MAED 314A Unit Plan Template

Name: Vincent Collins

Title of unit and grade/ course: Shape & Space – Math 10

1) Rationale and connections:

a) Why do we consider it important for students to learn this topic? Why is it included in the IRPs? (<>

spatial recognition and navigation is difficult for a lot of students who: can’t visualize shapes in their mind can’t describe a line segment in terms of an equation have difficulty with concepts like slope or rate can’t read maps easily being able to understand geometry and shapes helps with basic organizational and planning skills artists rely on concepts like perspective, size, shade & angle to render an image geometric principles are useful in all branches of mathematics and are necessary to progress in scientific subjects the subject unleashes a world of possible problems to solve and new explorations both in science and art to undertake

b) What are the historical origins and connections for this topic? (<100>

the Egyptians needed to use complex geometry and mathematics to determine the volume of a pyramid and to understand the physics behind building one the Babylonians used the movement and speed of the sun to set units of time and distance the Indians determined the relationship between the sides of right angled triangles, noting many ‘triples’ (e.g. 3,4,5 ; 5,12,13 ; etc.) the Greeks popularized and theorized concepts in geometry to expand the mathematics that described possible solids and volumes relationships between simple objects can be learned at any age and should be encouraged, especially for those who have the mathematical methods modern technology and structure is all derived from ancient principles of geometric relationships

c) How does this topic connect with life outside mathematics? (<100>

art relies heavily on principles of geometry & shapes architecture, construction, landscaping, athletics, exploration & more rely on these principles basic work in home improvement and organization can be more fruitful with a knowledge of geometry dividing a pie into 3 equal pieces can be done by sight, and dividing up any object into fair number of equal pieces is much better done with this knowledge concerns of safety when dealing with ladders, tire irons, shovels, etc. building stadiums so that every participant has a decent view

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

Unit Test: This will be a standard test aimed to familiarize the students with standardized tests (i.e. Provincials) and give a base mark for their progress. Section Test: This will be summative assessment of their progress to make sure they stay on task and to help let the students know where they are. Problem project: This is a task that will allow for more creativity and self-directed learning. The students will take ownership of a concept and present it. Class participation: This is a formative assessment of their progress and growth in my class. The marks are given based on contribution, attention, attendance, behaviour and positive attitude towards learning. Peer presentation: This think-pair-share approach to problem solving allows groups of students half an hour to solve a problem before sharing it with the class. Homework: This is practice of concepts learned and is useful for ensuring retention of information past the classroom door. It will be used sparingly however, because students may already be facing too much homework. In-class worksheets: These are fun activities that allow students to explore concepts at their own speed.

b) Project title and 50-word description CITY DESIGN The students will have to use their knowledge of shapes, space and geometry to design a city that has specific limits of construction. Mainly, there must be at least a certain number of buildings of certain heights that must be a maximum distance away from a water source, and receive a minimum amount of sunlight, given the surrounding buildings. This optimization problem will present many difficulties and be challenging and fun for students.

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

Lesson topic	Teaching strategies/ approaches used
1) Sine & Cosine Laws	-lecture & demonstration of laws -practice problems to work on -group work drawing of perfect circle with calculator and ruler(no protractor/compass)
2) Tangent Law & Pythagoras	-lecture & demonstration of law -proofs of Pythagorean Theorem -worksheet with sample problems
3) Identifying 2-D Objects	-clip from Flatland movie as ‘hook’ -nomenclature of n-sided shapes -problems & homework calculating circumferences and areas of shapes
4) Identifying 3-D Objects	-nomenclature of n-sided solids -problems & homework calculating surface area and volume of solids -section quiz
5) Perpendicular & Parallel Lines (Proofs and problems)	-introduction to basic proofs using deductive, ‘two-column’ problem solving -students working in groups to create their own proofs -proof problems for homework
6) Problems of accuracy and estimation	-measuring heights and distances of macroscopic, real-world objects -problems requiring estimations -problems where accuracy depends on where you’re measuring angles or lengths of a triangle
7) Calculating slope & solving multi-step problems	-lecture & demonstration of calculating the slope of a line -class participation on the board, involving problems with multi-step solutions -section quiz
8) Formalizing lines into equations	-f(x)=m*x+b -calculating intercepts -seat-work and homework calculating lines in 2 (& 3) dimensions
9) Review Day & seat-work time	-review of concepts learned -creation of review sheet to be handed in for marks and returned for the unit test -playing with manipulatives and spending time working on their group project
10) Unit Test	-unit test encompassing everything the students have learned, and challenging them to think outside the box

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).

