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Being in the dual degree program I was excited about this course for some time. I was finally going to learn all the magical curriculum and nitty gritty teacher things that everyone else seemed to know. However this is not what happened and I am so glad I was wrong. 

Throughout this course we were able to learn so much more than the BC ministries requirements. We were able to explore different ways a math class could look like and how to incorporate different elements into a math class. Its very reassuring to know that our future math classes do not have to be boring, sad places but rather fun places with activities and diverse styles of learning. I particularly enjoyed learning about how to incorporate other disciplines (art and dance for example) into math and as a way to learn math and demonstrate mathematical learning. 

Another interesting and unexpected element were the weekly problems. Although they were something I occasionally struggled with, it was refreshing to see the creativity and diversity of other people's solutions.

Overall, this course was a delightful way to expand my idea of what a math class could be and I am excited to see what will come next.

WURZELSCHNECKE

Being able to learn about the wurzelschnecke was really fun and exciting. It is a good way of showcasing how beautiful math can be. I particularly liked the idea of having the wurzelschnecke learning on different scales. The individual scale allowed me to have some time to get familiar with the idea and some experience with it one-on-one. Moving into smaller groups, I got to learn about some of the observations others had made while also getting some 3D experience. The large scale was very fun because we got to interact together as a class and collaborate in order to get all the pieces where we needed them. This was of particular interest because math is usually so isolating and consists of still work.

Off The Grid

This first stop I had was related to applying mathematics concepts, particularly grids, to real life situations. Although grids may appear to be a logic and systematic way in theory, they may not be the best fit for what we are trying to do. Especially with real life situations where the landscape can be challenging, it does not make sense to force the environment to change in order to adapt to the math. When in the classroom, I would not change the questions so that I could use my preferred method/concept. I would use the method best suited to the problem in order to solve it efficiently. 

This relates to the other stop I had which was in terms of scale. Just because an idea works really well on a small scale does not mean that it will work equally well at a larger scale. For example, if a method resonates well for one student, it might not have the same effect for the entire class. We need to be flexible as educators and able to adapt to the situations we are presented. Our unique landscape in Canada, varying from province to province and region to region should be the main factor in deciding how to live within that space. 

The last idea from the article I enjoyed was the notion of going back to old ideas rather than using technology to incorporate activities and bring life back into learning. Having this implies that our timeline is cyclic rather than linear. Being able to bring back older ideas rather than using new age technology to solve our problems could prove to be much more effective. It also opens up our education to more diverse cultural practices. Students will get a broader sense of what the world is and can be rather than a narrow view focused on one culture's perspective.

SNAP Math Fair Reflection

During our class on Monday Dec. 3 we had a visit from a grade 10 math class from Richmond. They shared their math projects done on a variety of subjects. I was very impressed by the some of the conflicting stories they created and enjoyed visiting the different projects and talking to the students about their processes. Being able to animate math into something fun and exciting rather than just stale questions is definitely a skill. The SNAP math fair really advocates for accessibility. Students were able to express their learning in a variety of ways (pictorial, orally, written, kinesthetic) while also being able to showcase their talents and interests. It allowed students to personalize these math concepts in ways that made sense to them.

One surprise I had was during a conversation with one of the students who expressed that they preferred word problems to rote exercise because it gave context to the problem and offered a reason to do the math (we want to figure something out rather than just practicing techniques). This is the complete opposite to what I normally hear students say. What I found interesting is that many of the problems presented were unrealistic, typical math word problems that would never actually happen in real life. I wonder if the SNAP math fairs could move towards students solving problems that they would actually have.

Unit Plan Draft - Take 2

Here is the link to my second draft unit plan.

10 Rats, 1000 Bottles of Wine

Assumptions:
1. Rats can and will drink a lot
2. This particular poison is also lethal to rats
3. Rats can be used more than once if needed and still alive
4. The poison is very strong and will not dilute when mixed with non-poisonous liquids

Phase 1: Picking the Best Half of Bottles
Pick 500 bottles to open and pour a tiny amount of each into a small vessel.
Allow the rat to drink from this and wait for it to possibly die (sorry rats :( ).


