Final Reflection

 When I first started this class (and the BEd in general) I was so nervous! I remember feeling so ill-prepared to become a math teacher and I was worried I made the wrong decision. From this class, I gained confidence and comfort in my skills, specifically through the group and individual micro-teaching. The biggest thing I learned this semester is that math isn't black and white, it can incorporate outdoor education, art, and even movement. My favourite articles to read this semester were the Skemp article, Flow ted talk, and Battleground schools. All three of these articles made me reconsider what mathematics education truly is, and I hope to incorporate these new concepts into my own classroom. 

Textbooks

 As a teacher, I understand why the examples are focused on who the textbook is aimed for, and specifically the language used in textbooks. A wordy textbook may be one that turns students away and is not accessible to students, especially students who are ELL. I also think the language of textbooks can either be engaging or disengaging for students. For example, if the textbook is full of problems that students cannot relate to, like the train is travelling east at 30 km/h example, students can become quite disengaged. Additionally, teachers should be previewing the textbooks given to students because let's say a textbook was written in Montreal and focuses on examples around Montreal, this can be disengaging for students that live in Metro Vancouver. 

As a student, I always used my textbook as a tool to study with. I remember from a young age my mom and I sitting with a math textbook and working through the problems so I could be prepared for tests. I always saw the textbook as a holy grail, holder of all knowledge, until I got to calculus in grade 12. In grade 12 I really disliked the textbook we were using in class and I supplemented what we were using in class with resources online. Additionally, as a student, I always found textbooks to be too wordy. I also always found myself losing steam by the time I got to the challenge questions and would often leave them, not necessarily because I did not understand them. I think as a student I would have benefitted from harder and easier questions being dispersed throughout the textbook. 

I think textbooks are a great tool for students to have something to fall back on. Textbooks are also great for parents who want to be involved in their child's learning and see what we are doing in the classroom. However, with our everchanging curriculum, textbooks must be supplemented with additional resources. Teacher made resources can often be tailored to student needs, and can provide students with more direct instruction than a textbook can. 

David Hewitt and Mathematical Awareness

 One stop I had while watching the video is the idea of powers of the mind. Hewitt brings up a good point that students are often relying solely on memory in the classroom rather than thinking and working out why math works the way it does. I think a lot of this comes from the traditional teaching technique of stand and deliver, students are able to take the backseat while the teacher is doing all the thinking and hard work. Using thinking classrooms can help put students in the driver's seat and take control of their learning. 

I also enjoyed that Hewitt did not explicitly tell students if they got the question correct but instead lets them figure it out on their own. I think this is important in the classroom because it moves away from the idea that the teacher is the holder of knowledge and rather encourages students to be independent. 

Finally, I also liked the idea of doing math verbally. I can see that working very well for certain students who are auditory learners. It can also work well for students who get overwhelmed by mathematics when it has been written down. However, I can also see how for certain students this method of learning can be inefficient because they get confused when things are not written down. 

In my own classroom I would love to try out thinking classrooms and fostering independence in students. I feel these methods are quite beneficial for students because it helps them take accountability for their learning. 

To solve the fraction problem of finding a number between 5/7 and 3/4 my first step was to make sure they both had the same common denominator. So the numbers would now be 20/28 and 21/28. Hence a number between 5/7 and 3/4 could be 41/56. 

I think Hewitt created these problems to be solved as a problem of day or as a way to start off an inquiry-based lesson. All the problems in the video are great examples of problems that have multiple correct answers which can help students understand that there are multiple ways to approach even one problem!    

Workplace Math 11 Unit Plan and Lesson Plans

Unit plan Slope and Rate of Change.docx 

Rise over Run Lesson Plan.docx

Review of Trig Ratios and the Pythagorean Theorem Lesson Plan.docx 

Rate of Change Lesson Plan.docx 

Flow

 I have felt flow in my math classes when I am particularly engaged or interested in a topic we are learning. I also feel flow during crocheting, singing, going on walks, and while exercising. I think being in flow depends on how interesting a topic is, or how it is presented to students. If a student is not interested they can never achieve flow. 

I think we can help students achieve flow by teaching math in engaging ways so students have an interest in the topic and are challenged by the content that is being taught to them. Teaching math through inquiry or project based learning can help students be more engaged and can help them reach a state of flow. I feel that achieving flow in math class is possible for all students, but we will need to cater to each students needs individually so they can meet flow in their own way. 