LESSON A – Math 10

Unit: Shape & Space

Topic: Identifying 2D Objects

I. PLO:

a. apply the sine and cosine laws to solve problems

II. SWBAT:

a. Calculate angles and lengths of regular 2D objects

b. Estimate solutions to optimization problems involving size

c. Calculate the area of regular and irregular objects

d. Understand the relationship between θ and n for n-sided shapes

III. Teaching Objectives:

a. Challenge the students to re-evaluate their intuition and understanding of ordinary objects with a mathematical perspective

b. Solve optimization problems involving size

IV. Hook: Show a scene from Flatland

a. This is the one where the ‘Sphere’ lifts A. Square off the plane of Flatland and into the realm of the 3D.

V. Lecture: Provide students formal definitions of 2D objects

a. Nomenclature of n-sided objects from n=1-10

b. Relationship between θ and n

c. Formalism for calculating the areas

VI. Participation: Provide the students with cookie cutters and pieces of paper

a. Challenge different groups to maximize the number of cookie cutter shapes they can fit on a pre-determined size piece of paper

b. Have the students determine which shape can be replicated in the least amount of space

c. Probe the students understanding of symmetry to determine if they know of shapes that replicate themselves with no space left behind (tessellations)

VII. Homework: Time given at the end of class to begin homework

a. Assign students homework that has them calculate (or at least estimate) the areas of difficult shapes (both regular and irregular)

VIII. Conclusion: Reiterate the power of sine & cosine in solving problems

a. Show the students how an understanding of mathematical formalism can help with solving simple problems like how many gingerbread men can I fit on one cooking sheet.

IX. Assessment: Homework, Observation, In-class worksheet, Unit Test

LESSON B – Math 10

Unit: Shape & Space

Topic: Problems of accuracy and estimation

I. PLO:

a. Solve problems involving distances between points in the coordinate plane

b. Solve problems involving midpoints of line segments

II. SWBAT:

a. Take measurements of objects with high degrees of accuracy

b. Make calculations and estimations of unfamiliar objects

c. Understand the importance of accuracy and measurement

d. Understand the difference in accuracy between measuring angles and measure lengths of a triangle

III. Teaching Objectives:

a. Gain an ability to estimate and calculate distances and sizes with a high degree of accuracy

b. Use geometry to make measurements of systems of lines and angles

IV. Hook: How many windows are in the Empire State building?

a. Show how one could estimate the number of windows based on the number of windows present in a given floor

b. Estimate the amount of light that the Empire State building emits as a function of the total area of the windows

V. Practical Activity: A practical calculation of the real-world

a. In groups of 3-4, the students are permitted to tour the school grounds estimating the total volume of the school

b. The students are permitted pencils, paper and calculators (no tape measures, protractors or rulers).

c. One group (bright students looking for an extra challenge), will be tasked with calculating the volume, but they are not allowed to exit the school to do it.

VI. Reporting: Students return to the classroom

a. Groups present and justify their answer for the class

b. The group that had to find their answer from within the school will present and justify their answer

c. Class will compare answers and discuss methods of accuracy

VII. Homework: Time given at end of class if possible:

a. Write up a list of possible improvements to the task of calculating the volume of the school

b. Write on the importance of accuracy in measurement, inference, estimation and calculation

VIII. Conclusion: Estimation is an important skill to develop and practice

IX. Assessment: Assessment will be based on problem solving ability, creativity, thoughtfulness, planning and results

LESSON C – Math 10

Unit: Shape & Space

Topic: Formalizing lines into equations

I. PLO:

a. Solve problems involving rise, run and slope of line segments

b. Determine the equation of a line, given information that uniquely determines the line

II. SWBAT:

a. Why the form f(x)=m*x+b is important and where it comes from

b. How to calculate or estimate slope

c. How to plot functions and determine intercepts

d. Model 3D examples of objects on a 2D page

III. Teaching Objectives:

a. Have students understand meaning of m & b in the equation

b. Have students understand how to navigate a graph based on information given

IV. Lecture: Do some examples of equation writing for unknown/unique line segments

a. Plot f(x) = -3x+6

b. Plot f(t) = 1.4t – 1.4

c. If f(3)=6 and f(6)=12, what is f(x)?

d. If f(1000)= 2 and f(2000)=4, what is f(x)?

