Math 11: Functions Unit Plan

MAED 314A Unit Plan Template
Name: Sam Douglas
Title of unit and grade/ course: Functions/Math 11


1) Rationale and connections:
a) Why do we consider it important for students to learn this topic? Why is it included in the IRPs? (< 150 words)
It is massively important to learn about functions. The idea of a mapping from one set to another can be used to describe a wide assortment of natural phenomena. Moreover, the inclusion of graphical elements allows for a conceptual transition from discrete computational ideas to a more continuous understanding of what a number can be. Additionally, even if a student decides not to take a path in life that explicitly requires a deep understanding of mathematics, a basic knowledge of functions will still be essential. Functions can be used to convey an understanding of causal relationships between data. This is essential for activities such as understanding the media’s use of statistics, or looking at a graph of stock market activity.
b) What are the historical origins and connections for this topic? (<100 words)
The notion of allowing an algebraic expression to vary over a set of values dates back to 12th century Persian mathematicians and their endeavours to solve cubic equations. Our modern understanding of a function as a curve, and the use of f(x) notation are due to Leibniz and Euler.
c) How does this topic connect with life outside mathematics? (<100 words)
Functions are utilized in a vast number of applications. Notable examples include engineering, biology, chemistry, physics, finance and sociology. The main reason for this is the importance of mathematically based models. Functions are used heavily in the physical and social sciences because they allow for easy extrapolation and interpolation when given a discrete set of data.

2) Balanced teaching, assessment and evaluation plan
a) Describe your balanced assessment and evaluation plan. Consider: •teacher, peer and self-assessment; •assessment of student learning, of teaching, and of the unit as a whole•the weighting of marks to take account of summative and formative assessment, instrumental and relational learning
Not having a great deal of experience with the different forms of assessment, I would be interested to try as many as possible to see what works best. In terms of formative assessment, I will mark homework strictly based on completion, and I will allow time in and outside of class for questions. I will also give 2 quizzes. One after the first 4 lessons, and the second after the 8th lesson. The quizzes will be peer-marked in class. Questions will be reviewed as needed. Together, I will count homework and the quizzes for 15% of the total grade. The modeling project will count for 20%, and it will be completed in groups of 4. The final unit test will count for 65%. If need be, I can tweak the homework/quiz mark, or the project mark if I feel that a student’s grade is not accurately reflected by the overall evaluation.

b) Project title and 50-word description

Mathematical Modeling.

To be done in groups of four. Students will be given a set of semi-realistic data. For this data, they will do three functional approximations: i) linear, ii) quadratic, iii) cubic. They will then do a short write-up on each approximation, describing their methods used and which one they felt was most accurate.




c) List of 10 lessons with brief topic outline and teaching strategies to be used.

Lesson topic
Teaching strategies/ approaches used
1)Operations With Functions



· Warm up
· Lecture/ hand out
· Group work / think pair / share
2) Composition of Functions



· Warm up/ activity
· Lecture/ discovery/ hand out
· Group work

3) Inverse Functions



· Introduction and examples
· Inverse pair matching game
· Recap.
4) Polynomial Functions and Inequalities



· Warm up/ hand out to discover
· Lecture/ Graphing
· Group activity
5) Absolute Value Functions 1



· Discovery by hand out
· Paired graphing activity
· Competition review
6) Equations and inequalities with absolute value functions.


· Hand out/ lecture
· Discovery/ examples / Graphing
· Group activity
7) Rational Functions 1




· Warm up / discovery
· Emphasize idea of division by zero and asymptotic behaviour.
8) Equations and inequalities with rational functions.



Graphical interpretation of solutions w/ aid of CAS.
Reason for stating restrictions.
Q&A session and collaborative problem solving.
9) Radical functions, equations and inequalities.



Relate to prior instances of functions.
Extraneous solutions.
Inverse of quadratics with restricted domain.
10) Review class
Focus on concepts relating to asymptotes and extraneous solutions.
Domain and range.


3) In detail:

a) Lesson plans for three lessons, showing a balanced instructional approach. (Note that you cannot use only lectures, homework, quizzes and tests to pass this assignment – that does not characterize a balanced approach!) Each lesson plan should be one page long.
b) Project plan for the unit project. Include a description, a rationale and a marking scheme (one page total).