Arbitrary vs Necessary

 Hewitt refers to arbitrary as something that is taught to be memorized, something that must be done exactly the way it is taught. For example, using the + sign for addition and the - sign for subtraction is purely convention, and students are required to memorize these signs to progress further in mathematics. Necessary is referred to as content where the students do not necessarily need to be informed. For example, students can learn how to add fractions with or without mixed numbers without the convention of writing it down. Necessary knowledge can come from within or be discovered through inquiry. Arbitrary must be taught to all students, necessary can become aware to certain students through other concepts. 

The concept of arbitrary vs necessary is new to me in theory, whoever, I feel that inherently I was always known about it in theory. Using a student led approach or inquiry based approach in the classroom can help students learn the necessary and can help all students understand the arbitrary. 

In my own classroom, I could create questions that teach all the students the arbitrary but leave room for certain students to learn the necessary. We want to move beyond teaching and having students regurgitate that information on tests without having any understanding of why the concept works. 

A personal example of this for me was in grade 12 when we learned the unit circle. I remember my teacher emphasizing that we must memorize the unit circle and even having quizzes where we would have to fill in the unit circle in 5 minutes. I purely did that, I sat down and I repeatedly practiced writing the unit circle until I knew with certainty that I could replicate it on the test. I had no idea how the values of the unit circle were derived or even what they meant, I just had them memorized. If this was my own classroom, I would have focused on the meaning behind the unit circle and I would have had students find connections between how different trig values create the unit circle.    

 

Giant Soup Can of Hornby Island

 I have put a picture of my calculations and following that there is a description of my rationale. 

I started this problem by considering the average size of a women's bike, after some research I realized that the height of an average bike is hard to find on the internet, and usually you can only find the length of the frame. The frame of a bike for a woman between 5'3 and 5'5 would be 71 cm (Hudson, 2022), from there I guessed that the total height could be 100 cm. 

From the picture, the diameter of the tank looked to be about 3 times the height of the bike, so I estimated that the diameter of the tank to be 300 cm. From some more research, I found that the average bike length is 175 cm (Ellis, 2022). In the picture, the height (or length as written in my work) of the tank looks to be about twice the length of the bicycle, so I estimated the length of the tank to be 350 cm. 

At this point, I wanted to make sure my calculations were comparable to that of a regular soup can, so I took ratios of both the tank and can (height to diameter) and compared them. They were very similar, only about .3 of a decimal difference!

My next step was to calculate the volume of the tank which ended up being 24.74 m^3. Now I wanted to see if this would be enough to put out a house fire. Through some research, I found out that you need 400 gallons per minute of water to fight a house fire (Smith, 2010). I then converted my volume into gallons to find out that the tank can fight a house fire for about 15 minutes. Once again, based off of some research, I found out that it takes about 20-40 minutes to fight a house fire (Chase, n.d.). So, if there was a small fire, the tank would have enough water to fight it. 

My student bird struggled with this problem because this was really new territory for me, while I understand all the mathematics behind the problem, I have never applied it in this manner and it was a new experience for me. 

However, my teacher bird enjoyed this problem because it made me think critically, problem solve, and apply my skills outside of the box. I think this would be an excellent puzzle for students who need an academic challenge or as a problem of the week where the class would discuss and solve it together. 

Extension: 

Here is a sculpture in downtown Calgary, a head you can stand inside of! A problem I would pose to my students could be, given my height 5'1 and the average size of a human skull, what are the dimensions of the sculpture? I would give the students a picture where I am standing inside the sculpture (I just don't have one at this moment!) so they could reference off my height.   

References: 

Chase. (n.d.). How Long Does It Take To Put Out a House Fire? https://firefighterinsider.com/how-long-put-out-house-fire/

Ellis, C. (2022, August 20). What size shed do I need for my bikes? | The Best Bike Lock. The Best Bike Lock. https://thebestbikelock.com/bike-storage-ideas/best-bike-storage-shed/what-size-shed-for-bikes/

Hudson, A. N. (2022). Rutland Cycling. Rutland Cycling. https://www.rutlandcycling.com/pages/sizing-guide/

P. Smith, J. (2010, December 1). Needed Fire Flow. Firehouse. https://www.firehouse.com/home/article/10465153/determining-how-much-water-is-needed-for-effective-fire-control

Eisner and Curriculum

I found it interesting that this article talks about null curriculum and how schools only focus on math, science, social studies, and English. It makes me wonder, are we teaching practical skills in our curriculum courses? Students are learning how to learn and critical thinking skills plus problem solving skills but is it unjust to our students to not have mandatory classes on cooking, cleaning, basic concepts about automotives, plumbing, and electrical work? While some might say those are skills that should be taught to children by parents, I really do believe we should be having mandatory classes in schools on these skills. These skills are critical for students to know and understand once they are adults and I feel it is the job of high school teachers to not only teach the curriculum but to prepare students for non-academic challenges they may face once high school is over.