V. Participation: A wayward traveler

a. A traveler has traversed a large city in a rather unorthodox pattern. If t represents the time in minutes that it took the traveler to walk about, graph x(t) and y(t) for the position in the x-direction and y-direction.

b. Have the students write their answer as step functions to describe the turning of a corner as going from the description of one line to the next

c. i.e. x(t) = 5t - 0<3 for round trip.

7t-6 - 3<6

-x+42 - 6<42

VI. Homework: Give seatwork time if possible

a. Complete the ‘wayward traveler’ problem

b. Complete problems from text book.

VII. Conclusion: This equation writing is very useful and has applications in science as well.

VIII. Assessment: Successful completion of homework and participation in class.

PROJECT – Math 10

Unit: Shape & Space

Topic: Planning & Optimization of Space

Premise: You have been asked by a developer to use you keen math skills to develop an ideal tourist destination off the coast of a local ocean front. The problem is that the developer won’t make money (hence he won’t want to hire you), unless you can get the conditions and demands of the hotels just right. The bigger (and therefore, more hotel space) you can make the properties, the more money you will make.

Conditions:

1. You are only allowed to build towers that are 100ft by 100ft and the developer can only afford to build on 150,000 sq. ft of land.
2. Each tower can have a maximum height of 500ft, but since each tower requires a minimum amount of sunshine, some towers can’t be built as high to avoid casting long shadows. At noon, at least 50% of each south-most tower face (side that is facing the sun) must have direct sunlight given that the sun shines down at 30° from the south horizon of the district.
3. Each tower must be beside a water source, whether it is the ocean or a channel that has been dug in the precious land to allow the water to access remote towers. For daily operation, each tower requires 5 cubic ft. of water for every cubic ft. of tower occupied.

Your task: Design your ideal vacation district, including a drawing of hotel locations, waterfront and the water channel.

Record the height, location, % of sunshine, and water consumption of each hotel tower.

What is the total volume (in cubic ft.) of the towers in your district?

Sample district:

MAED 314 – Assignment 3

MAED 314 – Assignment 3

• Vincent Collins
• Gregory Thiessen
• Rosemary Qi

November 23, 2009

Part 1 – (8) Develop a project of own choice based on the in-class library. Topics include Goldbach’s Conjecture (from novel), infinities, fractals, paradoxes, etc.

Grade: 11

Purpose:
This activity is both a history project as well as an activity to have the students think mathematically. These mathematical concepts can be very difficult to grasp and by doing this project, students will have to go beyond instrumental comprehension to fully understand the given topics and produce the desired results. Students will get the opportunity to work in small groups and present their findings to the rest of the class in a short presentation.

Description of Activity:
Project

In groups of 2 or 3 you will choose a topic from the provided list. Feel free to research and choose your own, however, please confirm your decision with me before proceeding.

• Goldbach’s Conjecture
• Infinities
• Fractals
• Paradoxes (Choose only one to work with)

A paradox is a statement that goes against our intuition but may be true, or a statement that is self-contradictory. Examples of some mathematical paradoxes are:

• Zeno’s paradoxes of motion (the dichotomy paradox, Achilles and the tortoise paradox, arrow paradox or the stadium paradox
• Russell paradox (Barbershop paradox)
• Greeling paradox
• Petersburg paradox
• Galileo’s paradox
• Etc.
Once you have made your decision, the following tasks must be completed:

1. Research your topic through the internet or by using books from the school library. I will provide a list of books that are available in the library.

2. Produce a visual representation of your topic. This can be in the form of a poster, tables, comic, performance, etc. Include a brief history of the topic including the person who developed the concept.

