Graphing Micro-teaching Lesson Plan

Bridge: Get the students to form pairs in order to work together on small graphing problems.

Teaching Objective: To interpret and comprehend a word problem and to revisit creating data tables from an equation and plotting the data points on a Cartesian plane.

Learning Objective: Students will be able to create lines of best fit for particular graphs and interpolate or extrapolate data points.

Pre-test: Ask if anyone remembers how to graph data points.

Participatory Activity: There are 3 different worksheets. One is about a non-zero y-intercept linear equation (Plumbing Problem), one is about a reciprocal graph (Retirement Gift) and one is about a quadratic function (Carpeting). The students are given 7 to 10 minutes to work on the problems.

Post-test: Representative students who completed one of each of the 3 worksheets will come up to the board, draw their graph and describe how they came to draw that graph. The students then will explain to their fellow students how they answered the corresponding questions.

Summary: Graphing data points on a Cartesian plane is a very useful in Math. It can allow the person to gain insight on mathematical systems that follow patterns. From tracking the trajectory of a ball in the air to predicting the revenue from ticket sales, graphing is important.

Division by Zero

The idea of splitting up a number into smaller parts
Analogy for division, working well in practical ways
Runs into problems with that integer between -1 and 1
Hyperbolas show us some insight into limits of 1/x
Left-hand limit as it gets closer to zero
And right-hand limit too, no agreement, no cooperation
Alas, we are left with inconsistencies
Two trains travelling in opposing directions, never to meet again
I ask my friend to shed some light
"Is it possible to divide something by nothing?"
His response: ERROR.

This poem was created with some inspiration from the 4 minute free-write sessions in class: one on divide and one on zero. After the free-write, I examined all the words written down and wrote this poem. Granted, many of the ideas in the poem were not in my free-write verbatim but I found inspiration in those words. One of the strengths of doing this free-write and poem are that one does not need to be an expert in math to be able to create a poem. Anyone can create the poem from whatever knowledge they have on the topic. Another strength is that it could let the instructor assess how students address a particular subject. If professional terminology is used, then maybe the student understands the importance of that terminology.

Some of the weaknesses of this process is that it is too broad. Students might be like me and write about anything that comes to mind but the content might not even be related to the subject, leading to distract the student from the task-at-hand. Students may conjure up words (like I did) that are completely unrelated and that does not help in understanding some aspect of the topic. Another small weakness in this exercise is that students who like math but not poetry may be thrown off the activity.

Something interesting that I found from the activity were the integration of an extremely academic subject like Math with a subject with openness and fluidity like English. I ended up writing some very un-Math-related things but it was very fun and allowed me the opportunity to write poetry, something that I have not done in a long time. (Aside: Actually, the last time I wrote a poem was for my Materials Chemistry course CHEM 427. I wrote 3 poems about 3 famous Materials Chemists.)

Reflection on Math Micro-teaching

Gigi, Min-Chee and I presented our group math micro-teaching lesson on graphing. It falls under the Math 9 Integrated Resource Package, Prescribed Learning Outcomes B2 (graph linear relations, analyze the graph, and interpolate or extrapolate to solve problems) and B3 (model and solve problems using linear equations of the form a/x = b, x≠0). Some of the positive areas in today's micro-lesson were that we had handouts for the students to use so that they could work through problems in groups of 2, the students successfully displayed their knowledge of graphing and assisted in teaching others, we utilized different types of graphs to introduce students to plotting data points and interpolate and extrapolate from the graph, and we went around the class during the activity and checked in with students that might have problems. The variety of problems, I believe, kept students engaged because the problems were practical and not just theoretically conjured up.

Some of the things which need improvement were our timing, our inability to keep control of explanation and discussion, the clarity of instructions in one of the problem sheets and our team-teaching ability. The timing was not good. The problems probably were not designed for 5 minutes of time to work. They required more time to answer and draw. Specifically, the "Plumbing Problem" had sub-questions that required thinking that would take a long time for a Math 9 student. With the student's presentation and explanation of his solution to the problem, it became very uncontrolled and the student was talking to the class and teaching them for a long period of time. Normally, student-teaching would be a good tool because students will have a better understanding of their fellow students' obstacles. This time, though, the student-teaching took up a large portion of time in our demo. I believe that I should have made the "Plumbing Problem" more explicit in the instructions. I saw a group graphing their points as a step-wise increase in cost. I should have stressed that the cost was gradually increasing from one hour to the next. The final point I have to make is that I spoke very little during the micro-lesson. Even though I did contribute to the preparation, I did not speak much during the micro-lesson and that was a bad move because I should have coordinated better with my team so that we could all give input.

Overall, I think this micro-teaching lesson gave me a lot to think about in terms of student participation and student understanding. I hope that I can come away from this, realizing how to get student involved and engaged so that they can come to appreciate math.

