Final Blog Reflection

At the start of this course, I didn't know what to expect or what I would be learning - all I really knew was the course was about math curriculum and pedagogy. Although I didn't expect I'd be doing art in a math class, it really changed my perspective of math and how there are so many ways we can teach math and integrate cross-curricular content! 

Throughout this course, I think the biggest thing I learned and took away was the idea of being more aware of how we as (future) teachers teach math to our students. There were a few readings in particular that really stood out to me such as the Skemp article and the Lockhart's Lament article. With Skemp's article, it made me more conscious of how and what we decide to teach students and has made me put a lot more thought into my lesson planning. As for Lockhart's Lament, it was a very engaging and enjoyable article to read and I really resonated with the ideas he put forth about having students try to come up with their solutions rather than the teacher give a demonstration and explanation beforehand and having students do problems afterwards - quite similar to a Thinking Classroom approach, I think. These articles have given me ideas and helped shape how I plan to teach my lessons in the future. 

Furthermore, I really enjoyed the math puzzles (I will definitely be keeping those for future math puzzles to give to my own students!) and I really am glad I had the chance to participate in the BCAMT math conference on Pro-D because it was a great place to get resources and a wonderful sharing space for teachers. 

Overall, I gained some new insights in regards to teaching math from this course. Thank you and happy holidays!

Unit Plan and 3 Detailed Lessons

Google Docs to my Unit Plan rationale and overview: 

https://docs.google.com/document/d/1-IBND0lGCl11GvzoP-w-e2ysdEJTPQjwOegYZUBkeCY/edit?usp=sharing 

My Lesson Plans: 

https://drive.google.com/file/d/1DevFu0eKLw8Z7nuAr3kNLSNWlLJ09_mb/view?usp=sharing 

FOM11 Unit Plan WIP

My Unit Plan Rationale and Overview:

https://docs.google.com/document/d/1-IBND0lGCl11GvzoP-w-e2ysdEJTPQjwOegYZUBkeCY/edit?usp=sharing 

Mathematics Textbooks

https://www.flickr.com/photos/46166795@N08/5504156024

I've never thought deeply about how math textbooks may shape how students (including my past self) perceive mathematics, but the article made an interesting in-depth analysis that made me rethink about my own experiences with my math textbooks as a student. Some points of interest to me were the mention of how most math textbooks use second person pronouns, especially structures such as (inanimate object + animate verb + "you"), and the use of verbs that express strong conviction and certainty. From these factors, it makes math textbooks feel very rigid and likely a reason they are not deemed very interesting to read by students (if students read them at all). By using second person pronouns, it feels very much like the textbook is controlling and directing the student to follow a set of rules and thinking processes. The "you" also makes it feel more directed to the student reading, having them work on the problems individually instead of discussing with a classmate. Meanwhile, the certainty of statements in the textbook may induce less creativity and questioning on the student's part. 

In this way, I don't particularly find mathematics textbooks appealing to use, at least not as the main source of learning for students. Textbooks give a good breakdown of chapters and topics that is useful for the teacher to use to structure and plan lessons. Furthermore, textbooks tend to have structured problems that increase in difficulty, so these are good resources as question banks for teachers or practice problems for students to work on. Nowadays, I see more workbook type of textbooks that contain problems for students to work directly in the book, individually or collaboratively, which I prefer over older textbooks I've seen/used. These workbooks also allow students to write their own personalized notes and understanding of the material directly in the book rather than having to stick to a rigid method given by a textbook. 

Flow

Japanese Logic Puzzle: Hashi
https://www.flickr.com/photos/davidmasters/3618249963

I tend to experience a state of flow whenever I work on puzzles (e.g. Sudoku, KenKen, 0hh1, Codewords, Nonogram, etc.). I especially love math-related puzzles because I like how they challenge my math skills/knowledge without feeling they are super difficult that I wouldn't be able to solve it and simply give up. They give me enough challenge that I won't get the answers right away and have to put effort into solving them. At the same time, I know they are doable and I have the ability to solve them, so I am focused, have a lot of patience and am persistent to solve them. 

I think it is possible to achieve a state of flow in secondary math classes, but it would definitely be difficult. Csikszentmihalyi mentions how being in a state of flow is related to the level of challenge and one's abilities, but given how many students are in a classroom and how diverse their skill levels can be, producing lessons that create a state of flow for the entire class would be difficult. As teachers, we could make modifications to accommodate different skill levels, but realistically, that would be a lot of work. However, I do want to work towards having a state of flow in the classroom. Taking inspiration from Peter Liljedahl's Thinking Classroom framework, I think providing mathematical puzzles/problems that are non-curricular for students to do at the start of class would help orient them into a flow state. Puzzles are interesting and fun and having them not be curricular means the students don't have that pressure to get the right answer or feel incompetent if they aren't able to solve it. Although achieving a flow state in every lesson for the entire duration is not possible, I think striving to get it for most lessons is a possible and good goal to aim for to keep students motivated and enjoy learning mathematics. 

Unit Plan Topic

UNIT PLAN TOPIC

Course: Foundations 11

Grade: 11

Unit: Mathematical Reasoning

Dave Hewitt Videos (Algebra, Awareness and Fractions)

One thing that stuck out to me in the video where Hewitt was teaching algebra to his class was his statement about not taking the role of the checker. As a teacher, we want our students to have confidence in their answer and check for themselves/with their classmates whether their answer makes sense. Along with this, in the video about awareness and fractions, Hewitt mentions a joke he heard about how a child learned adding 2 apples plus 3 apples is 5 apples, but is unable to answer what 2 bananas plus 3 bananas is. These two points stuck out to me because it reminded me that teachers should simple facilitate students' learning. We want our students to be independent learners, not dependent on teachers to tell them what the right answer is or if confirm whether their answer is right. In society, we won't have someone with us at all times to let us know how to do each step and if we're doing things correctly. Everyone can make mistakes, but what we want to foster in students is the ability to recognize them, identify why it happened and learn from them. 

A third stopping point for me was when Hewitt was combining 2 pencils and 4 brushes together. In my head, I said "2 pencils and 4 brushes," but Hewitt stated we might have said "6 things" which made me think that as a teacher, students may not give the answer you expect and there's many approaches/answer to the same problem. In my mind, I was thinking in terms of collecting like terms and how since pencils and brushes were different, they did not collect and simply were "2 pencils and 4 brushes." However, to Hewitt, he was considering the problem in terms of fractions (and like denominators) and giving the items the "same names" so they were able to be added together. I thought that this was really interesting. 

I really like the fractions problems Hewitt gives at the end because it starts off seeming simple (find a fraction between 5/7 and 3/4) because I've learned how to find equivalent fractions with like denominators. However, once I got 20/28 and 21/28, I actually really liked and appreciated how sneaky and tricky the problem actually is. To find a fraction between 20/28 and 21/28, you have to realize that there is no whole number between 20 and 21 and you can't put a decimal in the numerator. Students who understand the concept of fractions well will realize that you can actually simply divide further to get 40/56 and 42/56 and find 41/56 to be between 5/7 and 3/4. The following problems after also get more and more trickier and really makes you more aware of things that are true (and can discover are true) about fractions. I think teacher-created math problems are definitely more creative and interesting because it tests your thinking and understanding of a concept a lot more than the regular problems you'd get on assignments/tests. I actually have seen similar (I think) problems in a math class I volunteered for and I've always wanted to incorporate these types of questions in my own math classroom, perhaps as a warm-up thinking activity for students at the start of class or something to try and think about at the end of class.