3. Present your work to the class in a short presentation of no more than 5 minutes.

Marking Criteria:
Your grade will be dependent on your understanding of the mathematical paradox which will be conveyed through your final product and presentation. Effort and contribution to the team will affect your final mark. There will be a peer evaluation at the end.

Due Date: 2 weeks from day assigned.
This project is expected to be completed primarily outside of class.

Sources:

http://www.stormloader.com/ajy/paradoxes.html
http://www.suitcaseofdreams.net/Content_Paradox.htm
Mazur, Joseph. Zeno’s Paradox: Unravelling the Ancient Mystery Behind the Science of Space and Time.

Part 2 – Evaluation of the project

This project is an enrichment project designed to make students think mathematically about concepts not ordinarily encountered throughout life. Benefits and weaknesses of this project are listed below.

Benefits:

Students will learn of various mathematicians who have contributed to mathematical knowledge over the centuries.

Students will have to critically think about concepts of math outside of what they usually experience.

Students will have to work in groups to accomplish tasks.

Students will have to present their findings in a clear manner such that peers will understand.

Weaknesses:

Not all of the topics are connected to the grade 11 IRP’s. Some of the mathematical concepts may be more suited for grade 12.

Students may have a difficult time comprehending some of the paradoxes.

Students may not comprehend the purpose or necessity of this project and may view is as another mundane task they must work through.

Trying out the project as a sketch
Our group has decided to tackle this project by focusing on Zeno’s paradoxes of motion.

History of Zeno of Elea

The Greek philosopher Zeno of Elea, born around 490 B.C., proposed four paradoxes that are still discussed 2500 years later. He wanted to challenge the accepted notions of space and time that he encountered in various philosophical circles. His paradoxes confounded mathematicians for centuries, and it wasn't until Cantor's development (in the 1860's and 1870's) of the theory of infinite sets that the paradoxes could be fully resolved.
Zeno's paradoxes focus on the relation of the discrete to the continuous, an issue that is at the very heart of mathematics. His four famous paradoxes include the dichotomy paradox, Achilles and the tortoise paradox, arrow paradox and the stadium paradox.
http://mathforum.org/isaac/problems/zeno1.html

Part 3 – New Math Project Design

Grade: 11

Purpose:
The goal of this math project is for students to examine cyclical patterns in an artistic, fun and unique way. The students are presented with a classic problem that has been posed in an abstract way, to see if they can discover the numerical relationship. The students will be challenged to recreate and experiment with the ‘celtic knot’ pattern and hopefully they will begin to see that math exists is science and in art too. This is an enrichment project that promote deeper thinking in the students.

Description of activities:
1. Follow the guide (see below) for producing a ‘celtic knot’ and produce a 4 by 3 knot. Decorate it as necessary to show how the pattern loops.
2. If the knot has m vertices along the left side and n vertices along the top side, determine the formula/algorithm for determining the number of loops present in an m by n knot.
3. What other patterns of loops can be made by combining loops of various sizes. What are the limitations of this pattern, if any?

Guide:
The (1) blocks are placed at the four corners of the pattern (see below), each rotated 90 degrees from each other. The (2) blocks are placed next to them, cascading along the edge of the knot pattern to form a perimeter. The (3) blocks are placed in the middle to fill the space and complete the knot design. The picture below illustrates how this works. Under ‘Graphic’, there is a drawing of a completed knot for reference.

Sources:
There is no source for the creation of this knot pattern. This is original artwork that shall now be examined for mathematical inquisition.
Length of time that project will take:
Students are given 1 week from the date assigned to complete this project. If time permits, a small section of class time may be devoted to beginning the project. Otherwise, it is to be work upon outside of class in groups of 2-3.
What students are required to produce:
1. The students must draw a 4 by 3 ‘celtic knot’ and decorate it to make it easy to see.
2. The students must derive a formula for determining the number of loops in an m by n knot.
3. The students must create a new knot pattern by opening & closing loops to create their own knot pattern.

Graphic:
m=4, n=4, # of loops: 4 (highlighted as purple, red, green & blue)

Marking Criteria:
1. /4 - Completed drawing of 4 by 3 ‘celtic knot’.
2. /4 - Formula for the # of loops.
3. /2 - Originality of knot created.