Phase 2: Narrowing down the Bottles
If the rat died:
    -One of the 500 bottles is poisoned so the other 500 can chill
    -Take 250 from the original 500 and mix it into a different small vessel
    -Allow a different rat to drink from this and wait for it to possibly die

If the rat did not die:
    -The original 500 are safe for human consumption and one of the other is poisoned
    -Open 250 of the unopened bottles, mix and let another rat taste test
    -While waiting for rat to possibly die, allow servants to begin decantering the wine so that it will be ready for when the guests arrive (serve snacks that are thirsty quenching so that guests will not be too thirsty hence buying some time just in case, this wine is the better of the wines so guests can savor it while still sober).

Phase 3: Repeat
Keep this halving process until left with a choice of 2 bottles and thus can determine which is poisoned as shown below.

Phase 4: Safely discard the poison and enjoy the party

Method:

After discussing with others in the class, it was determined that mixing wines was allowed and possibly necessary to figuring out this problem. Being able to mix wines helps to figure out which bottles are safe since you can eliminate many bottles at once. Hopefully the rats metabolize the poison quickly so that it does not take extremely long to test. Testing should not take too long if you can avoid the poison until the end. This process will become problematic if you choose the poisoned bottle in every mixture, which will also mean all the rats will die. If you are lucky and avoid the poison until the end, only 1 rat has to die. Being able to combine quantities and then ruling out entire groups systematically rather than testing one by one is a much more efficient way of sorting and determining something. In this case, it also saves the lives of at least 9990 rats. 

Image Source: https://shawetcanada.files.wordpress.com/2017/06/ratatouille9.jpg?quality=80&strip=all&w=605

Unit Plan Draft

Here is a link to my draft unit plan (lessons included).

Also, a picture for your entertainment.

Image Source: https://encrypted-tbn0.gstatic.com/images?q=tbn%3AANd9GcQXrWYA0ZSgXLK_Gm2_00AhN-U-Gm-C6PQJkQoHgFxR5ms_PU11

Un-fair Math Fair at West Point Grey

This was a very fun and exciting experience. I was extremely impressed with the projects the students developed. Many of them were ready to share about their calculations and how they determined if their game was fair/unfair. It was refreshing to see the young students excited about math and eager to tell others about the projects they've done. While talking to some the students, they expressed how this project was much more enjoyable than taking a test and I think it was a wonderful way to introduce the students to the subject and apply their knowledge to a real life situation where they were able to work with others and communicate with people outside their class.

I am curious to see how they will discuss their findings. While walking around I noticed that many of the kids who were playing the games would listen to the explanations of the students and then choose the winning combinations which could skew the results when calculating the experimental probability. 

Math Textbooks

How you respond to the examples given here -- as a teacher and as a former studentThe first example of modeling a person's height given their leg length gives a real world application of when linear relations are useful. Unfortunately the book does not mention anything about how this relation was discovered, why it is important, who would need to know this information etc. As a student this was a common theme in my math classes. Although it is nice to see math being used for a purpose, the purpose needs to be clear and appeal to the student's interest and get them thinking rather than just having an application for the sake of applying math. By getting the students involved in the application process would also help solidify the idea for them. If they had a choice in what they could explore, discuss with their peers what types of data they would need and determine how the data is related to each other would be better use of their time and application. Having a more natural setting rather than a forced problem that is set up at the end is much more meaningful. As a teacher, I would rather implement "word problems" as projects that students can fully experience for themselves rather than just doing the same calculations as before but with words attached.

What are your thoughts about the reasons for using/ not using textbooks, and the changing role of math textbooks in schools?
I am not entirely convinced by the exclusive/inclusive imperatives suggested by Rotman. Although the authors can write a textbook however they see fit, teachers have some autonomy in how they present the material in the textbook to the students. For example, the teacher can make exclusive imperatives inclusive by having students work in groups to calculate, copy and write things down. Having multiple students perform a calculation in order to see if they both arrive at the same answer, compare the processes they took to arrive at the answer and then discuss what their answer means with each other. Furthermore, if students are working in groups they can each work independently to come up with a way to represent their understanding of a topic and then share their representation with others. 

Another confusing part of the article for me was near the end when the author mentioned the way mathematics textbooks function in school settings. I don't believe that depersonalizing mathematics for communication is the most effective way of communicating the content. Letting the students have a personal connection to mathematics would allow them to become interested and want to learn about it and communicate math with others. I'm not sure if a textbook will inspire students to learn more about math but it can help make the communication and example creating process easier for teachers. 