Mathematical Modeling Project
Step 1
Students will be split into groups of 4, and they will be given a set of data relating to some real life situation which they are to approximate.
Examples include: sales figures, pollution levels, etc.
Step 2
Students will utilize the data to create linear, quadratic and cubic approximations.
Students will use systems of linear equations to find coefficients which will fit the curves to the data.
Students will use their approximations to solve a problem related to extrapolation or interpolation of the data.
Step 3
Students will prepare a 6-8 page report including an introduction, each of their approximations (with full calculations), a write-up discussing which they felt was most accurate, and their findings related to their given problem.

Marking Scheme
The project will be marked mainly based on the accuracy and effective presentation of the calculations. 40% for the computations, 40% for effective choice of points to be used in the curve fitting, and 20% for overall presentation of the report (includes introduction, comparison, summary of findings and effective use of graphical aids).








Math 11 – Operations with Functions

Bridge:
- Introduction to Functions and some mathematical history. Sharaf al-Din al-Tusi, Euler and Leibniz.

Warm-up and Discovery:
- Ask students to simplify some Algebraic statements.
- Have students complete tables of values for selected f(x), g(x) and operations with each.
- Relate the idea to linear functions (already familiar by this point).

Learning Objective:
- Students will be able to do operations related to functions.
- Students will understand the idea of mappings between sets.
- Students will be able to identify domain and range for a given function.

Teaching Objective:
- To make them familiar with operations involving functions and how to use the graphing calculator.
Pre-test:
- Asked students during warm-up and discovery what they know about functions.
Participation:
- Students will be asked to answer questions all the time during the lecture. Asking students to participate and do group work activities.
Post-test:
- Exercises to be done by students in groups.
Summary:
- Summarizing how to use the graphing calculator and what functions are again.



Math 11 – Inverse Functions
Bridge:
- Introducing what inverse functions are.
Warm-up and Discovery:
- Get students to do some tables of values for some of the inverse functions, provided for them in a work sheet, and let them compare the graphs of these functions with their original functions.
- Idea: encourage intuitive development of what inverse functions are and how they relate to their original function.
Learning Objective:
- Inverse Functions and how to graph them with / without a graphing calculator.
Teaching Objective:
- To make the students familiar with the concept of “inverse functions”
- To teach students how to graph an inverse function.
- To make students realize the relationship between a function and its inverse.
Pre-test:
- Ask students during warm-up and discovery to answer questions related to this part.
Participation:
- Matching activity. Pass out function examples and have students find inverses.
- Continue activity by having students travel around the class to find their ‘inverse matches.’
Post-test:
- Group problem sets. Assess based on effective collaboration and unique solution methods.
- Emphasize the importance of trying multiple solution methods.
Summary
- Summarizing what inverse functions are and how to graph them with/ without a graphing calculator.







Math 11 – Absolute Value Functions 1.

Bridge:
- Introduction to how to graph functions involving an absolute value.

Warm-up and Discovery:
-Ask students to do the exercise provided for them related to absolute values (computational).
- Have students use tables of values to graph functions of form y = ax+b.
Learning Objective:
- Students will be able to graph absolute value functions.
- Students will be able to use graphing calculator and graph absolute value functions.

Teaching Objective:
- To make the students familiar with the absolute value functions.
- To teach students how to graph an absolute value function.
Pre-test:
- Ask students during warm-up and discovery to answer questions related to the above.
Participation:
- Paired graphing activity. Students will work with desk partners and collaborate to solve the graphing exercises.
Post-test:
- Number students into 4 teams and have them compete to complete different problem sets.
Summary:
- Summarizing what these functions look like when we graph them.

Short Practicum Stories

Story #1:

Story number one begins with food. On our first day at Prince of Wales, the admins for the school went to great effort to organize an orientation for all of the student teachers. This included a catered breakfast and lunch during which time, we all got to meet our sponsor teachers. This sort of a welcoming atmosphere was awesome considering that many of us were completely nervous, and we had no clear idea of what to expect.