Celtic Knots

Celtic Knots have been a fascination of mine for the longest time. Over the years, I have played around with creating simple knots and have developed my own, reliable pattern for producing them. As the more clever of you will soon see, the template for creating knots follows a couple pretty simple rules. The knots I produce rely on a basic weave in the center and wrap around clockwise at the edges.

There is a lot of fantastic math that can be derived from 'knot' problems and they are a curious phenomenon of spatial geometry. Take a look at the pattern below. This is one of my more complex creations that shows just how boggling these knots can be.

How many loops are in this knot? How are they connected?

Here I have highlighted the loops in colour to make it a little easier to see what is happening. Of course, it's still not ideal. But you can begin so see why these knots are so interesting. In fact, you can create knots that consist of a single (highly complex) loop, but those introduce an entirely knew concept. So, to get your feet wet, I've given you a knot that has a lot more symmetry and the pattern is more easily distinguishable.

For those of you who'd like a real clear visual, I have deconstructed this knot into it's basic loops. Essentially 'spilling the beans' on the question posed above, there are 14 loops. They all fit perfectly and snugly together to form a 'somewhat' symmetrical square. How can this be? Certainly, I didn't just stumble across this pattern. This took a lot of thinking and working with these loops and patterns to make it work. But the point is that it totally does work and it opens up a world of possibility for mathematical and creative exploration. Can you make an even better knot than this?

SEQUENCE PROBLEM: SOLUTION

The geometry is some what subtle, but I argue it's entirely possible to formalize a solution for incredibly large sequences based on this relationship. As you can see by the tiers of the pyramid (discrete triangular fractal), this graphical solution works for all sequences involving numbers of adjacent 1's in powers of 2 (i.e. 1, 11, 1111, 11111111, etc.). Identifying sequences becomes more difficult for sequences that have other numbers of adjacent 1's, but the principle is the same and the geometry is not impossible to navagate. The full solution will involve similar pyramids for all powers of prime numbers. However, a majority of sequences can be solved with the geometry from the simplest of these (e.g. powers of 2 - above). I encourage you to try making a pyramid of your own. Start with the sequence of all 0's, with a prime number of adjacent 1's in the middle. Carry the numbers downward, look at the pattern, and see just how many random sequences of 1's and 0's can be accounted for.

Problem Solved. (more or less) :o)

Two Column Solving: SEQUENCE PROBLEM

Gentlemen's Chess

Chess Invention: Gentlemen's (No Trading) Chess

Intro: The invention of Gentlemen's Chess spawned out of boredom of the tradition form of chess. The problem was that the regular rules for chess didn't inspire any creativity in game play and the games all seemed to go the same way. Bishop captures bishop, then pawn captures bishop back. Pieces of equal value were simply being 'traded', until the board was simplified and fewer (more valuable) pieces remained. It seemed too easy to lose your pieces and instead of fighting to protect a knight, we would just capture their knight in successive turns so that there was no net loss. Well clearly this wouldn't do. So we had to make a new rule. We said you couldn't trade pieces... what did that mean? It meant that a piece that was protected, couldn't be captured for whatever reason. It was permanently safe. A lovely notion. A piece that wasn't protected however, was fair game. We loved this idea and immediately started playing our new way. Almost immediately, we realized that this would create a stale-mate. Both players would move in such a way that all their pieces were protected all the time. This meant, that vast pawn-knight formations covered the entire board and it was impossible to move about. This lead to a second rule. Pawns can move in all directions. Boom! Suddenly, the board sprung to life and movement was possible, players started taking risks, pieces were lost and checkmates were performed. The game took on this whole new dimension. Strategies formed. Aggressive and defensive strategies were shown to each have their own pros and cons, and ultimately, Gentlemen's Chess was born.

Inventors:

Vincent Collins, Clarence Sng & Matt Hamm invented Gentlemen's Chess back in the 2003-2004 school year at UBC.

Rules:

1. If a piece, at the point of being captured by an opposing piece, is protected by a friendly piece, it cannot be captured.
   