Along with my opinion on null curriculum, I enjoyed what the article had to say about implicit curriculum, schools teach students soft skills. I would define soft skills as people skills such as how to converse, punctuality, respect, authority, learning to work with others and people who might be difficult to work with, prioritization, etc. We as teachers should be teaching our students to be good and productive members of society. It is important we model behaviour for our students because they look up to us.

This article expanded my views on curriculum because I have never stopped to think about what curriculum should be other than the explicit curriculum. After reading this article, I realized that it is so important to focus on soft skills and life skills in schools. This ties into the BC curriculum through the core competencies. The core competencies focus more on life skills and soft skills rather than just strictly curriculum. While I think we should have a class that explicitly teaches core competencies, they are taught implicitly in our classrooms. This article raised the question of how I can incorporate soft skills and life skills into my own classroom, this is something that I am still trying to figure out and answer. Part of the reason this question is so difficult for me to answer is because our focus in our classrooms is primarily on curriculum which makes it difficult to find the time to incorporate these crucial skills.  

Pro-D Day Reflection

 I attended the BCAMT conference in person. I had an amazing time at the conference and I feel like I learned so much about math education. The three sessions I attended were Helping Students Enjoy Pre-Calculus 11, 12 and Calculus Courses, Standard Based Assessment: Proficiency Levels and Exemplars in Practice, and Curricular Competencies in Secondary Mathematics. I chose these specific sessions (particularly the last two) because my goal is to have a firm grasp on standard based assessment. The idea of using standard based assessment is so new to me and I wanted the opportunity to learn more about it and see how other teachers are approaching standard based assessment. I was surprised to see that veteran teachers are also having a hard time using standard based assessment, which made me realize this is new for all teachers right now and we will all learn together! 

Group Microteaching Reflection

 

Overall, I feel that the group microteaching was successful. I really enjoyed lesson planning for the lesson and coming with my own resources to use while we were teaching. Lesson planning really helped me prepare for how I wanted the lesson to go. I felt like the lesson ended up coming off as a bit of review since we were asking the students for definitions, rather than teaching them the definitions. In the future I want to work on incorporating more interactive activities that are not just practice problems. I also want to work on my confidence in the classroom, because I doubted myself during the lesson when I was teaching how to calculate the median when there are two numbers in the middle, and because I doubted myself I messed up the instruction and taught something that was incorrect. 

Letters from Future Students

Dear Ms. Vaid,

I was in your class 5 years ago and I want to let you know how much I appreciated your teaching method! I liked how in class we focused on why mathematics works the way it does rather than just learning how to plug and chug. It made me view math in a different light and it help me have a deeper understanding of mathematics which helped me so much when I got to university! I also appreciated how encouraging you were to everyone in class, you really made me feel like I could succeed in math class even when I was struggling or doubting myself.

Thank you for everything!

 

Dear Ms. Vaid,

I am emailing you to tell you your class was a waste of time for me. I learned absolutely nothing about math from your class. You kept focusing on why this works and never showed us how to do the problems! I still struggle with math to this day, and I wish you would have focused more on how to solve the problems rather than making us appreciate the “beauty of math”. It was because of your class that I decided to not take any math in university and maybe if you had focused more on showing us how to get the answer as easily as possible, I would have taken math in university.

 

From writing these letters I can say that my biggest worry about becoming a math teacher is students who fall through the cracks. I worry that there will be a student in one of my classes who I cannot support the way I want to, or a student whose needs I cannot understand.  At the same time, I have hope that I can teach students to love and understand that mathematics is not just a set of algorithms and that everyone can succeed at mathematics if they try!

Microteaching Workplace 10 Lesson Plan

 Here is a link to our lesson plan: 

Lesson plan

Microteaching Reflection

 

My microteaching lesson did not go as I planned. I had chosen to teach everyone how to crochet a chain which looking back was too hard of a concept to cover in 10 minutes. I also had a hard time explaining to everyone how to hold and maneuver the hook, which I wasn't expecting to happen. I should have spent more time preparing for the lesson and looking up different ways to hold a hook so I could have scaffolded better. 

Battleground Schools

 The idea of "Math Wars" was interesting to me. It is interesting to see how politics influenced mathematics education. It never occurred to me that politics and mathematics could go hand in hand. It makes me wonder if teachers are involved in the decision of what curriculum should be, or is it just decided by politicians who might not have spent any time in classrooms. 

