Reflection on John Mason's Article

John Mason's article on questioning in Mathematics reminds me of what I have recently read in an award winning book "The Mathematical Mindset" that Mathematics is taught in a way that it seems we throw at students the answers to the questions they never asked. This perspective leads into asking why we don't let students ask questions before we begin answering their never-asked questions.

Questioning can be very effective in promoting inquiry-based learning, as it relies on students' explorations, their statements, and their ‘truth’. In inquiry-based learning, questions are genuine not rhetorical which is common in Math classroom, but not so useful in advancing students' learning as suggested by Mason.

I would like to extend the idea of questioning to dialogue, i.e. teaching Math via dialogue. I would love to have my students “talk” Math. Healthy back and forth discussions where the “talk” is based on understanding and exposure rather than facts and figures is a skill worth achieving.

In other words, creating a class atmosphere where learning is not about getting the right answer, but the procedure and the thought process, i.e. encouraging students to share their ideas, thoughts, and understanding without the fear of being wrong. The focus should be on justifying responses and backing them up with understanding rather than certainty, as Mason describes in the article.

The quickest response to incorporating Mason’s ideas into unit planning for the long practicum is asking the self, i.e. asking questions like what might go wrong in implementing a certain idea, how would the students react if presented a certain problem, and how can I make smooth transitions from one topic/activity to the other.

Lastly, ending the lesson in ways that genuinely raises students' curiosity and interest in the topic covered rather than having them answer forceful questions that seem to have no benefit in their eyes. Having them seeing me getting un-stuck when stuck, and allowing them to solve 'their-way' are other ways of having students see the creativity Math has to offer.

Microteaching !! Reflection

Based on self-reflection and the thoughtful feedback and comments we (me and my teammates) have recevied from our collesgues, there is definitely room for improvement in the lesson plan and its delivery. The participatory component of the lesson plan could have been better, as it would have been nice if we had incorporated a more intriguing and engaging “hook” into the lesson. It was a good opportunity to put into practice what we have been learning about student-centerd approach and problem based learning for teaching Mathematics.

However, it was suggested that we should have spent a few minutes solving and explaining the problem we posed via what we call the traditional teacher-centerd classrooom instead of asking the students to solve it. The “perfect” balance between the teacher-centered approach and the student-centered appraoch will become natural with practice and time. I look forward to implementing and improving my lesson plan and teaching procedures based on the feedback I’ve received!

Microteaching !!

Applications of Quadratic Functions

Subject: Math

Grade level: 11

Duration: 15 minutes

Prescribed Learning Outcomes:

 It is expected that students will be able to:

-C4: Analyze quadratic functions of the form y = ax² + bx + c to identify characteristics of the corresponding graph, including: vertex and to solve problems

-C5. Solve problems that involve quadratic equations

Objectives:

Students will be able to:

-Learn how to analyze and solve real-life problems involving quadratic functions

-Identify the relationships between maximum/minimum problems and quadratic functions

Materials/Resources:

-white boards in the classroom

-Handouts of the word problems

  

Lesson Plan
Introduction	5 minutes	-Divide the class into groups of three -Handout the “hook” question to each group and encourage the students to come up with the dimensions that would enclose the largest area without using algebra A rancher has 800m of fencing to enclose a rectangular cattle pen along a river bank. There is no fencing needed along the river bank. Can you guess which dimensions would enclose the largest area? (HINT: the dimensions are integers)
Entry	5 minutes	-Go over the rancher question on the board, using algebraic method Ask one or two students to  come to the board and present their solutions. Encourage those students that think they've made mistakes. This is to promote mathematical thinking and that mistakes are part of the learning process.
Development	~4 minutes	-Ask the students to work on a similar problem individually You are trying to build a rectangular dog fence in your backyard using 600cm of fencing material. Since your dog is relatively big, you goal is to make a fence with the largest area possible using the given material. What should the dimensions of the fence be?
Closing	1 minute	-Go over the solution together.