Story #2:

After the orientation day, we (the student teachers) were encouraged to go around and start observing classes. The first one I watched was a Math 9 class which was being taught by one of my sponsors. She had a tablet laptop with a customized worksheet uploaded, and she was modifying it in real time as the students took notes on their own personal copies. I thought this was just about the coolest thing possible, short of having my own smart board. The next week, I actually taught the same class using a set of notes that I had prepared. Although I was nervous for the first few minutes, I successfully hid my fear, and It went pretty well overall.

Group Micro-Teaching: Feedback and Reflection

Peer Feedback:

After reviewing the peer feedback forms from our micro-teaching activity, I found the general sentiment to be fairly positive. Most of the marks were in the 3-4 range with some above and some below. What I found most beneficial (and also most frustrating) were the written comments. Many of the comments indicated that the pacing of the lesson was too fast, and that more time needed to be spent on explication, especially with respect to development of the summation formula for the geometric series. After reflecting on the exercise, I do agree with this judgment, but I have some reservations which I will elaborate on in the full reflection to follow.

Reflection:

While I enjoyed working with my partners on this activity, I found some aspects of it to be somewhat frustrating. The most challenging obstacles for me were the division of labour during the delivery of the lesson, and the fifteen minute time constraint. With regard to the partitioning of the lesson material, I feel like too much time was spent on the introductory portion. As a result, I felt pressured to race through the main body of the lesson, and there was not enough time left for the experimental activity. This was fairly aggravating for me, as I had put quite a bit of thought into coming up with an interesting lesson plan which incorporated all of the elements required by the assignment. With better preparation and communication with the other members of my team, I feel that this difficulty could have been avoided, or at least minimized. The second main challenge I encountered with this assignment was trying to find a way to accomplish all of the goals of the lesson within the allotted time frame. In the end, it turned out to be impossible to do, and we had to scrap the independent problem-solving activity. While I found this to be slightly exasperating, I also found it to be a useful experience. I learned that it is necessary to be flexible in my teaching practices, and to ‘roll with the punches.’ Circumstances will rarely behave exactly as they are expected to. Thus, it is vital to be aware of this and to be able to adapt one’s strategies accordingly.

Zero Division Poem: Reflection

I was away from class on the day this exercise was assigned, and so I found out about it from my classmates the following Friday. My gut reaction upon hearing about it was something along the lines of: “Seriously? We have to write poetry? This is going to be lame.” I would describe my attitude as being slightly hostile. However, I soon began tossing around ideas, and I kept coming back to the metaphor of black holes, and tears in the fabric of reality. Also, I had been listening to a lot of music by a Swedish math-metal band called Meshuggah, and much of their lyrics deal with concepts like transcendence, insanity and the altering of consciousness. I then mixed these influences together and came up with the idea of zero division as an operation which is so powerful that it can tear the mind apart. Attempting to incorporate imagery was challenging, so I tried to convey the idea of a vortex by using sentence and line structure which got progressively shorter and more abrupt. After writing a few rough drafts, I realized that I was actually getting into the activity, and I really started to enjoy it. As a teaching exercise, I thought it was quite interesting, as it forced me to get very meta-cognitive. I really had to think about how to describe zero division and its relationship to the concept of ‘infinity’ in a very qualitative way. I think this could be an interesting activity to incorporate into a math class, especially to supplement the study of topics like asymptotic behaviour of functions. In the end, I found the experience to be quite rewarding and useful.

2 Column Problem Solving - Polygonal Numbers

They posted in reverse, so start reading from the bottom :)

Singularity

Wrenched. Hurtling through space and beyond the event horizon.
Titanic forces compress and rend thought.
Nothing splits my mind as I
Spiral ever deeper.
The Terminus approaches.
Faster, now.
Spinning.
The Crushing Vortex
Annihilates reality.
Oblivion gives
birth t
o the
inf
in
it
e
.