Also, textbooks seem to take out the rich culture and history behind the math making it seem like a very dry and boring subject that is purely about calculating answers when is much more than that. During word problems, some textbooks use names from different cultures and include women which is a good start but it is very surface level and does not offer substance to learning.

The Scales Problem

What are the values of the four weights?
At first, I thought maybe the weights would be in base two since computers use base two to describe all numbers. However, after doing some trial and error it did not seem possible to get 4 numbers to sum all numbers 1 to 40.

I then switched to equations by letting each number be a variable a, b , c, d. Assuming that the weights could be added to both sides of the scale and that one of the weights had to be 1 gram, I proceeded to try and solve the equations. After doing a few equations for 40 - 34, I noticed either we needed one of the weights to be 3 grams or the difference between two of the weights had to be 3. Since it was easier to assign one of the variables to 3, I decided to continue my calculations with two variables and two known values. Eventually, I got to a point where the two other variable needed to sum to 36.

I began to narrow down the possibilities by trying to find values that are not possible for each pair that sum to 36. One pair that I struggled to find any discrepancies was (9, 27). I confirm that this was a solution, I found representations for each value 1 to 40 using 1, 3, 9 and 27 which ended up working.

Are there several correct solutions?
I'm not sure if there are several correct solutions but I did notice that this solution had values that were all powers of 3.

How could you extend this puzzle to help your students understand the mathematics more deeply?
We could use an actual balance scale to physically see what is happening and how the scale would balance. We could also try to see if we could do this same exercise except with more weights for a larger range (1-100 grams perhaps?) and see if we could use powers of 3 or if the 40 grams was just a special case. We could see if there are other solutions if we allow the number of weights to increase and then discuss why merchants would want as few weights as possible.

Cleaned up version of my rough work:

Pro-D Day - CUEBC

I attended the CUEBC (Computer Using Educators of BC) pro-d day event and it was very interesting. I was always very curious as to what the teachers did during pro-d days and was excited to find out. This event was a little overwhelming because there were so many different sessions to choose from and it was difficult to pick just one. I ended up learning about different apps that Microsoft offers for teachers and classrooms such as FlipGrid, OneNote, Skype in the Classroom and premade STEM lesson plans. Although this conference was designed for all teachers, it was interesting to see what technology is available for classroom use and it got me thinking on how I could potentially use and adapt some of these technologies to fit a high school math classroom. I also attended a session that described a special education version of the game Minecraft that could be used in a class that could be adapted depending on the subject and grade level. For the last session, I attended a session on self-regulation. This was a very interesting session but I found it difficult to see how to transfer many of the suggestions offered to a math class. The presenter was a business teacher and that environment allowed for more self paced learning where students can more easily determine there plan and pace for the class than a math class were there is more pressure to finish content on time and to be more or less in sync with the other math classes. However, the online exit slips and self evaluation forms could be useful if the students all have access to technology because it will force them to reflect and answer the questions rather than loosing the paper version and skipping the long answer/reflection portions.

Elliot Eisner and The Three Curricula

Two or three 'stops' you have in this article

1. "One of the first things a student learns - and the lesson is taught throughout his or her school career - is to provide the teacher with what the teacher wants or expects."
This quote reminded me of the reading about grades for our inquiry class. By telling students what is going to be on the test, we are conditioning them to pay attention to these topics and revisit them later when studying. This type of school culture is persistent throughout an entire lifetime. Children try to find out what they parents want and expect, they then try to give their teachers what they are looking for and often ask other students or former students such as older siblings what that particular teacher's tests are like and what they need to do in order to get a certain grade. As the reading points out, students spend an awful lot of their lives in school exposed to this culture and it continues throughout their lives. Whether they go on to post secondary and try to do the assignments so as to please the professor or if they go into the workforce and do whatever is necessary to please their boss and maintain their employment. Students are stuck in this very survivalist culture of trying to make it to the next course with a particular grade, or trying to get into university.