Adaptations and Modifications:

-Allow students to work in groups

-Provide more time for work if needed

Assessment/Evaluation:

-Go around the classroom while the students solve the problems together in groups on the board to check their understanding

-Observe the students while they work independently and identify what the most commonly made mistakes are

2-Column Puzzle

Exit Slip - Mon, Nov 23

The video we watched reminds me of Silent Way, one of the methods used to teach pronunciation. It doesn't make sense when we first think about it, as pronunciation is teaching sounds, segmentals, and suprasegmentals that are to be taught by modeling and getting students exposed to the correct models, but it makes perfect sense when we begin to think beyond the surface. I made this link between Math and teaching English pronunciation via Silent Way, as I found the video did exactly that, i.e. the students were well engaged and actively participated without the typical and traditional pencil/paper approach. I look forward to using this approach with slight modifications depending on the topic/subject of course in my classroom.

Moreover, the video demonstrated long pauses when the teacher posed a question. This is something that I think all educators should consider to allow enough thinking time for students to get their cognitive processes working. 

Arbitrary vs. Necessary in the Math Curriculum

This article allows its readers to think about things that we generally tend to ignore under the umbrella of ‘that’s the way it is”, or ‘that’s the convention’, or even ‘true-because-teacher-says-so’. My question if what if I let the students break the convention. For example, what if I let them write the y-coordinate before they write the x-coordinate. Would it be looked upon as breaking the rules, or would it give the students some power or control over Math, I wonder. Moreover, I wouldn’t want to answer ‘the-why’ questions from the students via ‘that’s the convention’. I would like to support their curiosity and get them explore ‘why that’s the way it is’. However, there is a fine line between satisfying curiosity, and exploring for purpose and getting confused, i.e. as an educator, I’d need to be careful for not getting the students confused by sharing too much information and getting them exposed to it without support.

One of the ways for defining arbitrary and necessary is by considering what can/cannot be solved. In other words, ‘arbitrary’ is information (or “received wisdom”) that may not be be worked out and is in the "realm of memory", whereas ‘necessary’ includes the properties and relationships that are in the "realm of awareness", and can be worked out, as described in the article. The decision of what’s arbitrary and what's necessary can vary from topic to topic. The question of what arbitrary and necessary are under a given topic can be responded to by considering the answers. For instance, the aspects of the lesson that can be summarized as “symbols, notation, and convention” are arbitrary, and the properties and relationships that carry good explanation to satifsy the ‘why’ are necessary. However, if one excels at explaining the ‘why’, what’s arbitrary can become necessary, as Hewitt outlines why New York is New York.

Furthermore, the differentiation between arbitrary and necessary of course influences the lesson plans, as this decision would greatly impact how much time should be spent on what and why. As teachers, we don’t want to spent too much time to what's arbitrary. Instead, our focus should be on problem based and inquiry based learning within the given topic. It's related to what's we talked about previously regarding regarding instrumental and relational thinking. We want to teach the students to think mathematically under the given topic.

SNAP Math Fair Field Trip - Exit Slip

Firstly, thanks so much Susan for giving us the incredible opportunity to work with kids outside the classroom setting where the kids are the experts, and we’re the guests. Secondly, I’m more happy than surprised to see the expected outcomes of SNAP Math Fair becoming a reality. For instance, the students delivered and communicated their projects in such a professional way; they were confident, ready, and well-prepared and having fun at the same time.

Thirdly, I love the connections educators are establishing to make Math more relevant and fun. I personally never thought of anthropology as a source of creating Math fun problems/puzzles to solve. Me and my group members (Jordan and Mandeep) had a chance to ask the students where they got the ideas from for the problem/puzzle they were presenting. The ideas of course originated from the artifacts/displays at the Museum of Anthropology, but the connections the students made with Math was fascinating!

I’m glad to be part of the profession where there’s so much room for creativity and inquiry. However, making creativity and inquiry part of the Mathematics education for the general public to appreciate is yet to be achieved.