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.