Micro Teaching Lesson Plan

Geometric Series: Finite and Infinite cases
Bridge/Pretest: Recall the geometric sequence (consecutive terms have a constant ratio). Recall that a series is the sum of a geometric sequence. Give students the bouncy balls for initial investigation.
Learning Outcomes: Understand connection between sequences and series. Be able to compute finite and infinite sums. Begin to formulate an understanding of series convergence.
Teaching Outcomes: Incorporate multiple teaching modes. Prove the formula for calculating a geometric series.
Participatory: Have students bounce different balls. Investigate how successive bounces decrease in size. Lead into the idea that the heights of consecutive bounces form a constant ration = geometric progression. Discuss how to add the total length of 20 (or so) bounces. Prove formula for sum of geometric series. Raise the question: ‘what if the ball never stops bouncing, but continues forever with each subsequent bounce scaled by the same ratio?’ Discuss how powers of r, for r < 1, will approach zero if the exponent is sufficiently large. Segue into infinite series, and display how the formula for the sum will be modified.
Post-Assessment: Have students break into small groups (3-4) to collaborate to complete a problem set. Make some problems tricky by manipulating indices.
Summary: Recap relationship between sequences and series. Discuss infinite series, and the issue of convergence.

Citizenship Reflection

Citizenship Education in Mathematics: Reflection

I found Simmt’s article quite interesting, and I felt that it correlated quite well with the previous readings on problem posing. I also appreciated the list of goals for math education, as it highlights the importance of mathematics as a part of cultural heritage and a way to self-empower. I also found the description of mathematical modeling to be quite interesting. The way Simmt describes the ‘formatting power’ of mathematics, made it seem as though the relationship between math and societal phenomena is more intertwined than one might think. The image which came to my mind was of a feedback loop driven by information, where output of a model is utilized and reconfigured into new input. Although teaching to promote responsible citizenship has not been central to my thinking as a mathematics instructor, I do feel that attempting to mold young people into positive members of society is a sacred and important duty. Additionally, I like the idea of creating a classroom environment where students have the opportunity to collaborate and pose their own problems. In this way, students can feel like they are more involved in their personal learning processes. Moreover, by emphasizing the importance of multiple solution methods, students will be forced to reexamine their hypotheses and look at problems from different points of view. In general, I feel like education for citizenship in a math class would not be accomplished in an obvious way. Rather, it would be accomplished by stealth, and by properly manipulating the learning environment.

"What-if-Not" reflection

At the risk of sounding somewhat informal, I think that the ‘what-if-not’ strategy outlined in The Art of Problem Posing is really cool. From my initial reading of it, the strategy basically involves analyzing an existing problem, definition or object, making a list of its characteristics, and then omitting certain of them to see what might happen. I think this is a fabulous way to stimulate creative thinking among math students, and I already have some ideas about how I can apply it to the micro-teaching exercise next week. The idea I had for the presentation was to demonstrate a transition from arithmetic sequences to arithmetic series. I plan to do this by posing the problem of how to calculate the sum of the first 100 natural numbers, extending that to the sum of the first n natural numbers, and then applying the results to find the sum of a sequence of the form t(n) = a + (n-1)d. Basically, my plan is to ‘what-if-not’ two characteristics of the sum 1+2+3+…+ 100. Namely, ‘what if the sum does not end at 100, but continues to an arbitrary number,’ and ‘what if the initial term and difference are not 1.’ I think that ‘what-if-not’-ing is an excellent way to promote original and creative thinking. Particularly, I think it provides a great opportunity for teachers and students to collaborate in ways that could foster an atmosphere of discovery in a math class. From a teacher’s point of view, I think that the ‘what-if-not’ strategy is an effective method for generating multiple problems from a single mathematical object. Most importantly, the strategy allows students to experience discovery, which might lead them to be more curious about mathematics. One potential pitfall of using this strategy is that it seems like it might require quite a bit of effort on the part of the student. This is fine for more capable students, but if a student has a poor relationship with mathematics, implementation of the ‘what-if-not’ strategy may require a great deal of teacher intervention. Aside from this concern, I think ‘what-if-not’-ing is a strategy which I could use to great effect.

Art of Problem Posing: Entry #1

There were several aspects of this first reading that I found interesting:

1) I was pleased to see the description of problem posing as a method of inquiry, as it relates to some of the work being done in my other mathematics education course.

2) I have some reservations about how to directly apply this philosophy to math teaching in general.

3) I think that inquiry via problem solving is a useful tool for teaching, but I believe it would need to be supplemented with some conventional (ie: lecture based) methods.