2. Pawns are permitted to move in all 8 directions as long as they only capture in the normal forward-diagonal fashion.
   
3. Checkmate occurs when the king is threatened by a protected opposing piece.
   

Special circumstances:

3rd party protection: If a protected black rook is in between two white rooks, both white rooks are protecting each other. This is because if the black rook were to try to capture one of the white rooks, the other white rook, upon the black rook capturing, would be able to protect that space.

Protected mate: If a black queen is protected and on an adjacent square to a white king, checkmate is automatic, despite any aiding white pieces.

Removal of En Passant: Because pawns can move back and forth, En Passant no longer makes sense for pawn movement. Also, as inventors of the game, we've decided that En Passant doesn't add anything new or exciting to the game and is unnecessary.


Pawn promotion: Pawn promotion is a lot more common in this version of chess and therefore has the potential to sway strategy quite significantly. However, since pawns are so useful, some argue that when a pawn reaches the final row, it shouldn't have to promote (i.e. promote to a pawn). However, we maintain, that pawns must promote, even if that pawn only is sent to the final row to gain a better board position.


It is amazing that it has taken almost 6 years to finally start documenting the rules and history of this game. Please take the time to try playing chess this way. I guarantee you'll enjoy playing it and want to share it with others. :o)

Quick Write: Short-Practicum - Memorable Story

My sponsor teacher had dressed up as a navy officer for Hallowe'en and I had dressed up as a soldier. I swear it wasn't on purpose. The students all thought it was funny even though it was a little intimidating for them to have two 'officers' for teachers. Both my sponsor teacher and I volunteered to chaperone the high school dance and we looked like quite the pair. We were marching up and down the halls, guarding doors and looking tough. Well, then the fire alarm went off and we sprung into action like a pair of veterans. Hurrying toward the sound of distress! It turns out that the fog machine for the dance had been left on too long and the smoke alarm had been triggered. As the students began to file out into the sprinkling of rain upon the damp courtyard, the teachers stood in the doorway, huddled against the cold and watching the kids. All but my sponsor teacher and I who began marshalling students and warning them not to stray too far. Like sheep dogs we paced the perimeter, barking orders at stray students and keeping them safe. It was at this point that I realized my sponsor teacher was getting into character. He paced up and down the courtyard with his hands clasped behind his back. The street lamp casting him in shadow as he glared gloomily at the frolicking students. Finally, the alarm was turned off and we marched the students back into the building, stoically returning them to safety. It was quite an amusing sight. But we were a hit with the students and I think it made a great impression that will last with them until I see them in the long-practicum. :o)

Reflection: 'Free Write' & Math Poem Analysis

Exerpts from free-write:

"Zero is an ultimatim..."
"...it is the absense and it is comforting."
"It is the producer of uncertainty, goal of asymtotic behaviour"

Analysis:

I found my free-write to be too confusing to pull much from. I certainly didn't use any direct quotations from the free-write for my poem. But to give feedback on the effectiveness of this activity, I have created a pros & cons list.

Pros:

• Fun exercise that made me think a lot about pure numbers and their relations
• Good practice of applying knowledge of mathematics to an artistic medium
• The free-write did provide some useful ideas for future work

Cons:

• I needed some more inspiration before I did my free-write
• I would have prefered more variety in topic options
• I would have loved writing the poem if I had the freedom to rhyme

Math Poem

"Over Zero"

What do you give something that has
nothing? Only everything for which it
has the capacity, even if it's zero.

Is there a limit, a point where
we squeeze the answer up into it?
Rollerpin it flat and toss it on top.

Nothing has a specific size and
can be played with in this way.
That's how we found the answer.

Reflection: Micro-Teaching Presentation 14/10/09

For our presentation, we chose the topic of logarithms. We then had 15 min. to present the subject in an engaging fashion with our students. Given these conditions, it proved rather difficult to approach, as logarithms are such a large topic and we had to pick and choose what to present.

Our presentation started with a brief history of the inventor of logarithms and his motivation for creating them. I felt this was a very valuable aspects of teaching the subject, but in retrospect, it used up some precious time for teaching. Given a week to present logarithms to a class, I would devote a reasonable time to the history and more abstract qualities. But in 15 min. we really ought to have stuck to the basics. And so in our rush to fit in the rest of the material we had planned, we received some negative feedback regarding the organization, flow and fun of the presentation.