2. Implicit Curriculum
One thing that really stood out to me was the message that is conveyed immediately to the students by the physical set up of the classroom. If the desks are separate and in neat rows, the students know that it is an individualized learning environment with little opportunities to talk with others. On the other hand, if the desks are grouped together, students know to expect more group work and collaboration. This was less surprising to me than how the scheduling of classes affects how the subject is seen by a student. While I was in high school, our blocks were rotated throughout the year so that each subject was experienced at different times of the days. It was different in elementary school where art was typically reserved for afternoons. Many times while I volunteered at a kindergarten class the students were working on art or play in the afternoons while they had reading and math lessons in the mornings. This type of scheduling definitely contributed to the status of art and reading in the classroom. When a lesson was not finished on time, the math class or reading class would carry into the art class cutting it short. Certain students were unable to begin their artwork until after they completed their math/reading assignments. However, if an art activity was not finished it was seen as acceptable to just leave it for another day or the student could finish if they were able to complete the math/reading activity early. I was surprised that none of the students questioned why the math/reading was prioritized over art. Sure they were disappointed and complained but they never argued that art was as important as math/reading. It is amazing how early these subtle messages take affect.

3. Null Curriculum
As a high school student I always wondered how the duration of the subjects was determined (e.g why we needed 5 years of ELA, 4 years of math, and no mandatory art/music/dance/film studies etc.). What was so special about ELA, math and science and so dispensable about art and music?Having this conveys the message that ELA and math and science are superior subjects to art and music. Only students who are interested in pursuing art or music will immerse themselves in those subjects or enroll in special programs so as to get into a post secondary program for these areas. However, this implies that students other students will not benefit from having these types of experiences or classes throughout their schooling. Likewise, students entering art or music programs will see no need for math and science and determine it as a hurdle to jump over rather than something that could assist their studies and improve their learning. 

Another aspect of the null curriculum that struck me was the omission of certain topics within a subject. For example, conics was a section that was taken out of the curriculum while I was in high school yet it was something I encountered during my university studies. Why was this topic removed and what was put in its place (if anything)? What about conics did the curriculum decision makers feel was so unimportant that it was removed entirely?

Ways that this might expand our ideas about what is meant by 'curriculum'. How does the mandated BC Provincial Curriculum connect with Eisner's ideas?

Before this reading I equated curriculum with the explicit curriculum but I realize there is much more to curriculum then simply a list of requirements. Curriculum also encompasses what is left out, how the material is presented and how the environment is set up for the students among other things.

BC is helping to connect with Eisner's ideas by providing more opportunities for other types processes for students such as increasing collaboration, increasing ways of knowing such as Indigenous ways of knowing, using different modes of instruction, different modes of assessing. 
It also focuses on decreasing the null curriculum by adding financial literacy to the math curriculum. Students will leave high school with some basic understanding of finance such as knowing how they will be taxed and how to calculate their hours when completing a time sheet.

Curricular Micro Teaching Reflection

The lesson went better than expected. It was a good experience because we had a little set back and had to do some last minute adjustments. We had planned a fun, hands-on activity that would require the students to assemble a 3D object from a net so that they could see how the net is connected to the 3D object. Unfortunately the nets were left behind and we were unable to bring this activity to the students. We had to adjust our plans before going into the presentation by having the students draw out the nets by looking at the 3D object. In addition, we had planned an extra activity for showcasing different nets of the same object and showing what a correct net looks like compared to an incorrect net. This activity was planned to be supplemental in case we had extra time left over after our planned lesson. Although this activity worked out decently well I think letting the students physically hold and try to fold a net would be a better way for them to experience and understand how the nets are related to the object and why some nets do not work. It was a little difficult to figure out how loud to be during the lesson because of the other lesson going on, however I noticed that while teaching the lesson I completely forgot about the other group presenting and was entirely focused on communicating to my "students" and paying attention to my team mates so that we could effectively deliver our lesson and segue well so that the lesson felt cohesive. My main takeaway from this lesson was the importance of preparation. It is a good idea to have a plan for the lesson, have activities/experiences lined up, have extensions in case the lesson goes faster than anticipated and to be prepared to change the plan in case something happens.

Group Micro Teaching Feedback

Curricular Micro Teaching

Geometric Puzzle

What process did you use to work on and solve this puzzle?
This puzzle came a lot more naturally to me than the other before. Since there are an even number of points, each point would line up with another on the other side. Knowing this, I found the angle each segment would be and found that from the first point (1) to 7 is 84 degrees. Since we want the point exactly opposite, this point would be 180 degrees from 7. Doing some simple math I found the answer to be 22. I drew a picture to confirm my answer but it was not executed well (my compass broke) and it was difficult to follow the lines to their counterparts. If the circle had been perfectly drawn, it would have been easier to see. Also, if the number of points was smaller, say 6, one could easily find the diametric by simply tracing the lines back to the number.