4) I enjoyed how the author describes the significance of a problem as being proportional to its ingenuity and playfulness.

5) By promoting this idea of playfulness, I think it would be much easier to promote a sense of curiosity about mathematics.

6) Also, encouraging students to play around with problems and modify them could go a long way toward combating math phobia.

7) I was also very pleased to see the emphasis placed on shifting contexts and perspectives.

8) This idea of shifting perspectives was also brought up in our other math ed. class as a way of fostering a deeper understanding of a concept.

9) I am curious about the idea of internal vs. external thinking with regard to problem posing, and I hope to see this idea expanded upon in the later sections of the book.

10) I was also interested to read about how lay persons often have an easier time of creating interesting problems than those with more formal math training. It would seem to coincide with Ken Robinson's idea about formal education having a destructive influence on creativity.

2019: 10 years gone

Happy student:
Mr. Douglas always encouraged creativity and critical thinking. I felt that this was the aspect of his teaching that impacted me the most. He wasn't always the nicest guy, but you could tell that he cared deeply about the success of his students. Moreover, he tried to foster a sense of curiosity in his students, and this helped a great deal in maintaining my motivation.

Unhappy student:
Mr. Douglas was the worst teacher I ever had. I always dreaded going to class, because I knew it would just end up being 70 minutes of either boredom or anxiety. I felt like there was not enough variety in the structure of the lessons, and he seemed to teach more to the level of the best students without any concern for those of us who struggled.

Reflection: I think this is a pretty good indicator of some of the things I'm concerned with regarding my life as a teacher. The success of my students is very important to me, but I know it will be impossible to reach every single one. I suppose it will become an exercise in optimization.

Unorthodox Teaching Video

Dave Hewitt Video – Reflection

I found this film to be an interesting example of unorthodox pedagogical techniques in action, and I must readily admit that I would never have thought to engage an entire class in the way that Mr. Hewitt does. By focusing the entire class on the task of experimenting with a single skill, Mr. Hewitt accomplishes several things at once. In particular, he assesses the initial knowledge level of his students, fosters a participatory environment, and ensures that the students’ knowledge is expanded in a logical and organic manner. I was quite impressed with how Mr. Hewitt accomplished this last task. In one specific example, he segued from a discussion involving integers to a discussion of algebraic expressions, and he did it in a very sneaky way by simply stating that “I’ll just pretend this ‘x’ is like 15. I’ll put a number there later.” In all of my teaching experiences, the introduction of variables is almost always a dreadful shock to students. It was very interesting to see this task accomplished in such an efficient and effective way, and all without spooking the kids. I did have two minor reservations to Mr. Hewitt’s teaching style, and they stem from my own preferences as a student. Generally, I prefer lectures with a quicker rate of information delivery, so the pacing of Mr. Hewitt’s lessons might seem glacial and agonizing to a more capable student. Additionally, while I am impressed by his use of tactile/auditory teaching methods, it would be detrimental to cater exclusively to these styles of learners. Aside from these concerns, I found the video to be a good demonstration of an intriguing and innovative way to teach mathematics.