Next we reviewed many of the laws of logarithms with the students and derived the Multiplication and Division Laws. In retrospect, since we had agreed that the students would already know the basics of logarithms, we shouldn't have spent so much time reiterating them.

Lastly, we wanted to go through a typical logarithm problem and by having the students reply with the answers, slowly graph it. This would have worked well, because we were going to split a large logarithm problem into some shorter steps and have the class solve each one. Then we would simultaneously be plotting these answers to show the general trend. That way the students would understand the relationship between the mathematical and visual interpretations of the logarithm. Again, we received negative feedback because we ran out of time.

In future, we will work to be more careful with what amount of information can be transferred from teacher to student in the given time and prioritize accordingly.

BOOPPPS Lesson Plan - Exponentials & Logarithms

1. Bridge: We will begin our micro-teaching presentation with a 'hook', something to grab the students attention. In this case, we have a brief history of the invention of logarithms to show the class.
2. Teaching Objectives: We plan to cover the laws of exponentials and logarithms and then work through a problem along with the class. This will comprise of a number of smaller problems taken from the larger problem and posing them individually to the class for response. Finally, we take all those responses, put them together and graph the result.
3. Learning Objectives: We plan to have the students apply the various logarithm rules to solve a question and display their understanding.
4. Pre-test: We will go over the rules of exponentials and logarithms. We will then have the class answer a few simple questions. We will then show a graph comparing logarithms in various bases.
5. Participatory Activity: We will work through the long logarithm problem with the class and produce a result that everyone is happy with.
6. Post-test: Time permitting, we have a second (shorter) problem for the class to work on and solve.
7. Summary: We will briefly recap the equations and concepts that we have covered.

Citizenship & Democracy in Mathematics

Response to: "Citizenship Education in the Context of School Mathematics" ~ Elaine Simmt.

Elaine Simmt argues in her article that by studying mathematics, one will develop critical thinking and problem-solving skills, that are crucial for a democratic, informed member of society. I certainly see the potential validity of this statement. Having met, and studied under, many mathematicians, I know that people who study math passionately are conscious and critical thinking. There are many personality traits (reserved, patient, analytical, cautious) associated with mathematicians that would suggest that those who study math are affected positively by their extensive study.

But perhaps Simmt has missed a step between critical thinker and global citizen. As much as there are 'problem-solvers' in the world, there can be 'problem-posers', or in a social context, 'problem-causers'. Do we know any bad 'math people'? What about internet hackers, inside traders, card sharks, money lenders and ruthless businessmen? Have they not done the math? Having the skills to manipulate our highly quantized world doesn't pose any limit or moral barriers to those seek wealth or power.

I do agree with Simmt though, that math is very much a part of the human experience. And those who use math, do 'transform' our reality and not just 'transmit' it. Math is a double-edged sword. For every opportunity for constructive processes, there exists the possibility for destructive processes; for every problem solved, a problem posed. The beautiful symmetry of math means that everything has an inverse. Math gives us the skills to examine the world at its most fundamental levels, but the choice of good or bad still exists. All I can hope for, is that I can show my students the beauty, symmetry and good in mathematics and prepare them for the true problems that life will present them.

Review of "What-If-Not" strategy from "The Art of Problem Posing"

This review is in reference to pages 33-65 of the "The Art of Problem Posing" by Stephen I. Brown and comments on the strategy of "What-If-Not".

The term "What-If-Not", refers to a creative thinking process of examining mathematical/problem elements with a fresh perspective. With each concept, we identify key ideas & problems and then we try to expand on them in new directions, incorporating our own perspectives and thinking processes.

I really like this strategy for problem posing/solving. It's all about asking the right questions. Taking a theorem and changing a variable or condition to create a new situation and then examining it to see where it has taken us. This can be used extensively in the classroom, as we have begun to discover in our other classes. The students learn through inquiry and compound ideas and ultimately will have a more concrete understanding of the concept using this strategy.