It was also interesting to see the difference between "conservative" mathematics and "progressive" mathematics. To me, it seems like conservative mathematics focuses more on algorithms and teaching students computation based mathematics rather than teaching why mathematical concepts actually work. Progressive mathematics focuses more on exploratory thinking, it helps students understand mathematics for themselves and actually understand why mathematical concepts work. Additionally, it also helps teachers understand students skills past simple computation. 

Finally, I found the switches between mathematics education interesting. For me it is hard to pinpoint where we are currently in mathematics education, I feel that I was taught with a conservative lens in mathematics even though I graduated in 2018 from high school, which is relatively recent. But from my classes it feels we are moving towards a more progressive lens in mathematics. It is interesting to see this change happening right in front of me, and it makes me think about what mathematics is going to look like even 10 years from now!   

TPI Blog Post

 

From my TPI results I can see that my two strongest categories are Nurturing and Transmission. I am happy with the results I received. Nurturing means that I can understand the struggles that my students experience, and I can encourage them to work past difficulties they might face, additionally it also means that I challenge the students to do their very best. Transmission means I can deliver my content in a way that is concise and easy for the students to understand.

I was surprised that my development score was not higher. I feel that I am working towards understanding the student’s point of view and trying to approach my teaching in a way that is best suited for all my students.

How can I best support my students in their development and understanding of the content? How can I be more of a social reformative teacher?

Microteaching Topic

 I will be teaching the group how to crochet. Specifically the basics of how to hold a hook, create a slip knot, how to hold the yarn, how to yarn over and how to make a starting chain. 

Crochet Lesson Plan

Reflection on Assignment 1

 Our group presented on Serpinski Triangles. We focused on two aspects their design, Cellular Automoton and the use of different cells. Cellular Automoton was taught by showing the class the different rules that generate different triangles, while the concept of creating fractal shapes one level up was taught by having the class play around with physical cells that we cut out.

I really enjoyed this project because I got to see mathematics through a creative light, as someone who is a little terrified of art this project was really a chance for me learn and apply mathematics "outside of the box". I think the most challenging part of the project for me was understand how Cellular Automaton worked, the rules and how the new row is based on the previous row took a long time for me to get the hang of! When I finally understood the concept it was rewarding for me because I was able to translate my mathematical understanding into art which is not something I have experienced before. 

From this project, I learned to embrace the unknown! Sometimes, you might have to teach a concept that is really out of your element, and the best way to do that is to be willing to learn. Additionally I also learned that a lesson may not go the way you planned it to. Moving forward I am hoping to bring literature (specifically picture books) into my classroom, as well as trying to plan lessons that draw connections to topics such as art and science. 

Dishes Problem

 I had a hard time trying to solve the dishes problem without algebra because I am so used to using algebra when solving problems like this, so initially I tackled it with algebra. Following this I realized that the number of guests would have to be divisible by 2,3, and 4 so I made a list of all the numbers divisible by 2,3, and 4. The numbers would be 12, 24, 36, 48, 60, 72, and so on. From this list my next thought was to take each possibility and divide it by 2,3, and 4 and add up the answers. My final answer was that 60 guests were there because 60/2+60/3+60/4 = 65.

I think that it does make a difference to our students by offering problems from a variety of cultures. It can help students feel more included and heard. It can also help students see how math is presented in different cultures and societies. As well, it can help students be more interested in the problems we are asking them to solve. While this problem does not really involve any part of Chinese culture, it would be important to teach students about Chinese history in mathematics to go along with the problem.

Word problems and its imagery does matter. Word problems that are more applicable to our student’s everyday lives will help them be more interested in math!  

A Mathematician's Lament Blog Post

One thing that Lockhart mentioned that really resonated with me was that we are teaching our students to be “trained chimpanzees” (Lockhart, 2009), we are pumping our students full of algorithms and training them to study just for tests. Our students do not understand why math works the way it does or the reasoning behind why we study mathematics in the first place. This is like what Skemp said about the difference between relational vs instrumental mathematics. Our school system focuses on instrumental learning more than relational learning when we need a balance of both.

One thing I disagreed with was the idea that our curriculum is completely wrong and should be abolished. While our students are more likely to learn algorithms from the curriculum that we are presenting to them, it does not mean that they cannot learn critical thinking and problem-solving skills from the curriculum. There are aspects to our curriculum that are vital to teach to students wish to pursue STEM in university, we need to provide students with a basic understanding of mathematics before they can start approaching more theoretical aspects of mathematics such as a proofs and real analysis. A potential solution to this problem could be to introduce more theoretical aspects of mathematics in high school by moving some of the “basic” content down to the middle school level.

 

Works Cited

Lockhart, P. (2009). A Mathematician’s Lament. New York; Bellevue Literary Press.

Skemp Blackboard