Battleground Schools

Battleground Schools: Summary and Reflection

The chapter of Battleground Schools which focuses on mathematics education begins by highlighting the ideological schism which has been the source of many of the changes to math curricula in North America. The two main camps vying for influence in this conflict are the traditionalists (lecture, homework and assessment based) and the progressives (focused on understanding, inquiry and learning through experimentation). The article then gives an overview of the history of math education from the beginning of the 20th century to our present day. The Progressivist Reform (circa 1910 – 1940) sought to reinvigorate school math teaching by giving preference to the ‘why’ aspects of problem solving. Through this effort, they aimed to shift emphasis away from inexplicable procedural computation, and focus on improving students’ understanding. Following the launch of Sputnik in 1957, North Americans became intensely afraid of falling behind the Soviet Union technologically. The ‘New Math’ movement of the 1960s was a result of this insecurity, and its aim was to produce a generation of rocket scientists. What this led to was a massive overhaul of North American math curricula through the addition of many abstract topics, such as set theory and linear algebra. Ultimately, this approach was scrapped, as it neglected the importance of basic skills and the material was beyond the understanding level of many teachers. The article concludes with a section outlining the current conflicts which are raging due to the implementation of nationally standardized curricula, and it emphasizes the importance of working toward a consensus between the two ideological positions.
I really enjoyed this article. I found it to be a very good overview of the different directions math education has been taken in over the past century or so. Particularly, upon reading about the ‘New Math’ movement, I was reminded of an idea I had several years ago about potentially including an abstract algebra class in high schools. I think that exposing more gifted students to some of the elements of pure mathematics is an important idea, and worthy of consideration; however, I do not believe that these elements could ever serve as the bedrock for an entire set of K – 12 curricula. Concern for foundation skills would need to be addressed. Additionally, I was very interested to see the correlation between traditionalist views about math teaching, and evangelical Christian lobby groups. It would seem that even in math, there are rigid, dogmatic thinkers who attempt to push their views upon others. Overall, I feel like this article is an excellent source of background information for any prospective math instructor.

Assignment 1 Reflection

Assignment 1: Individual Reflection

I found this assignment to be quite interesting and useful to me. In particular, John Yamamoto’s input went a long way toward alleviating some of my concerns about specific aspects of teaching. Listening to the presentations of other groups was equally illuminating. Many groups asked their teachers intelligent and specific questions regarding how much homework to assign, or how to encourage discovery and discussion. These responses furnished me with some rough ideas which could help me during my first experiences in the classroom. Additionally, I noticed that there was a wide variety of different opinions represented among the surveyed teachers. Some seemed quite authoritarian, and they approached teaching in a very instrumental way; however, I got the impression that most teachers utilized a mixture of approaches, and they aimed to build rapport and appeal to different learning styles. On the other hand, I was slightly dismayed by the responses from the surveyed students. It seemed as though very few of them had any appreciation for mathematics beyond its use as a calculation tool. In general, I left with the feeling that most students would prefer a step by step method, or an algorithm that they could apply, so that they would be able to get an answer and move on. I hope that throughout my career as a teacher I will have the opportunity to force feed an appreciation for proper mathematics into at least a few of my students.

Conversation Summaries

Conversation Summaries

My group consisted of Rory, Amelia and myself. For our conversation, we collaborated to come up with a list of ten questions; five for a student, and five for a teacher. The student we interviewed was my sister, a 10th grade math student, and John Yamamoto was kind enough to participate during the teacher portion. For the teacher portion, our questions were:
1) What did you find to be your biggest challenges with your early teaching experiences?
John indicated that, initially, he had the greatest trouble managing his classroom effectively. He indicated that he would “work around the chatter,” rather than clamping down on it. His recommendation for dealing with this problem was to set very clear expectations, and be consistent about enforcing them.
2) What accommodations have you made to help students with learning difficulties?
With regard to students with learning difficulties, John stated that his work with deaf students at Burnaby South Secondary made him very aware of the pacing of his lessons. Most challenging for John, were the students who spoke little to no English, and he said that, if possible, he would liked to have had more experience working with ESL kids. Most importantly, John stated that it was necessary to be patient and mindful of kids’ histories, as “their problems [in the classroom] often have nothing to do with you.
3) How do you engage students with low motivation?
Here, John emphasized the importance of using humour to break down barriers. Also, he said that it was necessary to acknowledge that math was not their favourite subject, but hopefully they would learn to enjoy it a little bit. Additionally, John made a habit of taking an interest in his students’ interests, in order to relate to them more easily and build rapport. His main goal with these kids was to get them to come to class.
4) How do you vary your explanations when students have difficulty understanding?
Here, John’s main message was simply: “don’t worry about it too much.” Occasionally he would make use of colleagues when stuck with a particular lesson plan, or utilize the more gifted students to help collaborate on a different explanation. Mainly, he indicated that getting comfortable with different approaches to lesson planning was something that came with experience.
5) What do you enjoy most and least about teaching?
John said that it was amazing to form relationships with the students and to watch them grow, but it was often very difficult to see them leave and move on to new phases in life. Also, while he had some complaints regarding how certain topics were presented, and unbalanced curricula, he said that the good vastly outweighed the bad.
I asked my sister the next set of questions in a conversation over the phone:
1) Why do you think it is important to teach math in school?
After some initial uncertainty, she said that she felt math was important for many future careers, and that it helps to improve your reasoning power.
2) How do you develop your first impressions of a teacher?
She said that she judges mainly based on the teacher’s sense of humour and the level of organization in the classroom. She feels that a highly organized classroom is indicative of a strict teacher.
3) Think of a math lesson that you found particularly memorable. What made it unique?
In 7th grade, she had a student teacher give a lesson about Pi by dressing as a chef and feeding the class apple pie.
4) How would you feel about incorporating more group activities in your math class?
Initially, her reaction was quite negative. She indicated that explanations from other students often only contributed to her confusion. She did, however, seem open to the idea of doing group projects.
5) What is an effective way that a teacher has helped you to understand a tricky concept?
In 8th grade, her math instructor explained equation solving using a unique analogy involving negatives being bad and getting sent to the basement until they come out positive on the other side.
Overall, we found the experience to be quite positive and illuminating.