We first explored this in SCED 316a with the aptly named 'UnDemo' in which the students were in control of the experimental process and merely guided by the teacher. In this way, the students were allowed to ask "what-if" and then examine "what-if-not". If students can understand what something 'is not', then they have a better understanding of what 'it is'. We next explored this in our Principles of Teaching lecture in which we talked about what it means to define a chair. And then from that, looking at a plethora of objects that were 'chair-like', but not chairs (i.e. thrones, stools, lounges, couches, etc.). In this way, the pythagorean theorem is a chair, but the other equations that were plucked from it, where 'pythagorean-like' and equally valid for examination. And by understanding what the pythagorean theorem wasn't, a student would have a better understanding of where it fit in their theoretical framework.

The only limitation to this method of inquiry, however, is that fact that this type of exploration can often be off 'topic' or time-consuming and may even lead to confusion. We cannot use this strategy like a blunt tool, or it will damage the process. Instead, we must use it as a fine instrument that requires skill and precision in utilizing for best results.

10 Questions/Comments on "The Art of Problem Posing"

This response consists of 10 questions/comments regarding pages 1-32 of "The Art of Problem Posing" by Stephen I. Brown.

1. I valued most highly, the idea that problems are very much context-based. The difference between a lousy problem and a superb problem (on the same topic) lies almost solely in context. And I think a student can tell/appreciate when a truely good/unique/intriguing question has been asked (as opposed to a conventional one).
2. I often see my own contextual view of problem posing and solving as being quite narrow. What are some strategies or literature that might best help broaden my contextual view?
3. I like the notion of posing an idea and not a problem. The open-endedness of it is very freeing and allows the mind to explore new possibilities and solutions. I was amazed at how many answers there were to a given idea.
4. I also thought it was very interesting, the idea that the person being asked to respond to an idea like x^2 + y^2 = z^2, was, in a sense, posing and answering their own questions and that the problem-solver imposes their own views on an idea and becomes a unique problem-poser.
5. What do you do when the a student wants the proof to an equation or principle and the explanation of the proof would be too far above that student's current knowledge level? Often times, I think math teachers are compelled to have their students take their word for it, because the beautiful simplicity of an idea is too complex to explain to the student.
6. How strongly should I be emphasizing pattern recognition, rare sequences and phenomena in amongst the standard curriculum?
7. What license do I have as a teacher to assign value/marks to a student's answer that is a unique approach/solution to the problem for which they put the wrong answer? i.e. If I want the answer 5,12,13 and I get the answer -5,-12,-13?
8. I found it very funny in the hand-shake example of the jurors, because my mind immediately jumped to the problem of how many total handshakes. I guess that is my narrow contextual perspective.
9. How far can I probe into a simple concept before I run the risk of undoing the knowledge I taught and confusing my students?
10. What IS the use of examining phenomena?

Response to Timed-write 2

Well, I wrote this timed-write really fast, so I'm not sure how accurately it portrays my feelings. But I do know that it sheds some light on what I think about my future students and what my future performance might look like. A lot of what I wrote is stuff that I've already been thinking about and trying to problem solve. I sit on the bus dreaming up ways to approach topics with kids and how to best teach difficult concepts. Some of my ideas are way out there and that's why it was easy to write a positive and negative 3rd person perspective.

With the positive perspective, hypothetically written by a former student, I wrote it in much the same style that I might have written an actual letter to my former teacher. I admired many of his attitudes on the subject of learning math and I hope to embody some of those same teaching styles. Whether a student ever realizes the quality of my instruction is immaterial, so long as they receive the benefit from a well-suited instruction and curriculum.

With the negative perspective, again written by an imaginary former student, I try to shed light on some of my fears as a new teacher. I can imagine some of my creative teaching methods as being counter-productive, confusing or only benefiting a small portion of the class. So, I'm still struggling to find the right idea, or a least a good working-idea of what an effective (interactive) teaching approach is. Plus, this negative perspective mirrors some of the comments in the positive perspective and attempts to reveal that there can often be goods and bads to any approach. There will sometimes be disparity amongst students as to who really benefits from my teaching.

So, as you can see, my focus continues to be to find an effective and interactive teaching style. I like the picture I portrayed in the positive perspective and I'm going to try to find one-on-one strategies to avoid the picture I portrayed in the negative perspective.