Could you create other extended puzzles related to this one -- some possible, some impossible?  (Is there any value to giving your students impossible puzzles?)
This problem could easily be turned impossible by making the number of equally spaced points an odd number. Along the lines of geometry, you could present some ancient problems to students such as cube duplication and angle trisection. After students have struggled for a bit, you can tell them how there is no answer and people have worked for years and still have not been able to find a solution. I think there is some value in this because students become discouraged when they cannot figure out the answer. Learning that many mathematicians were unable to find solutions to these problems could be reassuring that just because an answer cannot be found, does not mean you cannot do math. Furthermore, I would bring up the Millennium Prize Problems and other unsolved problems in mathematics. Having some problems that are possible will also be worth exploring in class. In terms of geometry, I would try to bring in some other cultures geometric problems to get students practicing math while also learning some history. For example, bringing problems about volume of irregular shapes, approximations of pi and the Pythagorean Theorem from Ancient Babylonia, Ancient China and Ancient Indian.

What makes a puzzle truly geometric rather than simply logical?
This problem is purely geometrical because one does not need to do any computation. Knowing the properties of a circle and being able to see is a way to solve it. Once the diagram is set up, all one has to do is trace the line from 7 through the center and to the other side which will end up at 22. Problems were students do not need to compute and set up equations and do algebra to solve a problem, helps make it geometric. Although the problems could be solved logically, that is not the only way and typically the geometric way is easier.

Battleground Schools

1. Progressive Reform (1910 - 1940)
I was very surprised to learn that experimentation and inquiry based learning in math is not a new idea but rather a very old one. The idea was brought about by questioning and analyzing the education system and trying to improve it, similar to what has happened in the last few years in BC. This made me wonder how much the curriculum will change throughout my career and the impact I can have on its changing. If there are serious issues students are having with the curriculum, it should be revised in order to help the students. I worry that we will repeat the mistakes of the pass by not implementing the new curriculum and new ideas in all schools because teachers are more comfortable with their older systems and fixed viewpoints.

2. Attitudes toward math are contagious
Reading this part of the article I was reminded of how important a teacher's disposition is to the students. If a teacher does not like their subject or area in the curriculum it will have a negative impact on the students who might already have negative feelings towards that subject. Teachers should also be careful with how they discuss other subjects with their students. If students see teachers talking bad about other subjects, how can we expect them to come into that classroom with an open mind? I also think that math in particular has a bad reputation. Many students feel that math is something you are good at or you're not. Because of how math has been portrayed in the media, past experiences and friends/family experiences, it is almost cool to dislike math more that other subjects such as music or English.

3. The impact a test (TIMSS) had on the math wars in the US
I understand that a country would feel some disappointment in scoring low compared to other countries, however we must inquire why the scoring is what it is. Although it is natural to want to copy what the top ranked country is doing for their education system and pedagogical approaches, it is not that simple. The test does not account for the class size, which schools were selected to participate in the test, cultural differences in the attitudes towards schools, classroom demographics, and even the mental states of the students while they were taking the test are just a few things to consider when arguing for reformation. Yes we can learn a lot from other countries and the techniques they use in their classrooms, but we cannot simply copy them and expect the same result.

The Dishes Problem

Solution:
The dishes problem was a bit easier than I initially thought it would be. After rereading the problem and taking a few moments to think about it, I came to the conclusion that the number of guests must be a common multiple of 2, 3 and 4. I then started listing out the multiples for each and made another list containing the common multiples.

Common Multiples: 12, 24, 36, 48, 60

Using this list, I added the divisors to see if they would add up to 65. After a few tries, I found that 60 was the magic number.

(60/2) + (60/3) + (60/4) = 30 + 20 + 15 = 65

Therefore, there were 60 guests at this dinner.

Whether it makes a difference to our students to offer examples, puzzles and histories of mathematics from diverse cultures (or from 'their' cultures!)
I think the selection of puzzles and histories does make a difference to our students. By showing students that math was done by all kinds of civilizations during all time periods and all cultures will help to broaden their minds that math is not just for old white men. I do not want to discredit what these men discovered, but I would like the students to see that they were not the only ones coming up with amazing math. For me, I always became much more interested in a topic/subject when I discovered that there were ties to my culture. I felt like I could relate to it better and I had a deep sense of pride knowing the people of my culture were able to do such wonderful things. At the same time, I was liked to learn from diverse cultures too. Many times I was surprised at how similar they are to mine at time while also learning about the differences. 

