucial too. Many students will have a hard time following the steps if they just hear "this" and "that". It was mostly a stress issue from time constraints, but it was also my first time giving a formal presentation in front of a classroom for over four years. I have been tutoring and TA'ing, but those were less formal. I came into this program to improve my public speaking skills as well, and having some critiques during my UBC-presentations humbles me as I'm able to learn and improve for future presentations/practicum teachings. Teachers should also be taking advice and growing via feedback. For this artwork, I may construct it in front of the class (via screen projection) to let the students see how it relates to the Fibonacci sequence, but I probably will just let them try it out on their own time at home, since time is so limited and there is a rigorous curriculum to go through. I will mostly connect the Fibonacci numbers with the Golden ratio with the set of notes I have written. Visual proofs are very intuitive for students, and I use a lot of geometry to prove certain trigonometric properties, like why exactly is sin(30°) = 1/2. Physics is one direct way to apply math, and in a grade 12 calculus class on the integration topic, teachers can show students how their science 10 kinematics formulae are derived (from constant acceleration) using integration.

Sept 25 The Dishes Problem

The number of guests must be divisible by 2, 3, 4. You take the number of rice dishes (r) and double it and that gives you the total number of guests. You also get the total number of guests if you triple the number of broth dishes (b) or quadruple the number of meat dishes (m). So 2r=3b=4m. There is also the information that the sum of the dishes is 65, so r+b+m = 65. If we relate the broth and meat to rice, we get b = 2r/3 and m = r/2. Substituting those into the r+b+m expression, we get r + 2r/3 + r/2 = 65 = r(1+ 2/3 + 1/2) = r(6+4+3)/6 = 13r/6 ⇒r = 65(6/13) = 5(6) = 30 ⇒ total number of guests = 2r = 2(30) = 60.

Instead of using latin variables to set up an equation, one can use pictorial diagrams of the food dish that is being described in the question. One may also use words to describe the situation instead of using written equations. However there is an even better method to solve this problem without a system of equations that involves finding the total amount of dishes directly. If we know that 2r = 3b = 4m = n, where n is the total number of dishes, we can say n/2 + n/3 + n/4 = 65 = n(1/2 + 1/3 + 1/4) = n(6+4+3)/12 = 13n/12 ⇒ n = 65(12/13) = 5(12) = 60. This method is much more efficient. 

I believe using examples of puzzles from other cultures will spring interests into the students, so that they are working with something new. They may also feel better to see that mathematicians from their cultures were able to come up with problems as advanced as those studied here. Imagery is helpful for students to better visualize the problem and is more interactive. They're more familiar with whatever interesting character is introduced in the problem, for example a physics question involving R2-D2 catching Luke Skywalker's lightsaber. 

Sept 20 Letters from Future Students

Dear Mr. Zhang, 

I hope you're doing well. It has been ten years since you taught me high school math. I was happy to see you again at the restaurant and greet you briefly. However, I didn't get the chance to have a full conversation with you. Your math teachings were very helpful for me, since they prepared me well for the difficulties in university. Your rigor and proofs were genuine and didn't take too much time. They were intuitive and made me understand the topics on a deeper level. I loved your reminders that university does things a little differently. My friends from other classes were so shocked, but I wasn't! Your lessons were stepping stones to enter post secondary. Some topics are hard to grasp initially, but you broke it down very nicely. Thank you for your explanations!. . . 

Dear Mr. Zhang, 

I was happy to see you at the teachers conference. I'm teaching English/history right now. It was nice having a small chat with you regarding some of the pedagogical differences we have regarding math. To start, I'm not strong with math and not fluent. Sometimes your proofs are too long to follow, and I couldn't keep up. Your lessons on mathematical/symbolic logic was also not necessary, though it may be interesting to some. People like me are more interested in how and in which situations to use certain formulae. The tangents you went into about some cautions to take for the future weren't beneficial as I didn't take math afterwards. . . 

I'm hopeful that I can explain topics on a deeper level but I hope to not digress too far and focus on the main topics. The balance on how much rigor I should implement is something I have to keep in mind as well. Sometimes I may not have to introduce some concepts since it may not be useful.

Sept 20 Lockhart Article Response

I agree with Lockhart in that math taught in schools right now focuses too much on rote procedures and facts, an algorithm of sorts. Student's don't ask "why" or even care to ask the underlying principles behind such compacted formulae. A lot of it comes from very beautiful proofs that teachers don't bother spending time on, hence diminishing possible opportunities that gives rise to some intuition for the students. However I do think rigor is a necessary component in mathematics as it is the foundation block that builds up other fields around us. Treating it too much like an art is too flowery. While teacher enthusiasm and creativity is important, one can incorporate context to topics being taught, engaging in fun discussions while maintaining the rigor so that we know math is math and music is music. We can allow math and art to rhyme, but there is still a clear distinction between them. Compared with Skempt (who was relatively more neutral on this issue), Lockhart greatly emphasizes on the need for a more open, curiosity-driven approach, where students and teachers alike can experience the joy of mathematical discovery. Skempt gives an outline of relational vs. instrumental approaches to teaching. I believe Lockhart greatly emphasizes on the relational aspect, though not in the context of rigor but in creativity and personal exploration.

