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. 

Arbitrary and Necessary - Dave Hewitt

 From Hewitt's article, I think his ideas and arguments have connections to the discussion/reading about instrumental and relational mathematics as well as Thinking Classrooms. 

The idea of informing students about facts, especially if it's "necessary" knowledge, knowledge they could likely find out for themselves given time, is similar to the method of instrumental mathematics giving students a formula/method. Students simply accept, memorize and use it without knowing why it works. Even if the reasoning (proof) is given to them by the teacher, the students themselves are not the one doing the critical thinking and therefore they are still memorizing the process rather than exploring, learning, and understanding it themselves. Meanwhile, letting students use critical thinking, reasoning and recall of past knowledge to make their own discoveries is more meaningful. Making the discovery/achievement themselves makes the information they learned much more memorable than being spoon-fed the information. I've experienced forgetting the formula for the area of a circle many times because I simply learned what the formula is without knowing where it comes from. However, after figuring out the proof for the area of circle, things clicked and I actually remember it now. 

After reading this article, I will definitely be more mindful of "arbitrary" vs "necessary" knowledge when making lesson and unit plans. The interactive activities in my lessons I plan can focus on the "necessary" and have students try to reason by themselves or collaboratively on what the pattern is. For example, for graph transformations, students can try figuring out the pattern for each change of values in the general graph transformation formula. A vertical whiteboard (in Thinking Classrooms) approach would be very good in achieving having students figure out the problems together. I definitely also would give students a puzzle to try at the start of lessons to reason through to build their comfort critically thinking, problem-solving and collaborating with others in class. I will also be more mindful to not give hints too early or often and instead take a step back to let students think.

Giant Soup Can of Hornby Island Puzzle

Information I needed to research to try to solve the problem was:  

• normal soup can dimensions: 6in tall and 2.5in radius
• average adult bike height: 42in
• water needed to put out a house fire: 1 gallon of water puts out 3 square feet of fire
• dimensions of houses in BC: approx 2077 square feet; 20 ft high (2-story); 8 height roof
• conversions between inches, feet, gallons

Information I estimated/used reasoning (from the image):

• approximate height of the water tank: about 2-2.5 times the bike height (went with 2.5)
• approximated square house so approx 46ft side lengths (from 2077 square ft)

I ended up calculating the water tank to be about 126in long (upright cylinder height) and a radius of 52.5in. The volume was 1,091,035.86 cubic inches which was about 4,723.1 gallons. I calculated an average BC house to be 48,946 cubic feet which requires about 16,315 gallons to put out (if the entire house were on fire) so the water tank would definitely not have enough water to put out an entire house that was on fire. If only a portion of the house were on fire (say 1/4) then the water in the water tank could put it out. However, I did not take into account the building material which may affect the amount of water needed as well. 

As an extension to the puzzle, students could calculate how much paint might've been used to paint the entire exposed surface of the water tank. This would require their knowledge on surface area of cylinders and research on how much area a can of paint can cover. The difficulty could be increased by asking how much of each colour paint was used.