Whether the word problem/ puzzle story matters or makes a difference to our enjoyment of solving it?
The context/story make a large difference to me when solving a problem. One of the issues I had with this problem was the context. It made absolutely no sense to me how the official and the cook did not know the number of guests. Did the official not sign off on the guest list? How was the cook able to prepare enough food without knowing the number of people eating? I also wanted to know what type of gathering this was. Was it a celebration, and if so what were the people celebrating? What was life like in 4th century CE China? Knowing more about the story of the problem would have made it more enjoyable for me because I would be able to think of each of the 60 people in this problem as a real person who lived in China during the 4th century rather than just an answer to puzzle.

Micro Teaching Reflection

The planning process was interesting for me since I have been playing this game since I have been playing it for so long with other people who know the game well. I had to really stop and think about how to break down the rules and points system as well as explain the card values and suits to people who were not familiar with Italian card deck. It was difficult to estimate times for these explanations since I wasn't sure what people would find more confusing.

During the lesson I realized I should have left my lesson plan out to reference during the lesson. I was so preoccupied by making sure my group knew the values of the cards and would get a chance to play I completely forgot to state that the objective was for them. Taking that few seconds in the beginning to explain what I was going to present and what they should be able to by the end of the lesson would have been beneficial to my group so that they could know what to pay attention to during the lesson. I also found it difficult to juggle my attention between explaining the object of the game and how to play with how much time was left in class. I wanted to give as much time as possible to playing the game so I skipped giving multiple examples to give the group a chance to play a game with everyone's cards exposed so that they could have more time to play. In hindsight, doing more examples might have been better and made the game play a little easier for them. I also should have written out the rules and points for the group to reference while playing to make it easier for them to do strategic plays.

Micro Teaching Lesson Plan

Micro Teaching Topic

For my topic, I will be teaching the Italian card game scopa.

Wordy Puzzle

Word problems have always been a bit confusing for me and this one took me a while to think about. Since the words were confusing, I drew a picture to try to understand what is happening. 

Who is speaking?

I am speaking.

Who is "that man"?

That man is my son.

What makes this difficult/interesting?

This was difficult for me because there were no numbers. It was all words and we do not know who is speaking. The play on English words and usage of father adds to the confusion. It is interesting because one has to use logic and problem solving in a different way to figure out who is speaking and to determine that "that man's father" = "my father's son".

Math Art Reflection

After browsing the works on Bridges website, our group really liked Yvette's piece the best. It was visually interesting and also had mathematical concepts involved. I think my group did an amazing job by bringing more math into art by having set rules and systems (some with math history elements as well). It will be a good way to get students who are strong in art or learn well by creating to get a better understanding of some concepts. Furthermore, it will help all students to visualize what is happening with irrational/rational numbers and visually see the difference. 

Although we did not get to do this for our presentation, we had discussed using different textures to represent a digit or even incorporating braille in order to make this activity accessible for students who have visual impairments. It is definitely worth while to use a project like this in the classroom to get students to work together. 

The Locker Problem

After reading this problem, I tried to visualize it. Since it is not reasonable for me to visualize 1000 lockers at once, I decided to start by looking at only the first 10 lockers. After drawing it out and summarizing the findings, I tried to find a pattern. So far, the perfect squares less than 10 remain closed. I would try to test this by observing the next 10 lockers to see if this pattern holds. If it does, then the lockers that would remain closed would be {1,4,9,16}.

Letters From Past Students

Student 1:
I really enjoyed your class. You helped me understand math that was useful not only in class but in life. I felt like I mattered in your class even though I struggled with some of the math concepts. Thank you for always being there to listen to me and taking our thoughts and feelings into consideration when teaching.

Student 2:
Your class was the worst. It was boring and irrelevant and a waste of my time. You made no sense and math is more confusing now. I don't remember anything from your class and I never needed to use anything. I didn't feel like I belonged in your class or that I could do math.

My Thoughts:
I hope to create a classroom environment where students feel safe enough to talk to me about what they don't understand and how I can help them learn better, a place where they are not afraid of making mistakes and can learn with each other. Although it would be wonderful if everyone came out of my class loving math, I would be much happier if my students left feeling competent to handle challenges that came to them. I worry that I will fail in trying to create this environment or in my attempts to entice students into trying new things in math since many are used to a very basic approach.