Reflection on "Citizenship Education in the Context of School Mathematics"

After reading Elaine Simmt's "Citizenship Education in the Context of School Mathematics," I have more of an understanding as to the importance of mathematics in everyday life, not only for mathematicians or scientists but everyone. We have become a society of quantifiers, giving numerical values to many things in our lives (e.g. statistics, weather forecasts, hockey performance). Being able to understand a conversation or news report with these data present is becoming increasingly important for people to function in society. By showing real-life connections to math, students will be able to examine how math is presented and how to deal with those essential representations. Presenting problems to students in class would allow me to teach them not only how to solve the numerical problem but those skills applied during the solution can be transferred into their lives and applied as everyday problem-solving skills. Just as with science, I hope to encourage my students to seek answers and to explain themselves clearly. By explaining themselves, they will learn to be succinct and complete in their explanations, learning to interact with others with reasoning and solve the problems to come.

Reference
Simmt, E. (2005) Citizenship Education in the Context of School Mathematics, University of Alberta.

"What-If-Not" Approach

After reading pages 33 to 65 of "The Art of Problem Posing" by Stephen Brown and Marion Walter, I have a better idea of how to plan questions. The authors describe a 5 level process to posing problems.

• Level 0: Choosing a starting point
• Level I: Listing attributes
• Level II: What-If-Not-ing
• Level III: Question asking or problem posing
• Level IV: Analyzing the problem

This process of posing problems can be used by my group for next week's micro-teaching lesson. Our group chose a starting point: we will be talking about graphs, data points, and sets. More specifically, we decided to describe a linear relation. In being more specific, the linear equation will have a non-zero y-intercept and there will be a scenario setting up this word problem. The next step of "What-If-Not" set us up to develop two other graphs. In thinking about how the graph might look like if it wasn't linear, we formulated ideas of problems involving quadratic relations and reciprocal graphs. New questions could be asked in describing these new data points. Of course, analyzing the problems simply is a matter of graphing the data set and looking at the pattern.

I believe some strengths of the "What-If-Not" approach are that you are forced to think about alternatives (e.g. ways your question could go wrong) and you look at all sides of the problem. I believe a weakness of the "What-If-Not" approach is that it can be time-consuming, creating all the attributes and "What-If-Not"s for the situation.

Top 10 Questions/Comments for the Author

In no particular order of importance, here are my 10 questions/comments for the authors:

1. It is very true in math (and even sometimes in the sciences) that given the layout of information, certain questions or problems are assumed (following the x² + y² = z²).
2. Is it our job as future math educators to challenge students' preconceptions and ask them the "unexpected" question?
3. I think the list of possible patterns generated by the class studying the Pythagorean triples shows a higher level of thinking, trying to detect patterns.
4. Something that makes humans unique is our ability to detect patterns in data.
5. A study of the history behind questions and topics can give us background information on the topic and can frame a certain mindset for approaching the question.
6. In discussing the significance of questions and problems, I am reminded of the relational understanding versus instrumental understanding debate. Too often are people lost in plug-and-chug questions that they forget why they were doing them in the first place and lose sight of the meaning.
7. How often do you notice people stuck in internal exploration and don't explore externally?
8. Is it possible to switch gears in children to go from doing questions and accepting everything to learning to challenge the question?
9. "Merely looking for something to challenge will not guarantee that we will find it" (p 18). I find this quote to be very powerful and one of the reasons for doing research in science. If we do research into new areas, we are not sure of the results but that really is the reason for doing research.
10. Why did you choose x² + y² = z² as your first question?

Reference
Brown, S.I.; Walter, M.I. (2005). The Art of Problem Posing, Third Edition, Chapters 1-3, New York, NY, Routledge Taylor & Francis Group.

Letters from the Future

10 Years Teaching Experience

(Letter from student who disliked my teaching) Dear Mr. Siu,

I understand that you are trying to be a great Math teacher. Still, I find that your classes aren't that focused. You don't put an outline of what we're discussing during that class so I don't know what to expect. Also, you spend too much time lecturing during class. If we had more activities to make Math more interesting, or if we spent more time working on problems in-class (as opposed to you telling us the answers) then I think we could learn more. Also, you give way too much homework. My Chemistry teacher gives less homework! I think another way that you could make Math more interesting for us is to get us to do posters or projects on Math topics. That way, we could learn on our own. Also, if you were more available outside of the class hours, then I think I could ask more questions. Don't be discouraged. Thanks for trying to help us.

Kyle



(Letter from student who liked my teaching) Dear Mr. Siu,

I think you are one of the best teachers that I have ever had! You make Math interesting with your puzzles and real-life examples. Making real-life connections with games and toys was amazing! You don't spend the entire class lecturing to us. You give us time to work on problems in-class and you even let us do homework in-class! You're just as cool as my Chemistry teacher. The fact that you're very approachable makes it easy for me to ask questions. Your jokes are sometimes lame, but other times they're hilarious. Thanks for caring.

Sophie



From these two fabricated letters, I was able to verbalize some of my hopes and fears as a teacher. I am afraid of lecturing for too much time during the class or not keeping the students engaged in the class. I hope that I will seem approachable to the students and always devote enough time to my students, looking out for their needs.