Robinson Article Response

Robinson Article: Reflection/Response

I quite enjoyed this article because it gave some rough guidelines describing ways to go from an adequate math instructor, to an instructor who inspires. Robinson’s initial approach to teaching is reminiscent of just about every high school math teacher I’ve known. They are burdened by content – laden curricula, and they feel forced to teach in a very instrumental, skill based manner. Unfortunately, this leads to a situation where students are virtually unable to think critically, and they have great difficulty adapting their learning to unusual problems. This is precisely what Robinson experienced in her class. Initially, I was skeptical about the effectiveness of incorporating the types of activities suggested by Robinson. The only experience I have with group work in a math class is using integer tiles to solve worksheets in Math 8. I found this to be a colossal bore, as I was fluent in integer operations and I saw no need to ‘waste my precious time’ with these ridiculous blocks. However, after reading Robinson’s description of how she set up her function jigsaw groups, I found my attitude toward the idea of collaborative activities to be somewhat improved. I think I would be somewhat nervous about implementing such a strategy myself, until I had a firmer grasp of how to achieve all the PLOs for a given course; however, I would be interested to try it after becoming more acclimatized to my role as a teacher.

2 Memorable Teachers

Notes About Two Memorable Teachers

Mr. Norm Cheng (gr. 12 math teacher):
· Not the greatest teacher ever.
· Tended to be somewhat cranky, particularly in the mornings.
· Often made fun of his students, and was generally quite hilarious.
· Did a pretty good job of pointing out links between different concepts.
· Quite devoted to helping students succeed.
· Helped me teach myself calculus.
Dr. Joel Feldman (3rd year Real Analysis instructor):
· Possibly the most intelligent, well rounded math instructor in the history of math instruction.
· Genius at finding ways to explain very abstract concepts.
· Extremely challenging assignments that forced you to think creatively.
· Very helpful, and cared about student success.
· Conveyed an aura of total competence and mastery of his subject.

Reflection:
I think I admire both of these men. Mr. Cheng was the first instructor to teach me the awesome power of mathematics, and he gave me my first glimpse at what the subject was really about. Dr. Feldman was not typical of an instructor in an honours course, because he treated all of his students equally whether they were geniuses or not. Also, he would make time for you beyond the scope of his regular office hours. In addition to all this, he is a brilliant mathematician. The combination of genius and competent instructor is exceedingly rare, but that is exactly what Joel Feldman is. In my teaching, I would strive to be similar to both of these men by being considerate of my students’ needs, and encouraging them to think in creative ways.