Math & Me

-Ross and Rachel type of relationship 
-I couldn't remember which way the 5 went in kindergarden but forced myself to learn by not looking at the number line hanging by the doorway
-Hated math in grade 5 because I did not understand how to long divide but was helped by my dad and that "ah ha" moment was wonderful and I went back to loving math
-Grade 8 and 9 math was meh but it got very interesting in grade 10. My teacher was fantastic and there was a lot of overlap and connections to other subjects which made it more magical to learn (slopes in physics etc.)
-Grade 10 math made me want to learn more and pursue mathematics which is what I did
-I specifically wanted to teach because I went from hating to loving math because I had people in my life who helped me get the "ah ha" moments and I wanted to be one of those people for someone else
-It is very rewarding being able to help someone understand something they found so confusing

Mathematical Understanding & Multiple Representations

What convinces (or doesn't convince) you in the authors' argument?

I do believe students learn better when exposed to different techniques since they will be more flexible. It also opens up learning to students who learn better by visual/graphical processes which makes concepts accessible to other types of learners. That being said, I appreciated that the author used some visuals in his article to help make some of the points/examples clear. I would have benefited from some more examples and explicit definitions.

If we are viewing it as a process rather than an end product we should provide the students with the materials during an assessment. I think the author assumes that all students are able to connect the external and internal abstractions quickly and easily. Since we know that all students are different and have different strengths, they should all be allowed to have the materials that can help them succeed. 

What kinds of mathematical representations are included and excluded in this article? Can you think of an example of a mathematical representation of a particular math concept (from secondary or elementary school curricula) that is not included, but that might be helpful for students in developing understanding? Describe briefly how you might teach using this representation.

The article did not include technology based representations. If the school has access to a 3D printer students will be able to explore 3 dimensional shapes and can get a better understanding of surface area and volume by figuring out how much material is needed to create a hollow vs solid shape.

It also did not mention using students as a representation. Recalling my history of  mathematics course, I remember one group created a video where they performed proofs via dance. I think this is a great way to get students to physically understand what is happening since they are physically performing it themselves.

Furthermore, as mentioned in class, this type of physical movement could be used to graph parabolas and other types of graphs to get the roots and understand the shape and what is going on in the graph. They can examine the roots, areas of increase/decrease etc.

Relational Understanding vs. Instramental Understanding

Three things that made you "stop" as you read this piece, and why
1. "By many, probably a majority, his attempts to convince them that being able to use the rule is not enough will not be well received." 
Is this really true? Multiple times I have heard students question why what they're learning is relevent and when they would be able to use it. Clearly these students are wanting a deeper understanding. Furthermore, when students realize that the rule they learned isn't working anymore or if the context of the problem is changed, they will be confused and not know what to do or they will get the question wrong. 

2. "Over-burdened syllabi. Part of the trouble here is the high concentration of the information content of mathematics."
Although having a dense curriculum would make it difficult to go deeper into a topic and spend enough time to have relational understanding, I don't think it is completely to blame. If students have a firm grasp of the early concepts in early years, they will be confident in them going into the next year and can build upon them. I wonder if it would be easier for students to learn a topic in a year than move on to a new topic the next year. This way teachers could focus on one topic and build upon it at their own pace without having to do lots of recap to help the students remember the previous material.

3. "Difficulty of assessment of whether a person understands relationally or instrumentally."
I found it very interesting how Skemp proposes that mathematics students are examined orally. It would be beneficial for the students who find it difficult to get their thoughts on paper and are more comfortable with vocalizing the steps they are doing. I think this should be an option for those students who have an understanding of math but are unable to effectively communicate that understanding via pencil and paper. In order to have a record of the assessment, I would keep a recording of the conversation so that I could have something to reference. It would also be useful to the student to reflect back on their assessment.

Where you stand on the issue Skemp raises, and why
I believe students should be taught relationally initially. However, I am not opposed students learning and using "the rules" once they have relationally understood the concept. It is irresponsible mathematics to give students a formula/trick for particular problems without offering an explanation of why it works. Once students have a grasp on the concept, I don't see a problem with them using some short cuts or rules. Since mathematics is constantly building on prior concepts, it is to the students benefit to be comfortable with prior concepts. Furthermore, being able to complete certain problems quickly (such as factoring polynomials) will help the students focus on the new material being presented. I do think it would be beneficial to revisit the old material briefly before moving onto the new material so that the students could see the connection. 

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