Sept 18 Favourite and least favourite math teachers

My favourite math teacher is my grade 12 math teacher. He was very jolly and had a huge sense of responsibility. Being the head of athletics as well as teaching a computer science class, juggling between all of these tasks and maintaining a good teaching attitude made many people look up to him. He explained the math problems using different colours on the smartboard, so it captivates people's attention. Telling stories as well casually in class and jokingly says "thank you, I was just testing you" whenever someone points out his mistake. When things did get out of hand he would tell us to just do what we want, sit down, and we would realize we misbehaved over the limit. He comes back up and explains he genuinely cares for our learning, but if we don't give him the chance to speak/present, then we might as well do it ourselves - which isn't feasible. But this rarely happened and the class was in good shape most of the time. So he was a very reasonable person overall, caring for the well-being and the learning engagement of his students, which inspired me to continue in his work. My least favourite math teacher was a math professor. For many profs, they are very smart in their own eyes because they have had many years of training and accumulated lots of knowledge. He would speak demeaningly to students. When we struggled, he wouldn't really explain the concept to us, but just tells us to refer to the textbook he wrote. Overall he wasn't helpful and didn't build strong relationships with the students. This is a red flag for me since having good communication with students is a good start for further engagement and exploration for them.

Sept 13 chalk board photo

 

Sept 18 The Locker Problem (Updated Nov 6)

This problem was very intriguing, as I never really enjoyed these sorts of math puzzle problems. But after four years of doing math in university and taking many discrete math courses, this problem can't be harder than my math assignments right? To begin, I initially wanted to shorten this problem from 1000 to 10 and brute force my way through the 10 students. However brute forcing by using a smaller sample and then later extending it to a larger sample misses the point of being able to solve this problem without any brute force. So I thought about it a bit more and came up with a solution. Initially every locker is open. For each locker number n ranging from 1 to 1000, the question is how many times the state of each locker is changed after all 1000 students had their turn. That turns out to be the number of positive factors n has. For example, for locker 12, the students who will affect it will be student 1, 2, 3, 4, 6, and 12. So that will be 6 changes. Since 6 is even, the locker will in the end be open, since the doors alternate from open and shut, and here we initially started at open. Students with numbers greater than 12 will not affect locker 12 as they'll start off with locker numbers bigger than 12. So this method can be done with every number n from 1 to 1000. If the number of positive factors n has is even, locker n will be open in the end, and if odd, locker n will be shut in the end. I could go in more detail on how to find the number of positive factors n has, but not going into too much detail right now (the explanation requires prime factorization and the fundamental counting principle). In the end, the lockers with numbers that are perfect squares will be closed as when you write out the factors, the square root of that perfect square will be repeated twice. However that counts as one factor, which means in the end a perfect square will have an odd number of factors. Thus it will be closed. For example, the number 36 has factors 1,2, 3, 4, 6 (6 only counted once), 9,12,18,36. 36 has 9 (odd) distinct factors so it will be closed in the end.

Also interesting to note: my SA does math riddles at the start of her lessons, and on my second day of practicum she showed the class this riddle from Ted Talk.

Sept 11 Relational Understanding and Instrumental Understanding

    The words faux amis really resonated with me. Mathematics uses a lot of English words for topics that are nowhere close to the main definitions. When I took courses in Ring theory, in my head I always pictured actual golden rings. Whenever fields and ideals are mentioned, I picture a green flat field and a perfect suitable concept. Radicals too aren’t advocating complete political/social change. They’re just surds. I believe, however, there’s a reason for those words to be chosen. Another mention that stood out to me was how rigorous a student should understand what they’re learning. Personally I don’t enjoy just throwing out formulae. I always try to give a proof so that the students may see how the derivation utilizes previous/current/new (but not out of their scope) knowledge. Stating A = L×W [for a rectangle] without explaining the definition of area (how much two dimensional space is taken up by the object) won’t help students apply the concept of area to other non-rectangular shapes like circles, triangles, or area under graphs. Finally, I collect from the latter parts of the article how much rigor should be taught. It's important to have a balance between instrumental and relational learning. Personally I encourage students to ask "why" and have an intuition of what they're learning, without just glimpsing over statements taken for granted.

    I did enjoy reading this article since it addresses a problem I talked about with a smart junior back in high school. How much emphasis is needed on rigor in the curriculum? Where is that balance between proving statements and throwing them out at the students? Skemp gives pros and cons to both instrumental and relational learning. At times students don't necessarily understand the real meaning behind a concept/formula. Other times students will try to focus too much attention on the background derivation than the equation's formal usage in everyday life. I lean more towards explaining topics on a deeper level, proving some things, but not everything. At times a lot of the background knowledge comes from what is already taught in the current unit, so it is just another form of applying/extending. Skemp's goal is for the students to problem solve on their own, even when somewhere in the middle of the process things don't go their way. That was how I was taught and I hope to present myself in a similar way.