Micro Teaching: Summary & Reflection

Peer Evaluation Summary:
I found my classmates to be very receptive participants in my activity. They enjoyed my introductory anecdotes, saying that they were useful and provided a clear motivation for the specific technique I demonstrated. Additionally, they wrote that the practice with partners of disparate sizes was a good way to display the effectiveness of leverage and joint manipulation when neutralizing an opponent. They also indicated that I did a good job of monitoring their progress throughout the activity, and correcting their form as necessary. Although they found the initial technique to be too complicated, they were pleased at my ability to adapt the lesson to their skill levels. Some wrote that they would like to have seen some different ways to apply the technique, although they understood that the time limit was quite constraining. Finally, they stated that I needed to focus on improving the organization of my summary.
Self Evaluation and Reflection:
Overall, I thought the activity was carried out successfully. I felt like my colleagues understood the main points I was trying to get across; namely, some principles of self defense, and the idea of using joint locks to overcome a physically superior attacker. I was quite concerned by the difficulty my classmates were having with the technique I initially taught, but I felt like I did an effective job of modifying the lesson based on their lack of martial arts experience. Effectively summarizing a lesson is an area of my teaching that I have been working to improve, and I was not surprised that my classmates pointed it out. Also, I had wanted to demonstrate one or two other ways to apply the technique I showed, and with better time management I feel like this could have been accomplished. All in all, I’d give myself a B+.

Microteaching #1 - Wrist Grab Defense

Basic Single Wrist Grab Defense:
Symmetric Stance

Bridge:
First, I will survey my students to find out what prior martial arts experience they have. Also, I will explain that in feudal Japan it was common to attempt to prevent an opponent from drawing his weapon by seizing his wrist. As a result, many basic techniques in traditional styles of Japanese Jiu-Jitsu are initiated by wrist grabs.

Teaching Objectives:
· To improve my comfort level when teaching kinesthetically, and when supervising groups.
· To demonstrate that effective application of a technique can help compensate for physical limitations based on size, strength, etc.
· To introduce my students to some basic principles of self defense.

Learning Objectives:
Students will demonstrate an understanding of the historical motivation behind studying a wrist grab defense.
Students will display a basic understanding of how to utilize leverage and joint manipulation to overcome a physically superior adversary.

Pretest: To be done during bridge phase.

Participatory:
I will demonstrate the technique, and students will partner up into groups of 2-3 to practice with one another. I will encourage students to practice with someone similar to themselves in size and strength (group composition permitting).

Post test:
I will switch up the practice groups so that students can practice with opponents who have a significant size or strength difference, in order to help demonstrate the effectiveness of a properly applied technique.

Summary:
Give a quick rerun of the technique, and a brief glimpse at some similar defenses utilizing the same technique.

Skemp Reflection

Relational vs. Instrumental Understanding

After reading Skemp’s article, I am convinced of the general superiority of relational teaching in mathematics instruction. That being said, I believe there are some areas where an instrumental approach can have certain benefits.
Skemp initially describes instrumental knowledge as “rules without reasons.”
The first effect this statement had on me was to remind me of the vast majority of my experience as a math student. Virtually all of my teachers throughout high school taught mathematics in a very rigid, mechanical way, with little thought to deeper analysis or explication.
The second quotation which stood out to me was Skemp’s statement that some students just want “some kind of rule for getting the answer.”
What this reminded me of was some of the experiences I have had as an instructor at Sylvan. Many of my students have had learning difficulties in mathematics. The simple ‘rule to get the answer’ method is helpful for them, as it helps improve their confidence by making them feel successful. Unfortunately, I believe this approach also promotes intellectual laziness and the tendency to look for an ‘easy button’ whenever possible.
“… instrumental understanding […] usually involves a multiplicity of rules rather than fewer principles of more general application.”
Upon reading this, I thought of the analogy of a tool box. Instrumental teaching requires you to carry around a giant box, with many specialized tools. Conversely, an instructional method which promotes relational understanding lets you carry around a lighter box with fewer, more efficient tools.
Writing as a Devil’s Advocate in favour of instrumental instruction, Skemp writes: “They will at least acquire proficiency in a number of mathem-atical techniques which will be of use to them in other subjects …”
Upon initial inspection, I thought this point had some validity. However, after further reflection, I decided that it would be insulting to mathematicians everywhere. This would have us consider math as a mere tool for accomplishing tasks, rather than an independent discipline worthy of study in its own right.
The final quotation which struck me most was that relational instruction is “easier to remember.”
I agree wholeheartedly with this. Throughout my entire undergraduate career, I would always loathe to memorize formulae. Instead, I would learn to derive them, and pore over proofs in my notes until I fully understood how and why they worked. The memorization would then occur almost by accident.